ITERATIVE DESIGN OF STRUCTURED UNCERTAINTY MODELS AND ROBUST CONTROLLERS FOR LINEAR SYSTEMS

Transcription

1 Budapest University of Technology and Economics Department of Control Engineering and Information Technology Budapest, Hungary Computer and Automation Research Institute of Hungarian Academy of Sciences Systems and Control Laboratory Budapest, Hungary ITERATIVE DESIGN OF STRUCTURED UNCERTAINTY MODELS AND ROBUST CONTROLLERS FOR LINEAR SYSTEMS LINEÁRIS RENDSZEREK STRUKTURÁLT BIZONYTALANSÁGI MODELLJÉNEK ÉS ROBUSZTUS SZABÁLYOZÁSÁNAK ITERATÍV TERVEZÉSE Thesis by Gábor Rödönyi In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Supervisors: Prof. József Bokor Systems and Control Laboratory Computer and Automation Research Institute Hungarian Academy of Sciences Prof. Béla Lantos Dept. of Control Engineering and Information Technology Budapest University of Technology and Economics March, 21

2 Declaration Undersigned, Rödönyi Gábor, hereby state that this Ph.D. Thesis is my own work wherein I have used only the sources listed in the Bibliography. All parts taken from other works, either in a word for word citation or rewritten keeping the original contents, have been unambiguously marked by a reference to the source. Nyilatkozat Alulírott Rödönyi Gábor kijelentem, hogy ezt a doktori értekezést magam készítettem és abban csak a megadott forrásokat használtam fel. Minden olyan részt, amelyet szó szerint, vagy azonos tartalomban, de átfogalmazva más forrásból átvettem, egyértelműen, a forrás megadásával megjelöltem. Budapest, Rödönyi Gábor Az értekezésről készült bírálatok és a jegyzőkönyv a későbbiekben a Budapesti Műszaki és Gazdaságtudományi Egyetem Villamosmérnöki Karának Dékáni Hivatalában elérhetőek. The reviews of this Ph.D. Thesis and the record of defense will be available later in the Dean Office of the Faculty of Electrical Engineering and Informatics of the Budapest University of Technology and Economics.

3 Édesanyám emlékére Anna Mártának

4 Abstract In safety-critical applications guaranteed stability and performance against all disturbances and changes in the system s dynamics are expected. In the robust control theory of linear time-invariant (LTI) systems the notion of uncertain system has been introduced. An uncertain system is defined by a set of unknown systems with boundedh norm. The uncertain system is described by the feedback interconnection of a nominal model and an unknown system. The unknown system can be unstructured consisting of a full LTI system matrix or a structured block-diagonal matrix each block belonging to some system classes. The more details of the real plant can be taken into account the smaller can be the resulting model set and so the conservatism of the controller. In the dissertation, the notion of uncertainty is meant for both the set of neglected LTI dynamics and the set of deterministic disturbances. Usually, the structured uncertainty model is constructed based on physical assumptions. The sizes of disturbances and uncertainty blocks are characterized by frequency-dependent weighting functions. Their choice strongly influences the robustness of the designed system, since too small weights may lead to instability or too large ones to poor performance. The design of weighting functions involves much experimenting and heuristic solutions based on deep engineering insight. In the dissertation a method is presented that automatically designs structured uncertainty models based on measurement data and that improves robust performance by taking the performance specifications into account. In the first thesis an iterative algorithm is elaborated for LTI systems with the aim of optimizing robust performance by shaping uncertainty models and designing skew-ľ controllers. Advantageous properties are the handling of unstable experiments and the validation-based improvement of robust performance. In the second thesis the proposed algorithm is elaborated for linear parameter-varying (LPV) nominal models using integral quadratic constraints (IQCs). In the last thesis a control strategy, including the process of modelling, identification, uncertainty modelling and robust control design, is elaborated for steering a vehicle by using the electronic brake system. The proposed method can be applied in emergency situations.

5 CONTENTS Contents Notations 3 1 Introduction Robust control background and motivation Trends in system identification Iterative identification and control background Vehicle steering problems Layout of the thesis Basic notions Signal and system classes Linear matrix inequalities (LMIs) Integral quadratic constraints (IQCs) Analysis and synthesis of LTI and LPV systems Abstract stability and performance characterizations Analysis and synthesis in H /µ framework Robust control of uncertain LTI systems using IQCs Control of LPV systems using IQCs Summary Uncertainty modelling and robust control design for LTI systems Problem formulation Characterization of uncertainty Robust performance criterion An iterative algorithm for uncertainty modelling and control design Solutions with convex optimization Case study: Active steering for vehicle stability enhancement - LTI case Summary and Thesis Uncertainty modelling and robust control design for LPV systems Problem formulation Relation between skew µ analysis and IQC approach Output-feedback control of uncertain LPV systems Optimizing the uncertainty model

6 CONTENTS 5.5 Iterative design of controller and uncertainty model Case study: Active steering for vehicle stability enhancement - LPV case Summary and Thesis Vehicle safety enhancement by steer-by-brake control Experimental conditions Modelling for steer-by-brake control A strategy for trajectory design Robust control design based on D-K-W-iteration Analysis of model uncertainty Design results Summary and Thesis Conclusions 17 A Simple numerical example 111 A.1 Problem formulation A.2 Parametrization of the uncertainty model A.3 Computation of robust performance A.4 Numeric calculations B Derivation of a steer-by-brake vehicle model 117 B.1 Yaw dynamics B.2 Steering system B.3 Wheel model B.4 Derivation of the simplified model Acknowledgement 137 2

7 CONTENTS Notations R, C real and complex fields L Lebesgue-measurable functions L 2 finite energy Lebesgue-measurable functions RH space of proper and real-rational stable transfer matrices C imaginary axis of the complex plane A T,A transpose and conjugate transpose of matrix A n x length of vector x I,I 2,I x identity matrices possibly with indication of the size (I x : n x n x ) w 1,w 2,... blocks of a partitioned vector w σ(a) largest singular value of matrix A ρ(a) spectral radius of A ω frequency [rad/s] ρ(t) scheduling parameter of LPV systems s (ρ) scheduling matrix of LFT LPV systems u unknown LTI perturbation of a nominal model τ number of blocks in block-diagonal perturbation u S ρ convex set of all admissible scheduling parameters S s convex hull of all admissible scheduling matrices in LFT LPV parametrization S u set of all admissible LTI perturbations in RH S general perturbation set of causal, linear, finite L 2 -gain systems S P multiplier set for parametric, polytopic perturbations B unit ball of a set, e.g., BS = { S 1} >,<,, for matrices it denotes definiteness or semi-definiteness co{a 1,...,A n } convex hull of n matrices F U, F L upper- and lower linear fractional transformation () integral quadratic form specifying perturbation p () integral quadratic form specifying performace Π(jω) dynamic multiplier P real multiplier Ψ dynamic factor of a multiplier R,S,Q blocks of multipliers substitutes an entry in a symmetric matrix where it is unambiguous 3

8 CONTENTS 4

9 Chapter 1 Introduction The primary purpose of the present work is to introduce an optimization technique which facilitates both uncertainty modelling and robust feedback control design in order to achieve improved robust performance of the controlled system. The proposed technique is developed for both linear time-invariant and linear parameter-varying systems. The subject leads to the fields of research known as robust control and iterative identification and control (IIC). The contributions of the first two theses are introduced in Sections 1.1 and 1.3 approaching from these areas. The presented uncertainty modelling and robust control design method is motivated by safety critical control problems. The third thesis contributes on vehicle safety enhancement in a lateral control problem and is introduced in Section Robust control background and motivation The starting point of modern control theory is the mathematical description of the plant to be controlled. Using an accurate inverse model of the plant, perfect control would be possible by cancelling the plant s dynamics. There are many constraints, however, preventing perfect control, for example uninvertible modes, input or state limitations or model uncertainties. The mathematical model of a system can never be exact due to unmeasured disturbances and discrepancy between the physical plant and the mathematical model (neglected dynamics, unknown/varying parameters). The role of feedback control is to suppress the effects of uncertainties and enforce the dynamical system to behave in a prescribed manner. The so-named modern robust control is based on a model equipped with strict bounds of its uncertainties. This deterministic framework is motivated by strict demands on stability and performance where guarantees have to be provided for the worst-case occurrence of the uncertainties. The theory has its origin in the seminal paper of Zames [136], and come to be known as H control theory. The theory is well developed for linear time-invariant (LTI), [137, 43], and linear parameter-varying (LPV) systems [134, 76, 7, 6]. A control system is robust if it remains stable and achieves certain performance criteria in the presence of all possible uncertainties. In the H control theory, robust performance and robust stability problems are analyzed via Small-Gain Theorem in a system configuration where a stable system, M, (nominal model and controller) is 5

10 in feedback interconnection with an (uncertain) operator,, which belongs to a given bounded set of stable systems. If the uncertainty set is unstructured (it contains any full LTI matrix operators), then the interconnection is stable if and only if the H -norm of M is less than the reciprocal of the H -norm of any of the uncertainty set. This small-gain condition is conservative when the uncertainty set is structured (e.g. block diagonal matrix of full and repeated scalar blocks of LTI, LTV or constant operators), since the detailed specification defines only a subset of the general unstructured set. To get a sufficient and necessary condition for the stability and performance of systems with structured perturbation, the structured singular value µ [37, 115] should be computed. In control design point of view, involving information about the uncertainty into the uncertainty set (by structuring) allows an increased set of stabilizing controllers to be considered. Due to the additional information less conservative controllers (with improved performance) can be designed. An important task of the control designer is to specify the structure of the uncertainty model and the norm bounds of its elements, which used to be the part of a preceding modelling phase. The choice of these bounds is a nontrivial task but it may strongly influence the resulting stability and performance. The conventional way is based on the analysis of the physical system. The more details of the real plant (dynamic, time-varying, real parametric uncertainty [39, 29]) can be taken into account the smaller could be the resulting model set and so the conservatism of the controller. The sophisticated structures of perturbations, however, bear the price of increased computational complexity involving limitations in the computability of the achievable performance level. Only lower and upper bounds of the performance level can be calculated that are not necessarily tight in case of both dynamic and parametric perturbations, as it has been shown in the papers [135, 44, 45, 89]. The difficulty of defining the uncertainty bounds based on the physical equations of the system arises mainly from two sources. One is the absence of knowledge on the physical uncertainty, the other is the difficulty in evaluation of interactions between the uncertainties. These interactions may neutralize some uncertain effects implying the possibility of tightening the set of uncertainties. As we have seen structured uncertainty modelling is a heuristic step involving much engineering intuition. The goal of the thesis is to replace this step by a formalism that, based on measurement data and control related criteria, provides bounds of structured uncertainty. It is expected, that introducing more information into the design from closed-loop experiments, such that the control objective is taken into account, leads to improved robust performance as compared to any heuristic modelling methods. Note that there exist methods based on experimental data for unstructured uncertainty modelling. Some of the achievements in this field are mentioned in the next sections. The proposed method is elaborated first for LTI systems (Chapter 4) and then generalized to LPV systems (Chapter 5). The contribution is approached from a modelling aspect in the next sections. 6

11 1.2 Trends in system identification Before the 199s the identification and control developed separately. Controllers were designed according to the "certainty equivalence principle", i.e., the model represented the true system and no neglected dynamics was assumed. Early attempts to address the issue of parametric uncertainty were dual control [4, 133] and adaptive control [9], but still the model was assumed to be of full order, which means that the true system is in the selected model class. However, "any reasonable model structure gives a bias error" [62]. This recognition in the late 198s lead to approximate or restricted complexity modelling, where the quality of the model can be adapted to the intended application. The emergence of the powerful robust control theory and the experience that high performance control can often be achieved with very simple, crude nominal models turned the attention of identification experts to approximate modelling, where the purpose was to provide a nominal model together with an uncertainty set appropriate for robust control design. The topic known as modelling for control or control oriented system identification became one of the most active research area in system identification since the 199s. The motivation can be attributed to the recognition that efficient identification suited for control design should be performed based on criterion related to the actual end use. One framework in the field of robust control oriented modelling proceeds from the existing probabilistic identification methods, dominantly from the successful maximum likelihood and prediction error (PE) methods [81]. The gap in theory to overcome is quite large: 1. Characterize the parametric uncertainty of PE nominal models in a form suitable to match the general structure of H /µ methods. Instead of that general structure, only special forms with unstructured dynamic uncertainty have been considered in the literature. The reason is that ellipsoidal sets of PE model parameters correspond to very structured descriptions from the point of view of robust control. Based on the robustness results [11] on real parametric uncertainty, [112, 7] delivered stability tests for some simple nominal model structures. Frequency-domain bounds on the variance and gain error of PE models have been derived in many papers, e.g. [62, 5, 23, 59, 96]. The model error modelling [82, 83, 113, 24] approach should also be mentioned here. Mixed stochastic-deterministic methods [57, 35] obtain estimate for noise variance and bound on the decay rate of the system s linearly parameterized transfer function. Based on frequency domain error quantification, unstructured perturbation models (additive, multiplicative or coprime factor perturbations) are created from the parametric models. Such uncertainty sets can be well handled in certain robust control problems. 2. Define control relevant criterion for identification of the nominal model and for tuning of the uncertainty sets. The controller synthesis problem is non-convex for the general configuration with structured uncertainty, and practically solved by D-K iteration [13]. This might be the reason for considering simpler robust 7

