ADAPTIVE OUTPUT REGULATION OF NONLINEAR SYSTEMS WITH UNKNOWN HIGH-FREQUENCY GAINS
A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Sciences and Engineering
2020
Jingyu Chen Department of Electrical and Electronic Engineering School of Engineering Contents
List of Tables 6
List of Figures 7
Symbols 8
Abbreviations 9
Abstract 10
Declaration 12
Copyright Statement 13
Publications 14
Acknowledgements 15
1 Introduction 17
1.1 Background ...... 17
1.2 Research Problems ...... 19
2 1.3 Contributions and Organization ...... 20
2 Literature Review 23
2.1 Output Regulation Problem for SISO system ...... 23
2.1.1 Development of Output Regulation ...... 24
2.1.2 The Effects of the Knowledge of the High-Frequency Gain Sign . 28
2.2 Consensus Output Regulation of MASs ...... 30
2.2.1 Development of Consensus Control ...... 30
2.2.2 Control Directions in Consensus Output Regulation ...... 32
2.3 Summary ...... 34
3 Preliminary Studies and Problem Descriptions 35
3.1 Fundamentals of Nonlinear Systems ...... 35
3.1.1 Nonlinear Systems ...... 35
3.1.2 Stability Concepts for Nonlinear Systems ...... 37
3.1.3 Tools for Control Design ...... 39
3.2 Centre Manifold Theory ...... 40
3.3 The Output Regulation Theory ...... 41
3.4 Steady-State Generator ...... 45
3.4.1 Linear Immersion Assumption ...... 47
3.4.2 Nonlinear Immersion Assumption ...... 48
3.5 The Internal Model Principle ...... 49
3 3.6 Nussbaum Gain Method ...... 52
3.7 Graph Theory Basics ...... 53
3.8 State Feedback Consensus Protocols ...... 56
4 Adaptive Output Regulation of Output Feedback Nonlinear Systems with Unknown Nonlinear Exosystems and Unknown High-Frequency Gain Sign 59
4.1 Problem Formulation ...... 60
4.2 State Transformation ...... 62
4.3 The Internal Model Design ...... 64
4.4 Controller Design ...... 68
4.5 Stability Analysis ...... 71
4.6 Numerical Example ...... 73
4.7 Summary ...... 77
5 Adaptive Consensus Output Regulation of a Class of Heterogeneous Nonlinear Systems with Unknown Control Directions 78
5.1 Problem Formulation ...... 79
5.2 State Transformation and The Internal Model ...... 81
5.3 A Novel Nussbaum-Type Function for Consensus Control ...... 85
5.4 Consensus Output Regulation Design, ρ =1 ...... 87
5.5 Consensus Output Regulation Design, ρ > 1...... 90
5.6 Example ...... 93
4 5.7 Summary ...... 95
6 Distributed Adaptive Consensus Output Tracking of Nonlinear Sys- tems on Directed Graphs with Unknown High-Frequency Gains 96
6.1 Problem Statement ...... 97
6.2 Preliminary Results ...... 99
6.3 Control Design ...... 100
6.4 Simulation Example ...... 105
6.5 Summary ...... 106
7 Conclusions and Future Works 110
7.1 Final Conclusions ...... 110
7.2 Future Works ...... 112
Bibliography 114
5 List of Tables
3.1 Steady-state generators and internal models under different assumptions. 52
6 List of Figures
2.1 The control process of output regulation problem for a single system. . 24
4.1 The system output e and control input u for b = 1...... 75
4.2 Feedforward control η1 and its estimationη ˆ1 for b = 1...... 75
4.3 The system output e and control input u for b = −1...... 76
4.4 Feedforward control η1 and its estimationη ˆ1 for b = −1...... 76
5.1 The subsystem outputs...... 94
5.2 The subsystem control inputs...... 95
6.1 The subsystem outputs yi, i = 0,..., 4...... 107
6.2 The subsystem states xi,2, i = 0,..., 4...... 107
6.3 The adaptive gains ki, i = 1,..., 4...... 108
6.4 The tracking errors ζi, i = 1,..., 4...... 108
6.5 The control inputs ui, i = 1,..., 4...... 109
6.6 The control inputs ui, i = 1,..., 4 when t between 0-2s...... 109
7 Symbols
R set of real numbers Rn n-dimensional Euclidean space Rn×m set of n × m real matrices =: defined as ∀ for all ∈ belongs to ⊂ subsets of → tends to P summation Π product ⊗ the Kronecker product of matrices kxk the Euclidean norm of the vector x AT transpose of the matrix A AH conjugate transpose of the matrix A A−1 inverse of the matrix A diag(Ai) a block-diagonal matrix with Ai, i = 1,...,N, on the diagonal kAk the induced 2-norm of the matrix A
λmin(A)(λmax(A)) the minimum (maximum) eigenvalue of the matrix A sup supremum, the least upper bound inf infimum, the greatest lower bound
L∞ L∞ space
Lf h the lie derivative of h with respect to the vector field f f : S1 → S2 a function f mapping a set S1 into a set S2 N(·) Nussbaum function
8 Abbreviations
SISO Single-Input-Single-Output
MASs Multi-Agent Systems
HST Hubble Space Telescope
GRORP Global Robust Output Regulation Problem
ARE Algebraic Riccati Equation
9 Abstract
The University of Manchester
Jingyu Chen Doctor of Philosophy Adaptive Output Regulation of Nonlinear Systems with Unknown High- Frequency Gains October 31, 2020 Disturbance rejection and output regulation problem is an important topic in control design since disturbances are inevitable in practical systems. This thesis is concerned with this problem for a class of single-input-single-output (SISO) nonlinear systems and multi-agent nonlinear systems, especially when the high-frequency gains are un- known. Output regulation aims to design a control law to achieve asymptotic tracking in the presence of a class of disturbances while maintaining the stability of the closed- loop system. “Multi-agent systems (MASs)” is referred as a group of agents which are connected together by a communication network. In many applications, the subsys- tems or agents are required to reach an agreement, which is regarded as ”consensus control”. The main contribution of this thesis is to provide feasible methods to deal with the output regulation of SISO system and consensus output regulation for multi- agent systems.
The adaptive output regulation for SISO nonlinear output feedback system with un- known exosystem and unknown high frequency gain is first investigated. By transform- ing the system with ρ relative degree to 1, the augmented system is obtained. Then, using the invariant manifold theory, an adaptive internal model is proposed to tackle
10 the unknown feedforward input. In terms of the unknown high-frequency gain, a type of Nussbaum gain is presented, the control law is then obtained by adopting this gain and the adaptive internal model. To establish the closed-loop stability, the Lyapunov function is invoked, and this control scheme is finally validated by simulation example.
The adaptive consensus output regulation of multi-agent nonlinear systems with un- known control directions is then investigated. A new type Nussbaum gain with a potentially faster rate is proposed such that the boundedness of the system parame- ters can be established by an argument of contradiction even if the Nussbaum gain parameter for only one of the subsystems goes unbounded. The adaptive laws and control inputs with the information available from the subsystems and their neigh- bourhood are proposed, and therefore the adaptive laws and inputs are viewed as decentralized. The proposed control can deal with the subsystems with different dy- namics as long as the subsystems with the same relative degree. An example is finally included to demonstrate the proposed control design.
The consensus tracking problem for multi-agent nonlinear systems with unknown high- frequency gains is finally investigated for the subsystems connected over directed graph. An integral-Lyapunov function is proposed to tackle the asymmetry of the Laplacian matrices, then the new Nussbaum gain and the adaptive internal model are combined to design the controller. It can be shown that the control scheme and the adaptive laws are fully distributed, and the simulation result illustrates that the proposed control scheme guarantee the convergence of errors to zero asymptotically and the boundedness of the state variables.
11 Declaration
No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning.
12 Copyright Statement
i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes.
ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intel- lectual property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and com- mercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any rele- vant Thesis restriction declarations deposited in the University Library, The Univer- sity Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regul- ations) and in The University’s Policy on Presentation of Theses.
13 Publications
1). Chen, Jingyu, and Zhengtao Ding, ”Adaptive Output Regulation of Output Feed- back Systems with Unknown Nonlinear Exosystems and Unknown High-Frequency Gain Sign,” Annual American Control Conference (ACC), IEEE, pp. 4496-4501, 2018.
2). Chen, Jingyu, Zhenhong Li, and Zhengtao Ding, ”Adaptive output regulation of uncertain nonlinear systems with unknown control directions,” Science China Infor- mation Sciences, 62(8), pp. 1-3, 2019.
14 Acknowledgements
Undertaking this PhD has been a truly life-changing experience for me and this work would not have been possible without the support and guidance that I received from many people.
First and foremost, I would like to express my sincere gratitude to my super- visor Prof. Zhengtao Ding for his supervision during these past four years. He led me through the door of output regulation and consensus control, and also provided insightful discussions about my research field. Without his patience, motivation and constant feedback, this PhD would not have been achievable.
I would also like to thank all Control Systems Centre colleagues who helped me to solve the difficulties that I encountered during my studies.
My thanks also go to my husband, who has been by my side throughout my PhD research program, and without whom, I would not have had the courage to embark on this journey in the first place. Last but not least, I owe my deepest gratitude to my parents, who have provided me with unconditional love, encouragement and financial support for my Manchester life.
15 To My Loving Parents and Husband
16 Chapter 1
Introduction
1.1 Background
In general, all industrial systems are subjected to various sources of unknown dis- turbances and uncertainties, which may lead to poor control performance or even instability. To achieve the desired goals, a well-performed controller is expected to suppress undesirable effects of the disturbances and enhance the anti-interference abil- ity of the industrial systems. For instance, the Lorenz system, introduce in 1963 by Edward N.Lorenz, is a three-dimensional dynamic system, the state trajectory of the system displays a typical behaviour of what is now known as chaotic motion. Chaotic motion is a special phenomenon of an unstable nonlinear system. This phenomenon may be harmful to the operation of a real system and it may often be desirable to apply a feedback control law to eliminate the chaotic motion in the system [1]. The Hubble Space Telescope (HST) is an another example for illustration of the stabiliza- tion and regulation requirements in control system design. The HST whirls around earth at a speed of 7.5km/s at an altitude of 569km, inclined 28.5◦ to the equator. In order to take images of distant and faint objects, the pointing control system was thus designed for extremely steady and accurate attitude tracking. However, a compli- cated situation is that the pointing control system is experiencing unexpectedly large disturbances. Several potential sources of the HST disturbances include the reaction wheel assemblies, data recorders, fine guidance sensors, high gain antennas, magnetic
17 CHAPTER 1. INTRODUCTION 18 torques, and the solar arrays. It is thus necessary to design a pointing control system to reject of attenuate the disturbance [2]. Disturbances are unknown in reality, but for the theoretical research, we assume that they can be divided to various types, from random disturbances, wide-band, narrow-band disturbances, to deterministic distur- bances that include harmonic disturbances, i.e., sinusoidal functions, general periodic disturbances and other deterministic signals generated from nonlinear dynamics sys- tems such as limit cycles. The white noise is one of the typical disturbances in the academic research. It is not surprising that disturbance rejection and uncertainty at- tenuation is a crucial task in controller design. Briefly, the output regulation problem is concerned with designing a control law for the plant such that the closed-loop system satisfies two requirements. The first one is the stability of closed-loop system, and the second one is that the output of the closed-loop system can track the reference inputs asymptotically in the presence of a class of disturbances [3].
Compared to a single agent, with the development of computing, communication and sensing, having multiple control agents to work together to accomplish collec- tive group behaviours can significantly improve the operational effectiveness. Due to its potential applications in various areas such as satellite formation flying, robotics and electric power systems, cooperative output regulation of multi-agents has received great attention from the systems and control community. Cooperative output regu- lation of multi-agent systems is to have a group of agents connecting together via a network to track a prescribed trajectory asymptotically and/or maintain asymptotic rejection of disturbances [4]. In the formulation of cooperative control, there are two types of methods: centralised and distributed method. Due to the large number of agents and limited sensing capability of sensors, it is considered too expensive or even infeasible in practice to implement centralised controllers. The distributed method, which depends on information of the agents and their neighbours, brings more benefits.
As a result, the output regulation problems of single system and the multi-agent systems have drawn compelling attraction in recent years. For this kind of problem, the disturbances or the exosystem can be constant, time varying or unknown linear and nonlinear signals. The systems themselves are always uncertain, and the uncertainties can be general nonlinear functions and input-related and/or input unrelated. When CHAPTER 1. INTRODUCTION 19 there are unknown parameters in the systems, adaptive control algorithms are first developed for linearisable systems under the restrictions in the growth rate of non- linearities and matching conditions [5, 6]. One of the important tools in overcoming these difficulties is backstepping. The general adaptive algorithms for input-related uncertainties and backstepping are developed with the virtual coefficients equal to one or under the assumption that the coefficients are with known signs. These signs, called control directions or high-frequency gains, represent motion directions of system under any control, and knowledge of these signs makes robust control design much easier. However, when not all state variables are measurable, and when there exist large uncertainties in systems, it is difficult to detect the high-frequency gain directly. Thus, it is of interest to devise a method to remove the requirement on the sign of the high- frequency gain in the adaptive control and implement to output regulation problem. In comparison with the plentiful research results in the output regulation of SISO nonlinear systems, there are only a few results on this problem with both unknown exosystem and unknown control directions because of much greater technical difficulty in the control design, and this problem remains open. In terms of a network connected system with multiple subsystems, the existing control design for individual system with unknown high-frequency gain sign would not be able to establish the boundedness of all the variables in the adaptive consensus output regulation problem because each subsystem could move in different directions. Besides, the communication between agents can be undirected and directed, the asymmetric connection in a directed graph remained as an obstacle in the extension of adaptive schemes beyond undirected graphs to fully distributed adaptive consensus control, the adaptive consensus control problem of unknown nonlinear systems on directed graphs with unknown control directions thus draws our attention.
1.2 Research Problems
This thesis studies the adaptive output regulation problem for a class of SISO output feedback nonlinear system and consensus output regulation of multi-agent nonlinear systems through the tools from control theory and graph theory with emphasis on CHAPTER 1. INTRODUCTION 20 the unknown exosystems and unknown high-frequency gain signs. Specifically, we will focus on the following problems.
• For SISO nonlinear systems with unknown nonlinear exosystem and unknown high-frequency gain sign, the concept of internal model and the adaptive con- trol will be investigated, and how to convert the output regulation problem to stabilization problem will be also considered.
• For general nonlinear multi-agent systems with unknown control directions under undirected network connection, the decentralised protocols which only depend on the relative information of subsystem outputs will be investigated for the closed-loop systems to reach robust consensus output regulation.
• For leader-follower format of a class of network-connected uncertain nonlinear systems without knowing the control direction of each agent over directed graphs, the distributed consensus control based on the Lyapunov function analysis and the graph characteristics will be investigated.
1.3 Contributions and Organization
The main contributions are summarised as follows:
• In terms of the output regulation of SISO nonlinear systems with unknown non- linear exosystem and unknown high-frequency gain sign, a nonlinear internal model based on the centre manifold and the circle criterion is proposed to gener- ate the feedforward input term to tackle the unknown nonlinear exosystem, and some specific conditions are introduced for exosystems such that the proposed internal model can be applied. A new type of Nussbaum gain is then presented to deal with the unknown high-frequency gain sign.
• The consensus output regulation of a class of network-connected nonlinear dy- namic systems whose subsystems have all the system parameters completely CHAPTER 1. INTRODUCTION 21
unknown, including the high frequency gains over undirected graphs is investi- gated. Specific characteristics for the new type Nussbaum gain are identified for proposed consensus protocols to tackle the unknown control directions.
• The distributed consensus control in leader-follower format of multi-agent nonlin- ear systems with unknown control directions and directed communication graphs is investigated. The selection of appropriated Lyapunov function is derived to tackle the asymmetric feature of the Laplacian matrices of directed graphs. The new Nussbaum gain and the adaptive laws are then identified to guarantee the distributed consensus and output tracking of a group of nonlinear systems.
Overall, this thesis consists of seven chapters. The organization of each chapter is described in detail at the beginning of that chapter. To understand the whole thesis structure, a general overview of these chapters is given as follows.
In Chapter 1, we first introduce the basic idea of the disturbance rejection and output regulation problem of SISO nonlinear systems and the consensus output reg- ulation of multi-agent nonlinear systems. Afterwards, the problems of this project are put forward while the associated contributions are listed. In the end, for a better understanding of the following chapters, the structure of the thesis is presented.
In Chapter 2, the existing results about the development of the output regulation problem of SISO system and the multi-agent systems are reviewed, respectively. Then, we review the related work, including the overview of adaptive control, the knowledge of the high-frequency gain sign effects, uncertainties and external disturbances existing in control design.
In Chapter 3, some related preliminaries, including the fundamentals of nonlinear systems, the centre manifold theory, stability theory, matrix theory, basic algebraic graph theory, and preliminary results used in this thesis, are introduced.
In Chapter 4, we systematically investigate the adaptive output regulation prob- lem for a class of nonlinear systems with unknown nonlinear exosystem and unknown high-frequency gain sign. The circle criterion method is adopted to design the adaptive CHAPTER 1. INTRODUCTION 22 internal model such that the feedforward control input can be reproduced, and the so- called Nussbaum gain is applied to tackle the influence of the unknown high-frequency gain sign. Judicious analysis is carried out to tackle the high relative degree of system under the state transformation. By transforming the original system into the aug- mented system, the global output regulation problem of nonlinear system with relative degree ρ is then converted to the global stabilization problem of augmented nonlinear system with relative degree 1. In terms of the stability analysis, the Lyapunov-based function is established to demonstrate the proposed control scheme does ensure the boundedness of the state variables and the zero convergence of the error.
In Chapter 5, we consider the adaptive consensus output regulation problem for multi-agent nonlinear systems under undirected graph with both unknown exosystem and the unknown control directions. The Nussbaum gain designed for individual system to tackle the unknown control direction cannot be applied directly in the adaptive consensus control system, because the Nussbaum gain parameter for each subsystem could move in different directions. A new Nussbaum gain with a faster rate is then proposed to remove the knowledge of the high-frequency gains of the network- connected nonlinear systems. Moreover, the stability of the multi-agent systems is analysed by an argument of contradiction and the effectiveness of this control scheme is evaluated on a simulation example.
In Chapter 6, we systematically investigate the adaptive consensus output track- ing problem for uncertain multi-agent systems over directed graph with unknown high- frequency gain signs. The troublesome asymmetry of the Laplacian matrices involved in the control systems makes the design of the Lyapunov function and Nussbaum gain complicated. Sufficient conditions are derived for the multi-agent systems to guaran- tee the effectiveness of the proposed new type of integral-Lyapunov function, and thus achieve the consensus output tracking.
In Chapter 7, we summarize the thesis and provide suggestions for future work, including the possible extensions, and some interesting topics are also suggested for future research. Chapter 2
Literature Review
This chapter concludes two sections. In Section 2.1, it provides a review of existing related results, including the development of output regulation of SISO system, the knowledge of high-frequency gain effects on adaptive control and output regulation problem. In terms of the multi-agent systems, the general overview of consensus control is introduced, and then consensus output regulation and the corresponding key issues are given in Section 2.2.
2.1 Output Regulation Problem for SISO system
One of the most important problems in control theory is output regulation, which aims to design a feedback control law for a given controlled plant subject to some exogenous signal such that the closed-loop system is stable and, in addition, the out- put of the plant asymptotically approaches a given reference input determined by the exogenous signal, known as exosystem. The flow diagram of the control process is shown in Fig. 2.1. It illustrates that a plant subjected to external disturbances is con- trolled by a compensator processing certain plant measurements, a reference command signal r, and possibly the feedforward disturbance signal [7]. This problem generally arises from mathematically formulating practical control problems, for example, land- ing and taking-off of aircraft on carriers under severe weather condition, coordination
23 CHAPTER 2. LITERATURE REVIEW 24 and manipulation of robots, orbiting of satellites, motor speed regulation, and so forth. Study of this problem can be traced as far back as 1769, when James Watt devised a speed regulator for a steam engine. Yet the formulation of this problem in a modern state-space framework was not available until the 1970s. In contrast to similar prob- lems, such as trajectory tracking, where the trajectory to be tracked is assumed to be completely known, a distinctive feature of the output regulation problem is that the reference inputs and disturbances do not have to be known exactly so long as they are generated by a known, autonomous differential equation [8]. Thus, this problem has attracted attention of the control community for several decades and it has also been a driving force for the advancement of modern control theory and applications.
Fig. 2.1. The control process of output regulation problem for a single system.
2.1.1 Development of Output Regulation
The output regulation problem was first studied for the class of linear systems under various names, such as the robust servomechanism problem (Davison) or the struc- turally stable output regulation problem (Francis and Wonham). It was treated by several researches, including Davison [9], Francis and Wonham [7, 10], to name just a CHAPTER 2. LITERATURE REVIEW 25 few. In particular, the work [10] of Francis has shown that the solvability of a multi- variable linear regulator problem corresponds to the solvability of a system of two linear matrix equations which are called Sylvester equations. This is in turn equiva- lent to the location of the transmission zeros of the composite system formed from the plant and the exosystem, as illustrated by Hautus [11]. The work of Francis and Won- ham [7] has also showed that any regulator which solves the error feedback problem has to incorporate a model of the exogenous system generating the reference signal which is to be tracked and/or the disturbance that must be rejected. This property is commonly known as the internal model principle. This principle can intuitively be expressed as: “Any good regulator must create a model of the dynamic structure of the environment in the closed-loop system” [12]. From the control theoretic point of view, the significance of this principle is that a given plant can be converted into the stabilization problem of the augmented system composed of the given plant and a well-defined dynamic compensator called internal model, and the well-known classical PID (proportional-integral-derivative) control can be viewed as a special case of this principle when the exogenous signals are constant.
