Gain scheduling Robust Design and Automated Tuning of Automotive Controllers

Ir. G.J.L. Naus 2009.106 26th of October, 2009

WP3 - robust, low-complexity controller design

Supervisors: Prof.dr.ir. M. Steinbuch (TU/e) Dr.ir. M.J.G. v.d. Molengraft (TU/e) Dr.ir. R. Huisman (DAF) Ir. J. Ploeg (TNO)

TNO Science and Industry, Helmond (TNO) Business Unit Automotive Department of Advanced Chassis and Transport Systems

DAF Trucks N.V., Eindhoven (DAF) Product Development Technical Analysis group

University of Technology Eindhoven (TU/e) Department of Mechanical Engineering Division Dynamical Systems Design Control System Technology group 2 Contents

1 Introduction 1 1.1 Background and goals ...... 1 1.2 Outline ...... 2

2 Gain scheduling 3 2.1 Overview ...... 3 2.1.1 Classical gain scheduling ...... 3 2.1.2 LPV and LFT synthesis ...... 4 2.1.3 Fuzzy gain scheduling ...... 5 2.2 Gain scheduling vs Robust control ...... 5

3 Parameter-dependent plant models 7 3.1 linearization-based scheduling ...... 8 3.2 Off-equilibrium linearizations ...... 11 3.3 Quasi-LPV approach ...... 12 3.4 linear, parameter-dependent plant ...... 13 3.5 Linear fractional transformation description ...... 14 3.6 Conclusions / recap ...... 15

4 Classical gain scheduling 17 4.1 LTI controller design ...... 17 4.1.1 Direct gain scheduling controller design ...... 18 4.2 Gain scheduling controller design ...... 18 4.2.1 Linearization scheduling ...... 19 4.2.2 Interpolation methods ...... 21 4.2.3 Velocity-based scheduling ...... 23 4.3 Hidden coupling terms ...... 24 4.4 Stability properties ...... 25

5 LPV controller synthesis 27 5.1 General LPV controller synthesis setup ...... 28 5.1.1 Stability analysis ...... 29 5.1.2 Performance analysis ...... 30 5.1.3 General synthesis problem formulation ...... 31 5.2 Lyapunov-based LPV control synthesis ...... 32 5.2.1 Gridding ...... 32 5.2.2 Polytopic approximation ...... 32 5.3 Scaled small-gain or LFT synthesis ...... 33 5.3.1 LFT problem formulation ...... 33 5.3.2 Controller synthesis ...... 34 5.3.3 Pros and cons ...... 35 5.4 Mixed LPV-LFT approaches ...... 36

3 4 CONTENTS

5.4.1 Extended Kalman-Yakubovich-Popov (KYP) lemma ...... 36 5.4.2 LFT Lyapunov functions ...... 36 5.5 Conclusions ...... 36

6 Fuzzy gain-scheduling 39 6.1 Fuzzy modeling ...... 39 6.1.1 Weighting or scaling functions ...... 40 6.1.2 Approximation accuracy ...... 41 6.2 Fuzzy gain scheduling control ...... 43 6.2.1 Parallel Distributed Control ...... 43

7 Conclusions and future work 45 7.1 Conclusions ...... 45 7.2 Future work (tbd) ...... 47

Bibliography 52

A Longitudinal vehicle control 53

B Engine - idle governor 55 Chapter 1

Introduction

In this literature survey, an overview of gain-scheduling syntheses is put together. Many different notions can be viewed as Gain Scheduling (GS). For example switching or blending of gain values of controllers or models or switching or blending of complete controllers or model dynamics according to different operating conditions, or according to preset times. As the terms switching and blending already indicate, GS may either involve continuous or discrete scheduling of controllers or model dynamics. In general, gain-scheduling encompasses the attenuation of nonlinear dynamics over a range of operations, the attenuation of environmental time-variations or the attenuation of parameter vari- ations and uncertainties. Classical gain scheduling involves offline linearization of nonlinear system dynamics at multiple operating conditions, which are parameterized by a so-called scheduling vari- able θ, and the design of corresponding linear controllers at each point. Next, online scheduling of controller gains is established, based on θ = θ(t) to reflect the nearby operating condition. The in- tention is to extend the reach of a single linearization-based control design over an entire operating envelope. In general, the scheduling variable is time-varying and may either be an internal plant variable (or a function of internal plant variables) called endogenous variables, or an externally prescribed exogenous variable. Consequently, difficulties regarding stability and performance requirements may arise when the LTI controller designs are implemented for θ = θ(t). More recent LPV and LFT approaches hence take into account the time-varying nature of θ = θ(t) in the controller design. Guaranteed global stability and performance requirements can thus be given a priori. This survey encompasses classical gain-scheduling syntheses as well as more recent LPV and LFT approaches.

1.1 Background and goals

Background This report is based on the conclusions of (Naus 2007b, Naus 2007a), which discuss the controller design and tuning at Integrated Safety, Business Unit Automotive, TNO Helmond, and DAF Trucks N.V. respectively. Essentially, the main problems regarding controller design at DAF and TNO comprise i) a lack of proper system identification and corresponding modeling and ii) the lack of appropriate performance specifications. In some cases, application of system identification, subsequent modeling and corresponding local controller design is applied. However, scheduling of the resulting controller gains or switching between various controllers is commonly applied. This is done by experience and insight in the system. Stability issues as well as performance and robustness specifications are not considered. Hence, insight in gain scheduling and related control syntheses is demanded. At DAF, gain scheduling is adopted in many practical applications. In all cases, the scheduling is employed in an ad hoc manner, not considering stability issues or performance and robustness specifications. Besides employing gain scheduling as a suitable control synthesis for nonlinear

1 2 CHAPTER 1. INTRODUCTION systems, gain scheduling is also adopted for closed-loop performance improvements of linear plants by scheduling the controller gains on the basis of e.g. a tracking error. At TNO, discrete scheduling or switching as well as smooth scheduling is employed in an ad hoc fashion. The tuning and design of the controller (gains) corresponding to specific working conditions involves much trial-and-error. Furthermore, recent application of Model Predictive Control (MPC) in an explicit fashion (Naus, van den Bleek, Ploeg, Scheepers, van de Molengraft & Steinbuch 2008) yields a gain-scheduled controller. The interpretation of this controller and corresponding tuning require more insight in gain scheduling and related control syntheses.

Goals The main goals of this report involve two issues. • The report provides an overview of gain scheduling and related controller design syntheses. Available gain scheduling techniques, and guidelines for appropriate application of these techniques are discussed. • Based on the conclusions regarding the application of gain scheduling, goals for further research have to be defined within the project robust design and automated tuning of auto- motive controllers. Appropriate syntheses and research directions to elaborate on have to be defined, focusing on generic design and tuning methodologies for applications at DAF and TNO.

1.2 Outline

To start with, an overview of gain scheduling techniques is given (Chapter 2). Parameter- dependent plant models, which serve as a basis for gain scheduling controller design, are discussed in Chapter 3. Next, each of the main classes of gain scheduling techniques is discussed in detail; classical gain scheduling (Chapter 4), LPV and LFT control synthesis (Chapter 5) and fuzzy gain scheduling (Chapter 6). In Chapter ??, application examples, focusing on the automotive industry are discussed. Finally, conclusions and future work regarding the applicability of gain scheduling techniques in the remainder of this project are given (Chapter 7). The future work will be deter- mined in cooperation with DAF and TNO. Throughout this report, two example case studies are used; the problem of designing a longitudinal vehicle speed controller, i.e. cruise controller, and an engine example involving the design of a low idle governor. These case studies are defined in Appendix A and B respectively. Chapter 2

Gain scheduling

2.1 Overview

Different gain scheduling approaches can be distinguished, which may be classified in different ways. To start with, gain scheduling methods can be divided into i) methods decomposing non- linear design problems into linear sub-problems and ii) methods decomposing nonlinear design problems into simpler nonlinear sub-problems or affine sub-problems. (Leith & Leithead 2000) give an overview of the main theoretical results and design procedures relating to (continuous) gain-scheduling control in the sense of decomposing nonlinear design problems into linear sub- problems. Focus will lie on the former one, as linear system and corresponding control theory is well developed. Furthermore, gain scheduling methods decompose into i) continuous gain-scheduling methods and ii) discrete, i.e. hybrid or switched gain-scheduling methods. Discrete or hybrid in this sense involves the switching of a system or controller between dynamical regimes. E.g. friction models that have a clear distinction between stick and slip phases, backlash in gears, dead zones in cog wheels; phenomena like saturation, hysteresis, sensor and actuator failures. Hybrid systems in the sense of combining digital controllers with physical processes is not meant here (Schutter & Heemels 2004). Finally, three main approaches can be distinguished regarding gain-scheduling; i) so-called divide and conquer techniques, or the classical gain-scheduling approach, more recent LPV and LFT syntheses and thirdly, fuzzy techniques. A brief overview of each of these approaches is given next.

2.1.1 Classical gain scheduling Generally, classical gain scheduling involves the decomposition of the design of a nonlinear con- troller into the design of a number of linear controllers. Hence, well-established linear control design methods may be applied without restriction to a certain method, as opposed to nonlinear methods. No guarantees on the robustness, performance or even nominal stability of the closed- loop gain-scheduled design can be given however. Classical gain scheduling comprises four main steps (Rugh & Shamma 2000).

Step 1 A family of LTI approximations of a nonlinear plant at constant operating points (equilib- ria), parameterized by constant values of convenient plant variables or exogenous signals θ is computed using linearization-based scheduling. Jacobian linearization of the nonlinear plant about a manifold of constant equilibria, constant operating points or setpoints is employed. The linearizations have to correspond to zero error. This yields a parameterized family of linearized plants. Subsequent implementation of the controller requires θ to be a measurable variable. Other syntheses to derive a parameter-dependent model are

3 4 CHAPTER 2. GAIN SCHEDULING

• off-equilibrium or velocity-based linearizations, e.g. when zero error equilibrium points or working conditions are not present • a so-called quasi-LPV approach in which the plant dynamics are rewritten to distinguish nonlinearities as time-varying parameters that are used as scheduling variables • direct LPV modeling, based on a linear plant incorporating time-varying parameters, i.e. when no nonlinear plant is involved; this also includes black-box or data-based modeling methods

Step 2 LTI controllers corresponding to the previously derived set of local LTI models are de- signed, to achieve specified performance and stability at each operating point. Hence, the LTI controllers in combination with the corresponding LTI models have desired properties. The resulting set of controllers is also parameterized by θ. Although θ actually is time-varying, i.e. θ = θ(t), classical design methods are based on fixed or frozen scheduling parameter values θ. To enable subsequent scheduling, interpolation of e.g. controller coefficients, the set of LTI controllers requires fixed-structure controller designs. Two exceptions are

• in case direct derivation of a Linear Parameter Varying (LPV) controller for a corre- sponding LPV plant model is possible, subsequent scheduling, interpolation becomes superfluous. • when discrete or hybrid scheduling instead of continuous scheduling is demanded, the set of controller designs not necessarily need to be fixed-structured.

Step 3 Implementation of the family of LTI controllers such that the controller coefficients are scheduled according to the current value of the scheduling variable, e.g. by controller gain interpolation or scheduling. At this point, θ = θ(t) is implemented. At each operating point, the scheduled controller has to linearize to the corresponding linear controller design as well as provides a constant control value yielding zero error at these points. As mentioned in Step 2, in case of direct scheduling, this step becomes superfluous. Furthermore, in case of discrete scheduling, the implementation of the LTI controllers involves the design of a scheduled selection procedure that is applied to the set of LTI controllers, rather than the design of a family of scheduled controllers. The presence of hidden coupling terms is an important aspect, which yields various additional requirements to the scheduling procedure.

Step 4 Typically, local performance assessment can be performed analytically, whereas assess- ment of global performance and robustness has to be established by extensive simulations. Nonlocal performance of the gain scheduled controller is evaluated and checked by simula- tions.

2.1.2 LPV and LFT synthesis

LPV and LFT syntheses are based on LPV and LFT plant representations respectively. Both methods yield direct synthesis of a controller utilizing (l2-)norm based methods, with guarantees on the robustness, performance and nominal stability of the overall gain-scheduled design (Shamma & Athans 1990)). LPV and LFT syntheses essentially involve only two main steps.

Step 1 The first step corresponds to the classical approach. A family of LTI approximations of a nonlinear plant at constant operating points (equilibria), parameterized by constant values of convenient plant variables or exogenous signals θ is computed. Subsequent implementation of the controller requires θ = θ(t) to be a measurable variable. Besides the already mentioned methods, which all arrive at Linear Parameter-Varying (LPV) models, in specific cases a LFT description is possible. The LFT description serves as a basis for subsequent LFT controller synthesis. 2.2. GAIN SCHEDULING VS ROBUST CONTROL 5

Step 2 LPV and LFT control synthesis directly yield a gain-scheduled controller. Stability and performance specifications can be guaranteed a priori as the time-varying parameter θ(t) in- stead of its corresponding frozen-value θ is addressed in the design process. Only continuous gain-scheduling is considered.

2.1.3 Fuzzy gain scheduling Fuzzy gain scheduling should overcome the disadvantage of classical gain scheduling regarding the restriction of stability and performance analysis to local rather than global closed-loop behavior. The corresponding fuzzy modeling considers the transient dynamics of the original nonlinear model instead of local linearizations only. Fuzzy gain scheduling techniques may involve classical gain scheduling alike as well as LPV techniques. Focusing on the former one, four main steps have to be considered. Step 1 Analogous to classical gain scheduling, sets of local LTI models and corresponding LTI controllers have to be designed. Focus lies on the regions of the envelope of operating conditions for which these controllers assure stability and desired performance of the resulting (local) closed-loop systems. Step 2 To arrive at a fuzzy model, so-called weighting or scaling functions are designed, cor- responding to the before mentioned regions. Utilizing these weighting functions, the local models are blended. Specifying a specific approximation accuracy of the fuzzy model with respect to the original nonlinear model, yields the required number of local models. Step 3 The set of local controllers is blended analogous to the set of local models. The same weighting functions are utilized. The blending yields scheduling of the controller outputs rather than scheduling of the controllers or controller coefficients. Hence, members of the corresponding parameterized set of LTI controllers do not necessarily need to have fixed structure and dimension. Step 4 Stability and performance are established by extensive simulations, analogous to classical gain scheduling. However, in the case of fuzzy controller design, global as well as local specifications have to be derived from simulations, as the characteristic dynamics of the fuzzy model can not be related to the dynamics of the set of local models.