12 control problems in control oriented identification theory. The feedback configuration with the one degree of freedom controller used in H loop-shaping [49] is an example. For this structure simple sufficient conditions on robust stability and robust performance can be derived. It is shown, e.g., in [121, 48], that robust performance is implied by robust stability and a sufficient level of nominal performance. Performance expressions provide criterion for both the identification of the nominal model and tuning the uncertainty set [6]. The above formulation motivated the iterative identification and control design methods in both the probabilistic [2, 21, 75, 15, 9] and deterministic [121, 128, 34] frameworks. The repeated scheme of "modelling - control design - closed-loop experiments", as treated for example in [121], will not converge to the minimum of the robust performance criterion [61], however, the approach has been found useful in several applications. See, e.g., [11, 122]. It has been shown that caution is necessary when applying the controller updates, because the closed-loop system with the unknown true system is not guaranteed to be stable [5, 47]. 3. Design experiments that allow to identify optimal models and uncertainty sets. In case of restricted complexity models, the quality of the model is determined by the spectrum of the input. High quality is required within the bandwidth of the closed-loop system. It has been shown that the major improvement in closed-loop performance occurs when closed-loop identification data is applied [48]. For more details on closed-loop identification and experiment design, see [42, 128, 46]. Beyond the above problems, still a bridgeless gap remains: probabilistic identification can provide uncertainty bounds with some level of certainty, however, modern robust control requires strict bounds. See the survey papers [48, 6] and references therein for more insight on probabilistic control oriented identification. Identification methods with deterministic assumptions on the noise provide directly applicable hard bounds on the uncertainty for the robust control design. Similarly to the stochastic framework, special unstructured perturbation models are created. For example set membership methods [1, 91, 68] assume deterministic additive noise and additive unstructured perturbation. Based on time-domain model validation results of [18], Kosut [69] derived an uncertainty trade-off curve and gave hint on how dynamic versus noise uncertainty may be traded off for a set of unfalsified models. The trend of the research started with the objective of delivering control relevant nominal models [58, 92, 55, 12, 127]. Later, the focus shifted to control relevant uncertainty sets. The identification criterion and robust stability or robust performance can be directly connected and minimized by a joint identification and control design algorithm [53, 32, 56, 8]. Practically, the joint problem can be solved by iteration. 1.3 Iterative identification and control background Unfortunately, none of the existing IIC methods can be applied directly for structured uncertainty modelling and control design. 8

13 The windsurfer approach [77] gradually increases the bandwidth of the closed-loop by iteratively improving a parametric model based on internal model control relevant criterion. The unfalsified control concept has been introduced in [116] where the synthesis of "not demonstrably unrobust" controllers is addressed directly. This data-driven, modelfree control approach recursively falsifies controllers that fail to satisfy a performance requirement for some given measured data and a specified control law. Unfalsified control theory is employed, e.g., in [67, 27, 129]. The unfalsification concept has been applied for model parameter bounding in, e.g., [8, 68] where the criterion is the worstcase performance of the system. In [13] the parameter set is extended with scalar bounds on an unstructured additive perturbation term and an additive noise term, and global convergence has been proved. The concept of the unfalsification scheme presented in [13] is the closest to the concept presented in the dissertation, however, they differ significantly in their details. In a vast number of papers [33, 31, 121, 128] coprime factor plant models are identified iteratively with control design steps in a H loopshaping framework. The use of ν-gap metric [132] allows the characterization of all systems which are stabilized by the actual controller. If the plant is known, the metric allows the characterization of all stabilizing controllers. From this, a priori guarantees for the stability of the closed-loop can be derived. In [131] the iterative unfalsification and control design methods gradually exclude regions from the set of unfalsified models, thus, achieving improvement in the control performance. The method of the present work approaches the final model set from the opposing direction: the set of models gradually increases as the iteration evolves, thus, robustness is improved while performance might decrease. On the other hand, robust performance is optimized by shaping the uncertainty model with the help of a skew µ version of the D-K-W iteration of Ref. [Röd9]. In the approach of the present work the true plant is allowed to change (parameter variations and change in operating points) from experiment to experiment, while in the methods based on ν-gap metric a single plant is assumed. Another type of iterative schemes is called generic optimal controller scheme where the control and modelling errors are identical. The method can handle unstructured uncertainties and is applicable for nonlinear systems as well [22, 64]. In the dissertation, the problem of designing frequency-domain weighting functions, i.e. bounds on the uncertainty, for both perturbations and disturbances is placed into an IIC framework. In the proposed method of the thesis the following advantages of several IIC schemes are integrated. 1. The criteria of modelling and control design are identical, so modelling criterion depends on control purposes 2. By using closed-loop experimental data, the quality of the modelling is improved to the degree necessitated by the control 3. No a priori information is assumed on the disturbances, nor on the bounds of the neglected dynamics 9

14 4. Instability of an experiment is not discarded but its data are utilized to improve the robustness of the controller 5. Changes in the dynamics of the plant from experiment to experiment are allowed 1.4 Vehicle steering problems The industrial motivation of the iterative uncertainty modelling and control design approach of the thesis arises from control design problems of large, complex dynamic systems which require strict robust performance guarantees on the control, ensuring safe operation also for the worst-case occurrences of disturbances and system dynamics. Although there often exist high-fidelity simulators (CarSim/TruckSim [117, 118] for road vehicles and SIMONA [2], FlightGear, [1] and [111, 126] for aerial vehicles) which are able to imitate the real vehicle behavior, and can be used in final tests of controllers before implementation, they are inadequate to be the base of control design. Controllers of several sub-dynamics of the systems are designed based on highly simplified low order models in order to obtain low order implementable controllers. Many application examples show that simple linear, very crude models equipped with uncertainty description often suffice to give good closed-loop performance, see e.g. [14]. Simplified models of sub-dynamics, e.g., lateral motion of a road vehicle, are derived with the assumption of a limited impact of other dynamics. The effect of heave, roll and longitudinal dynamics can be considered as uncertain dynamics. The effect of disturbances, e.g., wind, road adhesion properties, slope and tilt angles of the road are also unknown during the control and both kind of effect are hard to model accurately in advance. Also, when one tries to measure the error of a given nominal model, it is often impossible to separate and identify the sources of uncertainty. A further reason for avoiding this physical uncertainty modelling is what we know from unfalsified modelling concept of [13]: there exist better models for control than the theoretically exact model. This is similar to one of the main contributions of restricted complexity modelling theory: restricted complexity models should depend on the actual control purposes. This idea is extended to structured uncertainty modelling and modern robust control design, and utilized in vehicle steering problems: instead of the tremendous work of exact uncertainty modelling the thesis proposes an automated data based modelling algorithm where a structured uncertainty model is shaped according to the control purposes. At the end of Chapter 4 and 5 an active steering control problem for power assistance is presented to illustrate the efficiency of the iterative scheme. The controller generates an additional steering angle/torque in order to follow a reference model for the yaw-rate. In Chapter 6 a separate thesis group is elaborated where steering is managed by using the front wheel brakes. This problem might emerge in emergency situations when the driver is incapable of steering and halting the vehicle safely, due to e.g. drowsiness or lipothymy. In the majority of the commercial heavy trucks with mechanic-pneumatic steering systems, the only device to automatically intervene into the motion is the electronic brake system. It is assumed in Chapter 6 that some smart sensor system 1

15 (e.g. camera) is available for determining vehicle position and a desired path on the road. The goal is to navigate and halt the vehicle to a safe position at the side of the road. For a yaw-rate reference tracking problem, appropriate nominal models are derived and identified, then uncertainty model and robust controller are designed with the proposed iterative methods of the dissertation. The topic fits the line of research on improving safety of road vehicles. Some examples of the latest developments in vehicle stability control problems are Anti-lock Braking System (ABS) and Anti-Slip Regulator (ASR) which prevent locking of the wheels when braking or driving, respectively; Electronic Stability Program (ESP) uses individual wheel braking in critical driving situations, see e.g. [78]. Advanced driver assistance systems utilize special sensors (radar, laser, GPS, video camera). For example, Adaptive Cruise Control System (ACC) helps preventing rear-end collisions in heavy traffic by automatically maintaining the correct distance from the preceding vehicle [19, Chapter 6.]; Lane Departure Warning System (LDW) warn the driver when the vehicle begins to move out of its lane, unless a turn signal is on in that direction; Driver Drowsiness Detection (DDD) systems learn driver patterns and can detect when a driver is becoming drowsy. LDW or DDD systems [16, 19, 72, 84, 93, 114] can be easily mounted also to commercial vehicles in order to detect the dangerous situations, when automatic intervention is necessary. In this case, the automatic control function is switched on. Desired path and desired yaw-rate can be generated and the yaw-rate reference tracking controller is activated. The focus in Chapter 6 is on the design process for the latter task. 1.5 Layout of the thesis The dissertation is organized as follows. After the list of notations the most important basic notions and theorems are summarized. The main achievements in control theory employed in the thesis chapters are summarized in Chapter 3 where analysis and synthesis methods of uncertain or LPV systems are presented. Three theses are discussed in chapters 4, 5 and 6, respectively. At the end of the chapters the contributions are summarized and the most important conclusions are drawn. In Chapter 4 the basic scheme of the proposed iterative algorithm is introduced for LTI systems. An important special case is also treated. In Chapter 5 the uncertainty modelling and robust control design results are extended for a broader class of systems with the help of Integral Quadratic Constraints (IQCs). To this end analysis and synthesis methods of uncertain LPV systems in LFT dependence on both the scheduling parameters and neglected dynamics are also presented. The last chapter is devoted to the emergency intervention system that applies the electronic brake system for steering a heavy vehicle. Modelling and identification aspects, robust control design and simulation results are detailed. Note, in the dissertation the own publications are separated from the other cited papers by using different citation formats. The own publications are cited by indicating 11

16 the initials of the authors and the year, e.g. [RGB9], while other works are simply ordered, e.g. [12]. 12

17 Chapter 2 Basic notions 2.1 Signal and system classes A system can be considered as a mapping between signal spaces. The performance of a controlled system is often specified in terms of induced norms defined on the mapping. The aim of control design is to modify the mapping by feedback in order to achieve predefined properties. This chapter presents the signal and system spaces and norms used in the dissertation. All signals and systems are in the continuous-time domain. The signals are assumed to be zero over the negative time axis. Let L n denote the set of all signals x : [, ) R n that are Lebesgue-measurable. All piece-wise continuous signals are contained in L n. A linear subspace of L n is the set of finite energy signals defined by L n 2 := {x L n : x 2 < }, where x 2 = x(t) 2 dt denotes L 2 -norm of signal x. This norm is called the energy of the signal. With the help of the truncation operator { P T : L n L n x(t) for t [,T], P T x(t) = for t (T, ) an extended signal space L n 2e is defined as L n 2e = {x Ln : P T x L n 2 for all T } Any signal that is bounded over a finite interval belongs to this signal space. A dynamical system M is a mapping M : L nu 2e Lny 2e. The system is causal if its output depends only on past inputs, i.e. P T M(u) = P T M(P T u) for all T, and all inputs u L nu 2e. A stable system maps any signal in L 2 into a signal that is also contained in L 2. Definition 2.1 (L 2 -gain) The L 2 -gain of the system M : L nu 2e M 2 := sup{ P TM(u) 2 P T u 2 Lny 2e u L nu 2e, T, P Tu 2 } is defined as = inf{γ R u L nu 2e, T : P TM(u) 2 γ P T u 2 } 13

18 CHAPTER 2. BASIC NOTIONS If M is causal then M 2 = sup u L2, u 2 > focus on linear, causal systems. Definition 2.2 (Linear time-invariant (LTI) systems) ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t). M(u) 2 u 2. In the dissertation we restrict our Throughout the dissertation [ A B M = C D ] with the separator lines denotes a system operator. For simplifying notations, symbol M may also denote the transfer function of the system M = M(jω) = C(jωI A) 1 B + D, which must everywhere be clear from the context. Separator lines are also applied for grouping input/output signals in order to improve clearness of formulas. The calligraphic symbol M represents the block matrix containing the state-space matrices of M [ ] A B M = C D The space L is a Banach space of matrix valued functions that are essentially bounded on jr, with the norm M = ess sup σ[m(jω)] ω R The space H is a closed subspace of L with functions that are analytic and bounded on the open right-half plane. The H norm defined as M = sup σ[m(jω)] ω R is equal to the L 2 -gain of the system. The real-rational subspace of H is denoted by RH and consists of all proper and real-rational stable transfer matrices. Definition 2.3 (Linear parameter-varying (LPV) systems) ẋ(t) = A(ρ(t))x(t) + B(ρ(t))u(t), y(t) = C(ρ(t))x(t) + D(ρ(t))u(t), where the state-space matrices depend on an on-line measurable time-varying scheduling parameter ρ(t) which takes values from a set S ρ R nρ. 14