At almost the same time that research on linear output regulation problem reached its peak, in the mid 1970s, Francis and Wonham considered this problem for a class of nonlinear systems for the special case when exogenous signals are constant [7]. They showed that a linear regulator design based on the linearised plant can solve the robust output regulation problem for a weakly nonlinear plant while maintaining the local stability of the closed-loop system. In 1984, Hepburn and Wonham [13] presented a complete extension of the internal model theory to the setting of nonlinear systems defined on differentiable manifolds. Desoer and Lin [14] then investigated conditions for the existence of regulators for the purposed of tracking constant reference signals. In the late 1980s, Huang and Rugh [15] presented a gain scheduling approach and related the solvability of the problem that mentioned above by Francis and Wonham [7] to solvability of a set of nonlinear algebraic equations.
To establish a general theory for the output regulation problem for uncertain nonlinear systems subject to time-varying exogenous signals, there are three important issues must be addressed: how to define and guarantee existence of the steady state CHAPTER 2. LITERATURE REVIEW 26 of the system, and hence characterize the solvability of the problem; how to tackle plant uncertainty when it is known that the linear internal model principle does not work for nonlinear systems in the general case; and how to achieve output regulation in a nonlinear system with arbitrarily large initial states of the plant, the exosystem, and the controller, in the presence of uncertain parameters that lie in an arbitrarily prescribed, bounded set. None of these three issues can be dealt with by a simple extension of the linear output regulation theory. Because of these challenges, the output regulation problem for nonlinear systems has attracted attention of numerous researchers from the world since the 1990s.
The difficulty associated with the first issue, existence of steady state, lies in the fact that the solution of a nonlinear system is not available. This issue was first addressed in 1990 by Isidori and Byrnes without considering the parameter uncertainty [16]. By introducing centre manifold theory, they found that it is possible to use a set of mixed nonlinear partial differential and algebraic equations, called regulator equations in what follows, to characterize the steady state of the system. It turns out that the regulator equations are a generalization of the Sylvester equations mentioned above. The solution of the regulator equations provided a feedforward control to cancel the steady-state tracking error. Based on the solution of the regulator equations, both state feedback and error feedback control laws can be readily synthesized to achieve asymptotic tracking and disturbance rejection for an exactly known plant while maintaining local stability of the closed-loop system.
The second issue is concerned with the plant uncertainty characterized by a set of unknown parameters. The feedforward control approach mentioned above cannot handle this case due to the presence of unknown parameters. The design approach based on the linear internal model principle does not work either, because that the steady-state tracking error in a nonlinear system is a nonlinear function of the exoge- nous signals. The key to achieve the output regulation problem is the solvability of the regulator equations. Thus it is necessary to develop approximation expansion to solving these equations. An approximation method based on Taylor series expansion was developed by Huang and Rugh in 1991 [17], and after that, Huang found that if the solution of the regulator equations is polynomial in the exogenous signals, then it CHAPTER 2. LITERATURE REVIEW 27 is possible to solve the output regulation problem for uncertain nonlinear systems by both state feedback and output feedback control. The robust version of the output regulation was further studied via the dynamic state feedback [18,19], dynamic output feedback [18,20–22], and the dynamic state/output feedback control with feedforward control [23, 24]. Various solvability conditions were given which impose assumptions on the solution of the regulator equations.
While the first two issues have been intensively addressed since the 1990s, the investigation of the third issue, the semiglobal or global output regulation problem for nonlinear systems with special structures was rapidly unfolding. In the original formulation of the output regulation problem, only local stability is required for the closed-loop system. For this case, the stability issue can be easily handled by Lya- punov’s linearisation method. When a global stability requirement is imposed on the closed-loop system, the situation becomes much more complicated. Khalil [21] studied the semiglobal robust output regulation problem for a class of feedback lin- earisable systems with known exosystem in 1994. His work was further extended to the class of lower triangular nonlinear systems by Isidori [25] in 1997. This problem for a general class of SISO nonlinear systems was then addressed by Serrani, Isidori and Marconi [26], they proposed a solution that does not require the assumption of input-to-state stability of the zero-dynamics of the plant. They then addressed the same problem for nonlinear systems driven by a linear, neutrally stable exosystem whose frequencies are not known a priori [27], where the adaptive internal model was proposed and a design solution provided. The output regulation problem with global stability was first solved for the class of output feedback nonlinear systems by Serrani and Isidori in 2000 [28]. For the same class of nonlinear systems considered in [28], but with unknown parameters, an adaptive version of global output regulation has been achieved using a dynamic swapping technique for state estimation in [29]. Ding [30] then introduced a new parameter-dependent state observer where the new formulation of internal model is used to generate the contribution of feedforward control compen- sation to a state variable for global disturbance rejections for nonlinear systems in output feedback form. An extension to solve the global output regulation for output CHAPTER 2. LITERATURE REVIEW 28 feedback nonlinear systems with unknown exosystem was proposed in [31]. The re- sult shown in [32] was able to tackle the unknown system parameters other than the high-frequency gain, as well as the unknown exosystem parameters, using adaptive ro- bust control method to dominate the certain nonlinear terms by known functions with adaptive coefficients. However, the solvability of the global robust output regulation problem in [28] is under assumption that a linear internal model exists. The same problem was studied in [33] without assuming the existence of a linear internal model, thus, this result can apply to nonpolynomial nonlinear systems in output feedback form. Huang and Chen [3] then established a general framework that converts the robust output regulation problem for a general nonlinear system into a robust stabi- lization problem for an augmented system in 2004. The global output regulation of a class of nonlinear systems with general known and unknown nonlinear exosystems was proposed in [34] and [35], respectively.
2.1.2 The Effects of the Knowledge of the High-Frequency Gain Sign
In the study of control theory, the adaptive control of uncertain nonlinear systems has been achieved using transformations based on differential geometry [5, 6, 36–39]. One of the important tools to develop adaptive control laws is backstepping. The un- known parameters in those systems can multiply general nonlinear functions and/or “control” variables, by which we mean actual control variables or virtual1 control vari- ables defined in backstepping design. The signs of unknown parameters multiplying “control” variables are usually assumed to be known a priori. These signs, called con- trol directions in [40], represent motion directions of the system under any control, and knowledge of these signs makes adaptive control design much easier. Many re- searchers have noticed that this assumption is very restrictive in general. It has been shown that the adaptive control of a linear system is possible without knowing the sign of the high-frequency gain (that is, control direction). The first result for first- order linear system was proposed by Nussbaum [41], where the adaptive control uses
1Virtual controls are actually state variables chosen in the backstepping design as controls, for which stabilizing functions are designed to stabilize corresponding subsystems. CHAPTER 2. LITERATURE REVIEW 29 the so-called Nussbaum-type gain, a transcendental function whose sign changes an infinite number of times as its argument tends to infinity. Later this Nussbaum-type gain was adopted on the adaptive control of linear systems with ρ 6 2 [42] and general linear systems with any relative degree [43, 44]. For nonlinear systems with unknown high-frequency gains, some results were first reported for first-order nonlinear systems without parameter estimation [45]. An alternative method called correction vector approach was proposed in [46] and has been extended to design adaptive control of first-order nonlinear systems with a restriction to global Lipschitz nonlinearities [47] and [48]. Then, the class of nonlinear systems studied in [40] is second-order, where the robust control laws provided here can be selected to be continuous in contrast with discontinuous adaptive control scheme formulated in [47, 48]. Later, the Nuss- baum gain technique has been further employed to various adaptive control problems for nonlinear systems in output-feedback form [49,50] or lower triangular form [51,52]. Compared to the Lipschitz condition of nonlinear terms assumed in [49], no growth restrictions are imposed on system nonlinearities in [50–52].
In terms of the output regulation problem for nonlinear systems with unknown high-frequency gain sign, Ding [53] extended the result in [29] to the case where the sign of the control direction is unknown. A breakthrough in tackling the uncertainty in the system model as well as rejecting the disturbances from an unknown exosystem concurrently was reported in [54, 55], where a new set of filters is proposed to tackle the unknown parameters in the exosystem and introduce a new auxiliary error through which the unknown parameters in the system and in the exosystem are decoupled in a way ready for adaptive backesteping design. A Nussbaum gain, which is used to tackle the unknown high-frequency gain sign, combined with the adaptive control to solve the output regulation nonlinear problem with completely unknown parameters. However, the unknown parameters mentioned above were assumed to be linearly parametrized and have naturally incorporated the Nussbaum gain technique into the standard adap- tive control scheme. To overcome the dilemma caused by the nonlinearly parametrized uncertainties and the unknown high-frequency gain sign, a Lyapunov direct method was adopted in [56], and applying this method and the Nussbaum gain technique to solve the global robust output regulation problem for output feedback systems without CHAPTER 2. LITERATURE REVIEW 30 the knowledge of the sign of the high-frequency gain. The introduction of the Lya- punov direct method is because that the approach used to handle the global robust output regulation in [33] was based on the small gain theorem, which cannot incorpo- rate the Nussbaum gain technique to produce an explicit Lyapunov function. Then, the global robust output regulation of lower triangular systems with unknown control direction was presented in [57] and a result on output regulation of lower triangular systems with both unknown linear exosystem and unknown high-frequency gain sign was obtained in [58].
2.2 Consensus Output Regulation of MASs
Consensus control is an another important and basic problem in the research of control theory, which studies the dynamics of multi-agent systems interconnected to each other by a communication topology. The topology represents the allowed information flow between the agents. The objective of consensus control is to devise the control proto- cols for the individual agent that guarantee a group of agents reaches an agreement. These control protocols must be distributed in the sense that each agent is allowed to depend only on information about that agent and its neighbours in the graph [59]. The communication restrictions imposed by graph topologies can severely limit what we can accomplished by local distributed control protocols at each agent. In fact, the graph topological properties complicated the design of consensus controllers and result it intriguing behaviours of multi-agent systems on graphs that do not occur in single- agent, centralized, or decentralized feedback control systems. However, compared to a single complex agent, consensus control can significantly improve the operational effectiveness, reduce the costs and provide additional degrees of redundancy.
2.2.1 Development of Consensus Control
The emerging interest in consensus control control in the last two decades was mainly stimulated by [60–62]. The most significant contribution in the control design was the eigenvalues of a Laplacian matrix based on graph theory. A general framework of the CHAPTER 2. LITERATURE REVIEW 31 consensus problem for networks of integrators was then proposed in [63]. Since then, a large number of works on consensus control have been extensively studied in different directions.
In the following, we briefly introduce some related works on consensus control. In the development of consensus control theory, works were generally done initially for simple systems including first [64] and second-order [65] integrator dynamics. These works showed that the consensus state of multi-agent systems with single integrator dynamics often converges to a constant value, meanwhile, consensus for second-order dynamics might converge to a dynamic final value. However, many practical physical systems cannot be feedback linearised as first or second-order dynamic model in reality. Hence, higher-order dynamic models may be needed. Consensus of networks of high- order integrators was studied in [66,67]. The consensus problem of multi-agent systems with general linear dynamics was then addressed in [68–72]. After that, the results were extended to nonlinear multi-agent systems. The difficulty of solving consensus control problem of nonlinear systems is mainly due to certain restrictions in using the information for individual systems. If a subsystem has all the information of its state, and the local control is allowed, the existing nonlinear control strategies may be applied to linearise the dynamics of each of the subsystems. Consensus control for second-order Lipschitz nonlinear multi-agent systems was investigated in [73,74]. The results shown in [71, 75–77] deal with high-order multi-agent systems with nonlinear dynamics. A common assumption in the previous results was that the dynamics of the agents are identical and precise known, which might not be practical in many circumstances. Due to the existence of the non-identical uncertainties, the consensus control of heterogeneous multi-agent systems was studied in [78,79].
The communication connections between the agents which are characterized by the Laplacian matrix are also playing an important role in the consensus control design, in particular, the second least value of the real parts of the Laplacian matrix eigen- values, which is commonly referred to as the connectivity of the network. The design of the consensus protocols in many existing works [69, 70, 76] requires the knowledge of the nonzero eigenvalue information of the Laplacian matrix. However, the nonzero eigenvalues of this matrix are global information in the sense that each subsystem has CHAPTER 2. LITERATURE REVIEW 32 to know the entire communication graph to compute them, and this kind of control strategy is often referred to as decentralized. Therefore, the consensus protocols given in the aforementioned papers were not completely distributed. Fully distributed adap- tive schemes were proposed in [71, 80], which are based on the observation that the control structure can be parametrized in such a way that a scalar constant (weight) can always be increased in value to achieve consensus control. Similar adaptive schemes were presented to achieve second-order consensus with nonlinear dynamics in [81,82]. Note that the protocols in [71, 80–82] were applicable to only undirected commu- nication graphs or leader-follower graphs where the sub-graphs among the followers are undirected. For a directed connection graph, the adaptive schemes in [83, 84] achieved consensus control at a price of sacrificing the asymptotic convergence. Fur- ther results on adaptive control under directed communication graph include adaptive coordination of multiple Lagrangian systems [85] and consensus tracking for state- feedback nonlinear systems in the strict feedback form [86], which were obtained using direct local measurements or certain parameters that still require the knowledge of the Laplacian matrix. Designing fully distributed adaptive consensus protocols for the case with general directed graphs were remained much more challenging, due to the asymmetry of the corresponding Laplacian matrices. By carefully implementing an integral-type Lyapunov function, a distributed adaptive scheme for linear systems with known parameters was constructed in [87] to achieve leader-follower consensus for any communication graph containing a directed spanning tree with the leader as the root node. An alternative distributed adaptive consensus protocol was presented in [88], which was further modified to be robust in the presence of bounded external disturbances. Moreover, in [80, 81, 87, 88], the subsystem dynamics were assumed to be precisely known. Further works were then reported in [89, 90] for fully distributed adaptive control for linear and nonlinear systems with unknown parameters and with uncertainty in the network connectivity, respectively.
2.2.2 Control Directions in Consensus Output Regulation
The consensus output regulation problem is closely related to the consensus problem as studied in [4] and the references therein. In recent years, many interesting results have CHAPTER 2. LITERATURE REVIEW 33 been reported on consensus output regulation, in particular, several state and output feedback control laws were proposed in [91, 92] and [93] to achieve consensus output regulation for multi-agent systems with heterogeneous but known linear subsystems. the robust consensus output regulation problem of uncertain linear multi-agent systems was studied in [94,95], where internal model based controllers were designed. However, the state observer used in [94] inevitably increases the dimension of the controller. Then, the output feedback control without state estimation was proposed in [96–98]. In [99], consensus global output regulation was discussed for several class of nonlinear multi-agent systems. The controllers given in the aforementioned papers were not fully distributed. The fully distributed consensus adaptive output regulation for a class of nonlinear uncertain multi-agent systems with unknown leader was handled in [100], where combines the adaptive internal model and the robust control to deal with the unknown parameters in the leader systems. To design fully distributed controllers to achieve consensus output regulation for heterogeneous multi-agent systems with general directed graphs is much more complicated and is still open, both the case with nominal and uncertain linear subsystems were first studied in [101]. In terms of the consensus output regulation problem for multi-agent systems with nominal linear subsystems, a distributed adaptive observer, which utilizes the observer states from neighbouring subsystems, was constructed for the subsystems to asymptotically estimate the state of the exosystem. The case with uncertain linear subsystems was further addressed by combining the internal model and distributed adaptive observer. Instead of using distributed adaptive observer, the method in [102] was to parametrise the exosystem in a specific from such that it has a skew-symmetric system matrix, whose property was further explored in the estimation of exosystem state, and in the estimation of the desired feedforward control inputs for output regulation.
In recent years, the consensus output regulation of a class of network-connected dynamics systems whose subsystems have all the system parameters completely un- known, including the high-frequency gains was intensively considered. From section 2.1.2, it is well known that Nussbaum gains can be used to tackle adaptive control with unknown control directions for SISO systems including the case of nonlinear output regulation. However, for a network connected system with multiple subsystems, the CHAPTER 2. LITERATURE REVIEW 34 existing Nussbaum gain designed for individual systems would not be able to establish the boundedness of all the variables in the adaptive consensus control system, as Nuss- baum gain parameters for different subsystems could move in different directions, and a usual contradiction could not be obtained. A result [103] proposed a Nussbaum gain for multi-agent systems with unknown control directions when the lower and upper bounds of the control coefficients are known. The consensus global output regulation problem of second-order nonlinear multi-agent systems subject to the unknown control directions was then presented in [104], the novel distributed controllers based on the Nussbaum-type dynamic gain, the adaptive control techniques can not only handle the unknown control direction but also the uncertain parameter that belongs to any unknown and non-compact set, and the arbitrary unknown control directions do not need to be identical. For the consensus output regulation of a class of general nonlinear systems with unknown high-frequency gains, a new Nussbaum gain with a potentially faster rate such that the boundedness of the system parameters can be established by an argument of contradiction even if the Nussbaum gain parameter for only one of the subsystems goes unbounded was proposed in [105], this remove the assumption of known lower and upper bounds of the control coefficients in [103]. However, the above works were based on undirected graph, the consensus output regulation for nonlinear systems with unknown high-frequency gains under directed graphs is still challenging, we will deal with this problem with a special case in Chapter 6.
2.3 Summary
In this chapter, we review the basic knowledge of the output regulation problem, and the development of this problem on both SISO system and multi-agent systems. In order to establish a general theory for this problem, three important issues are proposed, and the effects of the knowledge of the control directions are discussed. Chapter 3
Preliminary Studies and Problem Descriptions
In this chapter, some mathematical notions and fundamental definitions will be used in the remainder of this thesis are provided, such as fundamental stability concept of nonlinear systems, the centre manifold theory, the internal model principle, the output regulation theory, and the basic algebraic graph theory, etc.
3.1 Fundamentals of Nonlinear Systems
3.1.1 Nonlinear Systems
A general nonlinear dynamics system is described by [8,106]
x˙(t) = f(x(t), t), x(t0) = x0, (3.1)
n n n where x ∈ R , t ∈ [t0, ∞], and f : R × R → R . x is called the state of the system, n x0 ∈ R is the initial state, t0 ∈ R is the initial time and f is a nonlinear vector function. More specifically, the components of x and f are denoted by
T T x = [x1, . . . , xn] , f = [f1, . . . , fn] .
35 CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 36
The system (3.1) is called a nonautonomous system. If the time t does not explicitly appear in the function f, then (3.1) can be simplified as follows:
x˙(t) = f(x(t)), x(t0) = x0, (3.2)
A dynamic system of the form (3.2) is called an autonomous system. For convenience, the system (3.1) and (3.2) can be simplified asx ˙ = f(x, t) andx ˙ = f(x), respectively. A general nonlinear system is described by the following equations:
x˙ = f(x, u),
y = h(x, u), (3.3)
ym = hm(x, u), where x ∈ Rn is the plant state, u ∈ Rm is the plant input, y ∈ Rp is the performance q n m n n m p output, ym ∈ R is the measurement output and f : R ×R → R , h : R ×R → R . For many autonomous nonlinear control systems, the function f(x, u) is linear in the input u, and the function h(x, u) and hm(x, u) do not depend on the input u explicitly. In this case, (3.3) can be further simplified as follows:
x˙ = f(x) + g(x)u,
y = h(x), (3.4)
ym = hm(x), where g : Rn → Rn×m. We call (3.4) an affine nonlinear control system. A general nonlinear control law takes the following form
u = κ1(ν, ym), (3.5) ν˙ = κ2(ν, ym),
s s q m s q where ν ∈ R is the compensator state and κ1 : R × R → R and κ2 : R × R → Rs are two functions. The controller (3.5) is called a dynamic measurement output feedback controller if s > 0 and a static measurement output feedback controller if s = 0, and it is called a state feedback controller if ym = x and an output feedback controller if ym = y. CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 37
3.1.2 Stability Concepts for Nonlinear Systems
In this subsection, the stability concepts for the general autonomous system described by (3.2) are reviewed. The behaviour of the system around the equilibrium point can be used to determine whether the system is stable or not. Without loss of generality, one can assume that the origin of Rn, i.e., x = 0 is an equilibrium point. With respect to this equilibrium point, various stability concepts are defined below.
Definition 3.1 (Lyapunov stability, Definition 4.1 in [107]). For the system (3.2), the equilibrium point x = 0 is said to be Lyapunov stable if for any given positive real number R. there exists a positive real number r to ensure that kx(t)k < R for all t > 0 if kx(0)k < r. Otherwise, the equilibrium is unstable.
Definition 3.2 (Asymptotic stability, Definition 4.2 in [107]). For the system (3.2), the equilibrium point x = 0 is asymptotically stable if it is stable (Lyapunov) and furthermore limt→∞ x(t) = 0.
Definition 3.3 (Exponential stability, Definition 4.3 in [107]). For the system (3.2), the equilibrium point x = 0 is exponential stable if there exist two positive real numbers α and λ such that the following inequality holds:
kx(t)k < αkx(0)ke−λt, for t > 0 in some neighbourhood D1 ⊂ Rn containing the equilibrium point.
Definition 3.4 (Globally asymptotic stability, Definition 4.4 in [107]). If the asymp- totic stability defined in Definition 3.2 holds for any initial state in Rn, the equilibrium point is said to be globally asymptotically stable.
Definition 3.5 (Globally exponential stability, Definition 4.5 in [107]). If the exponen- tial stability defined in Definition 3.3 holds for any initial state in Rn, the equilibrium point is said to be globally exponentially stable.
Definition 3.6 (Positive definite function, Definition 4.6 in [107]). A function V (x) ∈
D ⊂ Rn is said to be locally positive definite if V (x) > 0 for x ∈ D except at x = 0 1Here D is used to denote a domain around the equilibrium point x = 0. This domain can be interpreted as a set with 0 as its interior point, or it can also be simplified as D = {x|kxk < r} for some positive r. CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 38 where V (x) = 0. If D = Rn, i.e., the above property holds for the entire state space, V (x) is said to be globally positive definite.
