Output Feedback Control of Nonlinear Systems with Unstabilizable/Undetectable Linearization

Home , Feedback

OUTPUT FEEDBACK CONTROL OF NONLINEAR SYSTEMS WITH UNSTABILIZABLE/UNDETECTABLE LINEARIZATION

BO YANG

Submitted in partial fulﬁllment of the requirements

For the degree of Doctor of Philosophy

Dissertation Advisor: Dr. Wei Lin

Department of Electrical Engineering and Computer Science

CASE WESTERN RESERVE UNIVERSITY

January, 2006 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

______

candidate for the Ph.D. degree *.

(signed)______(chair of the committee)

______

(date) ______

*We also certify that written approval has been obtained for any proprietary material contained therein. Copyright c 2006 by Bo Yang All rights reserved To my wife and my parents with love and gratitude TABLE OF CONTENTS

1. Introduction ...... 1 1.1 LiteratureReview...... 2 1.2 Motivations ...... 5 1.3 ContributionsofThisDissertation...... 7

2. Preliminaries ...... 11

2.1 StabilityTheory...... 11 2.2 HomogeneousSystemsTheory ...... 19 2.2.1 Homogeneous Functions and Vector Fields ...... 19 2.2.2 Stability of Homogeneous Systems ...... 21

2.3 UsefulInequalities ...... 24

3. Smooth Output Feedback Design ...... 26

3.1 Introduction...... 26 3.2 Output Feedback Design of Linear Systems — A Non-Separation Prin- cipleParadigm ...... 30 3.3 Smooth Output Feedback Stabilization of Homogeneous Systems. . . 35

3.4 Global Stabilization of a Class of Non-Homogeneous Systems via Out- putFeedback ...... 41 3.5 ExampleandDiscussion ...... 48 3.6 Summary ...... 56

4. Robust Control of Uncertain Systems by Smooth Output Feedback 58

iv 4.1 Introduction...... 58

4.2 Robust Output Feedback Design: the Case of p =1...... 62 4.3 Robust Output Feedback Design: the p-NormalFormCase ...... 67 4.4 Output Feedback Stabilization of Uncertain Cascade Systems..... 79 4.5 Summary ...... 87

5. Global Robust Stabilization by Nonsmooth Output Feedback . . . 89 5.1 Introduction...... 89

5.2 Robust Output Feedback Design: A Case Revisit ...... 93 5.3 Nonsmooth Output Feedback Stabilization of Uncertain Nonlinear Sys- tems ...... 97 5.4 ExtensionandDiscussion...... 104

5.5 Summary ...... 107

6. Global Output Feedback Control with Dynamical Rescaling .... 109

6.1 MainResultandDiscussion ...... 109 6.2 StateFeedbackDesign ...... 113 6.2.1 Dynamic Rescaling of the Original System ...... 113 6.2.2 StateFeedbackController ...... 116

6.3 OutputFeedbackDesign ...... 122 6.3.1 Reduced-OrderObserver ...... 122 6.3.2 Error Dynamics and Output Feedback Controller ...... 124 6.3.3 ObserverGainAssignment ...... 125

6.4 StabilityAnalysis ...... 130 6.5 Output Feedback Stabilization of Cascade Systems ...... 131 6.6 Summary ...... 135

7. Semi-Global Output Feedback Stabilization of Non-Uniformly Ob- servable and Nonsmoothly Stabilizable Systems ...... 137

v 7.1 Introduction...... 137

7.2 KeyTechnicalLemmas ...... 141 7.3 ASimplerParadigm ...... 142 7.4 ProofofMainResult ...... 149 7.5 Summary ...... 156

8. Conclusion ...... 157 8.1 Summary ...... 157

8.2 FutureWork...... 158

vi LIST OF FIGURES

3.1 Transient Responses of the closed-loop system (3.57)-(3.59) with x1(0) =

5, x2(0) = −3, xˆ1(0) = 3, xˆ2(0) = −5...... 51

3.2 Trajectories of the closed-loop system (3.60)-(3.61) with x1(0) = −2, x2(0) =

5, x3(0) = −3, xˆ2(0) = 3, xˆ3(0) = −5...... 52

4.1 Transient responses of the system (4.51)-(4.52) from x1(0) = 2, x2(0) =

−10, zˆ2(0) = −15...... 78 4.2 Transient responses of the closed-loop system (4.66)-(4.71) with θ =1

and the initial condition (ζ, η1, η2, zˆ2)=(1, 0.3, −6, −5) ...... 88

7.1 The level set Ωx on x-space and the saturation threshold M...... 144 7.2 The level set Ω on (x, zˆ)-space...... 148

vii Acknowledgment

I would like to ﬁrst acknowledge my profound debt of gratitude to my advisor Dr. Wei Lin. I could not have asked for a more encouraging and caring mentor than him.

Throughout these years, he has suggested many promising projects that I worked on in a very productive way. Without his constructive direction and invaluable advice, this work would not have been completed. I want to extend my gratitude to Dr. Kenneth A. Loparo, Dr. Marc R. Buchner and Dr. M. Cenk Cavusoglu for being on my advisory committee and reviewing my dissertation. I also wish to express my appreciation to Dr. Chunjiang Qian of the University of Texas at San Antonio, Dr. Qi Gong of Naval Postgraduate School, Dr. Radom Pongvuthithum of Chiang Mai University, and Dr. Xianqing Huang for their generous help on either a research level or a personal level. My thanks are also given to our research group members: Jianfeng Wei and Hao Lei with whom I share the research interests and friendship. Hao also helped me a lot complete paperwork at Case and make my defense possible when I was visiting visit Texas Tech University during the semester of Fall 2005. Same thanks to my oﬃcemates: Arsit Boonyaprapasorn, Jinbae Choi, Chunrong Dong, Yiming Huang, Ying Wang, and our secretary Marla Radvansky and student aﬀairs coordinator Eliz- abethanne Fuller-Murray for their kind assistance.

This career goal of mine would have never materialized had it not been for the sacriﬁces and support of my family. Words cannot adequately acknowledge their undying support. Finally, special thanks to my wife, Bei, for her unfailingly support and stood behind me in all my endeavors.

viii Output Feedback Control of Nonlinear Systems with Unstabilizable/Undetectable Linearization

Abstract

BO YANG

This dissertation addresses a number of fundamental and challenging output feedback control problems for a signiﬁcant class of uncertain nonlinear systems with unstabilizable/undetectable linearization, including: (1) global asymptotic stabilization via smooth output feedback under a high-order version of Lipschitz-like condition;

(2) robust control by smooth or nonsmooth output feedback with dynamic rescaling, under appropriate yet restrictive growth conditions; (3) semi-global asymptotic stabilization of non-uniformly observable and nonsmoothly stabilizable systems by nonsmooth output feedback, without imposing any growth condition.

First of all, we study the problem of global stabilization by smooth output feedback, for a class of n-dimensional systems whose Jacobian linearization is neither stabilizable nor detectable. A novel output feedback control scheme is proposed for the explicit design of both high-order observers and controllers in a recursive manner.

Its design philosophy is substantially different from that of the traditional “Luen- berger” observer in which the observer gain is determined by observability. Using the new output feedback design method, we then consider the problem of robust smooth or nonsmooth output feedback stabilization for several families of uncertain nonlinear systems with unstabilizable/undetectable linearization. To handle system uncertainties effectively, we introduce a novel rescaling transformation with an appropriate dilation and factor. Depending on the type of growth conditions, the rescaling factor can be either a sufficiently large constant or a time-varying function

ix x that needs to be tuned on-line through a Riccati-like diﬀerential equation. The con- structions of smooth or nonsmooth state feedback controllers and observers use only the knowledge of the bounding system rather than the uncertain system itself. The robust output feedback design approach thus developed is also extended to uncertain cascade systems beyond a strict-triangular structure.

In the last part of this thesis, we show that without imposing any growth condition, semi-global stabilization by nonsmooth output feedback can be achieved for a chain of odd power integrators perturbed by a triangular vector ﬁeld, which is in general not smoothly stabilizable nor uniformly observable. Chapter 1. INTRODUCTION

One of the most important problems in control theory is to make a controlled plant stable at its equilibrium point by state or output feedback in the presence of system uncertainties. The origin of its modern theory dates back to the year of 1868, when

J.C. Maxwell analyzed the stability of J. Watt’s ﬂyball governor. One and a half centuries later, extensive research in linear systems revealed the celebrated concepts such as controllability, stabilizability, observability, and detectability and their powerful applications. It has been known that for a linear system, a state or output feedback controller can be easily designed as long as the system is stabilizable and detectable. In the past decade, research has been focused on extending the linear output feedback control theory to nonlinear systems. Due to the nature of nonlinearity, most of nonlinear systems involve unstabilizable and undetectable linearization. In this case, output feedback control of nonlinear systems is a very challenging problem because the linear systems theory fails to be applied, and thus a truly nonlinear output feedback control theory must be developed.

This dissertation is focused on such a diﬃcult yet important topic — output feedback stabilization of nonlinear systems with unstabilizable and undetectable linearization. We address various problems in this area, including global asymptotic stabilization by smooth output feedback, global robust stabilization by smooth or nonsmooth output feedback with dynamic rescaling, for a number of classes of nonlinear systems with uncertainty, and semi-global asymptotic stabilization of non-uniformly observable and nonsmoothly stabilizable systems by nonsmooth output feedback.

1 2

1.1 Literature Review

In the 1990’s, with the aid of the diﬀerential geometric approach (Byrnes and Isidori (1991); Isidori (1995, 1999); Nijmeijer and Schaft (1990)), several systematic feedback design methodologies were developed towards global asymptotic stabilization, robust control and adaptive regulation of aﬃne systems in the strict-feedback form

x˙ 1 = x2 + f1(x1)

x˙ 2 = x3 + f2(x1, x2) . .

x˙ n = u + fn(x1, ··· , xn),

y = x1, where u ∈ IR, x ∈ IRn and y ∈ IR are the system input, state and output, respectively. Among them, a Lyapunov-based recursive design procedure known as adding an integrator (Byrnes and Isidori (1989); Tsinias (1999); Isidori (1995, 1999)), also called backstepping (Krstic et al. (1995); Marino and Tomei (1995)), has been proved to be a powerful tool for robust and adaptive control of the aﬃne system above. Due to the nature of adding an integrator, the systems under consideration are required to be fully or at least partially feedback linearizable, and linear in the control input.

Such a limitation has stimulated the development of new nonlinear feedback design techniques. For instance, for the three-dimensional system

3 x˙ 1 = x2

3 x˙ 2 = x3

x˙ 3 = u

y = x1, local and global asymptotic stabilization via smooth state feedback were investigated by Crouch and Irving (1983) and Byrnes and Isidori (1989), respectively. More re- 3 cently, a series of papers by Lin and Qian (2000b); Qian and Lin (2001a,b) have developed a synthesis technique called adding a power integrator which proves to be extremely eﬀective in achieving global asymptotic stabilization by state feedback for the following class of inherently nonlinear systems with unstabilizable linearization:

p1 x˙ 1 = x2 + f1(x1) . .

pn−1 x˙ n−1 = xn + fn−1(x1, ··· , xn−1)

x˙ n = u + fn(x1, ··· , xn)

y = x1, (1.1) where u ∈ IR, x ∈ IRn and y ∈ IR are the system input, state and output, respectively, and pi ≥ 1, i =1, ··· , n − 1 are odd integers. The state feedback controllers proposed in (Lin and Qian (2000b); Qian and Lin (2001a,b)) were constructed based on the homogeneous systems theory (see, Bacciotti (1992); Dayawansa (1992); Dayawansa et al. (1990); Hermes (1991a,b); Kawski (1989, 1990); Rosier (1992) and Chapter 2), the method of non-smooth analysis, and the feedback domination design philosophy. In contrast to the rapid development of state feedback stabilization, progress in its output feedback counterpart has been much slower. There are several reasons for this. Firstly, most of the existing observer design techniques are inherently linear

(e.g., Luenberger-like observers or high-gain observers), and only available for a restrictive class of nonlinear systems such as the triangular system with controllable and observable linearization. Secondly, even when a nonlinear observer is available, it may not be directly applied to the output feedback design because the so-called separation principle does not hold usually in the nonlinear case. Moreover, the lack of observability or detectability of the linearized system also makes the conventional output feedback design methods based on “Luenberger-type” observers (Gauthier et al. (1992); Krener and Isidori (1983); Khalil and Saberi (1987)) no longer applicable. 4

To the best of our knowledge, there were few results available in the literature, which were devoted to the problem of global stabilization by output feedback for nonlinear systems with unstabilizable/undetectable linearization such as (1.1) except the two recent papers (Dayawansa (2002); Qian and Lin (2002b)). Qian and Lin (2002b) ﬁrst studied the global stabilization of a class of 2-dimensional high-order triangular systems, using smooth output feedback. They proposed a reduced-order, one-dimensional nonlinear observer that is not based on the separation principle but rather, based on the idea of coupled controller-observer design. By combining the adding a power integrator technique in (Lin and Qian (2000b)) (for the design of state feedback) with a nonlinear-gain observer, a smooth dynamic output compensator was designed to achieve global asymptotic stabilization. According to the result obtained in (Qian and Lin (2002b)), we now know that global output feedback stabilization of the homogeneous system (1.1) is possible when n = 2. Independently,

Dayawansa showed in the personal communication (Dayawansa (2002)) that with the help of the theory of homogeneous systems (Zubov (1964); Hahn (1967); Bacciotti (1992); Bacciotti and Rosier (1992)) and some elegant techniques from (Dayawansa (1992); Dayawansa et al. (1990); Kawski (1989, 1990, 1995); Hermes (1991a,b)), it is possible to prove the existence of globally stabilizing, homogeneous output feedback controllers for system (1.1) when n = p1 = p2 = 3. However, in the case of n ≥ 4, global stabilization of the homogeneous system (1.1) via output feedback is still a challenging problem that remains unsolved and largely open. This dissertation will focus on this challenging problem and gives some global and semi-global asymptotic stabilization results via smooth or nonsmooth feedback for the system (1.1) under certain conditions. 5

1.2 Motivations

Many physical systems of practical importance are inherently nonlinear and diﬃcult to be controlled. For instance, a class of underactuated, weakly coupled, unstable mechanical systems (Rui et al. (1997); Qian and Lin (2001a)) can be represented, after a change of coordinates and pre-feedback, as

x˙ 1 = x2 g x˙ = x3 + sin x 2 3 l 1

x˙ 3 = x4

x˙ 4 = u

y = x1, (1.2) which is in the form of (1.1) and has a unstabilizable and undetectable linearization. Moreover, the uncontrollable mode is associated with an eigenvalue on the right-half plane. As a result, this type of nonlinear systems cannot be stabilized by any smooth feedback, even locally. Therefore, the problem of how to control the nonlinear system (1.1) is not only theoretically challenging but also very interesting and important from a practical point of view.

It should be noted that the nonlinear systems such as (1.2) and (1.1) all belong to the so-called Hessenberg normal form (Cheng and Lin (2003)). Indeed, a necessary and sufficient condition was first characterized by Cheng and Lin (2003) and then generalized by Respondek (2003), for the existence of a change of coordinates (diffeo- morphism) and a state feedback control law transforming a smooth affine system

ξ˙ = f(ξ)+ g(ξ)w and y = h(ξ),

into the nonlinear system (1.1) with a suitable form of fi(·). Thus, (1.1) can be viewed as a special case of the generalized normal form of aﬃne systems when exact feedback linearization is not possible. 6

In addition to the aforementioned motivation, the work presented in this thesis is also stimulated by the well known fact from linear systems theory, namely, the linear observer design can be regarded as a dual of the state feedback design. In the case of nonlinear systems, it is certainly natural to ask the following fundamental question: is it possible to find a recursive way for the construction of an output feedback compensator for the nonlinear system (1.1), which can be viewed as a dual of the adding a power integrator technique? This important issue will be addressed in Chapter 3-7, where affirmative answers are given. The key lies in the construction of nonlinear observers, which is not trivial and requires an in-depth analysis. Keeping in mind that the separation principle is usually not true for nonlinear systems, we shall couple the nonlinear observer design with the controller construction, instead of designing the observer and controller separately. However, such a couple design method usually makes the construction of output feedback control laws much more involved and complicated. To better understand when the problem of global stabilization is solvable by output feedback, we recall the following two counter examples from the papers by Mazenc et al. (1994); Qian and Lin (2002b). Specifically, consider the nonlinear systems

x˙ 1 = x2

3 x˙ 2 = u + x2

y = x1 and

3 x˙ 1 = x2

5 x˙ 2 = u + x2

y = x1.

It was shown in the work (Mazenc et al. (1994); Qian and Lin (2002b)) that without imposing extra growth conditions on the unmeasurable states of the system, global 7 stabilization of nonlinear systems via output feedback is usually impossible. Since then, much subsequent research work has been focused on the output feedback stabilization of nonlinear systems under various structural or growth conditions. In order to remove the restrictive growth conditions, one should pursue, other than global stabilization result, a less ambitious control objective such as semi-global asymptotic stabilization, as observed by Teel and Praly (1994, 1995). In fact, they found that for nonlinear systems, uniform observability and global stabilizability by smooth state feedback imply semi-global stabilizability by smooth output feedback. In conclusion, semi-global, instead of global, stabilization by output feedback is perhaps the most realistic control objective that can be achieved in the case of nonlinear systems.

1.3 Contributions of This Dissertation

In this dissertation, we will investigate the problem of output feedback stabilization in the aforementioned two directions — global output feedback stabilization under suitable growth assumptions and semi-global output feedback stabilization without imposing any growth condition.

In Chapter 3, we prove that for the system (1.1) with p1 = ··· = pn and f1(·) = ··· = fn(·) ≡ 0, it is indeed possible to achieve global asymptotic stabilization by smooth output feedback. The proof is constructive and relies on a new output feedback design method that enables one to explicitly construct a homogeneous observer as well as a smooth state feedback controller. While the state feedback law is designed by employing the tool of adding a power integrator (Lin and Qian (2000a,b)), the observer design is new and carried out by developing a machinery, which can be viewed as a dual of the adding a power integrator technique, making it possible to assign the gains of the homogeneous observer recursively. Such design overcomes not only the obstacle caused by unobservability of the Jacobian lineariza- 8 tion of the high-order system (1.1), but also provides an iterative way to tune the gains of the homogeneous observer. Notably, the new observer design is substantially different from the traditional “Luenberger” or “high-gain” observer design in which the observer gain is determined directly by observability. The objective of Chapter 4 is to construct, under a homogeneous growth condition, a single smooth dynamic output compensator, which globally robustly stabilizes the entire family of uncertain nonlinear systems in the form of (1.1) with p1 = ··· = pn−1. The design of such an output feedback controller uses only the knowledge of the homogeneous bounding system of the uncertain nonlinear systems. As a result, global output feedback stabilization will be achieved in a robust fashion, that is, in a manner which is not sensitive to perturbations and parametric uncertainty in the system. This is one of the major differences between Chapter 3 and Chapter 4. The key for achieving robustness is the introduction of a rescaling technique with a subtle dilation, which transforms the original system into a rescaled one for which a dynamic output compensator can be constructed using the output feedback design method in Chapter 3, with a suitable twist, in particular, by discarding the system uncertainty when designing homogeneous observers. With the help of the rescaling technique, the uncertain nonlinearities can be dominated easily by tuning the rescaling factor. In Chapter 5, we study the question of when the uncertain system (1.1) with different pi’s is robustly stabilizable by output feedback. Realizing that in the case when pi’s are distinct, it is usually not possible to deal with the nonlinear system (1.1) by smooth feedback, even locally, we developed a nonsmooth, rather than smooth, output feedback design method to tackle the robust output feedback stabilization problem for the uncertain system (1.1). In fact, we show that for a family of uncertain nonlinear systems (1.1) satisfying homogeneous growth conditions, global robust stabilization can be achieved by nonsmooth output feedback. To deal with the system uncertainty, we introduce a subtle rescaling transformation motivated by the work in 9

Chapter 4. The new rescaling technique integrated with the nonsmooth output feedback design method (Qian and Lin (2004b)) leads to a nonsmooth output feedback control scheme that solves the problem of global robust stabilization. Again, the construction of a nonsmooth state feedback controller and a homogeneous observer relies only on the knowledge of the bounding system rather than the uncertain system. The result obtained in this chapter is a natural generalization of the work by Qian and Lin (2004b). In Chapter 6, we consider a family of uncertain nonlinear systems with controllable/observable linearization. It is proved that if the uncertain system is dominated by a triangular system that satisﬁes a linear growth condition with an output dependent growth rate, global robust stabilization is still achievable by smooth output feedback. The contribution of this chapter is two-fold: 1) it generalizes the output feedback stabilization result of planar systems previously obtained in (Qian and Lin

(2002b)) to the n-dimensional uncertain nonlinear system (6.1); 2) it removes the restriction that the growth rate be a polynomial function of the output required in the paper by Praly and Jiang (2003). There are two key ingredients in our output feedback design method. The ﬁrst one is the introduction of a rescaling transformation with a dynamic factor, which is motivated by the work presented in Chapters 4-5, where the power of the rescaling technique has been demonstrated when dealing with the uncertainties. The rescaling factor is involved with the system output and tuned on-line via a Riccati-like diﬀerential equation (this owes an inspiration to the work by

Praly (2003)). It turns out that the rescaling transformation with a dynamic factor is an eﬀective tool for the analysis and synthesis of the uncertain system (6.1) satisfying the growth condition (6.2). The other ingredient is the development of a recursive observer design algorithm that can be viewed as a dual of adding a integrator design for the construction of smooth state feedback controllers. The algorithm allows one to assign the robust observer gains in a step-by-step fashion. 10

The main result of Chapter 7 is that without imposing any growth conditions as those required in Chapters 4 and 5, there exists a nonsmooth dynamic output compensator that semi-globally asymptotically stabilizes the triangular system (1.1). The signiﬁcance of this work over the existing results can be summarized as follows. On the one hand, it generalizes the semi-global output feedback stabilization results in (Teel and Praly (1994, 1995)) for nonlinear systems that are assumed to be uniformly observable and smoothly stabilizable, to a wider class of non-uniformly observable and nonsmoothly stabilizable systems such as (1.1). On the other hand, it shows that the local output feedback stabilization result in (Qian and Lin (2004b)) can be extended, without requiring any extra condition on (1.1), to the semi-global case, which is certainly a substantial progress from either a theoretical or practical viewpoint. Finally, compared with the global output feedback stabilization results in Chapters 3-6 and the paper by Qian and Lin (2004b), the restrictive conditions such as p1 = ··· = pn−1 and a high-order global Lipschitz-like condition in Chapter 3, or those growth requirements imposed on the nonlinear system (1.1) in (Qian and Lin (2004b)) have all been removed. The price we paid is that only semi-global rather than global stabilizability is achieved.

Finally, we present in Chapter 8 some concluding remarks and future research directions. Chapter 2. PRELIMINARIES

This chapter collects some basic concepts and results from the Lyapunov stability theory and the existing literature on feedback stabilization. Special focuses are on the theory of homogeneous systems, which will be frequently used in the sequel.

2.1 Stability Theory

We begin by reviewing some well-known facts on the stability of nonlinear systems. Consider a nonlinear system of the form

x˙ = f(x) (2.1) where f(x) is a Cr (r ≥ 2) vector ﬁeld deﬁned on a neighborhood U ⊂ IRn of the origin. Assume that x = 0 is an equilibrium point, i.e., f(0) = 0. Let

∂f F = ∂x x=0 h i denote the Jacobian of f at x = 0. Then, the linearized system of the autonomous system (2.1) at x = 0 is given by x˙ = F x. (2.2)

The following result, known as the Lyapunov’s ﬁrst approximation theorem, gives suﬃcient conditions under which the stability of nonlinear system (2.1) at the origin can be drawn based on the stability of the linearized system (2.2).

Theorem 2.1. If all the eigenvalues of the Jacobian matrix F have negative real part, then the origin of the nonlinear system (2.1) is a locally asymptotically stable

11 12 equilibrium. If one of the eigenvalues of F has a positive real part, then system (2.1) is unstable at the origin.

The theorem above can be used to investigate the stabilizability problem of a nonlinear control system.

Theorem 2.2. Consider a nonlinear control system of the form

x˙ = f(x, u) (2.3) with the state x, deﬁned on a neighborhood U of the origin in Rn, and the control input u ∈ IRm, and the map f : IRn × IRm → IRn being a Cr(r ≥ 2) map and f(0, 0) = 0. Its linearization at the origin (x, u)=(0, 0) is given by

x˙ = F x + Gu, (2.4)

∂f ∂f where F = (0, 0) and G = (0, 0). ∂x ∂u i) If (2.4) is asymptotically stabilizable by a linear feedback law u = Kx, then u = Kx is also a local asymptotic stabilizer for the control system (2.3);

ii) If the pair (F,G) is uncontrollable and the uncontrollable modes are associated with eigenvalues on the right-half plane, then there does not exist any linear or smooth state feedback law u = u(x) with u(0) = 0, which stabilizes the

nonlinear control system (2.3).

In the case when the Jacobian matrix F has some eigenvalues on the imaginary axis, the so-called critical case, the ﬁrst approximation theorem of Lyapunov is not applicable anymore and the stability analysis becomes much involved. Fortunately, center manifold theory (see, for instance, Carr (1981)) is useful and can employed, in many situations, to deal with the critical case. Suppose the matrix F has n0 eigenvalues with zero real part, n− eigenvalues with negative real part, and no eigenvalue with positive real part (otherwise, the origin 13 x = 0 would be an unstable equilibrium of (2.1)). It is well known from linear algebra that the domain of the linear map F can be decomposed into the direct sum of two invariant subspaces E0 and E− of dimension n0 and n−, respectively. If the linear map F , is viewed as a representation of the diﬀerential (at x = 0) of the nonlinear

n map f : U → IR , its domain is the tangent space T0U to U at x = 0, and the two subspaces in question can be viewed as subspaces of T0U satisfying

0 − T0U = E ⊕ E .

Deﬁnition 2.3. A Cr(r ≥ 2) submanifold S of U is said to be locally invariant for the nonlinear system (2.1), if for each x0 ∈ S, there exists t1 < 0 < t2 such that the trajectory x(t) of (2.1) satisfying x(0) = x0 is such that x(t) ∈ S for all t ∈ (t1, t2).

Deﬁnition 2.4. Let x = 0 be an equilibrium of the system (2.1). A manifold S, passing through x = 0, is said to be a center manifold for the system (2.1) at x = 0, if it is locally invariant and the tangent space to S at x = 0 is exactly E0.

For the nonlinear system (2.1), one can always choose coordinates in U such that it can be represented in the form

y˙ = Ay + g(y, z)

z˙ = Bz + h(y, z), (2.5) where A is an n− × n− matrix having all eigenvalues with negative real part, B is an n0 × n0 matrix having all eigenvalues with zero real part, and the functions g and h are Cr(r ≥ 2) functions vanishing at (y, z)=(0, 0) together with all their ﬁrst-order derivatives. Thus, there is no loss of generality in assuming that the system (2.1) is of the form (2.5) if appropriate coordinates are chosen. The existence of the center manifolds for the nonlinear system (2.5) is guaranteed by the following theorem. 14

0 Theorem 2.5. There exist a neighborhood V ⊂ IRn of z = 0 and a smooth map

− π : V → IRn such that

− S = (y, z) ∈ IRn × V : y = π(z) n o is a center manifold for (2.5).

By definition, a center manifold for the system (2.5) passes through (y, z)=(0, 0) and is tangent to the subset of points whose y-coordinate is equal to 0. Thus, the map π satisfies ∂π π(0) = 0, (0) = 0. (2.6) ∂z Moreover, this manifold is locally invariant for (2.5), which imposes on the map π the constraint ∂π (Bz + h(π(z), z)) = Aπ(z)+ g(π(z), z). (2.7) ∂z This relationship is deduced by differentiating with respect to time the system trajectory (y(t), z(t)) of (2.5) that belongs to the manifold, i.e., satisfies y(t)= π(z(t)).

In other words, a center manifold for (2.5) can be described as the graph of a map y = π(z) satisfying the partial differential equation (2.7), as well as the constraints specified by (2.6). Note that the previous statement describes only the existence, but not the uniqueness of a center manifold for (2.5). In fact, a nonlinear system may have infinite many center manifolds, as illustrated in the book (Carr (1981)). Moreover, if g and h are C∞ functions, the nonlinear system (2.5) has a Ck center manifold for any k ≥ 2, but not necessarily a C∞ center manifold.

Theorem 2.6. Suppose y = π(z) is a center manifold for (2.5) at (y, z)=(0, 0). Let

(y(t), z(t)) be a trajectory of (2.5). There exists a neighborhood U 0 of (y, z)=(0, 0) and real numbers M > 0 and K > 0 such that, if (y(0), z(0)) ∈ U 0, then

||y(t) − π(z(t))|| ≤ Me−Kt||y(0) − π(z(0))||, 15 for all t ≥ 0, as long as (y(t), z(t)) ∈ U 0.

This theorem shows that any trajectory of the system (2.5) starting at a point suﬃciently close to the origin (y, z) = (0, 0) converges exponentially to a trajectory which belongs to the center manifold.

The following statement shows to what extent center manifolds are useful in the analysis of the asymptotic properties of the system (2.5) near (y, z) = (0, 0). Recall that, by deﬁnition, any trajectory of (2.5) starting at a point y0 = π(z0) of a center manifold can be described in the form

y(t)= π(ζ(t)), z(t)= ζ(t) with ζ(t) the solution of the reduced system

ζ˙ = Bζ + h(π(ζ),ζ), (2.8) satisfying the initial condition ζ(0) = z0. The essence of the following result is that the asymptotic behavior of (2.5), for small initial conditions, is completely determined by its behavior for initial conditions on the center manifold, i.e., by the asymptotic behavior of (2.8).

Theorem 2.7. Suppose ζ = 0 is a stable (respectively, asymptotically stable, unstable) equilibrium of (2.8). Then (y, z)=(0, 0) is a stable (respectively, asymptotically stable, unstable) equilibrium of (2.5).

Thus, the stability problem of the (n0 + n−)-dimensional system (2.1) can be reduced to the problem of the n0-dimensional system (2.8). Although the center manifold theory has provided a powerful approach for the stability analysis of nonlinear systems in the critical case, it becomes useless in case when all the eigenvalues of the Jacobian matrix F are located on the imaginary axis, i.e., n− = 0 and n = n0. In this case, we may resort to some higher-order 16 approximations to deal with the critical case. Indeed, the following result known as

Malkin’s stability theorem (e.g., see the book by Hahn (1967)) provides a suﬃcient condition for the stability of the system (2.5). Rewrite the system (2.5) as follows:

y˙ = Ay + p(z,y)

z˙ = h(z,y), (2.9) where A is an (n− × n−) matrix, h(·) and p(·) are smooth functions with h(0, 0)=0 and p(0, 0) = 0. With respect to the z-subsystem, we need to introduce the following concept.

Deﬁnition 2.8. The equilibrium x = 0 of the system (2.1) is said to be asymptotically stable according to the N th order approximation if there exists an integer N > 0 such that the equilibrium x = 0 of any perturbed system of the following form

x˙ = f(x)+ O(||x||N+1)1, is always locally asymptotically stable.

Remark 2.9. It is clear that the equilibrium x = 0 of the system (2.1) is asymptotically stable according to the N th (N ≥ 1) order approximation if the Jacobian matrix F is a Hurwitz matrix. Moreover, the following one-dimensional system

x˙ = −xm, with m being an odd integer, is asymptotically stable according to the N th (N ≥ m) order approximation at x = 0.

Now, we are ready to introduce the so-called Malkin’s stability theorem which can be summarized in the following statement.

N N N N 1 The notation O(||x|| ) (or o(||x|| )) denotes the term O(||x|| ) (or o(||x|| )) satisfying ∃M > N N O(||x|| ) o(||x|| ) 0, N

Theorem 2.10. Consider the nonlinear system (2.9). Suppose that

1. A is an (n− × n−) matrix having all eigenvalues with negative real part,

2. The equilibrium z = 0 of the systemz ˙ = h(z, 0) is asymptotically stable according to the N th (N ≥ 1) order approximation,

3. p(z, 0) = O(||z||N+1) and p(0,y)= O(||y||2).

Then, the system (2.9) has a locally asymptotically stable equilibrium at (z,y) = (0, 0).

It should be pointed out that for a large number of highly nonlinear systems, it may not be easy to verify the asymptotic stability according to the N th order approximation. Therefore, a natural question is to find other higher-order approximations that may be easier to be computed. One of them is the so-called homogeneous approximation to be discussed extensively in the next section. Finally, we conclude this section by recalling Kurzweil’s stability results which generalize the Lyapunov stability theory to the non-Lipschitz continuous framework. It is well known that the classical Lyapunov stability theory and the well-known concepts such as stability and asymptotic stability in the sense of Lyapunov (e.g. see Hahn (1967)) can only be applied to a nonlinear differential equation whose solution from any initial condition is unique. However, if f in (2.1) is a non-Lipschitz continuous map, the continuous system (2.1) may have more than one trajectory starting from a given initial condition (e.g.x ˙ = x2/3 with x(0) = 0), it is therefore necessary to introduce different (from Lyapunov) notions of stability and asymptotic stability within the continuous framework. Kurzweil (1956) introduced the new notions of stability for continuous systems (2.1) and established Lyapunov’s second theorem as well as the converse theorem of Lyapunov on stability, without requiring uniqueness of the trajectories of (2.1) for any initial condition. In what follows, we recall Kurzweil’s 18 definition on global strong stability (see Kurzweil (1956); Rosier (1992)), which will be used later on.

Deﬁnition 2.11. The trivial solution x = 0 of the continuous system (2.1) is globally strongly stable (GSS) if there are two functions B : IR+ → IR+ and T : IR+ × IR+ →

+ IR with B being increasing and lims→0 B(s) = 0, such that for all β > 0 and ε> 0, for every solution x(t) of the continuous system (2.1) deﬁned on [0, t1) (0 < t1 ≤ +∞), with ||x(0)|| ≤ β, there exists a solution z(t) of the continuous system (2.1) deﬁned on [0, +∞) satisfying the following:

(1) z(t)= x(t), t ∈ [0, t1); (2) kz(t)k ≤ B(β), ∀t ≥ 0; (3) kz(t)k < ε, ∀t ≥ T (β,ε).

Remark 2.12. In the continuous framework, the reason why x = 0 is called the trivial solution, rather than the equilibrium, of the continuous system (2.1) is due to the following consideration: it can happen that the trajectories of the continuous system (2.1) starting at x(0) = 0 will not stay at x = 0 forever, because of non-

3 1/3 uniqueness of the solutions. For example, the continuous systemx ˙ = 2 x has more than one solutions starting at x(0) = 0. One of them is x(t) = t3/2 which tends to inﬁnity as t →∞. Thus, x = 0 is not an equilibrium of the system.

This deﬁnition is clearly a generalization of global asymptotic stability in the sense of Lyapunov for a system of the form (2.1) that has a unique solution. Indeed, with the help of the notion of globally strongly stability, Kurzweil proved the following result (e.g. see Kurzweil (1956); Qian and Lin (2001a)).

Theorem 2.13. Consider the system (2.1) with f being a continuous map and f(0) = 0. The trivial solution x = 0 of system (2.1) is globally strongly stable if and only if there exists a C∞ Lyapunov function V (x) : IRn → IR, which is positive 19 deﬁnite2 and proper3, such that

∂V f(x) < 0, ∀x ∈ IRn \{0}. ∂x

Obviously, Theorem 2.13 is analogous to Lyapunov’s second theorem as well as its converse theorem and contains globally asymptotic stability as its special case when the solution is unique. Intuitively speaking, Theorem 2.13, together with Deﬁnition 2.11, implies that one can deal with the stability problem via Lyapunov’s second method no matter whether the system has a unique solution or not.

2.2 Homogeneous Systems Theory

Aside from aesthetic aspects, mathematical objects that exhibit homogeneity often have useful properties in practice. For example, as in the case of linear systems, local asymptotic stability of homogeneous systems is equivalent to global asymptotic stability of homogeneous systems. Over the past decades, stability and feedback stabilization of homogeneous systems have been widely studied and some important results have been obtained (e.g. see Hermes (1991a); Kawski (1990); Dayawansa (1992); Dayawansa et al. (1990); Coron and Praly (1991); Rosier (1992); Tzamtzi and Tsinias (1999); Lin and Qian (2000a,b); Qian and Lin (2001a,b)). Moreover, the problem of stability and stabilization for a class of nonhomogeneous systems can also be solved easily if their corresponding homogeneous approximations can be found. In this section, we give a brief review about homogeneous systems theory.