2.2 Gain scheduling vs Robust control

An important issue in linear controller design involves the robustness and performance of the closed-loop system in the presence of uncertainty. System uncertainty essentially is of two types; i) dynamical uncertainty, which corresponds to neglected plant dynamics (high-frequency behaviour, nonlinearities, etc.), and ii) constant parametric uncertainty, which results either from inaccurate knowledge of the value of physical parameters or from variations of this value during operation. Robust H∞ theory may be adopted for the former one and µ-synthesis, Lyapunov-based techniques for the latter one (Apkarian & Gahinet 1995). LPV plants can be viewed as i) Linear Time-Invariant (LTI) plants subject to a time-varying parametric uncertainty θ(t) or ii) as models of linear time-varying plants, either derived as a non- linear model or a set of linear models describing a nonlinear plant. When θ(t) can be measured, the latter idea enables exploiting the available measurements of θ(t) in an appropriate control strategy. Assuming that the uncertainties are constant or only slowly varying, the former problem yields LTI robust control techniques (Apkarian, Gahinet & Becker 1995). Especially in case of large parameter variations, a single robust controller can be very conservative and plant stabilization by a single LTI controller may not even be feasible. Assuming that the varying parameter θ(t) can be measured during operation, gain-scheduling may provide less conservative solutions. Essentially, no a priori knowledge about θ(t) is required but for its range of operation and its rate of variation θ˙(t). 6 CHAPTER 2. GAIN SCHEDULING

Considering gain-scheduled controllers, the controllers are a function of the scheduling variable θ(t). A gain-scheduled controller thus can adjust itself according to the changes in the dynamics of the plant. As a result, a gain-scheduled controller usually is less conservative than a robust one. Combinations, i.e. robust gain-scheduled controllers exist as well. The parameter θ(t) is T assumed to be divided into θ(t) = [θm(t) θu(t)] where θm(t) is measured in real-time and θu(t) is uncertain. The gain-scheduling part schedules based on θm(t). The controller now adapts to changes in the plant dynamics due to θm(t) and is robust for changes in θu(t) (Bianchi, Mantz & Christiansen 2007). The other way to deal with θ(t) is to consider it as a time-varying uncertainty. In this sense, e.g. (Zhou, Khargonekar, Stoustrup & Niemann 1992) present the synthesis of a strongly robust H∞ performance criterion, i.e. robust performance for systems with time varying uncertainties. More recent LPV and LFT control synthesis techniques elaborate on this idea using i) parameter-varying Lyapunov functions and ii) scaled small-gain theorems. Chapter 3

Parameter-dependent plant models

In general, gain scheduling is considered as a controller synthesis to design controllers for a nonlin- ear plant. The general approach assumed in most literature is to start with a whitebox nonlinear model Σnl, describing the nonlinear plant dynamics.   x˙(t) = f(x(t), u(t), w(t)) Σnl : z(t) = g(x(t), u(t), w(t)) (3.1)  y = h(x(t), w(t)) where x(t) is the state of the system, u(t) is the input, z(t) denotes an error signal to be controlled, y(t) denotes the measured output, available to the controller, e.g. penalized variables, tracking commands, state variables, and w(t) represents external inputs such as reference commands, dis- turbances and noise. When such a model is not available however, a set of linear point-design black-box models may be used instead. The point-designs should cover the operating range of the nonlinear plant dynamics. Conceptually, gain-scheduling is based on a linear, time-invariant, parameter-dependent plant. A linear approximation Σl of the nonlinear model Σnl or the above mentioned set of linear point- designs can be used. The linear approximation is parameterized by the scheduling variable θ for n a corresponding operating envelope θ ∈ Θ ⊂ R θ , where nθ the length of the parameter vector θ. This yields a family of linear plant models Σl(θ), describing the nonlinear plant dynamics (furtheron, the subscript l, denoting a linear model will be omitted). The corresponding family of linear parameter-varying plant models Σ(θ) = Σl(θ) is given by   x˙ = A(θ)x + B1(θ)w + B2(θ)u Σ(θ): z = C1(θ)x + D11(θ)w + D12(θ)u θ ∈ Θ (3.2)  y = C2(θ)x + D21(θ)w

Throughout the controller synthesis, θ is regarded as a constant variable. However, when imple- menting the resulting gain scheduled controller, θ = θ(t) is used instead. The model Σ(θ) (3.2) represents a family of linear parameter-varying plant models rather than a single dynamic system. A true Linear Parameter Varying (LPV) model is defined as   x˙ = A(θ(t))x + B1(θ(t))w + B2(θ(t))u Σ(θ(t)) : z = C1(θ(t))x + D11(θ(t))w + D12(θ(t))u θ(t) ∈ Θ (3.3)  y = C2(θ(t))x + D21(θ(t))w

A LPV model comprises linear, parameter-dependent dynamics. The model matrices are fixed functions of some vector of varying parameters θ(t) (Shamma & Athans 1992). As these parameters are time-varying, the model becomes time-varying. Consider a parameterized family of Jacobian

7 8 CHAPTER 3. PARAMETER-DEPENDENT PLANT MODELS linearizations, i.e. local linear plant models. The synthesis of such a family is called linearization- based scheduling (Shamma & Athans 1992). Correspondingly implementing θ = θ(t), in practice yields an LPV model. Possible variations in the state, input and / or output transformations of the family members however, are neglected. This may yield closed-loop performance or even stability problems when implementing the corresponding controller. Different approaches enabling the design of true LPV models are exact linearization or quasi-LPV techniques (Shamma & Athans 1992, Rugh 1991), velocity-based scheduling techniques (Leith & Leithead 2000), or parameter- dependencies arising in a linear plant (Shahruz & Behtash 1990). In general, three types of linear, parameter-dependent models can be distinguished • Linear Parameter Varying (LPV) models Σ(θ(t))

• Linear Fractional Transformation (LFT) models ΣLF T (θ(t))

• Polytopic Linear Modeling (PLM) or fuzzy models ΣP LM (θ(t)) Correspondingly, three main methods to derive a linear parameter-dependent model of a nonlin- ear plant can be distinguished. In next sections, the above mentioned methods to derive LPV models and subsequently LFT models are presented. The fuzzy modeling is discussed separately in Chapter 6.

3.1 linearization-based scheduling

The classical approach, using Jacobian linearization of the nonlinear model about a manifold of constant equilibria, constant operating points or setpoints, is called linearization-based scheduling. When a corresponding scheduling variable θ is chosen appropriately to parameterize the set of linear models, a parameterized family of linearized models representing the original nonlinear model results.

Linearization theory

Consider the nonlinear plant dynamics, described by the nonlinear model Σnl (3.1). Define the x operating envelope of the plant R . An equilibrium or constant operating point (xo, uo, wo) ∈ o x R ⊂ R is defined by f(xo, uo, wo) = 0, where f(∗) as defined by (3.1). An equilibrium family and the corresponding error and output equilibrium families are defined as  0 = f(xo, uo, wo) o  R : zo = g(xo, uo, wo) (3.4)  yo = h(xo, wo)

o Assuming f(x, w, u) is continuously differentiable at (xo, uo, wo) ∈ R , the nonlinear model is approximated by   x˙ = fo + ∇xf|Ro (x − xo) + ∇wf|Ro (w − wo) + ∇uf|Ro (u − uo) + rf (x, w, u) Σo : z = go + ∇xg|Ro (x − xo) + ∇wg|Ro (w − wo) + ∇ug|Ro (u − uo) + rg(x, w, u)  y = ho + ∇xh|Ro (x − xo) + ∇wh|Ro (w − wo) + rh(x, w, u)

(3.5) where fo = f(xo, wo, uo), go = g(xo, wo, uo) and ho = h(xo, wo) respectively. The residual terms rf,g,h(x, w, u) satisfy r (x, w, u) lim f,g,h = 0 (3.6) p 2 2 2 (x,w,u)→(xo,wo,uo) |x − xo| + |w − wo| + |u − uo| and are assumed to be zero for simplicity. 3.1. LINEARIZATION-BASED SCHEDULING 9

Application of a coordinate transformation

x = xo +x, ˜ w = wo +w, ˜ u = uo +u, ˜ y = yo +y, ˜ z = zo +z ˜ (3.7) and definition of the time-invariant system matrices

A = ∇xf|Ro B1 = ∇wf|Ro B2 = ∇uf|Ro C1 = ∇xg|Ro D11 = ∇wg|Ro C12 = ∇ug|Ro (3.8) C2 = ∇xh|Ro D21 = ∇wh|Ro yields the linearization family Σ, locally describing the nonlinear model (3.1)

 ˙  x˜ = Ax˜ + B1w˜ + B2u˜ Σ: z˜ = C1x˜ + D11w˜ + D12u˜ (3.9)  y˜ = C2x˜ + D21w˜

Scheduling variable θ

Define a scheduling variable θ ∈ Θ, parameterizing the envelope of working conditions Rx → Θ and analogously Ro → Θo with Θo ⊂ Θ. This yields Ro = Ro(θ), where θ ∈ Θo. An equilibrium family and the corresponding error and output equilibrium families for any θ ∈ Θo are then defined by  0 = f(xo(θ), uo(θ), wo(θ)) o  o R (θ): zo(θ) = g(xo(θ), uo(θ), wo(θ)) θ ∈ Θ (3.10)  yo(θ) = h(xo(θ), wo(θ))

Corresponding to the parameterized equilibrium values (xo(θ), wo(θ), uo(θ), zo(θ), yo(θ)) fol- lowing from (3.10), a linear parameter-dependent linearization family exists, locally describing the nonlinear model (3.1).  x˜˙ = A(θ)˜x + B1(θ)w ˜ + B2(θ)˜u  o Σ(θ): z˜ = C1(θ)˜x + D11(θ)w ˜ + D12(θ)˜u θ ∈ Θ (3.11)  y˜ = C2(θ)˜x + D21(θ)w ˜

Due to the parameterization of the equilibrium values, the system matrices are now parameterized as well (compare to (3.8)). As discussed in the introduction, in general, (3.11) describes the nonlinear plant only locally. Hence, Σ(θ), θ ∈ Θo is a set or family of point-designs rather than a true LPV model, see Figure 3.1. A LPV model by definition, is valid for θ.

Θ ⊂ Rnθ

θ ∈ Θo

Figure 3.1: Operating envelope Θ ⊂ Rnθ and the set of equilibrium values or constant operating points Θo, parameterized by θ. 10 CHAPTER 3. PARAMETER-DEPENDENT PLANT MODELS

In specific cases, the set of point-designs immediately yields an LPV model Σ(θ(t)), θ(t) ∈ Θ. In practice, this is commonly established by implementation of θ = θ(t), yielding  ˙  x˜ = A(θ(t))˜x + B1(θ(t))w ˜ + B2(θ(t))˜u Σ(θ(t)) : z˜ = C1(θ(t))˜x + D11(θ(t))w ˜ + D12(θ(t))˜u θ(t) ∈ Θ (3.12)  y˜ = C2(θ(t))˜x + D21(θ(t))w ˜ Possible variations in the state, input and output transformations hence are neglected, which may result in closed-loop performance or even stability problems. In case subsequent classical gain scheduling controller design is applied, the set of linearized models Σ(θ), θ ∈ Θo may be used as a starting point. In this sense, a set of linear, point-design black-box models resulting from e.g. frequency-response or step-response measurements, yields the same result. In case subsequent LPV control synthesis is applied, an LPV model is required. A parameterized family of linearized models resulting from linearization-based scheduling or a number of black-box point-designs are only locally valid. In case an LPV model is based on such a set of linearized models, the accuracy of the resulting linear parameter-dependent model with respect to the original nonlinear model or plant is unknown and generally follows from extensive simulations.

Example 3.1 (Longitudinal vehicle control) Consider the nonlinear model (A.5) of the longitudinal vehicle control example (see A). Linearizing (A.5) around the equilibrium operating point (xe, ue, we) yields  0 = − 1 C0 x2 − g(sin w(1) + C cos w(1)) + 1 u  M d e e r e M e (2) Re : ze = we − xe (3.13)   ye = xe

The corresponding equilibrium families, and error and output equilibrium families are given by

 xd,e = ye = xe   ze = 0 T Re : h (1) (2)i T (3.14) we = we , we = [αe, xe]   0 2 (1) (1) ue = Cdxe + g(sin we + Cr cos we ) Define a scheduling variable θ = [α, x]T = [w(1), y]T ∈ Θ, where Θ represents the parameterized envelope of operating conditions. This yields

 (2) xd,e(θ) = ye(θ) := θ   ze(θ) = ze = 0  T Re(θ): we(θ) = [αe(θ), xe(θ)] := θ θ ∈ Θe (3.15)  0 2 (1) (1)  ue(θ) = Cdxe + g(sin we + Cr cos we )  0 (2)2 (1) (1) := Cdθ + g(sin θ + Cr cos θ )

where Θe ⊂ Θ defines the parameterized equilibrium operating points of the nonlinear system (A.5). The corresponding parameterized set of linear (LPV) models Σ(θ) is defined by  ˙  x˜(t) = A(θ)˜x(t) + B1(θ)w ˜(t) + B2u˜(t) Σ(θ): z˜(t) = C1x˜(t) + D11w˜(t) + D12u˜(t) θ ∈ Θe (3.16)  y˜(t) = C2x˜(t) + D21w˜(t) where the deviation variables x˜(t) = x(t) − x (θ), w˜(t) = w(t) − w (θ), u˜(t) = u(t) − u (θ) e e e (3.17) z˜(t) = z(t) − ze(θ), y˜(t) = y(t) − ye(θ) 3.2. OFF-EQUILIBRIUM LINEARIZATIONS 11

and the parameterized system matrices

A(θ) = − 2 C0 θ(2) B = g(C sin θ(1) − cos θ(1)) 0 B = 1 M d 1 r 2 M (3.18) C1 = −1 D11 = [0 1] D12 = 0 C2 = 1 D21 = 0

3.2 Off-equilibrium linearizations

A disadvantage of classical linearization-based scheduling is the restriction to equilibrium-point modeling and corresponding controller design. Using so-called velocity-based or off-equilibrium linearizations, (Leith & Leithead 1998b) enable linearization at every operating point. Rewriting the nonlinear dynamics (3.1) such that the nonlinearities are incorporated in θ(x, w, u) explicitly yields   x˙ = Ax + B1w + B2u + f(θ) Σnl(θ): z = C1x + D11w + D12u + g(θ) θ = θ(t) ∈ Θ (3.19)  y = C2x + D21w + h(θ) where A, B, C and D are constant matrices and f(θ), g(θ) and h(θ) are nonlinear functions ∂θ ∂θ ∂θ incorporating the model nonlinearities via θ(x, w, u), with ∂x , ∂x and ∂u constant. Considering an operating point (xo, wo, uo) at which θo = θ(xo, wo, uo), the solution to the corresponding velocity-based linearization becomes

 x˙ = ζ  o o  ˙ ∂f ∂f ∂f ζo = (A + ∂t (θo))ζo + (B1 + ∂t (θo))w ˙ o + (B2 + ∂t (θo))u ˙ o Σ(θo): ∂g ∂g ∂g (3.20)  z˙o = (C1 + ∂t (θo))ζo + (D11 + ∂t (θo))w ˙ o + (D12 + ∂t (θo))u ˙ o  ∂h ∂h y˙o = (C2 + ∂t (θo))ζo + (D21 + ∂t (θo))w ˙ o which is linear. (Leith & Leithead 1998b) show that (3.20) approximates the solution to the nonlin- ear model (3.19) and thus (3.1) locally to an arbitrary operating point (xo, wo, uo). Consequently, there is a velocity-based linearization associated with every operating point of a nonlinear sys- tem and the solutions may be pieced together. The resulting velocity-based linearization family, parameterized by θ, which captures the nonlinearities of the model, globally approximates the solution to the nonlinear model to an arbitrary degree of accuracy (Leith & Leithead 2000). As no restriction to equilibrium operating points is present, linear approximation of transient dynamics and operating points far from equilibrium operating points is enabled. Hence, the velocity-based linearization family Σ(θ) is an alternative representation of the nonlinear model (3.1).

 x˙ = ζ   ˙ ∂f ∂f ∂f ζ = (A + ∂t (θ))ζ + (B + ∂t (θ))w ˙ + (B2 + ∂t (θ))u ˙ Σ(θ): ∂g ∂g ∂g θ = θ(t) ∈ Θ (3.21)  z˙ = (C1 + ∂t (θ))ζ + (D11 + ∂t (θ))w ˙ + (D12 + ∂t (θ))u ˙  ∂h ∂h y˙ = (C2 + ∂t (θ))ζ + (D21 + ∂t (θ))w ˙ The velocity-based linearization family (3.21) may be adopted as a suitable basis for subsequent gain scheduling controller design (Leith & Leithead 1998c). Classical gain scheduling (Leith & Leithead 1998c), LPV synthesis (Leith & Leithead 2000) or fuzzy-alike blending techniques (Leith, Tsourdos, White & Leithead 2000) may be employed. The parameterized velocity-based linearization family (3.21) represents the entire dynamics of the nonlinear model (3.1) without loss of information. Hence, the velocity-based linearization fam- ily can be regarded as an alternative representation of the nonlinear model, globally approximating the solution to the nonlinear model to an arbitrary degree of accuracy. In practice however, a small set of velocity-based linearizations will be used. The corresponding parameterization θ ∈ Θo ⊂ Θ where Θ represents the total envelope of working conditions parameterized by θ, and corresponding interpolation functions are chosen such that a sufficiently accurate approximation of the nonlinear dynamics over the envelope of working conditions is ensured. The use of interpolation functions is comparable to fuzzy or Takagi-Sugeno modeling. However, in this case the dynamics of the 12 CHAPTER 3. PARAMETER-DEPENDENT PLANT MODELS interpolated models are directly related to the dynamic characteristics of the set of local models, which is not the case with fuzzy modeling (Leith & Leithead 1999). Due to the direct relation between the family of velocity-based linearizations and the original nonlinear model, stability of the nonlinear model (3.1) can be related to stability conditions concerning the velocity-based linearizations (Leith & Leithead 2000, Leith & Leithead 1998b). Provided that the members of the velocity-based linearization family corresponding to a certain nonlinear model are uniformly stable and the rate of evolution of the nonlinear model is sufficiently slow, the nonlinear model is BIBO stable for some restricted class of inputs and initial conditions. For systems where the slow variation condition is automatically satisfied, the class of allowable inputs and initial conditions is unrestricted and the stability analysis is global.