Definition 3.7 (Lyapunov function, Definition 4.7 in [107]). If in D ⊂ Rn containing the equilibrium point x = 0, the function V (x) is positive definite and has continuous partial derivatives, and if its time derivative along any state trajectory of system (3.2) is non-positive, i.e., V˙ (x) ≤ 0, then V (x) is a local Lyapunov function. If D = Rn, then V (x) is a said to be a global Lyapunov function for system (3.2).
Theorem 3.1 (Lyapunov theorem for local stability, Theorem 4.2 in [107]). Consider the system (3.2), if in D ⊂ Rn containing the equilibrium point x = 0, there exist a function V (x): D ⊂ Rn → R with continuous first-order derivatives such that
• V (x) is positive definite in D
• V˙ (x) is non-positive definite in D then the equilibrium point x = 0 is locally Lyapunov stable. Furthermore, if V˙ (x) is negative definite, then the equilibrium point x = 0 is locally asymptotically stable.
Definition 3.8 (Radially unbounded function, Definition 4.8 in [107]). A positive definite function V (x): Rn → R is said to be radially unbounded if V (x) → ∞ as kxk → ∞.
Theorem 3.2 (Lyapunov theorem for global stability, Theorem 4.3 in [107]). For the system (3.2) with D = Rn, if there exists a function V (x): Rn → R with continuous first order derivatives such that
• V (x) is positive definite
• V˙ (x) is negative definite
• V (x) is radially unbounded then the equilibrium point x = 0 is globally asymptotically stable. CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 39
3.1.3 Tools for Control Design
In this subsection, some tools for control design, including Barbalat’s lemma, Young’s inequality, etc., will be introduced.
Lemma 3.1 (Barbalat’s lemma, Lemma 7.1 in [107]). If a function f(t): R → R is R ∞ uniformly continuous for t ∈ [0, ∞) and 0 f(t)dt exists, then
lim f(t) = 0. t→∞
Lemma 3.2 (Young’s inequality [108]). Let p, q be real numbers greater than 1 such that 1 1 + = 1, p q then for any a, b ≥ 0: ap bq ab + . 6 p q The equality holds if and only if ap = bq.
Definition 3.9 (Lie derivative [107]). For any smooth function f : D ⊂ Rn → Rn n and a smooth function h : D ⊂ R → R, the Lie derivative Lf h, referred to as the derivative of h along f, is defined by ∂h(x) L h = f(x). f ∂x This notation can be used iteratively, that is
2 Lf (Lf h(x)) = Lf h(x),
k k−1 Lf (h(x)) = Lf (Lf h(x)). Definition 3.10 (Relative degree [107]). The dynamic system (3.4) has relative degree ρ at a point x if the following conditions are satisfied:
k LgLf h(x) = 0, for k = 0, . . . , ρ − 2,
ρ−1 LgLf h(x) 6= 0.
k Suppose that the relative degree for (3.4) is ρ, which implies that LgLf h(x) = 0 for k = 0, . . . , ρ − 2. Therefore, we have the derivatives of y expressed by
(k) k y = Lf h(x), for k = 0, . . . , ρ − 1,
(ρ) ρ ρ−1 y = Lf h(x) + LgLf h(x)u. CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 40
A partial state transformation is selected as
i−1 ξi = Lf h(x), for i = 1, . . . , ρ.
n−ρ For convenience, let z = [xρ+1, . . . , xn] ∈ R withz ˙ = f0(z, ξ). Then the system for ρ < n can be transformed under certain conditions to the normal form ˙ ξ1 = ξ2, . . ˙ ξρ−1 = ξρ, (3.6) ˙ ρ ρ−1 ξρ = Lf h(x) + LgLf h(x)u,
z˙ = f0(z, ξ). Definition 3.11 (Zero dynamics [107]). The last equation of system (3.6) is named as the system internal dynamics. If there exists a desired input with which output is imposed to be zero, the internal dynamics are then said to be zero dynamics. For example, for system (3.4), its zero dynamics arez ˙ = f0(z, 0).
3.2 Centre Manifold Theory
In this section, we present a few results from the centre manifold theory for the au- tonomous system (3.2) with the assumption that f(·) is a locally defined sufficiently smooth function vanishing at the origin; that is, f(·) is a Ck function for some suf-
ficiently large integer k defined in an open neighbourhood of the origin of Rn and f(0) = 0. The main references in this section are [8,109–111].
Definition 3.12. Let X be an open set of Rn. A set of the form
M = {x ∈ X | H(x) = 0}, (3.7)
n n1 ∂H(x) where H : R → R is a sufficiently smooth function and rank ∂x = n1 for all n x ∈ M is called an (n − n1)-dimensional hypersurface in R .
A hypersurface is a special type of a manifold in Rn. A set M as described in (3.7) is called a (locally) invariant manifold of (3.2) if the solution of (3.2) starting from x0 ∈ M remains in M for sufficiently small t > 0. CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 41
T Definition 3.13 (Jacobian matrix). Given a function f = [f1, . . . , fn] , the Jacobian matrix is defined as ∂f1 ∂f1 ∂x ... ∂x 1 n . .. . J = . . . . ∂fn ... ∂fn ∂x1 ∂xn
Now consider the nonlinear system (3.2), and let F ∈ Rn×n be the Jacobian matrix of f(x) at the origin. Assume F has 0 < n1 < n eigenvalues with nonzero real parts and n2 = n − n1 eigenvalues with zero real parts. Then there exists a nonsingular matrix T such that, in the new coordinates col(y, z) = T x where y ∈ Rn1 and z ∈ Rn2 , (3.2) can be written as follows:
y˙ = f1(y, z), z˙ = f2(y, z), (3.8) with ∂f1 ∂(y,z) (0, 0) AB = , ∂f2 ∂(y,z) (0, 0) 0 A1 where all the eigenvalues ofA have nonzero real parts and all the eigenvalues of A1 have zero real parts.
Theorem 3.3 (Theorem 2.25 in [8]). Consider the system (3.8). There exist an open neighbourhood Z ∈ Rn2 of z = 0 and a Ck−1 function y : Z → Rn1 with y(0) = 0, such that, for all z ∈ Z, ∂y(z) f (y(z), z) = f (y(z), z). (3.9) ∂z 2 1
Let
n M = {(y, z) ∈ R 1 × Z | y = y(z)}.
By Definition 3.12, M is an n2-dimensional invariant manifold for (3.8) passing through the origin.
3.3 The Output Regulation Theory
The system stabilisation and output tracking are treated as output regulation problem in control systems. The output regulation theory of nonlinear systems is generalised CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 42 in this section. Consider a nonlinear control system described by
x˙ = f(x, u, v, w),
e = he(x, u, v, w), (3.10)
ym = hm(x, u, v, w),
n m p where x ∈ R is the state with x(0) = x0, u ∈ R is the input, e ∈ R is the perfor-
q l1 mance output, ym ∈ R is the measurement output, v ∈ R represents disturbances or/and reference inputs, and w ∈ Rl2 represents an unknown constant vector whose nominal value is zero. It is assumed that v is generated by an autonomous differential equation
v˙ = a(v, σ), v(0) = v0, (3.11) where σ ∈ Rl3 is an unknown constant parameter vector. In what follows, we call the system (3.11) an exosystem. We will consider a control law of the following form:
u = κ1(ν, ym), (3.12) ν˙ = κ2(ν, ym), for some functions κ1 and κ2, and ν is the compensator state of dimension nν to be specified later. For simplicity, we assume all the functions in (3.10),(3.11), (3.12) are globally defined, vanishing at the respective origins, sufficiently smooth, and satisfy, for
l2 l3 all w ∈ R , f(0, 0, 0, w) = 0, he(0, 0, 0, w) = 0, hm(0, 0, 0, w) = 0, and for all σ ∈ R , a(0, σ) = 0. To guarantee the implementability of the controller, we assume that the function hm does not depend on u explicitly, i.e., ym = hm(x, v, w). Let xc = col(x, ν). Then, the closed-loop system composed of the plant (3.10), the controller (3.12), and the exosystem (3.11) is described as follows:
x˙ c = fc(xc, v, w), xc(0) = xc0,
e = hc(xc, v, w), (3.13) v˙ = a(v, σ), where f(x, κ1(ν, hm(x, v, w)), v, w) fc(xc, v, w) = , κ2(ν, hm(x, v, w))
hc(xc, v, w) = he(x, κ1(ν, hm(x, v, w)), v, w). CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 43
In terms of the closed-loop system, we can describe the problem as follows.
Global Robust Output Regulation Problem (GRORP): For any compact set
V ∈ Rl1 with a known bound, any compact set W ∈ Rl2 with a known bound, and any compact set S ∈ Rl3 with a know bound, find a controller of the form (3.12), such that the closed-loop system (3.13) has the following two properties.
Property 3.1. For all v ∈ V, w ∈ W and σ ∈ S, the trajectory of the closed-loop system (3.13) starting from any initial states xc(0) exists and is bounded for all t > 0.
Property 3.2. The trajectory described above satisfies limt→∞ e(t) = 0.
To have the problem well posed, the following assumptions are needed.
Assumption 3.1. There exists a class K function2 γ such that kv(t)k ≤ γ(kv(0)k),
∀t ≥ 0, for all σ ∈ S and all v(0) ∈ Rl1 .
A special case of Assumption 3.1 is as follows.
Assumption 3.2. The exosystem is of the form
v˙ = A1(σ)v, v(0) = v0, (3.14)
and, for any σ ∈ S, all eigenvalues of A1(σ) are simple with zero real parts.
Remark 3.1. Property 3.1 is guaranteed if the equilibrium point of the closed-loop system (3.13) at col(xc, v) = (0, 0) is stable in the sense of Lyapunov. Moreover, by Theorem 2.27 in [8] and Assumption 3.1 or Assumption 3.2, the equilibrium point of the closed-loop system (3.13) at col(xc, v) = (0, 0) is stable in the sense of Lyapunov if the closed-loop system has the following property.
Property 3.3. All the eigenvalues of the matrix
∂fc(0, 0, 0)
∂xc have negative real parts.
2A continuous function γ : [0, a) 7→ [0, ∞) is said to belong to class K if it strictly increasing and satisfies γ(0) = 0. CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 44
Lemma 3.3. Under Assumption 3.1 or Assumption 3.2, suppose the closed-loop sys- tem (3.13) has property 3.3. Then, it also has Property 3.2 if and if only there exists a sufficiently smooth function xc(v, w, σ) with xc(0, 0, 0) that satisfies, for all v ∈ V, w ∈ W and σ ∈ S, the following partial differential equations:
∂xc(v, w, σ) a(v, σ) = fc(xc(v, w, σ), v, w) ∂v (3.15)
0 = hc(xc(v, w, σ), v, w).
The first equation of (3.15) can be easily proved by Theorem 3.3, which means
(n+nν ) l1 that there exists a centre manifold M = {(xc, v) ∈ R × R | xc = xc(v, w, σ)} for the closed-loop system (3.13).
Based on Lemma 3.3, we will establish the solvability of the state feedback output regulation problem in terms of the given plant. Define a so-called composite system as follows: x˙ = f(x, u, v, w),
v˙ = a(v, σ), (3.16)
e = he(x, u, v, w).
Assumption 3.3. There exist sufficiently smooth functions x(v, w, σ) and u(v, w, σ) with x(0, 0, 0) = 0 and u(0, 0, 0) = 0 for all v ∈ V, w ∈ W and σ ∈ S that satisfy the following equations:
∂x(v, w, σ) a(v, σ) = f(x(v, w, σ), u(v, w, σ), v, w), ∂v (3.17)
0 = he(x(v, w, σ), u(v, w, σ), v, w).
Letx ¯ = x−x(v, w, σ) andu ¯ = u = u(v, w, σ). Then (3.17) implies thatx ¯ satisfies
x¯˙ = f¯(¯x, u,¯ v, w, σ), (3.18) ¯ e = he(¯x, u,¯ v, w, σ), where f¯(¯x, u,¯ v, w, σ) = f(¯x + x(v, w, σ), u¯ + u(v, w, σ), v, w)
− f(x(v, w, σ), u(v, w, σ), v, w), ¯ he(¯x, u,¯ v, w, σ) = he(¯x + x(v, w, σ), u¯ + u(v, w, σ), v, w). CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 45
¯ ¯ It can be verified that f(0, 0, v, w, σ) = 0 and he(0, 0, v, w, σ) = 0 for all v, w, σ. We call the system (3.18) the error system. Under Assumption 3.1, if there exists a control law of the formu ¯ = k(¯x) vanishing at the origin such thatx ¯ = 0 is globally asymptotically stable for any v ∈ V, w ∈ W and σ ∈ S, then, for anyx ¯(0),
lim x¯(t) = lim (x(t) − x(v, w, σ)) = 0, (3.19) t→∞ t→∞ and lim u¯(t) = lim (u(t) − u(v, w, σ)) = lim k(¯x(t)) = k(0) = 0. (3.20) t→∞ t→∞ t→∞ As a result,
¯ ¯ lim e(t) = lim he(¯x(t), u¯(t), v(t), w, σ) = he(0, 0, v(t), w, σ) = 0. (3.21) t→∞ t→∞
That is, the control law is defined as
u = u(v, w, σ) + k(x − x(v, w, σ)). (3.22)
Under the control law (3.22), the closed-loop system can be put in the form (3.13) with xc(v, w, σ) = x(v, w, σ), and
fc(xc(v, w, σ), v, w) = f(x(v, w, σ), u(v, w, σ), v, w) ∂x(v, w, σ) ∂x (v, w, σ) = a(v, σ) = c a(v, σ), ∂v ∂v
hc(xc(v, w, σ), v, w) = he(x(v, w, σ), u(v, w, σ), v, w) = 0, which is the same as (3.15). By Lemma 3.3, the controller (3.22) solves the nonlinear output regulation problem globally.
3.4 Steady-State Generator
From above section, the solution of the regulator equations (3.17) provides the neces- sary feedforward information for designing a control law to convert the output regula- tion problem of the given plant (3.16) to the stabilization problem of the well defined error system (3.18). A control law of the form (3.22) is called feedforward control law. It can be seen that the feedfoward control law works only if neither the given CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 46 plant nor the exosystem contains unknown parameter. However, this control law can- not be used if the unknown parameters w or/and σ appear. Therefore, this idea can be implemented only if we can reproduce the solution of the regulation equations by some dynamic compensator that does not rely on the unknown parameters. This idea motivates the so-called internal model design approach. Roughly, an internal model is a dynamic compensator independent of the uncertain parameters w and σ and can asymptotically reproduce the solution of the regulator equations. The composition of the internal model and the given plant constitutes a so-called augmented system. The internal model will be conceived such that the augmented system is stabilizable and the control law that stabilizes the augmented system together with the internal model will be the overall control law that solves the output regulation problem of the given plant. The concept of the steady-state generator is needed to introduce in order to ascertain the existence of an internal model.
Definition 3.14. Let c : Rn+m 7→ Rr be a mapping for some positive integer m ≤ r ≤ n + m. Under Assumption 3.3, the composite system (3.16) is said to have a steady- state generator with output c(x, u) ∈ Rr if there exists a triplet {θ, φ, ψ} where, for some integer %, θ : Rl1+l2+l3 7→ R%, φ : R%+l1+l3 7→ R%, and ψ : R% 7→ Rr are sufficiently smooth functions vanishing at the origin, such that, for any trajectory v ∈ V of the exosystem, w ∈ W and σ ∈ S,
θ˙(v, w, σ) = φ(θ(v, w, σ), v, σ),
c(v, w, σ) = ψ(θ(v, w, σ)), (3.23)
c(v, w, σ) : = c(x(v, w, σ), u(v, w, σ)).
Remark 3.2. Define an autonomous system as follows:
v˙ = a(v, σ), (3.24) y = c(v, w, σ).
Then, the composite system admits a steady-state generator with output c(x, u) ∈ Rr means that, for any σ ∈ Rl3 , the function can be reproduced by the following system:
θ˙ = φ(θ, v, σ), (3.25) y = ψ(θ), CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 47 which is independent of the unknown parameter w. If the exosystem does not contain the unknown parameter σ, then the system (3.25) is also independent of σ. The trans- formation from (3.24) to (3.25) is called system immersion [112]. If c(x, u) =col(x, u), then the solution of the regulator equations can be reproduced by the system (3.25). In
T many cases, assume that c(x, u) = [xi1 , xi2 , . . . , xin¯ , u] where 1 ≤ i1 ≤ i2 ≤ · · · ≤ in¯ ≤ n for some integern ¯ satisfying 0 ≤ n¯ ≤ n. With c(x, u) thus defined, a steady-state generator can be viewed as a dynamic system which reproduces the partial (¯n ≤ n) or whole (¯n = n) solution of the regulator equations. In the special case wheren ¯ = 0, only the control part of the solution of the regulator equations can be reproduced. Thus, (3.23) is called a partial (full) steady-state state generator if 0 < n¯ < n (¯n = n) or a steady-state input generator ifn ¯ = 0.
In what follows, various conditions under which the steady-state generator of (3.16) exists will be studied. For convenience, we focus on the case where r = 1, i.e., c(x, u) is a scalar function.
3.4.1 Linear Immersion Assumption
In this subsection, a simple case where the exosystem with the output c(v, w, σ) can be immersed to a linear system. For the purpose, let
∂c(v, w, σ) Lac(v, w, σ) : = a(v, σ), ∂v (3.26) ∂Li−1c(v, w, σ) Li c(v, w, σ) : = a a(v, σ), i = 2, 3,.... a ∂v
Assumption 3.4. There exist a positive integer χ and real scalars φi(σ), i = 1, . . . , χ, such that χ La c(v, w, σ) − φ1(σ)c(v, w, σ) − φ2(σ)Lac(v, w, σ) − · · · (3.27) χ−1 − φχ(σ)La c(v, w, σ) = 0, ∀v ∈ V, w ∈ W, σ ∈ S.
Lemma 3.4. Under Assumption 3.4, the composite system (3.16) admits a linear steady-state generator with output c(x, u):
θ˙(v, w, σ) = Φ(σ)θ(v, w, σ), (3.28) c(v, w, σ) = Ψθ(v, w, σ), CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 48
χ−1 where θ(v, w, σ) =col(c(v, w, σ),Lac(v, w, σ),...,La c(v, w, σ)), and T 0 1 0 ... 0 1 0 0 1 ... 0 0 ...... . Φ(σ) = . . . . . ,Ψ = . . (3.29) 0 0 0 ... 1 0 φ1(σ) φ2(σ) φ3(σ) . . . φχ(σ) 0
Moreover, the pair (Ψ,Φ(σ)) is observable.
Remark 3.3. Assumption 3.4 always hold if c(v, w, σ) is polynomial in v and As-
0 sumption 3.2 is satisfied. If the matrix A1 in (3.14) is known exactly, i.e., σ ∈ R , then the coefficients φi, i = 1, . . . , χ, are independent of σ. Hence, the steady-state generator (3.28) is also independent of σ.
3.4.2 Nonlinear Immersion Assumption
The validity of Lemma 3.4 relies on two assumptions, i.e., the function c(v, w, σ) is a polynomial in v, and the exosystem is linear. In this section, relaxing the first assumption is considered and the exosystem is linear and is known exactly.
Assumption 3.5. The exosystem is of the form
v˙ = A1v, v(0) = v0, (3.30) and all eigenvalues of A1 are simple with zero real parts.
With the operator La defined in (3.26), let
i i−1 Laπ(v, w) := col π(v, w),Laπ(v, w),...,La π(v, w) , i = 1, 2,..., for any sufficiently smooth scalar function π(v, w).
Assumption 3.6. There exist polynomials π1(v, w), . . . , πp(v, w) in v for some integer p > 0, and a sufficiently smooth function ψ vanishing at the origin such that
χ1 1 χp p c(v, w, σ) = ψ(La π (v, w),..., La π (v, w)), ∀v ∈ V, w ∈ W,
i where, χi, for i = 1, . . . , p, are the degree of the minimal zero polynomials of π (v, w). CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 49
Lemma 3.5. Under Assumption 3.5, if c(v, w, σ) satisfies Assumption 3.6, then the composite system (3.16) admits a steady-state generator of the following form:
θ˙(v, w) = Φθ(v, w), (3.31) c(v, w) = ψ(θ(v, w)), where θ = col(θ1, . . . , θp),Φ = diag(Φ1,...,Φp),
i χi i i χi×χi i with θ (v, w) = La π (v, w), Φ ∈ R being the companion matrix of π (v, w). Let Ψ = [Ψ 1,...,Ψ p] with Ψ i ∈ R1×χi being the Jacobian matrix of ψ at the origin. The steady-state generator (3.31) is linearly observable if the minimal zeroing polynomials of πi(v, w)’s are pairwise coprime [113], the pairs (Ψ i,Φi), i = 1, . . . , p, are observable.