2.2.1 Homogeneous Functions and Vector Fields

We begin with the deﬁnitions of homogeneous functions and homogeneous vector ﬁelds, and then review a number of important properties related to homogeneous

n 2 A function V : IR → IR is called positive deﬁnite if V (x) > 0, ∀x 6= 0 and V (0) = 0. n 3 A nonnegative function V : IR → IR is called proper if for each a > 0, the set V −1([0,a]) is n compact in IR . 20 systems. For a detailed discussion, the reader is referred to, for example, the references (Hahn (1967); Zubov (1964); Hermes (1991a); Kawski (1990, 1995); Dayawansa (1992); Bacciotti (1992); Bacciotti and Rosier (1992)).

n Deﬁnition 2.14. For a ﬁxed choice of coordinates x = (x1, ··· , xn) in IR , and ∆ positive real numbers (r1, ··· ,rn) = r, a one-parameter family of dilations is a map

r + n n r r1 rn ∆ : IR × IR → IR , deﬁned by ∆εx =(ε x1, ··· ,ε xn).

Deﬁnition 2.15. For a given dilation ∆r and a positive real number s, a continuous

n r function V : IR → IR is ∆ -homogeneous of degree s, denoted by V ∈ Hs, if V ◦ ∆r = εsV . A continuous vector ﬁeld f(x) = f (x)( ∂ ) is ∆r-homogeneous ε j ∂xj P r of degree s, denoted by f ∈ ns, if fj ∈ Hs+rj , j = 1, ··· , n. For p > 1, the ∆ -

n ∆ p/ri 1/p homogeneous p-norm of x ∈ IR is ||x||∆r,p =( i |xi| ) . P Homogeneous functions and homogeneous vector ﬁelds thus deﬁned have a number of important properties, as summarized below.

Theorem 2.16. Homogeneous functions and homogeneous vector ﬁelds have the following properties:

r 1. If V1 ∈ Hs1 ,V2 ∈ Hs2 with respect to ∆ε, then V1 × V2 ∈ Hs1+s2 .

n + n + r 2. If V1 : IR → IR ∈ Hs1 ,V2 : IR → IR ∈ Hs2 with respect to ∆ε, then s s s1 s2 V1 + V2 ∈ Hs.

n r 3. If V : IR → IR ∈ Hs with respect to ∆ε (where r = (r1, ··· ,rn)) and Wi :

m p IR → IR ∈ Hri with respect to ∆ε (where p =(p1, ··· ,pm)), i =1, ··· , n, then

p the composite function V (W1, ··· , Wn) ∈ Hs with respect to ∆ε.

r r 4. If f ∈ ns with respect to ∆ε and Wi ∈ Hri with respect to ∆ε, i = 1, ··· , n,

r then the composite vector ﬁeld f(W1, ··· , Wn) ∈ ns with respect to ∆ε.

1 ∂V 5. If V ∈ Hs ∩ C , then ∈ Hs−ri , i =1, ··· , n. ∂xi 21

∂V 6. If V ∈ H ∩ C1, f ∈ n , then L V = f ∈ H . s1 s2 f ∂x s1+s2

1 1 7. If g ∈ ns1 ∩ C , f ∈ ns2 ∩ C , then the Lie bracket [f,g] ∈ ns1+s2 .

r r 8. Euler’s formula: For a dilation ∆ε, deﬁne the ∆ -homogeneous Euler vector ﬁeld (of degree 0) as v(x)= n r x ∂ . Thus for a C1 function V , 1 i i ∂xi P ∂V V ∈ Hs ⇐⇒ LvV = rixi = sV ∂xi Xi 2.2.2 Stability of Homogeneous Systems

With the notions of homogeneous functions and homogeneous vector ﬁelds, one can introduce the following useful concept.

Deﬁnition 2.17. A system of the form (2.1) with f being continuous and homogeneous of degree s with respect to a given dilation ∆r and f(0) = 0, is called a ∆r-homogeneous system of degree s.

The following result can be proved straightforwardly.

Theorem 2.18. If x(t) is the trajectory of the ∆r-homogeneous system (2.1) starting

r s at x(0) = x0, then y(t) = ∆εx(ε t) is also the trajectory of (2.1) starting at y(0) =

r ∆εx0.

Remark 2.19. The result above means that a change of scale on time t can be compensated by a dilation on the state space and vice versa.

With the aid of Theorem 2.18, it is not diﬃcult to prove the following important conclusion on the stability of homogeneous systems.

Theorem 2.20. The trivial solution x = 0 of the ∆r-homogeneous system (2.1) is locally asymptotically stable if and only if the trivial solution x = 0 of (2.1) is globally asymptotically stable. 22

It is natural to ask if there exists a homogeneous Lyapunov function when the system under consideration is homogeneous. The following result gives an aﬃrmative answer even in the case that the homogeneous system does not have a unique solution.

Theorem 2.21. (Rosier (1992)) If the trivial solution x = 0ofthe∆r-homogeneous system (2.1) of degree s is globally strongly stable, there exists a ∆r-homogeneous

Lyapunov function for (2.1). More precisely, for any given integer p> 0, there exists

p n ∞ n a function V ∈ C (IR ) ∩ C (IR \{0}) ∩ Hk, which is positive deﬁnite and proper, such that ∂V V˙ | = f(x) < 0, ∀x =06 , (2.1) ∂x where k>p · max{r1, ··· ,rn} .

As a consequence of Theorem 2.21, the following result on the robustness of stable homogeneous systems can be easily obtained.

Theorem 2.22. (Hermes (1991a); Rosier (1992)) Consider the nonlinear system (2.1). Suppose the following assumptions hold.

1. f : IRn → IRn is a ∆r-homogeneous (of degree s) continuous vector ﬁeld and f(0) = 0;

2. For a real number k and an integer p such that p> 0 and k>p·max{r1, ··· ,rn}, a continuous vector ﬁeld g : IRn → IRn with g(0) = 0 satisﬁes

g (∆rx) lim i ε =0, i =1, ··· , n, ε→0 εs+ri

uniformly on x ∈ Σ, where

n Σ= x ∈ IR : ||x||∆r,k =1 n o is a compact manifold. That is, g(·) can be viewed as a vector ﬁeld whose homogeneous degree with respect to the dilation ∆r is larger than s; 23

3. The solution x = 0 of the homogeneous system (2.1) is locally asymptotically

stable.

Then, the solution x = 0 of the perturbed system

x˙ = f(x)+ g(x) (2.10) is locally asymptotically stable, too.

Example 2.23. Consider the following three-dimensional system

5 2 3 x˙ 1 = −x2 − x1x3 + x3

5 2 x˙ 2 = x1 − x2x3

3 2 3 x˙ 3 = −x3 + x3x1. (2.11)

5 2 5 2 3 3 2 3 In fact, let f(x) = col(−x2 −x1x3, x1 −x2x3, −x3) and g(x) = col(x3, 0, x3x1). It is easy to see that f(x) and g(x) are ∆r-homogeneous of degree 4 and 5, respectively, where r = (1, 1, 2). Now, we check the stability properties of the zero solution of the homogeneous ∂V systemx ˙ = f(x). Choose V (x)= x6 +x6 +x6. Clearly, V˙ (x)= f(x)= −6x2(x6 + 1 2 3 ∂x 3 1 6 6 x2 + x3). By the LaSalle’s invariant principle, the trivial solution x = 0 is locally asymptotically stable solution. It follows from Theorem 2.22 immediately that x =0 is also a locally asymptotically stable solution of (2.11). Had one used Taylor expansion, that is, had the common dilation been used

¯ 2 3 2 3 (where r = (1, 1, 1)), one would have f(x) = col(−x1x3 + x3, −x2x3, −x3) and 5 5 2 3 ¯ g¯(x) = col(−x2, x1, x3x1). Clearly, the zero solution ofx ˙ = f(x) is not locally asymptotically stable. Indeed, (x1, x2, x3)=(c1, c2, 0) is a solution for any constants c1, c2. In this case, the “standard” expansion provides no useful information. In other words, the choice of dilation plays a crucial role in the homogeneous approximation and the resulting stability analysis. 24

2.3 Useful Inequalities

In this section, we introduce several technical lemmas related to Young’s inequality, which will play an important role in the subsequent developments and will be used frequently in the following chapters. To begin with, let us recall the well-known Young’s inequality. For any real numbers x> 0 and y > 0,

xp yq xy ≤ + , (2.12) p q where p> 1 and q > 1 are H¨older conjugate numbers, i.e., 1/p +1/q = 1. Using the Young’s inequality (2.12), it is easy to prove the following two lemmas.

Lemma 2.24. Given positive real numbers x, y, m, n, a, b, the following inequality holds:

m n m+n n m + n − m m+n − m m+n ax y ≤ bx + ( ) n a n b n y . m + n m

This inequality indicates that the coeﬃcient of xm+n, i.e., b > 0 can be any arbitrarily small real number, as long as the coeﬃcient of ym+n is adjusted accordingly. This is a crucial property that will be used later.

Lemma 2.25. Given positive real numbers x, y, m, n, a, b, the following inequality holds:

m n m m+n m+n n m+n m+n abx y ≤ a m x + b n y . m + n m + n

Lemma 2.25 shows how the cross term abxmyn involving the parameters a and b, can be dominated by xm+n whose coeﬃcient depends only on a and by ym+n whose coeﬃcient depends only on b.

Lemma 2.26. Let x1, ··· , xn and p be positive real numbers. Then,

p p−1 p p (x1 + ··· + xn) ≤ max(n , 1)(x1 + ··· + xn). 25

Lemma 2.27. Let x and y be any real numbers and p> 0 an odd integer. Then,

(x − y)p+1 −(x − y)(xp − yp) ≤ − . 2p−1

Lemma 2.28. Suppose p ≥ 1 is an odd integer, then the following inequality holds:

|a − b|p ≤ 2p−1|ap − bp|, ∀a, b ∈ IR.

Lemma 2.29. For all x, y ∈ IR and any odd integer p ≥ 1,

|xp − yp| ≤ p|x − y|(xp−1 + yp−1).

The proofs of Lemmas 2.26—2.29 are straightforward and therefore left to the reader as an exercise. Chapter 3. SMOOTH OUTPUT FEEDBACK DESIGN

The purpose of this chapter is to study the problem of global stabilization by smooth output feedback, for a class of n-dimensional homogeneous systems whose Jacobian linearization is neither stabilizable nor detectable. A new output feedback control scheme is proposed for the explicit design of both homogeneous observers and controllers. While the smooth state feedback control law is constructed based on the tool of adding a power integrator, the observer design is new and carried out by developing a machinery, which makes it possible to assign the observer gains one-by-one, in an iterative manner. Such design philosophy is fundamentally different from that of the traditional “Luenberger” observer in which the observer gain is determined by observability. In the case of linear systems, our design method provides not only a new insight but also an alternative solution to the output feedback stabilization problem. For a class of high-order non-homogeneous systems, we further show how the proposed design method, with an appropriate modification, can still achieve global output feedback stabilization. Examples and simulations are given to demonstrate the main features and effectiveness of the proposed output feedback control schemes.

3.1 Introduction

In this chapter, we study the problem of global asymptotic stabilization by smooth output feedback for a class of homogeneous systems described by equations of the form

p x˙ 1 = x2

26 27

. .

p x˙ n−1 = xn

x˙ n = u

y = x1, (3.1) where u ∈ IR and y ∈ IR are the system input and output, respectively, and p ≥ 1 is an odd integer. Coordinate-free geometric conditions have been characterized recently in (Cheng and Lin (2003)) for the existence of a change of coordinates (diﬀeomorphism) and a state feedback law transforming a smooth aﬃne system

η˙ = f(η)+ g(η)v and y = h(η), (3.2) into a high-order Brunovsky canonical form (3.1) or system (3.37) (when exact feedback linearization is not possible), and hence need not to be repeated here. When n = 3 and p = 3, local and global asymptotic stabilization of system (3.1) via state feedback were investigated in (Crouch and Irving (1983); Byrnes and Isidori (1989)), respectively. In the n-dimensional case, a globally stabilizing homogeneous state feedback control law for (3.1) was constructed explicitly, for instance, by the adding a power integrator design (Lin and Qian (2000b)). The principal objectives of Chapter 3 are twofold: to prove the existence of a smooth dynamic output compensator

xˆ˙ = η(ˆx, y), xˆ ∈ IRn,

u = u(ˆx, y), (3.3) that globally asymptotically stabilizes the homogeneous system (3.1), and to develop a systematic design method for the explicit construction of such a globally stabilizing, output feedback controller. 28

When p = 1, system (3.1) reduces to a chain of linear integrators or the so-called

Brunovsky canonical form, which is controllable and observable. By the well-known separation principle, global asymptotic stabilization of (3.1) with p = 1 is always solvable by output feedback. However, in the case of p > 1, system (3.1) becomes a highly nonlinear system (indeed, a homogeneous system, e.g., see Hahn (1967);

Bacciotti (1992); Hermes (1991a); Kawski (1989, 1990); Dayawansa (1992); Rosier (1992)) whose Jacobian linearization is neither stabilizable nor detectable. For the n- dimensional homogeneous system (3.1), a fundamental question of whether it is possible to achieve global asymptotic stabilization by smooth output feedback has not been addressed in the literature, and remained unanswered until now. To the best of our knowledge, there are few results available, which are devoted to the problem of global output feedback stabilization for nonlinear systems with unstabilizable/undetectable linearization such as (3.1), except the two recent papers by Qian and Lin (2002b);

Dayawansa (2002). Qian and Lin (2002b) studied global stabilization of a class of high-order triangular systems in the plane, using smooth output feedback. Due to the lack of controllability and observability in the first approximation, output feedback control of nonlinear systems is a very difficult problem that has not been well studied and is much less understood. The essential obstacle is the lack of effective observer design methods for linearly unobservable systems (Krener and Xiao (2002)). In particular, the traditional “Luenberger-type” nonlinear observer suggested in (Krener and Isidori (1983)) or the “high-gain” observer proposed in (Khalil and Saberi (1987); Gauthier et al. (1992); Krener and Kang (2003)) cannot be applied directly. Thus, how to design a global observer for the high-order planar systems was a key issue that must be addressed. Qian and Lin (2002b) made a first attempt and proposed a reduced-order, one-dimensional nonlinear observer that is not based on the separation principle but rather, based on the idea of coupled controller-observer design. By combining the 29 adding a power integrator technique in (Lin and Qian (2000b)) (for the design of state feedback) with a nonlinear-gain observer, a smooth dynamic output compensator is designed achieving global asymptotic stabilization. Using the result obtained in (Qian and Lin (2002b)), we now know that global output feedback stabilization of the homogeneous system (3.1) is possible when n = 2. Independently, Dayawansa showed in the personal communication (Dayawansa (2002)) that using homogeneous systems theory (Hahn (1967); Bacciotti (1992)) and some elegant techniques from (Dayawansa (1992); Dayawansa et al. (1990); Kawski (1989, 1990); Hermes (1991a)), it is possible to prove the existence of globally stabilizing, homogeneous output feedback controllers for system (3.1) when n = p = 3. However, in the case of n ≥ 4, global stabilization of the homogeneous system (3.1) via output feedback is still a challenging problem that remains unsolved and largely open. At present there is no systematic design methods nor insight for tackling problems of this kind.

In this chapter, we shall address this question and provide a satisfactory answer. In particular, we prove that for the homogeneous system (3.1), it is indeed possible to achieve global asymptotic stabilization by smooth output feedback. The proof is constructive and relies on a new output feedback design method that enables one to explicitly construct a homogeneous observer as well as a smooth state feedback controller. While the state feedback law is designed by employing the tool of adding a power integrator (Lin and Qian (2000b,a)), the observer design is new and carried out by developing a machinery, which can be viewed as a dual of the adding a power integrator technique, making it possible to assign the gains of the homogeneous observer recursively. Such design overcomes not only the obstacle caused by unobservability of the Jacobian linearization of the high-order system (3.1), but also provides an iterative way to tune the gains of the homogeneous observer. Notably, the new observer design is substantially diﬀerent from the traditional “Luenberger” or “high-gain” observer design in which the observer gain is determined directly by 30 observability. In the case of linear observable systems, our design method provides a new insight and leads to an interesting alternative to the design of linear observers and output compensators. As a further development, we demonstrate how the new design method can be naturally generalized to a class of high-order nonlinear systems that are not necessarily homogeneous, achieving global asymptotic stabilization by smooth output feedback. This chapter is organized as follows. In Section 3.2, a non-separation principle based algorithm is presented, illustrating how a linear output feedback controller can be recursively designed, in a step-by-step fashion, for controllable and observable linear systems. The new ingredient is the development of an iterative design procedure for determining the observer gains. Using the insights gained from this study, we develop in Section 3.3 a novel output feedback control scheme that is nothing but a nonlinear enhancement of what we obtained in Section 3.2. The new feedback design approach enables one to construct both smooth state feedback controllers and homogeneous observers, in an iterative manner, resulting in a solution to the problem of global stabilization of the homogeneous system (3.1) by smooth output feedback. Section 3.4 discusses how the stabilization result obtained in Section 3.3 can be extended, under appropriate conditions, to a class of high-order, non-homogeneous systems with unstabilizable and undetectable linearization. Several examples and simulations are given in Section 3.5 to demonstrate the main features and eﬀectiveness of the proposed output feedback control schemes. Concluding remarks are drawn in

Section 3.6.

3.2 Output Feedback Design of Linear Systems — A Non-Separation Principle Paradigm

It is well known that output feedback stabilization of a linear system is solvable if the system is stabilizable and detectable. The linear output feedback control problem 31 can be solved by the so-called separation principle which allows the design of a state feedback control law to be separated from the observer design. In the nonlinear case, such a design philosophy does not work most of time because the separation principle usually does not hold. As a matter of fact, for many nonlinear control systems, global stabilizability via state feedback plus observability may not imply global stabilizability by output feedback. In this section we propose a new strategy, not based on the separation principle, to tackle the output feedback stabilization problem for a linear system in the Brunovsky form

x˙ = Ax + Bu, x ∈ IRn, u ∈ IR

y = Cx (3.4) where 0 1 ··· 0 0 ...... A =  . . . .  , B =  .  , C =[1 0 ··· 0],  0 0 ··· 1   0       0 0 ··· 0   1          which is controllable and observable. As we shall see, the new linear output feedback control scheme is not only interesting in its own right (as it provides a way to design a new type of high-gain observers) but also has a distinguished feature, namely, it has a valid nonlinear counterpart that proves to be very eﬀective, particularly, when dealing with a class of highly nonlinear systems such as (3.1) and (3.37). To see how an alternative solution to the output feedback stabilization of (3.4) can be derived, we take inspiration from the adding a power integrator design proposed in (Lin and Qian (2000a,b)), which is the starting point of our output feedback design. The main idea is quite simple—to design both controller and observer step-by-step, in an iterative manner.

A recursive design can be proceeded as follows. First, using the adding a integrator 32 design, one can construct easily a linear state feedback control law of the form

∗ xn+1 = −anξn = −b1x1 −···− bnxn, (3.5)

∗ where ξi = xi − xi , i =1, 2, ··· , n, and

∗ ∗ ∗ x1 =0, x2 = −a1ξ1, ··· , xn = −an−1ξn−1,

with ai and bi being known constants. The resulted closed-loop system (3.4)-(3.5) is such that ˙ 2 2 ∗ Un ≤ −2(ξ1 + ··· + ξn)+ ξn(u − xn+1), (3.6)

1 2 2 where Un = 2 (ξ1 + ··· + ξn) is a Lyapunov function. Since the state x(t) of system (3.4) is not measurable, to implement the stabilizing controller (3.5) we design a generalized high-gain observer of the form

xˆ˙ 1 =x ˆ2 + L1(x1 − xˆ1) . .

xˆ˙ n−1 =x ˆn + Ln−1 ··· L1(x1 − xˆ1)

xˆ˙ n = u + Ln ··· L1(x1 − xˆ1), (3.7)

where the gain parameters L1 > 0, ··· , Ln > 0 are to be determined later, step-by- step. Clearly, (3.7) reduces to a traditional high-gain observer when L1 = ··· = Ln ≡ L. By the certainty equivalence principle, the following controller can be implemented:

u = −b1xˆ1 −···− bnxˆn. (3.8)

In what follows, we show that by assigning recursively the observer gains L1, ··· , Ln, in a delicate manner, the dynamic output compensator (3.7)–(3.8) would stabilize a chain of integrators (3.4). 33

To this end, let ei = xi − xˆi be the estimate error. Then, the error dynamics is given by

e˙1 = e2 − L1e1 . .

e˙n−1 = en − Ln−1 ··· L1e1

e˙n = −Ln ··· L1e1. (3.9)

Using the completion of square, it is straightforward to deduce from (3.6), (3.5) and (3.8) that

˙ 2 2 Un ≤ −2(ξ1 + ··· + ξn)+ ξn(b1e1 + ··· + bnen)

2 2 2 2 ≤ −(ξ1 + ··· + ξn)+ c0(e1 + ··· + en), (3.10) where c0 > 0 is a ﬁxed constant. Now, consider the nonsingular transformation

e˜1 = e1, e˜2 = e2 − L2e1, ··· , e˜n = en − Lnen−1. (3.11)

In the coordinates of ξ ande ˜, the inequality (3.10) can be rewritten as

n ˙ 2 2 2 2 2 Un ≤ − ξi + nc0 e˜1 +(˜e2 + L2e˜1)+ ··· Xi=1 h 2 2 2 2 2 2 +(˜en + Lne˜n−1 + ··· + Ln ··· L2e˜1) (3.12) n i 2 2 2 2 ≤ − ξi + c1(L2, ··· , Ln)˜e1 + ··· + cn−1(Ln)˜en−1 + cne˜n Xi=1 where c1(·), ··· ,cn−1(·) are positive real constants dependent of Li’s, while cn > 0 is a real constant independent of Li’s. Note that in thee ˜–coordinate, the error dynamics becomes

e˜˙ 1 = −L1e1 + e2 . .

e˜˙ n−1 = −Ln−1en−1 + en

e˜˙ n = −Lnen. (3.13) 34

1 2 2 Choose the Lyapunov function Wn = 2 (˜e1 + ··· +e ˜n). Then, a direct calculation yields

˙ 2 2 Wn = [˜e1e2 − L1e˜1]+[˜e2e3 − L2e˜2 − L2e˜2(e2 − e˜2)] +

···

2 +[˜en−1en − Ln−1eñ−1 − Ln−1eñ−1(en−1 − eñ−1)]

2 −[Lneñ + Lneñ(en − eñ)]. (3.14)

By completing the square, it is easy to deduce from (3.11) that (for i =2, 3, ··· , n)

|e˜i−1ei| ≤ |e˜i−1| Li ··· L2|e˜1| + ··· + Li|e˜i−1| + |e˜i| 2 2 ≤ c¯1(L2, ··· , Ln)˜e1 + ··· +¯ci(Li+1, ··· , Ln)˜ei , (3.15) wherec ¯1(L2, ··· , Ln) > 0, ··· , c¯n−1(Ln) > 0 are constants dependent of Li’s, while the constantc ¯n > 0 is independent of Li’s. Similarly (for i =2, 3, ··· , n),

| − Lie˜i(ei − e˜i)| ≤ Li Li ··· L2|e˜1| + ··· + Li|e˜i−1| |e˜i| 2 2 ≤ cˆ1(L2, ··· , Ln)˜e1 + ··· +ˆci(Li+1, ··· , Ln)˜ei , (3.16) wherec ˆ1(L2, ··· , Ln) > 0, ··· , cˆn−1(Ln) > 0 are real constants dependent of Li’s, but cˆn > 0 is a constant independent of Li’s. Substituting (3.15)–(3.16) into (3.14), we arrive at

˙ 2 2 2 2 2 Wn ≤ −(L1e˜1 + ··· + Lneñ)+˜c1(L2, ··· , Ln)˜e1 + ··· +˜cn−1(Ln)˜en−1 +˜cneñ, (3.17) wherec ˜1(L2, ··· , Ln) > 0, ··· , cñ−1(Ln) > 0 are constants dependent of Li’s, but cñ > 0 is a constant independent of Li’s.

Finally, choose Vn(ξ, e˜) = Un(ξ)+ Wn(˜e) for the closed-loop system. Then, it follows from (3.12) and (3.17) that

˙ 2 2 2 Vn ≤ −[ξ1 + ··· + ξn] − [L1 − C1(L2, ··· , Ln)]˜e1 −···

2 2 − [Ln−1 − Cn−1(Ln)]˜en−1 − [Ln − Cn]˜en, (3.18) 35

where C1(L2, ··· , Ln) > 0, ··· ,Cn−1(Ln) > 0 are real constants, and the positive constant Cn is independent of all the gains Li’s.

From (3.18), it is clear that the gains Ln, Ln−1, ··· , L1 can be uniquely determined in a recursive manner, i.e.

Ln − Cn ≥ 1 ⇒ Ln ≥ 1+ Cn Ln−1 − Cn−1(Ln) ≥ 1 ⇒ Ln−1 ≥ 1+ Cn−1(Ln) . (3.19) . L1 − C1(L2, ··· , Ln) ≥ 1 ⇒ L1 ≥ 1+ C1(L2, ··· , Ln)

This, in turn, yields

˙ 2 2 2 2 Vn(ξ, e˜) ≤ −(ξ1 + ··· + ξn) − (˜e1 + ··· +˜en).

Thus, the linear system (3.4) is asymptotically stabilized by the output feedback controller (3.7)-(3.8).

3.3 Smooth Output Feedback Stabilization of Homogeneous Systems

The iterative output feedback design method presented in the previous section for the controllable/observable linear system (3.4) can be extended, perhaps surprisingly, to its nonlinear counterpart. In this section, we show how a similar output feedback control strategy can be developed for a class of homogeneous systems of the form (3.1) whose linearization is unstabilizable and undetectable. In particular, with the help of Lemmas 2.24—2.29, we are able to prove the following output feedback stabilization theorem which is one of the main results of Chapter 3.

Theorem 3.1. For the homogeneous system (3.1), there exists a smooth output feedback controller of the form (3.3), such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable at the equilibrium (x, xˆ)=(0, 0).

Proof: The proof is constructive and carried out by explicitly designing a homogeneous observer as well as a smooth state feedback control law. While the state 36 feedback controller is designed via the adding a power integrator technique (Lin and

Qian (2000a,b)), the construction of the homogeneous observer is new. The novelty lies in the development of a machinery that can be viewed as a dual of the adding a power integrator technique, enabling one to assign the observer gains in an iterative manner.

For the sake of convenience, we break up the proof in three parts. Part 1: State Feedback Design

If the state (x1, ··· , xn) of the homogeneous system (3.1) is available for feedback design, one can construct a set of smooth virtual controllers and Lyapunov functions, as done in the papers by Lin and Qian (2000a,b). In fact, following the design procedure of (Lin and Qian (2000a,b)), it is not diﬃcult to obtain

2 ∗ ∗ ξ1 x1 =0, ξ1 = x1 − x1, U1 = , 2 2 ∗ ∗ ξ2 x2 = −a1ξ1, ξ2 = x2 − x2, U2 = U1 + 2 , . . . (3.20) . . . 2 ∗ ∗ ξn xn = −an−1ξn−1, ξn = xn − xn, Un = Un−1 + 2 , and the smooth state feedback control law

∗ p p xn+1 = −(anξn) = −(b1x1 + ··· + bnxn) , (3.21) such that the closed-loop system (3.1)–(3.21) satisﬁes

˙ p+1 p+1 ∗ Un ≤ −4(ξ1 + ··· + ξn )+ ξn(u − xn+1), (3.22)

where all the parameters a1, ··· , an, b1, ··· , bn are known constants and the Lyapunov function ξ2 + ··· + ξ2 U (ξ , ··· ,ξ )= 1 n . n 1 n 2 Part 2: Design of a Homogeneous Observer

Since the state (x1, ··· , xn) of the homogeneous system (3.1) is not measurable, the smooth state feedback controller (3.21) is not implementable directly. In this case, a natural thing to do is to design an observer for the dynamic system (3.1). Motivated 37 by the classical theory of observation, we simply look at a candidate observer that consists of an exact copy of the homogeneous system (3.1), corrected by a term proportional to the “homogeneous error” between the true system output and the estimated output signal, i.e. a dynamic system of the form

˙ p p p xˆ1 =x ˆ2 + L1(x1 − xˆ1) . .

˙ p p p xˆn−1 =x ˆn + Ln−1 ··· L1(x1 − xˆ1) ˙ p p xˆn = u + Ln ··· L1(x1 − xˆ1), (3.23) where L1 > 0, ··· , Ln > 0 are the gain constants to be determined later. It should be pointed out that in the estimator above, we intentionally choose the

p p p p homogeneous function y − yˆ = x1 − xˆ1 as the error correction term, in order to take advantage of homogeneity of system (3.1). It turns out that such choice is crucial for a successful design. In particular, replacing the state (x1, ··· , xn) in the controller

(3.21) by its estimate (ˆx1, ··· , xˆn) generated from the homogeneous observer (3.23), it is possible to prove that with a delicate design of the gain parameters L1, ··· , Ln, the resulted dynamic output compensator (which is of the controller-observer type) globally stabilizes the homogeneous system (3.1).

Let ei = xi − xˆi, i =1, ··· , n, be the estimate errors. Then, the error dynamics is expressed as

p p p p e˙1 = (x2 − xˆ2) − L1(x1 − xˆ1) . .

p p p p e˙n−1 = (xn − xˆn) − Ln−1 ··· L1(x1 − xˆ1)

p p e˙n = −Ln ··· L1(x1 − xˆ1). (3.24)

Part 3: Certainty Equivalence Principle and Output Feedback Design Now, we apply the certainty equivalence principle to get an implementable output feedback controller. This is done by simply using the estimate (ˆx1, ··· , xˆn) produced 38 by the observer (3.23) to replace the state in the nonlinear controller (3.21), i.e.,

∆ p u = −(b1xˆ1 + ··· + bnxˆn) . (3.25)

The dynamic output compensator (3.23)-(3.25) designed in this way is of the form (3.3). In what follows, we shall prove that this dynamic output compensator does the job, if the observer gains L1, ··· , Ln are chosen appropriately. To this end, substituting (3.25) into (3.22) and using Lemmas 2.29-2.26 result in

n ˙ p+1 ∗ Un ≤ −4 ξi + ξn(u − xn+1) i=1 Xn p+1 p p ≤ −4 ξi + ξn − (b1xˆ1 + ··· + bnxˆn) +(b1x1 + ··· + bnxn) i=1 h i Xn p+1 ≤ −4 ξi + c|ξn|(|e1| + ··· + |en|) Xi=1 p−1 p−1 p−1 p−1 ·(x1 + ··· + xn + e1 + ··· + en ) for a suitable constant c> 0. In view of (3.20), Lemmas 2.26 and 2.24, the inequality above can be simpliﬁed as

˙ p+1 p+1 Un ≤ −4(ξ1 + ··· + ξn )+¯c|ξn|(|e1| + ··· + |en|)

p−1 p−1 p−1 p−1 ·(ξ1 + ··· + ξn + e1 + ··· + en ) n p+1 p+1 p+1 ≤ −3 ξi + c0(e1 + ··· + en ), (3.26) Xi=1 wherec> ¯ 0 and c0 > 0 are ﬁxed constants.

To facilitate the design of the observer gains L1, ··· , Ln, we introduce the change of coordinates

e˜1 = e1

e˜2 = e2 − L2e1 . .

e˜n = en − Lnen−1, (3.27) 39 whose inverse transformation is given by

e1 =e ˜1

e2 =e ˜2 + L2e˜1 . .

en =e ˜n + Lne˜n−1 + ··· + Ln ··· L2e˜1. (3.28)

Under the coordinates of ξ ande ˜, inequality (3.26) can be written as (by Lemma 2.26)

n ˙ p+1 p p+1 p+1 p+1 Un ≤ −3 ξi + n c0 e˜1 +[˜e2 +(L2e˜1) ]+ Xi=1 p+1 p+1 p+1 ··· + eñ +(Lneñ−1) + ··· +(Ln ··· L2e˜1) hn i p+1 p+1 p+1 p+1 ≤ −3 ξi + c1(L2, ··· , Ln)˜e1 + ··· + cn−1(Ln)˜en−1 + cneñ , (3.29) Xi=1 where c1(L2, ··· , Ln), ··· ,cn−1(Ln) are positive real constants, and cn > 0 is a known constant independent of Li’s. On the other hand, the error dynamics (3.24) in the coordinate˜e can be expressed as

˙ p p p p e˜1 = −L1(x1 − xˆ1)+(x2 − xˆ2) . .

˙ p p p p e˜n−1 = −Ln−1(xn−1 − xˆn−1)+(xn − xˆn) ˙ p p e˜n = −Ln(xn − xˆn). (3.30)

For this dynamic system, consider the Lyapunov function 1 W (˜e , ··· , e˜ )= (˜e2 + ··· +˜e2 ). n 1 n 2 1 n Then, a tedious but straightforward calculation yields

n n ˙ p p p p Wn = − Lie˜i (ˆxi +˜ei) − xˆi + e˜i−1(xi − xˆi ) i=1 h i i=2 Xn X p p − Lie˜i xi − (ˆxi +˜ei) . (3.31) Xi=2 h i 40

Now, we estimate each term on the right hand side of (3.31). First, using Lemmas

2.24–2.29, (3.20) and (3.28), one has (for i =2, 3, ··· , n)

p p p p p e˜i−1[xi − xˆi ] ≤ 2 |e˜i−1|(|xi| + |ei| )

p p−1 p p p p p p p p ≤ 2 · i |e˜i−1| |ξi| + |ai−1ξi−1| + Li ··· L2|e˜1| + ··· + Li |e˜i−1| + |e˜i| ξp+1 + ξp+1 ≤ i i−1 +¯c (L , ··· , L )˜ep+1 + ··· +¯c (L , ··· , L )˜ep+1, (3.32) n − 1 1 2 n 1 i i+1 n i wherec ¯1(L2, ··· , Ln), ··· , c¯n−1(Ln) are positive real constants, andc ¯n > 0 is a known constant independent of Li’s. Next, with the aid of Lemmas 2.24–2.29 and the relationship

xi − xˆi − e˜i = ei − e˜i = Li ··· L2e˜1 + ··· + Lie˜i−1, the following estimations can be obtained (for i =2, ··· , n):

p p − Lie˜i xi − (ˆxi +˜ei) ≤ c0Li|e˜i| Li ··· L2|e˜1| + ··· + Li|e˜i−1| p−1 p−1 p−1 p−1 · ξi +(ai−1ξi−1) +(Li ··· L2e˜1) + ··· +(Lie˜i−1) ≤c0 LiLi ··· L2|e˜1| + ··· + LiLi|e˜i−1| p p p p p p p p · |ξi| + |ξi−1| + Li ··· L2|e˜1| + ··· + Li |e˜i−1| + |e˜i| 1 ≤ (ξp+1 + ξp+1)+ˆc (L , ··· , L )˜ep+1 + ··· +ˆc (L , ··· , L )˜ep+1, (3.33) n − 1 i i−1 1 2 n 1 i i+1 n i wherec ˆ1(L2, ··· , Ln), ··· , cˆn−1(Ln) are positive constants, andc ˆn > 0 is a know constant independent of Li, 1 ≤ i ≤ n. Substituting (3.32) and (3.33) into (3.31), one can deduce from Lemma 2.27) that

L e˜p+1 + ··· + L e˜p+1 W˙ ≤ 2(ξp+1 + ··· + ξp+1) − 1 1 n n n 1 n 2p−1 p+1 p+1 p+1 +˜c1(L2, ··· , Ln)˜e1 + ··· +˜cn−1(Ln)˜en−1 +˜cne˜n (3.34)

wherec ˜1(L2, ··· , Ln), ··· , c˜n−1(Ln), are positive constants, andc ˜n > 0 is a known constant independent of Li, 1 ≤ i ≤ n. 41

Finally, choose the Lyapunov function

Vn(ξ, e˜)= Un(ξ1, ··· ,ξn)+ Wn(˜e1, ··· , eñ) for the closed-loop system in the coordinates (ξ, e˜). Then, it is straightforward to show that n L V˙ ≤ − ξp+1 − 1 − C (L , ··· , L ) e˜p+1 −··· n i 2p−1 1 2 n 1 Xi=1 Ln−1 p+1 Ln p+1 − p−1 − Cn−1(Ln) eñ−1 − p−1 − Cn eñ (3.35) 2 2 where C1(L2, ··· , Ln), ··· ,Cn−1(Ln) are positive constants, while Cn > 0 is a known constant independent of the gain parameters L1 > 0, ··· , Ln > 0.