3.3 Quasi-LPV approach

Quasi-LPV scheduling tries to overcome the general shortcomings of classical gain-scheduling regarding local validity of the resulting linearized model via transformation of a nonlinear model to an LPV form. The nonlinear terms are hidden by including them in the scheduling variable. As this involves a transformation rather than a linearization, the resulting LPV model exactly equals the original nonlinear model. LPV analysis and synthesis tools can be used, taking into account the presence of parameter variations. However, it is difficult to exploit the endogenous relation of the scheduling parameter y, which is the part of the state that accounts for the nonlinearities. Rewriting the nonlinear model (3.1) correspondingly yields

 y˙   y  = f(y) + A0(y) + B0 (y)w0 + B0 (y)u0 (3.22) x˙ 0 x0 1 2 where all nonlinearities are incorporated in f(y) and B1, B2 are assumed to be linear beforehand. Moreover, y is regarded as the scheduling variable. The corresponding equilibrium family is thus 0 0 0 parameterized by y, yielding (xo(y), wo(y), uo(y)) and is given by   0 y 0 0 0 0 0 = f(y) + A (y) 0 + B1(y)wo(y) + B2(y)uo(y) (3.23) xo(y) yielding

 A0(11)(y)   A0(12)(y)  f(y) + y = − x0 (y) − B0 (y)w0 (y) − B0 (y)u0 (y) (3.24) A0(21)(y) A0(22)(y) o 1 o 2 o where A0(ij) denotes the ijth element of A0. Substitution of (3.24) in (3.22) yields

 y˙   A0(12)(y)  = − x0 (y) − B0 (y)w0 (y) − B0 (y)u0 (y) + ... x˙ 0 A0(22)(y) o 1 o 2 o  A0(12)(y)  ... + x0 + B0 (y)w0 + B0 (y)u0 (3.25) A0(22)(y) 1 2  A0(12)(y)  = (x0 − x0 (y)) + B0 (y)(w0 − w0 (y)) − B0 (y)(u0 − u0 (y)) (3.26) A0(22)(y) o 1 o 2 o Application of a change of variables   y 0 0 0 x = 0 0 , w = w − wo(y), u = u − uo(y) (3.27) x − xo(y) with  y˙   y˙  x˙ = 0 0 = 0 0 (3.28) x˙ − x˙ o(y) x˙ − ∇yxo(y)y ˙ 3.4. LINEAR, PARAMETER-DEPENDENT PLANT 13 then yields  0(12)    0 A (y) 0 0 0 x˙ = 0(22) x + 0 + B1(y)w + B2(y)u (3.29) 0 A (y) −∇yxo(y)y ˙

= A(y)x + B1(y)w + B2(y)u (3.30) where  0 A0(12)(y)  A(y) = 0(22) 0 0(12) (3.31) 0 A (y) − ∇yxo(y)A (y) " # B0(1)(y) B (y) = 1 (3.32) 1 0(2) 0 0(1) B1 (y) − ∇yxo(y)B1 (y) " # B0(1)(y) B (y) = 2 (3.33) 2 0(2) 0 0(1) B2 (y) − ∇yxo(y)B2 (y) If all nonlinearities are incorporated in f(y) and no nonlinearities concerning x0 are present, the transformation is exact and (3.22) equals (3.30), taking into account the change of variables (3.27). If, more generally the nonlinearities aren’t exactly in y, a residual term rf (x) with dynamics remains, yielding

x˙ = A(y)x + B1(y)w + B2(y)u + rf (x) (3.34) where |r (x)| lim f = 0 (3.35) 0 0 0 0 x →xo(y) |x − xo(y)| The subsequent controller synthesis is based on all time-varying plants (3.30) where y = y(t). In practice, however, this introduces additional dynamics, which may not be present or even feasible in case of e.g. rapidly changing y(t), for the original model (3.22). Hence, the resulting controller synthesis will be conservative as it takes into account all y = y(t).

3.4 linear, parameter-dependent plant

When the starting point is a linear plant with one or several time-varying parameters rather than a nonlinear plant, appropriate selection of the scheduling variables directly yields an LPV model. Physical nonlinear modeling of the process may be omitted in case black-box models or measurements are available. Analogous to linearization-based scheduling, based on for example a set of frequency-response or step response measurements, which are parameterized by θ ∈ Θo, a family of linear, parameter-dependent models Σ(θ), θ ∈ Θ follows directly (e.g. (Nichols, Reichert & Rugh 1993)). Example 3.2 (Engine system) Given a model for the engine system (see Appendix B) G(T ) H (T ) = e−0.07s c T ∈ [300 400] K (3.36) e c 1.85s + 1 c

where G(Tc) = g1Tc + g2 is a temperature Tc-dependent gain of the engine as defined in (B.2). Defining θ = Tc shows that the model (3.36) in fact is a linear parameter model with one varying parameter. Hence, implementation of Tc = θ = θ(t) directly yields an LPV model. G(θ(t)) H (θ(t)) = e−0.07s θ(t) ∈ Θ (3.37) e 1.85s + 1 where Θ ∈ [300 400] K. 14 CHAPTER 3. PARAMETER-DEPENDENT PLANT MODELS

3.5 Linear fractional transformation description

In special cases, the parameter dependency in the LPV models resulting from a family of lineariza- tions or quasi-LPV modeling can be modeled or approximated as a Linear Fractional Transforma- tion (LFT). A LPV model, which may be formulated as a linear time-invariant system enclosed by a feedback loop with time-varying parameters θ(t), is regarded as an LFT model. The particular LFT structure enables specific approaches for the subsequent controller design. Constructing an LFT essentially is the same problem as the realization of a transfer function. Hence, representa- tions may not be unique or ‘minimal’. LFTs are a powerful and flexible approach to represent uncertainty in models by separating what is known, represented by M, from what is unknown, represented by ∆, in a feedback-like connection. In Figure 3.2, a lower LFT is depicted, where

 M (11) M (12)  M = (3.38) M (21) M (22)

The possible values of the unknown elements ∆ are bounded. z w

r M v

∆

Figure 3.2: uncertainty modeling using LFTs

This lower LFT is denoted by FL(M, ∆) and yields

z = F (M, ∆)w L (3.39) = (M (11) + M (12)∆(I − M (22)∆)−1M (21))w

Analogously, an upper LFT is defined as

(22) (21) (11) −1 (12) FU (M, ∆) = M + M ∆(I − M ∆) M (3.40)

In general, an LFT description enables representation of any polynomial or matrix function of a scalar variable. Hence, if the matrices of the model (3.3) depend rationally on the unknown, i.e. varying parameter θ, a LFT on a corresponding diagonal, θ-dependent ∆(θ) is enabled, where ∆ = diag(θ1In, . . . , θpIn) for θ having p components and M being of order n (Doyle, Packard & Zhou 1991, Zhou, Doyle & Glover 1996).

Example 3.3 (Engine system)

Consider the engine system He(Tc) = G(Tc)He,n, where G(Tc) = g1Tc + g2 and g1 = −3 8.7 · 10 , g2 = 2.1. Suppose Tc takes on values 300 ≤ Tc ≤ 400, which yields

4.71 ≤ G(Tc) ≤ 5.58. Rewriting this as a LFT yields G(Tc) = 5.145 + 0.57δG(Tc) with

δG(Tc) ∈ [−1, 1], i.e.

 5.145 0.57   G(T ) = , δ (3.41) c 1 0 G(Tc)

Figure 3.3 shows the corresponding block-diagrams.

The advantages of LFT modeling involve i) the flexibility to represent different system realiza- tions including uncertainties, e.g. state-space or transfer functions and corresponding parametric 3.6. CONCLUSIONS / RECAP 15

ωe Fv He,n G(Tc)

(a)

ωe Fv He,n 5.145 0.57 r 1 0 v

δG(Tc)

(b)

Figure 3.3: Modeling the dependence of G(Tc) on Tc as an uncertainty δG(Tc) using a lower LFT.

and / or model uncertainties and ii) the possibility to apply typical algebraic operations, e.g. cascade, parallel and feedback interconnections (Doyle et al. 1991). Hence, LFTs are very suitable for a general hierarchical representation of uncertainty and may be used as a suitable basis for subsequent gain-scheduling (v. Helvoort, Steinbuch, Lambrechts & v.d. Molengraft 2004, Wu & Dong 2006). In principle, LPV systems in which the state-space matrices are polynomial or rational func- tions of the parameters θ can be transformed into LFT form. However, this reformulation in general is nontrivial and may involve a considerable increase in the order of the system and cor- responding increase in the order of the controller (Leith & Leithead 2000).

3.6 Conclusions / recap

Linearization-based scheduling yields a set or family of locally valid, linear parameter-dependent plant models (3.2) rather than a true LPV model (3.3). This is comparable to the use of a set of black-box point-designs instead of a nonlinear model as a starting point. Direct gain scheduling as well as a quasi-LPV approach directly yield a true LPV model. Velocity-based scheduling allows for the implementation of θ = θ(t) and thus the design of a true LPV model as well. Subsequent LPV control synthesis requires a true LPV model as a basis, whereas a family of linear parameter- dependent plant models is sufficient for subsequent classical gain scheduling, which is based on local LTI controller designs. The distinction between LPV and LFT models follows from i) the different available techniques to analyze the system, ii) the allowable dependence of the state-space on the parameters and iii) the extent to which information about the parameter’s variation, i.e. the rate of variation θ˙(t), is exploited in the analysis. Regarding LPV models, the scheduling is limited to parameter vari- ations, whereas LFTs enable for all sorts of uncertainties. However, LFTs exhibit computational difficulties. Open question are how to select the scheduling variables θ appropriately and how to determine how many point-designs are required in case of linearization-based scheduling or black-box identi- fication. In practice, these issues are commonly resolved by insight in the system and experience. 16 CHAPTER 3. PARAMETER-DEPENDENT PLANT MODELS Chapter 4

Classical gain scheduling

As discussed in Section 2.1.1, a set of LTI controllers Λ(θ), parameterized by θ has to be de- signed. Subsequent implementation of the controllers requires the design of a family of parameter- dependent controllers Λ(θ(t)) based on this set of controllers. In the best case, a set of parameter- dependent controllers follows immediately, otherwise for example interpolation techniques may be applied to enable smooth scheduling or the set of controllers may be used for discrete scheduling. The derivation of a parameterized set of LTI controllers is discussed briefly after which the implementation of the resulting set of LTI controllers Λ(θ) is discussed. Various methods to derive a family of parameter-dependent controllers, e.g., via interpolation, and corresponding points of attention, e.g. hidden coupling terms, are discussed.

4.1 LTI controller design

The main advantage of applying classical gain scheduling techniques is the ability to use linear controller design synthesis without the restriction to a specific synthesis. Based on the modeling described in the previous chapter, a set of LTI controllers Λ(θ) is designed, which is parameterized by the same scheduling variable θ as used in the corresponding modeling. In case a parameterized set of linearized models is available instead of an LPV model or in case an LPV model is available, but different closed-loop dynamics are demanded for different operating points, a set of linear controllers has to be designed. A set of operating conditions i θi ∈ Θ ⊂ Θ is determined for which corresponding LTI controllers are designed. In case of a parameterized set of linearized models, typically the same operating conditions Θi = Θo are used. The corresponding set of linearized models or constant operating points of the LPV model enable model-based controller design. The resulting parameterized set of controllers hence is only locally valid. The parameterized set of linear controllers is defined by  ˙ ξ(t) = Ac(θ)ξ(t) + Bc(θ)˜z(t) i Λ(θi): θi ∈ Θ (4.1) u˜(t) = Cc(θ)ξ(t) + Dc(θ)˜z(t) with input signalz ˜(t), which is derived from the error signal z(t), including if appropriate a reference command r(t), e.g. z(t) = r(t)−y(t), the controller output signalu ˜(t) and the controller state ξ(t). Corresponding LTI controller design syntheses are not discussed in this report. For example standard loop-shaping or pole-placement techniques as well as robust H∞ and µ syntheses may be employed. Conceptually, the resulting set of linearized controllers Λ(θi) is used as a basis for a continuously scheduled family of parameter-dependent controllers Λ(θ(t)) lateron. In general, this enforces all controller designs to have the same structure and dimension. Notable exceptions are i) the appli- cation of discrete scheduling instead of continuous scheduling methods and ii) the application of fuzzy controller blending. Regarding discrete gain scheduling, a distinction between the scheduling

17 18 CHAPTER 4. CLASSICAL GAIN SCHEDULING of controller coefficients and the scheduling of complete controllers has to be made. Concerning the former, fixed controller structure and dimension obviously are demanded as well. Concerning the latter, a fixed controller structure and dimension is not demanded, however favorable nevertheless to be able to minimize differences in magnitude between controller transitions.

4.1.1 Direct gain scheduling controller design When a true LPV model is available (instead of a parameterized set of linearized models), and constant closed-loop behavior for the complete operating range is desirable, a set of parameter- dependent controllers can be determined immediately.

Example 4.1 (Engine system) Y (s) Consider the LPV engine model He(Tc) = U(s) (3.37) and the problem of designing a corresponding idle governor controller. The time-delay is not taken into account for the moment. For control purposes, an error z(t) = r(t) − y(t) is introduced, where r(t) is a certain reference trajectory to be followed by y(t). An LPV controller is designed directly using a (linear) parameter-dependent proportional controller U(s) k C(θ(t)) = = p (4.2) Z(s) G(θ(t)) where Z(s) = R(s) − Y (s). Combining (3.37) and (4.2) yields a closed-loop transfer function that is independent of θ(t) Y (s) k = p (4.3) R(s) 1.85s + 2

where kp is the tuning parameter (for kp = 23.31, suitable closed-loop behavior is achieved).

4.2 Gain scheduling controller design

i Based on the parameterized set of linear controllers Λ(θi), θi ∈ Θ , a family of linear, parameter- dependent controllers Λ(θ), θ ∈ Θ has to be derived. I.e. the operating point designs have to be extended to the entire operating envelope. To enable actual implementation, the scheduling variable θ is replaced by the measured variable θ(t), yielding θ = θ(t). This yields a nonlinear, scheduling controller of the form

 ξ˙(t) = f (ξ(t), z˜(t), w˜(t), θ(t)) Λ(θ(t)) : c θ(t) ∈ Θ (4.4) u˜(t) = hc(ξ(t), z˜(t), w˜(t), θ(t)) with input signalz ˜(t), which is derived from the error signal z(t), including if appropriate a reference command r(t), e.g. z(t) = r(t) − y(t), the controller output signalu ˜(t), the controller state ξ(t), external inputsw ˜(t) and the time-varying scheduling variable θ(t). In general, the controllers are scheduled utilizing the current value of the scheduling variable to interpolate or schedule the controller coefficients, e.g. the controller gains. In case of direct scheduling (see Section 4.1.1), a family of parameter-dependent controllers is present already and the implementation θ = θ(t) has to be considered only. Considering gain-scheduling and i interpolation of Λ(θi), θi ∈ Θ , the goal is to design a family of linear, parameter-dependent controllers Λ(θ), θ ∈ Θ, such that at each operating point θ ∈ Θi, the scheduled controller i) linearizes to the corresponding linear controller design and ii) provides a constant control value yielding zero error. The presence of hidden coupling terms will be shown to yield additional requirements to the scheduling procedure. The complexity of the scheduling or interpolation method depends on the structure of the set of LTI controllers. Consider e.g. fixed structured LTI controllers with only one or a few varying 4.2. GAIN SCHEDULING CONTROLLER DESIGN 19 gains for changing plant dynamics or operating conditions with respect to a set of LTI controllers with varying structure, possibly including different states and topology (Reichert 1992). The former often is the case when classical linear control techniques are applied to design the set of LTI controllers. The latter may be the result of application of multivariable state-space design procedures. In case of discrete scheduling, the implementation of the LTI controllers involves the design of a scheduled selection procedure that is applied to the set of LTI controllers, rather than the design of a family of scheduled controllers. The selection procedure or switching has the advantage of simplicity in defining the controller family. This comes down to the definition of regions for which the members of the set of LTI controllers are valid. However, discontinuities (jumps) may appear in the control output, or in the controller coefficients in case of discrete scheduling of the coefficients rather than the total controller, which may result in e.g. chattering behavior. In Figure 4.1 an example of discrete scheduling by switching between 2 controllers is shown. This works only in case Λ0 as well as Λ1 are open loop stable. Whereas this will be the case in general for classical controller designs, this is less obvious for controllers resulting from H∞ or µ-syntheses (Niemann & Stoustrup 1999).

u(t) System or y(t) Plant P

Λ0

Λ1 θ(t)

Figure 4.1: Gain scheduling by switching between two controllers.