3.5 The Internal Model Principle
The internal model principle was developed in the 1970s for solving the output regula- tion problem for uncertain linear systems and is one of the most remarkable methods for linear control system design. For the linear systems, an internal model is a dy- namic compensator determined by the exosystem. The internal model together with the given plant forms so-called augmented system. The internal model leads to a sta- bilizable augmented system and the stabilization solution of the augmented system leads to the solution of the robust output regulation problem of the given plant. For the nonlinear systems, the globally stabilizability problem is untractable. Therefore, the concept of internal model candidate, which is such that the stabilizibility of the augment system implies the solvability of the robust output regulation of the given plant, is introduced. It will be seen shortly that a steady-state generator itself is an internal model candidate, however, this internal model candidate does not lead to a stabilizable augmented system. If, for a class of nonlinear systems, an internal model candidate leads to a stabilizable augmented system, then this internal model candidate is further called an internal model for this class of nonlinear systems. First, an internal model candidate corresponding to an existing steady-state gen- erator is defined. CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 50
Definition 3.15. Under Assumption 3.3, suppose the composite system (3.16) admits a steady-state generator (3.23). Let γ : R%+r+l1 7→ R% be a sufficiently smooth function vanishing at the origin. Then the following system
η˙ = γ(η, c(x, u), v) (3.32) is called internal model candidate corresponding to (3.23) on a mapping τ if there exists a diffeomorphic mapping τ(θ(v, w, σ), σ): R%+l3 7→ R% such that ∂τ(θ(v, w, σ), σ) φ(θ(v, w, σ), v, σ) = γ(τ(θ(v, w, σ), σ), c(v, w, σ), v) (3.33) ∂θ for all v ∈ V, w ∈ W, σ ∈ S.
Remark 3.4. Under Assumption 3.3, suppose the composite system (3.16) admits a steady-state generator (3.23) and the corresponding internal model candidate (3.32) on a mapping τ(θ(v, w, σ), σ). Define a new vector function
θ0 = τ(θ(v, w, σ), σ).
Then, the system has an alternative steady-state generator
θ˙0(v, w, σ) = φ0(θ0(v, w, σ), v, σ), (3.34) c(v, w, σ) = ψ0(θ0(v, w, σ)), where 0 0 ∂τ(θ, σ) φ (θ , v, σ) = φ(θ, v, σ)|θ=τ −1(θ0,σ), ∂θ (3.35) ψ0(θ0) = ψ(τ −1(θ0, σ)). Here, τ −1(·, σ) is the inverse mapping of τ(·, σ), i.e., τ −1(τ(θ, σ), σ) = θ. As a result, the equation (3.33) implies
φ0(θ0(v, w, σ), v, σ) = γ(θ0(v, w, σ), c(v, w, σ), v).
Therefore, (3.32) is also an internal model candidate corresponding to the steady-state generator (3.34) on an identity mapping. In particular, if τ(θ, σ) = T (σ)θ for some nonsingular matrix T (σ), then the steady-state generator (3.34) is defined by
φ0(θ0(v, ω, σ), v, σ) = T (σ)φ(T −1(σ)θ0, v, σ),
ψ0(θ0(v, ω, σ)) = ψ(T −1(σ)θ0). CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 51
If the composite system (3.16) admits a steady-state generator with output c(x, u) and σ ∈ R0, then the steady-state generator itself is an internal model candidate on the identity mapping τ upon defining γ(η, c(x, u), v) = φ(η, v), however, this particular internal model candidate is uncoupled with the given plant, and cannot lead to a stabilizable augmented system. That is why a more general characterization of the internal model candidate is needed in Definition 3.15 which allows the internal model to be coupled to the given plant through the function c(x, u).
Some classes of internal model candidates corresponding to the steady-state gen- erators (3.28) and (3.31) are constructed as follows under various assumptions.
Example 3.1. Under Assumption 3.4, the composite system (3.16) admits a steady- state generator of the form (3.28) with the pair (Ψ,Φ(σ)) is observable. By Remark
3.3, if all the eigenvalues of the matrix A1(σ) in Assumption 3.2 have zero real part, so do the eigenvalues of the matrix Φ(σ). From Appendix A of [8], for any controllable pair (M,N) where M ∈ Rχ×χ with M Hurwitz and N ∈ R1×χ, the Sylvester equation
T (σ)Φ(σ) − MT (σ) = NΨ (3.36) has a unique nonsingular solution T (σ). Multiplying θ(v, w, σ) on each side of (3.36),
T (σ)Φ(σ)θ(v, w, σ) − MT (σ)θ(v, w, σ) = NΨθ(v, w, σ) = Nc(v, w, σ).
Then, it can be verified that the following system
η˙ = Mη + Nc(x, u) (3.37) is an internal model candidate on a mapping τ(θ, σ) = T (σ)θ(v, w, σ) corresponding to (3.28).
Example 3.2. Consider the steady-state generator of the form (3.31) and assume the pair (Ψ,Φ) is observable. Like the previous example, for any controllable pair (M,N) where M ∈ Rχ×χ with M Hurwitz and N ∈ R1×χ, the Sylvester equation
TΦ − MT = NΨ (3.38) has a unique solution T . Multiplying θ(v, w) on each side of (3.38),
T Φθ(v, w) − MT θ(v, w) = NΨθ(v, w). CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 52
Then, the system η˙ = Mη + N[c(x, u) − ψ(T −1η) + ΨT −1η] (3.39) is an internal model candidate on a mapping τ(θ, σ) = T θ(v, w) corresponding to (3.31) since
T Φθ(v, w) = MT θ(v, w) + N[c(v, w) − ψ(θ(v, w)) + Ψθ(v, w)]
= MT θ(v, w) + NΨθ(v, w).
In the above examples, various steady-state generators and the corresponding internal model candidates are constructed assuming c(x, u) ∈ R. The examples are summarized in Table 3.1 . In fact, all these results can be used to handle the more general case with c(x, u) ∈ Rr, r > 1. In particular, the steady-state generator and the corresponding internal model candidate can be constructed separately for each ci(x, u).
Exosystem Steady-state generator Internal model Mapping Assumption 3.4 (Linear immersion) θ˙(v, w, σ) = Φ(σ)θ(v, w, σ) τ(θ, σ) = T (σ)θ v˙ = c(v, w, σ) = Ψθ(v, w, σ) η˙ = Mη + Nc(x, u) T (σ)Φ(σ) A (σ)v 1 −MT (σ) = NΨ Assumption 3.6 (Nonlinear immersion) θ˙(v, w) = Φθ(v, w) η˙ = Mη + Nc(x, u) τ(θ, σ) = T θ v˙ = A v 1 c(v, w) = ψ(θ(v, w)) −N(ψ(T −1η) − ΨT −1η) TΦ − MT = NΨ
Table 3.1: Steady-state generators and internal models under different assumptions.
3.6 Nussbaum Gain Method
In this section, the following equations are considered [41]:
2 x˙ = ax + λx(y + 1)h(y), x(0) = x0, (3.40) 2 y˙ = x(y + 1), y(0) = y0.
Assume that h : R → R satisfies the following conditions:
(H) Then map h : R → R is even and differentiable. For every y0 ∈ R Z u sup h(s)ds = +∞, u>y0 y0 Z u inf h(s)ds = −∞. u>y0 y0 CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 53
Theorem 3.4. Assume that h(y) in equation (3.40) satisfies condition (H). Then for all real numbers a, λ, x0 and y0 such that λ 6= 0, the solutions x(t) and y(t) of equation (3.40) are defined for all t ≥ 0, lim x(t) = 0, t→∞
2 2 and limt→∞ y(t) exists and is finite, and h(y) can be set as y cos(y), y sin(y), and etc.
3.7 Graph Theory Basics
In this section, we recall some knowledge relates to graph theory, which is essential in the study of consensus control of multi-agent dynamic systems. The material in this section is from [4,114]. A team of agents interacts with each other via communication or sensing networks to achieve collective objectives. It is convenient to model the information exchanges among agents by directed or undirected graphs. A directed graph is a pair G = (V, E) with V = {v1, ··· , vn} being a set of nodes or vertices and E ∈ V × V a set of edges or arcs. A vertex represents an agent, and each edge represents a connection. A weighted graph associates a weight with every edge in the graph. We assume the graph is sim- ple, i.e., (vi, vi) ∈/ E, ∀i no self-loops, and no multiple edges between the same pairs of nodes. The edge (vi, vj) in the edge set E denotes that agent vj can obtain information from agent vi, but not necessarily vice versa. The edge (vi, vj) is said to be outgoing with respect to node vi and incoming with respect to vj; and vi is called the parent node, vj is the child node, and vi is a neighbour of vj. The set of neighbours of node vi is denoted as Ni, whose cardinality is called the in-degree of node vi. A graph is defined as being balanced when it has the same number of ingoing and outgoing edges for all the nodes. A graph with the property that (vi, vj) ∈ E implies
(vj, vi) ∈ E for any vi, vj ∈ V is said to be undirected, where the edge (vi, vj) denotes that agents vi and vj can obtain information from each other. Clearly, an undirected graph is a special balanced graph.
A directed graph is a sequence of nodes v0, v1, . . . , vr such that (vi, vi+1) ∈ E, i ∈
{0, 1, . . . , r − 1}. Node vi is said to be connected to node vj if there is a directed path CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 54
from vi to vj. A directed graph is strongly connected if vi, vj are connected for all dis- tinct nodes vi, vj ∈ V, that is, there is a directed path from every node to every other node. For undirected graphs, if there is a directed path from vi to vj, then there is a directed path from vj to vi, and the qualifier ’strongly’ is omitted. A (rooted) directed tree is a directed graph in which every node has exactly one parent except for one node, called the root, which has no parent and has directed paths to all other nodes. A directed tree is defined as spanning when it connects all the nodes in the graph. It can be demonstrated that this implies that there is at least one root node connected with a simple path to all the other nodes. A graph is said to have or contain a directed spanning tree if a subset of the edges forms a directed spanning tree. This is equivalent to saying that the graph has at least one node with directed paths to all other nodes. For undirected graphs, the existence of a directed spanning tree is equivalent to being connected. However, in directed graphs, the existence of a directed spanning tree is a weaker condition than being strongly connected. A strongly connected graph contains at least one directed spanning tree.
Given the edge weights aij, a graph can be represented by an adjacency ir con- nectivity matrix A = [aij], with weights (vj, vi) ∈ E and aij = 0 otherwise. Note that Pn aii = 0. The weighted in-degree and out-degree of i are defined as din(i) = j=1 aij Pn N×N and dout(i) = j=1 aji, respectively. The Laplacian matrix L = [lij] ∈ R is defined PN by lii = j=1 aij and lij = −aij when i 6= j. From the definition of the Laplacian matrix, it is easy to see that L is diagonally dominant and has non-negative diagonal entries. Since L has zero row sums, one has L1c = 0, with 1 = [1,..., 1]T ∈ RN being the vector of ones and c any constant. λ1 = 0 is an eigenvalue with a right eigenvector of 1c. That is, 1c ∈ N(L) the kernel of L. If the dimension of the kernel of L is equal to one, i.e., the rank of L is N − 1, then λ1 = 0 is non-repeated and 1c is the only vector in N(L). According to Gershgorin disc theorem [115], all nonzero eigenvalues of L are located within a disk in the complex plane centred at dmax and having radius of dmax, where dmax denotes the maximum in-degree of all nodes.
Lemma 3.6 ( [4,64,116]). The Laplacian matrix L of a directed graph G has at least zero eigenvalue with 1 as a corresponding right eigenvector and all nonzero eigenvalues have positive real parts. Furthermore, zero is a simple eigenvalue of L if and only if G has a directed spanning tree. In addition, there exists a non-negative left eigenvector CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 55
T T r = [r1, . . . , rN ] of L associated with the zero eigenvalue, satisfying r L = 0 and rT 1 = 1. Moreover, r is unique if G has a directed spanning tree. Suppose the graph
T is strongly connected, there exists RL + L R ≥ 0, where R = diag(r1, . . . , rN ).
Lemma 3.7 ( [76]). For a Laplacian matrix L that zero is a simple eigenvalue, there exists a similarity transformation T , with its first column being T1 = 1, such that
T −1LT = J, (3.41) with J being a block diagonal matrix in the real Jordan form 0 J1 .. . J = Jp , (3.42) Jp+1 .. . Jq
nk where Jk ∈ R , k = 1, 2, . . . , p are the Jordan blocks for real eigenvalues λk > 0 with the multiplicity nk in the form λk 1 λk 1 .. .. Jk = . . , λk 1 λk
2nk and Jk ∈ R , k = p + 1, p + 2, . . . , q, are the Jordan blocks for conjugate eigenvalues
αk ± jβk, αk > 0 and βk > 0, with the multiplicity nk in the form v(αk, βk) I2 v(αk, βk) I2 .. .. Jk = . . , v(αk, βk) I2 v(αk, βk)
2×2 with I2 being the identity matrix in R and αi βi 2×2 v(αk, βk) = ∈ R . −βk αi CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 56 3.8 State Feedback Consensus Protocols
In this section, consensus controllers using state variable feedback are reviewed, as- suming that the full states of each agent are available for feedback control design by itself and its neighbours in the communication graph. We require that any control protocols be distributed in the sense that the control for agent i depends only on its own information and that from its neighbours in the graph. It is shown that locally optimal design in terms of a local Riccati equation [117], along with selection of a suitable distributed control protocols, guarantees consensus on arbitrary communica- tion graphs that have a spanning tree. Before the introduction of the problem, the definition of Kronecker product is provided.
Definition 3.16. The Kronecker product of matrix A ∈ Rm×n and B ∈ Rp×q is defined as a11B . . . a1nB . . . A ⊗ B = . . . , am1B . . . amnB and they have following properties:
1.( A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD),
2.( A ⊗ B) + (A ⊗ C) = A ⊗ (B + C),
3.( A ⊗ B)−1 = A−1 ⊗ B−1,
4.( A + B) ⊗ C = (A ⊗ C) + (B ⊗ C),
5.( A ⊗ B)T = AT ⊗ BT ,
6. If A ∈ Rm×n and B ∈ Rp×q are both positive definite (positive semi-define), so is A ⊗ B, where the matrices are assumed to be compatible for multiplication.
Consider a network of N identical agents with general continuous time linear dynamics, which may also be regarded as the linearised model of some nonlinear CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 57 systems. The dynamics of i-th agent are described by
x˙ i = Axi + Bui, (3.43) yi = Cxi,
n p q for i = 1,...,N, where xi ∈ R is the state vector, ui ∈ R is the input, and yi ∈ R is the measured output. The objective of the consensus problem is to design distributed control laws using only local information to ensure that N agents described in (3.43) achieve consensus in the sense of limt→∞ kxi(t) − xj(t)k = 0, ∀i, j = 1,...,N. Intuitively, if the communication graph G can be decomposed into some discon- nected components, consensus is impossible to reach. Thus, the following assumption on graph topology is needed.
Assumption 3.7. The communication graph G contains a directed spanning tree.
If (A, B) is controllable, the feedback control law for each subsystem can be designed as ui = −Kxi such that A − BK is Hurwitz. For consensus control, one possible control law is N X ui = cK aij(xi − xj), (3.44) j=1 for i = 1,...,N, where c > 0 ∈ R is called the constant coupling gain or weight, p×n K ∈ R is the feedback gain matrix, and aij is the (i, j)-th entry of the adjacency matrix associated with G. Both c and K in (3.44) are to be determined. A distinct feature of the consensus protocols (3.44) is that it includes a positive scalar c, which can be regarded as a scaling factor or a uniform weight on the communication graph G. With the proposed control law, the closed-loop system is described by
N X x˙ i = Axi + cBK aij(xi − xj). (3.45) j=1
T T T If we use x to denote the all the subsystem states as x = [x1 , . . . , xN ] , we have
x˙ = (IN ⊗ A + cL ⊗ BK)x. (3.46)
Based on the vector r in Lemma 3.6, a state transformation is introduced
N X δi = xi − rjxj, (3.47) j=1 CHAPTER 3. PRELIMINARY STUDIES AND PROBLEM DESCRIPTIONS 58
T T T for i = 1,...,N. With δ = [δ1 , . . . , δN ] , the state transformation is given by
T δ = [(IN − 1r ) ⊗ In]x, (3.48)
T by definition of r, it is not difficult to see that 0 is a simple eigenvalue of IN − 1r with 1 as the corresponding right eigenvector and the rank of M is N − 1. Then it follows from (3.48) that δ = 0 if and only if x1 = ··· = xN . That is, the consensus problem is solved if and only if limt→∞ δ(t) = 0. We then refer to δ as the consensus error. By using the facts that L1 = 0 and rT L = 0, it follows from (3.46) and (3.48) that δ evolves according to the following dynamics:
˙ δ = (IN ⊗ A + cL ⊗ BK)δ. (3.49)
Theorem 3.5. Supposing Assumption 3.7 holds, let λi, i = 2,...,N be the eigenvalues of L. Then the consensus problem for the agents in (3.43) is solved if and only if all the matrices
A − cλiBK, i = 2,...,N (3.50) are asymptotically stable.
The next result shows how to select control gain K to guarantee stability of (3.49) by using a local Riccati design approach and a proper choice of the coupling gain c.
Theorem 3.6. Consider local distributed control protocols (3.44). Suppose (A, B) is stabilizable and let design matrices Q ∈ Rn×n and R ∈ Rp×p be positive definite. Design the feedback gain matrix K as
K = R−1BT P, (3.51) where P is the unique positive definite solution of the control algebraic Riccati equation (ARE) AT P + PA + Q − PBR−1BT P = 0. (3.52)
Then under Assumption 3.7, the error dynamics of (3.49) is asymptotically stable if the coupling gain satisfies 1 c ≥ . (3.53) 2 mini=2,...,N Re(λi) Then, all agents consensus to an agreement. Chapter 4
Adaptive Output Regulation of Output Feedback Nonlinear Systems with Unknown Nonlinear Exosystems and Unknown High-Frequency Gain Sign
In this chapter, we investigate the problem of adaptive output regulation for a class of output feedback systems subject to unknown disturbances generated from nonlinear exosystems and no knowledge of high-frequency gain sign. A nonlinear adaptive inter- nal model is proposed on a basis of circle criterion [118] to reproduce the feedforward input term. As for the nonlinear exosystems, some specific assumptions are determined such that the circle criterion can be used in the proposed internal model. Because of the lack knowledge of parameters in the system and the exosystem, a Nussbaum gain is then used to tackle the unknown high-frequency gain. The proposed internal model, Nussbaum gain and nonlinear adaptive backstepping technique together provide a so- lution to the output regulation with nonlinear exosystems. The control input is finally obtained by a recursive procedure. The contributions of this work can be summa- rized as follows. First, no work has addressed output regulation of nonlinear output
59 CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 60 feedback systems with both nonlinear exosystems and high-frequency gain sign being unknown so far. Second, a general theorem can be drawn from the main results of this work to relax the assumption [56] that the unknown exosystems have distinct eigenvalues of zero real parts. The rest of this chapter is organized as follows. Some notations and the prob- lem formulation are given in Section 4.1. Section 4.2 presents state transformation, which converts the original plant to an augmented plant. Based on the analysis of the augmented system in Section 4.2. Section 4.3 presents a new internal model, then a controller design method is proposed in Section 4.4 and the stability analysis is dis- cussed in Section 4.5. An example is finally included in Section 4.6 to illustrate the effectiveness of the control algorithm and Section 4.7 summarises this chapter.
4.1 Problem Formulation
We consider a single-input-single-output nonlinear system which can be transformed into the output feedback form
˙ ζ = Acζ + φ(y)a + E(ω) + bu, y = Cζ, (4.1)
e = y − q(ω), with
0 1 0 ... 0 0 T 1 . 0 0 1 ... 0 . 0 ...... Ac = . . . . . , b = bρ ,C = . , . . 0 0 0 ... 1 . 0 0 0 0 ... 0 bn where ζ ∈ Rn is the state vector, u ∈ R is the control input, y ∈ R is the output, e ∈ R is the measurement output, a ∈ Rm and b ∈ Rn are vectors of unknown parameters, bρ 6= 0 indicates that the nonlinear system has a constant relative degree of ρ, φ : R → Rn×m is a smooth nonlinear vector field with φ(0) = 0 and kφ(y)−φ(ˆy)k ≤ CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 61
Mσ(|y − yˆ|)|y| and M is an unknown positive constant, and σ(·) ∈ K is a known smooth function, q is an unknown polynomial of ω, and ω ∈ Rl are disturbances, and they are generated from an unknown nonlinear exosystem
ω˙ = s(ω). (4.2)
Remark 4.1. When the measurement, or the output, does not explicitly contain the disturbances, the control design is referred as asymptotic rejection problem for the boundedness of signals and asymptotic convergence to zero. For the output regulation problem, when the measurement contains the disturbance or exogenous signals, the control design is to ensure the measurement converge to zero asymptotically. When the measurement contains the exogenous signal, the output is normally different from the measurement, and in such a case, the output can be viewed as track an exogenous signal. The system (4.1) has a measurement e that is perturbed by an unknown polynomial of the unknown disturbance. This is the reason why the problem to be solved is an output regulation problem, rather than a disturbance rejection problem.
Remark 4.2. It is shown in [37] that if a system satisfies certain geometric conditions, then it can be transformed to system (4.1) by a global state-space diffeomorphism.
Pn n−i Assumption 4.1. The system is minimum phase, i.e., the polynomial B(s) = i=ρ bis is Hurwitz, and the sign of the high-frequency gain bρ is unknown.
Assumption 4.2. The flows of vector field s(ω) are bounded and converge to periodic solutions.
Remark 4.3. The system (4.1) is similar to the system considered in [56]. The main difference is that the exosystem is nonlinear.
The purpose of this chapter is to solve the adaptive output regulation problem, that is, to design an appropriate output feedback controller of the form
n δ˙ = f(δ, e(t)), δ ∈ R c (4.3) u = h(δ, e(t)) such that for any initial condition (ζ(0), ω(0), δ(0)) ∈ Rn × Rl × Rnc , the resulting trajectory of closed-loop system (4.1), (4.2) and (4.3) is bounded, and lim e(t) = 0. t→∞ CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 62
In [33], under certain assumptions, the output regulation problem for system (4.1) and (4.2) can be converted into a stabilization problem for some augmented systems. Before introducing these assumptions, a state transformation, based on the filtered transform in [37] is presented for system (4.1) with relative degree ρ > 1.