From inequality (3.35), it is clear that by choosing the gain parameters Ln, Ln−1, ··· , L1, one-by-one, in the order of Ln, Ln−1, ··· , L1, we immediately arrive at n n ˙ p+1 p+1 Vn(ξ1, ··· ,ξn, e˜1, ··· , e˜n) ≤ − e˜i − ξi . (3.36) Xi=1 Xi=1 This, in turn, implies that the homogeneous system (3.1) is globally asymptotically stabilized by the dynamic output compensator (3.23)–(3.25).

Remark 3.2. When n = p = 3, the homogeneous system (3.1) becomesx ˙ 1 =

3 3 x2, x˙ 2 = x3, x˙ 3 = u, y = x1, which has attracted considerable attention. Indeed, its asymptotic stabilization via state feedback was investigated in (Byrnes and Isidori (1989); Crouch and Irving (1983)), while Theorem 3.1 or Dayawansa (2002) showed how to achieve global stabilization by smooth output feedback. However, it was not clear whether the result in (Dayawansa (2002)) can be extended to n–dimensional homogeneous systems such as (3.1). Now, Theorem 3.1 has provided an aﬃrmative answer to this nontrivial question.

3.4 Global Stabilization of a Class of Non-Homogeneous Systems via Output Feedback

So far we have investigated the problem of global asymptotic stabilization via smooth output feedback, for the homogeneous system (3.1) whose linearization is neither sta- 42 bilizable nor detectable. The purpose of this section is to generalize, under appropriate conditions, the result obtained in the last section to a class of non-homogeneous systems described by equations of the form

p x˙ 1 = x2 + φ1(x, u) . .

p x˙ n−1 = xn + φn−1(x, u)

x˙ n = u + φn(x, u)

y = x1, (3.37)

n where x = (x1, ··· , xn) ∈ IR , u ∈ IR and y ∈ IR are the system state, input and

n output, respectively, p ≥ 1 is an odd integer, and the mappings φi : IR → IR, i =

1 1, ··· , n, are C with φi(0,u)=0, ∀u ∈ IR. As already pointed out in (Qian and Lin (2002b)), global smooth output feedback stabilization of the nonlinear system (3.37) with unstabilizable/undetectable linearization, even in the two-dimensional case, is a diﬃcult problem that may not be solvable without imposing suitable growth conditions. For instance, consider the planar system

3 x˙ 1 = x2

q x˙ 2 = u + x2

y = x1, (3.38) where q ≥ 1 is an odd integer. Clearly, the planar system above is of the form (3.37)

q with φ1(x, u) = 0 and φ2(x, u)= x2. According to (Lin and Qian (2000b)), (3.38) is q always globally stabilized by smooth state feedback, for example, by u = −x1 −x2 −x2. If q = 3, global stabilization of (3.38) can also be achieved by smooth output feedback (Qian and Lin (2002b)). However, there is no any smooth output feedback stabilizer

(even locally) for the planar system (3.38) when q =1or q = 5, as shown in (Qian and 43

Lin (2002b)). Therefore, a fundamental question arises: when can the n-dimensional non-homogeneous system (3.37) be globally stabilized by smooth output feedback? In what follows, we shall address this question and provide a partial answer under the following conditions.

Assumption 3.3. For the nonlinear system (3.37), there exists a real number K > 0 such that for i =1, 2, ··· , n,

i p−1 p−1 |φi(x, u) − φi(ˆx, u)| ≤ K(|x1 − xˆ1| + ··· + |xi − xˆi|) (xl +ˆxl ) . (3.39) h Xl=1 i Remark 3.4. Assumption 3.3 can be viewed as a high-order version of Lipschitz-like condition. In fact, (3.39) is degenerated to the global Lipschitz condition when p = 1:

|φi(x, u) − φi(ˆx, u)| ≤ L(|x1 − xˆ1| + ··· + |xi − xˆi|).

In the case when p> 1, the class of non-homogeneous systems (3.37) characterized by (3.39) encompasses, for example, the nonlinear system (3.57), or, the planar system (3.38) with q = 3 whose global stabilization problem by smooth output feedback was solved only recently (Qian and Lin (2002b)). In the higher dimensional case

(i.e. n ≥ 3), (3.37) includes the non-homogeneous system in Example 3.8 for which the output feedback stabilization problem remains, to the best of our knowledge, unsolved.

The main contribution of this section is the following result showing that As- sumption 3.3 guarantees the existence of a globally stabilizing, smooth reduced-order output feedback controller for the non-homogeneous system (3.37).

Theorem 3.5. Under Assumption 3.3, there exists a smooth dynamic output compensator

zˆ˙ = η(ˆz,y), zˆ ∈ IRn−1,

u = u(ˆz,y), (3.40) which globally asymptotically stabilizes the nonlinear system (3.37). 44

Proof: Since the output y = x1 is measurable and only unmeasurable states of

(3.37) are (x2, ··· , xn), an intuitive idea is to design a reduced-order (i.e., (n − 1) dimensional) observer, rather than a full-order observer proposed in the previous section, for the nonlinear system (3.37). This is the philosophy to be pursued in the design below.

Motivated by the reduced-order observer design method for linear systems, we shall build an (n − 1) − th order nonlinear observer to estimate, instead of the states

(x2, ··· , xn), the unmeasurable variables (z2, ··· , zn) deﬁned by

z2 = x2 − L2x1 . .

zn = xn − Ln ··· L2x1, (3.41) where the parameters L2 > 0, ··· , Ln > 0 are real constants to be assigned later. From (3.41), it follows immediately that

p p z˙2 = x3 + φ2(x, u) − L2[x2 + φ1(x, u)] . .

p p z˙n−1 = xn + φn−1(x, u) − Ln−1 ··· L2[x2 + φ1(x, u)]

p z˙n = u + φn(x, u) − Ln ··· L2[x2 + φ1(x, u)]. (3.42)

In view of (3.42), we construct the (n − 1)–dimensional nonlinear observer

˙ p ˆ p ˆ zˆ2 =x ˆ3 + φ2(·) − L2[ˆx2 + φ1(·)] . .

˙ p ˆ p ˆ zˆn−1 =x ˆn + φn−1(·) − Ln−1 ··· L2[ˆx2 + φ1(·)] ˙ ˆ p ˆ zˆn = u + φn(·) − Ln ··· L2[ˆx2 + φ1(·)], (3.43) where (ˆz2, ··· , zˆn) and (ˆx2, ··· , xˆn) are the estimates of the unmeasurable states

(z2, ··· , zn) and (x2, ··· , xn), respectively. Moreover,

ˆ φi(·) = φi(x1, xˆ2 ··· , xˆn,u), 1 ≤ i ≤ n, 45

xˆi =z ˆi + Li ··· L2x1, i =2, ··· , n. (3.44)

By construction, the reduced-order observer (3.43) is implementable.

Now, let ei = zi − zˆi, i =2, ··· , n, be the estimate errors. Using (3.41) and (3.44), it is clear that

ei = xi − xˆi, i =2, ··· , n.

Keeping this and Assumption 3.3 in mind, it is easy to see that the error dynamics can be written as

p p p p ˆ e˙2 = (x3 − xˆ3) − L2(x2 − xˆ2)+(φ2(x, u) − φ2(·)) . .

p p p p ˆ e˙n−1 = (xn − xˆn) − Ln−1 ··· L2(x2 − xˆ2)+(φn−1(x, u) − φn−1(·)) p p ˆ e˙n = −Ln ··· L2(x2 − xˆ2)+(φn(x, u) − φn(·)). (3.45)

Observe that by Assumption 3.3, the non-homogeneous system (3.37) satisﬁes the growth conditions (i =1, ··· , n)

i p−1 |φi(x1, ··· , xn,u)| ≤ K (|x1| + ··· + |xi|)( xl ) Xl=1 p p ≤ K¯ (|x1| + ··· + |xi| ),

This, in turn, implies that the assumptions in (Lin and Qian (2000a,b)) hold. Hence, applying the adding a power integrator technique (Lin and Qian (2000a,b)), we can construct the state feedback controller

∗ p p xn+1 = −(anξn) = −(b1x1 + ··· + bnxn) , (3.46)

1 2 2 and the Lyapunov function U(ξ1, ··· ,ξn)= 2 (ξ1 + ··· + ξn), such that

˙ p+1 p+1 ∗ U ≤ −4(ξ1 + ··· + ξn )+ ξn(u − xn+1), (3.47) where

ξ1 = x1, ξ2 = x2 + a1ξ1, ··· , ξn = xn + an−1ξn−1, 46

and all the ai’s and bi’s are known real constants.

Using the certainty equivalence principle, we replace the unmeasurable state (x2, ··· , xn) in the controller (3.46) by its estimate (ˆx2, ··· , xˆn) generated by the observer (3.43)– (3.44). In this way, the implementable feedback controller

p u = − (b1x1 + b2xˆ2 + ··· + bnxˆn) , (3.48) with

xˆi =z ˆi + Li ··· L2x1, 2 ≤ i ≤ n, can be obtained.

Similar to the estimation in (3.26), substituting (3.48) into (3.47) yields

˙ p+1 p+1 p+1 p+1 U ≤ −3(ξ1 + ··· + ξn )+ c0(e1 + ··· + en ), (3.49)

where c0 > 0 is a ﬁxed real constant. Next, we shall show that by assigning the appropriate observer gain parameters

L2, L3, ··· , Ln, the dynamic output compensator (3.43)–(3.48) thus constructed indeed does the job. That is, it globally asymptotically stabilizes the non-homogeneous system (3.37).

To prove this conclusion, consider the Lyapunov function

e2 +(e − L e )2 + ··· +(e − L e )2 V (ξ, e)= U(ξ)+ 2 3 3 2 n n n−1 2 for the closed-loop system (3.37)-(3.43)-(3.48). Then, a straightforward calculation gives

˙ p+1 p+1 p+1 p+1 V ≤ −3(ξ1 + ··· + ξn )+ c0(e1 + ··· + en ) n p p 2 p p + ei − Li(xi − xî )+(1+ Li+1)(xi+1 − xî+1) Xi=2 h p p 2 ˆ − Li+1(xi+2 − xî+2)+(1+ Li+1) φi(·) − φi(·) − Li+1 φi+1(·) − φî+1(·) − Li φi−1(·) − φî−1(·) . (3.50) i 47

Now, let us estimate each term on the right hand side of (3.50). First, it follows from Lemmas 2.29 and 2.25 that

2 p p p−1 2 p−1 p−1 p−1 (1 + Li+1)|ei(xi+1 − xˆi+1)| ≤ p 4 (1 + Li+1)|ei| |ei+1| ξi+1 +(aiξi) + ei+1 1 ≤ (ξp+1 + ξp+1)+ c (L )ep+1 + ep+1, (3.51) 5n i+1 i 1 i+1 i i+1 where c1(Li+1) > 0 is a real constant depending on Li+1.

Similarly, it can be shown that there exists a real constant c2(Li+1) > 0 such that

ξp+1 + ξp+1 L |e (xp − xˆp )| ≤ i+2 i+1 + c (L )ep+1 + ep+1. (3.52) i+1 i i+2 i+2 5n 2 i+1 i i+2

Using Assumption 3.3 and Lemma 2.26, one has

|φi(x, u) − φˆi(·)| i p−1 p−1 ≤ K(|e2| + ··· + |ei|) (xl +(xl − el) ) h Xl=1 i p−1 p−1 p−1 p−1 p−1 p−1 ≤ 2 K(|e2| + ··· + |ei|)(x1 + x2 + e2 + ··· + xi + ei ). (3.53)

With the help of (3.53) and the identity xi = ξi − ai−1ξi−1, it is easy to see that

i 2 ˆ ¯ 2 (1 + Li+1) ei(φi(·) − φi(·)) ≤ C(1 + Li+1)|ei|( |el|) Xl=2 p−1 p−1 p−1 p−1 ·(ξ1 + ··· + ξi + e2 + ··· + ei ) 1 i ≤ ξp+1 + c (L ) ep+1 + ··· + ep+1 (3.54) 5n l 3 i+1 2 i Xl=1 h i where C¯ > 0 is a ﬁxed real constant and c3(Li+1) > 0 is a real constant depending on Li+1. In a similar manner, we can prove that

1 i+1 L e φ (·) − φˆ (·) ≤ ξp+1 + c (L )ep+1 i+1 i i+1 i+1 5n l 4 i+1 2 Xl=1 p+1 p+1 + ··· + c4(Li+1)ei + ei+1 , (3.55) i−1 1 p+1 p+1 p+1 p+1 L e φ − (·) − φˆ − (·) ≤ ξ + c (L )e + ··· + c (L )e + e , (3.56) i i i 1 i 1 5n l 5 i 2 5 i i−1 i l=1 X where c4(Li+1) and c5(Li) are positive real constants. 48

Substituting (3.51)—(3.56) into (3.50), it is not diﬃcult to conclude that

n L V˙ ≤ − ξp+1 − 2 − C (L , ··· , L ) ep+1 i 2p−1 2 3 n 2 Xi=1 Ln−1 p+1 Ln p+1 −·· · − p−1 − Cn−1(Ln) en−1 − p−1 − Cn en , 2 2 where C2(L3, ··· , Ln), ··· ,Cn−1(Ln) are positive real constants, while Cn > 0 is a known constant independent of the gain parameters L2 > 0, ··· , Ln > 0.

Clearly, selecting the gain parameters in the order of Ln, Ln−1, ··· , L2 yields immediately ˙ p+1 p+1 p+1 p+1 V ≤ −(ξ1 + ··· + ξn ) − (e2 + ··· + en ), which implies global asymptotic stability of the closed-loop system (3.37)-(3.43)- (3.48).

Remark 3.6. Compared with the n-th order output feedback controller proposed in the last section, design of a reduced-order output compensator is less intuitive. How- ever, from an application viewpoint, using the reduced-order observer (3.43) makes both the construction and implementation of dynamic output controllers simpler, and hence practically feasible.

3.5 Example and Discussion

For the purpose of illustration, we now discuss, with the aid of several examples, how the smooth output feedback control schemes developed in the last two sections can be employed to achieve global asymptotic stabilization for a class of nonlinear systems with unstabilizable/undetectable linearization, which cannot be dealt with by existing output feedback design methods.

The ﬁrst example shows how a high-order nonlinear system in the plane can be globally stabilized by a full-order dynamic output compensator. 49

Example 3.7. Consider the planar system

3 3 x˙ 1 = x2 + x1 sin u

x˙ 2 = u

y = x1, (3.57) whose linearization is given by

0 0 0 (A,B,C)= , , [1 0] , " 0 0 # " 1 # ! which is unstabilizable and undetectable. However, this non-homogeneous system is of the form (3.37) and satisﬁes Assumption 3.3. By Theorem 3.5, global asymptotic stabilization of (3.57) is solvable by smooth output feedback. Next, we illustrate the procedure for the design of a globally stabilizing, smooth output feedback controller. As done in the proof of Theorem 3.1 or 3.5, we ﬁrst design

3 12 a globally stabilizing state feedback control law u = −(18ξ2) with ξ2 = x2 + 5 x1, via the adding a power integrator technique (Lin and Qian (2000a,b)). We then design the nonlinear observer

˙ 3 3 3 3 xˆ1 =x ˆ2 + x1 sin u + L1(x1 − xˆ1)

˙ 3 3 xˆ2 = u + L2L1(x1 − xˆ1), (3.58) where the gains L1 and L2 will be determined below.

Replacing the state (x1, x2) by its estimate (ˆx1, xˆ2), we obtain an output feedback controller that is composed of the observer (3.58) and

12 u = −[18(ˆx + xˆ )]3. (3.59) 2 5 1

Let e1 = x1 −xˆ1 and e2 = x2 −xˆ2 be the estimate errors. Then, the error dynamics are given by

3 3 3 3 e˙1 = (x2 − xˆ2) − L1(x1 − xˆ1)

3 3 e˙2 = −L2L1(x1 − xˆ1). 50

Choose the Lyapunov function

1 V (ξ, e)= x2 + ξ2 + e2 +(e − L e )2 , 2 1 2 1 2 2 1 for the closed-loop system (3.57)–(3.58). By Lemma 2.27 and the identity u =

12 3 −[18(ξ2 − e2 − 5 e1)] , it follows that 12 12 12 V˙ ≤ x4 + x (ξ − x )3 + ξ (ξ − x )3 1 1 2 5 1 2 5 2 5 1 12 h12 L L − [18(ξ − e − e )]3 + x3 sin u − 2 e4 − 1 e4 2 2 5 1 5 1 4 2 4 1 i 2 12 3 12 3 +(1+ L2) (ξ2 − x1) − (ξ2 − x1 − e2) e1. 5 5

Using Lemmas 2.24 and 2.25, one can choose L2 = 1 and a suﬃciently large gain

L1 > L2 = 1, such that all the cross terms in the inequality above are dominated by the negative deﬁnite terms. Indeed, a straightforward calculation similar to the proof of Theorem 3.1 or 3.5 gives

˙ 4 4 4 4 V ≤ −c(x1 + ξ2 + e1 + e2), c> 0.

Thus, the planar system (3.57) is globally asymptotically stabilizable by the dynamic output compensator (3.58)–(3.59). Simulation results shown in Fig. 3.1 demonstrates the eﬀectiveness of the output feedback controller (3.58)–(3.59).

Example 3.8. Consider the non-homogeneous system

3 x˙ 1 = x2

3 2 2 x˙ 2 = x3 + x2 ln(1 + x1)

x˙ 3 = u

y = x1, (3.60) with unstabilizable/undetectable linearization.

It should be pointed out that the output feedback stabilization problem of the three-dimensional system (3.60) is not solvable, even locally, by any existing output 51

6 10

4 5 2 x 1 ∧ x 0 1 x 0 2 ∧ −2 x 2 −4 −5 −6

−8 −10 0 10 20 30 0 10 20 30 Time Time

Fig. 3.1: Transient Responses of the closed-loop system (3.57)-(3.59) with x1(0) = 5, x2(0) = −3, xˆ1(0) = 3, xˆ2(0) = −5. feedback control scheme, due to the lack of eﬀective observer and/or output compensator design techniques for high-order systems such as (3.60).

On the other hand, a direct calculation shows that Assumption 3.3 holds. Indeed,

2 2 2 2 2 2 2 |x2 ln(1 + x1) − xˆ2 ln(1+x ˆ1)| ≤ (|x1 − xˆ1| + |x2 − xˆ2|)( (xl +ˆxl )). Xl=1 By Theorem 3.5, there is a reduced-order dynamic output compensator stabilizing the nonlinear system (3.60) globally. Following the design procedure in Section 3.4, one can easily get the reduced-order output feedback controller

˙ 3 2 2 3 zˆ2 =x ˆ3 +ˆx2 ln(1 + x1) − L2xˆ2

˙ 3 zˆ3 = u − L3L2xˆ2, (3.61)

wherex ˆ2 =z ˆ2 + L2x1, xˆ3 =z ˆ3 + L3L2x1, and L3 and L2 are positive real constants to be assigned later.

2 2 Observe that φ2(x, u) = x2 ln(1 + x1) satisﬁes the growth condition in (Lin and Qian (2000a,b)), i.e.,

1 2 |x2 ln(1 + x2)|≤|x x2| ≤ |x |3 + |x |3. 2 1 1 2 3 1 3 2 52

Using the tool of adding a power integrator (Lin and Qian (2000a,b)), it is easy to show that the high-order system (3.60) is globally asymptotically stabilized by the smooth state feedback controller

3 u = −(27x3 + 324x2 + 972x1) .

By the certainty equivalence principle, the implementable controller is given by

3 u = −(27ˆx3 + 324ˆx2 + 972x1) . (3.62)

Following the design procedure in Section 3.4, it can be verified that a suitable choice of the observer gains (for instance, L3 =0.005 and a sufficiently large L2) makes the output feedback controller (3.61)-(3.62) globally stabilize the three-dimensional system (3.60) at (x, zˆ)=(0, 0). Simulation results shown in Fig. 3.2 illustrate effectiveness of the reduced-order observer (3.61) and output feedback controller (3.62) as well as a transient response of the closed-loop system.

15 15

10 10

5 5 x ∧ 2 x 2 0 x 0 1 x 3 ∧ x −5 −5 3

−10 −10 0 2 4 6 8 10 0 2 4 6 8 10 Time Time

Fig. 3.2: Trajectories of the closed-loop system (3.60)-(3.61) with x1(0) = −2, x2(0) = 5, x3(0) = −3, xˆ2(0) = 3, xˆ3(0) = −5.

The ﬁnal example illustrates a possible extension of Theorem 3.5. From the proof of Theorem 3.5, it is not diﬃcult to see that Assumption 3.3 can be slightly relaxed 53 and Theorem 3.5 remains true, provided that the growth conditions (3.39) hold for

φ1(x, u), ··· ,φn−1(x, u) and the following condition is satisﬁed.

p p |φn(x1, ··· , xn,u)| ≤ K (|x1| + ··· + |xn| ) , (3.63) where K > 0 is a real constant. The condition (3.63) allows the output feedback control scheme developed in Section 3.4 can be applied to a class of nonlinear systems with uncertainty.

Example 3.9. Consider the three-dimensional uncertain system with unstabilizable /undetectable linearization, i.e.

3 x˙ 1 = x2

3 x˙ 2 = x3

3 2 x˙ 3 = u + x2 sin(x2u)+ x2x3 cos ωt

y = x1, (3.64)

3 where ω is an unknown frequency. Due to the presence of the term x2 sin(x2u), it is easy to check that Assumption 3.3 is not fulﬁlled. Therefore, Theorem 3.5 cannot be applied to the uncertain system (3.64). To the best of our knowledge, there are no output feedback design methods in the literature, capable of addressing the output feedback control problem for this type of uncertain nonlinear systems. On the other hand, it is easy to see that system (3.64) does satisfy the growth condition (3.63). It is thus concluded that system (3.64) is globally asymptotically stabilizable by smooth output feedback. Next, we show how a solution can be derived by the non-separation principle based output feedback design method. Using the adding a power integrator technique (Lin and Qian (2000a,b)), it is easy to ﬁnd the globally stabilizing state controller

∗ 3 x4 = −(16x1 + 34x2 +9x3) 54

1 2 2 2 and the Lyapunov function U3 = 2 (12ξ1 +50ξ2 +3ξ3), such that along the trajectories of (3.64), 1 U˙ ≤ − (ξ4 + ξ4 + ξ4)+ ξ (u − x∗) 3 10 1 2 3 3 4

ξ1 19 where ξ1 = x1, ξ2 = x2 + 2 , ξ3 = x3 + 5 ξ2. Following the proof of Theorem 3.1, one can design the homogeneous observer

˙ 3 3 3 xˆ1 =x ˆ2 + L1(x1 − xˆ1)

˙ 3 3 3 xˆ2 =x ˆ3 + L2L1(x1 − xˆ1)

˙ 3 3 xˆ3 = u + L3L2L1(x1 − xˆ1), (3.65)

where the gains L3, L2, L1 are positive real numbers to be determined later.

Let e1 = x1 − xˆ1, e2 = x2 − xˆ2, e3 = x3 − xˆ3 be the estimate errors. The error dynamic system can be expressed as

3 3 3 3 e˙1 = (x2 − xˆ2) − L1(x1 − xˆ1)

3 3 3 3 e˙2 = (x3 − xˆ3) − L2L1(x1 − xˆ1)

3 3 2 2 e˙3 = −L3L2L1(x1 − xˆ1)+ x2x3 sin(x2u)+ x2x3 cos ωt.

By the certainty equivalence principle, an implementable controller is given by

3 u = −(16ˆx1 + 34ˆx2 +9ˆx3) . (3.66)

Then, it is easy to show that

1 U˙ ≤ − (ξ4 + ξ4 + ξ4)+ ξ (u − x∗) 3 10 1 2 3 3 4 1 ≤ − (ξ4 + ξ4 + ξ4)+ c (e4 + e4 + e4) 15 1 2 3 0 1 2 3 where c0 > 0 is a known constant. Now, consider the Lyapunov function

1 W = e2 +(e − L e )2 +(e − L e )2 . 3 2 1 2 2 1 3 3 2 55 for the error dynamics. A simple calculation yields

L L W˙ ≤ − 1 e4 +3(1+ L2)|e e |(3x2 +2e2) − 2 e4 3 4 1 2 1 2 2 2 4 2 L +3(1 + L2)|e e |(3x2 +2e2) − 3 e4 3 2 3 3 3 4 3 2 2 2 +3L2|e1e3|(3x3 +2e3)+ |e3x2|x3

2 2 2 +L3|e2x2|x3 + |e3x3|x2 + L3|e2x3|x2.

With the aid of Lemma 2.24 and 2.25, the inequality above can be put into the form

L L W˙ ≤ − 1 − C (L , L ) e4 − 2 − C (L ) e4 3 4 1 2 3 1 4 2 3 2 hL 1i h i − 3 − C e4 + (ξ4 + ξ4 + ξ4), 4 3 3 30 1 2 3 h i where C3 > 0 is a constant independent of Li, i = 1, 2, 3, while C1(L2, L3) > 0 and

C2(L3) > 0 are constants dependent of Li.

From the inequality above, it is clear that by selecting L3 ≥ 4(C3 +1+ c0) and then

L2 ≥ 4 C2(L3)+1+ c0 , L1 ≥ 4 C1(L2, L3)+1+ c0 , we arrive at 1 W˙ ≤ −(c + 1) e4 + e4 + e4 + (ξ4 + ξ4 + ξ4). 3 0 1 2 3 30 1 2 3

Finally, choosing V3 = W3(e1, e2, e3)+ U3(ξ1,ξ2,ξ3) for the closed-loop system, we have 1 V˙ ≤ − (ξ4 + ξ4 + ξ4) − e4 − e4 − e4, 3 30 1 2 3 1 2 3 which implies that the closed-loop system (3.64)-(3.65)-(3.66) is globally asymptotically stable at the equilibrium (x, xˆ)=(0, 0).

Remark 3.10. Due to Assumption 3.3, the nonlinear systems considered in this chapter are required to be high-order systems. The assumption guarantees, on one hand, the existence of smooth output controllers. On the other hand, an obvious 56 disadvantage of the growth-condition (3.39) is that it cannot encompass a class of genuinely nonlinear systems such as

3 x˙ 1 = x2 + x1

x˙ 2 = u

y = x1, (3.67) or, a two degree of freedom unstable underactuated mechanical system (Rui et al.

(1997); Qian and Lin (2001a))

x˙ 1 = x2 g x˙ = x3 + sin x 2 3 l 1

x˙ 3 = x4

x˙ 4 = v

y = x1. (3.68)

Notably, both systems (3.67) and (3.68) involve an unstabilizable/undetectable linearization. Moreover, the uncontrollable mode has an eigenvalue on the open right- half plane. As a result, there are no smooth state/output feedback control laws stabilizing either (3.67) or (3.68) at the origin, even locally, and hence the smooth output feedback control schemes developed in Chapter 3 cannot be applied. To deal with inherently nonlinear systems such as (3.68) or (3.67), nonsmooth output feedback design approaches need to be developed, which will be shown in Chapter 5.

3.6 Summary

In this chapter, we have investigated the challenging problem of how to achieve global asymptotic stabilization by smooth output feedback, for a class of n-dimensional homogeneous systems whose linearization at origin is unstabilizable and undetectable. A new output feedback design approach that is not based on the separation principle has been developed, making it possible to recursively construct a homogeneous 57 observer as well as a state feedback controller. While the smooth state controllers was designed by the tool of adding a power integrator (Lin and Qian (2000a,b)), the observer design was based on a newly developed machinery that can be viewed as a dual of the adding a power integrator technique. A novelty of such an observer design is that the observer gains can be assigned in an iterative manner. The output feedback controller thus obtained is of the observer-controller type but the homogeneous observer by itself cannot recover the system states unless the sophisticated controller is in eﬀect. For a class of high-order non-homogeneous systems, we discussed how to generalize the proposed output feedback design approach for homogeneous systems so that global stabilization can be achieved by a reduced-order dynamic output compensator. Examples and simulations were given to demonstrate the main features and eﬀectiveness of the new output feedback control schemes. Chapter 4. ROBUST CONTROL OF UNCERTAIN SYSTEMS BY SMOOTH OUTPUT FEEDBACK

This chapter investigates the problem of robust output feedback stabilization for a family of uncertain nonlinear systems with unstabilizable/undetectable linearization. To achieve global robust stabilization via smooth output feedback, we introduce a rescaling transformation with an appropriate dilation, which turns out to be very effective in dealing with uncertainty of the system. Using this rescaling technique combined with the non-separation principle based design method (Qian and Lin (2002a)), we develop a robust output feedback control scheme for uncertain nonlinear systems in the p-normal form, under a homogeneous growth condition. The construction of smooth state feedback controllers and homogeneous observers uses only the knowledge of the bounding homogeneous system rather than the uncertain system itself. The robust output feedback design approach is then extended to a class of uncertain cascade systems beyond a strict-triangular structure. Examples are provided to illustrate the results of this chapter.

4.1 Introduction

In this chapter, we consider the problem of global robust stabilization via a single smooth output feedback controller, for a family of uncertain systems of the form

p η˙1 = η2 + φ1(t, η, v) . .

p η˙n−1 = ηn + φn−1(t, η, v)

η˙n = v + φn(t, η, v)

58 59

y = η1, (4.1) where v ∈ IR, η ∈ IRn and y ∈ IR are the system input, state and output, respectively,

n and p ≥ 1 is an odd integer. The mappings φi : IR × IR × IR → IR, i = 1, ··· , n, represent a class of C1 functions that involve uncertainty and may not be precisely known. For the class of aﬃne systems that is topologically equivalent to (4.1), interesting stabilization results have been obtained over the years. For example, local and global asymptotic stabilization of the system (4.1) with n = 3, p = 3 and φi(t, η, v) = 0, i = 1, 2, 3, using smooth state feedback, were investigated in (Crouch and Irving (1983); Byrnes and Isidori (1989)), respectively. In the n-dimensional case, a globally stabilizing smooth state feedback control law was explicitly designed by the tool of adding an power integrator (Lin and Qian (2000b)), for a class of nonlinear systems (4.1) under appropriate growth conditions that can be regarded as a high order version of feedback linearizable condition. Much of the literature on stabilization of nonlinear systems has focused on the design of state feedback. In the past two years, research eﬀorts towards the development of output feedback control schemes for nonlinear systems with unstabilizable/undetectable linearization have gained momentum. The paper (Qian and Lin (2002b)) studied the global stabilization of the nonlinear system (4.1) in the plane via smooth output feedback. Under suitable conditions imposed on φi(t, η, v), i =1, 2, a reduced-order nonlinear observer was designed in (Qian and Lin (2002b)), resulting in a globally stabilizing, smooth dynamic output compensator. Notably, the output feedback design in (Qian and Lin (2002b)) does not rely on the separation principle. Instead, it uses the idea of coupled controller-observer construction. Dayawansa (2002) proved the existence of a smooth output feedback stabilizer for the three- dimensional system (4.1) when p = 3 and φi(·) ≡ 0, i =1, 2, 3. His proof is based on the theory of homogeneous systems (Hahn (1967); Bacciotti (1992)) and some elegant 60 design techniques from (Dayawansa (1992); Dayawansa et al. (1990); Kawski (1989);

Hermes (1991a); Rosier (1992)). In the last chapter, we have shown that for the n-dimensional nonlinear system

(4.1) with φi(·)=0, i = 1, ··· , n, which is homogeneous, the problem of global stabilization is solvable by smooth output feedback. This was done by developing a new observe design technique for the construction of a homogeneous observer, combined with the method (Lin and Qian (2000b)) for the design of a smooth state feedback controller. When φi(·) =06 , i =1, ··· , n and satisfy a global Lipschitz-like condition, we further showed that global stabilization of the non-homogeneous system (4.1), which is not uniformly observable (Gauthier et al. (1992)), can still be achieved via smooth output feedback proposed in Chapter 3. A key ingredient of the output feedback control strategy in Chapter 3 is the development a recursive algorithm for the design of homogeneous observers, which makes it possible to assign the gains of the homogeneous observer one-by-one, in a step-by-step manner. Although such an observer design is substantially diﬀerent from the “Luenberger” or “high-gain” observer design (Khalil and Saberi (1987); Gauthier et al. (1992); Krener and Isidori (1983); Krener and Xiao (2002); Isidori (1999); Krener and Kang (2003)), it still uses a copy of the original system and hence requires the precise information of the controlled plant. In other words, the nonlinear functions φi(t, η, v), i = 1, ··· , n, in (4.1) must be independent of t and involve no uncertainty. As a result, the output feedback control scheme in the last chapter is not robust with respect to parametric or structural uncertainty. Moreover, it cannot be applied to uncertain nonlinear systems such as (4.1). The main purpose of this chapter is to address the robust issue discussed above, and to develop a robust output feedback control scheme for a family of uncertain systems (4.1) under suitable growth conditions. In the case of p = 1, such robust control problems have been studied, for instance, in (Qian and Lin (2002a)) under a 61 linear growth condition. In view of the work (Qian and Lin (2002a)), it appears to be natural to impose the homogeneous growth condition below, and to investigate the question whether global robust stabilization of the uncertain nonlinear system (4.1) is achievable by smooth output feedback.

Assumption 4.1. There exists a real constant C ≥ 0 such that ∀(t, η, v) ∈ IR × IRn × IR,

p p |φi(t, η, v)| ≤ C(|η1| + ··· + |ηi| ), i =1, ··· , n. (4.2)

One of the objectives of this chapter is to address this question and to provide an aﬃrmative answer by constructing, under Assumption 4.1, a smooth dynamic output compensator

xˆ˙ = θ(ˆx, y), xˆ ∈ IRn−1,

v = v(ˆx, y), (4.3) which globally robustly stabilizes the entire family of uncertain systems (4.1). Since

Assumption 4.1 is weaker than the higher-order type of the global Lipschitz condition given in Chapter 3 (see Remark 4.7), the class of nonlinear systems considered in this chapter is larger than those studied in Chapter 3. More signiﬁcantly, because the design of the dynamic output compensator (4.3) uses only the knowledge of the upper-bound of φi(·), i.e., the condition (4.2) instead of φi(t, η, v) itself, global output feedback stabilization will be achieved in a robust fashion, that is, in a manner which is not sensitive to perturbations and parametric uncertainty in the system. This is one of the major diﬀerences between Chapter 3 and Chapter 4.

The key for achieving robustness is the introduction of a rescaling technique with a subtle dilation, which transforms the original system (4.1) into a rescaled one for which a dynamic output compensator can be constructed using the output feedback design method in the last chapter, with a suitable twist, in particular, by discarding the system uncertainty when designing homogeneous observers. With the help 62

of the rescaling technique, the uncertain nonlinearities φi(·) in (4.1) can be dominated easily by tuning the rescaling factor. In the case of uncertain systems with controllable/observable linearization (i.e. p = 1), the new design method provides not only a deeper insight but also an interesting alternative solution to the robust output feedback stabilization problem considered in (Qian and Lin (2002a)).

The other goal of this chapter is to show how robust output feedback control strategies can be developed, under appropriate conditions, for a wider class of uncertain nonlinear systems with unstabilizable and undetectable linearization in the p-normal form (4.21) and cascade form (4.64), which go beyond a triangular structure. Several examples are given to demonstrate the applications of the robust output feedback design method proposed in this chapter.

4.2 Robust Output Feedback Design: the Case of p =1

To better understand how global robust stabilization of the uncertain nonlinear system (4.1) with unstabilizable/undetectable linearization can be achieved by smooth output feedback under Assumption 4.1, we revisit a simple situation of (4.1) where p = 1, i.e., the case when the ﬁrst approximation of (4.1) is controllable and observable. In this case, the uncertain system (4.1) can be rewritten as

η˙1 = η2 + φ1(t, η, v) . .

η˙n−1 = ηn + φn−1(t, η, v)

η˙n = v + φn(t, η, v)

y = η1, (4.4) and Assumption 4.1 reduces to the linear growth condition:

|φi(t, η, v)| ≤ C(|η1| + ··· + |ηi|), i =1, ··· , n. (4.5) 63

Qian and Lin (2002a) have shown that global robust stabilization of the uncertain system (4.4) satisfying (4.5) is solvable by a linear output dynamic compensator. The proof was not based on the separation principle but instead, relied on a coupled controller-observer design (Qian and Lin (2002a)). Due to the linear nature of the work (Qian and Lin (2002a)), it is, however, not easy to extend the output feedback design approach of (Qian and Lin (2002a)) to a family of uncertain nonlinear systems such as (4.1). In this section, we explore an alternative output feedback control strategy that takes advantage of homogeneity of the system, and hence might be naturally extended, in an intuitive and transparent manner, to the uncertain nonlinear system (4.1) with p ≥ 1. To this end, we introduce a rescaling transformation with a suitable dilation for the original system (4.4), which turns out to be crucial for dominating the uncertainty of (4.4). To be precise, let

η v x = i , i =1, ··· ,n, u = , (4.6) i M i−1 M n where M ≥ 1 is a rescaling factor to be determined later.