In general, continuous or smooth scheduling is required, which will be the focus of the re- mainder of this chapter. Issues involve how to schedule the controller, i.e. use interpolation, linearization scheduling or other methods, how many and which operating points to use as a basis for the scheduling and how to assess global stability and performance properties. Firstly, classical scheduling and various interpolation techniques are presented. Next, hidden coupling terms and closed-loop stability properties are discussed.

4.2.1 Linearization scheduling The objective of linearization scheduling is that the equilibrium family of the controller (4.4) matches the equilibrium family of the plant (3.1), such that i) the closed-loop system still can be tuned appropriately with respect to performance and robustness demands and ii) the linearization family of the controller equals the designed family of linear controllers. The former requirement requires the existence of a function ξe(θ) such that 0 = f (ξ (θ), z (θ), w (θ)) c e e e θ ∈ Θi (4.5) ue(θ) = hc(ξe(θ), ze(θ), we(θ)) The latter requirement yields combination of (4.1) and (4.4)

∇ξfc(ξe(θ), ze(θ), we(θ)) = Ac(θ) ∇ f (ξ (θ), z (θ), w (θ)) = B (θ) z˜ c e e e c θ ∈ Θ (4.6) ∇ξhc(ξe(θ), ze(θ), we(θ)) = Cc(θ) ∇z˜hc(ξe(θ), ze(θ), we(θ)) = Dc(θ) 20 CHAPTER 4. CLASSICAL GAIN SCHEDULING

Both conditions are satisfied by the general controller family

ξ˙(t) = A (θ)(ξ(t) − ξ (θ)) + B (θ)(z(t) − z (θ)) c e c e θ ∈ Θ (4.7) u(t) = Cc(θ)(ξ(t) − ξe(θ)) + Dc(θ)(z(t) − ze(θ)) + ue(θ) Implementation of θ = θ(t) in (4.7) then yields the (nonlinear) gain-scheduled controller

ξ˙(t) = A (θ(t)) (ξ(t) − ξ (θ(t))) + B (θ(t)) (z(t) − z (θ(t))) c e c e θ(t) ∈ Θ (4.8) u(t) = Cc(θ(t)) (ξ(t) − ξe(θ(t))) + Dc(θ(t)) (z(t) − ze(θ(t))) Example 4.2 (Longitudinal vehicle control) Consider the set of parameterized LPV models (3.16). Given a corresponding set of parameterized LTI PI-controllers ( ˜˙ Λ(θ): ξ(t) =z ˜(t) θ ∈ Θ (4.9) ˜ e u˜(t) = ki(θ)ξ(t) + kp(θ)˜z(t)

locally yielding the closed-loop dynamics  " # "   # x˜˙(t) A(θ)˜x(t) + B (θ)w ˜(t) + B k (θ)ξ˜(t) + k (θ)(˜z(t))  = 1 2 i p  ˜˙  ξ(t) z˜(t)        A(θ) − B2kp(θ) B2ki(θ) x˜(t) B1(θ) + [0 B2kp(θ)] Σcl : = + w˜(t)  −1 0 ξ˜(t) 0 1       x˜(t)  y˜(t) = 1 0  ξ˜(t)

(4.10)

Using linearization scheduling, a nonlinear, gain-scheduled controller Λnl is to be de- signed.  ˙ ξ(t) = fc(ξ(t), w(t), y(t)) Λnl : (4.11) u(t) = gc(ξ(t), w(t), y(t))

The controller equilibrium family ξe(θ), θ ∈ Θe should be such that the equilibrium conditions of the original (nonlinear) plant and the controller match. Furthermore, at equilibrium, the linearized controller should correspond to the corresponding original point-design controller. Substituting the equilibrium values (3.15) for the deviation variables in (4.9) yields the controller equilibrium ξe(θ) 1 ξe(θ) = ue(θ) θ ∈ Θe (4.12) ki(θ)

1  0 (2)2 (1) (1)  = Cdθ + g(sin θ + Cr cos θ ) θ ∈ Θe ki(θ)

Adopting the general controller family (4.7) yields

 ξ˙(t) = z(t) − z (θ) Λ(θ): e (4.13) u(t) = ki(θ)(ξ(t) − ξe(θ)) + kp(θ)(z(t) − ze(θ)) + ue(θ) Again substituting the equilibrium values then yields the LPV controller family

 ξ˙(t) = z(t) Λ(θ): θ ∈ Θ (4.14) u(t) = ki(θ)ξ(t) + kp(θ)z(t) 4.2. GAIN SCHEDULING CONTROLLER DESIGN 21

By replacing the parameter θ by its measured value θ(t) = [w(1)(t), y(t)]T , the non- linear gain-scheduled controller results

 ξ˙(t) = z(t) Λnl : (1) (1) (4.15) u(t) = ki(w (t), y(t))ξ(t) + kp(w (t), y(t))z(t)

4.2.2 Interpolation methods Most of the literature on continuous gain scheduling assumes that linearization scheduling is available and possible. However, in the case of isolated point-designs, corresponding interpolation of the controller (coefficients) is required. In practice, ad hoc interpolation of local point-design controllers is gladly adopted to arrive at a gain-scheduled LPV controller. However, in general no global or even local stability and performance guarantees can be given. Hence, more theoretically justified methods are proposed, which guarantee local stability. Practical implementation of these methods however, is difficult.

Ad hoc interpolation methods Although ad hoc interpolation methods lack a theoretical basis, many practical applications are present in literature. In general, these examples perform satisfactorily for the application consid- ered. However, as they are based on intuition rather than a theoretical basis, application in other cases often is difficult or even prohibited. The lack of a theoretical basis implies that no perfor- mance guarantees can be given at intermediate operating points and even closed-loop instability may occur. Ad hoc interpolation methods in general require a set of local LTI controller designs with fixed structure as well as dimension as a basis. I.e. when focusing on transfer functions, all controllers should have the same number of poles and zeros. (Nichols et al. 1993) interpolate point-design H∞ controllers by means of interpolating the corresponding poles, zeros and gains. The number of operating points involved, depends on the rate and size of migration of poles, zeros and gains. Although no guarantees can be given regarding global stability and performance specifications of the resulting LPV controller, more recent research by (Paijmans, Symens, v. Brussel & Swevers 2008) is based on the same principle. The main advantage of the method is the practical applicability. Stability and performance is validated by means of extensive simulations, thereby specifically focusing on intermediate operating points. (Reichert 1992) design a set of LTI H∞ controllers using Riccati equations. Linear interpola- tion of the Riccati equations yields a family of parameter-dependent controllers. No guarantees regarding performance, robustness or even stability regarding the nonlinear plant can be given, locally nor globally. All point-design controllers have to be of the same structure and dimension, which is not obvious for a set of H∞ controllers. Other ad hoc interpolation methods, such as for example presented by (Kellet 1991, Ko¸c, Knittel, de Mathelin & Abba 2002) will not be discussed further at this point. Finally, con- troller blending sometimes is referred to as a special case of an ad hoc interpolation technique (e.g. (Niemann & Stoustrup 1999)). As this can be interpreted as fuzzy scheduling, this will be considered separately in Chapter 6.

More theoretically justified methods The main goal of theoretically justified interpolation techniques for the synthesis of gain scheduling controllers with respect to the before presented ad hoc interpolation methods, comprises a stability preserving condition. A true LPV model Σ(θ), θ ∈ Θ of the original plant is assumed to be available. Analogous to ad hoc methods, the interpolation of a set of LTI controllers Λ(θi), i θi ∈ Θ , which yield different closed-loop behavior for different operating points is considered. When constant closed-loop behavior is required, a gain-scheduled family of LPV controllers often follows immediately from the corresponding LPV model (see Section 4.1.1). The resulting LPV controller Λ(θ), θ ∈ Θ is called stability preserving if (Stilwell & Rugh 2000) 22 CHAPTER 4. CLASSICAL GAIN SCHEDULING

• the coefficients of Λ(θ) are continuous functions of θ

• Λ(θ) = Λ(θi) for θ = θi

• Λ(θ) stabilizes Σ(θ) for all θ ∈ Θ

i The interpolation methods are based on a stability covering condition for Λ(θi), θi ∈ Θ with respect to Σ(θ), θ ∈ Θ. The stability covering condition demands the existence of at least one i S controller Λ(θ ), θ ∈ Θ that stabilizes Σ(θ), for all θ ∈ Θ. I.e. if Θ ⊂ i U where U ⊂ Θ i i θi∈Θ θi θi i an open neighborhood containing θi ∈ Θ , such that Λ(θi) stabilizes Σ(θ) for all θ ∈ Uθi , the set i of local LTI controllers Λ(θi), θi ∈ Θ is stability covering for Σ(θ), θ ∈ Θ. This is a sufficient i condition on the selection of parameter values Θ for which locally valid LTI controllers Λ(θi), i θi ∈ Θ are designed.

Uθjk Θ ⊂ Rnθ

θj,k,l

Uθj,k,l

i θi ∈ Θ

nθ i Figure 4.2: Operating envelope Θ ⊂ R with the set Θ and three example stability regions Uθj,k,l corresponding to θj,k,l. The intersection or overlapping region Uθjk of Uθj and Uθk is indicated in grey.

Define an intersection or overlap of multiple regions Uθi as Uθijk... , e.g. Uθjk represents the intersection of Uθj and Uθk (see Figure 4.2). The proposed interpolation method generates corre- ˆ sponding stability preserving controllers Λ(Uθijk... ), Uθijk... ∈ UΘ, where UΘ the set comprising all overlapping regions, that follow from interpolation of the controllers Λ(θi) of the corresponding i regions Uθi , θi ∈ Θ . The resulting gain-scheduled controller Λ(θ) for the complete envelope of operating conditions θ ∈ Θ then becomes  Λ(θi), θ ∈ Uθi , θ∈ / UΘ Λ(θ): ˆ θ ∈ Θ (4.16) Λ(Uθijk ), θ ∈ Uθijk ∈ UΘ

The method actually involves a particular form of controller blending. The resulting controller yield switching between controllers with guaranteed smoothness. Additional slow-variation argu- ments yield local exponential stability of the gain-scheduled nonlinear closed-loop system. Various approaches are presented in literature. (Shahruz & Behtash 1990) describes a method to linearly interpolate the controller gains of a state feedback controller resulting from eigenvalue placement. (Stilwell & Rugh 1999) elaborate on this idea introducing an observer-based state feedback and linearly interpolating the observer gains. Combining this with a Youla parame- terization subsequently ((Stilwell 1999)), yields a basis for state-space interpolation of a set of observer-based point-design controllers (Stilwell & Rugh 1998). Analogously, a transfer function interpolation method is proposed. Both methods are presented by (Stilwell & Rugh 2000), who also determine an upper bound on the rate of variation of the scheduling variable. Regarding state- space interpolation, the method proposed by (Stilwell & Rugh 2000) is restricted to full-order controllers. (Pellanda, Apkarian & Alazard 2000) generalize the approach to augmented order state-space controllers, targeting at the application of H∞ and µ-synthesis. (Claveau, Chevrel 4.2. GAIN SCHEDULING CONTROLLER DESIGN 23

& Knittel 2007) present a practical example of linear interpolation of a special observer-based realization. An analogous approach from a mathematical point of view is provided by (McNichols & Fadali 2003). Interval mathematics are applied to determine regions for which closed-loop poles remain within specified bounds. Correspondingly, the required number of point-designs is deter- mined. The methodology is restricted to proper transfer functions with time-varying parameters (coefficients) where Θ is known and θ˙ is known to be small. Global stability has to be inferred from extensive simulations.

4.2.3 Velocity-based scheduling

The velocity-based linearization family (3.21) may be adopted as a suitable basis for subsequent gain scheduling controller design (Leith & Leithead 1998c). Classical gain scheduling (Leith & Leithead 1998c), LPV synthesis (Leith & Leithead 2000) or fuzzy-alike blending techniques (Leith et al. 2000) may be employed. A velocity-based linearization family by definition has infinitely many constant operating points as linearizations are assigned to every operating point instead of equilibrium operating points only. The velocity-based linearization family thus broadens the application range of classical or divide-and-conquer gain scheduling. In practice, LTI controller design will be applied for a grid of operating points in the total operating envelope. Subsequent scheduling or interpolation techniques (see Section 4.2.1, 4.2.2) can be adopted to design a parameter-varying controller. (Leith & Leithead 1999, Leith & Leithead 2000) propose to use a blended multiple model representation, which has close similarity to Takagi-Sugeno or fuzzy modeling. The velocity-based linearization family is approximated by a small number of local models. The difference with respect to fuzzy modeling is that the blended local models directly equal the velocity-based linearization model at the corresponding operating point. The blended model is approximated by the weighted combination of the local model solutions. Difficulties arise when practical implementation is considered due to the resulting velocity- based controller, which requiresy ˙ rather than y as input, see Figure 4.3. The corresponding differentiator and integrator may be absorbed in the controller in special cases (Leith & Leithead 1998a). The corresponding velocity-based controller is given by

w(t) w˙ (t) Velocity- w(t) z(t) Nonlinear d based dt y˙(t) u˙(t) R u(t) plant y(t) controller

Figure 4.3: Implementation of a velocity-based controller design.

 ˙  ξ = η Λ(%): η˙ = Ac(%)η + B1c(%)w ˙ + B2c(%)y ˙ (4.17)  u˙ = C(%)η + D(%)w ˙

Stability and performance of velocity-based gain-scheduled systems is established using general nonlinear system theory such as small gain or Lyapunov theories. Hence, generally conservative results are obtained, since, except in special cases, only sufficient conditions are known for the stability of nonlinear systems. In contrast to conventional gain scheduling approaches, the resulting nonlinear controller is valid throughout the operating envelope of the plant, not just in the vicinity of the equilibrium operating points. 24 CHAPTER 4. CLASSICAL GAIN SCHEDULING

4.3 Hidden coupling terms

The fact that the scheduling variable θ(t) is time-varying, whereas the controller design generally i is based on corresponding fixed values θi ∈ Θ introduces so-called hidden coupling terms. These terms introduce differences between (4.7) and the corresponding linearization of (4.8). Due to the (endogenous) variable θ(t), additional terms with respect to the linearization of (4.7), which is given by (4.1), may appear. Define g(t) = θ(t). Linearization of (4.7) then yields (the time-dependencies are omitted for reasons of clarity)

˜˙ ˜ ξ = Ac(θ)ξ + Bc(θ)˜z − ... n ∂ξo(θ) ∂zo(θ) o n ∂g(zo(θ),wo(θ)) ∂g(zo(θ),wo(θ)) o ... − Ac(θ) ∂θ + Bc(θ) ∂θ × ∂z z˜ + ∂w w˜ ˜ u˜ = Cc(θ)ξ + Dc(θ)˜z + ... n ∂uo(θ) ∂ξo(θ) ∂zo(θ) o n ∂g(zo(θ),wo(θ)) ∂g(zo(θ),wo(θ)) o ... + ∂θ − Cc(θ) ∂θ − Dc(θ) ∂θ × ∂z z˜ + ∂w w˜ (4.18)

Comparing (4.18) to (4.1) reveals the additional hidden coupling terms that result due to imple- mentation of θ = θ(t). These terms represent additional feedback terms in case of components involvingz ˜(t) or exogenous disturbances in case of components involvingw ˜(t). Hence, local sta- bility as well as local performance of the resulting closed-loop differs from the original local LTI designs. Correspondingly, the issue of preventing hidden coupling terms revolves around the no- tion that the linearized closed-loop system at any operating condition, parameterized by fixed values of the scheduling variable, should precisely match the interconnection of the corresponding linearized plant and linear controller design at that specific point. (Lawrence & Rugh 1995, Nichols et al. 1993, Rugh & Shamma 2000) discuss the phenomenon of hidden coupling terms, presenting sufficient conditions to prohibit the existence of these terms in case of a SISO or decoupled problem. Nevertheless, typically hidden coupling terms can not be prevented in all cases. No hidden coupling terms will be present when, in addition to (4.5) and (4.6), the following conditions are fulfilled (Rugh & Shamma 2000, Nichols et al. 1993)

∇ f (ξ (θ), z (θ), w (θ)) = 0 θ c o o o θ ∈ Θ (4.19) ∇θhc(ξo(θ), zo(θ), wo(θ)) = 0 which yield

A (θ) ∂ξo(θ) + B (θ) ∂zo(θ) = 0 c ∂θ c ∂θ θ ∈ Θ (4.20) ∂uo(θ) ∂ξo(θ) ∂zo(θ) ∂θ − Cc(θ) ∂θ − Dc(θ) ∂θ = 0

Practical application of (4.19) generally requires insight in the system at hand and comes down to physical interpretation of the corresponding terms. The extent to which hidden coupling terms can be prevented is discussed by (Lawrence & Rugh 1995).