19 2.2. LINEAR MATRIX INEQUALITIES (LMIS) Definition 2.4 (Affine LPV systems) The LPV system is called affine if the statespace matrices depend affinely on the scheduling parameter which belongs to a convex set S ρ. [ ] A(ρ(t)) B(ρ(t)) M(ρ(t)) := = M C(ρ(t)) D(ρ(t)) + ρ 1 M ρ nρ M nρ, ρ S ρ Definition 2.5 (Linear fractional transformation [ (LFT)) The ] LFT is a feedback M11 M interconnection of two systems M and. Let M := 12 be appropriately M 21 M 22 partitioned. The lower LFT is defined by F L (M, ) := M 11 + M 12 (I M 22 ) 1 M 21 provided that the inverse exists. The upper LFT is defined by provided that the inverse exists. F U (M, ) := M 22 + M 21 (I M 11 ) 1 M 12 Definition 2.6 (LFT LPV systems) LFT LPV system F U (M, s ) is defined by a LTI system A B 1 B 2 M := C 1 D 11 D 12 C 2 D 21 D 22 and time-varying scheduling matrix s (t) which belongs to a convex set S s : s S s. Alternative notion can be that s (ρ(t)), ρ S ρ. The L 2 -gain of the LPV system is defined by M(ρ(t)) := M(ρ) 2 := sup [ A(ρ(t)) B(ρ(t)) C(ρ(t)) D(ρ(t)) sup ρ S ρ u L 2, u 2 > 2.2 Linear matrix inequalities (LMIs) ] M(ρ)u 2 u 2 Many problems in system analysis and control design can be composed by linear matrix inequalities. Definition 2.7 (LMI) The linear matrix inequality in the variables x i R is an expression of the form F(x) := F + x 1 F x m F m > (2.1) where F,...,F m are real symmetric matrices, F i = F T i R n n, i = 1,...,m. 15

20 CHAPTER 2. BASIC NOTIONS The linear matrix inequality (2.1) expresses the positive definiteness of the affine matrix function F. The set of variables x satisfying the condition constitute a convex set. Three of the basic problems related to LMIs will reappear in the dissertation. The feasibility problem is to find a solution x that satisfies (2.1). The optimization problem is to minimize a linear functional of x subject to the constraint (2.1). The generalized eigenvalue problem is to minimize λ subject to A(x) < λb(x), B(x) > and C(x) <, where A,B and C are affine matrix functions of x. The following lemma provides conditions on linearizing a special kind of nonlinear matrix inequality by applying Schur complements: Lemma 2.1 (Schur complement) Let an affine mapping F(x) be partitioned as [ ] Q(x) S(x) F(x) = S(x) T R(x) If R(x) >, then the nonlinear matrix inequality is equivalent to F(x) >. If Q(x) >, then is equivalent to F(x) >. Q(x) S(x)R(x) 1 S(x) T >, R(x) S(x) T Q(x) 1 S(x) > In the above Lemma symbol > can be replaced everywhere by < and still the Lemma holds. 2.3 Integral quadratic constraints (IQCs) The integral quadratic constraints are general forms for characterizing uncertainty, stability and performance of systems. An IQC is an inequality of the form Σ(x), where Σ(x) is an integral quadratic function that can be defined in the frequency-domain as Σ(x) = 1 2π x(jω) Π(jω)x(jω)dω via a measurable Hermitian bounded mapping called multiplier: Π : jω C Π(jω) C nx nx, Π(jω) c, where c is a constant, and x L 2. The time-domain counterpart Σ(x) = z(t) Pz(t)dt can be derived using the Parseval-theorem with z = Ψx, where Ψ is a real rational proper stable system defined by the factorization Π(jω) = Ψ(jω) PΨ(jω) (2.2) The following lemma facilitate the formulation of IQCs, e.g. performance specifications, as LMI conditions. 16

21 2.3. INTEGRAL QUADRATIC CONSTRAINTS (IQCS) Lemma 2.2 (Frequency-domain inequality (FDI)) Suppose Π is a measurable bounded Hermitian valued mapping on C. Then the following statements are equivalent There exists an ǫ > such that 1 2π There exists an ǫ > such that x(jω) Π(jω)x(jω)dω ǫ 2π Π(jω) ǫi for all ω R Π(jω) < for all ω R { } x(jω) x(jω)dω The following lemma is used in the dissertation to provide equivalent stability or performance conditions in the time- and frequency-domain, respectively. Lemma 2.3 (Kalman-Yacubovich-Popov) Suppose, the LTI system ẋ = Ax + Bu, y = Cx + Du, [ ] A B is controllable. Let M :=. Let P be a real symmetric matrix. Then the C D following statements are equivalent There exists a nonnegative function V : R nx R such that t1 V (x(t )) + s(u(t),y(t))dt V (x(t 1 )), (2.3) t [ ] T [ ] u u s(u,y) = P y y for all t t 1 and all trajectories (u,x,y) of system M. For all ω R with det(jωi A) there holds [ ] [ ] I I P M(jω) M(jω) There exists X = X T such that [ A T ] [ X + XA XB I B T + X C D ] T P [ I C D ] The system with the so-named quadratic supply function s and satisfying (2.3) is called dissipative. All non-strict inequalities can be replaced by strict inequalities and the system is called strictly dissipative. 17

22 CHAPTER 2. BASIC NOTIONS 18

23 Chapter 3 Analysis and synthesis of LTI and LPV systems This chapter is devoted to uncertain LTI and nominal LPV systems. In both cases the system is perturbed by an operator via LFT and excited by disturbances. As for LTI systems, the operator represents uncertainty and belongs to a given, possibly structured, bounded set. As for LPV systems, the operator is a time-varying scheduling parameter, which is on-line available for measurement and takes values from a bounded, possibly structured set. Analysis of stability and performance and controller synthesis problems are handled in a common framework based on IQCs. In Section 3.1 abstract stability and performance notions are defined and conditions are provided. Sections discuss special cases of the operator sets. The presented results provide design criteria and build the fundamental tools of controller design for the thesis chapters 4-6. The following collection of notions is based mainly on [12], [119] and [137]. 3.1 Abstract stability and performance characterizations The common setup for analyzing the stability of uncertain and LPV systems is depicted in Figure 3.1, where the perturbation belongs to set S := { : L nz 2e Lnw 2e, is a causal and linear system with finite L 2-gain} and nominal model M : L nw 2e Lnz 2e which is a causal, linear system with finite L 2-gain. z z + + M w + w Figure 3.1: The general uncertain feedback configuration for analyzing robust stability 19

24 CHAPTER 3. ANALYSIS AND SYNTHESIS OF LTI AND LPV SYSTEMS An example for the perturbation set is the set of structured full-block dynamic uncertainties representing neglected/unmodelled dynamics. The neglected dynamics is assumed to belong to set S u S, where S u := { u RH u = diag{ 1,..., τ }, i (jω) C n wi n zi, i = 1,...,τ} LPV systems can also be handled in robust control framework. In a LFT LPV system the time-varying on-line measurable scheduling matrix is contained in set S s S, defined by S s := { s (t) : R R nw nz, s (t) co{ s1,..., sκ }}, where κ is the number of matrices spanning the convex hull. Set S s is assumed to be star-shaped, i.e., s S s r s S s for all r [,1]. Note, that both S u and S s are subsets of S. The following stability and performance theorems are stated for the general configuration depicted in Figure 3.1 with the assumption that S. The interconnection can be characterized by the relation [ w z ] [ w = L M ( ) z ] [ I, where L M ( ) = M I Definition 3.1 ((Uniform) robust stability (RS)) The interconnection in Figure 3.1 is uniform robust stable if L M ( ) has a causal inverse (well-posedness) L M ( ) 1 has finite L 2 -gain for all S (robust stability) there exists a common bound on L M ( ) 1 2 for all S (uniformity) The following theorem is the combination of Theorem 3.7 and Theorem 3.8 of reference [12] applied to the case of linear perturbations. It provides conditions for robust stability in case of stable linear systems with feedback interconnection to linear causal bounded (L 2 -gain) perturbations. Theorem 3.1 (An abstract stability characterization) Let Σ : L nw+nz 2 R be an integral quadratic function. Suppose that all S satisfy ([ z Σ z ]) ]. for all z L nz 2. (3.1) The interconnection in Figure 3.1 is uniform robust stable for all S if and only if there exists an ǫ > with ([ ]) w Σ ǫ w 2 2 for all w Lnw Mw 2. (3.2) 2

25 3.1. ABSTRACT STABILITY AND PERFORMANCE CHARACTERIZATIONS The philosophy of applying IQCs for the analysis of uncertain systems is as follows. Instead of checking (3.1) for all S, which would be an infinite dimensional problem, a manageable set of IQCs is determined so that each IQC in the set implies S. Then, condition (3.1) can be omitted and condition (3.2) is sufficient to be satisfied by one element of the set of IQCs. For a given set of uncertainty S, one tries to find all IQCs of form (3.1) that are satisfied by all uncertainties in set S. This practically means that integral quadratic function Σ ([ w z ]) = 1 2π [ w(jω) z(jω) ] Π(jω) [ w(jω) z(jω) ] dω with w = z [ ] Qω (jω) S is parameterized through multiplier Π(jω) = ω (jω) S ω (jω) by specifying all R ω (jω) Q ω,s ω and R ω that satisfy (3.1). The more multipliers fulfill (3.1) the smaller is the conservatism of the analysis results, since (3.2) is enough to be satisfied by one of all the multipliers. In Refs. [88, 12] several uncertainty types are characterized via IQCs and summarized in the following list. (When [ multiplier ] Π is restricted to be a real Q S symmetric matrix, it is denoted by P = S T throughout the dissertation in R accordance with the notation in (2.2)). Structured linear causal mappings = diag{ 1,..., τ } with 1 fulfill (3.1) for the class of multipliers R = diag{d 1 I,...,d κ I}, S =, Q = R, (3.3) where d i >, i = 1,...,κ, are real scalars. Structured linear time-varying uncertainties (t) = diag{ 1,..., κ } with (t) 1 form a special class of linear causal mappings and fulfill not only (3.1) but quadratic constraint [ ] T [ ] (t) (t) P (3.4) I I for any P in form (3.3). The quadratic constraint still holds if P is time-varying. Repeated structured time-varying uncertainties (t) = diag{δ 1 (t)i,...,δ κ (t)i} with δ(t) 1 fulfill the quadratic constraint (3.4) for the class of multipliers parameterized as R = diag{r 1,...,R κ } >, Q = R, S = diag{s 1,...,S κ }, S + S T =, (3.5) where R i,s i, i = 1,...,κ, are real matrices. Again, we can generalize to timevarying multipliers. 21

26 CHAPTER 3. ANALYSIS AND SYNTHESIS OF LTI AND LPV SYSTEMS z w z p M w p Figure 3.2: The general uncertain feedback configuration for analyzing robust performance Using indirect parametrization a larger class of multipliers can be defined for parametric uncertainties, as compared to parametrization (3.3) or (3.5), which reduces the conservatism of the uncertainty characterization and consequently the stability result. The polytopic uncertainty set S s fulfill (3.4), if P S P, where [ ] Q S S P = {P = S T R Q <, [ j I ] T P [ j I ] >, j = 1,...,κ} (3.6) Further advantage of indirect parametrization is that we do not need to bother about the specific structure of the uncertainties and derive the corresponding structure of multipliers. In contrast to (3.4), indirect parametrization admit a numerically tractable description in terms of finitely many LMIs, however, at the expense of conservatism. Structured uncertain causal LTI dynamics S u with gain 1 satisfies IQCs (3.1) with parametrization R ω (jω) = diag{d 1 (ω)i,...,d κ (ω)i}, S ω (jω) =, Q ω (jω) = R ω (jω), with < d i (ω) R. Robust performance [ can be analyzed ] on the setup depicted in Figure 3.2 where M Mu M is partitioned as M = up. Performance specifications can be characterized M pu M p as follows: there exists an ǫ > with ([ ]) wp Σ p ǫ w z p 2 2 p for all w p L 2, (3.7) where Σ p is an arbitrary mapping Σ p : L 2 R satisfying Σ p ([ z p ]). The conditions of meeting the general performance specification (3.7) are given in the following theorem, [12, Theorem 3.16]. Theorem 3.2 (An abstract performance characterization) Suppose that all S satisfy the IQC ([ ]) w Σ (3.8) z 22