Under the new coordinates xi’s, the uncertain system (4.4) can be expressed as

x˙ 1 = Mx2 + f1(t, x, u) . .

x˙ n−1 = Mxn + fn−1(t, x, u)

x˙ n = Mu + fn(t, x, u)

y = x1. (4.7)

By the hypothesis (4.5), the uncertain functions fi(·), i = 1, ··· , n, also satisfy the linear growth condition:

|f1(·)| = |φ1(·)| ≤ C|η1| ≤ C|x1|,

φ2(·) C(|η1| + |η2|) |f2(·)| = ≤ ≤ C (|x1| + |x2|) , M M

. .

φn(·) C(|η1| + ··· + |ηn|) |fn(·)| = ≤ ≤ C (|x1| + ··· + |xn|) . (4.8) M n−1 M n−1

For the rescaled uncertain system (4.7) with the constraint (4.8), it is straightforward to design recursively, in a fashion similar to the one in (Qian and Lin (2002a)), a linear state feedback controller

∗ xn+1 = −βnξn = −(b1x1 + ··· + bnxn), (4.9) such that ˙ 2 2 ∗ Un(ξ) ≤ −M 3(ξ1 + ··· + ξn) − ξn(u − xn+1) , (4.10) h i 1 2 2 ∗ where Un(ξ) = 2 (ξ1 + ··· + ξn) is a quadratic Lyapunov function, ξi = xi − xi , i = 1, ··· , n, and

∗ ∗ ∗ x1 =0, x2 = −β1ξ1, ··· , xn = −βn−1ξn−1 with βi and bi being known constants independent of M. Next, we shall design a linear observer for the rescaled system (4.7). Because y = x1 is measurable and only unmeasurable states of (4.7) are (x2, ··· , xn), it is natural to design an (n − 1)-dimensional observer rather than a full-order observer. Motivated by the reduced-order observer design for linear systems, we shall build an

(n − 1) − th order linear observer to estimate, instead of the states (x2, ··· , xn), the unmeasurable variables (z2, ··· , zn) deﬁned by

zi = xi − Li ··· L2x1, i =2, ··· , n, (4.11) where Ln, ··· , L2 > 0 are gain constants to be assigned later. From (4.11) it is clear that

z˙2 = M(x3 − L2x2)+ f2(·) − L2f1(·) . . (4.12)

z˙n−1 = M(xn − Ln−1 ··· L2x2)+ fn−1(·) − Ln−1 ··· L2f1(·)

z˙n = M(u − Ln ··· L2x2)+ fn(·) − Ln ··· L2f1(·). 65

In view of (4.12), we construct the (n − 1)-th order linear observer (regardless of the uncertain terms fi(·), 1 ≤ i ≤ n)

zˆ˙ 2 = M[(ˆz3 + L3L2x1) − L2(ˆz2 + L2x1)] . . ˙ zˆn−1 = M[(ˆzn + Ln ··· L2x1) − Ln−1 ··· L2(ˆz2 + L2x1)]

zˆ˙ n = M[u − Ln ··· L2(ˆz2 + L2x1)] (4.13)

where (ˆz2, ··· , zˆn) is the estimates of the unmeasurable state (z2, ··· , zn).

Consequently, the unmeasurable state (x2, ··· , xn) of (4.7) can be estimated as follows:

xˆi =z ˆi + Li ··· L2x1, i =2, ··· , n. (4.14)

By construction, the reduced-order observer (4.13)-(4.14) is implementable.

Let ei = xi − xˆi = xi − Li ··· L2x1 − zˆi, 2 ≤ i ≤ n, be the estimate errors. Then, the error dynamics is given by

e˙2 = M(e3 − L2e2)+ f2(·) − L2f1(·) . . (4.15)

e˙n−1 = M(en − Ln−1 ··· L2e2)+ fn−1(·) − Ln−1 ··· L2f1(·)

e˙n = −MLn ··· L2e2 + fn(·) − Ln ··· L2f1(·).

Inspired by the certainty equivalence principle, we replace the unmeasurable state

(x2, ··· , xn) in the controller (4.9) by its estimate (ˆx2, ··· , xˆn) generated by the observer (4.13)-(4.14). In this way, we get the following implementable controller

u = − (b1x1 + b2xˆ2 + ··· + bnxˆn) . (4.16)

Substituting (4.16) into (4.10) yields

˙ 2 2 2 2 Un ≤ −M 2(ξ1 + ··· + ξn) − K(e2 + ··· + en) , (4.17) h i 66 where K > 0 is a constant independent of M.

Now, consider the Lyapunov function

1 2 2 2 V = U (ξ)+ e +(e − L e ) + ··· +(e − L e − ) n n 2 2 3 3 2 n n n 1 h i for the closed-loop system (4.16)-(4.15)-(4.7). A simple calculation results in

V˙n = U˙n(ξ)+ Me2(e3 − L2e2)+ e2 f2(·) − L2f1(·) n hn i −M Li(ei − Liei−1)ei − (ei−1 − Li−1ei−2)ei (4.18) h i=3 i=4 n X X fi(·) fi−1(·) − (e − L e − ) − L . i i i 1 M i M Xi=3 i Using the completion of square, together with the linear growth condition (4.8), it is not diﬃcult to deduce from (4.18)-(4.17) that

n K L2 V˙ ≤ −M ξ2 + L − c (L , ··· , L ) − 2 2 e2 n i 2 2 3 n M 2 h Xi=1 2 Kn−1Ln−1 2 + ··· + L − − c − (L ) − e n 1 n 1 n M n−1 K L2 + L − c − n n e2 , (4.19) n n M n i where c2(L3, ··· , Ln), ··· ,cn−2(Ln−1, Ln),cn−1(Ln) are positive constants independent of M, while cn > 0 and Ki > 0, 2 ≤ i ≤ n are known constants independent of M and all Li’s.

Choosing the gain parameters Li and M one-by-one, in the order of Ln, ··· , L2, M as follows:

Ln ≥ 1+ cn + Kn

Ln−1 ≥ 1+ cn−1(Ln)+ Kn−1 . . (4.20)

L2 ≥ 1+ c2(L3, ··· , Ln)+ K2

2 2 M ≥ max{L2, ··· , Ln} ≥ 1, 67 we have ˙ 2 2 2 2 Vn ≤ −M (ξ1 + ··· + ξn)+(e2 + ··· + en) . h i Hence, the uncertain system (4.4) is globally robustly stabilizable by the linear output feedback controller (4.13)-(4.14)-(4.16).

Remark 4.2. Notably, the reduced-order observer (4.13) is diﬀerent from the one in (Qian and Lin (2002a)) in two respects: 1) it involves n − 1 gain parameters

L2, L3, ··· , Ln to be designed. Only when M = 1 and L2 = L3 = ··· = Ln ≡ L, (4.13) reduces to a traditional reduced-order high-gain observer; 2) it contains a rescaling factor M that is introduced to create an extra design freedom and turns out to be very useful in dealing with uncertainty of system (4.4), under the linear growth condition (4.5).

4.3 Robust Output Feedback Design: the p-Normal Form Case

Interestingly, the robust output feedback control scheme developed so far for the uncertain system (4.4) with controllable/observable linearization can be carried over, in a parallel manner, to its high order counterpart. In this section, we show that in spite of the lack of controllability and observability in the ﬁrst approximation, a robust output feedback control method can be developed for a family of uncertain systems such as (4.1) satisfying the growth condition (4.2). To this end, we ﬁrst present a robust stabilization result for the following class of nonlinear systems

p p−1 η˙1 = η2 + η2 φ1,p−1(t, η, v)+ ··· + η2φ1,1(t, η, v)+ φ1,0(t, η, v) . .

p p−1 η˙n−1 = ηn + ηn φn−1,p−1(t, η, v)+ ··· + ηnφn−1,1(t, η, v)+ φn−1,0(t, η, v)

η˙n = v + φn,0(t, η, v)

y = η1, (4.21) 68 called p-normal form (Cheng and Lin (2003)), where v ∈ IR, η ∈ IRn and y ∈ IR are the system input, state and output, respectively, and p ≥ 1 is an odd integer. The

n 1 mappings φi,j : IR × IR × IR → IR, i = 1, ··· , n, j = 0, ··· ,p − 1 are C , involve uncertainty and may be unknown. As shown in (Cheng and Lin (2003); Respondek (2003)), every smooth aﬃne system is, under appropriate conditions, feedback equivalent to (4.21). To achieve global robust stabilization of (4.21) by output feedback, we introduce the following hypothesis which is a natural generalization of the homogeneous growth condition (4.2).

Assumption 4.3. There exists a constant C > 0 such that ∀(t, η, v) ∈ IR × IRn × IR,

p−j p−j |φi,j(t, η, v)| ≤ C(|η1| + ··· + |ηi| ), (4.22) where j =0, ··· ,p − 1 when i =1, ··· , n − 1. Moreover, j = 0 when i = n.

With the aid of the growth condition (4.22), it is possible to establish the following output feedback stabilization theorem which is one of the main results of this chapter.

Theorem 4.4. Under Assumption 4.3, there exists a smooth dynamic output compensator (4.3) making the uncertain system (4.21) globally asymptotically stable.

Proof: Similar to the philosophy of the previous section, the proof of this theorem is carried out by explicitly designing a robust smooth state feedback controller, and a homogeneous observer that does not require the knowledge of the system uncertainty, i.e., φi,j(t, η, v), i =1, ··· , n. The construction of the observer is substantially diﬀer- ent from the one in the previous chapter in the sense that here no copy of φi,j(t, η, v) is used for the design of a robust observer, while the nonlinear observer in Chapter 3 did involve a copy of φi,j(t, η, v). For this reason, φi,j(t, η, v) in the last chapter must be known precisely. Another new ingredient of our output feedback design is the 69 development of a rescaling technique for handling the uncertain terms in the system

(4.21). In particular, a higher-order rescaling transformation with a suitable dilation is employed to deal with the system uncertainty eﬀectively. For the convenience of the reader, we break up the proof into three parts. (i) Rescaling of the p-Normal Form

Observe that by the homogeneous systems theory (see, for instance, Hahn (1967); Kawski (1989); Hermes (1991a); Rosier (1992)), system (4.21) is homogeneous with dilation (1, ··· , 1; p) and degree p − 1 when φi,j(t, η, v)=0, ∀i, j. Keeping this in mind and motivated by (4.6), we introduce the following rescaling transformation

1 ··· 1 p + + i−1 x1 = η1; xi = ηi/(M p ), i =2, ··· , n;

1 1 1+ +···+ − u = v/(M p pn 1 ) (4.23)

1 1 1 1 1 with dilation (0, p , ··· , p + ··· + pn−1 ;1+ p + ··· + pn−1 ), where M ≥ 1 is a rescaling factor to be assigned later.

In the rescaled coordinates xi’s, the uncertain system (4.21) can be represented as

p x˙ 1 = Mx2 + g1(t, x, u) . .

p x˙ n−1 = Mxn + gn−1(t, x, u)

x˙ n = Mu + gn(t, x, u)

y = x1, (4.24) where

p−1 j( 1 +···+ 1 ) p pi j M xi+1φi,j(·) j=0 gi(·) = X 1 1 , 1 ≤ i ≤ n − 1; +···+ − M p pi 1 φn,0(·) gn(·) = 1 1 . +···+ − M p pn 1 70

Using the condition (4.22) and the fact that M ≥ 1, it is easy to see that

p−1 1 ··· 1 − 1 ··· 1 j( p + + i ) ( p + + i−1 ) j |gi(·)| ≤ CM p p |xi+1| jX=0 − 1 ··· 1 p−j (p j)( p + + i−1 ) p−j ×(|x1| + ··· + M p |xi| ) − p 1 p−j 1− i j p−j p−j ≤ CM p |xi+1| (|x1| + ··· + |xi| ) jX=0 − 1 p 1 1− i j p−j p−j ≤ CM p |xi+1| (|x1| + ··· + |xi| ) jX=0 for i =1, ··· , n − 1. (4.25)

Applying Lemma 2.24 to (4.25), we obtain the following estimations (i =1, ··· , n−1)

1− 1 1 |g (·)| ≤ M pi |x |p + C (|x |p + ··· + |x |p) i 2 i+1 0 1 i h 1 i 1− n p p |gn(·)| ≤ C0M p (|x1| + ··· + |xn| ) (4.26) where C0 > 0 is a known constant independent of M. In this way, a new parameter – the rescaling factor M – is introduced for the design of dynamic output compensators. It creates an extra freedom and plays an important role in dealing with the system uncertainty, i.e., gi(·), 1 ≤ i ≤ n, in (4.24). (ii) State Feedback Design For the rescaled system (4.24) satisfying the growth condition (4.25), one can construct a robust state feedback controller via the smooth feedback design method (Lin and Qian (2000b)).

∗ ∗ 1 2 Let ξ1 = x1 − x1 with x1 = 0 and choose the Lyapunov function U1 = 2 ξ1 . Then, it is easy to deduce from (4.26) that 1 U˙ ≤ M ξ xp + |ξ |( |xp| + C |xp|) 1 1 2 1 2 2 0 1 h 1 i 1 ≤ M ξ x∗p + |ξ ||x∗p| + C ξp+1 + ξ (xp − x∗p)+ |ξ |(|xp|−|x∗p|) . 1 2 2 1 2 0 1 1 2 2 2 1 2 2 h i ∗ Keeping |a|−|b| ≤ |a − b| in mind, the virtual controller x2 = −β1ξ1, with β1 =

1/p [2(C0 + n + 5)] being a constant independent of M, results in 1 U˙ ≤ M − (n + 5)ξp+1 + ξ (xp − x∗p)+ |ξ ||xp − x∗p| (4.27) 1 1 1 2 2 2 1 2 2 h i 71

3 ≤ M − (n + 5)ξp+1 + |ξ ||xp − x∗p| . 1 2 1 2 2 h i ∗ Next, let ξ2 = x2 − x2 = x2 + β1x1 and choose the Lyapunov function U2 =

1 2 U1(ξ1)+ 2 ξ2 . Using (4.24) and (4.26)-(4.27), we have 3 U˙ ≤ M − (n + 5)ξp+1 + |ξ ||xp − x∗p| 2 1 2 1 2 2 h 1 +ξ x∗p + |ξ ||x∗p| + C |ξ |(|xp| + |xp|) 2 3 2 2 3 0 2 1 2 1 +β |ξ ||xp| + β |ξ |( |xp| + C |xp|) (4.28) 1 2 2 1 2 2 2 0 1 1 +ξ (xp − x∗p)+ |ξ |(|xp|−|x∗p|) . 2 3 3 2 2 3 3 i

With the help of Lemma 2.29 and x2 = ξ2 − β1ξ1, a direct computation gives

3 3p ξp+1 |ξ ||xp − x∗p| ≤ |ξ ||ξ |(xp−1 + x∗p−1) ≤ 1 + α ξp+1, (4.29) 2 1 2 2 2 1 2 2 2 2 1 2 where α1 > 0 is a constant independent of M. Substituting (4.29) into (4.28), we deduce from Lemmas 2.24 and 2.26 that

1 3 U˙ ≤ M − (n + 4)ξp+1 + ξ x∗p + |ξ ||x∗p| + α ξp+1 + |ξ ||xp − x∗p| , (4.30) 2 1 2 3 2 2 3 2 2 2 2 3 3 h i where α2 > 0 is a constant independent of M.

∗ 1/p Thus, the virtual controller x3 = −β3ξ2, with β2 = [2(α2 + n + 4)] being a constant independent of M, is such that

3 U˙ ≤ M − (n + 4)(ξp+1 + ξp+1)+ |ξ ||xp − x∗p| . (4.31) 2 1 2 2 2 2 2 h i Using an inductive argument similar to the one in (Lin and Qian (2000b)), one can ﬁnd a set of virtual controllers, transformations and Lyapunov functions:

2 ∗ ∗ ξ1 x1 =0, ξ1 = x1 − x1, U1 = , 2 2 ∗ ∗ ξ2 x2 = −β1ξ1, ξ2 = x2 − x2, U2 = U1 + 2 , . . . (4.32) . . . 2 ∗ ∗ ξn xn = −βn−1ξn−1, ξn = xn − xn, Un = Un−1 + 2 , and a smooth state feedback control law

∗ p p xn+1 = −(βnξn) = −(b1x1 + ··· + bnxn) , (4.33) 72 such that n 3 U˙ (ξ , ··· ,ξ ) ≤ M − 6( ξp+1)+ |ξ ||u − x∗ | (4.34) n 1 n i 2 n n+1 h Xi=1 i where all the constants β1, ··· , βn and b1, ··· , bn are known and independent of M. (iii) Output Feedback Design

Since (x2, ··· , xn) of the rescaled system (4.24) are unmeasurable but y = x1 is measurable, we need only to design an (n − 1)-dimensional observer for (4.24). However, the reduced-order observer design method in Chapter 3 cannot be applied to the rescaled system (4.24) because it uses a copy of gi(·), i =1, ··· , n, which are time- varying and not precisely known. Motivated by the robust observer design in Section

4.2, in what follows we shall construct an (n − 1)-dimensional robust homogeneous observer to estimate, instead of the states (x2, ··· , xn), the unmeasurable variables

zi = xi − Li ··· L2x1, i =2, ··· , n. (4.35) where the parameters Ln, ··· , L2 ≥ 1 are gain constants to be determined later. From (4.35), it follows that

p p z˙2 = Mx3 + g2(·) − L2(Mx2 + g1(·)) . . (4.36)

p p z˙n−1 = Mxn + gn−1(·) − Ln−1 ··· L2(Mx2 + g1(·))

p z˙n = Mu + gn(·) − Ln ··· L2(Mx2 + g1(·)).

In view of (4.36), one can construct the (n − 1)-dimensional homogeneous observer

p p zˆ˙ 2 = M(ˆz3 + L3L2x1) − ML2(ˆz2 + L2x1) . .

p p zˆ˙ n−1 = M(ˆzn + Ln−1 ··· L2x1) − MLn−1 ··· L2(ˆz2 + L2x1)

˙ p zˆn = Mu − MLn ··· L2(ˆz2 + L2x1) , (4.37) which does not involve the uncertain functions g1(·), ··· ,gn(·) in (4.24). This is substantially diﬀerent from the homogeneous observer proposed in the previous chapter. 73

By construction, the reduced-order observer (4.37) is implementable. Moreover, the estimates of xi’s can be obtained based on the relationships:

xˆi =z ˆi + Li ··· L2x1, i =2, ··· , n. (4.38)

Let ei = xi − xˆi = zi − zˆi, 2 ≤ i ≤ n, be the estimate errors. Then, the error dynamics is given by

p p p p e˙2 = M(x3 − xˆ3)+ g2(·) − ML2(x2 − xˆ2) − L2g1(·) . . (4.39)

p p p p e˙n−1 = M(xn − xˆn)+ gn−1(·) − MLn−1 ··· L2(x2 − xˆ2) − Ln−1 ··· L2g1(·)

p p e˙n = gn(·) − MLn ··· L2(x2 − xˆ2) − Ln ··· L2g1(·).

By the certainty equivalence principle, the unmeasurable state (x2, ··· , xn) in the controller (4.33) can be replaced by its estimate (ˆx2, ··· , xˆn) generated by the nonlinear observer (4.37)–(4.38). In this way, one obtains the implementable feedback controller

p u = − (b1x1 + b2xˆ2 + ··· + bnxˆn) . (4.40)

Substituting (4.40) into (4.34) and using Lemmas 2.29 and 2.26, we have

3 U˙ ≤ M |ξ | (b x + b xˆ + ··· + b xˆ )p n 2 n 1 1 2 2 n n h n p p+1 −(b1x1 + b2x2 ··· + bnxn) − 6( ξi ) i=1 i n n X n p−1 p−1 p+1 ≤ M |ξn|(K1 |ei|)[ (xi + ei )] − 6( ξi ) , h Xi=2 Xi=2 Xi=1 i where K1 > 0 is a real constant related to bi’s and independent of Ln, ··· , L2, M. With the aid of (4.32), Lemmas 2.26 and 2.24, the inequality above can be sim- pliﬁed as n n ˙ p+1 p+1 Un ≤ M − 5( ξi )+ K2( ei ) , (4.41) h Xi=1 Xi=2 i where K2 > 0 is a constant independent of Ln, ··· , L2, M. 74

To determine the observer gains Ln, ··· , L2, consider the change of coordinates

e˜2 = e2, e˜3 = e3 − L3e2, ··· , e˜n = en − Lnen−1. (4.42)

In the coordinates of ξ ande ˜, (4.41) can be represented as (by Lemma 2.26)

n ˙ p+1 p+1 Un ≤ M − 5( ξi )+ c2(L3, ··· , Ln)˜e2 h Xi=1 p+1 p+1 + ··· + cn−1(Ln)˜en−1 + cne˜n , (4.43) i where c2(L3, ··· , Ln), ··· ,cn−2(Ln−1, Ln),cn−1(Ln) are positive real constants independent of M, and cn > 0 is a known constant independent of M and all the Li’s. The error dynamics (4.39) in the coordinatee ˜ can be rewritten as

˙ p p p p e˜2 = M(x3 − xˆ3)+ g2(·) − ML2(x2 − xˆ2) − L2g1(·) . . (4.44)

˙ p p p p e˜n−1 = M(xn − xˆn)+ gn−1(·) − MLn−1(xn−1 − xˆn−1) − Ln−1gn−2(·) ˙ p p e˜n = gn(·) − MLn(xn − xˆn) − Lngn−1(·)

Now, consider the Lyapunov function

1 W (˜e , ··· , e˜ )= (˜e2 + ··· +˜e2 ). n 2 n 2 2 n

A direct computation gives

n n ˙ p p p p Wn = M − Lie˜i[(ˆxi +˜ei) − xî ]+ e˜i−1(xi − xî ) i=2 i=3 n X n X n p p − Lie˜i[xi − (ˆxi +˜ei) ] + e˜igi(·) − Lie˜igi−1(·). (4.45) Xi=2 Xi=2 Xi=2 Similar to Chapter 3, it is not difficult to get the following estimations for each term on the right hand side of (4.45).

1 −e˜ [(ˆx +˜e )p − xˆp] ≤ − e˜p+1 i i i i 2p−1 i p+1 p+1 p p ξi +ξi−1 p+1 p+1 e˜ − (x − xˆ ) ≤ +ˆc (L , ··· ,L )˜e + ··· +ˆc (L , ··· ,L )˜e i 1 i i n−1 2 3 n 2 i i+1 n i

1 L e˜ [xp − (ˆx +˜e )p] ≤ (ξp+1 + ξp+1) i i i i i n − 1 i i−1

p+1 p+1 +ˆc2(L3 , ··· , Ln)˜e2 + ··· +ˆci−1(Li, ··· , Ln)˜ei

1 i−1 1 i−1 1− i p 1− i p |Lie˜igi−1(·)| ≤ CM p Lie˜i( |xj| ) ≤ KM˜ p Lie˜i( |ξj| ) (4.46) jX=1 jX=1 i−1 1 M p+1 1− − p+1 p+1 ≤ ( ξ )+ KM˜ pi 1 L e˜ n − 1 j i i jX=1 M i |e˜ g (·)| ≤ ( ξp+1)+ MK˜ e˜p+1, i i n − 1 j i jX=1 wherec ˆ2(L3, ··· , Ln), ··· , cˆn−2(Ln−1, Ln), cˆn−1(Ln) are positive constants independent of M. Substituting (4.46) into (4.45), a tedious but straightforward calculation leads to

n 1 n W˙ ≤ M 4( ξp+1) − ( L e˜p+1) n i 2p−1 i i Xi=1 Xi=2 − 1 p p+1 p+1 + c˜2(L3, ··· , Ln)+ K2M L2 e˜2

h 1 i − − pn 2 p+1 p+1 + ··· + c˜n−1(Ln)+ Kn−1M Ln−1 e˜n−1

h 1 i − − pn 1 p+1 p+1 + cñ + KnM Ln eñ , (4.47) h i wherec ˜2(L3, ··· , Ln), ··· , cñ−1(Ln−1, Ln), cñ−1(Ln) are positive constants independent of M, whilec ñ > 0 and Ki > 0, 2 ≤ i ≤ n are known constants independent of Li’s and M. Finally, choose the Lyapunov function

Vn(ξ1, ··· ,ξn, e˜2, ··· , e˜n)= Un(ξ1, ··· ,ξn)+ Wn(˜e2, ··· , e˜n) for the closed-loop system in the coordinates (ξ, e˜). Then,it follows from (4.43) and

(4.47) that

˙ p+1 p+1 L2 K2 p+1 p+1 Vn ≤ −M ξ1 + ··· + ξn + p−1 − C2(L3, ··· , Ln) − 1 L2 e˜2 2 p h i h M i Ln−1 Kn−1 p+1 p+1 + ··· + − Cn−1(Ln) − 1 Ln−1 e˜n−1 p−1 − 2 pn 2 h M i Ln Kn p+1 p+1 + − Cn − 1 Ln e˜n , (4.48) p−1 − 2 pn 1 h M i 76

where C2(L3, ··· , Ln), ··· , Cn−2(Ln−1, Ln),Cn−1(Ln)are positive constants independent of M, while Cn > 0 and Ki > 0, 2 ≤ i ≤ n are positive constants independent of Li’s and M.

From (4.48), it is easy to conclude that if the gain parameters Li’s and M are assigned one-by-one, in the following manner:

p−1 Ln ≥ 2 1+ Cn + Kn p−1 Ln−1 ≥ 2 1+ Cn−1(Ln)+ Kn−1 . . (4.49)

p−1 L2 ≥ 2 1+ C2(L3, ··· , Ln)+ K2

2 (p+1)p (p+1)p (p+1)pn−1 M ≥ max{L2 , L3 ··· , Ln } ≥ 1, we have ˙ p+1 p+1 p+1 p+1 Vn ≤ −M (ξ1 + ··· + ξn )+(˜e2 + ··· +˜en ) . (4.50) h i This, in turn, implies that the uncertain high order system (4.21) is globally asymptotically stabilized by the dynamic output compensator (4.37)–(4.40).

As a consequence of Theorem 4.4, we have the following important result that provides a solution to the global stabilization problem of system (4.1).

Corollary 4.5. For a family of uncertain systems (4.1) satisfying Assumption 4.1, there exists a smooth output feedback controller of the form (4.3), such that the closed-loop system (4.1)–(4.3) is globally asymptotically stable at the equilibrium (x, xˆ)=(0, 0).

Proof: Corollary 4.5 follows immediately from Theorem 4.4 if one observes that system (4.1) is a special case of the uncertain system (4.21) and Assumption 4.3 reduces to Assumption 4.1 when φi,j(t,η,u)=0 for i =1, ··· , n − 1, j =1, ··· ,p − 1. The reader is referred to (Lin and Yang (2004)) for further details. 77

The application of Corollary 4.5 and the main features of the robust smooth output feedback control scheme developed so far can be illustrated by the following example.

Example 4.6. Consider the uncertain planar system

2 3 d(t)η2η1 2 η˙1 = η2 + 2 ln(1 + η1) 1+ η2

η˙2 = v

y = η1, (4.51) where d : IR → IR, is a continuous time-varying function satisfying |d(t)| ≤ 1.

Clearly, global output feedback stabilization of the uncertain system (4.51) is a diﬃcult problem for two reasons: 1) it requires the design of a single output feedback controller to stabilize a family of nonlinear systems, due to the presence of the time- varying parameter d(t); 2) the output feedback control schemes proposed in Chapter

3 cannot be applied to the uncertain system (4.51), because of the lack of eﬀective design methods for the construction of robust observers and/or output compensators for uncertain nonlinear systems with unstabilizable/undetectable linearization. On the other hand, a simple calculation shows that the uncertain system (4.51) satisﬁes the homogeneous growth condition (4.2). Indeed,

d(t)η η2 2 1 ln(1 + η2) ≤|η |3, ∀d(t) ∈ [−1, 1]. 1+ η2 1 1 2

By Corollary 4.5, a reduced-order dynamic output compensator can be designed as follows:

˙ 3 zˆ2 = 0.32v − 90(ˆz2 +3y)

3 v = −91.4(5.7ˆz2 + 73.4y) , (4.52) which globally robustly stabilizes system (4.51). It should be pointed out that unlike in the last chapter, the design of the output feedback controller (4.52) does not require the knowledge of the uncertain term 78

2 d(t)η2 η1 2 2 ln(1 + η ). This is substantially diﬀerent from Chapter 3, where a copy of the 1+η2 1 2 d(t)η2η1 2 term 2 ln(1 + η ) must be used for the construction of a nonlinear observer. As 1+η2 1 a result, the output feedback control scheme proposed in Chapter 3 is not robust and cannot be employed to control uncertain systems such as (4.51). A simulation result is given in Fig. 4.1 with d(t) = sin t, illustrating eﬀectiveness of the output feedback controller (4.52) and a transient response of the closed-loop system.

x 0 1 x 2 ∧ z 2 −5

−10

−15

−20

−25

−30

−35 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Time

Fig. 4.1: Transient responses of the system (4.51)-(4.52) from x1(0) = 2, x2(0) = −10, zˆ2(0) = −15.

Remark 4.7. It is worth pointing out that even in the absence of uncertainty, Corol- lary 4.5 is new. Moreover, it has incorporated and generalized the output feedback stabilization results obtained previously in Chapter 3. Speciﬁcally, Corollary 4.5 can 79 be applied to a larger class of nonlinear systems such as

3 3 3 η˙1 = η2 + η1 sin(η2v) η˙1 = η2 η˙ = η3 η˙ = η3 + η η sin η 2 3 and 2 3 1 2 2 η˙3 = v η˙3 = v y = η1, y = η1, because both of them satisfy the homogeneous growth condition (4.2). However, none of them satisﬁes the higher-order version of global Lipschitz-like condition, i.e., Assumption 3.3. As a result, Theorem 3.5 cannot be employed here to solve the output feedback stabilization problem for the above nonlinear systems.

4.4 Output Feedback Stabilization of Uncertain Cascade Systems

The purpose of this section is to investigate how the robust output feedback stabilization result obtained in the previous section can be extended to a family of uncertain cascade systems of the form

ζ˙ = F0(t,ζ,η,v)

p p−1 η˙1 = η2 + η2 ψ1,p−1(t,ζ,η,v)+ ··· + η2ψ1,1(t,ζ,η,v)+ ψ1,0(t,ζ,η,v) . .

p p−1 η˙n−1 = ηn + ηn ψn−1,p−1(t,ζ,η,v)+ ··· + ηnψn−1,1(t,ζ,η,v)+ ψn−1,0(t,ζ,η,v)

η˙n = v + ψn,0(t,ζ,η,v)

y = η1, (4.53) where v ∈ IR and y ∈ IR are the system input and output, respectively, ζ ∈ IRr and η ∈ IRn are the system states, and p ≥ 1 is an odd integer. The functions

n+r r n+r F0 : IR×IR ×IR → IR and ψi,j : IR×IR ×IR → IR, i =1, ··· , n, j =0, ··· ,p−1 are C0. To tackle the problem of global robust stabilization by smooth output feedback for the cascade system (4.53), we make the following assumptions: 80

2 Assumption 4.8. There is a C Lyapunov function U0(ζ), which is positive deﬁnite and proper, such that ∀(t,ζ,η,v) ∈ IR × IRn+r × IR,

∂U 0 F (t,ζ,η,v) ≤ −||ζ||p+1 + K ηp+1, (4.54) ∂ζ 0 0 1 where K0 > 0 is a real constant.

Assumption 4.9. For i =1, ··· , n − 1 and j =0, ··· ,p − 1,

p−j p−j p−j |ψi,j(t,ζ,η,v)| ≤ C(||ζ|| + |η1| + ··· + |ηi| )

p p p |ψn,0(t,ζ,η,v)| ≤ C(||ζ|| + |η1| + ··· + |ηn| ). (4.55)

Clearly, Assumption 4.8 is a sort of ISS-like condition, while Assumption 4.9 is an extension of the homogeneous growth condition (4.22). With the help of these two conditions, we can prove the following result on global output feedback stabilization of the uncertain cascade system (4.53).

Theorem 4.10. Under Assumptions 4.8-4.9, the uncertain cascade system (4.53) is globally asymptotically stabilizable by smooth output feedback.

Proof: The proof of this result is similar to that of Theorem 4.4. A key difference lies in the design of a partial-state rather than full-state feedback controller for the uncertain cascade system (4.53). For this reason, we give only a sketch of the proof highlighting the difference. As done in the proof of Theorem 4.4, we first introduce a rescaling transformation that is composed of ζ = ζ and (4.23) for the uncertain system (4.53). Such a transformation gives

˙ ζ = F0(t,ζ,η,v)

p x˙ 1 = Mx2 + G1(t,ζ,x,u) . . 81

p x˙ n−1 = Mxn + Gn−1(t,ζ,x,u)

x˙ n = Mu + Gn(t,ζ,x,u)

y = x1. (4.56)

In view of Assumption 4.9, it is straightforward to show that the uncertainty of system (4.56) satisﬁes the constraint

1 1− 1 p p p |G (·)| ≤ M p |x | + C (||ζ|| + |x | ) 1 2 2 0 1 . h i . (4.57)

1 n−1 1− n−1 1 p p p |G − (·)| ≤ M p |x | + C (||ζ|| + |x | ) n 1 2 n 0 j h jX=1 i 1 1− n p p p |Gn(·)| ≤ C0M p (||ζ|| + |x1| + ··· + |xn| )

where C0 > 0 is a known constant independent of M. For the rescaled system (4.56) with the constraint (4.57), it can be proved that Assumptions 4.8 implies the existence of a globally stabilizing, partial-state feedback controller u(x1, ··· , xn). To see why this is the case, consider the Lyapunov function

Uˆ0(ζ)=2nMU0(ζ). (4.58)

Then, it follows from (4.54) that

ˆ˙ p+1 p+1 U0 ≤ M − 2n||ζ|| +2nK0x1 . (4.59) h i

Let ξ1 = x1 and choose the Lyapunov function 1 U (ζ,ξ )= Uˆ (ζ)+ ξ2. 1 1 0 2 1 Using the fact M ≥ 1 and (4.59), one deduces from (4.56)-(4.57) and Young’s inequality that 1 U˙ ≤ M − 2n||ζ||p+1 +2nK xp+1 + ξ x∗p + |ξ ||x∗p| + C |ξ | ||ζ||p + C ξp+1 1 0 1 1 2 2 1 2 0 1 0 1 h 1 +ξ (xp − x∗p)+ |ξ | |xp − x∗p| 1 2 2 2 1 2 2 1 i 3 ≤ M − (2n − 1)||ζ||p+1 + ξ x∗p + |ξ ||x∗p| + α ξp+1 + |ξ ||xp − x∗p| , 1 2 2 1 2 0 1 2 1 2 2 h i 82

where α0 > 0 is a constant independent of M.

∗ 1/p Clearly, the virtual controller x2 = −β1ξ1, with β1 = [2(α0 +2n − 1)] being a constant independent of M, yields

3 U˙ ≤ M − (2n − 1)(||ζ||p+1 + ξp+1)+ |ξ ||xp − x∗p| . 1 1 2 1 2 2 h i Using a similar inductive argument we conclude at the n-th step that there exist a set of transformations

∗ ∗ ξi = xi − xi , xi = −βi−1ξi−1, i =1, ··· , n, (4.60) a Lyapunov function

1 U (ζ,ξ , ··· ,ξ )=2nMU (ζ)+ (ξ2 + ··· + ξ2), n 1 n 0 2 1 n and a partial-state feedback control law of the form

∗ p xn+1 = −βnξn = −(b1x1 + ··· + bnxn) , (4.61) such that

3 U˙ ≤ M − (n + 1)(||ζ||p+1 + ξp+1 + ··· + ξp+1)+ |ξ | |u − x∗ | , (4.62) n 1 n 2 n n+1 h i where all the parameters β1, ··· , βk and b1, ··· , bk > 0 are known constants independent of M. Note that inequality (4.62) reduces to (4.34) in the absence of ζ-dynamics.