Example 4.3 (Longitudinal vehicle control) Consider the nonlinear, gain-scheduled controller (4.15). Define g(t) = θ(t) = g(w(t), y(t)) = [w(1)(t), y(t)]T . Linearization of (4.15) around an equilibrium point yields a local, pa- 4.4. STABILITY PROPERTIES 25

rameterized controller Λ(θ), θ ∈ Θe

 ˙ ∂z (θ)  ∂g(y (θ),w (θ)) ∂g(y (θ),w (θ))  ξ˜ =z ˜ − e e e y˜ + e e w˜  ∂θ ∂y ∂w   =z ˜   = [0 1]w ˜ − x˜     u˜ = k (θ)ξ˜+ k (θ)˜z + ∂ue(θ) − k (θ) ∂ξe(θ) − k (θ) ∂ze(θ) ...  i p ∂θ i ∂θ p ∂θ   Λ(θ): ... ∂g(ye(θ),we(θ)) y˜ + ∂g(ye(θ),we(θ)) w˜  ∂y ∂w      ˜ ∂ue(θ) 1 ∂ue(θ) ∂ 1  = ki(θ)ξ + kp(θ)˜z + − ki(θ) + ue(θ) (˜y + [1 0]w ˜(t))  ∂θ ki(θ) ∂θ ∂θ ki(θ)  ˜ ∂ 1  = ki(θ)ξ + kp(θ)˜z − ki(θ)ue(θ) (˜y + [1 0]w ˜(t))  ∂θ ki(θ)  ˜ h ∂ 1 i  ∂ 1   = ki(θ)ξ + −ki(θ)ue(θ) kp(θ) w˜ − kp(θ) + ki(θ)ue(θ) x˜ ∂θ ki(θ) ∂θ ki(θ) (4.21)

Hence, the corresponding closed-loop system, combining (3.16) and (4.21) becomes

 " # "   # x˜˙ ∂ 1    −B2 kp(θ) + ki(θ)ue(θ) ∂θ k (θ) B2ki(θ) x˜  ˙ = i ˜ + ...  ξ˜ −1 0 ξ   " h ∂ 1 i # B1(θ) + −ki(θ)ue(θ) kp(θ) Σcl(θ): ... + ∂θ ki(θ) w˜  0 1    x˜   y˜ =  1 0   ξ˜ (4.22)

Comparing the linearization result (4.22) to the original, locally designed closed-loop dynamics (4.10) shows additional terms. These terms are the result of substituting θ by its corresponding measured values g(t). By doing so, additional (feedback) loops are created in the closed-loop system, which might influence stability and performance at the corresponding equilibrium points. In this case, the additional constraints (4.19) are easily fulfilled by choosing another parameterization of the controller. Rewriting (4.9) as ( ˜˙ Λ(θ): ξ(t) = ki(θ)˜z(t) θ ∈ Θ (4.23) ˜ e u˜(t) = ξ(t) + kp(θ)˜z(t)

does not change the controller as it just another non-unique parameterization. How- ever, the controller equilibrium becomes ξe(θ) = ue(θ), as opposed to (4.12). Using this equilibrium value, the additional conditions (4.19) indeed are fulfilled. Hence, no hidden coupling terms will be present anymore when implementing θ = θ(t).

4.4 Stability properties

Stability and performance properties of the closed-loop system incorporating the original nonlinear plant or model (3.1) and the parameter-dependent controller (4.8) form an important issue in the design and application of gain-scheduled controllers. Regarding classical gain-scheduling synthesis as described in this chapter, difficulties arise due to • the equilibrium-operating-point-based controller designs, which disregard off-equilibrium op- erating points • the use of a frozen-valued scheduling variable θ instead of θ(t) in the controller design, which disregards transient dynamics 26 CHAPTER 4. CLASSICAL GAIN SCHEDULING

Stability analysis of nonlinear systems and hence closed-loop LPV systems is difficult. A classi- cal result by (Desoer 1969) state that LPV systems are stable for sufficiently small θ˙(t). (Shamma & Atans 1991) extend this result considering robust performance for unmodeled dynamics as well. (Shamma & Athans 1990) derive sufficient conditions for robust stability and performance of quasi-LPV systems, which again comes down to a sufficiently small θ˙(t). The conditions also hold for non-quasi-LPV systems if the nonlinearities in the state variables other than the scheduling variables are small. The corresponding rule-of-thumb ’the plant nonlinearities should be captured by the scheduling variables’ is often referred to. However, the conditions typically are conser- vative as they are based on classical Lyapunov and small-gain stability analyses. Furthermore, implementation of θ = θ(t) to derive a nonlinear, parameter-dependent controller may introduce additional dynamics, so-called hidden terms, as the controller design is based on the frozen-valued scheduling parameter θ. Hidden terms may influence stability (see Section 4.3). A priori assessment of guaranteed stability and performance properties of classical gain-scheduling regarding i) non-local or off-equilibrium operating points and ii) fast variations of the scheduling variable, is impossible. As this is not considered in the controller design process, guaranteed prop- erties of the overall design cannot be established. (Shamma & Athans 1992) present an example in which the frozen parameter designs are based on stable plant models, but the actual plant may be unstable. The linear models, therefore, are an inaccurate representation of the true plant. Stability and performance specifications thus are typically inferred from extensive simulations. In the vicinity of equilibrium operating conditions, the stability of a nonlinear system may be related to the stability of the corresponding frozen-input nonlinear system. Following (Rugh 1991, Lawrence & Rugh 1995), the deviation of a (nonlinear) system from its equilibrium surface remains small if 1. the initial conditions lie in the neighborhood of an equilibrium point, which yields     xi(θ(t)) xe(θ(t)) − < γ, γ > 0, θ(t) ∈ Θ (4.24) ξi(θ(t)) ξe(θ(t)) The inputs and initial conditions are thus confined to lie in the neighborhood of the corre- sponding equilibrium operating points (Leith & Leithead 1998b, Leith & Leithead 1998c). 2. a slow variation requirement is present, which yields

˙ θ(t) < µ, µ > 0 (4.25) This condition restricts the class of allowable inputs and initial conditions to remain suffi- ciently close to the equilibrium operating conditions. 3. the linearization of the nonlinear system equals the local point-designs, i.e. no hidden cou- pling terms are present, and is stable, i.e. <(λ) ≤ −ε < 0, ∀ θ(t) ∈ Θ If all three conditions are fulfilled, the deviation from the equilibrium surface remains small and the stability of the nonlinear system may be related to the stability of a set of LTI models representing the nonlinear system. More recent controller design syntheses, which stem from classical gain scheduling synthe- sis such as LPV and LFT syntheses incorporate θ(t) in the controller design process. As the time-varying nature of the scheduling parameter is taken into account in the controller design syn- thesis explicitly, global stability and performance properties can be derived. In general however, these syntheses guarantee stability and performance ∀ θ instead of for θ(t). Hence, conservatism is introduced. Quasi-LPV design suffers the same drawback (see Section 3.3). Typically, this conservatism is reduced by defining explicitly a range and rate of variation corresponding to the scheduling parameter. Taking these parameters into account in the controller design, decreases the conservatism. Chapter 5

LPV controller synthesis

The term Linear Parameter Varying (LPV) control synthesis is commonly used to indicate con- troller synthesis based on either LPV or Linear Fractional Transformation (LFT) models. LPV control synthesis yields parameter-dependent controllers with a priori guaranteed stability and performance properties. The available real-time information of the parameter variation is used in the control synthesis. The time-varying nature of the corresponding LPV dynamics is thus incor- porated in the LPV control synthesis as opposed to classical gain-scheduling methods (Shamma & Atans 1991). The a priori stability and performance guarantees prohibit the need for validation of stability and performance characteristics afterwards by means of extensive simulations. LPV control synthesis focuses on an LPV model rather than a nonlinear model of the plant, which explains the naming. LPV modeling is discussed in Chapter 3. The main advantages of LPV control synthesis are i) there exist a solid theoretical foundation guaranteeing a priori stability and performance for all θ(t) given a corresponding range and rate of variation of θ(t), ii) the corresponding controller design is global with respect to the parame- terized operating envelope Θ, whereas classical gain scheduling techniques focus on local system properties, and iii) a controller is synthesized directly, rather than its construction from a family of local linear controllers. The main disadvantages are i) with respect to classical gain scheduling techniques, the controller synthesis is much more involved, which results in focusing on appro- priate problem formulation rather than the actual controller design, ii) generally, conservatism has to be introduced to arrive at a feasible and convex problem, and iii) with respect to classical gain scheduling, allowing for arbitrary linear controller design techniques, a predefined controller design synthesis has to be adopted. As the latter point already indicates, LPV syntheses constitutes a specific performance evalu- ation framework, whereas classical gain scheduling provides an open framework. Typically, LPV syntheses employ the induced L2-norm as a performance measure, which is directly related to linear H∞-techniques. As a result, an LPV control synthesis applied to a time-invariant system is equivalent to a standard H∞-approach. In general, LPV control syntheses can be categorized into

• techniques utilizing a Lyapunov-based approach, also known as (genuine) LPV-techniques

• techniques exploiting the specific structure of systems with LFT parameter dependence, utilizing a small-gain approach, which are also determined as LFT approaches

• a combination of the two preceding points, which will be referred to as ‘mixed’ LPV-LFT approaches

The LPV paradigm was introduced circa 1990 as a framework to analyse gain scheduling (Shamma & Atans 1991, Shamma & Athans 1992) and has seen considerable activity since (Rugh & Shamma 2000). By now, application examples are present in literature, e.g., (Wang, Nishimura & Shimogo 2007, Dinh, Scorletti, Fromion & Magarotto 2005, Dettori & Scherer 2002). However, a general

27 28 CHAPTER 5. LPV CONTROLLER SYNTHESIS solution for the synthesis of an LPV controller still has not been determined. Until now, research has come up with solutions for special extensions rather than genuine generalizations of linear H∞-designs. Correspondingly, in general, conservatism is introduced, which is one of the main drawbacks of LPV controller synthesis. Current research hence focuses on solving the problem for true generalizations (Rieber, Scherer & Allg¨ower 2008, Wu & Dong 2006, Dinh et al. 2005). In the next sections an overview of the general LPV synthesis problem is given. Next, the different approaches are detailed: LPV, LFT and ‘mixed’ LPV-LFT approaches. Finally, a con- clusion with guidelines for application of each approach in practice is given. Elaboration of all matrices, algorithms and LMIs is saved for a case study.

5.1 General LPV controller synthesis setup

Consider the LPV model Σ(θ(t)) (3.3) (see Figure 5.1). The parameter values θ(t) are measured in real-time and time variations are not known in advance. The parameter vector θ(t) may comprise exogenous as well as endogenous variables. Conceptually, θ(t) is regarded as an exogenous variable when LPV control synthesis is considered, i.e. θ(t) = θ(w(t)). When θ(t) is an endogenous variable, θ(t) = θ(x(t)), this is called quasi-LPV (see Section 3.3). In a more complete model, θ(t) naturally will always be endogenous as is depicted in Figure 5.1.

θ(t) w(t) z(t) LPV u(t) y(t) model

Figure 5.1: LPV model

A general LPV controller is defined as

 ξ˙(t) = A (θ(t))ξ(t) + B (θ(t))y(t) Λ(θ(t)) : c c (5.1) u = Cc(θ(t))ξ(t) + Dc(θ(t))y(t)

The rate of variation of the parameter vector, θ˙(t), can be regarded as a scheduling variable as well, yielding  ˙ ˙ ˙ ξ(t) = Ac(θ(t), θ(t))ξ(t) + Bc(θ(t), θ(t))y(t) Λ(θ(t), ψ(t)(t)) : ˙ ˙ (5.2) u = Cc(θ(t), θ(t))ξ(t) + Dc(θ(t), θ(t))y(t)

In this way, θ˙(t) can be accounted for explicitly in the controller design. Assume the time-variation T of the parameter vector θ(t) = [θ1(t), . . . , θnθ (t)] is known to be constrained by n θ(t) ∈ Θ ⊂ R θ (5.3) n ψ(t) = θ˙(t) ∈ Ψ ⊂ R θ (5.4) where nθ the length of θ(t). Define correspondingly

2n φ(t) ∈ {θ(t), ψ(t)} ∈ Θ × Ψ ⊂ R θ (5.5) Consider a closed-loop LPV system combining an LPV model Σ(θ(t)) (3.3) and a corresponding LPV controller Λ(θ(t)) (5.1). The corresponding closed-loop system state vector equals ζ = [x, ξ]T and the closed-loop system becomes

 ζ˙(t)   A(θ(t)) B(θ(t))   ζ(t)  = (5.6) z(t) C(θ(t)) D(θ(t)) w(t) 5.1. GENERAL LPV CONTROLLER SYNTHESIS SETUP 29 with  A + B D C BC B + B D D   AB  2 c 2 c 1 2 c 21 = B C A B D (5.7) CD  c 2 c c 21  C1 + D12DcC2 D12Cc D11 + D12DcD21 where the parameter-dependency of the matrices is omitted for clarity. Moreover, it has to be remarked that the closed-loop matrices in (5.6) may be dependent on ψ(t) as well. LPV controller synthesis is based on stability and performance analyses considering the closed-loop system (5.7).

5.1.1 Stability analysis Regarding Linear Time-Invariant (LTI) systems, asymptotic stability is well-known to be charac- terized by the set of LMIs

AT X + X A ≺ 0, X 0 (5.8) where A the (closed-loop) system matrix. Considering the time-varying-parameter-dependent closed-loop system (5.7), exponential stability of the LPV system is defined analogous to (5.8) by

A(θ(t))T X + XA(θ(t)) ≺ 0, X 0, ∀θ(t) ∈ Θ (5.9)

Again, (5.6) may be dependent on ψ(t) as well, changing the stability criterion (5.9) accordingly. The LMI (5.9) guarantees stability for all θ(t) ∈ Θ, not considering the rate of variation ψ(t) of θ(t). Hence, (5.9) guarantees stability for all possible rates of variation ψ(t) ∈ Rnθ , e.g. for infinitely fast variations, which in practice obviously is not possible and thus introduces conservatism. Conservatism of (5.9) may be reduced, using a parameter-dependent Lyapunov function X = X (θ(t)) instead. In this case, additional information on ψ(t) may be taken into account as well. However, in general, constant θ(t) = θ is considered, i.e., ψ(t) ∈ ∅, when a parameter-dependent Lyapunov function X = X (θ(t)) is adopted. Consider a closed-loop LPV system combining the LPV model Σ(θ(t)) (3.3) and a corresponding LPV controller Λ(θ(t), ψ(t)) (5.2). The correspond- ing closed-loop system becomes

 ζ˙(t)   A(θ(t), ψ(t)) B(θ(t), ψ(t))   ζ(t)  = (5.10) z(t) C(θ(t), ψ(t)) D(θ(t), ψ(t)) w(t)

A classical result regarding stability of this time-varying system is that it is stable if i) the cor- responding frozen-time, time-invariant system with system matrix A(θo, 0) is stable and ii) the rates of time variation θ˙(t) are sufficiently slow. This is captured in the following theorem (e.g. (Rugh & Shamma 2000)).