27 3.1. ABSTRACT STABILITY AND PERFORMANCE CHARACTERIZATIONS Suppose there exists an ǫ > such that ([ w Σ z ]) + Σ p ([ wp z p ]) ǫ ( w w p 2 ) 2 (3.9) for all w,w p L 2. Then I M u has a causal inverse whose L 2 -gain is bounded uniformly for S (uniform robust stability) and uncertain system F U (M, ) satisfies the performance criterion (3.7). For example, quadratic performance is defined by the choice Σ p ([ wp z p ]) = [ wp z p ] T P p [ wp where performance index P p is a fixed symmetric matrix that satisfies As a special case, strict passivity [ Qp S P p = p z p (t) T w p (t)dt ǫ S T p R p z p ] dt, (3.1) ], R p (3.11) w p (t) T w p (t)dt for all w p L 2, [ 1 can be specified by P p = 2 I ] 1 2 I. Another case is the induced L 2 -norm of a system being less than a given number γ which can be expressed with Q p = γ p I, S p =, R p = γ 1 p I, (3.12) This is proved in the next section. Theorem 3.2 provides a systematic procedure for analysis and synthesis of many types of systems, uncertainties and performance problems. The closed-loop system is formulated in structure -M plotted in Figure 3.2. There can be more than one uncertainty and performance channels. For all perturbation channels, IQCs of the form (3.2), characterizing uncertainty classes, are parameterized by multipliers Π j (and, possibly, by additional constraints), which can be organized in one large block-diagonal multiplier Π = diag{π 1,Π 2,...}. Similarly, multiple performance channels (in mixed performance problems) are specified by IQCs whose performance indices P p,j are collected in one block-diagonal multiplier P p = diag{p p,1,p p,2,...}. Uncertainty and performance are handled in a common setup where a list of IQCs specifying uncertainty and performance is formed in inequality (3.9). The application of Theorem 3.2 is demonstrated in Sections 3.3 and 3.4, and the synthesis procedure is also presented. The IQC formulation for structured LTI uncertainties and induced L 2 -norm performance criterion leads to the classical µ-synthesis, which is summarized in the following section. 23

28 CHAPTER 3. ANALYSIS AND SYNTHESIS OF LTI AND LPV SYSTEMS 3.2 Analysis and synthesis in H /µ framework In standard H /µ control, robust stability (stability for each system with S u ) is analyzed by the structured singular value (SSV) denoted by µ. It is known that the satisfaction of robust performance ( F U (M, ) 1 for all BS u ) is equivalent to a robust stability problem where the performance output is fed back to the inputs through a fictive perturbation block p RH nwp n zp : z p w p, p 1. See also [36, 37, 137, 12, 13] for more details. The SSV of a complex matrix µ a (M(jω)) := 1 min a { σ( a (jω)) : det(i M(jω) a (jω)) = } is defined as the reciprocal of the norm of the smallest destabilizing structured perturbation a = diag{, p }. Using µ, the small-gain theorem is generalized to the case of structured perturbations [137, Theorem 11.9]. Lemma 3.1 For all S u with 1 β loop F U(M, ) is well-posed, internally stable and F U (M, ) β if and only if µ a (M) < β. The theory of µ-analysis has been developed for mixed full LTI, repeated scalar LTI and repeated real scalar constant perturbation blocks as well, but in the subsequent chapters only full LTI blocks are assumed. The computation of µ a is NP-hard in general, however, for guaranteeing robust performance, it is satisfactory to compute a tight upper-bound that can be accomplished by solving LMIs. Up to 3 full complex blocks in a this upper-bound is exact. Consider the system depicted in Figure 3.2. Define stable, stable invertible scaling functions D := {d i d i,d 1 i RH, i = 1,...,τ} and repeated diagonal matrices D L := diag{d 1 I z,1,...,d τ I z,τ,i zp } and D R := diag{d 1 I w,1,...,d τ I w,τ,i wp }. By using the upper-bound µ a (M(jω)) inf D σ(d L(jω)M(jω)D 1 R (jω)) the sufficient and necessary condition µ a (M(jω)) < β in Lemma 3.1 can be replaced by the computable but only sufficient condition inf D σ(d L (jω)m(jω)d 1 R (jω)) < β In the µ-synthesis problem, an upper-bound of the system gain γ is iteratively minimized. A nominal H control design step and a µ analysis step are iterated as follows. 24 Initialization: D L := I, D R := I Nominal H -controller design via bisection algorithm: a scalar γ is minimized in the controller parameters subject to σ(d L MD 1 R ) < γ, where M = F L (G,K) with G denoting the nominal augmented plant. In each iteration the feasibility of a H -control problem is tested by the solution of two Riccati equations. Finally, controller K is constructed.

29 3.3. ROBUST CONTROL OF UNCERTAIN LTI SYSTEMS USING IQCS µ analysis. Scalars γ k are minimized over a frequency grid ω k, k = 1,...,n ω, in scaling matrices D L,k and D R,k, which are parameterized by set D k = {d i,k d i,k R, i = 1,...,τ} as before. inf σ(d L,k M(jω k )D 1 D R,k ) < γ k k This is an LMI problem. Then, stable, stable invertible transfer functions (d i RH ) are fitted in magnitude to the positive real scalars d i,k, i = 1,...,τ, which results in updated values of D L and D R. In Lemma 3.1, the sizes of the perturbation blocks and the performance block are all related with each other through a common scalar β. A scaled version of the lemma allows to test scenes when some of the blocks have different size. In the next lemma, [119, Lemma 34], the H -norm of the perturbation blocks are bounded by γ 1 γ 3 while the L 2 -gain of the closed-loop system (performance) by γ 3 γ 2. Lemma 3.2 For all S u with γ 1 γ 3 loop F U (M, ) is well-posed, internally stable and F U (M, ) γ 3 γ 2 if and only if µ a (Mdiag{γ 1 I,γ 2 I}) < γ 3. A special case of scalings with γ 1 = γ 3 = γ s and γ 2 = 1 results in the scaled SSV called skew SSV that has been used for analysis purposes in [38, 41, 63]. The following lemma plays fundamental role in Chapter 4. Lemma 3.3 For all BS u loop F U (M, ) is well-posed, internally stable and for all ω: σ (F U (M(jω), (jω))) γ s (jω) if and only if µ a (M(jω)diag{γ s (jω)i w,i wp }) < γ s (jω) (3.13) It is shown in Chapter 4 how the skew µ analysis results can be used for controller synthesis. Details of an iterative method, similar to the above D-K iteration, are elaborated as part of a joint uncertainty modelling and control design algorithm. It is also shown in Section 5.2 that µ analysis (or skew µ analysis) can be carried out based on IQCs. 3.3 Robust control of uncertain LTI systems using IQCs This and the next sections are the basis of the robust LPV controller synthesis presented in Chapter 5. This section presents, based on Ref. [12], the main steps of the control synthesis solving the robust quadratic performance problem for LTI systems with LFT uncertainty. The problem is formulated by IQCs Problem formulation The closed-loop system in consideration is plotted in Figure 3.3. Augmented plant G is an LTI system ẋ A B u B p B x z z p = C u D u D up E u w C p D pu D p E p w p y K C F u F p u K 25

30 CHAPTER 3. ANALYSIS AND SYNTHESIS OF LTI AND LPV SYSTEMS z w z p G w p y K u K K Figure 3.3: The general -G-K setup in robust control theory whose uncertainty is characterized by the feedback through an unknown mapping w = z, S, where admissible uncertainty set S consists of linear causal systems of finite L 2 -gain and described indirectly by IQC (3.8) in the form 1 [ w (jω) 2π z (jω) ] [ w (jω) Π(jω) z (jω) [ ] Qω (jω) S Π(jω) = ω (jω) S ω (jω) R ω (jω) The controller K is an LTI system, u K = Ky K where [ ] Ac B K = c C c D c ] dω, (3.14) (3.15) Robust quadratic performance of the controlled system is specified by (3.7) with (3.1) and (3.11). The analysis problem is to test the robust quadratic performance condition (3.7) for all admissible uncertainty in set S. The synthesis problem is to find a controller K that renders the closed-loop system robustly stable and satisfies robust quadratic performance for all S Analysis Let the closed-loop nominal system be denoted by M = F L (G,K) and partitioned according to 26 [ z z p ] [ ] Mu M = up M pu M p }{{} M [ w w p ] (3.16)

31 3.3. ROBUST CONTROL OF UNCERTAIN LTI SYSTEMS USING IQCS The state-space matrices of M are affine functions of the controller: A B u B p M := C u D u D up C p D pu D p A B u B p B [ ][ = C u D u D up + I Ac B c I E u C c D c C F u F p C p D pu D p E p ] Σ The evaluation ]) of ([ condition ]) (3.9) for robust performance requires the computation of wp + Σ p. For this reason, Σ p is transformed to frequency-domain ([ w z z p by Parseval-theorem, thus, (3.9) can be rewritten as 1 2π w z w z Q ω S ω Sω R ω Q p S p Sp T R p w z w z dω ǫ ( w 2 2 2π + w p 2 2) Permuting the rows and columns of the extended multiplier, using (3.16) and applying Lemma 2.2, the condition reappears as I Q ω S ω I I Q p S p I M u M up Sω R ω M u M up <, (3.17) M pu M p } Sp T {{ R p } M pu M p P e for all ω R { }. At this point we could finish dealing with analysis and continue with the synthesis section, but analysis results play an important role also in the technical derivation of the synthesis formulas. Frequency-domain analysis result (3.17), however, is directly not applicable in the synthesis problem. Analogous condition in time-domain is required because the controller is parameterized in state-space. These conditions can be obtained by applying Lemma 2.3. Two cases must be distinguished. The one case is when the multiplier characterizing uncertainty is real: Π = P. Then, the time-domain counterpart of (3.17) is the starting point of the synthesis procedure. In the other case with dynamic multipliers, some preceding preparations are necessary for being able to apply KYP lemma: in order to get rid of the frequency dependency of multiplier Π, it is factorized, as in Section 2.3, and the outer factor of (3.17) is multiplied by factor Ψ(jω) to obtain Ψ 1 I Ψ 2 M u Ψ 2 M up M pu M p Q S Q p S p S R Sp T R p Ψ 1 I Ψ 2 M u Ψ 2 M up M pu M p <, (3.18) 27

32 CHAPTER 3. ANALYSIS AND SYNTHESIS OF LTI AND LPV SYSTEMS where Π(jω) = Ψ(jω) PΨ(jω), Ψ = [ Ψ1 Ψ 2 ] [ Q S, P = S R with real matrix P. Although inequality (3.18) can be transformed to time-domain by Lemma 2.3, the resulting formula loses the nice structure which can be well employed in the synthesis procedure presented in the next section. It is shown in Chapter 5 that in the special block diagonal case of Ψ(jω) = diag{ D(jω), D(jω)} with some invertible D(jω), (3.18) can be transformed to the favourable form of (3.17). The analysis results of this section are summarized in the following lemma where the time-domain condition is presented only for the case of Π = P for the above-mentioned reasons. Theorem 3.3 (Analysis inequalities) Suppose that any S satisfies IQC (3.14) for multiplier (3.15). The closed-loop system in Figure 3.3 is robustly stable and satisfies robust quadratic performance defined by (3.7), (3.1) and (3.11) if for all ω R { } with Re{λ(A)} < there holds [ I M(jω) ] P e [ I M(jω) ] <, (3.19) where P e is defined by (3.17). Now, suppose that Π = P, i.e., P e is real. Then, (3.19) holds if and only if there exists X = X T > such that [ ] T [ ] I I P M M <, (3.2) M where P M = X Q S Q p S p X S R Sp T R p ] Testing stability and robust quadratic performance of a given system M is carried out by either searching for multiplier Π that renders (3.19) satisfied, or searching for multiplier Π and a Lyapunov matrix X which render (3.2) satisfied. The former task is potentially performed sequentially over a frequency grid, while the latter requires the solution of one LMI. Both concepts may have legitimacy. Suppose that the performance specification can be parameterized by a real scalar, say γ, expressing, e.g., induced L 2 - gain of the system. The frequency-domain test can be used for the minimization of γ, individually at every frequency point on the grid, to have a picture about the frequency distribution of the achievable performance level. This test does not guarantee the validity of any deduction for all frequencies, only locally at the tested frequency points. On the contrary, test (3.2) provides a global result, but without detailed frequency information. 28