Because the states (x2, ··· , xn) are unmeasurable, the controller (4.61) cannot be directly implemented. To obtain an implementable controller, we design an (n − 1)- dimensional observer for recovering (x2, ··· , xn) of the rescaled system (4.56). Mo- tivated by the robust observer design in the last section, we ignore the uncertain terms Gi(t,ζ,x,u), i = 1, ··· , n, in system (4.56) and construct the dynamic output compensator

p p zˆ˙ 2 = M (ˆz3 + L3L2x1) − L2(ˆz2 + L2x1) h i 83

. .

˙ p zˆn = M u − Ln ··· L2(ˆz2 + L2x1) h i p u = − (b1x1 + b2xˆ2 + ··· + bnxˆn)

xˆi =z ˆi + Li ··· L2x1, 2 ≤ i ≤ n, (4.63)

where Ln, ··· , L2 are the observer gains to be assigned.

The remaining part of the proof is to determine the parameters Ln, ··· , L2 as well as the rescaling factor M, which is analogous to that of Theorem 4.4 and therefore left to the reader as an exercise. In conclusion, one can prove that by suitably choosing the gain constants Ln, Ln−1 ··· , L2 and M one-by-one, the closed-loop system (4.56)– (4.63) can be rendered globally asymptotically stable at the origin. Clearly, in the case when an uncertain system is of the form

ζ˙ = F0(t,ζ,η,v)

p η˙1 = η2 + φ1(t,ζ,η,v) . .

p η˙n−1 = ηn + φn−1(t,ζ,η,v)

η˙n = v + φn(t,ζ,η,v)

y = η1, (4.64)

Assumption 4.9 reduces to

Assumption 4.11. There exists a constant C > 0 such that ∀(t,ζ,η,v) ∈ IR ×

IRn+r × IR,

p p p |φi(t,ζ,η,v)| ≤ C(||ζ|| + |η1| + ··· + |ηi| ), i =1, ··· , n. (4.65)

Then, we have the following useful corollary.

Corollary 4.12. Under Assumptions 4.8 and 4.11, the uncertain cascade system

(4.64) is globally robustly stabilizable by smooth output feedback. 84

We conclude this section with an example that illustrates how Theorem 4.10 can be employed to solve the diﬃcult problem of global robust stabilization by smooth output feedback, for uncertain cascade systems beyond a triangular structure.

Example 4.13. Consider the uncertain cascade system

ζ˙ = −ζ + η1 cos(ζv)

3 η˙1 = η2 + θη2 sin(ζη1)

η˙2 = v

y = η1, (4.66) where θ is an unknown constant bounded by a known constant, for instance, by one.

Note that this nonlinear system has three key features that make global output feedback stabilization of (4.66) subtle. First of all, system (4.66) is not in a lower- triangular form due to the second dynamic equation. Secondly, the linearized system of (4.66) is given by

−1 1 0 0 A =  0 00  , B =  0  , C = [1 0 0], 0 00 1         which is neither stabilizable nor detectable. While the latter makes the current output feedback design methods hard to be applied to system (4.66), the former prevents an application of the output feedback control scheme developed in Chapter 3 to the cascade system (4.66). Finally, the presence of the unknown constant θ requires that a robust output feedback control scheme be used, and therefore the output feedback design method proposed in Chapter 3 cannot be applied to (4.66), due to the nature of the non-robust design. On the other hand, it is easy to see that the three-dimensional cascade system (4.66) is of the form (4.53) with n = 2 and p = 3, and satisﬁes the growth condition

(4.55). Moreover, the ISS-like inequality (4.54) holds for the ζ-subsystem of (4.66). 85

4 As a matter of fact, using the Lyapunov function U0(ζ)= ζ and Lemma 2.24 yields ∂U 0 (−ζ + η cos(ζv)) ≤ −3ζ4 + 27η4. ∂ζ 1 1

Moreover, the uncertain term θη2 sin(ζη1)= η2ψ1,1(ζ,η,v). Thus, 1 1 |ψ (ζ,η,v)| = |θ sin(ζx )| ≤ |ζ|2 + |x |2, ∀|θ| ≤ 1. 1,1 1 2 2 1 That is, Assumption 4.9 holds.

By Theorem 4.10, there exists a dynamic output compensator of the form (4.3) such that the closed-loop system is globally asymptotically stable. In what follows, a detailed design procedure is presented for the purpose of illustration.

η2 First, we introduce the rescaling transformation ζ = ζ, x1 = η1, x2 = M 1/3 , u = v M 4/3 , where M ≥ 1 is a rescaling factor to be determined later. Such a transformation results in

4/3 ζ˙ = −ζ + x1 cos(M ζu)

3 1/3 x˙ 1 = Mx2 + M θx2 sin(ζx1)

x˙ 2 = Mu

y = x1. (4.67)

Using the smooth state feedback design method in (Lin and Qian (2000b)), one can ﬁnd a Lyapunov function of the form 1 U(ζ, x ,ξ )= MU (ζ)+ (x2 + ξ2), ξ = x + a x , 1 2 0 2 1 2 2 2 1 1

∗ 3 3 and a partial-state feedback controller x3 = −(a2ξ2) = −[a2(x2 + a1x1)] , such that

˙ 4 4 4 ∗ U ≤ −M[2(ζ + x1 + ξ2 ) − ξ2(u − x3)], where both a1 and a2 are positive constants independent of M.

Next, let z2 = x2 − L2x1 with L2 > 0 being a gain constant to be assigned later. Since

3 1/3 z˙2 = Mu − L2[Mx2 + M θx2 sin(ζx1)], 86 we design the reduced-order observer

˙ 3 zˆ2 = Mu − L2Mxˆ2, wherex ˆ2 =z ˆ2 + L2x1, (4.68)

1/3 which is a copy of (4.68) without the uncertain term M θx2 sin(ζx1).

Usingx ˆ2 thus obtained and the certainty equivalence principle, we deduce from

∗ x3 that

3 3 u = −[a2(ˆx2 + a1x1)] = −[a2(ˆz2 + L2x1 + a1x1)] . (4.69)

Finally, we show that the dynamic output compensator (4.68)-(4.69) globally robustly stabilizes the uncertain cascade system (4.67) for all θ ∈ [−1, 1], if L2 and M are chosen suitably.

To this end, Let e2 = x2 − xˆ2 ≡ z2 − zˆ2 be the estimate error. The error dynamics is

3 3 1/3 e˙2 = −L2M(x2 − xˆ2) − L2M θx2 sin(ζx1).

2 e2 Choose the Lyapunov function W = 2 . Then,

˙ 3 3 1/3 W = e2 − L2M(x2 − xˆ2) − L2M θx2 sin(ζx1) h L i ≤ M − 2 e4 + L M −2/3|ζx x e | , ∀θ ∈ [−1, 1]. 4 2 2 1 2 2 h i Selecting

3/2 M ≥ L2 (4.70) yields

L L 1 W˙ ≤ M − 2 e4 + |ζx x e | ≤ M − 2 e4 + (ζ4 + x4 + ξ4)+ k e4 , 4 2 1 2 2 4 2 2 1 2 0 2 h i h i where k0 > 0 is a constant independent of M.

Now, consider the Lyapunov function V (ζ, x1,ξ2, e2)= U(ζ, x1,ξ2)+W (e2) for the closed-loop system (4.67)-(4.68)-(4.69). Using Lemmas 2.24-2.29, it is not diﬃcult to prove that L V˙ ≤ M − ( 2 − K)e4 − (ζ4 + x4 + ξ4) . 4 2 1 2 h i 87

In view of the relationship (4.70) and M ≥ 1, it is clear that the choices L2 = 4(K +1)

3/2 and M = max(L2 , 1) result in

˙ 4 4 4 4 V ≤ −M(ζ + x1 + ξ2 + e2), which implies global asymptotic stability of the closed-loop system (4.67)-(4.68)-(4.69) for all θ ∈ [−1, 1].

The aforementioned design procedure leads to, for instance, the dynamic output compensator

˙ 3 zˆ2 =0.4v − 80(ˆz2 + 24x1)

3 v = −34[24(ˆz2 + 11.4x1)] (4.71) that does the job. The simulation shown in Fig. 4.2 demonstrates the property of robust stability of the closed-loop system (4.66)-(4.71).

4.5 Summary

This chapter has proved that under appropriate homogeneous growth conditions, global robust stabilization by smooth output feedback can be achieved for a family of uncertain nonlinear systems that are not uniformly observable (Gauthier et al. (1992)) and have unstabilizable and undetectable linearization. A robust output feedback design approach has been developed based on a rescaling technique and the idea of non-separation principle design, enabling one to recursively construct a robust state feedback controller and a homogeneous observer that does not depend on the uncertainty of the system. The main result of Chapter 4 has incorporated and generalized the robust output feedback stabilization theorem in (Qian and Lin (2002a)), where global robust stabilization was shown to be possible for a family of uncertain systems with controllable/observable linearization. For high order uncertain systems in a cascade form or in the so-called p-normal form (which are beyond a 88

0.8

0.6 ζ

0.4

0.2 η 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time

0 ∧ z 2 η 2 −1

−2

−3

−4

−5

−6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time

Fig. 4.2: Transient responses of the closed-loop system (4.66)-(4.71) with θ =1 and the initial condition (ζ, η1, η2, zˆ2)=(1, 0.3, −6, −5) strict-triangular structure), we have also identiﬁed suitable conditions for the problem of global robust stabilization to be solvable by smooth output feedback. The applications of the proposed robust output feedback control schemes have been illustrated by several examples (also, see Remark 4.7). Chapter 5. GLOBAL ROBUST STABILIZATION BY NONSMOOTH OUTPUT FEEDBACK

We present in this chapter additional results on global robust stabilization by nonsmooth output feedback, for a class of n-dimensional uncertain systems that are not smoothly stabilizable. To handle the system uncertainty eﬀectively, a rescaling transformation with a suitable dilation is introduced. The rescaling technique, together with the nonsmooth output feedback design method developed recently, leads to a robust output feedback control scheme which achieves, under a homogeneous growth condition, global stabilization for a family of uncertain nonlinear systems with unstabilizable/undetectable linearization. The design of nonsmooth state feedback controllers and homogeneous observers does not require the precise information of the uncertain system and depends only on the knowledge of the bounding system.

5.1 Introduction

The purpose of this chapter is to investigate the problem of global robust stabilization by nonsmooth output feedback, for a family of uncertain nonlinear systems of the form

p1 η˙1 = η2 + φ1(t, η, v) . .

pn−1 η˙n−1 = ηn + φn−1(t, η, v)

η˙n = v + φn(t, η, v)

y = η1, (5.1)

89 90

T where v ∈ IR, η = (η1, ··· , ηn) and y ∈ IR are the system input, state and output,

n respectively. For i =1, ··· , n − 1, pi ≥ 1 is an odd positive integer, and φi : IR × IR ×

1 IR → IR, is a C mapping with φi(0, ··· , 0) = 0.

In (Cheng and Lin (2003)), it was shown that system (5.1) with suitable φi(·) is a generalized normal form of affine systems when exact feedback linearization is not possible. Indeed, a necessary and sufficient, coordinate-free geometric condition was given in (Cheng and Lin (2003)) and then extended in (Respondek (2003)), under which the affine system (3.2) is feedback equivalent to the nonlinear system (5.1) with

pi−1 k φi = k=0 ηi+1ai,k(η1, ··· , ηi). ForP a nonlinear system (5.1) in the p-normal form (Cheng and Lin (2003); Re- spondek (2003)), global stabilization results by nonsmooth state feedback have been obtained under some appropriate conditions; see, for instance, Qian and Lin (2001a) and the references therein. Despite the rapid development of stabilization theory by state feedback, progress in its output feedback counterpart has been slow. There are many reasons for this, such as the lack of the separation principle even assuming that the nonlinear system has a simple structure, the diﬃculty of designing nonlinear observers, the trouble caused by unstabilizable/undetectable linearization, etc. All of these makes the output feedback control of the nonlinear system (5.1) a diﬃcult problem, and the conventional output feedback design methods based on “Luenberger- type” or “high-gain” observers (Gauthier et al. (1992); Krener and Isidori (1983); Krener and Kang (2003)) are no longer applicable.

In the past few years, some preliminary results have been reported on global output feedback stabilization of the nonlinear system (5.1). For example, in (Qian and Lin (2002b, 2003)) it was shown that in the two-dimensional case, global stabilization of (5.1) can be achieved by smooth or nonsmooth output feedback, if the planar system (5.1) has a strict-triangular structure and satisﬁes suitable growth conditions. In Chapter 3, we have addressed systematically the problem of global stabilization 91

via smooth output feedback, for the n-dimensional system (5.1) with φi(·)=0, 1 ≤ i ≤ n. This was accomplished by employing a new observer design technique for the construction of homogeneous observers, combined with the tool of adding an integrator for the design of smooth state feedback controllers. In addition, it was proved in that global output feedback stabilization of the nonlinear system (5.1) is also possible, as long as (5.1) satisﬁes a high-order version of global Lipschitz-like condition. The main contribution of Chapter 3 was the development of a recursive algorithm for the design of homogeneous observers, enabling one to assign the observer gains step-by-step.

There are two limitations in the work of Chapter 3: 1) it requires that p1 = ··· = pn−1; 2) it cannot be used to deal with a larger class of non-uniformly observable and non-smoothly stabilizable systems. To overcome these shortcomings, a nonsmooth output feedback control scheme was proposed in (Qian and Lin (2004b)), in which a recursive, non-smooth observer design algorithm was developed for the nonlinear system (5.1) with diﬀerent pi’s. The non-smooth observer thus obtained, together with the nonsmooth state feedback controllers derived in (Qian and Lin (2001a)), resulted in new results on global stabilization of a subclass of systems (5.1) with diﬀerent pi’s. In the local case, the paper (Qian and Lin (2004b)) also showed that, by means of the homogeneous systems theory, local stabilization of the nonlinear system (5.1) in the p-normal form is achievable by nonsmooth output feedback, without imposing any growth condition.

Although the observer designs in Chapter 3 and the paper (Qian and Lin (2004b)) are substantially diﬀerent from the traditional ones (Gauthier et al. (1992); Khalil and Saberi (1987); Krener and Isidori (1983); Krener and Kang (2003); Krener and Xiao (2002)), they still use a copy of the original system and hence require the precise information of the controlled plant. As a result, the output feedback control scheme proposed in Chapter 3 and the paper (Qian and Lin (2004b)) is not robust with 92 respect to parametric uncertainty and cannot be applied to nonlinear systems with uncertainty. Such a robust output feedback control problem was addressed in (Qian and Lin

(2002a)) for the uncertain system (5.1) with p1 = ··· = pn−1 = 1 (i.e., with controllable/observable linearization), and later on in (Yang and Lin (2005a)), i.e. Chapter

4, for the uncertain high order system (5.1) when p1 = ··· = pn−1 > 1, under a homogeneous growth condition (4.22). In this chapter, we shall study the question of when the uncertain system (5.1) with diﬀerent pi’s is robustly stabilizable by output feedback, and present some preliminary results. Realizing that in the case when pi’s are distinct, it is usually not possible to deal with the nonlinear system (5.1) by smooth feedback, even locally, we shall develop a nonsmooth, rather than smooth, output feedback design method to tackle the robust output feedback stabilization problem for the uncertain system (5.1).

The goal of this chapter is to design a nonsmooth dynamic output compensator of the form

xˆ˙ = θ(ˆx, y), xˆ ∈ IRn−1,

v = v(ˆx, y), (5.2) achieving global robust stabilization for a family of uncertain systems (5.1). Throughout this chapter, we assume that the uncertain nonlinear system (5.1) satisﬁes the following condition:

Assumption 5.1. There exists a real constant C ≥ 0 such that ∀(t, η, v) ∈ IR × IRn × IR (i =1, ··· , n),

1/(p1···pi−1) 1/pi−1 |φi(t, η, v)| ≤ C(|η1| + ··· + |ηi| + |ηi|). (5.3)

Clearly, Assumption 5.1 reduces to the linear growth condition in (Qian and Lin

(2002a)) when p1 = ··· = pn−1 = 1. In the case of pi > 1, it is weaker than the 93 condition (3.9) in (Qian and Lin (2004b)), which is nothing but a homogeneous type of the global Lipschitz condition. Moreover, due to the growth condition (5.3), the family of nonlinear systems considered here is larger than those considered in (Qian and Lin (2004b)), as illustrated in Examples 5.3, 5.11 and 5.12. In this work, we show that for a family of uncertain nonlinear systems (5.1) satisfying Assumption 5.1, global robust stabilization can be achieved by nonsmooth output feedback. To deal with the system uncertainty, we introduce a subtle rescaling transformation motivated by the work in the last chapter. The new rescaling technique integrated with the nonsmooth output feedback design method (Qian and Lin (2004b)) leads to a nonsmooth output feedback control scheme that solves the problem of global robust stabilization. The construction of a nonsmooth state feedback controller and a homogeneous observer relies only on the knowledge of the bounding system rather than on the uncertain system. This is a substantial diﬀerence between this chapter and the work (Qian and

Lin (2004b)). Three examples are given to illustrate the applications of the robust output feedback control scheme developed in this chapter.

5.2 Robust Output Feedback Design: A Case Revisit

To tackle the problem of global robust output feedback stabilization for the uncertain nonlinear system (5.1) whose linearization is neither stabilizable nor detectable, we revisit in this section the case where p1 = ··· = pn−1 = 1 in (5.1). In this case, system (5.1) is simpliﬁed as

η˙1 = η2 + φ1(t, η, v) . .

η˙n−1 = ηn + φn−1(t, η, v)

η˙n = v + φn(t, η, v)

y = η1. (5.4) 94

Obviously, the linearized system of (5.4) is controllable and observable. Moreover, the homogeneous growth condition (5.3) reduces to

|φi(t, η, v)| ≤ C(|η1| + ··· + |ηi|), i =1, ··· , n. (5.5)

Under the linear growth condition (5.5), it has been proved in (Qian and Lin (2002a)) that global robust stabilization of the uncertain system (5.4) can be achieved by a linear output feedback controller. The proof was done by means of a coupled controller-observer design (Qian and Lin (2002a)) which is not based on the separation principle. However, the output feedback control scheme proposed in (Qian and Lin (2002a)) cannot be easily extended to the nonlinear case, for instance, the case when pi ≥ 1, because of the linear nature of the design in (Qian and Lin (2002a)). In what follows, we seek an alternative output feedback design method that is capable of taking advantage of homogeneity of the uncertain system (5.4). As we shall see later, the robust linear output feedback scheme developed in this section has a valid nonsmooth counterpart that turns out to be eﬀective in dealing with a family of uncertain nonlinear systems (5.1) with pi ≥ 1. To begin with, we ﬁrst introduce the rescaling transformation for the uncertain system (5.4):

η v x = i , i =1, ··· , n, and u = , (5.6) i M i−1 M n where M ≥ 1 is a rescaling constant to be assigned later.

Under the new coordinates xi’s, the uncertain system (5.4) is represented as

x˙ 1 = Mx2 + f1(t, x, u) . .

x˙ n−1 = Mxn + fn−1(t, x, u)

x˙ n = Mu + fn(t, x, u)

y = x1, (5.7) 95

where the uncertain functions fi(t, x, u), i = 1, ··· , n, satisfy the linear growth condition (by (5.5))

|φ (t, η, v)| C(|η | + ··· + |η |) |f (t, x, u)| = i ≤ 1 i i M i−1 M i−1

≤C (|x1| + ··· + |xi|) . (5.8)

For the rescaled uncertain system (5.7) with the constraints (5.8), it is easy to design recursively, in a fashion similar to (Qian and Lin (2002a)), a linear state feedback controller

∗ u = −βnξn = −(bnxn + ··· + b1x1), (5.9) such that ˙ 2 2 ∗ U ≤ M − 3(ξ1 + ··· + ξn)+ ξn(u − u ) , (5.10) h i 1 2 2 ∗ where U = 2 (ξ1 + ··· + ξn), ξi = xi − xi , i =1, ··· , n, and

∗ ∗ ∗ x1 =0, x2 = −β1ξ1, ··· , xn = −βn−1ξn−1

with βi > 0 and bi > 0 being known constants independent of M. Next, we shall design a linear observer for the rescaled system (5.7). Because y = x1 is measurable and only unmeasurable states of (5.7) are (x2, ··· , xn), it is natural to design an (n − 1)-dimensional observer rather than a full-order observer. Motivated by the reduced-order observer design for linear systems, we build an (n − 1) − th order linear observer to estimate, instead of the states (x2, ··· , xn), the unmeasurable variables (z2, ··· , zn) deﬁned by

z2 = x2 − L2x1, ··· , zn = xn − Lnxn−1, (5.11)

where Li’s> 0 are gain constants to be determined later. From (5.11) it is clear that

z˙2 = Mx3 + f2(·) − ML2x2 − L2f1(·) 96

. . (5.12)

z˙n−1 = Mxn +fn−1(·)−MLn−1xn−1 −Ln−1fn−2(·)

z˙n = Mu + fn(·) − MLnxn − Lnfn−1(·).

In view of (5.12), we construct the (n − 1)-th order linear observer (regardless of the uncertain terms fi(·), 1 ≤ i ≤ n): ˙ zˆ2 = Mxˆ3 − ML2xˆ2, xˆ2 = z2 + L2x1 . .x ˆ = z + L xˆ 3 3 3 2 (5.13) ˙ . zˆn−1 = Mxˆn − MLn−1xˆn−1, . zˆ˙ n = Mu − MLnxˆn, xˆn = zn + Lnxˆn−1, where (ˆz2, ··· , zˆn)and(ˆx2, ··· , xˆn) are the estimates of the unmeasurable state (z2, ··· , zn) and (x2, ··· , xn).

Let ei = zi − zˆi, 2 ≤ i ≤ n, be the estimate errors. It follows that xi − xˆi = ei + Li−1ei−1 + ··· + Li−1 ··· L2e2. Thus, the error dynamics are

e˙2 = M(e3 + L2e2)+ f2(·) − ML2e2 − L2f1(·) . . (5.14)

e˙n = fn(·) − MLn(en + Ln−1en−1 + ··· + Ln−1 ··· L2e2) − Lnfn−1(·).

To obtain an implementable controller, we simply replace the unmeasurable states

(x2, ··· , xn) in (5.9) by its estimates (ˆx2, ··· , xˆn), which are generated by the linear observer (5.13). In this way, we arrive at

u = −(bnxˆn + ··· + b2xˆ2 + b1x1). (5.15)

Now, consider the Lyapunov function 1 V (x, e)= U(x)+ e2 + ··· + e2 . 2 2 n Substituting (5.15) into (5.10) and using the completion of square, it can be shown from the linear growth condition (5.8) that L + ··· + L K L V˙ ≤ −M 2 − 2 n (ξ2 + ··· + ξ2)+ L − c (L , ··· , L ) − 2 2 e2 M 1 n 2 2 3 n M 2 nh i h i 97

Kn−1Ln−1 2 KnLn 2 + ··· + L − − c − (L ) − e + L − c − e , n 1 n 1 n M n−1 n n M n h i h i o (5.16)

where c2(L3, ··· , Ln), ··· ,cn−1(Ln−1, Ln),cn−1(Ln) are positive constants independent of M, cn > 0 and Ki > 0, 2 ≤ i ≤ n are known constants independent of M and all

Li’s.

Choosing the gain parameters Li and M one-by-one, in the following manner:

Ln ≥ 2+ cn

Ln−1 ≥ 2+ cn−1(Ln) . . (5.17)

L2 ≥ 2+ c2(L3, ··· , Ln)

M ≥ max{L2 + ··· + Ln,K2L2, ··· ,KnLn}, we have ˙ 2 2 2 2 V (·) ≤ −M (ξ1 + ··· + ξn)+(e2 + ··· + en) . h i Therefore, the uncertain system (5.4) is globally robustly stabilized by the output feedback controller (5.15)-(5.13).

5.3 Nonsmooth Output Feedback Stabilization of Uncertain Nonlinear Systems

The robust output feedback design method presented in the last section for the uncertain system (5.4) with controllable/observable linearization can be generalized, with a suitable twist, to system (5.1). In this section, we show that under Assumption 5.1, a robust output feedback control strategy can be developed for a family of uncertain nonlinear systems (5.1) whose linearization is neither stabilizable nor detectable. In particular, with the aid of Lemmas 2.24—2.29, we can prove the following important result. 98

Theorem 5.2. Under Assumption 5.1, the uncertain nonlinear systems (5.1) can be globally robustly stabilized by a nonsmooth dynamic output compensator of the form (5.2).

Proof: This result will be proved by explicitly designing a nonsmooth state feedback controller and a robust homogeneous observer, both of them do not depend on the uncertain functions φi(t, η, v). Note that the construction of the nonsmooth observer is diﬀerently from the one in (Qian and Lin (2004b)). In particular, no copy of φi(t, η, v) is needed here for the design of a robust observer, while the observer in

(Qian and Lin (2004b)) involved a copy of φi(t, η, v). Thus, φi(t, η, v) in (Qian and Lin (2004b)) must be time independent and precisely known. That is, no system uncertainty is allowed. We begin with the proof by observing that according to the homogeneous system theory (Dayawansa (1992); Hermes (1991a); Kawski (1989)), system (5.1) with φi(·)= 0 is a homogeneous system with the dilation (1, 1 , ··· , 1 ; 1 ) and degree p1 p1···pn−1 p1···pn−1 0. With this in mind, we introduce the rescaling transformation

ηi v xi = , i =1, ··· ,n, u = (5.18) M qi M qn+1 q1 +1 qn−1 +1 q1 = 0, q2 = , ··· , qn = , qn+1 = qn +1. p1 pn−1 which is a natural extension of (5.6) proposed in the last section. The rescaling transformation (5.18) has a dilation (q1, ··· , qn; qn+1), and M ≥ 1 in (5.18) is a rescaling factor to be determined later.

In the new coordinate (x1, ··· , xn), the uncertain nonlinear system (5.1) is rewritten as

p1 x˙ 1 = Mx2 + f1(t, x, u) 99

. .

pn−1 x˙ n−1 = Mxn + fn−1(t, x, u)

x˙ n = Mu + fn(t, x, u) (5.19)

qi where fi(·)= φi(·)/M , i =1, ··· , n. Using Assumption 5.1 and the fact that M ≥ 1, it is easy to see from (5.18) that for i =1, ··· , n,

1/(p1···pi−1) 1/pi−1 |fi(·)| ≤ C(|x1| + ··· + |xi−1| + |xi|). (5.20)

Next, we use the nonsmooth feedback design method (Qian and Lin (2001a)) to derive a robust state feedback controller for the rescaled system (5.19) under the growth condition (5.20). In fact, using an inductive argument as done in (Qian

1 and Lin (2001a)), there are a C Lyapunov function U(x1, ··· , xn), which is positive

0 ∗ ∗ ∗ deﬁnite and proper, and a set of C virtual controllers x1, ··· , xn,u , deﬁned by

∗ ∗ x1 =0, ξ1 = x1 − x1 = x1 ∗p1 p1 ∗p1 p1 x2 = −β1ξ1, ξ2 = x2 − x2 = x2 + β1x1 . . . . ∗p1···pn−1 p1···pn−1 ∗ p1···pn−1 xn = −βn−1ξn−1, ξn = xn − xn

1 ∗ ··· − p1 ··· p1 pn 1 p1 pn−1 u = − (bnxn + ··· + b2x2 + b1x1) (5.21) with constants βi, bi > 0 are constants independent of M such that

˙ 2 2 2−1/(p1···pn−1) ∗ U(ξ) ≤ M[ − 4(ξ1 + ··· + ξn)+ ξn (u − u )]. (5.22)

∗ Because the states (x2, ··· , xn) are unmeasurable, the virtual controller u in (5.21) cannot be directly implemented. To obtain an implementable controller, a natural thing to do is to design an (n − 1)-dimensional observer for recovering (x2, ··· , xn) of the rescaled system (5.19). Notably, the observer design method proposed in (Qian and Lin (2004b)) is not applicable to the uncertain system (5.19) since it requires a copy of fi(·), i =1, ··· , n, which are unknown in the current case. Inspired 100 by the robust observer design approach reviewed in the last section, we build an

(n − 1)-dimensional robust homogeneous observer to estimate, instead of the states

(x2, ··· , xn), the unmeasurable variables (Qian and Lin (2004b)):

p1 pn−1 z2 = x2 − L2x1, ··· , , zn = xn − Lnxn−1, (5.23)

where Li’s> 0 are gain parameters to be determined later. Using (5.23), it is easy to see that

p1−1 p2 p1 z˙2 =p1x2 (Mx3 + f2(·)) − L2(Mx2 + f1(·)) . . (5.24)

pn−1−1 pn−1 z˙n =pn−1xn (Mu + fn(·)) − Ln(Mxn + fn−1(·)).

As done in the last section, we ignore the uncertain terms fi(t, x, u) in system (5.24) and simply design the following (n − 1)–dimensional homogeneous observer similar to (Qian and Lin (2004b)):

˙ p1 p1 zˆ2 = −ML2xˆ2 , xˆ2 = z2 + L2x1 ˙ p2 p2 zˆ3 = −ML2xˆ3 , xˆ3 = z3 + L3xˆ2 . . (5.25) . . ˙ pn−1 pn−1 zˆn = −MLnxˆn , xˆn = zn + Lnxˆn−1 where (ˆz2, ··· , zˆn) and (ˆx2, ··· , xˆn) are the estimates of the unmeasurable states

(z2, ··· , zn) and (x2, ··· , xn).

Let ei = zi − zˆi, i =2, ··· , n, be the estimate errors. Note that

pi pi ei+1 = xi+1 − Li+1xi − xˆi+1 + Li+1xˆi. (5.26)

Thus,

pi pi xi+1 − xˆi+1 = ei+1 + Li+1(xi − xˆi). (5.27)

Therefore, the error dynamics are given by

p1−1 p2 e˙2 = p1x2 (Mx3 + f2(·)) − ML2e2 − L2f1(·) 101

p2−1 p3 e˙3 = p2x3 (Mx4 + f3(·)) − ML3(e3 + L3(x2 − xˆ2)) − L3f2(·) . . (5.28)

pn−1−1 e˙n = pn−1xn (Mu + fn(·)) − MLn(en + Ln(xn−1 − xˆn−1)) − Lnfn−1(·).

Now, consider the Lyapunov function in (Qian and Lin (2004b))

2 2p1 2p1···pn−2 e2 e3 en V (e2, ··· , en)= + + ··· + , (5.29) 2 2p1 2p1 ··· pn−2 which is positive deﬁnite and proper. Clearly,

˙ 2p1···pn−2−1 pn−1−1 ∗ 2p1···pn−2−1 pn−1−1 ∗ V = pn−1Men xn (u − u )+ pn−1en xn Mu + fn(·) n−1 n 2p1···pi−2−1 pi−1−1 pi 2p1···pi−2 + pi−1ei xi Mxi+1 + fi(·) − M Liei (5.30) i=2 i=2 X n n X 2 2p1···pi−2−1 2p1···pi−2−1 − M Li ei (xi−1 − xˆi−1) − Liei fi−1(·). Xi=3 Xi=2 In order to estimate the terms on the right-hand side of (5.31), we introduce the following propositions whose proofs are similar to (Qian and Lin (2004b)) and involve tedious calculations but nevertheless can be carried out straightforwardly by using Lemmas 2.24–2.29 and are omitted.

Proposition 5.3. There is a real constant K > 0 independent of M such that

n−1 2p1···pn−2−1 pn−1−1 ∗ 2p1···pi−2−1 pi−1−1 pi pn−1en xn Mu + fn(·) + pi−1ei xi Mxi+1 + fi(·) i=2 X n 2p1···pi−2 2 2 ≤M K( ei )+(ξ1 + ··· + ξn) . (5.31) h Xi=2 i

Proposition 5.4. There exist constants ci(Li+1, ··· , Ln) > 0, i =2, ··· , n − 1, independent of M and cn > 0 independent of all the Li’s and M such that

n 2 2p1···pi−2−1 2 Li ei (xi−1 − xˆi−1) ≤ c2(L3, ··· , Ln)e2

Xi=3 ··· − 2p1 pn 3 2p1···pn−2 + ··· + cn−1(Ln)en−1 + cnen . (5.32) 102

Proposition 5.5. There exist constants Ki > 0, i = 2, ··· , n, independent of M such that

n n 2p1···pi−2−1 2p1···pi−2 2 2 Liei fi−1(·) ≤ ( KiLiei )+(L2 + ··· + Ln)(ξ2 + ··· + ξn).

i=2 i=2 X X (5.33)

With the help of Propositions 5.3–5.5, the following inequality can be obtained

L + ··· + L V˙ ≤ M 1+ 2 n (ξ2 + ··· + ξ2) M 1 n nh i K L − L − K − c (L , ··· , L ) − 2 2 e2 −··· 2 2 3 n M 2 h i Kn−1Ln−1 2p1···pn−3 − L − − K − c − (L ) − e n 1 n 1 n M n−1 h i KnLn − L − K − c − e2p1···pn−2 n n M n h i 2p1···pn−2−1 pn−1−1 ∗ +pn−1en xn (u − u ) . (5.34) o Now, we apply the certainty equivalence principle to obtain an implementable output feedback controller. Observe that the reduced-order observer (5.25) has provided an estimation for the unmeasurable states (x2, ··· , xn). Keeping this in mind, we simply replace (x2, ··· , xn) in the controller (5.21) by its estimate (ˆx2, ··· , xˆn) generated from the observer (5.25). Thus,

1 ··· − p1 ··· p1 pn 1 p1 pn−1 u = − (bnxˆn + ··· + b2xˆ2 + b1x1) . (5.35)

The next proposition gives an estimation for the terms involving with u − u∗ in

(5.22) and (5.34). Its proof is similar to (Qian and Lin (2004b)) and omitted.

Proposition 5.6. There exist constantsc ¯i(Li+1, ··· , Ln) > 0, i = 2, ··· , n − 1, independent of M and a real constantc ¯n > 0 independent of all the Li’s and M such that

2−1/(p1···pn−1) 2p1···pn−2−1 pn−1−1 ∗ (ξn + pn−1en xn )(u − u )

2 2 2 2p1···pn−3 2p1···pn−2 ≤ (ξ2 + ··· + ξn)+¯c2(L3, ··· , Ln)e2 + ··· +¯cn−1( Ln)en−1 +¯cnen−1 (5.36). 103

Immediately, the inequality (5.36), together with (5.22) and (5.34), yields

L + ··· + L U˙ + V˙ ≤ −M 2 − 2 n (ξ2 + ··· + ξ2) M 2 n nh i K L + L − K − c (L , ··· , L ) − c¯ (L , ··· , L ) − 2 2 e2 2 2 3 n 2 3 n M 2 h i KnLn + ··· + L − K − c − c¯ − e2p1···pn−2 . (5.37) n n n M n h i o

From (5.37), it is clear that choosing the gain parameters Li one-by-one, in the following manner:

Ln ≥ K + cn +¯cn +2

Ln−1 ≥ K + cn−1(Ln)+¯cn−1(Ln)+2 . . (5.38)

L2 = K + c2(L3, ··· , Ln)+¯c2(L3, ··· , Ln)+2,

M ≥ max{L2 + ··· + Ln,K2L2, ··· ,KnLn}, we immediately have

n ˙ ˙ 2 2 2p1···pi−2 U + V ≤ −M (ξ1 + ··· + ξn)+ ei , h Xi=2 i which is negative deﬁnite. Therefore, the closed-loop system (5.1)-(5.25)-(5.35) is globally stable in the sense of Kurzweil (see Kurzweil (1956); Qian and Lin (2001a)). As a consequence, we have the following important corollary.

Corollary 5.7. If there exists a constant C > 0 such that

|φi(·)| ≤ C(|ηi| + ··· + |ηi−ki |), i =1, ··· , n, (5.39)

where ki = max{j ≥ 0 | pi−1 = ··· = pi−j = 1}, system (5.1) is globally stabilizable by nonsmooth output feedback.

Consider the uncertain nonlinear system

η˙1 = η2 104

3 2 η˙2 = η3 + η2 + d(t) ln(1 + η1)

η˙3 = η4

η˙4 = u + η4 + η3

y = η1, (5.40)

where d(t) is a C0 time-varying parameter satisfying |d(t)| ≤ 1. It is easy to verify that (5.40) is of the form (5.39). In fact, it is not diﬃcult to see that

k2 = k4 = C = 1. By Corollary 5.7, there is a nonsmooth output feedback controller globally stabilizing (5.40). Notably, system (5.40) is non-homogeneous

and involves the time-varying parameter d(t). Thus, it cannot be dealt with by the output feedback scheme suggested in (Qian and Lin (2004b)).