Theorem 5.1 (Lyapunov stability for time-varying-parameter dependent systems) Suppose there exists X (θ(t)) = X T (θ(t)) > 0 such that

X (θ(t))A(θ(t), ψ(t)) + AT (θ(t), ψ(t))X (θ(t)) + ∂X (θ(t), ψ(t)) < 0 (5.11) for all (θ(t), ψ(t)) ∈ Θ × Ψ, where

n Xθ ∂X(θ(t)) ∂X (θ(t), ψ(t)) ≡ ψ (t) (5.12) ∂θ (t) i i=1 i Then the closed-loop system (5.10) is exponentially stable for all parameter trajectories (θ(t), ψ(t)) ∈ Θ × Ψ, where

V (x(t), θ(t)) = x(t)T X (θ(t))x(t) (5.13) is a corresponding parameter-dependent quadratic Lyapunov function. 30 CHAPTER 5. LPV CONTROLLER SYNTHESIS

Equation (5.11) yields a set of LMIs

 ∂X (θ(t), ψ(t)) X (θ(t))   I  I A(θ(t), ψ(t))T  ≺ 0 (5.14) X (θ(t)) 0 A(θ(t), ψ(t))

Lyapunov functions are widespread as a tool for stability analysis of nonlinear systems. Stan- dard Lyapunov theory for exponential stability of nonlinear systems is established in (Khalil 1992). Adopting particular classes of Lyapunov functions, e.g. quadratic Lyapunov functions (5.13), in- troduces conservatism. Subsequent control synthesis might hence be conservative as (5.11) is a sufficient condition for stability only. Although research to non-quadratic Lyapunov functions is present as well, focus is mainly on quadratic Lyapunov functions.

5.1.2 Performance analysis Performance analysis of the closed-loop system combining an LPV model Σ(θ(t)) (3.3) and a corresponding LPV controller Λ(θ(t), θ˙(t)) (5.2), is commonly related to the induced norm of the exogenous signals w(t) to the error signals z(t), the so-called performance channel. The energy gain over the performance channel w(t) 7→ z(t) is defined by the induced L2-gain as

||z(t)|| sup 2 (5.15) w(t)6=0 ||w(t)||2

nx where w(t) ∈ L2, L2 = {x : [0, ∞) → R | kxk2 < ∞}, and

sZ ∞ T kxk2 = x(t) x(t)dt (5.16) 0

Good performance of the corresponding closed-loop system is guaranteed if (5.15) is small. Defining z(t) = Tzww(t) and accounting for (5.3) and (5.4), this yields (e.g., (Dettori 2001))

||z(t)||2 kTzwk∞ = sup sup (5.17) (θ,ψ)∈Θ×Ψ w(t)6=0 ||w(t)||2

Introducing a performance value γ, closed-loop performance is guaranteed by

||Tzw||2 < γ (5.18)

Combination of (5.17), (5.16) and (5.18) yields

Z ∞  T     Z ∞ w(t) −γI 0 w(t) T 1 dt ≤  w(t) w(t)dt (5.19) 0 z(t) 0 γ I z(t) 0

More generally, the quadratic performance specification is given by

Z ∞  T     Z ∞ w(t) Qp Sp w(t) T T dt ≤  w(t) w(t)dt (5.20) 0 z(t) Sp Rp z(t) 0 with Rp ≥ 0. Combining (5.14) and (5.20) guarantees stability as well as quadratic performance of the cor- responding closed-loop system. This leads to the following theorem (Dettori 2001):

Theorem 5.2 (Closed-loop quadratic performance) Quadratic performance and exponential stability is achieved if there exists

X (θ(t)) = X T (θ(t)) > 0 (5.21) 5.1. GENERAL LPV CONTROLLER SYNTHESIS SETUP 31 such that

T  1 0   ∂X (θ(t), ψ(t)) X (θ(t)) 0 0   1 0   A(θ(t)) B(θ(t))   X (θ(t)) 0 0 0   A(θ(t)) B(θ(t))        < 0  0 1   0 0 Qp Sp   0 1  T C(θ(t)) D(θ(t)) 0 0 Sp Rp C(θ(t)) D(θ(t)) (5.22)

m P ∂X (θ(t),ψ(t)) for all θ(t) ∈ Θ and ψ(t) ∈ Ψ, where ∂X (θ(t), ψ(t)) = ∂θ(t) ψ(t) and x˙ = Ax + Bw, i=1 z = Cx + Dw for x(0) = 0.

5.1.3 General synthesis problem formulation The LPV control synthesis is based on (5.21) and (5.22). The goal is, given a parameter-dependent LPV system Σ(θ(t), ψ(t)) (3.3), to design a parameter-dependent controller

K = {Ac(θ(t), ψ(t)) Bc(θ(t), ψ(t)) Cc(θ(t), ψ(t)) Dc(θ(t), ψ(t))} (5.23) using the constraints (5.21) and (5.22). Considering the constraints (5.21) and (5.22), three inherent problems are present. Firstly, (5.22) is nonlinear in the decision variables; for example, A(θ(t)) includes Ac(θ(t)) and hence a product between two decision variables X (θ(t)) and A(θ(t)). However, employing a nonlinear change of variables ((Masubuchi, Ohara & Suda 1995, Scherer, Gahinet & Chilali 1997)), this problem can be solved efficiently: Introduce the new set of variables

V (θ(t)) = {X(θ(t)) Y (θ(t)) K(θ(t), ψ(t)) L(θ(t)) M(θ(t)) N(θ(t))} (5.24) and apply the nonlinear transformation

{X (θ(t)) | Ac(θ(t), ψ(t)) Bc(θ(t)) Cc(θ(t)) Dc(θ(t))} 7→ V (θ(t)) (5.25) to the original search variables X(θ(t)), determining the Lyapunov function, and Ac(θ(t), ψ(t)), Bc(θ(t)), Cc(θ(t)) and Dc(θ(t)), the feedback controller matrices. This yields a convex optimization problem. The original search variables are derived from V (θ(t)). The second problem is the fact that the parameter-dependent decision variables V (θ(t)) yield functional inequalities, i.e., the dependency of the decision variables on the varying parameter θ(t) is unknown a priori. Standard LMI algorithms do not allow solving functional inequalities. To overcome this problem, several approaches have been proposed in literature. Until now, how- ever, no general solution is present. One might adopt a fixed, a priori chosen structure for the decision variables, e.g., affine of polytopic; this is proposed in both LPV and LFT approaches. Furthermore, one might assume an LFT parameter-dependency of the controller in combination with parameter-independent Lyapunov functions, i.e., the LFT approach, or various variations on this, i.e., ‘mixed’ LPV-LFT approaches. In all cases, a feasible solution is achieved at the cost of additional conservatism. Finally, the problem is parameterized by (θ(t), ψ(t)) ∈ Θ × Ψ, which implies that it constitutes infinitely many inequalities. Again, no general solution to this problem is present, although in specific cases nonconservative solutions are presented. The most commonly adopted approaches, in practice, are to use gridding of the parameter range or to use a polytopic approximation. In literature, is often referred to as (genuine) LPV controller synthesis. Exploiting the fact that the system has rational dependence on the varying parameters enables a transformation to a finite set of LMIs. This is referred to as the LFT approach. More recently, ‘mixed’ LPV-LFT approaches have been proposed to transform the infinite set of LMIs to a finite one. All approaches, however, arrive at a set of finite dimensional LMIs at the cost of additional conservatism. 32 CHAPTER 5. LPV CONTROLLER SYNTHESIS

5.2 Lyapunov-based LPV control synthesis

When the LPV paradigm was first introduced, research focused on quadratic Lyapunov functions. (Shahruz & Behtash 1990) present the synthesis of a continuous time state feedback controller using a common quadratic Lyapunov functions V (x) = xT x. (Becker, Packard, Philbrick & Balas 1993) extend this to output feedback controllers, adding specific L2-norm performance bounds as well, i.e. V (x) = xT Xx, X = XT > 0, where x the closed-loop system state. (Apkarian & Gahinet 1995) extend the previous results to discrete time systems. To reduce conservatism, parameter-dependent Lyapunov functions can be adopted.

5.2.1 Gridding

⇒ consider a parameterized subset (p, q) ∈ (Πs × Π˙ s) ⊂ (Π × Π);˙ compute for these fixed points a solution (Apkarian et al. 1995, Becker & Packard 1994, Wu, Yang, Packard & Becker 1996, Zhou et al. 1996)

+ widely used in LPV synthesis; often suitable for practical applications

+ rate of variation can be accounted for

− interpolation required for intermediate points

− dense grid / large m yields large LMI

The number of required gridding points increases very fast for an increasing number of param- eters. The more dense the grid, the larger the optimization problem. However, to assure global properties, a dense grid is required. Gridding is commonly adopted in LPV control design to arrive at a finite dimensional set of LMIs. It enables to account for bounded rates of variation of the parameters. E.g., (Wu, Packard & Balas 1995, Wu et al. 1996) present a LPV control synthesis yielding a LPV controller with a ψ(t)-dependent controller matrix Dc(θ(t), ψ(t)). Considering this specific example, however, measurement of ψ(t) in practice is difficult, i.e. implementation of ψ(t)-dependent controllers is difficult.

5.2.2 Polytopic approximation

⇒ i) assume affine or polytopic parameter dependencies for V (p), i.e., plant and controller, ii) con-

sider a convex polytope (Πo × Π˙ o) ⊃ (Π × Π);˙ compute a solution for extreme points of the polytope (Gahinet, Apkarian & Becker 1996, Apkarian & Gahinet 1995, Yu & Sideris 1995)

+ no need for gridding

+ solution for functional inequalities

− nonlinear parameter dependencies are not considered / all parameters vary independently (conservative)

− polytope is (generally) a (conservative) overestimation

example: in (5.22), q enters affinely via ∂X (p, q), hence q ∈ Π˙ 0, where Π˙ 0 the extreme points of Π˙ 5.3. SCALED SMALL-GAIN OR LFT SYNTHESIS 33

An affine LPV model is defined as an affine dependency of e.g. the system matrix A(θ(t)) on θ(t), i.e. A(θ(t)) = Ao + θ(t)A1. Considering affine LPV models only, implies that nonlinear parameter dependencies are not considered. Hence, all parameters are assumed to vary inde- pendently, which introduces conservatism. Furthermore, if Φ is not a convex polytope, a larger polytope Φ ⊂ Φ ⊂ Rnθ , which is convex may be considered. However, this introduces parameter combinations, which may not be possible in the original model and hence introduces conservatism.

5.3 Scaled small-gain or LFT synthesis

Exploiting LFT parameter-dependency of LPV systems enables application of a generalized H∞ control synthesis using the optimally scaled small-gain theorem. The parameter variations are temporarily regarded as unknown perturbations (Packard 1994, Apkarian & Gahinet 1995). Mul- tipliers or scalings describing the nature of the unknown, time-varying parameter θ(t) are intro- duced to decrease conservatism with respect to LPV methods using a constant, common quadratic Lyapunov function only. In the LFT approach, a constant, common quadratic Lyapunov function is adopted as well, hence, infinitely fast parameter variations are accounted for.

5.3.1 LFT problem formulation An LFT model (see Section 3.5) is used as a basis. An LFT model in fact is a special case of a LPV model, which is transformed using an upper LFT, yielding a constant, known part M and a corresponding time-varying, unknown part ∆(θ(t)). To enable an LFT representation, the system should have rational dependence on the parameter θ and no poles in zero. Analogously, transformation of the LPV controller Λ(θ(t)) (5.1) using a lower LFT, yields a constant, known part K and a corresponding unknown part ∆K (θ(t)), where ∆K (θ(t)) represents a possibly nonlinear, controller scheduling function, see Figure 5.2. In general, ∆K (θ) 6= ∆(θ(t)). For the controller to be represented in LFT form, the same restrictions hold as for the system. The closed-loop interconnection of the resulting LFTs is transformed, again using a lower LFT, yielding the closed- loop interconnection T over the channel w 7→ z

T = FL ( FU (M, ∆(θ(t))),FL(K, ∆K (θ(t))) ) (5.26)

Rewriting this yields

T = FU ( FL(M,K), ∆ ) (5.27) where ∆ = diag(∆(θ(t)), ∆K (θ(t))). The closed-loop system matrix is defined as follows:       ζ˙(t) ABu Bp ζ(t)  zu(t)  =  Cu Du Dup   wu(t)  (5.28) z(t) Cp Dpu Dp w(t) where w(t) 7→ z(t) the performance channel and zu(t) = ∆wu(t). In Figure 5.2, the correspond- ing interconnection structures are shown. The closed-loop system has repeated linear fractional parameter dependence. The use of a constant Lyapunov function and the required rational dependence of the controller on the scheduling parameters introduces conservatism. However, the resulting problem is a gener- alized robust performance problem for the augmented system M with respect to the uncertainty ∆ = ∆(θ(t)), and the theoretical background covering analysis and controller synthesis for these problems is extensive. Assume that ∆(θ(t)) is block-diagonal with diagonal terms |∆i(θ(t))| ≤ 1, which is not restrictive. An LTI controller K has to be found that renders the augmented system M stable such that robust quadratic performance is achieved for the performance channel w 7→ z. After the design, the corresponding parameter variations are incorporated in the controller design again, yielding an LPV controller, also with LFT parameter-dependence. For θ(t) = θ, a genuine robust control problem results. 34 CHAPTER 5. LPV CONTROLLER SYNTHESIS

∆ r ∆ v r v z M w z M w y u y u K K rK vK vK rK

∆K

(a) (b)

Figure 5.2: LPV LFT interconnections, (a) represents FL ( FU (M, ∆(θ(t))),FL(K, ∆K (θ(t))) ), (b) represents FU ( FL(M,K), ∆ ).

5.3.2 Controller synthesis Analogous to the previously presented LPV controller synthesis, the scaled small-gain control synthesis is based on the robust performance analysis of the closed-loop system. Consider the closed-loop LFT interconnection system which is represented by T (5.27), (5.28). The corre- sponding analysis involves three main conditions regarding the closed-loop model T , comprising (Aangenent 2008) • well-posedness, i.e. for all initial conditions corresponding to the augmented system M and all inputs w(t) to the system, the system should have a unique solution. • exponential stability of the closed-loop system, which is analyzed via the small-gain theory, which states that a closed-loop system is stable provided that the loop-gain is less than unity • performance; for example quadratic performance, i.e. asymptotic convergence of the per- formance channel w(t) 7→ z(t), or l2-performance, i.e., the loop-gain over the performance channel w(t) 7→ z(t) is smaller than γ Combining all three conditions yields a set of LMIs, which is directly derived from the general LPV synthesis constraints (5.21) and (5.22), e.g., (Dettori 2001).

Theorem 5.3 (LFT system analysis) If there exists a symmetric matrix X and scalings Q = QT , S and R = RT such that

X > 0, (5.29) and T  I 0 0   0 X 0 0 0 0   I 0 0   ABp Bu   X 0 0 0 0 0   ABp Bu         0 I 0   0 0 Qp Sp 0 0   0 I 0     T    < 0, (5.30)  Cp Dp Dpu   0 0 S Rp 0 0   Cp Dp Dpu     p     0 0 I   0 0 0 0 QS   0 0 I  T Cu Dup Du 0 0 0 0 S R Cu Dup Du and T  ∆(θ(t))   QS   ∆(θ(t))  > 0, ∀θ(t) ∈ Θ, (5.31) I ST R I 5.3. SCALED SMALL-GAIN OR LFT SYNTHESIS 35 then the system

 x˙   A(θ(t)) B(θ(t))   x  = (5.32) zp C(θ(t)) D(θ(t)) wp is uniformly exponentially stable and achieves quadratic performance over the channel w(t) 7→ z(t).

Analogously, for     Qp Sp −γI 0 T = 1 (5.33) Sp Rp 0 γ I the L2-performance criterion results, achieving an L2-gain over the performance channel w(t) 7→ z(t), smaller than γ. In order to arrive at this result, specific scalings or so-called multipliers   Qp Sp P = T (5.34) Sp Rp are employed. This explains the naming of scaled small-gain control synthesis. Solving the LMIs yield the corresponding Lyapunov functions and the multipliers from which the controller can be reconstructed. Again, a nonlinear transformation is required to transform (5.30) into a convex LMI, see Section 5.1.3. However, (5.31) is nonconvex as well, which is a result of the use of multipliers. Additional conditions on the multipliers, relaxations, are required to arrive at a standard LMI problem, which can be solved using existing algorithms (Scherer 1999, Scherer 2001). Several approaches have been proposed in literature:

• Assuming the matrix ∆(θ(t)) to be diagonal, a corresponding diagonal structure of the multiplier P yields (5.31) automatically true, see, e.g., (Apkarian et al. 1995):

Qp = diag(Q1,...,Qm),Sp = 0,Pp = Q (5.35)

However, conservatism is introduced as less freedom is left to solve (5.30).