33 3.3. ROBUST CONTROL OF UNCERTAIN LTI SYSTEMS USING IQCS Remark. 3.1 In case of LTI uncertainty (3.19) provides a suitable analysis tool for robust performance: (3.19) is potentially less conservative as compared to (3.2), since it contains the set of IQCs defined by (3.2) as a subset. An equivalent characterization of the time- and frequency-domain forms is discussed in Chapter 5. Regarding controller synthesis, (3.2) can be transformed to equivalent conditions consisting of LMIs of controller data, as presented in the next section Synthesis The robust controller synthesis problem is based on the analysis results: search multiplier, controller parameters and a Lyapunov matrix X > which render (3.2) satisfied. This problem is nonlinear and non-convex in all of the variables and, unfortunately, there exist no efficient solution for nonlinear matrix inequalities of this kind. It has been shown in [87] that, using a nonlinear transformation, conditions X > and (3.2) can be rewritten as where X(v) = M(v) = = P v = X(v) >, [ I M(v) [ ] Y I I X A(v) B u (v) B p (v) C u (v) D u (v) D up (v) C p (v) D pu (v) D p (v) AY A B u B p XA XB u XB p C u Y C u D u D up + C p Y C p D pu D p I Q S Q p S p I S R Sp T R p ] T P v [ I M(v) B I E u E p [ ] Qv S = v Sv T R v ] <, (3.21) [ ][ ] Av B v I C F u F p Thus, instead of X,A c,b c,c c,d c new variables v = [X, Y, A v, B v, C v, D v ] have been introduced. The advantage of this transformation is that new condition (3.21) is quadratic in variables v. The nonlinear transformation can be described by the equations [ ] 1 [ ] Y V I X =, (3.22) [ Ac B c C c D c ] = I [ U XB I C v C v D v D v X U ] 1 [ ] [ ] Av XAY B v V T 1, (3.23) CY I 29

34 CHAPTER 3. ANALYSIS AND SYNTHESIS OF LTI AND LPV SYSTEMS where U and V are arbitrary nonsingular matrices satisfying I XY = UV T. The resulted conditions can be transformed to LMIs. One approach uses the Linearization Lemma [12, Lemma 4.1] which applies Schur complement to get rid of the nonlinear term. Lemma 3.4 (Synthesis by linearization) Suppose that R v can be factorized as R v = H T U(R)H such that H is a constant matrix and U(R) > is an affine function. Suppose that S is constant. Then, the nonlinear constraint (3.21) in variables Q, R and v is equivalent to [ Qv + S X(v) >, U(R) >, v M(v) + M(v) T Sv T M(v) T H T ] HM(v) U(R) 1 <, which is a system of LMIs in the variables. Note, Linearization Lemma keeps all of the variables. Another approach is the elimination of several parameters whose advantage is that the problem reduces to smaller dimensional problems of smaller number of variables, which reduces computation time. A disadvantage is that a nonlinear equality constraint appears between the multiplier Π and its dual variable defined by Π := Π 1. This problem emerges in case of perturbed systems where no information is available on the perturbation (uncertainty) and, usually, solved by iteration similar to what is applied in µ-synthesis. Fortunately, the nonlinear equality constraint disappears when the perturbation is measurable and can be utilized by the controller (LPV scheduling parameters). The control synthesis method using elimination of variables is presented below in more details. Some useful lemmas are enounced first, the proof of them can be found in [12]. Lemma 3.5 (Projection Lemma) For arbitrary A,B and symmetric P, the LMI in the unstructured X has a solution if and only if P + A T XB + B T X T A < (3.24) Ax = or Bx = imply x T Px < or x =. (3.25) If A and B denote arbitrary matrices whose columns form a basis of ker(a) and ker(b) respectively, (3.25) is equivalent to A T PA < and B T PB < Inequality (3.24) is called basic LMI for which numerical solvers have been implemented under Matlab, [13], to provide a feasible solution X. [ ] Q S Lemma 3.6 (Elimination Lemma) Let P = S T with R have the inverse P R [ ] Q S := S T with R Q and A,B,C and X be real matrices of appropriate size. The quadratic inequality [ ] T [ ] I I A T P XB + C A T < (3.26) XB + C 3

35 3.3. ROBUST CONTROL OF UNCERTAIN LTI SYSTEMS USING IQCS has a solution for X if and only if B T [ I C ] T P [ I C ] [ B < and A T C T I ] T [ C T P I ] A > hold true. Proof. Only the main steps of the proof of sufficiency ( ) is presented, because it is constructive and applied when computing the controller parameters. Since R = R T, it can be factorized as R = H T UH with U >. Quadratic inequality (3.26) equals [ I C ] T P [ I C ] + (A T XB) T (S T + RC) + (S T + RC) T (A T XB)+ (A T XB) T H T UH(A T XB) < Since U > the Schur-complement can be applied and so (3.26) is equivalent to [ I C ] T [ ] I P C U 1 [ (S + T + RC) T A T HA T [ B T ] X [ B ] + ] X T [ A(S T + RC) AH T ] < which is in the form of basic LMI (3.24) in Projection Lemma. Note, that Lemma 3.6 can be applied for the second inequality in (3.21). Let Υ denote the basis matrix (matrix of basis vectors) of Ker[C [ F u F p ] and Φ a basis matrix of Ker[B T Eu T Ep T]. Let the inverse of multiplier Qp S p Sp T be denoted by R [ ] p Qp Pp 1 Sp :=. R p S T p Lemma 3.7 (Synthesis inequalities) Quadratic matrix inequality (3.21) [ has a ] solution v if and only if there exist symmetric X, Y and multipliers P = Q S S T and R 31

36 CHAPTER 3. ANALYSIS AND SYNTHESIS OF LTI AND LPV SYSTEMS [ ] Q S P = S T which satisfy R [ ] Y I >, (3.27) I X T I X I Q S Ψ T I Q p S p [ ] A B u B p X Ψ < (3.28) C u D u D up S T R C p D pu D p Sp T R p Φ T A T Cu T Cp T Bu T Du T Dpu T Bp T Dup T Dp T I I I [ ] 1 [ ] Q S Q S S T = R S T R T Y Q S Qp Sp Y ST R ST p Rp [ ] Φ > (3.29) (3.3) Because of nonlinear equality constraint (3.3), the multipliers cannot be obtained together with X and Y at the same time, therefore, an iteration similar to D-K-iteration in µ-synthesis is applied. In one step, multiplier P is fixed and synthesis inequalities (3.27)-(3.29) are solved for X and Y. Once this feasibility problem has been solved, transformed controller parameters A v, B v, C v, D v can be computed according to the proof of Elimination Lemma. Then, original state-space matrices of the controller and Lyapunov matrix X of the closed-loop are calculated based on (3.22)-(3.23). In the second step of the iteration, controller parameters are fixed and the multipliers are obtained by solving the analysis inequality (3.2). 3.4 Control of LPV systems using IQCs Linear parameter varying systems are linear systems where the state-space matrices are scheduled by on-line measurable time-varying parameters. LPV systems are able to model nonlinear and/or time-varying dynamics as well, which is a reason why LPV modelling is favored and widely used in control design [52, 86, 15, 16, 17]. In synthesis and analysis this approach uses LMIs for which efficient numerical algorithms appeared [94] and made it possible to guarantee global stability and fulfill performance requirements on the whole region of operation. One of the research paths uses Lyapunov methods and gridding techniques for LPV systems with general parameter dependence [17, 134, 76, 18]. Another path of research is preferred in the thesis. It involves LPV systems and controllers whose parameter-dependence can be expressed as linear fractional transformation [7, 12]. This approach relies on scaled small-gain methods and has the 32

37 3.4. CONTROL OF LPV SYSTEMS USING IQCS s z s w s z p G w p y K u K K w c z c c Figure 3.4: Closed-loop nominal LPV LFT control configuration advantageous property that a frequency-domain characterization of robust performance is available allowing the formulation of criterion for frequency-domain uncertainty modelling methods. This section presents, based on Ref. [12], the main steps of induced L 2 -gain control synthesis for nominal LPV systems with LFT dependence on scheduling parameters. The concept is similar to that in the previous section, but the synthesis problem here comes convex due to the additional information on the perturbation, so it does not require an iterative solution Problem formulation The closed-loop system in consideration is plotted in Figure 3.4. Augmented plant G is an LTI system ẋ z s z p y K = A B s B p B C s D s D sp E s C p D ps D p E p C F s F p The scheduling parameter dependence is characterized by a time-varying on-line measurable scheduling matrix s as w s (t) = s (t)z s (t), s S s, where admissible scheduling parameter set S s is defined by the convex polytope x w s w p u K S s := { s (t) : R R nws n zs, s (t) co{ s1,..., sκ }} (3.31) with some fixed generator matrices sj. 33

38 CHAPTER 3. ANALYSIS AND SYNTHESIS OF LTI AND LPV SYSTEMS z s s c ( s ) w s z c G a w c z p G w p y K u K w c K z c Figure 3.5: -G-K reformulation of the closed-loop LPV system The controller K is an LPV system with LFT dependence on scheduling parameter c ( s ): ẋ c [ ] x u Ac B c = c y, C z c D c c w c w c (t) = c ( s (t))z c (t) Scheduling parameters of the controller depend on the measured scheduling parameters of the system. Robust quadratic performance of the controlled system is specified by (3.7) with (3.1) and (3.11). Given an LPV controller, the analysis problem is to test condition (3.7) for all admissible scheduling parameters s S s. The synthesis problem is to find a controller K with function c that renders the closed-loop system internally stable and satisfies robust quadratic performance (3.7) for all s S s Analysis The system in Figure 3.4 is redrawn into -G-K form for analysis, see Figure 3.5. The closed-loop system can, alternatively, be described by scheduling augmented LTI system G a 34

39 3.4. CONTROL OF LPV SYSTEMS USING IQCS ẋ z s z c z p y w c = A B s B p B C s D s D sp E s I C p D ps D p E p C F s F p I } {{ } G a x w s w c w p u z c and augmented perturbation a [ ws w c ] = [ s ] } c ( s ) {{} a [ zs z c ], In section 3.1 a numerically tractable characterization of polytopic parameter set S s was introduced: set S P defined in (3.6) consists of multipliers [ satisfying ] quadratic Qs S constraints at the vertices of the polytope. Let P s = s S P denote the multiplier associated with scheduling matrix s S s. For the characterization of augmented perturbation a, we augment multiplier P s as S T s R s [ Qa S P a := a S T a R a ] = Q s Q 12 S s S 12 Q 21 Q 22 S 21 S 22 R s R 12 R 21 R 22, with Q a <, R a > (3.32) With c ( s ) = the parametrization with full-block multiplier P a renders the IQCs to the polytopic set S s. Equipped with P a we turn back to the general characterization (3.4) of LTV perturbations where multiplier P a must satisfy s c ( s ) I I T P a s c ( s ) I I > for all s S s (3.33) It is shown in Section that this condition can be guaranteed by a finite set of LMIs and an appropriate choice of scheduling function c (). The state-space matrices of closed-loop nominal system M = F L (G a,k) are affine 35

40 CHAPTER 3. ANALYSIS AND SYNTHESIS OF LTI AND LPV SYSTEMS functions of the controller. M := = A B s B c B p C s D s D sc D sp C c D cs D c D cp C p D ps D pc D p A B s B p C s D s D sp C p D ps D p + B I E s I E p [ Ac B c C c D c ] I C F s F p I The following theorem states the conditions of robust quadratic performance of the closed-loop system. Theorem 3.4 (Analysis inequalities) The closed-loop system in Figure 3.5 is exponentially stable and quadratic performance defined by (3.7), (3.1) and (3.11) is satisfied if there exists a Lyapunov matrix X = X T > and multiplier P a with (3.32)-(3.33) satisfying (3.2) where P M = X Q s Q 12 S s S 12 Q 21 Q 22 S 21 S 22 Q p S p X Ss T S21 T R s R 12 S12 T S22 T R 21 R 22 Sp T R p General LPV systems cannot be transformed to frequency-domain. The IQC approach, however, have the advantage that stability and performance of LFT LPV systems can be analyzed in the frequency-domain as well, by applying KYP lemma for (3.2), once the parameter dependence is described by multipliers. This fact is exploited in Chapter 5 where robust performance criterion of the control design is minimized also by variables defined in the frequency-domain Synthesis The similar elimination procedure can be applied in the synthesis of LPV controllers as can be carried out for robust controllers, see Lemma 3.6. The resulted conditions are the same with one important exception: due to the augmented structure of nominal plant G a, the additional rows and columns of augmented multiplier P a are multiplied by zero in the projected LMI conditions. As a consequence, the synthesis of the LPV problem leads to the same inequalities as in the robust LTI problem, but without the nonlinear equality constraint. 36

41 3.4. CONTROL OF LPV SYSTEMS USING IQCS Theorem 3.5 (Synthesis inequalities) The following statements are equivalent. There exists a controller K and a scheduling function c ( s ) such that the controlled system as described in Figure 3.5 admits a Lyapunov matrix and a multiplier (3.32)-(3.33) that satisfy (3.2) with M and P M defined in Theorem 3.4. There exist X, Y and multipliers P S P and P S P with [ Q S P = { P S = S T R ] R >, [ I T j ] T [ P I T j ] <, j = 1,...,κ}(3.34) that satisfy the LMIs of the form (3.27)-(3.29), where index u of state-space matrices are replaced by s. Due to the elimination of controller parameters one does not need to take care on the augmented multiplier P a and the infinite dimensional constraint (3.33) when calculating a feasible solution in the projected variables. Instead, a finite set of LMIs has to be solved. Once the feasibility problem has been solved for X,Y,P and P, multiplier P a and scheduling function c ( s ) are constructed as in the proof of Theorem 5.4 in [12]. The extension of the multiplier is performed so that the following conditions must be satisfied: [ ] 1 [ ] P P =, Q a <, with the notation [ ] I P = I I I R a > Q s Q 12 S s S 12 Q 21 Q 22 S 21 S 22 R s R 12 R 21 R 22 } {{ } P a I I I I With full knowledge of P a, c is constructed to satisfy (3.33) as follows. Constraint (3.33) is equivalent to U 11 U 12 (W 11 + ) T W T 21 U 21 U 22 W12 T (W 22 + c ( )) T W 11 + W 12 V 11 V 12 > W 21 W 22 + c ( ) V 21 V 22 by Schur-complement where U = R a S T a Q 1 a S a > V = Q 1 a > W = Q 1 a S a 37