5.4 Extension and Discussion

In this section, we discuss brieﬂy how the output feedback stabilization result for the uncertain system (5.1) can be extended to a family of C1 uncertain cascade systems of the form

ζ˙ = F (t,ζ,η,v), ζ ∈ IRn−r,

p1 η˙1 = η2 + φ1(t,ζ,η,v) . .

pr−1 η˙r−1 = ηr + φr−1(t,ζ,η,v)

η˙r = v + φr(t,ζ,η,v)

y = η1, (5.41) where v ∈ IR, (ζ, η) ∈ IRn and y ∈ IR are the system input, state and output, respectively, and pi ≥ 1 are odd integers. To achieve global robust stabilization by nonsmooth output feedback, we make the following assumptions. 105

Assumption 5.8. For i =1, ··· ,r,

1 1 1 p1···p −1 p1···p −1 p −1 |φi(t,ζ,η,v)| ≤ C(||ζ|| i + |η1| i + ··· + |ηi−1| i + |ηi|). (5.42)

2 Assumption 5.9. There is a C Lyapunov function U0(ζ), which is positive deﬁnite and proper, such that

∂U 0 F (t,ζ,η,v) ≤ −||ζ||2 + K η2, (5.43) ∂ζ 0 1 where K0 > 0 is a ﬁxed known constant.

Then, the following output feedback stabilization result can be proved by means of an argument similar to that of Theorem 5.2 and are omitted accordingly.

Theorem 5.10. Under Assumptions 5.8 and 5.9, the uncertain cascade system (5.41) is globally robustly stabilized by a nonsmooth output feedback controller of the form

(5.2).

For the purpose of illustration, we present two examples to demonstrate how the nonsmooth output feedback control scheme developed so far can be employed to achieve global robust stabilization for uncertain nonlinear systems with unstabilizable/undetectable linearization.

Example 5.11. Consider the uncertain planar system

1 η˙ = η3 + η ed(t) sin η2 1 2 3 1

η˙2 = v

y = η1, (5.44) where d(t) is a C0 time-varying parameter with |d(t)| ≤ 1.

Clearly, global output feedback stabilization of the uncertain system (5.44) is a nontrivial problem for two reasons: 1) it requires the design of a single output 106 feedback controller to stabilize a family of nonlinear systems, due to the presence of the unknown parameter d(t); 2) most of the existing output feedback control schemes including the one proposed recently in (Qian and Lin (2004b)) is hard to be applied to the uncertain system (5.44), because of the lack of eﬀective design methods for the construction of robust observers and/or output compensators for uncertain nonlinear systems with unstabilizable/undetectable linearization. However, a simple calculation shows that the uncertain system (5.44) satisﬁes the homogeneous growth condition (5.3). Indeed, 1 | η ed(t) sin η2 |≤|η |, ∀d(t) ∈ [−1, 1]. 3 1 1 By Theorem 5.2, a reduced-order dynamic output compensator can be designed as

zˆ˙2 = −106.3(ˆz2 +3y)

1/3 v = −100(5.4ˆz2 + 82y) , (5.45) which globally robustly stabilizes the uncertain system (5.44). It should be pointed out that unlike in (Qian and Lin (2004b)), the design of the output feedback controller (5.45) does not use the knowledge of the uncertain term

d(t) sin η2 η1e . This is substantially diﬀerent from the work (Qian and Lin (2004b)),

d(t) sin η2 where a copy of the term η1e is needed for the construction of a nonlinear observer. As a result, the output feedback control scheme proposed in (Qian and Lin

(2004b)) is not robust and cannot be applied to uncertain systems such as (5.44).

Example 5.12. Consider a cascade system of the form

˙ 2 ζ = −ζ + ln(1 + η1)

3 η˙1 = η2 + θη1 cos(ζη2)

η˙2 = v

y = η1. (5.46)

1 where θ is an unknown constant satisfying θ ∈ [ 2 , 1]. 107

Note that this nonlinear system has a signiﬁcant features that make global output feedback stabilization of (5.46) diﬃcult. Indeed, the linearized system of (5.46) is given by −1 0 0 0 A =  0 θ 0  , B =  0  , C = [1 0 0], 0 00 1        1 which is neither stabilizable nor detectable ∀θ ∈ [ 2 , 1]. Moreover, the uncontrollable mode is associated with a positive eigenvalue θ. For this reason, system (5.46) must be stabilized by nonsmooth feedback. However, due to the presence of the unknown parameter θ, the nonsmooth output feedback control scheme proposed in (Qian and Lin (2004b)) is inapplicable to the uncertain system (5.46). On the other hand, it is easy to see that the cascade system (5.46) is of the form

(5.41) with r = 2 and p1 = 3, p2 = 1, and satisﬁes Assumption 5.8. Moreover, the

z2 ISS-like inequality (5.43) holds for the z-subsystem of (5.46) by choosing U0(z)= 2 . By Theorem 5.10, one can design the dynamic output compensator

zˆ˙2 = −45.5(ˆz2 + 52y)

1/3 v = −29[12(ˆz2 + 25y)] , (5.47)

1 such that the closed-loop system (5.46)-(5.47) is globally robustly stable ∀θ ∈ [ 2 , 1].

5.5 Summary

In this chapter, we have presented a robust nonsmooth output feedback design method, which integrates the nonsmooth state feedback design in (Qian and Lin (2001a)) and the recursive algorithm for the construction of nonsmooth nonlinear observers (Qian and Lin (2004b)). The new ingredient is the introduction of a rescaling technique combined with the idea of the non-separation principle design, making it possible to recursively construct both robust state feedback controllers and homogeneous observers that do not depend on the uncertainty of the system. Using this new output 108 feedback control strategy that is nonsmooth in nature, we have identiﬁed appropriate conditions under which the problem of global robust stabilization is solvable by nonsmooth output feedback for a class of uncertain nonlinear systems, although they cannot be stabilized, even locally, by any smooth feedback. The signiﬁcance of the nonsmooth output feedback control schemes proposed in this chapter have been illustrated by three examples, which cannot be dealt with by the non-smooth output feedback design method in (Qian and Lin (2004b)). Chapter 6. GLOBAL OUTPUT FEEDBACK CONTROL WITH DYNAMICAL RESCALING

In Chapter 6, for a family of uncertain nonlinear systems dominated by a triangular system that satisfies linear growth condition with an output dependent growth rate, we prove that global robust stabilization can be achieved by smooth output feedback. This conclusion has incorporated and generalized the recent output feedback stabilization results, for instance, the work (Qian and Lin (2002b)) where the same conclusion was already shown to be true for planar systems, and the work (Praly and Jiang (2003)) where the growth rate is required to be a polynomial function of the system output. There are two key ingredients in the present contribution. One of them is the introduction of a rescaling transformation with a dynamic factor that is tuned on-line through a Riccati-like differential equation, which turns out to be extremely effective in dealing with the system uncertainty. The other one is the development of a recursive observer design algorithm making it possible to assign the robust observer gains in a step-by-step fashion. Both a smooth state feedback controller and a robust observer are explicitly constructed for the rescaled system using only the knowledge of the bounding system.

6.1 Main Result and Discussion

In this chapter, we investigate the problem of global output feedback stabilization for a family of uncertain nonlinear systems of the form

η˙1 = η2 + φ1(t, η, v)

η˙2 = η3 + φ2(t, η, v)

109 110

. . (6.1)

η˙n = v + φn(t, η, v)

y = η1,

n where η =(η1, ··· , ηn) ∈ IR , v ∈ IR,y ∈ IR are the system states, input and output,

n respectively. The functions φi : IR × IR × IR → IR, i =1, ··· , n, are continuous with respect to all the variables and represent the system uncertainty. Throughout this

n chapter, we assume that the uncertain term φi(t, η, v) satisﬁes ∀(t, η, v) ∈ IR×IR ×IR,

|φi(t, η, v)| ≤ C(y)(|η1| + ··· + |ηi|), i =1, ··· , n, (6.2) where C : IR → (0, +∞) is a known smooth function. The purpose of this chapter is to prove that under the growth condition (6.2), the entire family of uncertain nonlinear systems (6.1) is globally robustly stabilizable by a single, smooth dynamic output compensator

w˙ = θ(w,y), w ∈ IRm, m ≤ n,

v = v(w,y). (6.3)

Formally, the main result of this chapter can be summarized as follows:

Theorem 6.1. For a family of uncertain nonlinear systems (6.1) satisfying the growth condition (6.2), there is a smooth output feedback controller (6.3) rendering the closed-loop system (6.1)–(6.3) globally uniformly bounded and (η1, ··· , ηn) uniformly convergent to the origin.

Remark 6.2. The growth condition (6.2) essentially says the bounding system of

(6.1) must be linear in its unmeasurable states η2, ··· , ηn but can be nonlinear with respect to the system output. The necessity of imposing the linear growth requirement on the unmeasurable states of (6.1) has been explained, for instance, in (Mazenc et al. (1994)). In fact, counter-examples given in (Mazenc et al. (1994); Qian and Lin 111

(2002a)) have indicated that if the system nonlinearity φi(t, η, v) grows faster than a quadratic function with respect to the variables (η2, ··· , ηn) — unmeasurable states of (6.1), it may not be possible to globally stabilize the nonlinear system (6.1) by any continuous output feedback.

Remark 6.3. It is worth pointing out that the assumption of C(y) being a smooth function of y involves no loss of generality. Indeed, if C(y) is only continuous (not necessarily locally Lipschitz continuous), it is straightforward to deduce, using the fact that C(0) > 0, the existence of a smooth function C˜ : IR → IR+, such that C˜(y) ≥ C(y) > 0, ∀y. In other words, Theorem 6.1 still holds even if C(y) is only a positive continuous function.

The following discussions in order explain connections between this chapter and some of closely related works in the literature.

• Theorem 6.1 was ﬁrst proved to be true in (Qian and Lin (2002b)) for the two- dimensional uncertain system (6.1) (see Corollary 3.1 by Qian and Lin (2002b)

for details). The novelty of the paper by Qian and Lin (2002b) lies in the explicit construction of a one-dimensional nonlinear observer combined with a feedback domination design. However, the output feedback control scheme proposed in (Qian and Lin (2002b)) is an ad hoc one and only works for planar systems. It

is very diﬃcult to be extended to the n-dimensional case.

• For the n-dimensional uncertain system (6.1), Theorem 6.1 was also shown to be true when C(y) ≡ C (Qian and Lin (2002a)), where C > 0 is a real

constant. This was made possible by using the traditional high-gain observer in (Khalil and Saberi (1987); Gauthier et al. (1992)) and by developing a coupled observer-controller design method that is not based on the separation principle. An advantage of such an output feedback design is that no precise knowledge

of the system uncertainty is required. 112

• Recently, Theorem 6.1 has been proved to hold under the extra requirement

that the output dependent growth rate C(y)isa polynomial function of y (Praly and Jiang (2003)). This result is strongly reminiscent of the previous work by Praly (2003), where the idea of using dynamic gain was introduced and global output feedback stabilization was shown achievable for a triangular system sat-

isfying global Lipschitz-like condition (precisely, the Lipschitz “constant” is a non-negative smooth function of the output y). Thus, Praly and Jiang (2003) has improved the result of (Praly (2003)) and relaxed the global Lipschitz-like condition by the weaker condition (6.2). This was done by modifying the scal-

ing technique introduced in (Praly (2003)) and by imposing the restriction that C(y) be a polynomial function of y, together with some useful matrix inequalities introduced in (Praly (2003); Praly and Jiang (2003); Krishnamurthy and Khorrami (2002)).

In summary, the contribution of Theorem 6.1 is two-fold: 1) it generalizes the output feedback stabilization result of planar systems previously obtained in (Qian and Lin (2002b)) to the n-dimensional uncertain nonlinear system (6.1); 2) it removes the restriction that C(y) be a polynomial function of y required in the paper by Praly and Jiang (2003). In the remainder of this chapter, we shall give a constructive proof of Theorem 6.1. In particular, we shall develop a new robust output feedback control scheme that provides an iterative method for the design of both robust observers and controllers.

There are two key ingredients in our output feedback design method. The first one is the introduction of a rescaling transformation with a dynamic factor, which is motivated by the work presented in the last two chapters, where the power of the rescaling technique has been demonstrated for uncertain systems. The rescaling factor is involved with the system output and tuned on-line via a Riccati-like differential equation (this owes an inspiration to the work by Praly (2003)). It turns out that 113 the rescaling transformation with a dynamic factor is an effective tool for the analysis and synthesis of the uncertain system (6.1) satisfying the growth condition (6.2). The other ingredient is the development of a recursive observer design algorithm that can be viewed as a dual of adding a integrator design for the construction of smooth state feedback controllers. The algorithm allows one to assign the robust observer gains in a step-by-step fashion. Compared with the output feedback design methods proposed in (Praly (2003); Praly and Jiang (2003); Krishnamurthy and Khorrami (2002)), where the construction of dynamic output compensators relies crucially on several matrix inequalities (see the appendices of Krishnamurthy and Khorrami (2002); Praly and Jiang (2003)), a substantial difference is that in this chapter, both robust controller and observer are recursively constructed, step-by-step, for the rescaled system without involving any matrix inequality. This feature makes it possible to further extend Theorem 6.1 to a much wider class of nonlinear systems with unstabilizable/undetectable linearization, for instance, a nonlinear system in the p-normal form (Yang and Lin (2005a); Cheng and Lin (2003)). This part of research is beyond the scope of this chapter and will be reported in Conclusion.

6.2 State Feedback Design

6.2.1 Dynamic Rescaling of the Original System

In the last two chapters, a rescaling transformation with a suitable dilation has been introduced, which turns out to be rather eﬀective in handling the system uncertainty. Speciﬁcally, a constant M was used as a rescaling factor to deal with a family of uncertain systems (6.1) satisfying the linear growth condition (6.2) with C(y) ≡ C, where C > 0 is a real constant. In this chapter, since the growth rate in (6.2) is not a constant but a function of the output y, it is natural to expect that a dynamic rescaling factor M(t) instead of a constant needs to be used for dealing with 114 the uncertain system (6.1). This kind of thinking leads to the following rescaling transformation η η v x = η , x = 2 , ··· , x = n , and u = , (6.4) 1 1 2 M n M n−1 M n where M is, in contrast to a constant in Chapters 4-5, a dynamic rescaling factor that is updated by a dynamic system

M˙ = ζ(M,y) with a ﬁxed M(0), (6.5) to be determined later.

Remark 6.4. The idea of tuning the rescaling factor M(t) through the dynamic system (6.5) is inspired by the work (Praly (2003)), in which it was shown how an adapted high-gain observer can be designed and how the observer gain can be updated on-line via a Riccati-like diﬀerential equation. The reader is referred to (Krishnamurthy and Khorrami (2002)) for additional applications of adapted high- gain observers.

Using the transformation (6.4), the original system (6.1) can be written as

x˙ 1 = Mx2 + f1(·) M˙ x˙ = Mx − x + f (·) 2 3 M 2 2 . . (6.6) M˙ x˙ = Mu − (n − 1) x + f (·) n M n n

y = x1,

φi(t,η,v) where fi(·)= M i−1 , i =1, ··· , n. If M ≥ 1, it is straightforward to show that the rescaled system (6.6) also satisﬁes the output dependent growth condition. As a matter of fact,

|f1(·)| ≤ C(y)|y| (6.7) |η | + |η | + ··· + |η | |y| |f (·)| ≤ C(y)( 1 2 i ) ≤ C(y)( + |x | + ··· + |x |), i =2, ··· , n. i M i−1 M 2 i 115

Notably, the introduction of the dynamic rescaling transformation (6.4)-(6.5) has created a new design parameter — the rescaling factor M, which can be used to dominate the uncertainties of the rescaled system (i.e. fi(·), 1 ≤ i ≤ n, in (6.6)), if ζ(M,y) is appropriately chosen. With this in mind and in view of (6.5)–(6.6), a natural candidate of ζ(M,y) seems to be

M˙ − = −∆(y) ⇔ M˙ = M∆(y), M M(0) = 1, (6.8) where ∆(y) > 0 is a smooth function to be designed.

For a ﬁxed ∆(y), it is clear that (6.8) is an implementable “linear” dynamic system with respect to M driven by y. By construction, M˙ > 0 when M = 1, which implies M(t) ≥ 1, ∀t ≥ 0. Moreover, substituting (6.8) into (6.6) yields

y˙ = Mx2 + f1(·)

x˙ 2 = Mx3 − ∆(y)x2 + f2(·) . .

x˙ n = Mu − (n − 1)∆(y)xn + fn(·)

Because of the presence of the terms −(i − 1)∆(y)xi (i = 2, ··· , n), it is possible to choose ∆(y) > 0 appropriately to dominate the uncertain functions f2(·), ··· , fn(·) under the growth condition (6.7). Solving the diﬀerential equation (6.8), one has

t M(t) = exp 0 ∆(y(s))ds . Hence, M(t) is bounded ∀t ∈ [0, +∞) as long as y(t) is uniformly boundedR over [0, +∞). However, (6.8) doesn’t warrant uniform boundedness of M(t) over [0, +∞) even if ∆(y) is set and y(t) is uniformly bounded on [0, +∞). This issue can be resolved by adding a remedy term into the updated law (6.8), as shown hereafter. 116

Based on the discussion above, we choose ζ(M,y) as follows (instead of (6.8)):

M˙ − = M − ∆(y) ⇔ M˙ = M∆(y) − M 2 =∆ ζ(M,y), M M(0) = 1, (6.9) where ∆(y) > 1 is a C∞ function to be determined later. Clearly, once ∆(y) is ﬁxed, (6.9) is an implementable Riccati diﬀerential equation driven by y. From (6.9) it follows that

> 0 whenever M =1 M˙ ( < 0 whenever M > ∆(y), which implies 1 ≤ M(t) ≤ ∆(supt≥0 |y(t)|), ∀t ≥ 0. Using (6.9), (6.6) can be expressed as

y˙ = Mx2 + f1(·)

x˙ 2 = Mx3 + Mx2 − ∆(y)x2 + f2(·) . . (6.10)

x˙ n = Mu +(n − 1)Mxn − (n − 1)∆(y)xn + fn(·)

y = x1.

From (6.10), it becomes clear that one may take advantage of the extra freedom created by the dynamic rescaling, particularly, by suitably choosing ∆(y) in (6.10) so that the uncertain terms fi(·), i = 1, ··· , n, can be dominated under the growth condition (6.7). This is exactly the philosophy to be pursued below for the design of robust controllers and observers.

6.2.2 State Feedback Controller

With the help of (6.9), in what follows we shall show how a robust state feedback controller can be constructed recursively for the rescaled system (6.10). 117

y2 Step 1. Consider the Lyapunov function V1 = 2 . Its derivative along the trajectories of (6.10) is

˙ ∗ ∗ 2 V1 = y Mx2 + f1(·) ≤ M yx2 + y(x2 − x2) + C(y)y . h i h i Clearly, the virtual controller

β (y)+ L (y) C(y)+ n +1+ L (y) x∗ = − 1 1 y = − 1 y 2 M M yields ˙ 2 ∗ V1 ≤ −[n +1+ L1(y)]y + My(x2 − x2), (6.11) where L1(y) > 0 is to be determined in the observer design.

∗ Step 2. In the previous step, the virtual controller x2 with the non-constant

β1(y)+L1(y) gain M was constructed to dominate the system uncertainty f1(·). However, the same trick cannot be used to handle the uncertain terms f2(·), ··· , fn(·). This is because if the gains of the virtual controllers were involved with y, one would have to take the derivative of the gains repeatedly at each step of the recursive design. As a result, some messy terms would appear making the feedback design, particularly, the observer design extremely complicated.

To overcome such a diﬃculty, we shall construct from now on a set of virtual

∗ ∗ ∗ controllers x3, ··· , xn with constant gains (in contrast to x2 derived in step 1, whose gain is non-constant and depends on y). To this end, we introduce, similar to the work by Krishnamurthy and Khorrami (2002), the transformation

β (y)+ L (y) ξ = x − x∗ = x + 1 1 y (6.12) 2 2 2 2 M

β1(y)+L1(y) whose inverse transformation is given by x2 = ξ2 − M y.

In the coordinate (y,ξ2, x3, ··· , xn), system (6.10) can be written as

y˙ = Mx2 + f1(·) ˙ ˜ ξ2 = Mx3 + Mξ2 − ∆(y)ξ2 + f2(·) 118

. . (6.13)

x˙ n = Mu +(n − 1)Mxn − (n − 1)∆(y)xn + fn(·),

˜ ∂[(β1(y)+L1(y))y] β1(y)+L1(y) f1(·) where f2(·)= f2(·)+ ∂y [ξ2 − M y + M ] satisfying ∂[(β (y)+ L (y))y] |y| |f˜ (·)| ≤ 1 1 + C(y) β (y)+ L (y)+ C(y)+1 + |ξ | 2 ∂y 1 1 M 2 h i |y| ≤ ψ (y)( + |ξ |), (6.14) M 2 where ψ(y) > 1 is a smooth function depending on L1(y) and becomes known once

L1(y) is determined.

Similarly, the growth condition (6.7) in the coordinate (y,ξ2, x3, ··· , xn) becomes

|y| |f (·)| ≤ ψ¯(y)( + |ξ | + |x |) 3 M 2 3 . . |y| |f (·)| ≤ ψ¯(y)( + |ξ | + |x | + ··· + |x |), (6.15) n M 2 3 n where ψ¯(y) > 1 is a smooth function depending on L1(y). Now, we continue our recursive design for the uncertain (6.13)-(6.15). Consider the Lyapunov function

2 V2(y,ξ2)= V1(y)+ Mξ2 .

Then,

˙ 2 2 2 V2 ≤ −[n +1+ L1(y)]y + Myξ2 +(M∆(y) − M )ξ2 ˜ +2Mξ2 Mx3 + Mξ2 − ∆(y)ξ2 + f2(·) h i 2 2 ∗ ∗ 2 2 2 ≤ −[n +1+ L1(y)]y +2M ξ2[x3 +(x3 − x3)] + M ξ2 − M∆(y)ξ2

+Mξ2(y +2f˜2(·)). (6.16)

By (6.14), the following estimate can be obtained:

˜ 2 |Mξ2(y +2f2(·))| ≤ |y||Mξ2| +2ψ(y)|ξ2||y| +2Mψ(y)ξ2

2 2 2 2 ≤ y + M ξ2 + Mψ2(y)ξ2, (6.17) 119

where ψ2(y) > 1 is a smooth function depending on L1(y). Using (6.16) and (6.17), one has

˙ 2 2 ∗ ∗ 2 V2 ≤ −[n + L1(y)]y +2M ξ2[ξ2 + x3 +(x3 − x3)] − M[∆(y) − ψ2(y)]ξ2.

Choosing the virtual controller

n x∗ = −β ξ with β = +1 ≥ 1, 3 2 2 2 2 yields

˙ 2 2 2 2 ∗ 2 V2 ≤ −[n + L1(y)]y − nM ξ2 +2M ξ2(x3 − x3) − M[∆(y) − ψ2(y)]ξ2.(6.18)

Step 3. Let

∗ x3 − x3 x3 x3 ξ3 = = + ξ2 or = ξ3 − ξ2, (6.19) β2 β2 β2 and choose the Lyapunov function

2 V3(y,ξ2,ξ3)=2V2(y,ξ2)+ Mξ3 .

From (6.18) and (6.13), it is deduced that

˙ 2 2 2 2 ∗ 2 V3 ≤ 2 − [n + L1(y)]y − nM ξ2 +2M ξ2(x3 − x3) − M[∆(y) − ψ2(y)]ξ2 1 +(M∆(y) − M 2)ξ2 +2Mξ [Mx +2Mx − 2∆(y)x 3 3 β 4 3 3 2 ˜ +f3(·)]+[Mx3 + Mξ2 − ∆(y)ξ2 + f2(·)] . (6.20) In ξ-coordinate, (6.20) can be rewritten as

˙ 2 2 2 2 2 ∗ ∗ V3 ≤ −[n + L1(y)]y − nM ξ2 + M ξ3[x4 +(x4 − x4)] β2 2 +M ξ3 4β2ξ2 +2β2(ξ3 − ξ2)+3ξ3 − 2ξ2 h i 2 2 2 f3(·) ˜ −M∆(y)[2ξ2 +3ξ3 − 2ξ3ξ2]+2Mψ2(y)ξ2 +2Mξ3 + f2(·) . " β2 # 120

Using (6.15)-(6.14), (6.19) and the facts that β2 ≥ 1, it is easy to prove the existence of a smooth function ψ3(y), which depends on L1(y) and satisﬁes ψ3(y) >

ψ2(y) > 1, such that

f (·) 1 ψ (y) 2Mξ 3 ≤ y2 + M 3 (ξ2 + ξ2) 3 β 2 4 2 3 2 1 ψ (y) 2Mξ f˜ (·) ≤ y2 + M 3 (ξ2 + ξ2). 3 2 2 4 2 3

This, in turn, leads to

2 2 2 f3(·) ˜ −M∆(y)[2ξ2 +3ξ3 − 2ξ3ξ2]+2Mψ2(y)ξ2 +2Mξ3 + f2(·) " β2 # 2 2 2 ≤ y − M[∆(y) − ψ3(y)](ξ2 + ξ3 ). (6.21)

Observe that there is a constant B2 ≥ 1 satisfying

2 2 ξ3 4β2ξ2 +2β2(ξ3 − ξ2)+3ξ3 − 2ξ2 ≤ ξ2 +2B2ξ3 . (6.22) h i

Putting (6.21) and (6.22) together, we have

x∗ x − x∗ V˙ ≤ −[n − 1+ L (y)]y2 − (n − 1)M 2ξ2 +2M 2ξ B ξ + 4 + 4 4 3 1 2 3 2 3 β β h 2 2 i 2 2 −M[∆(y) − ψ3(y)](ξ2 + ξ3 ).

Then, the virtual controller

n − 1 x∗ = −β ξ with β = β ( + B ) ≥ 1, 4 3 3 3 2 2 2 results in

∗ ˙ 2 2 2 2 2 x4 − x4 2 2 V3 ≤ −[n−1+L1(y)]y −(n−1)M (ξ2 +ξ3 )+2M ξ3 −M[∆(y)−ψ3(y)](ξ2 +ξ3 ). β2 (6.23) Step n. Using an inductive argument, one can prove that a similar conclusion holds at the n-th step. In fact, let

∗ xn − xn xn x3 xn ξn = = + ··· + + ξ2 or = ξn − ξn−1, βn−1 βn−1 β2 βn−1 121 and choose the Lynapunov function

2 n−2 2 n−2 2 n−3 2 2 Vn =2Vn−1 + Mξn =2 y + M(2 ξ2 +2 ξ3 + ··· + ξn).

A direct computation gives ∗ ˙ 2 2 2 2 2 xn − xn Vn ≤ 2 − [3 + L1(y)]y − 3M (ξ2 + ··· + ξn−1)+2M ξn−1 β − n 2 2 2 2 2 −M[∆(y) − ψn−1(y)](ξ2 + ··· + ξn−1) +(M∆(y) − M )ξn 1 +2Mξn [Mu +(n − 1)Mxn − (n − 1)∆(y)xn + fn(·)] β − n 1 + ··· + [Mx3 + Mξ2 − ∆(y)ξ2 + f˜2(·)] , which, in the coordinate of ξ, can be represented as 2 ˙ 2 2 2 2 2M ξn ∗ ∗ Vn ≤ −[3 + L1(y)]y − 3M ξ2 + ··· + ξn−1 + xn+1 +(u − xn+1) β − n 1 h i 2 βn−1 βn−1 +M ξn 4 ξn−1 + 2 (ξn − ξn−1)+ ··· +2β2(ξ3 − ξ2) β − β − h n 2 n 2 +(2n − 3)ξn − 2(ξ2 + ··· + ξn−1) i 2 2 2 −M∆(y) 2(ξ2 + ··· + ξn−1)+(2(n − 2)+1)ξn − 2ξn(ξ2 + ··· + ξn−1) h i 2 2 fn(·) ˜ +2Mψn−1(y)(ξ2 + ··· + ξn−1)+2Mξn + ··· + f2(·) . " βn−1 #

Note that the constants βi ≥ 1, i = 2, ··· , n − 1 and ξ2, ··· ,ξn are linear combi- nations of ξ2, x3, ··· , xn. Similar to step 3, it is easy to show that

2 2 2 −M∆(y) 2(ξ2 + ··· + ξn−1)+(2(n − 2)+1)ξn − 2ξn(ξ2 + ··· + ξn−1) h i 2 2 fn(·) ˜ +2Mψn−1(y)(ξ2 + ··· + ξn−1)+2Mξn + ··· + f2(·) " βn−1 # 2 2 2 ≤ y − M[∆(y) − ψn(y)](ξ2 + ··· + ξn), (6.24) where ψn(y), with ψn(y) > ψn−1(y) > ··· > ψ2(y) > 1, is a smooth function depending on L1(y). Similarly, it can be shown as done in step 3 that

βn−1 βn−1 ξn 4 ξn−1 + 2 (ξn − ξn−1)+ ··· +2β2(ξ3 − ξ2) β − β − h n 2 n 2 +(2n − 3)ξn − 2(ξ2 + ··· + ξn−1) i 2 2 2 ≤ ξ2 + ··· + ξn−1 +2Bn−1ξn, (6.25) 122

where Bn−1 ≥ 1 is a constant. In view of (6.24)-(6.25), we arrive at

˙ 2 2 2 2 Vn ≤ −[2 + L1(y)]y − 2M (ξ2 + ··· + ξn−1) ∗ ∗ 2 xn+1 u − xn+1 2 +2M ξn Bn−1ξn + + − M ∆(y) − ψn(y) ||ξ|| . β − β − h n 1 n 1 i h i Then, it is clear that a state feedback controller of the form

∗ ∆ xn+1 = −βnξn = −d2ξ2 − d3x3 −···− dnxn (6.26)

with βn = βn−1(1 + Bn−1) ≥ 1 and dn, ··· ,d2 being ﬁxed constants, is such that

∗ 2 2 2 2 u − xn+1 2 V˙n ≤ −[2 + L1(y)]y − 2M ||ξ|| +2M ξn − M ∆(y) − ψn(y) ||ξ|| . βn−1 h i (6.27)

∗ Remark 6.5. As pointed out in step 1, L1(y) is introduced in x2 to create an extra design freedom that will be used in the observer design (see step n in the next section).

In the state feedback case, L1(y) ≡ 0 and hence ψn(y) is a known smooth function.

∗ As a consequence, setting ∆(y) = ψn(y) > 1 and u = xn+1 in the inequality (6.27) yields

2 2 2 V˙n ≤ −2y − 2M ||ξ|| , which implies that the state feedback controller (6.26) globally stabilizes (6.10) or (6.1) at the origin.

6.3 Output Feedback Design

6.3.1 Reduced-Order Observer

Since (ξ2, x3, ··· , xn) of the rescaled system (6.13) is not measurable, the feedback controller (6.26) cannot be implemented. To obtain a realizable controller, a natural thing to do is to design an (n − 1)-dimensional observer generating an estimation of 123

(ξ2, x3, ··· , xn). Motivated by the reduced-order robust observer design in Chapters 3-5, we introduce the nonsingular transformation

z2 = x2 − L2y ⇔ x2 = z2 + L2y z3 = x3 − L3x2 ⇔ x3 = z3 + L3x2 . (6.28) . zn = xn − Lnxn−1 ⇔ xn = zn + Lnxn−1 where Li ≥ 1, 2 ≤ i ≤ n, are gain constants to be assigned later. Then, it is deduced from (6.10) and (6.28) that

z˙2 = [Mx3 +(M − ∆(y))x2 + f2(·)] − L2[Mx2 + f1(·)] . . (6.29)

z˙n = [Mu +(n − 1)(M − ∆(y))xn + fn(·)]

−Ln[Mxn +(n − 2)(M − ∆(y))xn−1 + fn−1(·)].

Using the idea of robust observer design in the last two chapters, we build an (n − 1)- dimensional observer of the form

zˆ˙2 = [Mxˆ3 +(M − ∆(y))ˆx2] − L2Mxˆ2 . . (6.30) ˙ zˆn = [Mu +(n − 1)(M − ∆(y))ˆxn] − Ln[Mxˆn +(n − 2)(M − ∆(y))ˆxn−1] for the uncertain system (6.29), which does not involve the system uncertainties fi(·), i =1, ··· , n.

Consequently, the estimates of the unmeasurable state (ξ2, x3, ··· , xn) of (6.13) are generated by

y xˆ =z ˆ + L y ⇒ ξˆ =x ˆ + [β (y)+ L (y)] 2 2 2 2 2 1 1 M

xˆ3 =z ˆ3 + L3xˆ2 . .

xˆn =z ˆn + Lnxˆn−1. (6.31) 124

By construction, the nonlinear observer (6.30)-(6.31) is realizable once Ln, ··· , L2, ∆(y) are chosen.

6.3.2 Error Dynamics and Output Feedback Controller

According to the discussion in the previous subsection, we deﬁne the estimated errors as follows: ˆ z2 − zˆ2 x2 − xˆ2 ξ2 − ξ2 e2 = = = L2 L2 L2

z3 − zˆ3 x3 − xˆ3 x2 − xˆ2 e3 = 2 = 2 − (6.32) L3L2 L3L2 L3L2 . . zn − zˆn xn − xˆn xn−1 − xˆn−1 en = 2 2 = 2 2 − 2 2 Ln ··· L3L2 Ln ··· L3L2 LnLn−1 ··· L3L2 or, equivalently,

ξ2 − ξˆ2 x2 − xˆ2 = = e2 L2 L2

x3 − xˆ3 e2 2 = e3 + L3L2 L3 . . (6.33)

xn − xˆn en−1 e2 2 2 = en + + ··· + . Ln ··· L3L2 Ln Ln ··· L3 With this in mind, the error dynamics can be written as

e − e f (·) e˙ = (n − 1)(M − ∆(y))(e + n 1 + ··· + 2 )+ n n n L L ··· L L2 ··· L2L h n n 3 n 3 2 i en−2 e2 − M(Lnen + en−1 + + ··· + ) Ln−1 Ln−1 ··· L3 h e − e f − (·) +(n − 2)(M − ∆(y))( n 1 + ··· + 2 )+ n 1 L L ··· L L L2 ··· L2L n n 3 n n−1 3 2 i 2 Ln Ln e˙n−1 = M(Lnen + Lnen−1 + en−2 + ··· + e2) L − L − ··· L h n 1 n 1 3 e2 fn−1(·) +(n − 2)(M − ∆(y))(en−1 + ··· + )+ 2 2 L − ··· L L ··· L L n 1 3 n−1 3 2 i e2 − M(Ln−1en−1 + ··· + ) L − ··· L h n 2 3 125

en−2 e2 fn−2(·) +(n − 3)(M − ∆(y))( + ··· + )+ 2 2 L − L − ··· L L − L ··· L L n 1 n 1 3 n 1 n−2 3 2 i . .

2 f2(·) e˙2 = M(L3e3 + L3e2)+(M − ∆(y))e2 + − ML2e2 + f1(·) . (6.34) L2 h i h i

By the certainty equivalence principle, we replace the states ξ2, x3, ··· , xn in (6.26) by its estimates ξˆ2, xˆ3, ··· , xˆn. In this way, we obtain the implementable controller

ˆ u = −d2ξ2 − d3xˆ3 −···− dnxˆn. (6.35)

Substituting (6.35) into (6.27) leads to (note that Li ≥ 1, i =2, ··· , n)

2 2 2 2 V˙n ≤ −[2 + L1(y)]y − 2M ||ξ|| − M(∆(y) − ψn(y))||ξ|| 2 2M ξn 2 2 en−1 e2 + d2L2e2 + ··· + dnLn ··· L3L2(en + + ··· + ) β − L L ··· L n 1 h n n 3 i 2 2 2 2 4 4 2 2 2 ≤ −[1 + L1(y)]y − M ||ξ|| − M(∆(y) − ψn(y))||ξ|| + KLn ··· L3L2M ||e|| ,

(6.36) where K > 0 is a suﬃciently large constant.

6.3.3 Observer Gain Assignment

Motivated by the work of Chapters 3-5, next we shall show that by a delicate design of the observer gains Ln, ··· , L2 and ∆(y) in the Riccati equation (6.9) step-by-step, Theorem 6.1 can be proved and the output feedback controller (6.9)-(6.30)-(6.31)- (6.35) does the job. The recursive design procedure below for the assignment of the observer gains can be regarded as a dual of the adding a integrator design given in section 6.2.