• Following the previous item, however, taking S 6= 0 but block diagonal and skew-symmetric yields the same results; (5.31) is automatically true at the cost of introducing conservatism in solving (5.30), see, e.g., (Helmersson 1995, Scorletti & Ghaoui 1995).

• Finally, full-block scalings may be applied; either with additional inertia or (partial) concav- ity constraints on the multipliers, e.g., Q < 0, R > 0 (Dettori 2001, Scherer et al. 1997), or with general full-block multipliers, also known as the full-block S-procedure (Scherer 2001). In the latter case, the final synthesis problem comprises many parameters, which makes it a (very) complex problem. The former approach is complex as well, and, besides, still imposes structural requirements on P , introducing conservatism.

5.3.3 Pros and cons The main advantages of the LFT approach are the possibility to use the optimally scaled small-gain theorem, which is well-established in literature, and the possibility to arrive at a finite dimensional set of LMIs using multipliers. However, the general full-block multiplier approach is complex, whereas relaxations on the multipliers introduce (much) conservatism. Furthermore, the use of a constant, common quadratic Lyapunov function, implying that infinitely fast parameter variations are accounted for, introduces conservatism, especially in the case of large parameter variations. Finally, the controller inherits the LFT parameter dependency, which might be conservative, and possible realness of the parameter θ(t) is not exploited, introducing conservatism as well. 36 CHAPTER 5. LPV CONTROLLER SYNTHESIS

5.4 Mixed LPV-LFT approaches

5.4.1 Extended Kalman-Yakubovich-Popov (KYP) lemma

• KYP lemma: equivalence of infinite dimensional FDIs and finite dimensional LMIs (Rantzer 1996)

⇒ replace infinite dimensional (LFT) LMI by single LMI via introduction of additional multiplier (comparable to full-block S-procedure (Scherer 2005))

+ parameter-dependent Lyapunov function X (p)(q ∈ ∅)

− no general synthesis method (Iwasaki & Hara 2005, Iwasaki, Meinsma & Fu 2000)

Specific extension

• one parameter, rational decision variables (Dinh et al. 2005)

5.4.2 LFT Lyapunov functions

⇒ replace infinite dimensional (LPV) LMI by single LMI via quadratic LFT Lyapunov functions and full-block multipliers

+ implicit model to account for q bounds (analysis (Helmersson 1996, Iwasaki & Shibata 2001))

+ general parameter-dependent controller (Wu & Dong 2006)

− quadratic LFT Lyapunov functions: difficult to choose (no general procedure)

5.5 Conclusions

As already mentioned in the introduction, current LPV control syntheses comprise solutions to special extensions rather than genuine generalizations of linear H∞-designs. In all approaches, conservativeness has to be introduced to arrive at a standard LMI problem. Hence, current research focuses on the definition of an appropriate set of LMIs, which can be solved by standard LMI algorithms, introducing the least possible conservatism. The main advantages and disadvantages of LPV control synthesis with respect to classical gain scheduling are discussed in the introduction of this chapter. Focusing on the class of systems the LPV control synthesis may be applied to, a true LPV model rather than, for example, a family of linearizations is used as a basis. Furthermore, the open control framework enabled by classical gain scheduling, allows accounting for possible modeling errors intuitively. LPV control synthesis, on the other hand, constitutes a predefined control synthesis relying on the LPV model extensively. The preceding LPV modeling hence has to be very adequate. Whereas classical gain scheduling requires extensive simulation-based testing as in general, stability and performance can only be guaranteed locally, this is not required for LPV control syntheses. However, in practice, extensive testing is still critical as LPV control syntheses relies extensively on the preceding LPV modeling. 5.5. CONCLUSIONS 37

LPV control synthesis Focusing on gridding, which is a commonly employed technique when solving an LPV control problem, a parameterized family of controllers is generated rather than a true LPV controller. The family of controllers has to be interpolated to arrive at an LPV controller, thus neglecting transient behavior. In this case, classical gain scheduling techniques suffer the same disadvantage, however exhibit the advantage of employing an open control design framework, which may yield less stringent modeling demands. Especially in the case of many parameters, the problem size may explode when gridding is adopted. Nevertheless, LPV control synthesis enables to incorporate parameter trajectories θ(t), as well as ψ(t), and design the controllers in one shot. Considering affine or polytopic parameter-dependent systems, gridding is not required. The corresponding LPV controller synthesis problem becomes relatively easy. For truely affine or polytopic systems, the only conservatism lies in the corresponding structural restrictions on the resulting controller.

LFT control synthesis In standard small-gain control synthesis, robustness to parameter variations is ensured. How- ever, at the cost of overestimating admissible parameter trajectories due to the use of a constant Lyapunov function. This yields conservative results. The use of multipliers, i.e., scaled small- gain control synthesis, decreases conservativeness, although at the cost of numerical difficulties. Multiplier relaxations are introduced to resolve these difficulties, however, again at the cost of conservatism.

Mixed LPV-LFT syntheses More recent approaches target genuine generalizations of the LPV and / or LFT approaches. Two examples are the use of an extended KYP-lemma and the use of Lyapunov functions with LFT parameter dependency. Until now, all results still correspond to special extensions rather than genuine generalizations (Scherer 2005). Nevertheless, for a rational dependence of the sys- tem matrices on θ(t), (Wu & Dong 2006) propose a quite general solution using LFT Lyapunov functions. Furthermore, for one varying parameter, i.e., θ(t) ∈ R1, and rational decision vari- ables, (Dinh et al. 2005) present a solution with the leas possible conservatism using the extended KYP-lemma.

LPV vs LFT Focusing on application of LPV control synthesis in practice, some guidelines on when to use either LPV or LFT techniques are, see Figure 5.5. • In case the parameter vector θ(t) constitutes only one parameter and relevant rate bounds for ψ(t) are known, LPV techniques seem most suitable. Parameter-dependent Lyapunov functions are a suitable way of taking into account ψ(t). • In case θ(t) constitutes many quickly varying parameters, or if the system exhibits a nice dependence on θ(t) allowing for a LFT representation, LFT techniques seem most suitable. The LFT representation may be restrictive as parameter-rate-of-bounds are not explored. • More recent ‘mixed’ LPV-LFT syntheses are still evolving. No general control synthesis is present yet, although some solutions to specific extensions are present. In general, the LMI problems resulting from LFT control synthesis are less computational de- manding than the corresponding LMI problems resulting from LPV control synthesis. Including robustness for unmeasured uncertainties is the easiest using an LFT approach. The LPV approach is most suitable for introducing practical controller modifications, e.g., adding an anti-windup. 38 CHAPTER 5. LPV CONTROLLER SYNTHESIS dependence # parameters affine rational non-rational q ∈ Rm: LFT, many X else X∗ ∗else few LPV LPV q ∈ Rm: LFT, one else LPV∗∗ rate of variation q ∗∗p ∈ R1: KYP, LPV zero (constant p) else LPV bounded LPV LFT / infinitely fast LFT LPV Chapter 6

Fuzzy gain-scheduling

The restriction of classical gain scheduling techniques to near-operating point system behavior, not taking into account transient system dynamics, and additionally imposing a slow variation re- quirement, has lead to the development of fuzzy gain scheduling techniques. Fuzzy gain scheduling originates from the idea to relax these restrictions, while remaining closely related to the classical gain scheduling approach (Leith & Leithead 2000). Literature and research to fuzzy modeling and controller design methods is abundant and extensive, see e.g. (Guerra & Vermeiren 2001). It is not the intention to give an overview of fuzzy modeling at this point. In this chapter, a brief overview of fuzzy methods leading to gain scheduling alike techniques is given. The basis of fuzzy gain scheduling is a fuzzy model or so- called blended multiple model representation, which is discussed in Section 6.1. The corresponding controller design is discussed in Section 6.2.

6.1 Fuzzy modeling

A blended multiple model representation or fuzzy modeling involves the blending of a set of models describing some nonlinear dynamics via point-designs. Examples of multiple model representations are Takagi-Sugeno (fuzzy) models (e.g. (Sugeno & Kang 1988)), local model networks (Shorten, Murray-Smith, Bjorgan & Gollee 1988) or Polytopic Linear Models (PLMs) (Schulte 2005, Angelis 2000). Targeting at subsequent application of gain scheduling alike techniques, i.e. to retain the possibility of applying linear controller design techniques, commonly linear local models are used, e.g. (Gawthrop 1995, Tanaka & Sano 1994). i i i Consider a set of Nm local, indexed models Σ(θ ), where i = 1,...,Nm, and θ ∈ Θ ⊂ Θ parameterizing e.g. the system’s equilibrium operating points or a set of constant operating points covering the nonlinear, changing system dynamics. The local models are thus parameterized by θ = θi (compare to (3.11)). The equilibrium or constant operating points corresponding to the o i i i i i i indexed models are defined by R (θ ) = (xo(θ ), wo(θ ), uo(θ ), zo(θ ), yo(θ )). i Scalar weighting or validity functions α (θα) are defined, where θα represents the dependence of the weighting functions on the working conditions, i.e. θα = θα(x, w, u). Hence, by definition θ differs from θα. Assuming all models have the same states, inputs and outputs, the nonlinear dynamics (3.1) are approximated by a weighted combination of the models Σ(θi), θi ∈ Θi, yielding a nonlinear fuzzy model

 Nm  x˙(t) = P αi(θ ) A(θi)˜xi + B (θi)w ˜i + B (θi)˜ui  α 1 2  i=1  Nm P i  i i i i i i Σnl(θα): z(t) = α (θα) C1(θ )˜x + D11(θ )w ˜ + D12(θ )˜u (6.1)  i=1  Nm  P i  i i i i  y(t) = α (θα) C2(θ )˜x + D21(θ )w ˜ i=1

39 40 CHAPTER 6. FUZZY GAIN-SCHEDULING

The indexing of the states x˜˙ i(t), inputsw ˜i(t),u ˜i(t), and outputsz ˜i(t),y ˜i(t) of the local mod- els results from the model specific equilibrium values Ro(θi). Rewriting (6.1) according to the coordinate transformation (3.7) yields

 Nm  x˙(t) = P αi(θ ) A(θi)x + B (θi)w + B (θi)u + f (θi)  α 1 2 o  i=1  Nm P i  i i i i Σnl(θα): z(t) = α (θα) C1(θ )x + D11(θ )w + D12(θ )u + go(θ ) (6.2)  i=1  Nm  P i  i i i  y(t) = α (θα) C2(θ )x + D21(θ )w + ho(θ ) i=1 where

i i i i i i i fo(θ ) = −A(θ )xo(θ ) − B1(θ )wo(θ ) − B2(θ )uo(θ ) i i i i i i i i go(θ ) = zo(θ ) − C1(θ )xo(θ ) − D11(θ )wo(θ ) − D12(θ )uo(θ ) (6.3) i i i i i i ho(θ ) = yo(θ ) − C2(θ )xo(θ ) − D21(θ )wo(θ ) which is a time-variable, weighted combination of linear and / or affine state-space systems. Gen- i i erally, appropriate modeling assures the latter two terms go(θ ) and ho(θ ) to equal zero. Besides linear modeling, which is considered throughout this report, literature on fuzzy modeling in gen- eral considers affine modeling as well. In case affine instead of linear local models are used, an i i i i additional term appears in fo(θ ) due to f(xo(θ ), wo(θ ), uo(θ )) 6= 0, see e.g. (Schulte 2005). The main disadvantage of fuzzy modeling comprises the fact that the nonlinear, blended model dynamics (6.2) can not be related to the original set of local point-design models, i.e. to the original nonlinear plant dynamics. This is shown when considering the corresponding indexed linearizations. Assuming the set of local models contains models of the form (3.11), the indexed linearizations of (6.2) are given by

 ˙ i i i i i i x˜ = (A + ∇xα (θα))˜x + (B1 + ∇wα (θα))w ˜ + (B2 + ∇uα (θα))˜u i  i i i i i i Σ (θα): z˜ = (C1 + ∇xα (θα))˜x + (D11 + ∇wα (θα))w ˜ + (D12 + ∇uα (θα))˜u (6.4)  i i i i y˜ = (C2 + ∇xα (θα))˜x + (D21 + ∇wα (θα))w ˜

i i i where Ai = A(θ ), B1 = B1(θ ), etc. With respect to the original set of local models (3.11), the indexed linearizations (6.4) contain cross-terms involving derivatives of the weighting func- i tion α (θα). Regarding subsequent controller design, the time-varying, operating-point-dependent i nature of α (θα) thus has to be taken into account or blending should be minimal. Moreover, anal- ogous to classical gain scheduling, a slow-varying condition has to be fulfilled, however now with i respect to α (θα) Furthermore, off-equilibrium points need not be stable, nor can performance be guaranteed (Shorten et al. 1988).

6.1.1 Weighting or scaling functions

i The blending, utilizing the weighting or scaling functions α (θα) is essential to the fuzzy gain PNm i scheduling approach. Typically, the weighting functions are normalized such that i=1 α (θα) = 1. Furthermore, the functions are parameterized such that the local models switch over into each other i smoothly. Two example sets of scaling functions α (θα) are shown in Figure 6.1. The design of these functions will not be discussed further to this point. A distinction can be made between the case of local models describing the original, nonlinear plant dynamics i) at distinct operating points and ii) at operating regions of the operating envelope of the nonlinear plant dynamics rather than distinct points. In the latter case, each local model is valid in an extended operating region and blending correspondingly is confined to small transitions at the boundaries between these operating regions. Blending thus plays a minor role with respect to the former case in which blending should provide smooth interpolation between the local models. 6.1. FUZZY MODELING 41

α1 α2 α3 α4 α5 1

0 θα(x, w, u)

α1 α2 α3 α4 α5 1

0 θα(x, w, u)

i Figure 6.1: Two example distributions of the scaling functions α (θα), i = 1,..., 5.

6.1.2 Approximation accuracy One of the main advantages of fuzzy modeling as a basis for fuzzy gain scheduling involves the fact that the original nonlinear model may be approximated arbitrarily close by a fuzzy model. Moreover, the fuzzy model is a global approximation of the nonlinear model, thus taking into account transient dynamics as well, contrary to a linearisation-based parameter-dependent set of local models as adopted in classical gain scheduling. In (Schulte 2005) an upper bound on the number of models Nm that is sufficient to construct a fuzzy model with predefined accuracy with respect to the original nonlinear plant model is derived. These results are based on (Angelis 2000). The upper bound depends on an upper bound of the error, the size of the operating space and the estimation of the largest nonlinearity of the plant model.

Theorem (approximation accuracy) (Schulte 2005) Consider the dynamicsx ˙ = f(x, w, u) of the nonlinear model (3.1). Let f(x, w, u) = T F [x, w, u] + fnl(x, w, u) be the corresponding partitioning of the linear and nonlinear dynamics and let ε ∈ R+ be an error bound. Determine the distance between the original nonlinear dynamics f(x, w, u) and a corresponding nonlinear fuzzy model approximation fˆ(x, w, u) as the supremum of the Euclidian error norm ˆ ˆ d(f, f) := sup ||f(x, w, u) − f(x, w, u)||2 (6.5) (x,w,u)

Then the equation d(f, fˆ) ≤ ε is satisfied, i.e. the original nonlinear model is approximated with ε-accuracy by the corresponding fuzzy approximation, if Nm local models are employed, with

nnl   Y βi q √ Nm = ceil √ λH nnl n (6.6) i=1 2 2ε where nnl = dim(fnl(x, w, u)) the number of nonlinear terms, βi the range of the corresponding nonlinear term i = 1, . . . , nN , n = dim(F ) the order of the model and λH = max(λHj ) the Gb,j maximum eigenvalue of all eigenvalues corresponding to the Hessian matrices that are derived from the nonlinear terms.