4.2 State Transformation
In this subsection, we present backstepping design with filtered transformation [107] for the system (4.1). Define an input filter
˙ ξi = −λiξi + ξi+1, for i = 1, . . . , ρ − 2, (4.4) ˙ ξρ−1 = −λρ−1ξρ−1 + u, where λi > 0 for i = 1, . . . , ρ are the design parameters. Define the filtered transfor- mation ρ−1 X ¯ x¯ = ζ − diξi, (4.5) i=1 ¯ n where di ∈ R for i = 1, . . . , ρ − 1 and they are generated recursively by
¯ dρ−1 = b, ¯ ¯ di = (Ac + λi+1I)di+1, for i = 1, . . . , ρ − 2.
We also denote ¯ d = (Ac + λ1I)d1.
From the filtered transformation, we have
ρ−2 X ¯ ¯ x¯˙ = Acζ + bu + φ(y)a + E(ω) − di(−λiξi + ξi+1) − dρ−1(−λρ−1ξρ−1 + u) i=1 ρ−1 ρ−1 ρ−2 X ¯ X ¯ X ¯ = Acx¯ + Acdiξi + φ(y)a + E(ω) + diλiξi − diξi+1 i=1 i=1 i=1 ρ−1 ρ−2 X ¯ X ¯ = Acx¯ + (Ac + λiI)diξi + φ(y)a + E(ω) − diξi+1 i=1 i=1
= Acx¯ + φ(y)a + E(ω) + dξ1. CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 63
The output under the coordinate ζ¯ is given by
ρ−1 X ¯ y = Cx¯ + Cdiξi, i=1 = Cx,¯
¯ ¯ because Cdi = 0 for i = 1, . . . , ρ, from the fact that di,j = 0 for i = 1, . . . , ρ, 1 ≤ j ≤ i. Hence, under the filtered transformation, the system (4.1) is then transformed to
x¯˙ = Acx¯ + φ(y)a + E(ω) + dξ1, (4.6) y = Cx.¯
Let us find a bit more information of d. From the definition
¯ dρ−2 = (Ac + λρ−1I)b, we have n n X ¯ n−i X n−i dρ−2,is = (s + λρ−1) bis . i=ρ−1 i=ρ Repeating the process iteratively, we can obtain
n ρ−1 n X n−i Y X n−i dis = (s + λi) bis , i=1 i=1 i=ρ which implies that d1 = bρ, and d is Hurwitz if b Hurwitz. In the special form of Ac and
C used here, b and d decide the zeros of the linear systems characterised by (Ac, b, C) and (Ac, d, C) respectively as the solutions to the following polynomial equations:
n X n−i bis = 0, i=ρ n X n−i dis = 0. i=1
Hence, the invariant zeros of (Ac, d, C) are the invariant zeros of (Ac, b, C) plus λi for i = 1, . . . , ρ − 1. For the transformed system, ξ1 can be viewed as the new input. In this case, the relative degree with ξ1 as the input is 1. The filtered transformation lifts the relative degree from ρ to 1. As the filtered transformation may have its use independent of backstepping de- sign shown here, we summarise the property of the filtered transformation in the following lemma. CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 64
Lemma 4.1. For a system in the form (4.1) with relative degree ρ, the filtered trans- formation defined in (4.5) transforms the system to (4.6) of relative degree 1, with the same high frequency gain. Furthermore, the zeros of (4.6) consist of the zeros of the original system (4.1) and λi for i = 1, . . . , ρ − 1.
We then introduce another state transformation to extract the internal dynamics of (4.6) with x ∈ Rn−1 given by
d2:n x =x ¯2:n − y, (4.7) d1 where x ∈ Rn−1 forms the state variable of the transformed system together with y, the notation (·)2:n refers to the 2nd row to the nth row of the corresponding vector or matrix. The system (4.6) is put in the coordinate (x, y) as ˜ x˙ = Dx + Ξy + Ω1(y, d)a + E(ω, d), (4.8) d2 y˙ = x1 + y + φ1(y)a + E1(ω) + bρξ1, d1 where D is the companion matrix of d given by −d2/d1 1 0 ... 0 −d3/d1 0 1 ... 0 . . . . . D = . . . . . , (4.9) −dn−1/d1 0 0 ... 1 −dn/d1 0 0 ... 0 and d Ξ = D 2:n , d1 d2:n Ω1(y, d) = φ2:n(y) − φ1(y), d1 ˜ d2:n E(ω, d) = E2:n(ω) − E1(ω). d1
Notice that Ω1(y, d) is dependent on φ(y), thus it is easy to check that Ω1(0, d) = 0.
4.3 The Internal Model Design
The invariant manifold theory and internal model principle play a crucial role in es- tablishing the solvability of the nonlinear output regulation problem [8]. Motivated CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 65 by these results, the adaptive output regulation problem is solved if the following assumption is satisfied.
Assumption 4.3. There exists an invariant manifold π(ω) ∈ Rn−1 satisfying ∂π(ω) s(ω) = Dπ(ω) + ψ(q(ω), ω, µ), (4.10) ∂ω
˜ T T T where ψ(q(ω), ω, µ) = Ξq(ω) + Ω1(q(ω), d)a + E(ω, d), µ = [a , b ] .
Based on above assumption, we have
∂q(ω) s(ω) = π + ψ (q(ω), ω, µ) + b α(ω), (4.11) ∂ω 1 y ρ
d2 where ψy(q(ω), ω, µ) = q(ω) + φ1(q(ω))a + E1(ω), α is the feedforward control input d1 for disturbance suppression, and it is refer to
∂q(ω) α(ω) = b−1 s(ω) − π − ψ (q(ω), ω, µ) . (4.12) ρ ∂ω 1 y
Letz ˜ = x − π(ω), then (4.10) and (4.11) imply thatz ˜ satisfies
z˜˙ = Dz˜ + Ξe + Ω(y, ω, d)a,
d2 e˙ =z ˜1 + e + (φ1(y) − φ1(q(ω)))a + bρ(ξ1 − α(ω)), d1 (4.13) ˙ ξi = −λiξi + ξi+1, for i = 1, . . . , ρ − 2, ˙ ξρ−1 = −λρ−1ξρ−1 + u, where d2:n Ω(y, ω, d) = φ2:n(y) − φ2:n(q(ω)) − (φ1(y) − φ1(q(ω))) . d1 The output regulation problem of system (4.1) degenerates to the stabilization problem of system (4.13). However, the unknown feedforward control input, α(ω), makes it difficult for us to achieve stabilization. To tackle the problem, an assumption on the structure of exosystem is needed.
Assumption 4.4. For the system
ω˙ = s(ω),
α = α(ω), CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 66 there exists an immersion of the exosystem
η˙ = F η + Gγ(Hη), (4.14) α = Jη, where η ∈ Rnr , J = [1, 0,..., 0], the pair (J, F ) is detectable, γ(·) is locally Lipschitz
nr and is nondecreasing, that is, for all %1,%2 ∈ R , it satisfies [118]
T (%1 − %2) [γ(%1) − γ(%2)] > 0, (4.15) and G, H are some appropriate dimensional matrices.
Remark 4.4. Assumption 4 is inspired by [34, 35], this immersion system and the circle criterion are exploited for the design of the internal model and the estimated feedforward term α.
Remark 4.5. Conditions have been identified for the existence of an internal model, sometimes even a nonlinear internal model as shown in [3], when the exosystems are linear. However, it is not clear at the moment what general conditions can be specified to guarantee the existence of such an internal model for nonlinear systems with nonlinear exosystems. This is the reason why Assumption 4.4 is needed for the proposed algorithm. An illustrative example is included in this chapter to demonstrate that such an immersion of (4.14) exists for an uncertain nonlinear system, and the proposed algorithm can be used to solve the output regulation problem.
Since the feedforward term α is unknown, based on (4.14), we design the following internal model, which produces the estimated feedforward term α,
˙ −1 −1 ηˆ = (F − KJ)(ˆη − bρ Ke) + Gγ(H(ˆη − bρ Ke)) + Kξ1, (4.16) where K ∈ Rnr is chosen such that F − KJ is Hurwitz. We define the auxiliary error
−1 η˜ = η − ηˆ + bρ Ke, (4.17) then
˙ −1 η˜ = (F − KJ)˜η + G(γ(Hη) − γ(H(ˆη − bρ Ke)) (4.18) −1 d2 + bρ K(˜z1 + e + (φ1(y) − φ1(q(ω)))a). d1 CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 67
We rewrite the system (4.18) as
η˜˙ = (F − KJ)˜η + Gϕ(t, ψ) + Υ, (4.19) Ψ = Hη,˜
−1 d2 where Υ = bρ K(˜z1 + e + (φ1(y) − φ1(q(ω)))a), and d1
ϕ(t, ψ) = γ(v1) − γ(v2),
v1 : = Hη,
−1 v2 : = H(ˆη − bρ Ke),
Ψ = v1 − v2,
T T −1 andη ˜ H (γ(Hη)−γ(H(ˆη−bρ Ke))) > 0. According to the circle criterion [118], there exist a positive definite matrix Pη and a semi-positive definite matrix Q satisfying
T (F − KJ) Pη + Pη(F − KJ) = −Q (4.20) T PηG + H = 0.
And
T 2 η Qη ≥ γ0|η1| , γ0 > 0, (4.21) span(PηK) ⊆ span(Q).
T Let Vη =η ˜ Pηη˜, then
˙ T T −1 Vη = −η˜ Qη˜ + 2˜η PηG(γ(Hη) − γ(H(ˆη − bρ Ke))
T −1 d2 + 2˜η Pηbρ K(˜z1 + e + (φ1(y) − φ1(q(ω)))a) (4.22) d1 T T −1 d2 ≤ −η˜ Qη˜ + 2˜η Pηbρ K(˜z1 + e + (φ1(y) − φ1(q(ω)))a). d1
Lemma 4.2. There exist nondecreasing known function σ1(·) and an unknown con- stant N, which is dependent on the initial state of ω0, such that
|φ1(y) − φ1(q(ω))| ≤ N|e|σ1(|e|).
Proof. From the assumption of φ, we have
|φ1(y) − φ1(q(ω))| ≤ Mσ(|e|)|q(ω)|. CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 68
Because the exosystem is bounded, there exists a smooth nondecreasing known func-
0 tion σ1(·) and M (·) such that
σ(|e|) ≤ |e|σ1(e),
0 |q(ω)| ≤ M (|ω0|).
2 2 2 Then using Young’s equality and σ1(|e|) ≤ σ1(1+e ), there exist unknown positive real constants Λ1,Λ2,Λ3, such that
1 V˙ ≤ − γ |η˜ |2 + Λ z˜ 2 + Λ e2 + Λ e2σ2(1 + e2), (4.23) η 4 0 1 1 1 2 3 1 noting that 1 1 2˜ηT P b−1Kz˜ ≤ η˜T Qη˜ + 4b−2z˜ 2 ≤ η˜T Qη˜ + Λ z˜ 2, η ρ 1 4 ρ 1 4 1 1 and
T −1 d2 1 T −2 d2 2 2 1 T 2 2˜η Pηbρ K e ≤ η˜ Qη˜ + 4bρ ( ) e ≤ η˜ Qη˜ + Λ2e , d1 4 d1 4 and 1 2˜ηT P b−1K(φ (y) − φ (q(ω)))a ≤ η˜T Qη˜ + Λ e2σ2(1 + e2), η ρ 1 1 4 3 1 −2 2 T where Λ3 = 4bρ N a a.
4.4 Controller Design
If the system (4.1) is of relative degree 1, then ξ1 in (4.13) is the control input. For system with higher relative degrees, adaptive backstepping is then applied. Indeed, in the backstepping design, ξi for i = 1, . . . , ρ − 1 can be viewed as virtual controls. Define
z1 = e,
zi = ξi−1 − βi−1, for i = 2, . . . , ρ, (4.24)
zρ+1 = u − βρ, where βi for i = 1, . . . , ρ are stabilizing functions to be designed. Since the sign of the high-frequency gain bρ is unknown, a Nussbaum gain N(κ) [41] should be employed CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 69
in the stabilizing function β1. Let
¯ β1 = N(κ)β1, (4.25) ¯ κ˙ = eβ1, κ(0) = 0, where the Nussbaum gain N is a function (e.g., N(κ) = κ2 cos κ) that satisfies the two-sided Nussbaum properties [50,54]
1 Z κ lim sup N(s)ds = +∞, κ→±∞ κ 0 (4.26) 1 Z κ lim inf N(s)ds = −∞, κ→±∞ κ 0 where κ → ±∞ denotes κ → +∞ and κ → −∞ respectively. From (4.13) and (4.25), we have
d2 ¯ ¯ e˙ =z ˜1 + e + (φ1(y) − φ1(q(ω)))a + (bρN(κ) − 1)β1 + β1 + bρz2 − bρη1. (4.27) d1
We now design the virtual control β1 as, with c1 > 0,
¯ ˆ 2 2 ˆ β1 = −c1e − k1e(1 + σ1(1 + e )) + bρηˆ1 − K1e. (4.28)
Using (4.17), the resultant error dynamics are obtained as
ˆ 2 2 d2 e˙ = −c1e − k1e(1 + σ1(1 + e )) +z ˜1 + e d1 ¯ (4.29) + (φ1(y) − φ1(q(ω)))a + (bρN(κ) − 1)β1 ˆ ˜ ˜ + bρz2 + bρz2 − bρηˆ1 − bρη˜1,
˜ ˆ ˆ where bρ = bρ − bρ, k1 is an adaptive coefficient, and the adaptive law is given by
ˆ˙ 2 2 2 k1 = e (1 + σ1(1 + e )). (4.30)
If the relative degree ρ = 1, then u = β1. For ρ > 1, the final control u will be obtained in ρ − 1 steps by using adaptive backstepping. In the second step, we have
˙ ˙ z˙2 = ξ1 − β1 (4.31) ∂β1 ˙ ∂β1 ˙ ∂β1 ∂β1 ∂β1 = −λ ξ + z + β − kˆ − ˆb − ηˆ˙ − κ˙ − e,˙ 1 1 3 2 ˆ 1 ˆ ρ 1 ∂k1 ∂bρ ∂ηˆ1 ∂κ ∂e CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 70
d2 wheree ˙ =z ˜1 + e + (φ1(y) − φ1(q(ω)))a + bρ(ξ1 − η1). Based on (4.29) and (4.31), we d1 design β2 as
∂β 2 β = λ ξ − ˆb e − c z − k 1 z 2 1 1 ρ 2 2 2 ∂e 2 (4.32) ∂β1 ˙ ∂β1 ∂β1 ∂β1 ∂β1 + kˆ + τ + ηˆ˙ + κ˙ + (ˆb (ξ − ηˆ ) + K e), ˆ 1 ˆ 2 1 ρ 1 1 1 ∂k1 ∂bρ ∂ηˆ1 ∂κ ∂e where the notations τi for i = 2, . . . , ρ are referred to as tuning functions. The resultant dynamics of z2 are 2 ∂β1 ∂β1 ˙ z˙ = −ˆb e − c z − k z − (ˆb − τ ) 2 ρ 2 2 2 2 ˆ ρ 2 ∂e ∂bρ ∂β1 d2 (4.33) − z˜1 + e + (φ1(y) − φ1(q(ω)))a ∂e d1 ∂β ∂β − 1 ˜b ξ + 1 (˜b ηˆ + b η˜ ) + z . ∂e ρ 1 ∂e ρ 1 ρ 1 3
Then the adaptive backstepping can be carried on for zi with 3 ≤ i ≤ ρ. The resultant dynamics of zi can be written as
˙ ˙ z˙i = ξi−1 − βi−1
∂βi−1 ˙ ∂βi−1 ∂βi−1 = −λ ξ + z + β − ˆb − ηˆ˙ − κ˙ i−1 i−1 i+1 i ˆ ρ ∂bρ ∂ηˆ ∂κ (4.34) i−2 X ∂βi−1 ∂βi−1 − (−λ ξ + ξ ) − e.˙ ∂ξ j j j+1 ∂e j=1 j
The design of βi is given by
∂β 2 ∂β ∂β ∂β β = λ ξ − z − c z − k i−1 z + i−1 τ + χ + i−1 ηˆ˙ + i−1 κ˙ i i−1 i−1 i−1 i i i i ˆ i i ∂e ∂bρ ∂ηˆ ∂κ i−2 X ∂βi−1 ∂βi−1 + (−λ ξ + ξ ) + (ˆb (ξ − ηˆ ) + K e), ∂ξ j j j+1 ∂e ρ 1 1 1 j=1 j (4.35)
ˆ˙ where χi for i = 3, . . . , ρ are functions to be designed later to tackle the terms (bρ − τi) in stability analysis, ci and ki, i = 2, . . . ρ are positive real design parameters. The resultant dynamics of, 3 ≤ i ≤ ρ, are obtained as
2 ∂βi−1 ∂βi−1 ˆ˙ ∂βi−1 ˜ z˙i = −zi−1 − cizi − ki zi + zi+1 − (bρ − τi) + χi − bρξ1 ∂e ∂ˆb ∂e ρ (4.36) ∂βi−1 d2 ∂βi−1 ˜ − z˜1 + e + (φ1(y) − φ1(q(ω)))a + (bρηˆ1 + bρη˜1). ∂e d1 ∂e CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 71
When i = ρ, the control input appears in the dynamics of zi, through the term zρ+1. We design the control input by setting zρ+1 = 0, which gives
u = βρ. (4.37)
In such a case, the adaptive law and tuning functions are given by
i X ∂βj−1 τ = (z − ηˆ )e − z (ξ − ηˆ ), for i = 1, . . . , ρ, i 2 1 ∂e j 1 1 j=2 i−1 X ∂βj−1 ∂βi−1 (4.38) χ = z (ξ − ηˆ ), for i = 3, . . . , ρ, i j ˆ ∂e 1 1 j=2 ∂bρ ˆ˙ bρ = τρ.
4.5 Stability Analysis
In this section, the stability of the closed-loop system will be established, which ensures the boundedness of all the variables and the zero convergence of the error output.
Theorem 4.1. The output regulation problem of system (4.1) can be globally solved by the combination of the ξ-filters (4.4), nonlinear internal model (4.16), control input (4.37) and adaptive laws (4.30) (4.38).
T Proof. We begin with the analysis of internal dynamics of (4.13). Define Vz˜ =z ˜ Pz˜z˜, T where D Pz˜ + Pz˜D = −I, then similar to the derivation of (4.21) and (4.23), there exist positive real constant Λ4,Λ5 such that
˙ T T Vz˜ = −z˜ z˜ + 2˜z Pz˜(Ξe + Ω(y, ω, d)a) 1 (4.39) ≤ − z˜T z˜ + Λ e2 + Λ e2σ2(1 + e2). 2 4 5 1 In terms of the composite system (4.13), we define the following Lyapunov function candidate
ρ ! 1 X V = γ V + γ V + z2 + (kˆ − k )2 + ˜b2 , (4.40) 1 η 2 z˜ 2 i 1 1 ρ i=1 where γ1, γ2 are two positive real numbers. From (4.23), (4.39) and adaptive laws CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 72 designed in the previous section, we have the derivative of V as
1 V˙ ≤ γ − γ |η˜ |2 + Λ z˜ 2 + Λ e2 + Λ e2σ2(1 + e2) 1 4 0 1 1 1 2 3 1 1 + γ − z˜T z˜ + Λ e2 + Λ e2σ2(1 + e2) 2 2 4 5 1 2 ˆ 2 2 2 d2 2 − c1e − k1e (1 + σ1(1 + e )) + ez˜1 + e d1 ¯ + (φ1(y) − φ1(q(ω)))ea + (bρN(κ) − 1)β1e ρ 2 ! X ∂βi−1 − b η˜ e + −c z2 − k z2 ρ 1 i i i ∂e i i=2 ρ X ∂βi−1 d2 − z˜ + e + (φ (y) − φ (q(ω)))a z ∂e 1 d 1 1 i i=2 1 ρ X ∂βi−1 + b η˜ z + (kˆ − k )e2(1 + σ2(1 + e2)). ∂e ρ 1 i 1 1 1 i=2 For the cross terms in above equation, we have
2 1 2 |ez˜1| ≤ Q1e + z˜1, 4Q1 2 1 2 2 2 2 |(φ1(y) − φ1(q(ω)))ea| ≤ Q2e + N e σ1(1 + e ) 4Q2 2 1 2 2 |bρη˜1e| ≤ Q3e + bρη˜1, 4Q3 2 ∂βi−1 2 1 ∂βi−1 2 z˜1zi ≤ Q4,iz˜1 + zi , ∂e 4Q4,i ∂e 2 2 ∂βi−1 d2 2 1 ∂βi−1 d2 2 ezi ≤ Q5,ie + 2 zi , ∂e d1 4Q5,i ∂e d1 2 ∂βi−1 2 2 1 ∂βi−1 2 bρη˜1zi ≤ Q6,ibρη˜1 + zi , ∂e 4Q6,i ∂e 2 ∂βi−1 2 2 2 2 2 1 ∂βi−1 2 (φ1(y) − φ1(q(ω)))azi ≤ Q7,ia N e σ1(1 + e ) + zi , ∂e 4Q7,i ∂e where Q1, Q2, Q3, Qi,j, i = 4,..., 7 and j = 2, . . . , ρ are real positive reals, which satisfy the following conditions
ρ 1 X 1 b2 + Q b2 ≤ γ γ , 4Q ρ 6,i ρ 4 0 1 3 i=2 ρ 1 X 1 + Q + γ Λ ≤ γ , 4Q 4,i 1 1 2 2 1 i=2 2 1 1 d2 1 1 + 2 + + ≤ ki, i = 2, . . . , ρ, 4Q4,i 4Q5,i d1 4Q6,i 4Q7,i CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 73 and choose
k1 = γ1Λ2 + γ1Λ3 + γ2Λ4 + γ2Λ5 ρ ρ d2 1 X X + Q + + Q + N 2 + Q + Q + Q a2N 2. 1 d 2 4Q 3 5,i 7,i 1 2 i=2 i=2 Based on above discussion, we can conclude that there exists a positive real con- stant ν, such that ρ ! ˙ T 2 2 X 2 V ≤ (bρN(κ) − 1)κ ˙ − ν z˜ z˜ +η ˜1 + e + zi . (4.41) i=2 By integrating (4.41), we have Z t ρ ! T 2 2 X 2 V (t) + ν z˜ z˜ +η ˜1 + e + zi dt 0 i=2 (4.42) Z κ(t) ≤ bρ N(s)ds − κ(t) + V (0). 0 Since the left-side of (4.42) is nonnegative, if the above inequality always holds, κ(t) must be bounded, which implies the boundedness of V (t). The boundedness of V (t) ˆ ˆ further implies the boundedness ofη, ˜ z,˜ e, k1, bρ and zi, for i = 2, . . . , ρ. Because the disturbance ω is bounded, e, z,˜ η˜ ∈ L∞ imply the boundedness of x, y, ηˆ. According to equation (4.25) and (4.28), β1 is bounded, which implies the boundedness of ξ1. The ˆ ˆ boundedness of β1, ξ1, bρ, e, k1, ηˆ1, κ imply the boundedness of β2 and ξ2. The bound- edness of βi and ξi for 3 ≤ i ≤ ρ can then be established recursively. Therefore, u is bounded and we can conclude that all the variables of system (4.13) are bounded. The boundedness ofη, ˜ z,˜ e and zi further imply the boundedness of η,˜˙ z,˜˙ e˙ andz ˙i. Finally, applying Barbalat’s lemma, we have limt→∞ z˜ = 0, limt→∞ η˜ = 0, limt→∞ e(t) = 0 and limt→∞ zi = 0 for i = 2, . . . , ρ.