2 Step 1. Choose the Lynapunov function Un = Men. Clearly,

e − e U˙ = (M∆(y) − M 2)e2 +2Me (n − 1)(M − ∆(y))(e + n 1 + ··· + 2 ) n n n n L L ··· L h n n 3 fn(·) en−2 e2 + 2 2 − M(Lnen + en−1 + + ··· + ) Ln ··· L3L2 Ln−1 Ln−1 ··· L3 e e f (·) −(n − 2)(M − ∆(y))( n−1 + ··· + 2 ) − n−1 L L ··· L L L2 ··· L2L n n 3 n n−1 3 2 i 126 which, in turn, leads to

2e U˙ ≤ −2M 2L e2 + M 2e (2n − 3)e + n−1 + ··· n n n n n L h n 2e2 2e2 + − 2en−1 −···− L ··· L L − ··· L n 3 n 1 3 i 2 en−1 e2 −M∆(y) (2n − 3)en +2en( + ··· + ) Ln Ln ··· L3 h i 2Men fn(·) + − f − (·) . L L2 ··· L2L L n 1 n n−1 3 2 h n i

Keeping in mind that Li ≥ 1, i =2, ··· , n, it is not diﬃcult to show that

e e 2e ( n−1 + ··· + 2 ) ≤ (n − 2)e2 +(e2 + ··· + e2) n L L ··· L n n−1 2 n n 3

2Men fn(·) 2 2 2 2 2 − fn−1(·) ≤ y + Mωn(y) ||ξ|| +(n − 1)en LnL ··· L L2 Ln n−1 3 h i

where ωn(y) > 1 is a smooth function depending on L1(y). Consequently,

2 en−1 e2 −M∆(y) (2n − 3)en +2en( + ··· + ) Ln Ln ··· L3 h i 2Men fn(·) + − f − (·) L L2 ··· L2L L n 1 n n−1 3 2 h n i n−2 2 2 2 2 ≤ y + Mωn(y)||ξ|| − (n − 1)M[∆(y) − ωn(y)]en + M∆(y)( ei ). (6.37) Xi=2 Similarly, it can be proved that

2en−1 2e2 2e2 en (2n − 3)en + + ··· + − 2en−1 −···− L L ··· L L − ··· L h n n 3 n 1 3 i 2 2 2 ≤ 2 Dnen +(en−1 + ··· + e2). (6.38)

where Dn ≥ 1 is a constant. Using the estimations (6.37) and (6.38), we have

˙ 2 2 2 2 2 2 2 Un ≤ y + Mωn(y)||ξ|| +2M en(−Ln + Dn)+ M (en−1 + ··· + e2)

2 2 2 −(n − 1)M[∆(y) − ωn(y)]en + M∆(y)(en−1 + ··· + e2). 127

n−1 Choosing Ln = 2 + Dn ≥ 1 yields

˙ 2 2 2 2 2 2 Un ≤ y + Mωn(y)||ξ|| + M [−(n − 1)en +(en−1 + ··· + e2)]

2 2 2 −(n − 1)M[∆(y) − ωn(y)]en + M∆(y)(en−1 + ··· + e2). (6.39)

2 Step 2. Consider the Lyapunov function Un−1 = Un +2Men−1. Then,

˙ 2 2 2 2 2 2 Un−1 ≤ y + Mωn(y)||ξ|| + M [−(n − 1)en +(en−1 + ··· + e2)]

2 2 2 −(n − 1)M[∆(y) − ωn(y)]en + M∆(y)(en−1 + ··· + e2)

2 2 2 Ln +2(M∆(y) − M )en−1 +4Men−1 M(Lnen + ··· + e2) Ln−1 ··· L3 h e2 fn−1(·) +(n − 2)(M − ∆(y))(en−1 + ··· + )+ 2 2 Ln−1 ··· L3 Ln−1 ··· L3L2 e2 −M(Ln−1en−1 + ··· + ) Ln−2 ··· L3 en−2 e2 fn−2(·) −(n − 3)(M − ∆(y))( + ··· + ) − 2 2 . L − L − ··· L L − L ··· L L n 1 n 1 3 n 1 n−2 3 2 i

After a tedious calculation, U˙ n−1 can be estimated as

˙ 2 2 2 2 Un−1 ≤ −(n − 1)M en − 4M Ln−1en−1

2 2 Lne2 e2 +M 4en−1(Lnen+Lnen−1+···+ −en−2−···− ) L − ···L L − ···L h n 1 3 n 2 3 n−2 2 en−2 e2 2 +(4n − 9)en−1 +4en−1( + ··· + )+( ei ) Ln−1 Ln−1 ··· L3 Xi=2 i 2 2 −M∆(y) (n − 1)en + (4(n − 3)+1)en−1 h n−2 en−2 e2 2 2 2 +4en−1( + ··· + ) − ( ei ) + y + Mωn(y)||ξ|| Ln−1 Ln−1 ··· L3 Xi=2 i 2 4Men−1 fn−1(·) +(n − 1)Mωn(y)en + 2 2 − fn−2(·) . Ln−1Ln−2 ··· L3L2 Ln−1 !

Since Li ≥ 1, 2 ≤ i ≤ n, the following estimations are not diﬃcult to obtain:

en−2 e2 2 2 2 4en−1( + ··· + ) ≤2(n − 3)en−1 + 2(en−2 + ··· + e2) L − L − ··· L n 1 n 1 3

4 Men−1 fn−1(·) 2 ωn−1(y) 2 2 2 2 2 −fn−2(·) ≤y +M ||ξ|| +(n−2)(en+en−1) , Ln−1L ··· L L2 Ln−1 ! 2 n−2 3

where ωn−1(y) > 1 is a smooth function depending on L1(y). 128

In view of the last two inequalities and the fact that 2(n − 3)+1 ≥ n − 2, ∀n ≥ 3, it is not diﬃcult to obtain the following estimation:

2 2 en−2 e2 −M∆(y) (n − 1)en + (4(n − 3)+1)en−1 +4en−1( + ··· + ) L − L − ··· L h n 1 n 1 3 n−2 2 2 2 2 −( ei ) + y + Mωn(y)||ξ|| +(n − 1)Mωn(y)en Xi=2 i 4Men−1 fn−1(·) + 2 2 − fn−2(·) Ln−1Ln−2 ··· L3L2 " Ln−1 # 2 2 2 2 ≤ 2y + Mωn−1(y)||ξ|| +3M∆(y)(en−2 + ··· + e2)

2 2 −(n − 2)M[∆(y) − ωn−1(y)](en + en−1) (6.40)

where ωn−1(y) >ωn(y) > 1 is a smooth function depending on L1(y).

Since Ln was ﬁxed in step 1 and Li ≥ 1, it can be shown that

2 Lne2 e2 4en−1(Lnen + Lnen−1 + ··· + − en−2 −···− ) L − ··· L L − ··· L n 1 3 n 2 3 2 en−2 e2 2 2 +(4 n − 9)en−1 +4en−1( + ··· + )+(en−2 + ··· + e2) L − L − ··· L n 1 n 1 3 2 2 2 2 ≤ en +4Dn−1en−1 + 3(en−2 + ··· + e2), (6.41)

where Dn−1 ≥ 0 is a ﬁxed constant. From (6.40)-(6.41) it follows that

˙ 2 2 2 2 2 2 Un−1 ≤ 2y + Mωn−1(y)||ξ|| − (n − 2)M en +4M en−1[−Ln−1 + Dn−1]

2 2 2 2 2 +3M (en−2 + ··· + e2) − (n − 2)M[∆(y) − ωn−1(y)](en + en−1)

2 2 +3M∆(y)(en−2 + ··· + e2).

n−2 Set Ln−1 = 4 + Dn−1 ≥ 1. Thus,

˙ 2 2 2 2 2 2 2 Un−1 ≤ 2y + Mωn−1(y)||ξ|| + M − (n − 2)(en + en−1)+3(en−2 + ··· + e2) h n−2 i 2 2 2 −(n − 2)M[∆(y) − ωn−1(y)](en + en−1)+3M∆(y)( ei ). (6.42) Xi=2 By an inductive argument, it is not diﬃcult to reach the following conclusion at step n − 1. 129

Step n − 1. There is a Lynapunov function of the form

n−2 2 2 2 n−2 2 U2 = U3 +2 Me2 = M(en +2en−1 + ··· +2 e2), such that

˙ 2 2 2 2 2 n−2 2 U2 ≤ (n − 2)y + Mω3(y)||ξ|| + M − 2(en + ··· + e3)+(2 − 1)e2 n h i 2 n−2 2 n−2 2 2 −2M[∆(y) − ω3(y)]( ei )+(2 − 1)M∆(y)e2 +2 (M∆(y) − M )e2 Xi=3 f (·) +2n−1Me M(L2e + L e )+(M − ∆(y))e + 2 − ML e − f (·) . 2 3 3 3 2 2 L 2 2 1 h 2 i (6.43)

Similar to step 2, (6.43) can be represented as

n ˙ 2 2 n−1 2 2 2 n−1 2 n−1 U2 ≤ −2M ( ei ) − 2 M L2e2 + M e2 2 (L3e3 + L3e2)+(2 − 1)e2 Xi=3 h i n−1 2 2 2 2 −2 Me2f1(·) − M∆(y) 2(en + ··· + e3)+ e2 +(n − 2)y h i 2 2 2 n−1 f2(·) +Mω3(y)||ξ|| +2Mω3(y)(en + ··· + e3)+2 Me2 . L2

Note that at this step L3 has been ﬁxed and L2 ≥ 1. Similar to step 2, it is easy to show that

2 2 2 2 n−1 f2(·) (n − 2)y + Mω3(y)||ξ|| +2Mω3(y)(en + ··· + e3)+2 Me2 L2 2 2 2 2 ≤ (n − 1)y + Mω2(y)||ξ|| − M[∆(y) − ω2(y)](en + ··· + e2), (6.44)

2 n−1 2 n−1 n−1 M e2 2 (L3e3 + L3e2)+(2 − 1)e2 − 2 Me2f1(·) 2 2 2 2 n−1 2 2 ≤ C (y)y + M e3 +2 M D2e2, (6.45)

where D2 ≥ 1 is a ﬁxed constant and ω2(y), with ω2(y) > ω3(y) > ··· > ωn(y) > 1, is a smooth function depending on L1(y). Using (6.44) and (6.45), we arrive at

˙ 2 2 2 2 2 2 U2 ≤ [n − 1+ C (y)]y + Mω2(y)||ξ|| − M (en + ··· + e3)

n−1 2 2 2 +2 M e2(−L2 + D2) − M(∆(y) − ω2(y))||e|| . 130

1 Select L2 = 2n−1 + D2 ≥ 1. Hence,

2 2 2 2 2 2 U˙ 2 ≤ [n − 1+ C (y)]y + Mω2(y)||ξ|| − M ||e|| − M[∆(y) − ω2(y)]||e|| . (6.46)

Step n. Finally, consider the composite Lyapunov function

˜ ˜ 4 4 2 W = Vn(y,ξ2, ··· ,ξn)+ KU2(en, ··· , e2), with K = KLn ··· L3L2 +1 > 0, (6.47) for the closed-loop system. Using (6.36) and (6.46), one has

2 2 2 2 W˙ ≤ − 1+ L1(y) − K˜ (n − 1+ C (y)) y − M ||ξ|| h i −M ∆(y) − Ω(y) ||ξ||2 − M 2||e||2 − M ∆(y) − Ω(y) ||e||2, h i h i where Ω(y)= ψn(y)+ Kω˜ 2(y) > 1. Now, set

2 L1(y)= K˜ (n − 1+ C (y)). (6.48)

Consequently, all the smooth functions ψi(y),ωi(y), i = 2, ··· , n, and Ω(y) can be determined accordingly and becomes known. Therefore, any smooth function ∆(y) satisfying ∆(y) ≥ Ω(y) > 1 is such that

W˙ ≤ −y2 − M 2||ξ||2 − M 2||e||2. (6.49)

6.4 Stability Analysis

We now prove global stability and attractivity of the closed-loop system. First of all, observe that M ≥ 1, ∀t ≥ 0. With this in mind, it is straightforward to deduce from (6.49) and (6.47) that there is a constant µ> 0, such that

W˙ ≤ −µW. 131

1/2 1/2 1/2 1/2 This, in turn, implies that y, M ξ2, ··· , M ξn and M e2, ··· , M en exponentially converge to zero. In fact, for a(t) which is any one of them,

|a(t)| ≤ W (t) ≤ W (0)e−µt, ∀t ≥ 0.

Consequently, there exists a constant λ > 0 such that |y(t)| ≤ λW (0), ∀t ≥ 0.

Hence, M(t) is uniformly bounded over [0, +∞) because 1 ≤ M(t) ≤ ∆(λW (0)) ∀t ∈ [0, +∞) (see the discussion after equation (6.9) in section 6.2). The boundedness of M(t) implies that ξ(t) and e(t) are globally uniformly bounded and converge to the origin as t → +∞.

Finally, using the deﬁnition of ξi, 2 ≤ i ≤ n, the transformation (6.4) and (6.31)- (6.32), it is straightforward to conclude global uniform boundedness and convergence of (η2(t), ··· , ηn(t)) and (ˆz2(t), ··· , zˆn(t)) of the closed-loop system. This completes the proof of Theorem 6.1.

Remark 6.6. From the proof of Theorem 6.1, it is clear that the dynamic output compensator (6.9)-(6.30)-(6.31)-(6.35), which is of the form (6.3), achieves not only global attractivity of the system state (η1, ··· , ηn) but also all the states of the dynamic compensator, except for the component M. In other words, the resulted closed-loop system is globally uniformly bounded and almost globally uniformly attractive.

6.5 Output Feedback Stabilization of Cascade Systems

In this section we brieﬂy illustrate under an appropriate ISS-like condition, how the robust output feedback stabilization result obtained so far can be extended to a family of uncertain systems with zero-dynamics

Z˙ = F0(t,Z,η,v)

η˙1 = η2 + φ1(t,Z,η,v) 132

. .

η˙n−1 = ηn + φn−1(t,Z,η,v)

η˙n = v + φn(t,Z,η,v)

y = η1, (6.50) where v ∈ IR and y ∈ IR are the system input and output, respectively, Z ∈ IRr

n n+r r and η ∈ IR are the system states. The functions F0 : IR × IR × IR → IR

n+r 1 and φi : IR × IR × IR → IR, i = 1, ··· , n, are C with F0(t, 0, 0, 0) = 0 and

φi(t, 0, ··· , 0) = 0.

1 Remark 6.7. Since φi(·) are C with respect to all the variables and φi(t, 0, ··· , 0) =

0, we can always separate φi(·) into the following form as done in (Lin and Pongvuthithum (2002)) (see Lemma 2.5 in Lin and Pongvuthithum (2002))

|φi(t,Z,η,v)| ≤ α¯(t, Z)||Z|| + C¯i(t, η1, ··· , ηi, v)(|η1| + ··· + |ηn| + |v|)

1 withα ¯(·) > 0 and C¯i(·) > 0 being C functions.

In view of Remarks 6.2, 6.7 and the assumption of Theorem 6.1, it is natural to assume that system (6.50) satisﬁes the following conditions.

r Assumption 6.8. There exists smooth functions α : IR → IR+ and C : IR → IR+ such that ∀(t,Z,η,v) ∈ IR × IRn+r × IR,

|φi(t,Z,η,v)| ≤ α(Z)||Z|| + C(y)(|η1| + ··· + |ηi|), i =1, ··· , n. (6.51)

2 Assumption 6.9. There is a C Lyapunov function U0(Z), which is positive deﬁnite and proper, such that for all (t,Z,η,v) ∈ IR × IRn+r × IR,

∂U 0 F (t,Z,η,v) ≤ −||Z||2 + K (y)η2, (6.52) ∂Z 0 0 1 where K0(y) ≥ 0 is a known smooth function. 133

Clearly, Assumption 6.8 is a natural generalization of the growth condition (6.2), while Assumption 6.9 is a sort of ISS-like condition (Sontag and Wang (1999); Lin and Pongvuthithum (2002)). With the help of these two conditions, the following global output feedback stabilization result can be established for the uncertain cascade system (6.50).

Theorem 6.10. Under Assumptions 6.8-6.9, the uncertain cascade system (6.50) is globally robustly stabilizable by smooth output feedback.

Proof: The proof can be carried out by using the iterative design method proposed in Theorem 6.1 combined with the technique of changing supply rate (Sontag and Teel (1995); Lin and Pongvuthithum (2002)). In what follows, we give only a sketchy proof with an emphasis on the major diﬀerence. As done in the proof of Theorem 6.1, we ﬁrst introduce a rescaling transformation that is composed of Z = Z and (6.4) for the uncertain system (6.50). Such a transformation results in

Z˙ = F0(t,Z,η,v)

x˙ 1 = Mx2 + f1(t,Z,x,u) M˙ x˙ = Mx − x + f (t,Z,x,u) 2 3 M 2 2 . . M˙ x˙ = Mu − (n − 1) x + f (t,Z,x,u) n M n n

y = x1, (6.53) where the system uncertainty satisﬁes the constraints:

|f1(·)| = |φ1(·)| ≤ α(Z)||Z|| + C(y)|y| |φ (·)| α(Z)||Z|| |y| |f (·)| = 2 ≤ + C(y)( + |x |) 2 M M M 2 . . (6.54) |φ (·)| α(Z)||Z|| |y| |f (·)| = n ≤ + C(y)( + |x | + ··· + |x |). n M n−1 M M 2 n 134

For the rescaled system (6.53) with the constraint (6.54), it is not diﬃcult to see that Assumptions 6.8-6.9 imply the existence of a globally stabilizing, partial-state feedback controller. This conclusion can be proved in a manner similar to (Lin and Pongvuthithum (2002)), as illustrated below. Consider the Lyapunov function

U0(Z) V0(Z)= ρ(s)ds (6.55) Z0 where ρ(s) > 0, ∀s ≥ 0 is a C0 nondecreasing function to be assigned. It is not

1 difficult to prove that V0(Z) thus defined is C , positive definite and proper. Using the idea of changing supply rate (Sontag and Teel (1995)), it follows from Assumption 6.9 that given any smooth function α(Z) > 0, it is possible to find a nondecreasing function ρ(·), such that

∂U V˙ = ρ(U (Z)) 0 F (t,Z,η,v) ≤ −||Z||2 − 2nα2(Z)||Z||2 + Kˆ (y)η2 (6.56) 0 0 ∂Z 0 0 1 where Kˆ0(y) ≥ 0 is a smooth function.

y2 Consider the Lyapunov function V1(Z,y) = V0(Z)+ 2 . Its derivative along the trajectories of (6.10) is

˙ 2 2 2 ˆ 2 V1 = −||Z|| − 2nα (Z)||Z|| + K0(y)η1 + y Mx2 + f1(·) h i 2 2 ∗ ∗ ˆ 2 ≤ − 1 − (2n − 1)α (Z) ||Z|| + Myx2 + My(x2 − x2) + [C(y)+ K0(y)]y . Clearly, the virtual controller

β (y)+ L (y) C(y)+ n +1+ L (y)+ Kˆ (y) x∗ = − 1 1 y = − 1 0 y 2 M M yields

˙ 2 2 2 ∗ V1 ≤ − 1 − (2n − 1)α (Z) ||Z|| − [n +1+ L1(y)]y + My(x2 − x2) (6.57) where L1(y) > 0 is to be determined in the observer design. 135

By a similar inductive argument, at the n-th step, we conclude that there exist a set of transformations

∗ ∗ xi − xi ∗ ξ2 = x2 − x2, ξi = , xi = −βi−1ξi−1, i =3, ··· , n, (6.58) βn−1 a Lyapunov function

U0(Z) n−2 2 n−2 2 n−3 2 2 Vn(Z,y,ξ2, ··· ,ξn)= ρ(s)ds +2 y + M(2 ξ2 +2 ξ3 + ··· + ξn), Z0 and a partial-state feedback control law of the form

∗ xn+1 = −βnξn (6.59) such that

2 2 2 2 2 V˙n ≤ −(n − 1)α (Z)||Z|| − [2 + L1(y)]y − 2M ||ξ|| ∗ 2 u − xn+1 2 +2M ξn − M ∆(y) − ψn(y) ||ξ|| . (6.60) β − n 1 where all the parameters β1, ··· , βk are known constants independent of M. The rest of the proof is similar to that of Theorem 6.1. In fact, one can also build the reduced-order observer (6.30)-(6.31) independent of Z, estimating the partial states (η2, ··· , ηn) of (6.50), or, equivalently, (x2, ··· , xn) of (6.53). By the certainly equivalence principle, an implementable controller (6.35) can thus be obtained. Fi- nally, taking advantage of the negative term −(n−1)α2(Z)||Z||2 in (6.60) and using it to dominate all the Z-terms generated from the observer design, a recursive observer gain assignment procedure, which is parallel to what has been done in section 6.3, shows that there still exists a dynamic output compensator of the form (6.9)-(6.30)- (6.31)-(6.35), rendering all the states of the uncertain cascade system (6.50) system globally uniformly bounded and attractive.

6.6 Summary

The main result of this chapter is Theorem 6.1 which has generalized the output feedback stabilization result previously obtained in (Qian and Lin (2002b)) (see Corollary 136

3.1) for two-dimensional nonlinear systems to the n-dimensional case. It has also removed the restriction that the growth rate in (6.2) be a polynomial function of the system output — a crucial condition required in (Praly and Jiang (2003)). All of these were made possible by developing a new robust output feedback control scheme that enables one to design both robust observers and controllers recursively for the rescaled system, using only the knowledge of the bounding system instead of the uncertain system itself. The rescaled system was obtained by introducing a rescaling transformation with a dynamic factor that is tuned on-line via a Riccati differential equation. The rescaling transformation created an extra design freedom that has proved to be extremely effective in handling the system uncertainty. The iterative algorithm for the assignment of the observer gains developed in this chapter is nothing but a dual of the adding a integrator design. Its significance lies in that it can be further extended and used to design nonlinear observers for nonlinear systems with unstabilizable/undetectable linearization, as already illustrated in the last two chapters. It should be pointing out that as shown in section 6.5, without much effort, The- orem 6.1 can be easily extended to a family of uncertain cascade systems under appropriate ISS conditions on the zero-dynamics. Assumptions 6.8-6.9 can be further relaxed if a weaker ISS-condition than (6.52), such as those in (Sontag and Wang (1999); Isidori (1999)), is employed. Details are left to the reader as an exercise. Chapter 7. SEMI-GLOBAL OUTPUT FEEDBACK STABILIZATION OF NON-UNIFORMLY OBSERVABLE AND NONSMOOTHLY STABILIZABLE SYSTEMS

The objective of Chapter 7 is to investigate the problem of semi-global output feedback stabilization for nonlinear systems. The main contribution of this chapter is to prove that without imposing any growth condition, it is possible to achieve semi-global stabilization by nonsmooth output feedback for a chain of odd power integrators perturbed by a triangular vector ﬁeld, which is in general not smoothly stabilizable nor uniformly observable.

7.1 Introduction

The purpose of this chapter is to address the problem of semi-global stabilization by output feedback, for a class of highly nonlinear systems that are neither uniformly observable nor smoothly stabilizable. Speciﬁcally, we are interested in the question of when semi-global stabilization by nonsmooth output feedback can be achieved for the triangular system

p1 x˙ 1 = x2 + f1(x1) . .

pn−1 x˙ n−1 = xn + fn−1(x1, ··· , xn−1)

x˙ n = u + fn(x1, ··· , xn)

y = x1, (7.1)

137 138

T n where x = (x1, ··· , xn) ∈ IR , u ∈ IR and y ∈ IR are the system state, input and

i 1 output, respectively. The mappings fi : IR → IR, i =1, ··· , n, are C functions with fi(0, ··· , 0)=0 and p1, ··· ,pn−1 are odd positive integers. The problem of semi-global stabilization by nonsmooth output feedback can be formulated as follows. Given a bound r > 0, ﬁnd, if possible, a nonsmooth dynamic output compensator, which may depend on r, of the form

zˆ˙ = η(ˆz,y), zˆ ∈ IRn−1

u = u(ˆz,y) (7.2) such that the following two properties hold:

• Local Stability: The closed-loop system (7.1)-(7.2) is locally asymptotically stable at the origin (x, zˆ)=(0, 0);

• Semi-Global Attraction: All the trajectories of the closed-loop system starting

∆ n n−1 n n−1 from the compact set Γx × Γzˆ = [−r, r] × [−r, r] ⊂ IR × IR converge to the origin.

It has been known that for nonlinear control systems, global stabilizability by state feedback plus global observability is usually not suﬃcient for achieving global stabilizability by output feedback. As a matter of fact, counter-examples were given in (Mazenc et al. (1994)) illustrating that even for a simple feedback linearizable or minimum-phase system in the plane, which is uniformly observable and stabilizable by smooth state feedback, global output feedback stabilization is still not possible.

The impossibility of this kind indicates that in the nonlinear case, semi-global, in stead of global, stabilization by output feedback is perhaps the more realistic control objective to be pursued. While there are numerous papers in the literature devoted to the topic of semi- global output feedback stabilization, a major progress was reported in the work (Teel 139 and Praly (1994)), where it was shown that for nonlinear systems, uniform observability (Gauthier et al. (1992)) and global stabilizability by smooth state feedback implies semi-global stabilizability by smooth output feedback. As a consequence, semi-global stabilization by smooth output feedback was shown to be possible for minimum- phase nonlinear systems including system (7.1) with p1 = ··· = pn−1 = 1 (Teel and Praly (1995)), as well as for a class of non-minimum-phase nonlinear systems (Isidori (2000)).

Note that when pi ≡ 1, i = 1, ··· , n − 1, the triangular system (7.1) is feedback linearizable and hence globally stabilizable by smooth state feedback (Isidori (1999)).

In addition, according to the characterization given in (Gauthier et al. (1992)), the system is also uniformly observable. These two conditions are, however, violated when some of pi are larger than one. Indeed, the triangular system (7.1) is no longer uniformly observable (Gauthier et al. (1992); Teel and Praly (1994)), simply because the system state of (7.1) can only be represented as a nonsmooth rather than smooth function of the system input, output, and their derivatives (see, for instance, Example 7.8). Furthermore, system (7.1) is not smoothly stabilizable, even locally, for the reason that its linearized system may have uncontrollable modes associated with eigenvalues on the right-half plane, as illustrated by the simple planar systemx ˙ 1 =

3 x2 + x1, x˙ 2 = u,y = x1. Although the nonlinear system (7.1) is neither uniformly observable nor smoothly stabilizable, it is globally stabilizable by nonsmooth state feedback, as shown in (Qian and Lin (2001a)). Recently, it has been further proved in (Qian and Lin (2004b)), among the other things, that the local stabilization of the triangular system (7.1) is achievable by nonsmooth output feedback. This was accomplished by means of the theory of homogeneous systems (Bacciotti (1992)), particularly, using the homogeneous approximation (Dayawansa (1992); Dayawansa et al. (1990); Kawski (1989, 1995)) and the robust stability of homogeneous systems (Hermes (1991a); Rosier (1992)). In 140 view of the results obtained in (Qian and Lin (2001a, 2004b)), one might naturally make the conjecture that the triangular system (7.1) is semi-globally stabilizable by nonsmooth output feedback, without requiring any growth condition. It turns out that this conjecture is true and semi-global stabilization by nonsmooth output feedback is indeed possible, for a signiﬁcant class of non-uniformly observable and nonsmoothly stabilizable systems such as (7.1). The main contribution of this chapter is the following theorem.

Theorem 7.1. For the triangular system (7.1), there exists a nonsmooth dynamic output compensator of the form (7.2), such that the closed-loop system (7.1)-(7.2) is semi-globally asymptotically stable.

The signiﬁcance of Theorem 7.1 over the existing results can be summarized as follows. On the one hand, it generalizes the semi-global output feedback stabilization results in (Teel and Praly (1994, 1995)) for nonlinear systems that are require to be uniformly observable and smoothly stabilizable, to a wider class of non-uniformly observable and nonsmoothly stabilizable systems such as (7.1). On the other hand, it shows that the local output feedback stabilization result in (Qian and Lin (2004b)) (see Theorem 3.5) can be extended, without requiring any growth condition on (7.1), to the semi-global case, which is certainly a substantial progress from either a theoretical or practical viewpoint. Finally, compared with the previous global output feedback stabilization results in Chapters 3-6 and the paper (Qian and Lin (2004b)), all the restrictive conditions such as p1 = ··· = pn−1 and a high-order global Lipschitz- like condition in Chapter 3, or those growth requirements imposed on the nonlinear system (7.1) in (Qian and Lin (2004b)) have been removed. The price we paid is that only semi-global rather than global stabilizability is achieved. In the remainder of this chapter, we shall prove Theorem 7.1 by explicitly constructing a nonsmooth, dynamic output feedback compensator of the form (7.2) for the triangular system (7.1). While the design of H¨older continuous state feedback 141 control laws is adopted from (Qian and Lin (2001a)), the nonsmooth observer construction is new and carried out in a subtle manner. It integrates the idea of the recursive observer design (Qian and Lin (2004b)) with the technique of the saturated state estimates (Khalil and Esfandiari (1995); Teel and Praly (1994)). The proof of semi-global stabilizability is motivated by the work (Teel and Praly (1994)) yet much simpler. In fact, it is strong reminiscent of the one given in (Yang and Lin (2006)), where a simple and intuitive argument was developed without involving intricate Lyapunov functions (Teel and Praly (1994)). The use of simple Lyapunov functions, together with a delicate choice of the level sets, makes it possible to simplify the analysis and synthesis of our semi-global, nonsmooth output feedback control scheme. In the feedback linearizable case, the result presented in this chapter provides an alternative yet simpler solution to the semi-global output feedback stabilization problem considered, e.g., in (Teel and Praly (1995)).

7.2 Key Technical Lemmas

In this section, we list several useful lemmas that will be used in the sequel.

Deﬁnition 7.2. A saturation function with the threshold M ≥ 0 is deﬁned as

−M if x< −M satM (x) : IR → IR =  x if |x| ≤ M (7.3)  M if x>M.  Clearly, the saturation function thus deﬁned is continuous, bounded by M and has the following property.

Lemma 7.3. Assume that p ≥ 1 is an odd integer and a ∈ [−M, M]. Then,

p p 1/p |a − satM (b)| ≤ 2 min{|a − b | , M}, ∀b ∈ IR.

The following lemma characterizes a useful property of smooth functions on a compact set. 142

Lemma 7.4. Let f : IRn → IRn be a smooth mapping and Γ = [−N, N]n a compact

n set in IR , with N > 0 being a real number. Then, for σi ∈ (0, 1], i = 1, ··· , n, there exists a constant K ≥ 1 depending on N, such that ∀(x1, ··· , xn) ∈ Γ and

∀(y1, ··· ,yn) ∈ Γ,

σ1 σn |f(x1, ···,xn) −f(y1, ···,yn)| ≤K(|x1 − y1| + ··· + |xn − yn| ).

The proofs of Lemmas 7.3 and 7.4 are straightforward and left to reader as an exercise.

7.3 A Simpler Paradigm

In order to tackle the semi-global stabilization problem for the highly nonlinear system (7.1) by nonsmooth output feedback, it is essential to understand how the problem was solved in the feedback linearizable case, i.e., pi = 1, 1 ≤ i ≤ n in (7.1). In that case, system (7.1) becomes smoothly stabilizable and uniformly observable. As a consequence of the work (Teel and Praly (1994)), semi-global stabilization of (7.1) can be achieved by smooth output feedback. This was done based on the high-gain observer (Gauthier et al. (1992)), the idea of saturating the estimated state (Khalil and Esfandiari (1995)), and the technique of dynamic extension. Due to the use of uniform observability and smoothness of state feedback, the semi-global design method of (Teel and Praly (1994, 1995)) is, however, hard to be extended to the nonlinear system (7.1) that is non-uniformly observable and nonsmoothly stabilizable. Furthermore, the proof of semi-global stability of the closed-loop system in (Teel and Praly (1994)) involved a subtle construction of control Lyapunov functions only deﬁned on compact sets, making the stability analysis less intuitive and diﬃcult to be adopted for system (7.1) with pi ≥ 1. In this section, we revisit the feedback linearizable case, i.e.

x˙ 1 = x2 + f1(x1) 143

. .

x˙ n = u + fn(x1, ··· , xn),

y = x1. (7.4)

The objective is to develop an alternative semi-global output feedback control strategy involving ideas that do not seem to have been fully exploited yet. Speciﬁcally, we shall generalize the simple and intuitive argument proposed in (Yang and Lin (2006)) to the highly nonlinear system (7.1) with pi ≥ 1. As we shall see in a moment, the output feedback control scheme developed here is reminiscent of (Yang and Lin (2006)) and simpler than those in (Teel and Praly (1994, 1995)). More importantly, it can be carried over, with a subtle twist, to a class of inherently nonlinear systems such as (7.1), as shown in section 7.4. To begin with, we observe that by adding an integrator, it is easy to get recursively a smooth state feedback controller

∗ u (x1, ··· , xn)= −ξnβn(x1, ··· , xn) (7.5) such that the closed-loop system (7.4)-(7.5) satisﬁes

˙ 2 2 ∗ Vc(x) ≤ −3(ξ1 + ··· + ξn)+ ξn(u − u (x1, ··· , xn)), (7.6)

1 2 2 ∗ where Vc(x)= 2 (ξ1 + ··· + ξn), ξi = xi − xi , i =1, ··· , n,

∗ ∗ ∗ x1=0, x2 = − ξ1β1(x1), ··· , xn = − ξn−1βn−1(x1, ··· , xn−1)

with βi(·) > 0 being known smooth functions.

Since Vc(·) is positive deﬁnite and proper, one can deﬁne the level set Ωx =

n n {x ∈ IR |Vc(x) ≤ 2r0}, where r0 > 0 is a constant such that Γx = [−r, r] ⊂ {x ∈

n IR |Vc(x) ≤ r0}. Then, denote

M = max ||x||∞ > 0 x∈Ωx 144

− x M Γ M 1 x

≤ {Vc () x r0}

Ω x −M

Fig. 7.1: The level set Ωx on x-space and the saturation threshold M.

as a saturation threshold, where ||·||∞ stands for l∞ norm of vectors. The relations among the compact sets thus introduced are illustrated in Fig. 7.1.

Because the states (x2, ··· , xn) of (7.4) are not measurable, the controller (7.5) is not realizable. To get an implementable controller, we shall design an (n−1)−th order observer to estimate, instead of the states (x2, ··· , xn), the unmeasurable variables

(z2, ··· , zn) deﬁned by

z2 = x2 − L2x1, ··· , zn = xn − Lnxn−1, (7.7)

where Li ≥ 1, i =2, ··· , n are gains to be assigned later. From (7.7) it follows that

z˙2 = [x3 + f2(·)] − L2[x2 + f1(·)] . .

z˙n = [u + fn(·)] − Ln[xn + fn−1(·)]. (7.8) 145

In view of (7.8), we design the implementable dynamic compensator

zˆ˙2 = [ˆx3 + fˆ2(·)] − L2[ˆx2 + f1(·)]

zˆ˙ 3 = [ˆx4 + fˆ3(·)] − L3[ˆx3 + fˆ2(·)] . . ˙ ˆ ˆ zˆn = [u + fn(·)] − Ln[ˆxn + fn−1(·)], (7.9) where

xˆ2 =z ˆ2 + L2x1, ··· , xˆn =z ˆn + Lnxˆn−1 (7.10)

∆ fˆi(·) = fi(x1, satM (ˆx2), ··· , satM (ˆxi)), i =2, ··· , n. (7.11)

Using the certainty equivalence principle, we obtain the realizable controller

∗ ∆ ∗ u =u ˆ (·) = u (x1, satM (ˆx2), ··· , satM (ˆxn)). (7.12)

Let ei = zi − zˆi, i =2, ··· , n be the estimate errors. Then,

xi − xˆi = ei + Liei−1 + ··· + Li ··· L3e2. (7.13)

Consequently, the error dynamics are given by

e˙2 = [e3 + L3e2 +(f2(·) − fˆ2(·))] − L2e2. . . (7.14)

ˆ 2 e˙n = [fn(·) − fn(·)] − [Lnen + Lnen−1 + ···

2 ˆ +LnLn−1 ··· L3e2 + Ln(fn−1(·) − fn−1(·))].