Example 6.1 (Longitudinal vehicle control) Consider the nonlinear model (A.5), focusing on the dynamicsx ˙ = f(x, w, u), where 42 CHAPTER 6. FUZZY GAIN-SCHEDULING

x ∈ Rnx the state, u ∈ Rnu the input and w ∈ Rnw as defined in (A.4). Rewriting this, explicitly discriminating between the linear and nonlinear parts, yields

 x(t)    (1) 1  w (t)  1 0 2 (1) (1) x˙(t) = 0, 0, 0,  +− C x(t) − g(sin w (t) + Cr cos w (t) (6.7) M  w(2)(t)  M d | {z } | {z } u(t) T F fnl=[fnl,1, fnl,2]

where the order of the system dynamics nx = 1 and the number of nonlinear terms nx nw nu nnl = 2. The bounds for the working space G = R × R × R are determined via

(1) π 0 < x(t) ≤ xmax, 0 ≤ w (t) ≤ 4 , (2) (6.8) 0 < w (t) ≤ xmax, 0 < u(t) ≤ umax

where xmax the maximum allowable velocity of the vehicle and umax the maximum drive force. This yields a working envelope Gb. Using (6.8), the working ranges of the parameters, βi = Gb,i ⊂ Rnl,i, i = 1, 2 corresponding to the nonlinear terms fnl,i, are given by β1 = xmax, β2 = π/4. The Hessian matrices from the terms fnl,i, i = 1, 2 are calculated using

Hi = 2.dfnl,ix, i = 1, 2 (6.9)

which yields the corresponding eigenvalues λHi

2 0 λH1 = − M Cd (1) (1) (6.10) λH2 = g(sin w + Cr cos w )

Using the parameter values for a general passenger car, the maximal eigenvalue then is

λH = max {λHi } , i = 1, 2 (6.11) (x,w,u)∈Gb   2 0 (1) (1) = max − Cd, g(sin w + Cr cos w ) (6.12) (x,w,u)∈Gb M = max −5.1 · 10−4, 7.0 (6.13)

Demanding an ε-accuracy of ε = 10 and using a maximal vehicle velocity xmax = −1 150/3.6 m s , requires the use of Nm local models as a basis for the nonlinear fuzzy model, with

nnl   Y βi q √ Nm = ceil √ λH nnl nx (6.14) i=1 2 2ε  x √   π/4 √  = ceil √max 7.0 · 2 · ceil √ 7.0 · 2 (6.15) 2 20 2 20 = 18 · 1 = 18 (6.16)

Assuming that the models with system matrices as defined in (3.18) are locally valid, (1) equally spaced triangular functions are defined in the domain (x, w ) ∈ Gb,1 × Gb,2. The fuzzy model approximating the original nonlinear model (A.5) then is defined by (6.2) and (eq:TSmodelRes). Implementation of the nonlinear fuzzy model using Nm = 16 linear models, yields the results shown in Figure 6.2. 6.2. FUZZY GAIN SCHEDULING CONTROL 43

0.4 40

0.2 30 fuzzy fuzzy 0 20 and x and f nl nl f −0.2 x 10

−0.4 0 0 200 400 600 0 200 400 600

−3 x 10 10 0.4

0.2 fuzzy

fuzzy 5 −f −x

nl nl 0 0 error f error x −0.2

−5 −0.4 0 200 400 600 0 200 400 600 time [s] time [s]

Figure 6.2: Comparison of simulation results corresponding to the original nonlinear model (A.5) (solid black) and the approximated model using Nm = 16 equally distributed LTI models (dotted grey).

6.2 Fuzzy gain scheduling control

i i Consider the fuzzy model Σnl(θα) (6.2), based on a set of local, indexed models Σ(θ ), where θ ∈ i i Θ ⊂ Θ parameterizing e.g. the equilibrium operating points Θ = Θe or a set of constant operating points Θi = Θo covering the nonlinear, changing system dynamics appropriately. Utilizing Σ(θi) as a basis, either blended, local controller design techniques, or LMI-based LPV methods may be applied to arrive at a parameter-dependent controller. As no direct relationship between the dynamic characteristics of the fuzzy model and the characteristics of the original set of local models is present, the analysis and design of corresponding controllers with guaranteed stability and performance specifications is difficult. In the remainder of this section, the Parallel Distributed Control (PDC) synthesis (Guerra & Vermeiren 2001), which is an often applied specific form of blended, local controller designs, is briefly explained. LMI-based LPV methods, such as described in e.g. (Tanaka & Sano 1994, Guerra & Vermeiren 2001, Angelis 2000), will not be discussed further.

6.2.1 Parallel Distributed Control Parallel Distributed Control (PDC) synthesis involves the blending of a set of LTI controllers Λ(θi), i i corresponding to Σ(θ ). The same weighting or scaling functions α (θα) as for the modeling are utilized, yielding a nonlinear gain-scheduled controller immediately. The number of local controller designs equals the number of models, which is determined on the basis of the desired approximation accuracy (see Section 6.1.2). PDC is strongly related to classical gain scheduling controller design techniques. However, the fixed-structure LTI controller design requirement to enable appropriate interpolation or scheduling when employing classical gain scheduling techniques, may be relaxed significantly. The controller outputs are blended (scheduled) rather than scheduling e.g. the controller coefficients. 44 CHAPTER 6. FUZZY GAIN-SCHEDULING

The PDC control law comprises a feedback and a feedforward alike part. The feedback part i Λfb(θα) directly follows from the local controller designs Λ(θ ).

Nm X i i Λfb(θα) := α (θα)Λ(θ ) (6.17) i=1

Focusing on the nonlinear fuzzy model Σnl(θα) (6.2), the feedback part Λfb(θα) only holds for the parameterized, linear part. I.e. the corresponding offsets (6.3) are not compensated. Accordingly, a feedforward alike part, Λff (θα) is determined. As this compensator will depend on θα(x, u, w), which may consist of measured states, it is no feedforward in its strict sense. Consider the case i i when go(theta ) = ho(theta ) = 0. The corresponding feedforward compensator then becomes

Nm  T −1 T X i i Λff (θα) := − B2(θα) B2(θα) B2(θα) α (θα)fo(θ ) (6.18) i=1 where

Nm X i i B(θα) := α (θα)B2(θ ) (6.19) i=1

The resulting gain-scheduled controller then equals Λ(θα) = Λfb(θα) + Λff (θα), where θα = θα(x, u, w) is the scheduling vector. Chapter 7

Conclusions and future work

7.1 Conclusions

Modeling and scheduling variable Most literature on gain scheduling focuses on the design of a scheduled controller rather than the corresponding modeling. Commonly, it is assumed that a parameterized set of local models or even a LPV model is present. In that case, this report discusses various techniques and guidelines how to design a corresponding scheduled controller. However, focusing on continuous gain scheduling of a parameterized set of local models, several issues involving the design of this set have not explicitly been considered; How to select the number of models to design, how to select the corresponding operating points and how to select the scheduling variables θ? As said, in general these issues are not considered in gain scheduling. In this sense, fuzzy gain scheduling, more specifically the theorem considering the approxima- tion accuracy of a parameterized set of local models with respect to the original nonlinear model, might provide a good starting point. In general, research on fuzzy modeling is abundant, contrary to research to the design of a parameterized set of LTI models. A required number of LTI models is determined. The corresponding operating points are assumed to be equally distributed over the envelope of operating conditions. If these points do not represent constant operating points, equilibrium operating conditions have to be considered instead. Otherwise, velocity-based gain scheduling might resolve the problem. Typically, determination of the scheduling variable θ is based on the parameterized set of LTI models. This requires insight in the system as well as experience. An often used rule- of-thumb states that the scheduling variable should incorporate the main nonlinearities of the system. Changing operating conditions typically are a result of nonlinearities. Furthermore, the parameterized set of LTI models or the LPV model should reflect the nonlinear system behavior for the complete envelope of operating conditions. Hence, the scheduling variable should capture the system’s nonlinearities. Quasi-LPV techniques are based on this idea an may thus provide a suitable solution. Besides that, the scheduling variable θ obviously should be measurable or at least be observable to enable actual implementation of the scheduled controller.

Application of gain scheduling All gain scheduling techniques involve firstly parameter-dependent modeling and subsequently parameter-dependent, i.e. scheduled controller design. Analysis of the modeling involves stability analysis as well as analysis of the approximation accuracy. The controller design requires stability and performance analysis. Taking in mind that gain scheduling is employed in an ad hoc manner at DAF as well as at TNO (see 1), focus lies on the stability and performance analysis corresponding to the controller design. I.e. the main question is not whether a system can be approximated sufficiently close with a scheduling model based on a set of linear models, but whether gain

45 46 CHAPTER 7. CONCLUSIONS AND FUTURE WORK scheduling of the controller gains is required to assure stability and performance specifications or even enable tightening of these specifications over the entire operating envelope. The before mentioned fuzzy theorem regarding approximation accuracy of the parameterized set of LTI models, can be interpreted as a measure of the nonlinearity of the system. Hence, is might be used as a measure of the need to apply a gain scheduling controller design. Obviously, this does not hold in case gain scheduling is adopted to improve closed-loop performance by employing a nonlinear controller to a linear system, e.g. by scheduling the controller parameters according to the error rather than a system nonlinearity. In general, this is an open issue in control theory (Aangenent 2008). In general, application of gain scheduling in case of a system with slowly varying dynamics is assumed to be beneficial regarding the resulting closed-loop performance. When a measurable scheduling variable, reflecting the changing dynamics appropriately, can be determined, application of gain scheduling instead of robust control should be considered. Typically, nonlinear systems are considered, whereas robust control focuses on linear system representations with immeasurable parameter variations or disturbances.

Gain scheduling methods Classical gain scheduling and various related gain scheduling techniques, LPV and LFT synthesis and fuzzy gain scheduling are discussed in this report. The main advantage of classical gain scheduling is that it inherits the benefits of linear controller design methods, including intuitive classical design tools and time as well as frequency domain performance specifications. PID control is the most used control strategy in industrial applications due to its relatively simple and intuitive design, hence, this is a major advantage with respect to other nonlinear controller design syntheses. The approach thus enables the design of low computational effort controllers. Conceptually, gain scheduling involves an intuitive ’simplification’ of the problem into parallel decompositions of the total system. The major drawback of classical gain scheduling involves the lack of guaranteed global ro- bustness, performance and especially stability. The robustness, performance and even nominal stability properties of the global gain scheduled controller are not addressed explicitly in the de- sign process. Such properties are rather inferred from extensive simulations. Only in the case of slowly varying parameters, these designs can guarantee stability. The key issues are i) the fact that transitions among operating conditions are not addressed in the design and ii) shortcomings of the linearizations. Hence, extensive offline testing is required to establish global stability and performance guarantees. An often used rule-of-thumb states that the scheduling variable should only vary slowly to prohibit the introduction of additional dynamics. Generally, the scheduling is based intuitively on physical variables in the plant. In case of the quasi-LPV however, a coordinate transformation has to be applied. As this approach is conservative, this might lead to infeasible controller design problem, i.e. no feasible controller may be available. Quasi-LPV techniques on the other hand, enable global stability as well as performance and robustness analysis. Furthermore, in practice, classical gain scheduling often is applied in an ad hoc manner, which is not suited for more difficult problems. LPV and LFT synthesis require a true LPV model as a basis. In general however, gain scheduling may be employed in the absence of an analytical model, e.g. on the basis of a collection of plant linearizations. Consequently, controller design based on whitebox as well as blackbox and even data-based ’modeling’ is possible. If the possibility of fast parameter variations is not addressed in the design process, guaranteed properties of the overall gain-scheduled design cannot be established. The main advantage of LPV and LFT control synthesis is that they do account for parameter variations in the controller design, which results in a priori guarantees regarding stability and performance specifications. The main drawback of LPV and LFT control synthesis involves conservativeness, which has to be introduced to enable solving the resulting LMIs. As a result of that, current LPV and LFT syntheses comprise specific extensions of robust control techniques rather than true generalizations. However, current and future research still provides and will provide less conservative solutions. 7.2. FUTURE WORK (TBD) 47

Furthermore, LPV and LFT control syntheses are relatively involved, i.e. designing a SISO LPV controller for a LPV model of order 4 has to be regarded as computationally complex already. Fuzzy gain scheduling is only considered briefly in this report. Focus lies on fuzzy modeling and controller design following classical gain scheduling methods. The main drawback of fuzzy gain scheduling involves the lack of a relation between the dynamic characteristics of the original nonlinear model and the fuzzy model. Even locally, the dynamics of the fuzzy model can not be related to the original nonlinear model. Whereas some stability and performance specifications can be derived in special cases for classical gain scheduling and even a priori stability and performance can be guaranteed when LPV or LFT control synthesis is adopted, this is thus not the case for fuzzy gain scheduling.

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Longitudinal vehicle control

Consider a vehicle with mass M, which is subject to the following forces (see Figure A.1) 1 F (t) = − ρC A q˙(t)2 air 2 d d 0 2 = −Cdq˙(t) (A.1)

Fslope(t) = −Mg sin α(t) (A.2)

Froll(t) = −CrMg cos α(t) (A.3) where Fair(t), Froll(t) and Fslope(t) are the air drag, the rolling resistance and road slope resultant force respectively, and ρ, Cd, Ad, g and Cr appropriate vehicle and environmental characteristics. The slope α(t) is regarded as an external disturbance and the vehicle speedq ˙(t) is regarded as the system output. Corresponding parameter values for a passenger car are given in Table A.1.

Fslope q Fair

F α Froll roll

Figure A.1: A vehicle and its corresponding environmental disturbances.

The control goal is to achieve a desired vehicle speedq ˙d(t), which is established by control of the throttle and brakes, yielding a resulting control force Ft/b(t). Define the external input vector w(t)

T h (1) (2) i T w(t) = w (t), w (t) = [α(t), xd(t)] (A.4)

The nonlinear model for longitudinal vehicle control is hence given by

 x˙(t) = − 1 C0 x(t)2 − g(sin w(1)(t) + C cos w(1)(t)) + 1 u(t)  M d r M (2) Σnl : z(t) = w (t) − x(t) (A.5)  y(t) = x(t)

53 54 APPENDIX A. LONGITUDINAL VEHICLE CONTROL

Table A.1: General passenger car characteristic parameter values Quantity Symbol Value Unit mass M 1400 kg 2 frontal area Ad 2 m air drag coefficient Cd 0.3 − rolling resistance Cr 0.015 − wheel radius wr 0.3 m final drive ratio fr 4.0 − gear ratio gr 3.4-2.1-1.4-1.0-0.77 − −1 idle speed ωi 73.3 rad s air density ρ 1.2 kg m−3 gravity g 9.81 m s−2

where u(t) = Ft/b the control input, x(t) =q ˙(t) the state of the model and correspondingly xd(t) =q ˙d(t) the desired vehicle velocity. Appendix B

Engine - idle governor

The engine is regarded as a system with fuel value F v in mg as input and the rotational speed ωe in rpm as output (see Figure B.1). Depending on the present load, the operating point of the engine and the temperature of the engine, the plant He exhibits different characteristics. In a gain-scheduling context only the temperature dependency is taken into account for the sake of simplicity. The temperature of the coolant is measured and adopted as an approximation for the temperature of the engine, i.e. Te ≈ Tc. Furthermore, the temperature dependence is regarded as an exogenous variable.

Te K

F v mg Engine ωe rpm = He(Te)

Figure B.1: The engine system He

Based on step-response as well as frequency-response measurements, a nominal LTI model He,n is determined for a heated engine (Naus & Huisman 2007). 250 H (T ) = e−0.07s for T = 355 K (B.1) e,n c,n 1.85s + 1 c,n Identification of the temperature dependency of the engine response is based on testbench measurements (Naus & Huisman 2007). These measurements indicate that only the system gain is temperature-dependent. The higher the engine temperature Te, the higher the viscosity of the lubricant and thus the less friction will be present. The effect of expanding parts is assumed to be minor with respect to this phenomenon. Assuming the differences between the testbench and the experimental measurement results can be related to a different static gain only, the resulting temperature-dependent engine gain equals

G(Tc) = 2.2Tc(t) − 529.8,Tc ∈ [300 400] K (B.2) where gaine,tb(t) in rpm/% and Tc the temperature of the cooling water in Kelvin. The % indicates the percentage of the maximal possible injection and corresponds to about 3 mg per percent. The resulting model for the engine system is given by G(T ) H (T ) = e−0.07s c T ∈ [300 400] K (B.3) e c 1.85s + 1 c where G(Tc) = g1Tc + g2 is the temperature-dependent gain of the engine as defined in (B.2).

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