4.6 Numerical Example
We use an example to demonstrate the application of the proposed control design. Consider a first order system
˙ 3 3 ζ1 = (θ − 1)y − θω1 + 2ω1 + ω2 + bu,
y = ζ1, (4.43)
e = y − ω1, CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 74 where θ is an unknown parameter, and the disturbance ω is generated by
3 ω˙ 1 = ω1 + ω2 − ω1, (4.44) 3 ω˙ 2 = −ω1 − ω2.
ω1 It is easy to observe that q(ω) = ω1, π(ω) = ω1 and α(ω) = − . From the feedforward b ω input α(ω), it can be seen that Assumption 4 is satisfied with η = − , and η is in the b 3 format of (4.14), with γ1(t) = γ2(t) = t and 1 1 −b2 0 1 0 F = ,G = ,H = . −1 0 0 −b2 0 1
3 In order to obtain the internal model in the format of (4.16), let K = . Then 0 −2 1 F − KJ = and −1 0
˙ −1 −1 3 ηˆ1 = −2(ˆη1 − 3b e) +η ˆ2 − (ˆη1 − 3b e) + 3u, (4.45) ˙ −1 3 ηˆ2 = −(ˆη1 − 3b e) − ηˆ2.
Finally, the control input is given by
2 ¯ u = κ cos κβ1, ¯ κ˙ = eβ1. ¯ ˆ 2 2 ˆ β1 = −c1e − k1e(1 + (1 + e ) ) + bηˆ1 − 3e. (4.46) ˆ˙ 2 2 2 k1 = e (1 + (1 + e ) ), ˆ˙ b = −ηˆ1e.
In the simulation studies, we set c1 = 37, θ = 0.5, while the initial condition of the system is chosen as ω1(0) = 2, ω2(0) = 2 and y(0) = 1. The initial condition of dynamic controller is zero. The system error output, the control input, the feedforward term and its estimation for b = 1 and b = −1 are shown in Figs. 4.1 to 4.4 respectively. As shown in the figures, the feedforward control produced from internal model is equal to its estimation as time increases, and the system error is regulated to zero. CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 75
0.2
0
-0.2
-0.4 Tracking error e -0.6
-0.8
-1 0 5 10 15 Time (sec)
40
35
30
25
20
15 Control u
10
5
0
-5 0 5 10 15 Time (sec)
Fig. 4.1. The system output e and control input u for b = 1.
2
η1 ηˆ1 1.5
1
0.5 1 η 0 and ˆ 1 η -0.5
-1
-1.5
-2 0 5 10 15 Time (sec)
Fig. 4.2. Feedforward control η1 and its estimationη ˆ1 for b = 1. CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 76
0.2
0
-0.2
-0.4
-0.6
-0.8 Tracking error e
-1
-1.2
-1.4 0 5 10 15 Time (sec)
30
20
10
0
-10
-20 Control u
-30
-40
-50
-60 0 5 10 15 Time (sec)
Fig. 4.3. The system output e and control input u for b = −1.
2
η1 ηˆ1 1.5
1 1
η 0.5 and ˆ 1
η 0
-0.5
-1
-1.5 0 5 10 15 Time (sec)
Fig. 4.4. Feedforward control η1 and its estimationη ˆ1 for b = −1. CHAPTER 4. OUTPUT REGULATION OF SISO NONLINEAR SYSTEMS 77 4.7 Summary
In this chapter, the adaptive output regulation problem for nonlinear systems in output feedback form proposed in [35] has been considered without the knowledge of the high- frequency gain sign. The auxiliary filters have been designed to achieve the system transformation. The proposed internal model, which is under some conditions, is designed by the circle criterion. The success of the introduction of the Nussbaum gain together with the internal model makes it possible to deduce the adaptive laws, and by using adaptive backstepping, the control input has been obtained. Finally, the example demonstrates the effectiveness of the proposed adaptive output regulation algorithm for nonlinear output feedback systems, which ensures the boundedness of all variables and the asymptotic convergence of the measurement output to zero. The proposed scheme also provides innovation for the future research in output regulation of multi-agent systems with unknown control directions. Chapter 5
Adaptive Consensus Output Regulation of a Class of Heterogeneous Nonlinear Systems with Unknown Control Directions
In this chapter, we deal with the adaptive output regulation of a class of network- connected nonlinear systems with completely unknown parameters, including the high frequency gains of the subsystems. A new type of Nussbaum gain is proposed to deal with adaptive consensus control of network-connected systems without any knowledge of the high frequency gains. Adaptive laws and internal models are designed for the subsystem to deal with the unknown system parameters and the unknown exosystem. In the control design, the regulation error of one of the subsystem is available and only the relative information of subsystem outputs are used, and the proposed scheme is decentralised. The contributions of this chapter are two folds. Firstly, a new Nussbaum gain with a potentially faster rate is presented such that the boundedness of the system parameters can be established. Secondly, the proposed control can deal with the subsystems with different dynamics as long as the subsystems with the same relative degree, and the adaptive laws and control inputs are still viewed as decentralized even with higher relative degrees.
78 CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 79
The remainder of this chapter is organized as follows. Section 5.1 presents some notations and the problem formulation. The state transformation and the internal model design are shown in Section 5.2. Section 5.3 presents a new Nussbaum gain for consensus control. The control design and the stability analysis are given in Section 5.4 and Section 5.5 respectively. Simulation results are illustrated in Section 5.6. Section 5.7 summarises this chapter.
5.1 Problem Formulation
In this chapter, a set of N nonlinear subsystems are considered, of which the subsys- tems are described by
x˙ i = Acixi + φi(yi, ω, ai) + biui, (5.1) yi = Cixi, with
0 1 0 ... 0 0 T 1 . 0 0 1 ... 0 . 0 ...... Aci = . . . . . , bi = bi,ρ ,Ci = . , . . 0 0 0 ... 1 . 0 0 0 0 ... 0 bi,n
ni for i = 1,...,N, where xi ∈ R is the state vector, ui ∈ R is the control input and
qi ni yi ∈ R is the output of the i-th subsystem, ai ∈ R , bi ∈ R are vectors of unknown parameters, bi,ρ 6= 0 indicates that the system has a constant relative degree of ρ,
m qi ni φi : R × R × R → R is a smooth nonlinear vector field with φi(0, ω, ai) = 0, and ω ∈ Rm are disturbances, and they are generated from an unknown exosystem
ω˙ = S(σ)ω, (5.2) with unknown σ ∈ Rs, of which, S ∈ Rm×m is a constant matrix with distinct eigen- values of zero real parts. We define the output regulation errors as
ei = yi − g(ω), (5.3) CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 80 with g : Rm → R being polynomials of its variables, for i = 1,...,N. In our setup, not every subsystem has access to g(ω), and the consensus output regulation will be achieved through the network connections among the subsystems. The interactions among the subsystems are specified by an undirected graph G which consists of a set of vertices denoted by V = {1,...,N} and a set of edges de- noted by E. A vertex denotes a subsystem and each edge represents a connection.
Associated with the graph, its adjacency matrix A with elements aij denotes the con- nections such that aij = 1 if there is a path from subsystem j to subsystem i, and T aij = 0 otherwise. Since the connection is undirected, we have A = A . The Laplacian PN matrix L is defined as lii = j=1 aij and lij = −aij when i 6= j. Not all the subsystems have the access to the function value of g(ω). We use a diagonal matrix ∆ to denote the access to g(ω) in the way that if δi = 1, the i-th subsystem has access to the value of g(ω) for the control design, and δi = 0 otherwise. At least one subsystem has the access. The subsystems which do not have access to the desired output rely on the network connections to achieve the consensus tracking. The adaptive consensus output regulation problem considered in this chapter is to design an adaptive control strategy using the relative output information yi −yj, i 6= j, provided by the network connection to each subsystem to ensure the convergence to zero of output regulation errors ei for i = 1,...,N under any initial condition of the system in the state space, i.e., the convergence of the subsystem outputs yi to the common function g(ω). We make several assumptions about the dynamics of the subsystems, the exosys- tem and the connections between the subsystems.
Assumption 5.1. The invariant zeros of (Aci, bi,Ci) are stable, for i = 1,...,N, and all the subsystems have the same sign of the high-frequency gains.
Assumption 5.2. The eigenvalues of S are distinct and on the imaginary axis.
Assumption 5.3. The adjacency matrix A is irreducible.
Remark 5.1. Assumption 5.2 on the eigenvalues of the exosystem dynamics is com- mon in the formulation of output regulation, as the stable modes in the exosystem do not have an impact asymptotically, and the adjacency matrix is irreducible if there exists a path between any two subsystems. CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 81 5.2 State Transformation and The Internal Model
Similar to Section 4.2 and 4.3, we first introduce filtered transformation with the filter for the subsystem i, for i = 1,...,N,
˙ ξi,1 = −λ1ξi,1 + ξi,2, ··· (5.4) ˙ ξi,ρ−1 = −λρ−1ξi,ρ−1 + ui, where λj > 0 for j = 1, . . . , ρ − 1 are the design parameters, and the filtered transfor- mation ¯ ¯ z¯i = xi − [di,1 ... di,ρ−1]ξi, (5.5)
T ¯ ni where ξi = [ξi,1 . . . ξi,ρ−1] , di,j ∈ R for j = 1, . . . , ρ − 1 and they are generated recursively by
¯ di,ρ−1 = bi, ¯ ¯ di,j = (Aci + λj+1I)di,j+1, for j = 1, . . . , ρ − 2.
Hence, under the filtered transformation, the system (5.1) is then transformed to
z¯˙i = Aciz¯i + φi(yi, ω, ai) + diξi,1, (5.6) yi = Ciz¯i, ¯ where di = (Aci + λi,1I)di,1. It can be shown that di,1 = bi,ρ and
ni ρ−1 ni X n−j Y X ni−j di,js = (s + λj) bi,js . (5.7) j=1 j=1 j=ρ
With ξi,1 as the input, the system (5.6) is with relative degree one and minimum phase. We then introduce another state transformation to extract the internal dynamics of
ni−1 (5.6) with zi ∈ R given by
di,2:ni zi =z ¯i,2:ni − yi, (5.8) di,1 where the notation (·)2:n refers to the 2nd row to the nth row of the corresponding vector or matrix. The system (5.6) is put in the coordinate (zi, yi) as
z˙i = Dizi + ψi(yi, ω, θi), (5.9) y˙i = zi,1 + ψi,y(yi, ω, θi) + bi,ρξi,1, CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 82
T T T where the unknown parameter vector θi = [ai , bi ] , and Di is the left companion matrix of di given by −di,2/di,1 1 0 ... 0 −di,3/di,1 0 1 ... 0 . . . . . Di = . . . . . , (5.10) −di,ni−1/di,1 0 0 ... 1 −di,ni /di,1 0 0 ... 0 and di,2:ni di,2:ni ψi(yi, ω, θi) = Di yi + φi,2:ni (yi, ω, ai) − φi,1(yi, ω, ai), di,1 di,1 di,2 ψi,y(yi, ω, θi) = yi + φi,1(yi, ω, ai). di,1
Note that Di is Hurwitz, and that the dependence of di on bi reflected in the pa- rameter θi in ψi(yi, ω, θi) and ψi,y(yi, ω, θi), and it is easy to check that ψi(0, ω, θi) = 0 and ψi,y(0, ω, θi) = 0. The solution of the output regulation problem depends on the existence of certain invariant manifold and feedforward input. From the structure of exosystem, the distur- bances are sinusoidal functions. Polynomials of sinusoidal functions are still sinusoidal functions, but with some high frequency terms. Since all the nonlinear functions in- volved in the system (5.1) are polynomials of their variables. According to Lemma 3.4, the following results can be obtained.
Lemma 5.1. For a subsystem i of (5.1), for i = 1,...,N, there exists an invariant
ni−1 manifold πi(ω) ∈ R satisfying ∂π (ω) i S(σ)ω = D π (ω) + ψ (g(ω), ω, θ ). (5.11) ∂ω i i i i Then there exists an immersion for the feedfoward control input ∂τ (ω, θ , σ) i i S(σ)ω = Φ (σ)τ (ω, θ , σ), ∂ω i i i
αi(ω, θi, σ) = Γiτi(ω, θi, σ), where ∂g(ω) α (ω, θ , σ) = b−1( S(σ)ω − π − ψ (g(ω), ω, θ )). i i i,ρ ∂ω i,1 i,y i Furthermore, this immersion can be converted to
−1 T η˙i = (Fi + Gibi,ρ li )ηi, (5.12) −1 T αi = bi,ρ li ηi, CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 83
where (Fi,Gi) is a controllable pair with compatible dimensions, ηi = Miτi and li = −1 ΓiMi with Mi satisfying
Mi(σ)Φi(σ) − FiMi(σ) = GiΓi. (5.13)
For a controllable pair (Fi,Gi), Mi(σ) is an invertible solution of (5.13) if (Φi(σ), Γi) is observable, which is guaranteed by the immersion.
We now introduce the last transformation based on the invariant manifold with
z˜i = zi − πi(ω). (5.14)
Finally we have the model for the control design
˜ z˜˙i = Diz˜i + ψi, ˜ T e˙i =z ˜i,1 + ψi,y + bi,ρξi,1 − li ηi, ˙ ξi,1 = −λ1ξi,1 + ξi,2, (5.15) ··· ˙ ξi,ρ−1 = −λρ−1ξi,ρ−1 + ui,
˜ ˜ where ψi = ψi(yi, ω, θi) − ψi(g(ω), ω, θi), ψi,y = ψi,y(yi, ω, θi) − ψi,y(g(ω), ω, θi).
Since the state in the internal model ηi is unknown, we design the adaptive internal model ˙ ηˆi = Fiηˆi + Giξi,1. (5.16)
If we design the auxiliary error
−1 η˜i = ηi − ηˆi + bi,ρ Giei, (5.17) it can be shown that
˙ −1 −1 −1 ˜ η˜i = Fiη˜i − bi,ρ FiGiei + bi,ρ Giz˜i,1 + bi,ρ Giψi,y. (5.18)
If the system (5.1) is of relative degree 1, then ξi,1 in (5.15) is the control input for the subsystem i. For the systems with higher relative degrees, adaptive backstepping will be used to find the final control input ui from the desirable value of ξi,1. Before we introduce the control design, we need a result on the Laplacian matrix. CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 84
Lemma 5.2. [89,116] If the adjacency matrix A is irreducible, and the non-negative diagonal matrix ∆ has at least one positive diagonal element, there exists a positive diagonal matrix G = diag{g1, . . . , gN } such that
T G(L + ∆) + (L + ∆) G ≥ r0I, (5.19) for some positive real number r0, where L is the Laplacian matrix corresponding to A.
T Let us denote e = [e1, . . . , eN ] , and the consensus regulation error
ζ = Qe, (5.20) where Q = L + ∆, with ∆ = diag{δ1, . . . , δN }. Since Q is invertible, the control objective is equivalent to limt→∞ ζ(t) = 0. It is worth noting that (5.20) implies that N N X X ζi = qijej = aij(yi − yj) + δi(yi − g(ω)), (5.21) j=1 j=1 for i = 1,...,N, where qij are the elements of Q. Clearly, ζi is available to the control design for the i-th subsystem. We have another result relating ζ and e that is needed for the stability analysis.
Lemma 5.3. With ζ = Qe, the following inequality holds for any positive integer p, N N X 2p p−1 2p X 2p ei ≤ U λmax(Q) ζi , (5.22) i=1 i=1 H where λmax(Q) denotes the square root of the maximum eigenvalue of Q Q.
Proof. 1 p N " N # p X 1 X ζ2p = U (ζ2)p i U i i=1 i=1 " N #!p 1 X ≥ U (ζ2) U i i=1 = U 1−p(kζk2)p = U 1−p(kQek2)p
1−p −2p 2p ≥ U λmax(Q)kek N !p 1−p −2p X 2 ≥ U λmax(Q) ei i=1 N 1−p −2p X 2p ≥ U λmax(Q) ei , i=1 from which (5.22) is obtained. CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 85 5.3 A Novel Nussbaum-Type Function for Consen- sus Control
When high-frequency gains are completely unknown, Nussbaum gains are used in adaptive control. The basic idea of Nussbaum gain design is to construct a function N(κ) where κ is a variable, and the control input takes the form u = N(κ)¯u. Then the control design is continued withu ¯ such that a condition in the following form is obtained, for a single-input system, Z κ V (t) ≤ V (0) + (bρN(s) − 1)ds + r(t), (5.23) 0 where V is a positive definite function, κ(t) is a continuous function with κ(0) = 0, and r(t) is a bounded function and bρ is the unknown high-frequency gain. The bound- edness of κ and subsequently the boundedness of V can be established by seeking a contradiction using (5.23) if the Nussbaum function satisfies (4.26). Commonly used Nussbaum-type functions include κ2 cos(κ), κ2 sin(κ), ek2 cos(κ) [119–122]. For consen- sus control, there are N unknown high-frequency gains, and we aim at a condition N X Z κi V (t) ≤ V (0) + (bi,ρN(si) − 1)dsi + r(t), (5.24) i=1 0 similar to (5.23), but with multiple continuous functions κi’s. However, it is not clear how the existing Nussbaum-type functions can be used to solve the consensus problem for multi-agent systems with unknown control directions. The reason is that multiple Nussbaum-type function terms would coexist in the same conditional inequality and
κi’s are independent. Thus, we expect a function which grows faster such that one of the κi’s is dominant for the positive definite condition for consensus control in (5.24). We consider 2 κ 2 N(κ) = e 2 (κ + 2) sin κ, (5.25) and indeed, it can be used as a Nussbaum gain for consensus control. The equation (5.25) is obtained by experience, and it can be shown in the following lemma that this kind of Nussbaum gain can be used to for consensus control to prove that one of the
κi’s can be dominant for the positive definite condition of Lyapunov function.
Lemma 5.4. With the Nussbaum gain shown in (5.25), the boundedness of κi’s and V can be established from (5.24). CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 86
R κ Proof. Let M(κ) = 0 N(s)ds, it can be obtained that
κ2 M(κ) = e 2 (κ sin(κ) − cos(κ)) + 1. (5.26)
Furthermore, it can be shown that M(κ) takes local minimums at κ = 2nπ and local maximums at κ = (2n + 1)π, with j ∈ N0, a natural number. Hence, for 2nπ < κ ≤ 2(n + 1)π, we have
(2(n+1)π)2 ((2n+1)π)2 −e 2 + 1 ≤ M(κ) ≤ e 2 + 1.
In order to seek a contradiction, suppose that at least one of κi’s becomes unbounded.
Then, on the time interval [0, tf ), there exists an increasing sequence {tn}, n = 0, 1,..., defined by min1≤i≤N {t : |κi(t)| = (2n + 1)π} , if sgn(bi,ρ) = −1, tn = (5.27) min1≤i≤N {t : |κi(t)| = 2(n + 1)π} , if sgn(bi,ρ) = 1. clearly, limn→∞ tn = tf . Since the sign of bi,ρ is the same, the analysis can be divided into two parts, i.e., sgn(bi,ρ) = 1 and sgn(bi,ρ) = −1.
For the case sgn(bi,ρ) = 1, substituting (5.27) into (5.24), together with (5.26), the value of V at time t = tn satisfies N N X X V (tn) = V (0) + bi,ρM(κi(tn)) − κi(tn) + r(tn), i=1 i=1 (5.28) 2 2 (2(n+1)π) ¯ ((2n+1)π) ≤ V (0) + b(−e 2 + 1) + (N − 1)b(e 2 + 1) + r(tn),
N ¯ N where b = mini=1{bi,ρ} and b = maxi=1{bi,ρ}. With
(2(n+1)π)2 ((2n+1)π)2 − be 2 + (N − 1)¯be 2 (5.29) ((2n+1)π)2 (4n+3)π2 (N − 1)¯b = −be 2 e 2 − , b we have ((2n+1)π)2 (4n+3)π2 (N − 1)¯b V (t ) ≤ −be 2 e 2 − +r ¯(t ), (5.30) n b n
(4n+3)π2 wherer ¯(tn) is bounded. As e 2 will dominate any bounded functions with suffi- cient large n, we can conclude from (5.30), V (tn) < 0 for sufficient large n’s. This is a contradiction, as V (t) is a positive definite function. Hence, none of the κi’s becomes unbounded, and therefore boundedness of κi’s and V are established.