In view of the fact that |satM (ˆxi)| ≤ M and Lemmas 7.3-7.4, there exists a constant K ≥ 1, which depends on M and is independent of Li’s, so that on the set

n−1 ∆ 2n−1 n BM × IR = {(x, zˆ) ∈ IR |(x1, ··· , xn) ∈ [−M, M] }, the following estimations hold (i =2, ··· , n,). i ˆ K |fi(·) − fi(·)| n−1 ≤ |xi − satM (ˆxi)| BM ×IR n j=2 X ≤ K |ei| + KLi|ei−1| + ··· + KLi ··· L3|e2| (7.15)

∗ ∗ |uˆ (·) − u (·)| n−1 ≤ K min{|en| + Ln|en−1| + ··· + Ln ··· L3|e2|, 1}, (7.16) BM ×IR

146

where the notation f(·)|Γ denotes the restriction of a function f(·) on the set Γ. Using (7.6) and (7.16) yields

˙ 2 2 Vc n−1 ≤ − 3(ξ1 + ··· + ξn)+ K|ξn| min{|en| + Ln|en−1| + ··· + Ln ··· L3|e2|, 1} BM ×IR

n 2 2 2 2 ≤ − 2( ξi ) + 2 min{Cnen + Cn−1(Ln)en−1 + ··· + C2(Ln, ··· , L3)e2,Cn}, (7.17) Xi=1 where Cn ≥ 1 is a constant independent of Li’s, Cn−1(Ln) ≥ Cn, ··· ,C2(Ln, ··· , L3) ≥

Cn are polynomial functions of their arguments. They can be obtained by completing the square, as done in Chapter 3.

1 2 2 Now, choose Ve(e) = 2 (en + ··· + e2) for the error dynamics (7.14). On the set n−1 BM × IR , we have

˙ ˆ ˆ Ve n−1 ≤|en[fn(·)−fn(·)]|+|enLn[fn−1(·)−fn−1(·)]| BM ×IR

2 2 2 − Lnen + |en(Lnen−1 + ··· + LnLn−1 ··· L3e2)|

ˆ 2 + ··· + |e2[f2(·) − f2(·)]| + |e2(e3 + L3e2)| − L2e2

¯ 2 ¯ 2 ≤ − (Ln − Cn)en − (Ln−1 − Cn−1(Ln))en−1

¯ 2 −···− (L2 − C2(Ln, ··· , L3))e2, (7.18) where C¯n ≥ 1 is a constant independent of Li’s, C¯n−1(Ln) ≥ 1, ··· , C¯2(Ln, ··· , L3) ≥ 1 are ﬁxed polynomial functions of their own arguments. They can be calculated by the completion of the square, as done in Chapter 3.

From (7.18), it is easy to see that the gain assignments

∆ Ln = Ln(L) = C¯n + LCn ≥ 1

∆ Ln−1 = Ln−1(L) = C¯n−1(Ln)+ LCn−1(Ln) ≥ 1 . . (7.19)

∆ L2 = L2(L) = C¯2(Ln, ··· , L3)+ LC2(Ln, ··· , L3) ≥ 1 with L> 0 being a parameter to be determined later, render

˙ Ve n−1 ≤ −LWe, (7.20) BM ×IR

147

∆ 2 2 2 where We = Cnen + Cn−1(L)en−1 + ··· + C2(L)e2 and Cn−1(L) ≥ Cn, ··· ,C2(L) ≥ Cn are ﬁxed positive polynomial functions of L. Motivated by (Teel and Praly (1994)), we deﬁne

n 1 2 µ(L)= (2r + Li(L)r) ≥ max Ve(x, zˆ) > 0 2 (x,zˆ)∈Γx×Γzˆ Xi=2 and choose the following Lyapunov function, which is diﬀerent from the complicated one used in (Teel and Praly (1994)), for the closed-loop system (7.4)-(7.12)-(7.9)- (7.10)-(7.11): V (x) r ln(1 + V (e)) V (x, zˆ)= c + 0 e . (7.21) 2 2 ln(1 + µ(L)) Moreover, deﬁne the corresponding level set

2n−1 Ω= {(x, zˆ) ∈ IR |V (x, zˆ) ≤ r0}. (7.22)

Then, it is easy to verify the following facts (see Fig. 7.2): 1) For every L > 0, V (·) is a positive definite and proper function and Ω is a compact set in IRn × IRn−1 (i.e., (x, zˆ)-space). Once L> 0 is fixed, V and Ω are fixed too;

2) ∀L> 0, Ω ⊃ Γx × Γzˆ;

n−1 3) ∀L> 0, BM × IR ⊃ Ω. To prove the semi-global asymptotic stability, it remains to show that one can take advantage of the uniform boundedness of Ω with respect to L and choose L> 0, ˙ such that V |Ω ≤ 0. n−1 In view of the relationship BM × IR ⊃ Ω, we deduce from (7.20) and (7.17) that ∀L> 0,

1 r0 V˙e V˙ = V˙c + (7.23) Ω 2 Ω 2 ln(1 + µ(L)) 1+ V Ω e n 2 r0L We ≤ − ( ξi )+ min{We,Cn}− ln(1+ µ(L)) 2+2Ve Xi=1 Observe that Ci(L) ≥ Cn ≥ 1, i =2, ··· , n implies

We 1 ≥ min{We,Cn}. 2+2Ve 3 148

zˆ

Γ x

−M Γ M zˆ x

Ω (depending on L )

Fig. 7.2: The level set Ω on (x, zˆ)-space.

We have, n 2 r0L V˙ ≤ −( ξ ) − − 1 min{We,Cn}. Ω i 3 ln(1 + µ(L)) i=1 h i X By construction, µ(L) > 0 is a ﬁxed polynomial function of L. Thus, there is a constant L∗ > 0 such that

r L 0 ≥ 2, ∀L ≥ L∗. 3 ln(1 + µ(L))

Choosing L = L∗, we have immediately,

n 2 V˙ ≤ −( ξ ) − min{We,Cn}. Ω i i=1 X

That is, the uniformly observable, feedback linearizable system (7.4) is semi-globally stabilizable via smooth output feedback. 149

7.4 Proof of Main Result

Now we are ready to prove the main result of this chapter — Theorem 7.1. The proof is constructive and carried out by generalizing the semi-global output feedback design method illustrated in the previous section, with a subtle twist, to the non- uniformly observable and nonsmoothly stabilizable system (7.1). In particular, we construct explicitly a nonsmooth dynamic output compensator by integrating the tool of adding a power integrator (Qian and Lin (2001a)), the recursive nonsmooth observer design algorithm (Qian and Lin (2004b)), and the idea of saturating the estimated states (Khalil and Esfandiari (1995)). Proof of Theorem 7.1: First of all, using the nonsmooth state feedback control scheme in (Qian and Lin (2001a)), we can design a globally stabilizing state feedback controller as follows.

1 2 1 Let ξ1 = x1 and choose V1(x1)= 2 ξ1 . Since f1(x1)isa C function with f1(0) = 0, there exists a smooth function ρ1(·) ≥ 0 such that |f1(x1)|≤|x1|ρ1(x1). Hence,

˙ ∗p1 p1 ∗p1 2 V1 ≤ ξ1x2 + ξ1(x2 − x2 )+ ρ1(x1)ξ1 .

∗p1 ∆ Setting x2 = −ξ1β1(x1) = −ξ1[(n +2)+ ρ1(x1)] yields

˙ 2 p1 ∗p1 V1 ≤ −(n + 2)ξ1 + ξ1(x2 − x2 ).

p1 ∗p1 Next, let ξ2 = x2 − x2 and choose

x2 1 ∗p1 2− p1 p1 V2(x1, x2)= V1(x1)+ ∗ (s − x2 ) ds Zx2 1 which is C , positive deﬁnite and proper (Qian and Lin (2001a)). Note that |f2(x1, x2)| ≤

(|x1| + |x2|)¯ρ2(x1, x2) withρ ¯2(·) ≥ 0 being a smooth function. With this in mind, it follows that

2− 1 ˙ 2 ∗p2 p1 p2 ∗p2 2 V2 ≤ −(n + 1)ξ1 + ξ2x3 + ξ2 (x3 − x3 )+ ρ2(x1, x2)ξ2 ,

where ρ2(·) ≥ 0 is a smooth function. 150

Then, it is not diﬃcult to show the existence of a smooth function β2(·, ·) ≥ 0, such that

p1 β2(x1, x2 ) ≥ (n +1)+ ρ2(x1, x2), ∀(x1, x2).

∗p2p1 p1 Then, setting x3 = −ξ2β2(x1, x2 ) yields

2− 1 ˙ 2 2 p1 p2 ∗p2 V2 ≤ −(n + 1)(ξ1 + ξ2 )+ ξ2 (x3 − x3 ).

Following the inductive argument in (Qian and Lin (2001a)), we can obtain a set of virtual controllers

pi−1···p1 ∗pi−1···p1 ξi = xi − xi , i =1, ··· , n, (7.24)

∗ where x1 = 0 and

∗p1 x2 = −ξ1β1(x1) . . (7.25)

∗pn−1···p1 p1 pn−2···p1 xn = −ξn−1βn−1(x1, x2 , ··· , xn−1 )

∗pn−1···p1 p1 pn−1···p1 xn+1 = −ξnβn(x1, x2 , ··· , xn )

∆ ∗ p1 pn−1···p1 = u (x1, x2 , ··· , xn ) (7.26)

i + ∗ n with βi : IR → IR , i = 1, ··· , n, and u : IR → IR being smooth functions, and a

1 C Lyapunov function Vc(x), which is positive deﬁnite and proper (whose form can be found in (Qian and Lin (2001a))), such that

n 1 2− ··· ˙ 2 pn−1 p1 ∗ Vc ≤ − 3( ξi )+ ξn (u − u ) (7.27) Xi=1

Similar to the feedback linearizable case, we deﬁne the level set Ωx = {x ∈

n n IR |Vc ≤ 2r0}, where r0 > 0 is a constant such that Γx ⊂ {x ∈ IR |Vc ≤ r0}.

Moreover, denote M = maxx∈Ωx ||x||∞ as a saturation threshold. As done in the last section, to get an implementable controller a reduced-order observer must be designed for the estimation of the unmeasurable states (x2, ··· , xn)of 151 system (7.1). Motivated by the nonsmooth observer design in (Qian and Lin (2004b)), next we construct a reduced-order observer to estimate, instead of (x2, ··· , xn), the unmeasurable variables (z2, ··· , zn) deﬁned by

p1 p1 z2 = x2 − L2x1 ⇔ x2 = z2 + L2x1 . . (7.28)

pn−1 pn−1 zn = xn − Lnxn−1 ⇔ xn = zn + Lnxn−1, where Li ≥ 1, 2 ≤ i ≤ n are gain constants to be assigned later. By (7.28), the z-dynamics can be described by

p1−1 p2 p1 z˙2 = p1x2 [x3 + f2(·)] − L2[x2 + f1(·)] . . (7.29)

pn−1−1 pn−1 z˙n = pn−1xn [u + fn(·)] − Ln[xn + fn−1(·)].

In view of (7.29), we design, similar to what we did in the last section (see (7.8)-(7.9)), the realizable observer

˙ p1 zˆ2 = −L2[ˆx2 + f1(·)] . .

˙ pn−1 ˆ zˆn = −Ln[ˆxn + fn−1(·)] (7.30) where for i =2, ··· , n,

pn−1 pi−1 xî =z î + Lixî−1 ⇔ zî =x î − Lixî−1 (7.31) ˆ ∆ fi(·) = fi(x1, satM (ˆx2), ··· , satM (ˆxi)). (7.32)

By the certainty equivalence principle, we replace the unmeasurable state (x2, ··· , xn)

∗ in the virtual controller xn+1 by the saturated state estimate (ˆx2, ··· , xˆn), which is generated by the observer (7.30)-(7.32). In doing so, we obtain the realizable controller

pn···p1 ∗ ∆ ∗ p1 pn−1···p1 u =ˆu (·) = u (x1, [satM (ˆx2)] , ···, [satM (ˆxn)] ). (7.33) 152

For i =2, ··· , n, deﬁne the estimate errors

pi−1 ei = zi − zˆi = xi − Lixi−1 − zˆi. (7.34)

pi−1 pi−1 Note that xi − xˆi = ei + Li(xi−1 − xˆi−1), i =2, ··· , n. Thus, the error dynamics can be expressed as

p1−1 p2 e˙2 = p1x2 [x3 + f2(·)] − L2e2 . . (7.35)

pn−1−1 ˆ e˙n = pn−1xn [u + fn(·)] − Lnen − (xn−1 − xˆn−1) − Ln[fn−1(·) − fn−1(·)].

To analyze the error dynamics, we now introduce several useful propositions that can be proved by direct but tedious calculations.

The following notation is used in the remainder of this section.

n−1 ∆ 2n−1 n BM × IR = {(x, zˆ) ∈ IR |(x1, ··· , xn) ∈ [−M, M] }.

Proposition 7.5. There exists a generic constant K ≥ 1, which depends on M and is

n−1 independent of all the Li’s, such that on the set BM × IR the following estimations hold:

1 ˆ p1 |f2(·)−f2(·)| n−1 ≤ K|e2| BM×IR

. . (7.36)

1 1 1 p −1 ˆ pn−1 n pn−1pn−2 |fn(·)−fn(·)| n−1 ≤ K(|en| +Ln |en−1| BM×IR 1 1 1 p −1 p −1···p2 ··· n n pn−1 p1 + ··· + Ln ··· L3 |e2| ).

Proposition 7.6. There is a generic constant K ≥ 1, which depends on M and is

n−1 independent of Li’s, so that on the set BM × IR ,

n 1 1 1 1 ∗ ··· p −1 pn−1 p1 pn−1 n pn−1pn−2 |u − u | n−1 ≤ Kmin ( |ξi| )+|en| +Ln |en−1| BM ×IR n Xi=1 1 1 1 p −1 p −1···p2 ··· n n pn−1 p1 + ··· + Ln ··· L3 |e2| , 1 . (7.37) o 153

Proposition 7.7. There is a generic constant K ≥ 1, which depends on M and is

n−1 independent of all the Li’s, such that on the set BM × IR the following inequalities hold (i =2, ··· , n):

1 1 ··· ··· pi−1 p1 pi−1 p1 |xi| n−1 ≤ K(|ξ1| + ··· + |ξi| ) (7.38) BM ×IR 1 1 ··· ··· pi−1 p1 pi−1 p1 |fi(·)| n−1 ≤ K(|ξ1| + ··· + |ξi| ) (7.39) BM ×IR n 1 1 ··· pn−1 p1 pn−1 |u| n−1 ≤ K ( |ξi| )+ |en| (7.40) BM ×IR h Xi=1 1 1 1 1 1 p −1 p −1 p −1···p2 ··· n pn−1pn−2 n n pn−1 p1 +Ln |en−1| + ··· +Ln ···L3 |e2| . i Using the Young’s inequality, it is not diﬃcult to deduce from (7.27) and (7.37) that

n 2 2pn−2···p1 V˙c ≤ −2( ξ ) + 2 min Cne (7.41) Ω i n i=1 n X 2pn−3···p1 2 + Cn−1(Ln)en−1 + ··· + C2(Ln, ··· , L3)e2, Cn , o where Cn ≥ 1 is a constant independent of Li’s, while Cn−1(Ln) ≥ Cn, ··· ,

C2(Ln, ··· , L3) ≥ Cn are ﬁxed polynomial functions of their own arguments. They can be obtained in a manner similar to the one in (Qian and Lin (2004b)).

For the error dynamics, consider the Lyapunov function

1 V = (e2pn−2···p1 + ··· + e2) (7.42) e 2 n 2

n−1 whose derivative along the trajectories of (7.35) on the set BM × IR satisﬁes

n ˙ 2 ¯ 2pn−2···p1 Ve n−1 ≤ K( ξi ) − [Ln − Cn]en (7.43) BM ×IR Xi=1 ¯ 2pn−3···p1 ¯ 2 −[Ln−1 −Cn−1(Ln)]en−1 −···−[L2 −C2(Ln, ···,L3)]e2,

where K and C¯n ≥ 1 are positive constants independent of Li’s, while C¯n−1(Ln) ≥

1, ··· , C¯2(Ln, ··· , L3) ≥ 1 are ﬁxed polynomial functions of their own arguments, which can be computed using a similar argument as done in (Qian and Lin (2004b)). 154

Now, it is clear that by choosing the gain constants

∆ Ln = Ln(L) = C¯n + LCn ≥ 1

∆ Ln−1 = Ln−1(L) = C¯n−1(Ln)+ LCn−1(Ln) ≥ 1 . .

∆ L2 = L2(L) = C¯2(Ln, ··· , L3)+ LC2(Ln, ··· , L3) ≥ 1 (7.44) with L> 0 being a parameter to be determined later, one has

˙ 2 2 Ve n−1 ≤ K(ξ1 + ··· + ξn) − LWe, (7.45) BM ×IR

− ∆ 2pn−2···p1 2pn 3···p1 2 where We = Cnen +Cn−1(L)en−1 +···+C2(L)e2 and Cn−1(L) ≥ 1, ··· ,C2(L) ≥ 1 are ﬁxed positive polynomial functions of L. For the closed-loop system (7.1)-(7.33)-(7.30)-(7.31)-(7.32), we choose the Lya- punov function V (x) r ln(1 + V (e)) V (x, zˆ)= c + 0 e , (7.46) 2 2 ln(1 + µ(L)) where

n ∆ 1 pi−1 2pi−2···p1 µ(L) = (r +Li(L)r +r) ≥ max Ve > 0. 2 (x,zˆ)∈Γx×Γzˆ Xi=2 Associated with V (x, zˆ), deﬁne the level set

2n−1 Ω= {(x, zˆ) ∈ IR |V (x, zˆ) ≤ r0}.

As shown in Section 7.3, similar properties of V and Ω still hold according to Fig. 7.2.

1. For every L> 0, V (·) is a positive definite and proper function and Ω is a compact set in IRn × IRn−1. Once L> 0 is fixed, both V and Ω are also fixed;

2. ∀L> 0, Ω ⊃ Γx × Γzˆ;

n−1 3. ∀L> 0, BM × IR ⊃ Ω. 155

n−1 By construction, BM × IR ⊃ Ω. With this in mind, it follows from (7.45) and (7.41) that ∀L> 0,

1 r0 V˙e V˙ = V˙c + Ω 2 Ω 2 ln(1 + µ(L)) 1+ V Ω e Kr ≤ − 1 − 0 (ξ2 + ··· + ξ2) 2 ln(1 + µ(L))(1 + V ) 1 n h e i r L − 0 − 1 min{W ,C }. (7.47) 3 ln(1 + µ(L)) e n h i ˙ The remaining part of the proof is to ﬁnd a suitable constant L such that V |Ω ≤ 0. Recall that µ(L) is a ﬁxed polynomial function of L and the constant K is independent of L. Hence, there exists a constant L∗ > 0 such that ∀L ≥ L∗,

Kr 1 r L 0 ≤ and 0 ≥ 2. 2 ln(1 + µ(L)) 2 3 ln(1 + µ(L))

∗ ˙ Choosing L = L , we can estimate V |Ω as follows:

1 2 2 V˙ ≤ − (ξ + ··· + ξ ) − min{We,Cn}. Ω 2 1 n

In summary, system (7.4) is semi-globally stabilizable by the nonsmooth dynamic output compensator (7.33)-(7.30)-(7.31)-(7.32), although it is non-uniformly observable and nonsmoothly stabilizable.

Example 7.8. Consider the nonsmoothly stabilizable system

3 x1 x˙ 1 = x2 + x1e

2 x˙ 2 = x3 + x1x2

x˙ 3 = u

y = x1 (7.48)

y 1/3 which is not uniformly observable, because x2 = (y ˙ − ye ) and

y y d 1 y¨ − ye˙ − yye˙ 2 x = [(y ˙ − yey) 3 ] − yx2 = − y(y ˙ − yey) 3 . 3 dt 2 3(y ˙ − yey)2/3 156

Thus, the method in (Teel and Praly (1994, 1995)) is invalid. On the other hand, the work (Qian and Lin (2004b)) gives only a local output feedback stabilization result

x1 2 due to the non-homogeneous terms x1e and x1x2. By Theorem 7.1, we now know that system (7.48) is semi-globally stabilizable via output feedback.

7.5 Summary

In this chapter, we have proved that without requiring uniform observability and smooth stabilizability by state feedback, it is still possible to achieve semi-global stabilization via nonsmooth output feedback, for a signiﬁcant class of nonlinear systems such as (7.1). This was made possible by developing a nonsmooth semi-global output feedback control scheme, which extended the output feedback design approach in (Teel and Praly (1994); Yang and Lin (2006)) and integrated the recursive nonsmooth observer design algorithm (Qian and Lin (2004b)) with the idea of saturating the estimated state (Khalil and Esfandiari (1995); Teel and Praly (1994)). In the case when the nonlinear system is uniformly observable and smoothly stabilizable, the result of this chapter has provided an alternative yet simpler semi-global output feedback design method. Chapter 8. CONCLUSION

8.1 Summary

In this dissertation, we have investigated several important and diﬃcult problems on how to achieve global or semi-global asymptotic stabilization by smooth or nonsmooth output feedback, for a class of nonlinear systems whose linearization at the origin may be unstabilizable and undetectable. A new output feedback design approach that is not based on the separation principle has been developed, making it possible to recursively construct a nonlinear observer as well as a state feedback controller.

While the state controller was designed by the tool of adding a power integrator (Lin and Qian (2000a,b)), the observer design was based on a newly developed machinery that can be viewed as a dual of the adding a power integrator technique. A novelty of such an observer design is that the observer gains can be assigned in an iterative manner. The output feedback controller thus obtained is of the observer-controller type but the nonlinear observer by itself cannot recover the system states unless the corresponding controller is in eﬀect. In Chapters 4-6, global robust stabilization by smooth or nonsmooth output feedback was shown to be possible for a family of uncertain nonlinear systems under appropriate growth conditions. A robust output feedback design approach has been developed based on a (static or dynamic) rescaling technique and the idea of non- separation principle design, enabling one to recursively construct a robust state feedback controller and a nonlinear observer that does not depend on the uncertainty of the system. For a class of uncertain systems in a cascade form or in the so-called p-normal form (which are beyond a strict-triangular structure), we have also identi-

157 158

ﬁed suitable conditions for the problem of global robust stabilization to be solvable by smooth or nonsmooth output feedback. The applications of the proposed robust output feedback control schemes have been illustrated by several examples. In Chapter 7, we have proved that without requiring uniform observability and smooth stabilizability by state feedback, it is still possible to achieve semi-global stabilization via nonsmooth output feedback, for a signiﬁcant class of nonlinear systems such as (7.1). This was made possible by developing a nonsmooth semi-global output feedback control scheme, which extended the output feedback design approach in (Teel and Praly (1994); Yang and Lin (2006)) and integrated the recursive nonsmooth observer design algorithm (Qian and Lin (2004b)) with the idea of saturating the estimated state (Khalil and Esfandiari (1995); Teel and Praly (1994)).

8.2 Future Work

In what follows, we brieﬂy describe some future research directions which, we believe, are promising and important.

• In Chapter 7, we have obtained an important result on semi-global stabilization by nonsmooth output feedback. Unlike the work reported in Chapters 4-6, the

stabilization result of Chapter 7 is not a robust one because it requires the

precise information of the controlled plant. In fact, the functions fi(·)’s must be precisely known and are used to construct the observer (7.30). Therefore, a natural question arises: is it possible to design a nonsmooth output feedback

control law that robustly semi-globally stabilizes the nonlinear system (7.1) at the origin, using only the bounding functions of system uncertainties? To the

best of our knowledge, even when p1 = ··· = pn−1 = 1, i.e. the system has controllable and observable linearization, this question is still open and worth

for future investigation.

• In Chapter 6, it was proved that for a family of uncertain nonlinear systems 159

dominated by a triangular system that satisﬁes linear growth condition with an

output dependent growth rate, robust global stabilization can be achieved by smooth output feedback. Naturally, the next step of our study is to consider the problem of global output feedback stabilization for a more general class

of uncertain nonlinear systems in the form of (1.1) where pi’s are diﬀerent and system uncertainties satisfy some homogeneous growth condition with an output dependent growth rate. We are currently working on this problem and have already made substantial progress. Indeed, we can prove if the growth rate C is changed into the known function C(y) in Assumption 5.1, the system is

still robustly globally stabilizable by a nonsmooth output feedback. This result will be reported elsewhere soon.

• It should be pointed out in Chapters 3-7, the proposed nonlinear observer was shown to be able to estimate the unmeasured system state, with the help of

the speciﬁc controller we constructed. This observation raises an interesting issue: under what condition, can a nonlinear observer be constructed for an autonomous system with undetectable linearization? Clearly, the method given in Chapters 3-7 and the traditional “Luenberger-type” observer cannot be ap-

plied directly. Any progress in this direction will be a signiﬁcant step for the development of nonlinear estimation theory.

• Last but not least, it has been known that robust feedback stabilization in the large plays an important role in achieving some other control goals such as

adaptive control, global output tracking or regulation, disturbance decoupling or attenuation, etc. With the help of the new output feedback design tools developed in this dissertation, we expect to be able to address these control problems by output feedback in the near future. BIBLIOGRAPHY

Bacciotti, A. Local stabilizability of nonlinear control systems, World Scientific, Sin- gapore, 1992. A. Bacciotti and L. Rosier, Lyapunov functions and stability in control theory, Lectrue Note in Control and Info. Sciences, Vol. 267, Springler, 2001. Byrnes, C.I. and Isidori, A. “New results and examplse in nonlinear feedback stabilization”, Syst. Contr. Lett., Vol. 12, pp. 437-442, 1989. Byrnes, C.I. and Isidori, A. “Asymptotic stabilization of minimum phase nonlinear systems”, IEEE Trans. Automat. Contr., Vol. 36, pp. 1122-1137, 1991. Carr, J. Applications of Center Manifold Theory, Springer-Verlag, New York, 1981. Cheng, D. and Lin, W. “On p-normal forms of nonlinear systems”, IEEE Trans. Automat. Contr., Vol. 48, pp. 1242-1248, 2003. Coron, J.M. and Praly, L. “Adding an integrator for the stabilization problem”, Syst. Contr. Lett., Vol. 17, pp. 89-104, 1991. Crouch, P.E. and Irving, M. “On sufficient conditions for local asymptotic stability of nonlinear systems whose linearization is uncontrollable”, Control Theory Centre Report, No. 114, Univ. Warwick, 1983. Dayawansa, W. P. “Recent advances in the stabilization problem for low dimensional systems”, Proc. 2nd IFAC Symp. on Nonlinear Contr. Syst. Design, Bordeaux, pp. 1-8, 1992. Dayawansa, W.P. Personal Communication, 2002. Dayawansa, W.P.; Martin, C. F.; Knowles, G. “Asymptotic stabilization of a class of smooth two dimensional systems”, SIAM. J. Contr. Optimiz., Vol. 28, pp. 1321- 1349, 1990. Gauthier, J.P.; Hammouri, H.; Othman, S. “A simple observer for nonlinear systems, applications to bioreactocrs”, IEEE Trans. Automat. Contr., Vol. 37, pp. 875-880, 1992. Hahn, W. Stability of Motion, Springer-Verlag, New York, 1967. Hermes, H. “Homogeneous coordinates and continuous asymptotically stabilizing feedback controls”, in Differential Equations: Stability and Control, Ed. Elaydi, S., Marcel Dekker, New York, pp. 249-260, 1991.

160 161

Hermes, H. “Nilpotent and high-order approximations of vector ﬁeld systems”, SIAM. Review, Vol. 33, pp. 238-264, 1991. Huang, X; Lin, W.; Yang, B. “Global ﬁnite-time stabilization of a class of uncertain nonlinear systems”, Automatica, Vol. 41, pp. 881-888, 2005. Isidori, A. Nonlinear Control Systems, 3rd ed., Springer-Verlag, New York, 1995.

Isidori, A. Nonlinear Control Systems II, Springer-Verlag, New York, 1999. Isidori, A. “A tool for semiglobal stabilization of uncertain non-minimum-phase nonlinear systems via output feedback”, IEEE Trans. Automat. Contr., Vol. 45, pp. 1817-1827, 2000. Kawski, M. “Stabilization of nonlinear systems in the plane”, Syst. Contr. Lett., Vol. 12, pp. 169-175, 1989. Kawski, M. “Homogeneous stabilizing feedback laws”, Control Theory and Advanced Technology, Vol. 6, pp. 497-516, 1990. Kawski, M. “Geometric homogeneity and applications to stabilization”, Proc. 3rd IFAC Symp. Nonlinear Contr. Syst., Lake Tahoe, pp. 164-169, 1995. Khalil, H. Nonlinear systems, Macmillan, New York, 1992. Khalil, H. and Esfandiari, F. “Semi-global stabilization of a class of nonlinear systems using output feedback”, IEEE Trans. Automat. Contr., Vol. 38, pp. 1412-1415, 1995. Khalil, H. and Saberi, A. “Adaptive stabilization of a class of nonlinear systems using high-gain feedback”, IEEE Trans. Automat. Contr., Vol. 38, pp. 1031-1035, 1987. Krener, A.J. and Isidori, A. “Linearization by output injection and nonlinear observer”, Syst. Contr. Lett., Vol. 3, pp. 47-52, 1983. Krener, A.J. and Kang, W. “Locally convergent nonlinear observers”, SIAM J. Contr. Optimiz., Vol. 42, pp. 155-177, 2003. Krener, A.J. and Xiao, M. “Observers for linearly unobservable nonlinear systems”, Syst. Contr. Lett., Vol. 46, pp. 281-288, 2002. Krishnamurthy, P. and Khorrami, E. “Generalized adaptive output feedback form with unknown parameters multiplying high output relative-degree states”, Proc. 41st IEEE Conf. Decision Contr., Las Vages, pp. 1503-1508, 2002.

Krstic, M.; Kanellakopoulos, I.; Kokotovic, P.V. Nonlinear and Adaptive Control Design, John Wiley, 1995. Kurzweil, J. “On the inversion of Lyapunov’s second theorem on the stability of motion”, Amer. Math. Soc. Trans., Vol. 24, pp. 19-77, 1956. 162

Lin, W. and Pongvuthithum, R. “Global stabilization of cascade systems by C0 partial state feedback”, IEEE Trans. Automat. Contr., Vol. 47, pp. 1356-1362, 2002. Lin, W. and Qian, C. “Robust regulation of a chain of power integrators perturbed by a lower-triangular vector ﬁeld”, Int. J. of Robust and Nonlinear Contr., Vol. 10, pp. 397-421, 2000. Lin, W. and Qian, C. “Adding one power integrator: a tool for global stabilization of high-order triangular systems”, Syst. Contr. Lett., Vol. 39, pp. 339-351, 2000. Lin, W. and Yang, B. “Robust stabilization of uncertain high-order nonlinear systems by output feedback”, in Proc. 6th IFAC Symp. Nonlinear Contr. Syst., Stuttgart, pp. 67-74, 2004.

Marino, R. and Tomei, P. Nonlinear control design, Prentice Hall International, UK, 1995. Mazenc, F.; Praly, L.; Dayawansa, W.P. “Global stabilization by output feedback: examples and counterexamples”, Syst. Contr. Lett., Vol. 23, pp. 119-125, 1994.

Nijmeijer, H. and van der Schaft, A.J. Nonlinear dynamical control systems, Springer- Verlag, 1990. Praly, L. “Asymptotic stabilization via output feedback for lower-triangular systems with output dependent incremental rate”, IEEE Trans. Automat. Contr., Vol. 48, pp. 1103-1108, 2003. Praly, L. and Jiang, Z.P. “On global output feedback stabilization of uncertain nonlinear systems”, Proc. 42nd IEEE Conf. Decision Contr., Maui,1544-1549, 2003. Qian, C. and Lin, W. “A continuous feedback approach to global strong stabilization of nonlinear systems”, IEEE Trans. Automat. Contr., Vol. 46, pp. 1061-1079, 2001. Qian, C. and Lin, W. “Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization”, Syst. Contr. Lett., Vol. 42, pp. 185- 200, 2001.

Qian, C. and Lin, W. “Output feedback control of a class of nonlinear systems: a non-separation principle diagram”, IEEE Trans. Automat. Contr., Vol. 47, pp. 1710-1715, 2002. Qian, C. and Lin, W. “Smooth output feedback stabilization of planar systems without controllable/observable linearization”, IEEE Trans. Automat. Contr., Vol. 47, pp. 2068-2073, 2002. Qian, C. and Lin, W. “Nonsmooth output feedback stabilization of a class of genuinely nonlinear systems in the plane”, IEEE Trans. Automat. Contr., Vol. 48, pp. 1824- 1829, 2003. 163

Qian, C. and Lin, W. “New results on nonsmooth output feedback stabilization of nonlinear systems”, New Directions in Control Theory and Applications, Lecture Notes in Contr. & Info. Sciences, W. P. Dayawansa, A. Lindquist and Y. Zhou eds., Vol. 321, pp. 305-320, Springer-Verlag, Berlin (2005). Qian, C. and Lin, W. “Recursive observer design and nonsmooth output feedback stabilization of inherently nonlinear systems”, Proc. of 43rd IEEE Conf. Decision Contr., Nassau, pp. 4927-4932, 2004. Respondek, W. “Transforming a single-input system to a p-normal form via feedback”, Proc. 42nd IEEE Conf. Decision Contr., Maui, pp. 1574–1579, 2003. L. Rosier, “Homogeneous Lyapunov functions for homogeneous continuous vector ﬁeld”, Syst. Contr. Lett., Vol. 19, pp. 467-473, 1992. Rui, C.; Reyhanoglu, M.; Kolmanovsky, I.; Cho S.; McClamroch, N.H. “Nonsmooth stabilization of an underactuated unstable two degrees of freedom mechanical system”, Proc. 36th IEEE Conf. Decision Contr., San Diego, pp. 3998-4003, 1997.

Sontag, E. and Teel, A. “Changing supply functions in input/state stable systems”, IEEE Trans. Automat. Contr., Vol. 40, pp. 1476-1478, 1995. Sontag, E. and Wang, Y. “On characterizations of the input-to-state stability property”, Syst. Contr. Lett., Vol. 24, pp. 351-359, 1999.

Teel, A. and Praly, L. “Global stabilizability and observability imply semi-global stabilizability by output feedback”, Syst. Contr. Lett., Vol. 22, pp. 313-325, 1994. Teel, A. and Praly, L. “Tools for semiglobal stabilization by partial state and output feedback”, SIAM J. Contr. Optimiz., Vol. 33, pp. 1443-1488, 1995.

Tsinias, J. “Suﬃcient Lyapunov-like conditions for stabilization”, Math. Contr. Sig- nals Syst., Vol. 2, pp. 343-357, 1989. M. Tzamtzi and J. Tsinias, Explicit formulas of feedback stabilizers for a class of triangular systems with uncontrollable linearization, Syst. Contr. Lett., Vol. 38 (1999), 115-126. Yang, B. and Lin, W. “Output feedback stabilization of a class of homogeneous and high-order nonlinear systems”, Proc. 42nd IEEE Conf. Decision Contr., Maui, pp. 37-42, 2003. Yang, B. and Lin, W. “Homogeneous observers, iterative design and global stabilization of high order nonlinear systems by smooth output feedback,” IEEE Trans. Automat. Contr., Vol. 49, pp. 1069-1080, 2004. Yang, B. and Lin, W. “A furthur result on global stabilization of uncertain nonlinear systems by output feedback”, Proc. 6th IFAC Symp. Nonlinear Contr. Syst., Stuttgart, Germany, pp. 95-100, 2004. 164

Yang, B. and Lin, W. “Robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization”, IEEE Trans. Au- tomat. Contr., Vol. 50, pp. 619-630, 2005. Yang, B. and Lin, W. “Further results on global stabilization of uncertain nonlinear systems by output feedback”, Int. J. Robust Nonlinear Contr., Vol.15, pp. 247-268, 2005.

Yang, B. and Lin, W. “Finite-time stabilization of nonlinear systems with uncontrollable unstable linearization”, Proc. 16th IFAC World Congress, Prague, paper code: Tu-M22-TO/4, 2005. Yang, B. and Lin, W. “Robust stabilization of uncertain nonlinear systems by nonsmooth output feedback”, Proc. 2005 American Contr. Conf., Portland, pp. 4072- 4707, 2005. Yang, B. and Lin, W. “Semi-global output feedback stabilization of non-uniformly observable and nonsmoothly stabilizable systems”, to appear in Proc. 44th IEEE Conf. Decision Contr., Seville, 2005. Yang, B.; Lin, W.; Dayawansa, W.P. “Finite-time stabilization of nonsmoothly stabilizable systems with uncertainty”, submitted to SIAM J. Contr. Optimiz., 2005. Yang, B. and Lin, W. “On semi-global stabilizability of MIMO nonlinear systems by output feedback”, to appear in Automatica. Zubov, V.I. Mathematical Methods for the Study of Automatic Control Systems, No- ordhoﬀ, Groningen, 1964.