42 CHAPTER 3. ANALYSIS AND SYNTHESIS OF LTI AND LPV SYSTEMS One possible solution is c ( ) = W 22 + [ W 21 V 21 ] [ U 11 (W 11 + ) T W 11 + V 11 ][ U12 W 12 ] Finally, controller parameters A c, B c, C c, D c can be computed in the same way as in case of robust controllers in the previous section. 3.5 Summary The object of this chapter is to provide methods regarding analysis and synthesis of linear systems with structured uncertainty or parametric scheduling. The analysis conditions provide the criterion of uncertainty modelling in the remaining chapters while the synthesis methods are included as parts of the iterative control design algorithms. The abstract performance theorem 3.2 establishes a general approach for studying robust performance of perturbed systems. The concept is that performance is specified by one IQC (3.7) and the perturbation set is characterized by a set of IQCs, as large as possible, all satisfying (3.8). For certifying robust performance of the perturbed system, one has to find one IQC in the set that satisfies the joint IQC (3.9). Based on this scheme many control problems can be analyzed in both time- and frequency-domain. Two of them has been detailed: LTI systems with structured dynamic perturbation (sections 3.2 and 3.3) and LPV systems with polytopic LFT parametrization (Section 3.4). Controller synthesis is performed by the following general scheme: take the analysis inequalities in time-domain (of form (3.2)) and apply a nonlinear transformation to get rid of terms with both controller parameters and Lyapunov matrices and arrive to a quadratic matrix function (3.21) in the controller parameters; then, eliminate controller parameters using the Elimination Lemma 3.6 and solve the inequalities for the projected variables. The construction of the controller matrices needs the solution of a basic LMI (3.24). In case the perturbation can be measured, this additional information can be utilized to schedule the controller and, furthermore, a nonlinear constraint disappears and the synthesis constraints reduce to a system of LMIs. The analysis and synthesis methods for the combined perturbation structure of uncertainty and gain-scheduling parameters are presented in Chapter 5. 38

43 Chapter 4 Uncertainty modelling and robust control design for LTI systems The analysis and synthesis procedures of robust controllers summarized in Sections 3.2 and 3.3 start from an a priori fixed augmented nominal model (denoted by G in Figure 3.3). So far we have not discussed on how to obtain this augmented model. A typical procedure starts with the nominal plant model (denoted by U 23 in the subsequent sections) that can be a result of some identification method or physical modelling. Based on physical considerations, the nominal model is augmented to describe the relation to neglected dynamics and effects of disturbances (thus we get U of subsequent sections). The structure of neglected dynamics is also specified by defining S u. The most risky task is to specify possibly frequency-dependent bounds for the perturbation blocks (denoted by W ) and disturbances (denoted by W d ). Without a careful choice of the uncertainty bounds, the set of assumed uncertainties can be too large causing conservatism of the designed controller, or too small bearing the risk of closed-loop instability. Determination of disturbance sets is a similarly effortful task and requires much physical insight into the system and the control problem. The next step is to further augment uncertain system U to define the closed-loop interconnection. New performance inputs (e.g. reference signals) can be added and performance outputs, which should be small, must be defined. All new inputs are normalized by weighting functions which are joined to augmented system G. The next effortful task is to chose penalizing weighting functions for the performance outputs. These weighting functions influence e.g. control effort and its frequency power distribution, tracking error and its frequency distribution, disturbance attenuation etc. In this way the control objectives and the model-reality relation can be specified and built into the generalized plant G. There exist approaches supporting the selection of performance weighting functions when all other parameters are fixed, see e.g. [74], however, the design problem of structured uncertainty weighting functions W,W d is not automated yet. The modelling of uncertainty can be one of the most time-consuming task: it involves many experiments, trial and error settings of all weighting functions. In this chapter a design method is presented for uncertainty weighting functions with the assumption of fixed performance weighting functions. The presented algorithm can drastically decrease the necessary intuitive and heuristic design steps. 39

44 CHAPTER 4. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LTI SYSTEMS A natural idea for determining uncertainty sets is to use measurement data. One possibility to support weight selection is based on model validation methods [18, 123, 25, 28, 79]. One approach is that several weighting functions are fixed (e.g. by assuming known disturbance bounds) and the others of equal bounds are tuned to get a consistent model with the given data. Many approaches appeared in the literature to identify unstructured uncertainty sets applicable for robust control. Recently, the criterion of modelling is connected to the achievable control performance. In the chapter, a concept is presented for modelling structured uncertainties directly applicable for the synthesis techniques of the former chapter. No a priori information is assumed on the disturbances. Both W and W d are designed based on a criterion which is exactly the same as for the control design. The inputs of the presented algorithm are the nominal model of the system, the structure of the uncertainty model, and the specification of robust performance. The algorithm is to deliver the weighting functions of the uncertainty model and disturbances, and a controller satisfying robust performance specifications. The overall algorithm reveals a double iterative solution. Since the design variables are involved in a non-convex optimization problem both in robust output-feedback control and in joint modelling and control problems, the search for the variables is usually solved by iteration of convex steps. These steps compose the inner iteration loop. The outer loop involves measurement data of experiments. New controller may generate data which invalidate the actual model. A controller based on an invalid model may destabilize the plant. For that reason new experiments are required to test new controllers and to gain more information on the system. The proposed method is built on the following three pillars: the uncertainty set is constrained by model validation conditions the uncertainty set is shaped based on robust performance specifications the uncertainty set and the robustness of the controller are iteratively improved based on new experiments The first two items serve to ensure the minimality and optimality of uncertainty sets when a controller is given. The third item is responsible for handling unstable experiments and improving robustness. A by-product of the algorithm that may have a self-contained value is the elaboration of a skew µ synthesis procedure that can easily be implemented based on existing µ synthesis methods available in Matlab [13]. Note that skew µ analysis tools are already available in Skew Mu Toolbox for Matlab, [41]. A simple numerical example presented in Appendix A is referred to in this chapter in order to demonstrate the key steps of the proposed algorithm through analytic calculations. 4

45 4.1. PROBLEM FORMULATION 4.1 Problem formulation System configuration The true plant to be controlled and which generates the data is described by operator T y = T ( d,u), T S T, d S d L 2, (4.1) where y and u denote the measured outputs and control inputs, respectively, d is a physical disturbance vector of an undetermined set and physical system T is a multivariable stable system affected by variations of physical parameters and operating point changes which is expressed by the undetermined set of stable systems S T. No a priori information is assumed on the size of the disturbances and the variation of the system s dynamics. Note that T is not required to be linear, it is only assumed that in finite z u w u ŷ U = [ ] U11 U 12 U 13 U 21 U 22 U 23 w u d W W d d u Figure 4.1: Uncertain system model time experiments the input-output behavior can be described with the help of a fictive disturbance d L n d 2 and by an LFT of a given LTI model U RH nzu +n y n wu +n d +n u and a structured dynamic uncertainty in the form [ ] [ ] w zu U11 U = 12 U u 13 d ŷ U 21 U 22 U, w u = z u, (4.2) 23 }{{} u U see also Figure 4.1. The unknown dynamics S u belongs to a set of stable perturbations S u = { RH nwu n zu = diag{ 1,..., τ }, i RH nw u,i nz u,i, i = 1,...,τ}(4.3) Block U 23 RH ny nu is the nominal model, which can be the result of an identification method for restricted complexity models. Other blocks U ij describe the interconnection structure of the nominal model and the structured uncertainty. The uncertainty can be normalized by weighting functions W = diag{w,1 I wu,1,...,w,τ I wu,τ } RH nwu n wu W d = diag{w 1,...,w nd } RH n d n d 41

46 CHAPTER 4. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LTI SYSTEMS such that for all S u there exists a perturbation and for all d L 2 there exists a normalized disturbance d such that = W, BS u, d = W d d, d BL 2 The feedback signals w u := z u = [w u,1,...,w u,τ ] T and z u = [z u,1,...,z u,τ ] T are partitioned according to block structure S u Experiments and model consistency Information about the true system is gained by taking experiments: the system is excited by input signal u(t) which can be either a predefined signal (open-loop experiment) or the output of a controller (closed-loop experiment). The inputs and outputs of the true system are measured and stored. We assume that the sampling is fast enough to have a good sampled data representation of the signals, i.e., the Shannon conditions hold. Then, Discrete Fourier Transform (DFT) can be applied, possibly with a window function, to compute the frequency-domain representation of the signals. Many textbooks are dealing with the topic of sampling and Fourier transform, in the sequel we does not concern signal processing problems of this kind. Suppose that the input-output data of system T in the lth experiment are given in the frequency domain by E l = {(y l (jω k ),u l (jω k )) nω k=1 }, where n ω is the number of frequency samples. The data containing N experiments are denoted by E N = N l=1 E l. Definition 4.1 (Consistency) The uncertain system (4.2) with weighting functions W and W d is consistent with respect to data set E l if the following condition holds. There exists a perturbation l BS u and a disturbance d l BL n d 2 such that measured output y l is exactly reproduced by model output ŷ l, i.e., for all frequencies k = 1,...,n ω ŷ l (jω k ) = y l (jω k ) (4.4) where [ Wd (jω ŷ l (jω k ) := F U (U(jω k ),W (jω k ) l (jω k )) k )d l (jω k ) u l (jω k ) ] The model is unfalsified if it is consistent with all experiments E N. It is required of the model structure to be able to represent the true system in all experiments. Certain rank conditions on system U, detailed in Section 4.2, ensure the satisfaction of Assumption 4.1 There exist a W and a W d such that the model is consistent with all experiments. 42

47 4.1. PROBLEM FORMULATION z d p r G T K y K u K Figure 4.2: Closed-loop system configuration with true plant Control performance In modern control theory, the requirements on the control performance are defined by drawing the closed-loop control configuration. New inputs, outputs and weighting functions are introduced to build an augmented closed-loop system whose input-output mapping is to minimize in some chosen norm. In order to specify the objectives of the design, true system T is augmented and placed in a closed-loop configuration as in Figure 4.2. New inputs, denoted by r, containing known or measurable signals, e.g., reference inputs, can be added. Performance outputs, denoted by z p, which should be small during the control, must be defined. Signals y K and u K are defined as the inputs and outputs of controller K. They are not necessarily equal to y and u, respectively. Augmented plant G T contains system T and weighting functions, which specify performance signal z p and normalize outer signals r and d. The control objective is to minimize γ(k) := sup z p 2, z p = F L (G T,K)( d, r). (4.5) T S T, d S d, r BL 2 Since true plant T and sets S T and S d are not known in advance, the controller will be designed based on model (4.2) which must be augmented in the same way, resulting in augmented plant G and the configuration in Figure 4.3. Without loss in generality, system G and input r can be scaled such that we can calculate with wp T := [dt r T ] BL n d+n r 2 instead of d BL n d 2, r BLnr 2. In order to minimize the control objective (4.5) by calculations based on model (4.2), we will need the following assumption on performance specification. Assumption 4.2 The consistency of the model (4.2) with respect to closed-loop data (y l (jω k ),u l (jω k )) implies ([ the consistency ] [ of the augmented ]) plant model in Figure 4.3 zp,l (jω with respect to data k ) rl (jω, k ) and that z y K,l (jω k ) u K,l (jω k ) p,l (jω k ) = z p,l (jω k ). Assumption 4.2 expresses that plant augmentation preserves the consistency property. We can summarize the problem statement as follows. Problem 1. Given an unknown system T, an LTI plant model U of fixed structure for uncertainty and control specification defined by G. Provide an iterative uncertainty model unfalsification and robust control design procedure that 1. constructs a consistent 43