For the case sgn(bi,ρ) = −1, the proof can be carried out in the same way as for the case sgn(bi,ρ) = 1, and is omitted here. CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 87 5.4 Consensus Output Regulation Design, ρ = 1
For the subsystems with relative degree ρ = 1, the filtered transformation is not needed and we have ξi,1 = ui in (5.15). The output dynamics of the subsystem i are
˜ z˜˙i = Diz˜i + ψi, (5.31) ˜ T e˙i =z ˜i,1 + ψi,y + bi,ρui − li ηi.
Note that ei is not available for control design. The control design is based on ζ = Qe.
Since the high-frequency gain bi,ρ is completely unknown, we use the Nussbaum gain proposed in the previous section for adaptive control
ui = γN(κi)¯ui, (5.32) κ˙ i = ζiu¯i, whereκ ˙ i(0) = 0, the Nussbaum gain N is defined in (5.25), and γ is a positive real design parameter. From (5.31) and (5.32), we have
˜ T e˙i =z ˜i,1 + ψi,y + (γbi,ρN(κi) − 1)¯ui +u ¯i − li ηi. (5.33)
We now designu ¯i as ˆ 2p−1 ˆT u¯i = −c1ζi − ki0(ζi + ζi ) + li ηˆi, (5.34) ˆ ˆ where c1 ≥ 2 is a constant design parameter, li is an estimate of li, ki0 is an estimate of unknown positive constant ki0 andη ˆi is generated from (5.16) with ξi,1 = ui. Using (5.18), we have the resultant error dynamics
ˆ 2p−1 ˜ e˙i = −c1ζi − ki0(ζi + ζi ) +z ˜i,1 + ψi,y + (γbi,ρN(κi) − 1)¯ui (5.35) T ˜ −1 T − li η˜i − liηˆi + bi,ρ li Giei,
˜ ˆ where li = li − li. The adaptive laws are given by
ˆ 2 2p ki0 = ζi + ζi , (5.36) ˆ li = −ηˆiζi.
Note that the control design and adaptive laws for subsystem i only use ζi that is available via network connection to the subsystem, and therefore the control design and adaptive laws are decentralised. CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 88
Theorem 5.1. The distributed control inputs (5.32) and adaptive laws (5.36) solve the distributed adaptive consensus output regulation problem when the relative degree ρ = 1, and the regulation error e converges to zero asymptotically.
Proof. Define N 1 X V = eT Qe + (˜lT ˜l + k˜2 ), (5.37) e 2 i i i0 i=1 ˜ ˆ where ki0 = ki0 − ki0. It can be obtained that
N ˙ X ˜T ˆ˙ ˜ ˆ˙ Ve = (ζie˙i − li li − ki0ki0). (5.38) i=1 From (5.35), (5.36) and Young’s inequality in Lemma 3.2, we have
N ˙ X 2 2 2p ˜ Ve = −c1ζi − ki0(ζi + ζi ) + ζiz˜i,1 + ζiψi,y i=1 T −1 T + (γbi,ρN(κi) − 1)ζiu¯i − ζili η˜i + ζibi,ρ li Giei N (5.39) X 1 ≤ −k (ζ2 + ζ2p) + (γb N(κ ) − 1)ζ u¯ + kz k2 i0 i i i,ρ i i i 2 i i=1 1 1 1 + kl k2kη˜ k2 + |b−1lT G |2|e |2 + kψ˜ k2 . 2 i i 2 i,ρ i i i 2 i,y
To analyse the dynamics ofz ˜i, let
N X T Vz = z˜i Pi,zz˜i, (5.40) i=1 where Pi,z is a positive definite matrix that satisfies
T Di Pi,z + Pi,zDi = −3I.
From (5.31), we have
N ˙ X 2 ˜ Vz = (−3kz˜ik + 2˜ziPi,zψi) i=1 (5.41) N X 2 2 ˜ 2 ≤ (−2kz˜ik + kPi,zk kψik ). i=1
Then, we consider the stability ofη ˜i. Let
N X 2 Vη = η˜i Pi,ηη˜i, (5.42) i=1 CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 89
where Pi,η is a positive definite matrix that satisfies
T Fi Pi,η + Pi,ηFi = −5I.
From (5.18), we obtain that
N ˙ X 2 −1 2 2 Vη ≤ −2kη˜ik + kbi,ρ Pi,ηFiGik |ei| i=1 (5.43) −1 2 2 −1 2 ˜ 2 +kbi,ρ Pi,ηGik kz˜ik + kbi,ρ Pi,ηGik |ψi,y| .
Finally, define a Lyapunov function candidate
V = Ve + β1Vz + β2Vη, (5.44) where 1 β ≥ kl k2, 2 4 i 1 1 β ≥ + β kb−1P G k2. 1 4 2 2 i,ρ i,η i From (5.39), (5.41) and (5.43), we have
N ˙ X 2 2p V ≤ −ki0(ζi + ζi ) + (γbi,ρN(κi) − 1)ζiu¯i i=1 1 −1 T 2 −1 2 2 + |b l Gi| + β2kb Pi,ηFiGik |ei| 2 i,ρ i i,ρ (5.45) 1 + + β kb−1P G k2 |ψ˜ |2 2 2 i,ρ i,η i i,y i 2 ˜ 2 + β1kPi,zk kψik .
˜ ˜ ˜ Since the nonlinear functions involved in ψi and ψi,y are polynomials with ψi(0, ω, θi, σ) ˜ = 0 and ψi,y(0, ω, θi, σ) = 0, ω is bounded, and the unknown parameters are constants, it can be shown that ˜ 2 2 2p kψik ≤ r¯z(ei + ei ), (5.46) ˜ 2 2 2p kψi,yk ≤ r¯y(ei + ei ), ˜ ˜ where p is a known positive integer, depending on the polynomials in ψi and ψi,y, and r¯z andr ¯y are unknown positive real constants.
From (5.46), there exists a positive constant r0 such that
N ˙ X 2 2p 2 2p V ≤ −ki0(ζi + ζi ) + (γbi,ρN(κi) − 1)ζiu¯i + r0(ei + ei ) . (5.47) i=1 CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 90
From Lemma 5.3, we can set
2 p−1 2p ki0 ≥ r0 max{λmax(Q),U λmax(Q)}, (5.48) which results in N ˙ X V ≤ (γbi,ρN(κi) − 1)κ ˙ i. (5.49) i=1 The boundedness of V can be established based on the Nussbaum gain properties (4.26) via an argument of contradiction. In fact, integrating (5.49) gives
N N X Z κi X V (t) ≤ γbi,ρN(si)dsi − κi(t) + V (0). (5.50) i=1 0 i=1 PN If i=1 κi(t) is not bounded from above, if can be shown that the right-hand side of (5.50) will be negative at some instants of time, which is a contradiction, since the left-hand side of (5.50) is non-negative. Therefore, κi’s are bounded, which imply the boundedness of V , and we can further conclude that the boundedness of all variables in the adaptive control system and limt→∞ e(t) = 0 from Barbalat’s Lemma.
5.5 Consensus Output Regulation Design, ρ > 1
For system (5.15), when ρ > 1, we cannot assign values of ξi,1, but a desired value of ˆ ξi,1, denoted by ξi,1, can be determined in a similar way to the control design for the case of ρ = 1 shown in the previous section. The final control ui can be obtained by adaptive backstepping, following the dynamics of the filters in ρ − 1 steps. ˆ We first design the desired values ξi,1 as ˆ ¯ ξi,1 = γN(κi)ξi,1, (5.51) ¯ κ˙ i = ζiξi,1, whereκ ˙ i(0) = 0, the Nussbaum gain N is defined in (5.25). From (5.15) and (5.51), we have
˜ ¯ ¯ ˜ ˜ ˆ ˜ T e˙ =z ˜i,1 + ψi,y + (γbi,ρN(κi) − 1)ξi,1 + ξi,1 + bi,ρξi,1 + bi,ρξi,1 − li ηi, (5.52)
ˆ ˜ ˆ ˜ ¯ where bi,ρ is an estimate of bi,ρ, bi,ρ = bi,ρ − bi,ρ, and ξi,1 = ξi,1 − ξi,1. We design ¯ ˆ 2p−1 ˆT ξi,1 = −c1ζi − ki0(ζi + ζi ) + li ηˆi, (5.53) CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 91
ˆ ˆ where li is an estimate of li, ki0 is an estimate value of unknown positive constant ki0.
From (5.15) and (5.52), the resultant dynamics of ei are obtained as
ˆ 2p−1 ˜ ¯ e˙i = −c1ζi − ki0(ζi + ζi ) +z ˜i,1 + ψi,y + (γbi,ρN(κi) − 1)ξi,1 (5.54) ˜ ˜ ˆ ˜ T ˜ −1 T + bi,ρξi,1 + bi,ρξi,1 − li η˜i − liηˆi + bi,ρ li Giei.
In the second step, we have
˜˙ ˙ ˆ˙ ξi,1 = ξi,1 − ξi,1 (5.55) ˆ˙ = −λ1ξi,1 + ξi,2 − ξi,1.
ˆ ˆ ˆ Note that ξi,1 is a function of ζi, ki0, li, ηˆi and κi. Hence, we have
˜˙ ξi,1 = −λ1ξi,1 + ξi,2 ˆ ˆ ˆ ˆ ∂ξi,1 ˙ ∂ξi,1 ˙ ∂ξi,1 ∂ξi,1 − kˆ − ˆl − ηˆ˙ − κ˙ ˆ i0 ˆ i i i ∂ki0 ∂li ∂ηˆi ∂κi (5.56) ˆ N ∂ξi,1 X − q (˜z + ψ˜ + b ξ − lT η ). ∂ζ ij j,1 j,y j,ρ j,1 j j i j=1
ˆ Based on (5.54) and (5.56), we design ξi,2 as
ˆ !2 ˆ ˆ ˜ ∂ξi,1 ˜ ξi,2 = λ1ξi,1 − bi,ρζi − c2,1ξi,1 − c2,2 ξi,1 ∂ζi ˆ ˆ ˆ ˆ ∂ξi,1 ˙ ∂ξi,1 ˙ ∂ξi,1 ∂ξi,1 + kˆ + ˆl + ηˆ˙ + κ˙ (5.57) ˆ i0 ˆ i i i ∂ki0 ∂li ∂ηˆi ∂κi ˆ N ∂ξi,1 X + q (ˆb ξ − ˆlT ηˆ ), ∂ζ ij j,ρ j,1 i j i j=1 where c2,1 and c2,2 are positive constant parameters which are designed later. The ˜ resultant dynamics of ξi,1 are obtained as
ˆ !2 ˜˙ ˆ ˜ ∂ξi,1 ˜ ˜ ξi,1 = −bi,ρζi − c2,1ξi,1 − c2,2 ξi,1 + ξi,2 ∂ζi (5.58) ˆ N ∂ξi,1 X − q (˜z + ψ˜ − ˜b ξ − lT η˜ − ˜l ηˆ + b−1lT G e ). ∂ζ ij j,1 j,y j,ρ j,1 j j j j j,ρ j j j i j=1
˜ If the relative degree ρ = 2, we have ξi,2 = 0 and we set
ˆ ui = ξi,2. (5.59) CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 92
In such a case, we can design the adaptive laws as
N ˆ ! ˙ X ∂ξj,1 ˆb = ζ ξ˜ − q ξ˜ ξ , i,ρ i i,1 ji ∂ζ j,1 i,1 j=1 j (5.60) N ˆ ! ˙ X ∂ξj,1 ˆl = −ζ ηˆ + q ξ˜ ηˆ . i i i ji ∂ζ j,1 i j=1 j
Note that the control inputs in (5.57) and adaptive laws in (5.60) are still decentralised, because they only use the information of the subsystems that are connected to the subsystem i. Clearly, control design can be carried out for ρ > 2 by following the same design procedures, but more tedious. We will not show the details of the design here. Instead, we show the result of the stability analysis for the case for ρ = 2.
Theorem 5.2. The decentralised control inputs (5.59) and adaptive laws (5.60) solve the adaptive consensus output regulation problem when the relative degree ρ = 2, and the regulation errors ei converge to zero asymptotically.
Proof. Let N 1 X V = eT Qe + (˜b2 + ˜lT ˜l + k˜2 + ξ˜2 ), (5.61) e 2 i,ρ i i i0 i,1 i=1
From (5.36), (5.54), (5.58) and (5.60), with c1 ≥ 2 and c2,2 ≥ 2, we have
N ˙ X 2 2 2p ˜ ¯ Ve = (−c1ζi − ki0(ζi + ζi ) + ζiz˜i,1 + ζiψi,y + (γbi,ρN(κi) − 1)ζiξi,1 i=1 ˆ !2 T −1 T ˜2 ∂ξi,1 ˜2 − ζili η˜i + ζibi,ρ li Giei − c2,1ξi,1 − c2,2 ξi,1 ∂ζi ˆ N ∂ξi,1 X − ξ˜ q (˜z + ψ˜ − lT η˜ + b−1lT G e ) ∂ζ i,1 ij j,1 j,y j j j,ρ j j j i j=1 N ˆ !2 X ∂ξi,1 ≤ (−k (ζ2 + ζ2p) + (γb N(κ ) − 1)ζ ξ¯ − (c − 2) ξ˜2 i0 i i i,ρ i i i,1 2,2 ∂ζ i,1 (5.62) i=1 i 1 − c ξ˜2 + (kz˜ k2 + kψ˜ k2 + kl k2kη˜ k2 + |b−1lT G |2|e |2) 2,1 i,1 2 i i,y i i i,ρ i i i N 1 X + q2 (kz˜ k2 + kψ˜ k2 + kl k2kη˜ k2 + |b−1lT G |2|e |2)) 2 ij j j,y j j j,ρ j j j i=1 N X 2 2p ¯ ˜2 ≤ (−ki0(ζi + ζi ) + (γbi,ρN(κi) − 1)ζiξi,1 − c2,1ξi,1 i=1 1 + (1 + kQk2)(kz˜ k2 + kψ˜ k2 + kl k2kη˜ k2 + |b−1lT G |2|e |2). 2 i i,y i i i,ρ i i i CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 93
It can be seen that the expression in (5.62) is similar to (5.39) when ρ = 1, and the dynamics ofz ˜i andη ˜i can be analysed in the same way as in the proof of Theorem 5.1 by using the same Vz and Vη. The rest of the proof follows similarly to the proof of Theorem 5.1.
5.6 Example
In this section, we use an example to demonstrate the application of the proposed control design, which works for the subsystems of different orders as long as the relative degrees are the same. We will consider a network connected systems with 5 subsystems of different order. For i = 1, 3, 5, each subsystem is described by a third-order nonlinear model
x˙ i,1 = xi,2,
3 x˙ i,2 = xi,3 + µi,1yi ω1 + bi,2ui,
3 x˙ i,3 = µi,2yi + bi,3ui,
yi = xi,1, and for i = 2, 4, each subsystem is described as a second-order nonlinear model
2 x˙ i,1 = xi,2 + µi,1yi ω1,
3 x˙ i,2 = µi,1xi,2 + µi,3ω2 + bi,2ui,
yi = xi,1, with the exosystem 0 σ h i ω˙ = ω, g(ω) = 1 0 ω. −σ 0 Note that the exosystem generates sinusoidal functions with frequency σ. The desired trajectory q(ω) = ω1 which is only available to the second subsystem. The adjacency matrix and the result Q are given by 0 1 1 0 1 3 −1 −1 0 −1 1 0 1 1 0 −1 4 −1 −1 0 A = 1 1 0 1 0 ,Q = −1 −1 3 −1 0 . 0 1 1 0 1 0 −1 −1 3 −1 1 0 0 1 0 −1 0 0 −1 2 CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 94
It can be seen that the system considered here satisfies Assumption 5.2 and 5.3. In order to satisfy Assumption 5.1, we can choose suitable unknown parameters in the simulation. By calculation, the desired feedforward controls αi can be shown to contain the sinusoidal functions of frequencies σ and 3σ. The frequency components of 3σ are due to the nonlinear functions in the system. For the internal model design,
4×4 we need to choose Fi ∈ R . It can also shown that p = 3 is enough. The design of control inputs then follows the exact steps shown in Section 5.5.
In the simulation study, we set the unknown parameters µi,j = 0.1, and the other unknown parameters as 1, including bi,2 and σ. Then we used 0 1 0 1 0 0 0 1 0 0 Fi = ,Gi = , 0 0 0 1 0 −16 −32 −24 −8 10 for the internal models, and the control parameters were set as 1 except γ = 0.002 and c2,2 = 5. Simulation results for the output and the control inputs of the subsystems are shown in Fig. 5.1 and Fig. 5.2 , which shows that the outputs for subsystems 1, 3, 5 and subsystems 2, 4 converge to the same value, in other words, reach the consensus.
2.5
y1 2 y2 y3 y4 y5 1.5
1
0.5
0
-0.5 outputs of subsystems
-1
-1.5
-2
-2.5 0 2 4 6 8 10 12 14 16 18 20 t
Fig. 5.1. The subsystem outputs. CHAPTER 5. CONSENSUS OUTPUT REGULATION OF MASS 95
50
0
u1 -50 u2 u3 u4 u5 -100
-150 control inputs of subsystems
-200
-250 0 2 4 6 8 10 12 14 16 18 20 t
Fig. 5.2. The subsystem control inputs.
5.7 Summary
In this chapter, we have proposed a new Nussbaum gain which one of the subsystem Nussbaum function grows faster such that it can dominant for the positive definite condition for the consensus control. This newly Nussbaum gain then used together with designed adaptive laws and internal models for adaptive consensus output regula- tion of a class of network-connected output feedback nonlinear systems with different nonlinear dynamics. The adaptive laws depend only on the information available from the subsystems in the neighbourhood, which are decentralised. The proposed exam- ple demonstrates that the control scheme ensures the convergence of the subsystem outputs to a common trajectory and the output regulation errors converge to zero asymptotically. Chapter 6
Distributed Adaptive Consensus Output Tracking of Nonlinear Systems on Directed Graphs with Unknown High-Frequency Gains
In this chapter, we consider the consensus output tracking problem of a class of network-connected uncertain nonlinear systems by output feedback. Each subsystem is a minimum-phase SISO system with relative degree one and unknown parameters and unknown control direction, and the connection graph among the subsystems is directed. A distributed adaptive controller together with a novel Nussbaum gain and an adaptive internal model are presented to achieve consensus output tracking in the sense that the subsystem outputs asymptotically follow a reference signal. The proposed adaptive control only uses relative output measurements and the local in- formation of the connection to each subsystem, and hence the proposed controller is fully distributed. The contributions of this chapter is at least two-fold. First, contrary to previous works, the parameters of the subsystems in this chapter are completely unknown and the connection between subsystems is direct, which make the design of Nussbaum gain, internal model and the Lyapunov function becomes much more chal- lenging because of the asymmetry of the Laplacian matrices. Second, the adaptive
96 CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 97 protocols proposed in this chapter depend on only the relative output information, which is much more difficult in contrast to the adaptive protocols that rely on the relative states of neighbouring subsystems. The rest of this chapter is organized as follow. The distributed adaptive consensus output tracking problem of a group of unknown nonlinear subsystems with unknown control directions over directed graphs is formulated in Section 6.2. The state trans- formation is shown in Section 6.3. Section 6.4 presents a new adaptive internal model, the design of consensus controllers and the stability analysis. Simulation examples are demonstrated in Section 6.5. Section 6.6 summaries this chapter.
6.1 Problem Statement
In this chapter, we consider a group of N unknown nonlinear subsystems, of which the dynamics of the i-th subsystem are described by
x˙ i = Acxi + φ(yi, d) + bui, (6.1) yi = Cxi, with φ, b, CT ∈ Rn and 0 1 0 ... 0 1 0 0 1 ... 0 b1 0 ...... . T Ac = . . . . . , b = . ,C = . , . 0 0 0 ... 1 bn 0 0 0 0 ... 0 n for i = 1,...,N, where xi ∈ R is the state vector, with n being a known positive constant integer denoting the order of the subsystems yi, ui ∈ R are the output and input of the ith subsystem, and d ∈ Rq and b ∈ Rn are vectors of unknown parameters, with b being a Hurwitz vector with b1 6= 0, which implies the relative degree of the system is 1, φ : R × Rq → Rn contains unknown nonlinear functions with each element being polynomials of its variables and satisfies φ(0, d) = 0. We take the following system as the reference signal, which is described by
x˙ 0 = Acx0 + φ(y0, d) + bu0, (6.2) y0 = Cx0, CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 98
with a constant control input u0. The output tracking error ei is then defined as
ei = yi − y0, i = 1,...,N. (6.3)
To describe the communications among the subsystems, a directed graph G is introduced that consists of a set of vertices denoted by V = {1,...,N} and a set of edges denoted by E. A vertex represents a subsystem, and each edge represents a con- nection. A directed graph is strongly connected if there is a directed path from every vertex to every other vertex. For the graph G, its adjacency matrix A has elements aij = 1 if there is a path from subsystem j to subsystem i, and aij = 0 otherwise. PN The Laplacian matrix L is defined as lii = j=1 aij and lij = −aij when i 6= j. In our setup, not every subsystem has access to y0, and the consensus output tracking will be achieved through the network connections. We use a diagonal matrix ∆ = diag(δ1, . . . , δN ) to denote the access to y0 in the way that if δi = 1, the ith subsystem has access to the value of y0 for the control design, and δi = 0 otherwise. The distributed adaptive consensus output tracking problem considered in this chapter is to design a distributed adaptive controller using the relative output infor- mation yi − yj, i 6= j of neighbouring subsystems to ensure the convergence to zero of output tracking errors ei for i = 1,...,N under any initial condition of the system in the state space, which implies the convergence of the subsystem outputs yi to the common function y0. Regarding the interactions among the subsystems and the exosystem, we make the following assumptions.