48 CHAPTER 4. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LTI SYSTEMS z u w u w u W z p G d W d d r } wp y K u K K Figure 4.3: Closed-loop system configuration with the model for robust control design model (U,W,W d ) for the true plant, based on closed-loop experiments, and 2. designs a H controller K that minimizes objective (4.5). 4.2 Characterization of uncertainty In this section all consistent models are parameterized with respect to given measurement data. The frequency-domain model validation technique and affine parametrization given in [123] are applied for the constant matrix problem at a certain frequency point. Then consistency constraints are defined for the normalizing weighting functions of the model. Consistency equation (4.4) is satisfied if and only if there exist w u,l (jω k ) and d,l (jω k ) solving e l (jω k ) = [ U 21 (jω k ) U 22 (jω k ) ] [ w u,l (jω k ) d,l (jω k ) where nominal model error is defined by e l (jω k ) := y l (jω k ) U 23 (jω k )u l (jω k ). For solvability of the equation, the nominal model error must satisfy e l (jω k ) Im{ [ U 21 (jω k ) U 22 (jω k ) ] }, which ensures the satisfaction of Assumption 4.1. In order to avoid trivial solutions dim(ker[u 21 (jω k ) U 22 (jω k )]) > is also assumed. In this case all solutions can be formalized as [ wu,l (jω k ) d,l (jω k ) ] [ [ = U21(jω k ) U 22(jω k ) ] [ Bu (jω k ) B d (jω k ) ], ] ][ el (jω k ) [ ] Bu (jω where k ) forms a basis for the kernel of [ U B d (jω k ) 21 (jω k ) U 22 (jω k ) ] and θ lk C nwu +n d n y is any free parameter at frequency ω k and experiment l. The parametrization of weighting functions in W and W d can be given indirectly through parameters θ lk and some inequality constraints as follows. 44 θ lk ],

49 4.3. ROBUST PERFORMANCE CRITERION Theorem 4.1 Given stable and stable invertible weighting functions W and W d as defined in Section and given the set E N of measurement data. Then, for every experiment l = 1,...,N there exists a perturbation l BS u and a disturbance d l BL n d 2 that satisfy consistency condition (4.4) for all k and l, if and only if there exist θ lk k = 1,...,n ω and l = 1,...,N such that W,i (jω k ) w u,l,i(jω k,θ lk ), i = 1,...,τ (4.6) z u,l,i (jω k,θ lk ) W d,i (jω k ) d,l,i (jω k,θ lk ) i = 1,...,n d (4.7) The theorem follows from the results of [79, Lemma 3], [25] and [28]. The solutions W and W d of Problem 1 are characterized by set Θ N := {θ lk C n θ, l = 1,...,N, k = 1,...,n ω } with n θ = n wu +n d n y and constraints (4.6) and (4.7). Whenever parameters θ lk are fixed, any low-order stable, stable invertible weighting functions W,i (jω k ) or W d,i (jω k ) can be chosen that tightly over-bound in magnitude the right hand sides of (4.6) and (4.7), respectively. The weighting functions determined by the choice of parameters θ lk, l = 1,...,N, k = 1,...,n ω and the over-bounding method are denoted by W (Θ N ) and W d (Θ N ). For simplifying notation W := diag{w,w d,i} is introduced where the dimension of the unity matrix I depends on the context. The nominal closed-loop system M := F L (G,K), the weighted nominal closed-loop system M := M W and the weighted augmented plant G := G W are defined where W is of appropriate size. An uncertainty model which is consistent with experiments E N can be represented by W(Θ N ). For a simple example on the parametrization of weighting functions, see Section A.2 in Appendix A. The following section defines the robust performance measure which is a criterion for tuning the weighting functions and designing the controller. 4.3 Robust performance criterion Worst case performance level γ(k) defined on the true closed-loop system cannot be computed, since S T and S d are not known. A computable upper-bound of γ(k) will be derived in this section based on model (4.2). Define the worst-case performance level based on the model as γ s (K,W) := sup z p 2 = sup F L (F U (G W, ),K). w p BL 2, BS u BS u If the model is consistent with the true plant and Assumption 4.2 holds, then there exist w p BL 2, BS u for all T S T, d S d such that z p = z p. As a result, we can conclude Lemma 4.1 If W is consistent with the data of all experiments with controller K, then γ s (K,W) γ(k). 45

50 CHAPTER 4. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LTI SYSTEMS Remark. 4.1 A more stringent statement also holds: If W is consistent with the data of an experiment, say E l, generated in closed-loop by controller K, then γ s (K,W) z p,l 2. This is also a simple consequence of Assumption 4.2 and definition of γ s (K,W). Clearly, we are interested in the smallest upper-bound denoted by γ s (K) := inf γ s (K,W), sought among all consistent models. For the purpose of computing the robust performance of systems with consistent uncertainty models, the bounds for perturbations must be fixed to 1, since the weighting functions capture the size information. The measure that scales only the performance channel is the skewed SSV introduced in Section 3.2. Given a controller K, the sought upper-bound of γ(k) is γ s (K) := γ, where γ is the solution of the following optimization problem. subject to γ γ s (jω), ω min γ (4.8) D,W,γ s(jω) σ(d L (jω) F L (G (jω),k(jω)) D 1 R (jω) W(jω) diag{γ s(jω)i wu,i nd +n r }) < γ s (jω), where W satisfies (4.6) and (4.7) in all experiments with K and D,D L,D R are defined in Section 3.2. In the iterative algorithm presented in the next section, the theoretic consistency constraint on W must be replaced by the one based on a finite set of experiments performed so far. In each stage of the iteration new information on the system is gained by means of new closed-loop experiments with the actual controller. It turns out that a new experiment is always such that it either remains consistent with the actual model and holds the achieved (a priori) performance bound or has a worst performance, and in the same time, falsifies the uncertainty model. This fortunate property is due to the chosen criterion skew µ, since BS u is guaranteed by both consistency and the control design. This is justified by the following theorem: Theorem 4.2 Given experimental data set E N and a controller K N designed for the consistent model characterized by W(Θ N ). The guaranteed performance level skew µ is γ s (K N ). A new experiment is performed with the controller on the true system defined in (4.1). The gathered data set is E N+1, and the realized performance γ N+1 := z p,n+1 2. Then, the following implications hold. 1. W(Θ N ) is consistent with E N+1 = γ N+1 γ s (K N ) 2. γ N+1 > γ s (K N ) = W(Θ N ) is not consistent with E N+1. Proof. The first implication follows from Lemma 4.1. For proving the second one, suppose, in contrast to the claim, that z p,n+1 2 > γ s (K N ) and W(Θ N ) is consistent with E N+1. Then, there exists a parameter set Θ N+1 such that W(Θ N+1 ) = W(Θ N ) and, for all k = 1,...,n ω, ŷ N+1 (jω k ) = y N+1 (jω k ). By Assumption 4.2, z p,n+1 (jω k ) = z p,n+1 (jω k ) also holds. But bound (4.8) guarantees for the model with W(Θ N ) that z p,n+1 (jω k ) γ s (jω k ) γ s (K N ), which contradicts the initial supposition. 46

51 4.4. AN ITERATIVE ALGORITHM FOR UNCERTAINTY MODELLING AND CONTROL DESIGN The first implication expresses that if a new experiment does not falsify the model with W(Θ N ), then the performance of the control is, as expected, below the guaranteed level γ s (K N ). The second implication states that if the realized performance in the new experiment exceeds the guaranteed level, then the model is necessarily falsified. Based on these statements the following suggestions can be offered. According to implication 2., the performance should be measured during experiments. A degradation of performance by an unacceptable level can, generally, be noticed before instability would damage the system. In this case the experiment should be halted and the measured data can be used to improve the model. Consistency of the model should always be checked after experiments. Note that both implications of the theorem holds with classical criterion µ only if β := µ a (M) 1, and it provides a conservative performance measure if β < 1. The reason is that performance is guaranteed for 1 β BS u. If β > 1, a valid model may destabilize the system or an experiment of poor performance may be consistent with the actual model. 4.4 An iterative algorithm for uncertainty modelling and control design In the following iterative algorithm basically two aims of opposing directions are integrated. Aim 1. is to increase robustness based on new data by decreasing the set of unfalsified models (increasing uncertainty weighting functions), which implies the degradation of achievable performance. Aim 2. is to chose an optimal unfalsified model, thus minimizing the robust performance level. Actually, all steps in D-K-W iteration minimize the upper-bound (4.8) of performance criterion skew µ. The subsequent sections detail the steps of the algorithm. Algorithm 1. A. Initialization B. D-K-W iteration 1) Scaled H controllersynthesis(k) 2) Optimization of uncertainty model(w) 3) Skew µ analysis by finding scalings(d) C. Closed-loop experiments D. Update of the uncertainty model Step A. Initialization Let N = 1. Initial data set E 1 may consist of open- and closed-loop experiment data. Construct initial uncertainty bounds W(Θ 1 ). Let D 1 L = I, D1 R = I. 47

52 CHAPTER 4. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LTI SYSTEMS Step B. D-K-W iteration The following three steps are iterated until the improvement of performance level γ s (K N ) is less than a given number in the last iteration steps of a given number. During the iteration only the stable controller with improved performance is stored. For notational brevity index N is omitted. Step B1. Controller synthesis Let the generalized plant be denoted by G D := D L G WDR 1. A scaled γ iteration is applied where in each step the solvability of F L (G D diag{γi wu,i},k) < γ is tested by searching for an appropriate controller. This is a bisection algorithm with standard H controller synthesis problem for scaled plant G D diag{γi wu,i}. Step B2. Optimization of the uncertainty model Let M D := F L (D L G DR 1,K) denote the generalized closed-loop model with scalings. Let v ik >, i = 1,...,τ + n d, be the candidate magnitudes of the uncertainty weighting functions at frequency ω k v ik = W,i (jω k ), i = 1,...,τ, (4.9) v τ+i k = W d,i (jω k ), i = 1,...,n d, (4.1) and let V I,k = diag{v 1k I wu,1,...,v τk I wu,τ,v τ+1 k,...,v τ+nd k,i r }. For each frequency point ω k, k = 1,...,n ω, minimize γ k subject to σ( M D (jω k )V I,k diag{γ k I wu,i nd +n r } ) < γ k, (4.11) v ik w u,l,i(jω k,θ lk ), i = 1,...,τ, l = 1,...,N (4.12) z u,l,i (jω k,θ lk ) v τ+i k d,l,i (jω k,θ lk ), i = 1,...,n d, l = 1,...,N (4.13) in variables θ lk, l = 1,...,N, and v ik, i = 1,...,τ + n d. Then, point-wise bounds v ik are over-bounded by stable minimal-phase transfer function by using e.g. the log-chebychev magnitude design algorithm of the reference [12, fitmagfrd.m]. The orders of the transfer functions are increased until the fitted weighting functions, denoted by W, satisfy γ s (jω k ) < εγ k, k = 1,...,n ω, where ε > 1 is a given tolerance and γ s (jω k ) := ρ(z 22 (jω k ) Z 12 (jω k ) (Z 11 (jω k ) I wu ) 1 Z 12 (jω k )), [ ] Z11 Z Z = 12 Z12 := W M Z DM D W, 22 For the existence of γ s (jω k ), Z 11 (jω k ) < I wu must hold. To ensure this a weighting w fit (ω k ) := ρ(z 11 (jω k )) is applied in the fitting procedure in order to force higher accuracy at frequencies where ρ(z 11 (jω k )) is near to 1. 48

53 4.4. AN ITERATIVE ALGORITHM FOR UNCERTAINTY MODELLING AND CONTROL DESIGN Step B3. Skew µ analysis by finding the scalings Let M := F L (G W,K), Minimize γ k frequency-wise subject to D L,k := diag{d 1k I zu,1,...,d τk I zu,τ,i z }, D R,k := diag{d 1k I wu,1,...,d τk I wu,τ,i nd +n r }. σ( D L,k M(jω k )D 1 R,k diag{γ ki wu,i nd +n r } ) < γ k in variables d ik >, i = 1,...,τ. This convex problem is equivalent to a generalized eigenvalue problem. Then, point-wise scalings d ik are fitted in magnitude by stable, stable invertible transfer functions where the selection of order and the weighting of the fitting criterion are similar to those in the previous step. Step C. Closed-loop experiments In reference [131] the iterative control design schemes are classified as stability assured (all experiments are stable), instability tolerant (all experiments are stable after an a priori known iteration number) and stability uncertain (all experiments are stable after an a priori unknown iteration number). It is shown in [131] that the first two properties can be achieved only if strict a priori assumptions are made on the system. The approaches based on ν-gap metric, provide schemes with guaranteed stability, but the system generating the data is not allowed to change its dynamics, i.e. set S T is a singleton. This is not the case in this chapter. We can, however, borrow the idea of Ref. [13] on handling instability and saturation. Assumption 4.3 Saturation levels U and Y can be defined such that, for all t, any i = 1,...,n u and any j = 1,...,n y, u i (t) > U or y j (t) > Y infer the degradation of the performance beyond a certain level c: z p 2 > c. Closed-loop experiments are carried out by monitoring the saturation conditions. As soon as one of them is satisfied, the possibly destabilizing controller (K N ) is switched off and, instead, a known stabilizing controller or control input is switched on for a period long enough for the transients to decay. After this period K N can be switched on again. Theorem 4.2 guarantees that data E N+1 of the experiment invalidates model W(Θ N ), the weighting functions will be increased and in the next iteration the robustness of the controller will be improved. Practically, it can be useful to perform the first few iterations on a high-fidelity simulator program, while there is chance for saturation. Step D. Update of the uncertainty model The consistency of model W(Θ N ) with E N+1 is checked as follows. Solve the following optimization problem point-wise for each ω k. Let v ik := W,i (jω k ), i = 1,...,τ and 49