Assumption 6.1. The invariant zeros of {Ac, b, C} are stable, for i = 1,...,N, and all the subsystems have the same sign but completely unknown high-frequency gains.
Assumption 6.2. The directed graph G among the N subsystems is strongly con- nected and at least one subsystem has access to y0.
Assumption 6.3. For the nonlinear function φ, the following condition holds:
2 2 2p kφ(yi, d) − φ(y0, d)k ≤ γφ(ei + ei ), (6.4)
where γφ is a positive real number, and p is a known positive integer. CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 99
Remark 6.1. Note that the subsystem (6.1) is in the standard nonlinear output feedback form. The geometric conditions have been identified in [123] that any general nonlinear systems can be transformed to such a structure.
Remark 6.2. Assumption 3 is clearly satisfied for linear systems with unknown pa- rameters. The nonlinear functions involved in φi are polynomials with φ(0, d) = 0, and the unknown parameters are constant. In such a case, Assumption 3 is then satisfied.
6.2 Preliminary Results
n−1 We introduce a state transform to extract the internal dynamics of (6.1) with zi ∈ R given by b2:n zi = xi,2:n − yi, (6.5) b1 where (·)2:n refers to the vector or matrix formed by 2nd row to the nith row. With the coordinates (zi, yi), (6.1) is rewritten as
z˙i = Bzi + ψ(yi, θ), (6.6) T y˙i = h zi + ψy(yi, θ) + b1ui, where the unknown parameter vector θ = [dT , bT ]T , h = [1, 0, ··· , 0] ∈ Rn−1, and B is the left companion matrix of b given by −b2/b1 1 ... 0 −b3/b1 0 ... 0 . . . . B = . . . . , (6.7) −bn−1/b1 0 ... 1 −bn/b1 0 ... 0 and b2:n b2:n ψ(yi, θ) = B yi + φ2:n(yi, d) − φ1(yi, d), b1 b1 b2 ψy(yi, θ) = yi + φ1(yi, d). b1 Note that B is Hurwitz from Assumption 1, and it is easy to check that ψ(0, θ) = 0 and ψy(0, θ) = 0. We have the following results on the Laplacian matrix before the introduction of CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 100 control design. For notational convenience, we let Q = L + ∆. Under Assumption 6.2, it is not difficulty to verify that Q is a nonsingular M-matrix, which satisfies the following property.
Lemma 6.1. [87,116] There exists a positive diagonal matrix G with G = diag{g1, ··· , gN } such that
T GQ + Q G ≥ r0I, (6.8) for some positive real number r0.
We define the consensus regulation error
N X ζi = aij(yi − yj) + δi(yi − y0), i = 1, ··· ,N. (6.9) j=1
Using the notation qij to denote the (i, j)-th entry of the matrix Q, then
N X ζi = aij(ei − ej) + δiei j=1 (6.10) N X = qijej, j=1
N or in the vector form, ζ = Qe where ζ, e ∈ R are the vectors with ζi and ei as elements, respectively. Clearly, ζi is available to the control design for the ith subsystem. For the leader, the internal dynamics have the same formulation as (6.6), only change index i to 0. Then, letz ˜i = zi − z0, the subsystem dynamics in (6.1) can be rewritten as ˜ z˜˙i = Bz˜i + ψi, (6.11) T ˜ e˙i = h z˜i + ψi,y + b1(ui − u0), ˜ ˜ where ψi = ψ(yi, θ) − ψ(y0, θ), ψi,y = ψy(yi, θ) − ψy(y0, θ).
6.3 Control Design
In this section, we will design a distributed adaptive output feedback control law that globally stabilizes the augmented system (6.11) under the assumption that the high-
2 ki /2 2 frequency gain b1 is completely unknown. Denote N(ki) = e (ki + 2) sin(ki), which is a type of Nussbaum function proposed in last chapter. CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 101
Lemma 6.2. Consider the closed-loop system composed of (6.11) and the following control law:
ui = γN(ki)¯ui + ξi, (6.12) ˙ 2p−1 ki = r0(fi + ρi)(ζi + ζi )¯ui, ki(0) = 0, for i = 1,...,N, where r0 is illustrated in (6.8), γ is a positive real design parameter, and 2p−1 u¯i = −(fi + ρi)(ζi + ζi ), (6.13)
2 with ρi = ζi and fi, ξi are generated by
˙ 2 2p fi = γf (ζi + ζi ), fi(0) = f0, (6.14) ˙ ξi = −ξi + ui, with γf , f0 being any known positive constants. Note that fi can be viewed as an adaptive gain. Then, there exists a Lyapunov function candidate V (t) such that, along the trajectory of the closed-loop system
N ˙ X ˙ V ≤ (γb1N(ki) − 1)ki + c(t) (6.15) i=1 where c(t) is a bounded function.
Proof. Let
−1 η˜i = u0 − ξi + b1 ei, (6.16) from (6.11) and (6.14), it can be shown that
˙ −1 −1 T −1 ˜ η˜i = −η˜i + b1 ei + b1 h z˜i + b1 ψi,y. (6.17)
The closed-loop subsystem dynamics of ei can be obtained as
T ˜ 2p−1 e˙i = h z˜i + ψi,y − (fi + ρi)(ζi + ζi ) (6.18) + (γb1N(ki) − 1)¯ui − b1η˜i + ei.
Next, we define the Lyapunov function candidate as
N 2p 2p+2 X ζ2 ζ ζ4 ζ V = 2g f i + i + i + i ζ i i 2 2p 4 2p + 2 i=1 (6.19) N 1 X + (f − f ∗)2, 2γ i f i=1 CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 102
∗ where gi is defined as in Lemma 6.1 and f is a constant to be determined later. Using (6.11) and (6.12), we have
N N ˙ X 2p−1 X Vζ = 2gi(fi + ρi)(ζi + ζi ) qije˙j i=1 i=1 N 2p N X ζ 1 X + g ζ2 + i f˙ + (f − f ∗)f˙ i i p i γ i i i=1 f i=1 N N X 2p−1 X = 2gi(fi + ρi)(ζi + ζi ) qij (γb1N(kj) − 1)u ¯j i=1 j=1 (6.20) T p−1 T ˜ + 2ζ (F + ρ)(IN + ρ )GQ (IN ⊗ h )˜z + Ψy
T p−1 T p−1 − ζ (F + ρ)(IN + ρ )(GQ + Q G)(IN + ρ )(F + ρ)ζ N ρp X + γ ρ + G(ρ + ρp) + ζT F(I + ρp−1)ζ − f ∗ (ζ2 + ζ2p) f p N i i i=1 T p−1 + 2ζ (F + ρ)(IN + ρ )G (−b1Qη˜ + Qe) ,
T T T T T T T T T T T T ˜ where e = [e1 , . . . , eN ] , ζ = [ζ1 , . . . , ζN ] , z˜ = [˜z1 ,..., z˜N ] , η˜ = [˜η1 ,..., η˜N ] , Ψy = ˜T ˜T T [ψ1,y,..., ψN,y] , F = diag(f1, . . . , fN ), ρ = diag(ρ1, . . . , ρN ). Note that ζ = Qe, and from (6.8), (6.12) and Young’s inequality, we have
N X 8 V˙ ≤ (γb N(k ) − 1)k˙ − r k(F + ρ)(I + ρp−1)ζk2 ζ 1 i i 12 0 N i=1 12 + kGQk2kz˜k2 + kGQk2kΨ˜ k2 + kb GQk2kη˜k2 r y 1 0 (6.21) ρp + kGk2kζk2 + γ ρ + G(ρ + ρp) f p N T p−1 ∗ X 2 2p + ζ F(IN + ρ )ζ − f (ζi + ζi ), i=1 Similar to [90], we can obtain that ρp r γ ρ + G(ρ + ρp) ≤ 0 k(F + ρ)(I + ρp−1)ζk2 + ν(r )ρ, (6.22) f p 12 N 0 where ν : R+ → R+ is a function that depends on unknown parameters, and we used
T p−1 r0 p−1 2 3 2 ζ F(IN + ρ )ζ ≤ k(F + ρ)(IN + ρ )ζk + kζk , (6.23) 12 r0 and from Assumption 6.3, it can be shown that there exists a positive real constant
νy such that N N 12 X X kΨ˜ k2 ≤ γ (e2 + e2p) ≤ ν (ζ2 + ζ2p). (6.24) r y y i i y i i 0 i=1 i=1 CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 103
From (6.21), (6.22), (6.23) and (6.24), we have
N X r0 V˙ ≤ (γb N(k ) − 1)k˙ − k(F + ρ)(I + ρp−1)ζk2 ζ 1 i i 2 N i=1 12 2 2 2 2 ∗ 3 + kGQk kz˜k + kb1GQk kη˜k − f − (6.25) r0 r0 N 12 X −kGQk2ν − kGk2 − ν(r ) (ζ2 + ζ2p). y r 0 i i 0 i=1
To analyse the dynamics ofz ˜i, let
N X T Vz = z˜i Pzz˜i. (6.26) i=1
Since B is Hurwitz, there exists a positive definite matrix Pz such that
T PzB + B Pz = −3I.
From (6.12), it can be obtained that
N N ˙ X 2 X T ˜ Vz = −3 kz˜ik + 2 z˜i Pzψi i=1 i=1 (6.27) N 2 2 X 2 2p 6 −2kz˜k + kPzk µψ (ζi + ζi ), i=1 where µψ is a positive real constant. Then, we consider the stability ofη ˜i. Let
N X 2 Vη = η˜i , (6.28) i=1 from (6.17), it can be obtained that
N N ˙ X 2 X −1 −1 T −1 ˜ Vη = −2 η˜i + 2 η˜i(b1 ei + b1 h z˜i + b1 ψi,y) i=1 i=1 N N N ! X X 1 khk2 1 X − η˜2 + 3 e2 + kz˜ k2 + µ (ζ2 + ζ2p) (6.29) 6 i b2 i b2 i b2 ψy i i i=1 i=1 1 1 1 i=1 N khk2 1 1 X −kη˜k2 + 3 kz˜k2 + 3 kQ−1k2 + µ (ζ2 + ζ2p), 6 b2 b2 b2 ψy i i 1 1 1 i=1
where µψy is a positive real constant. Finally, let
V = Vζ + 2β1Vη + β2Vz, (6.30) CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 104
where β1 and β2 are positive constants satisfying
12 2 β1 = kb1GQk , r0 2 β1khk 12 2 β2 = 6 2 + kGQk b1 r0 and setting ∗ 2 12 2 3 f = β3 + kGQk νy + kGk + ν(r0) + r0 r0 2 1 −1 2 1 + β2kPzk µψ + 6β1 2 kQ k + 2 µψy b1 b1 where β3 is a positive constant. Then, we can obtain that
N N ˙ X ˙ X 2 2p V ≤ (γb1N(ki) − 1)ki − β3 (ζi + ζi ) i=1 i=1 (6.31) 2 2 − β2kz˜k − β1kη˜k .
The proof is completed based on above analysis.
Now we will show that, with the Lyapunov-like function V (t) and inequality (6.15), the stability of the closed-loop multi-agent system can be established. For this purpose, a lemma for is given below for convenience.
Lemma 6.3. Let V (t) and ki(t), i = 1,...,N, be smooth functions defined on [0, tf )
2 ki /2 2 with V (t) ≥ 0 and ki(0) = 0. Also let N(ki) = e (ki + 2) sin(ki). If the following inequality holds for any t ∈ [0, tf ):
N Z t X ˙ V (t) ≤ V (0) + γb1N(ki(τ))ki(τ)dτ 0 i=1 (6.32) N Z t X ˙ − ki(τ)dτ + r(t), i=1 0 where r represents some suitable constant, then V (t), ki(t) for i = 1,...,N and PN R t ˙ i=1 0 γb1N(ki(τ))ki(τ)dτ are bounded on [0, tf ).
Use Lemma 6.2 and Lemma 6.3, we can conclude that, for any given initial con- dition, all ki(t), 1 ≤ i ≤ N, in the closed-loop system are bounded on [0, tf ). More- PN R t ˙ over, V (t), i=1 0 γb1N(ki(τ))ki(τ)dτ are bounded on [0, tf ). Since V (t) is a proper positive definite function in ζi, z˜i andη ˜i, i = 1,...,N, ζi, z˜i andη ˜i, i = 1,...,N, are CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 105
bounded on [0, tf ). Therefore, finite-time escape cannot occur and tf = ∞, that is ζi, z˜i andη ˜i, i = 1,...,N, are bounded for all t ≥ 0. As a result, from (6.18),e ˙i, i = 1,...,N are bounded for all t ≥ 0. Using Barbalat’s lemma, we can show that limt→∞ ei(t) = 0 for i = 1,...,N.
6.4 Simulation Example
In this section, an example is provided to validate the effectiveness of the proposed control design for adaptive consensus output tracking. The system under consideration is a connection of 4 subsystems, each of them is described by a second-order state space model as 3 x˙ i,1 = xi,2 + d(yi − 0.3y ) + b1ui, y (6.33) x˙ = − i + b u , i,2 d 2 i with yi = xi,1, where d, b1 and b2 are unknown positive real parameters. Note that when ui = 0, the system is a van der Pol oscillator, and its trajectories are bounded. Hence, it can be shown that Assumption 6.3 is satisfied with p = 3. For the reference signal, the formation is the same as (6.33) but with u0 = 2. Then, we assume interaction graph among the subsystems is 0 0 1 0 1 0 0 0 A = , 0 0 0 1 0 1 0 0 and only subsystem 1 and 3 have access to y0, and thereby the result Q is given by 2 0 −1 0 −1 1 0 0 Q = . 0 0 2 −1 0 −1 0 1
According to Lemma 6.2, the distributed adaptive controller is designed as the format (6.12),(6.13) and (6.14) for i = 1,..., 4.
Simulation study has been carried out with the parameters b1 = b2 = 1, and CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 106
r0 = 1, γ = 1 and γf = 5. The parameter d is set as 0.2 for 0 ≤ t ≤ 30, d = 2 for t ≥ 30, so that two different limit cycles are used as the trajectories of the reference signal. The simulation results of the subsystems outputs and states are shown in Fig. 6.1,
Fig. 6.2. Fig. 6.3 and Fig. 6.4 show the adaptive gains ki and the tracking errors. It can be seen that both the output and the states converge to the reference signal trajectory, which are shown in Fig. 6.1, and Fig. 6.2 with the initial values [2, −1]T , and the Fig. 6.3 shows that the adaptive gains are bounded, as well as the tracking errors in Fig. 6.4. The control inputs are shown in Fig. 6.5 and the specific one is shown in Fig. 6.6. It is also noted that the trajectories are different after 30s in the simulation, due to the change in the value of d.
6.5 Summary
In this chapter, we have proposed a new distributed adaptive control design to solve the consensus output tracking problem for strongly connected nonlinear multi-agent systems with unknown high-frequency gains. The newly proposed internal models are used to generate the contribution of the desired input compensation to state variable, which is then used in the control design. These internal models together with the adaptive laws and the Nussbaum gains tackle both the unknown connectivity and the unknown parameters in the subsystem dynamics. The presented protocols do not use any global information of the graph connection, and a distinct feature is that they can be designed by each agent in a fully distributed manner. Finally, the adaptive laws and the control design ensure the asymptotic zero convergence of the regulation errors of all the subsystems. Further research could be carried out to relax some of the assumptions made in the proposed design. One direction would be to allow subsystems to have different dynamics. It is not expected to be straightforward for the class of nonlinear output regulation with nonlinear exosystem over directed graphs. However, it is a good chance to extend results for uncertain linear subsystems somehow. CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 107
4 y0 y1 y2 3 y3 y4
2
1
0 outputs of subsystems -1
-2
-3 0 50 100 150 200 250 300 t
Fig. 6.1. The subsystem outputs yi, i = 0,..., 4.
30
25 x02 x12 x22 x32 20 x42
15 i,2 10 state x
5
0
-5
-10 0 50 100 150 200 250 300 t
Fig. 6.2. The subsystem states xi,2, i = 0,..., 4. CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 108
4.5
4
k1 k2 3.5 k3 k4
3 i 2.5 gain k
2
1.5
1
0.5 0 50 100 150 200 250 300 t
Fig. 6.3. The adaptive gains ki, i = 1,..., 4.
2.5
1
2 2 3
4 1.5
1
i 0.5
0
-0.5
-1
-1.5 0 50 100 150 200 250 300 t
Fig. 6.4. The tracking errors ζi, i = 1,..., 4. CHAPTER 6. DISTRIBUTED CONSENSUS OUTPUT TRACKING 109
50
X 246.8 Y 2 0
u1 u2 u3 -50 u4
-100
-150 control inputs of subsystems
-200
-250 0 50 100 150 200 250 300 t
Fig. 6.5. The control inputs ui, i = 1,..., 4.
10
5
0 u1 u2 u3 u4
-5 control inputs of subsystems
-10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t
Fig. 6.6. The control inputs ui, i = 1,..., 4 when t between 0-2s. Chapter 7
Conclusions and Future Works
Since summaries and detailed conclusions have been made at the end of each chapter, in this chapter we give some general conclusions, and some potentially future works will be discussed.
7.1 Final Conclusions
In this thesis, the Nussbaum gain based approaches are employed in the output regula- tion of a variety of nonlinear systems consisting of various type exogenous signals with unknown high-frequency gains. All systems considered are generally in the assumption of output feedback form, which facilitates our research.
In the inspiration of the well-established previous publications mentioned in Chap- ter 1, 2 and 3, with the assumption on the existence of an invariant manifold and internal model, the primary concerns of the output regulation are generalised as:
1. The closed-loop system influence from the uncertain parameters of the exosystem to the system dynamics themselves.
2. The ability of the proposed Nussbaum gain that tackles the effect of the unknown control directions.
110 CHAPTER 7. CONCLUSIONS AND FUTURE WORKS 111
3. The output tracking and the stability of the entire systems.
In Chapter 4, a novel control strategy has been developed for adaptive output regulation problem for SISO nonlinear output feedback systems with unknown high- frequency gain sign which are subjected to unknown nonlinear exosystems. In this case, based on the circle criterion, the newly internal model can be applied to reproduce the feedforward input. Moreover, some certain assumptions are presented such that this criterion can be used. In terms of the unknown high-frequency gain, a novel Nussbaum gain is then introduced. The combination of adaptive backstepping, adaptive internal model and the Nussbaum gain is finally to tackle the output regulation problem and the stability analysis based on Lyapunov function is presented.
Similar to Chapter 4, a novel control algorithm has been developed for a class of network connected multi-agent nonlinear output feedback systems with unknown control directions in Chapter 5, where a new type Nussbaum gain with a faster rate such that one of them dominant for the positive definite condition for the consensus control and the stability characteristics are analysed. It has been shown that the controller can be designed for a decentralised way, which means only use the relative information of subsystem outputs, and the controller can be applied to higher relative degree nonlinear systems using the backstepping techniques.
Chapter 6 shows a new idea to describe the consensus output tracking of multi- agent nonlinear system with unknown high-frequency gains over directed graphs. The asymmetry of the Laplacian matrices becomes the obstacle in the control design. In order to tackle this problem, a new type integral-Lyapunov function together with the adaptive internal model and the Nussbaum gain is then proposed. Moreover, the pro- posed schemes only depend on the relative outputs information of subsystems, rather than the relative states, which provide a stumbling block to our research. Simulations eventually have been employed to demonstrate the validity of the theoretical results. CHAPTER 7. CONCLUSIONS AND FUTURE WORKS 112 7.2 Future Works
Besides the completed work in this thesis, there are still some open questions to answer in future works.
1. The non-periodic disturbances will be taken into account. Actually, the distur- bances are always unpredictable in practice. They might have a sudden change or be discontinuous sometimes. In this regard, it seems that the internal model principle is hardly helpful. From the point of the switched control, it might be a way to achieve the disturbance rejection theoretically.
2. In Chapter 6, consensus output tracking for a class of nonlinear systems under directed graph has been studied. The reference signal with constant input is much more conservative. How to design a distributed adaptive consensus con- troller if the reference is a periodic signal with unknown parameters is more challenging because of the asymmetry of Laplacian matrices, the construction of the Lyapunov function becomes more difficult. The σ-modification method might work. Besides, the adaptive event-triggered control based on the frequency of data transmission proposed in [124, 125] might be a way to achieve the dis- turbance rejection theoretically. Further analysis to tackle these problems is a topic of future research.
3. Time delays widely exist in practical multi-agent systems due to the time taken for transmission of signals, transport of material, etc. The presence of time delays, if not considered in the controller design, may seriously degrade the performance of the controlled systems, even may cause the loss of stability. One basic idea for tackling an input delay is to predict the evolution of state variable for the delay period and then use the predicted state for control. The state prediction is based on the explicit solution of the state equation, which consist of the zero input and the zero state solutions. However, the zero state solution involves the integral of the past control input and causes difficulty in control implementation. An alternative method based on the prediction is to ignore the troublesome zero state solution, and use the zero input solution as the prediction, CHAPTER 7. CONCLUSIONS AND FUTURE WORKS 113
which is referred to as the truncated prediction [126, 127]. By transforming the Laplacian matrix into the real Jordan form, sufficient conditions are needed such that the proposed control algorithms can achieve the consensus. Therefore, by using the truncated prediction feedback for consensus output regulation of nonlinear multi-agent systems with input delay draws our attention.
4. The theory in this thesis will be applied in some practical cases, such as power energy field or medical equipment design. Bibliography
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