54 CHAPTER 4. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LTI SYSTEMS v ik := W d,i (jω k ), i = 1,...,n ω. Minimize γ k subject to v ik + γ k w u,n+1,i(jω k,θ N+1 k ) z u,n+1,i (jω k,θ N+1 k ), i = 1,...,τ, v τ+i k + γ k d,n+1,i (jω k,θ N+1 k ), i = 1,...,n d, If for all k = 1,...,n ω γ k holds, then, the algorithm is finished with controller K N uncertainty model W(Θ N ) and performance level γ s (K N ). Otherwise, let W(Θ N+1 ) be constructed by the over-bounding method presented in Step B2 of Algorithm 1 for points v ik + max(,γ k ). Let E N+1 = E N E N+1, N = N + 1, and continue from Step B. 4.5 Solutions with convex optimization An important special case of uncertainty model structure is when U 12 = and U 11 is block-diagonal or zero. In this case signal z u is independent on parameters θ lk and, therefore, (4.6) and (4.7) define LMI constraints. To show this first suppose that U 11 is block-diagonal, i.e., w u,i =,i z u,i z u,i = U 11,i w u,i + U 23,i u for i = 1,...,τ. It follows that w u,i =,i (I U 11,i,i ) 1 U 23,i u. It has been assumed that and U are stable systems, consequently, a new additive uncertainty block can be introduced without loss in generality: Hence U 11 = can be assumed.,i :=,i (I U 11,i,i ) 1 d W d w u W w u z u d U 21 U 22 U 13 y + + U 23 u Figure 4.4: Additive uncertain system model Let the particular solution of consistency equation (4.4) be denoted by [ au,lk a d,lk ] := [ U 21 (jω k ) U 22 (jω k ) ] el (jω k ) 5

55 4.5. SOLUTIONS WITH CONVEX OPTIMIZATION Define real diagonal matrices V k = diag{ W (jω k ), W d (jω k ) } R nwu +n d n wu +n d, (4.14) diagonal of complex vectors { } au,lk,1 A lk = diag z u,l,1 (jω k ),..., a u,lk,τ z u,l,τ (jω k ),a d,lk,1,...,a d,lk,nd C nwu +n d τ+n d, and diagonal of complex matrices { } Bu,1 (jω k ) B lk = diag z u,l,1 (jω k ),..., B u,τ (jω k ) z u,l,τ (jω k ),B d,1(jω k ),...,B d,nd (jω k ) C nwu +n d n θ (τ+n d ), and let Θ lk = diag{θ lk,...,θ lk } C n θ(τ+n d ) τ+n d be the τ + n d times repeated blockdiagonal matrix. The consistency conditions formulated in Theorem 4.1 are provided for the case of additive uncertainty structure of Figure 4.4 in the following theorem. Theorem 4.3 Given stable and stable invertible weighting functions W and W d as defined in Section and given set E N of measurement data. Then, for every experiment l = 1,...,N there exists a perturbation l BS u and a disturbance d l BL n d 2 that satisfy consistency condition (4.4) for all k and l, if and only if there exist θ lk k = 1,...,n ω and l = 1,...,N such that [ Vk 2 (A lk + B lk Θ lk ) ] (4.15) A lk + B lk Θ lk I nwu +n d In Step B2 of Algorithm 1, constraints (4.12) and (4.13) should be replaced by (4.15). Note that V k defined by (4.14) is nothing but V k = diag{v 1k I wu,1,...,v τk I wu,τ,v τ+1,...,v τ+nd }, (4.16) where v ik > are the candidate magnitudes of the uncertainty weighting functions at frequency ω k. Recall that V I,k = diag{v k,i r } has already been defined in Step B2 of Section 4.4. Criterion (4.11) can always be rewritten to the complex LMI of the form [ ] Inw γk 2I VI,k 2 VI,k 2 M D(jω k ) n d +n r > (4.17) M D (jω k )VI,k 2 I by applying Schur complement. Minimization of γk 2 subject to constraints (4.15) and (4.17) is a generalized eigenvalue problem in variables θ lk, vik 2 and γ2 k. Step D of Algorithm 1 (update of the uncertainty model) gives also a convex problem. The consistency of model W(Θ N ) with E N+1 is checked as follows. Solve the following optimization problem point-wise for each ω k. Let v ik := W,i (jω k ), i = 1,...,τ and v ik := W d,i (jω k ), i = 1,...,n d. Minimize γ k subject to [ Vk 2 + γ2 k I (A N+1 k + B N+1 k Θ N+1 k ) ] A N+1 k + B N+1 k Θ N+1 k I nwu +n d 51

56 CHAPTER 4. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LTI SYSTEMS Finally, we can conceive theoretical conditions for the model based upper-bound γ s (K) being equal to the true performance level γ(k). For avoiding exactness problems due to sampling, and for simplifying notation, the conditions are stated in terms of continuous-frequency representation. Suppose that experimental data consist of inputoutput pairs (y(jω), u(jω)), ω R, of the worst-case experiment E wc = {(y(jω),u(jω)) (T, d, r) = arg sup z p 2 } T S T, d S d, r BL 2 i.e. the measured performance of the worst-case experiment equals to γ(k). Theorem 4.4 Given a controller K and closed-loop experimental data set E l = E wc generated by K. Suppose, the structured uncertainty has at most two blocks: u S u with τ 2, and the uncertainty structure is additive, i.e., U 11 = and U 12 =. Suppose that Assumption 4.2 holds. Then γ(k) = max γ(ω) ω where γ(ω) is the optimum of the following minimization problem: Minimize γ(ω) subject to (4.17) and (4.15), where all variables and transfer matrices are taken at ω instead of ω k. Proof. From τ 2 it follows that extended perturbation block a in the robust performance problem has at most three complex blocks, which implies that µ upper-bound is exact: inf D σ(d L MD 1 R ) = µ a (M). Assumption 4.2 ensures that z p = z p which implies that the calculated bound max ω γ(ω) is at least γ(k), i.e. it is an upper-bound. However, this upper-bound is exact, since the additive structure allows a convex characterization of the optimization problem, for this reason there exist Θ l and v i such that z p 2 is the worst-case performance of the model. A simple illustrative example for the D-K-W iteration with standard (non-skew) µ-synthesis is presented in Appendix A. Remarks on Algorithm 1 On model validation methods The measurement data define a set of consistency constraints on the uncertainty model. Time-domain model validation approaches, e.g. [18] and [124], have the drawback that they cannot handle long data series. This is because in case of dynamic uncertainty, the number of variables is at least as high as the length of the experiment. Frequencydomain methods [123, 25, 28] and [79] have the advantage that the problem reduces to a series of small tractable feasibility problems. The number of variables increases linearly with the number of experiments. A further problem arises from the structure of the uncertainty model. Structured uncertainty model in LFT interconnection leads to a model validation problem, which is a NP-hard problem in general. There are, however, many approaches for computing 52

57 4.5. SOLUTIONS WITH CONVEX OPTIMIZATION approximate solutions in the model validation context. Methods based on S-procedure [124, 98], scaling-approach [28] and skew µ approach [3, 73, 95] are some examples. In the latter approach, the model validation problem is reformulated as a skew µ analysis problem and there is no direct connection to the skew µ synthesis of the dissertation. Concerning control synthesis, the available robust design methods have their limits in case of too sophisticated uncertainty descriptions. The upper bounds on the SSV, in case of mixed parametric and dynamic uncertainties, are not necessarily tight [45, 89]. Based on the above discussion, the perturbation model in the dissertation is restricted to contain only LTI components. The special case of η(θ) = η is particularly advantageous in computational point of view, because the algorithm consists of only convex optimization steps subject to LMI constraints. Implementation The controller synthesis step and the search for the scalings can be implemented by the minor modification of the existing routines in the Robust Control Toolbox for Matlab [12] designed for D-K iteration. The computational burden of steps D and K is the same as in the µ-synthesis. Step W requires approximately the same complexity of calculations as step D. We note that it is numerically more advantageous to introduce a variable for the last block of D L and D R, and the search for scalings is followed by a normalization step. Generalization to other class of systems Frequency-domain consistency constraints are applicable in case of LTI perturbation models. Nominal model G 23, however, does not play a role in the constraints, it can be replaced, therefore, by LPV or nonlinear models. The only requirement is that a robust controller can be designed for the structured uncertainty model. A formulation of the algorithm for LPV nominal models is presented in Chapter 5. For the sake of simplicity the plant is restricted to be stable in the present work. This assumption is necessary for the used frequency-domain model validation method. The proposed scheme can be extended, however, to unstable systems along the idea presented in [54]. The extension can be the subject of future research. Convergence of iterations Standard D-K iteration is not guaranteed to converge to the global optimum, but it has been shown to work on practical problems with LTI perturbations [99]. The same can be said about the presented D-K-W iteration [RGB9, RGB1, RG1]. Regarding the whole iterative algorithm, the uncertainty set increases with every new falsifying experimental data. In the presented applications (Sections 4.6, 5.6 and Chapter 6) the algorithm stopped after 2-3 iterations. 53

58 CHAPTER 4. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LTI SYSTEMS 4.6 Case study: Active steering for vehicle stability enhancement - LTI case The example of this section has been constructed to demonstrate the proposed iterative algorithm. A frequent problem in automotive control is to design a steering controller for yawrate reference tracking. This problem emerges for example when the objective is to improve ride quality and steerability by making the steering robust against variations in road conditions, wind effects and vehicle parameters (e.g. mass and load distribution) [85, 97, 13]. Figure 4.5: Concept of steering control In the presented example, the driver force on the steering column is manipulated by an additional motor drive mounted on the steering rack. The reference for the manipulation is generated by a reference model of the vehicle which produces a desired yaw-rate signal based on the measured steering-wheel angle (see Figure 4.5). The controller is to generate an additional steering force/angle to enforce the vehicle to track the yaw-rate reference while attenuating effects of road and wind. The only sensors used by the proposed control algorithm are yaw-rate and steering wheel angle which are available on commercial vehicles. The performance is defined in terms of L 2 norm of the tracking error and the control signal. True system T consists of a linearized single-track model and a steering system dynamics plotted in Figures They approximate the dynamics of a MAN/Actros heavy truck. The model equations can be derived based on [RB5a] and [Röd7]. The LTI system is given by the state-space description T : [ ẋ y ] [ A Bd B = u C D d D u ] x d u with state vector x T = [β ψ δ δ R R δ m δm ], disturbance d = vwind 2, control input 54

59 4.6. CASE STUDY: ACTIVE STEERING FOR VEHICLE STABILITY ENHANCEMENT - LTI CASE u = T d and output y = ψ. The system matrices are given by A= c f+c r mv c rl r c f l f J z n RKc f J sl c f 1 + crlr c fl f mv 2 mv crl2 r +c fl 2 f c f l f J zv J z 1 K sl ǫ J sl + nrkc fl f J sl v K sl+n RKc f J sl B sl J sl K sl N LJ sl 1 bǫ M R b M R a R pm R b N LM R BR M R 1 C S J swr p k m J swr p a M R CS J sw km J sw B d = w β w r 6 1, B u = [ cj sw ], C = [ ], Dd =, D u = The state and input variables are defined in Table 4.1 and the physical parameters in Figure 4.6: Single-track model Figure 4.7: Neglected dynamics: a stock steering system. Table 4.2. Adhesion coefficient µ and vehicle velocity v are constant uncertain parameters which define set S T : in each iteration of the algorithm, Step C consists of three experiments with (µ,v) {(.2,1),(1,3),(1, 1)}. 55

60 CHAPTER 4. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LTI SYSTEMS Table 4.1: State and input variables β ψ δ R δ m v wind T d vehicle side-slip angle yaw-rate (measured output) steering angle rack position steering wheel angle side-wind velocity (disturbance) applied torque on the steering wheel (control input) Table 4.2: Physical parameters of the vehicle model l f = m, l r = 2.856m m = 142 kg c f = µ , c r = µ J z = Nm/(rad/sec 2 ) J sw =.34 Nm/(rad/sec 2 ) K sc = 427 Nm rad, B sc =.3642 Nm rad/s) K tb = 83 Nm rad, B tb =. Nm rad/s) M R = 6 kg R p =.7367 m B R = 352 Nm/(rad/s) N L = m K sl = Nm rad, B sl = 16 Nm rad/s) ǫ =.1 η F =.985, η β =.985 η ps =.95 G ps = 3 N/rad J sl = 6.1 Nm/(rad/sec 2 ) n RK =.61 m c =.39 w β =.12, w r = COG-front and COG-rear axis distances Mass of vehicle Front and rear cornering stiffness Inertia of vehicle around the vertical axis Inertia of steering wheel Steering column stiffness and damping Torsion bar stiffness and damping Mass of rack Radius of pinion Rack damping Steering linkage ration Steering linkage stiffness and damping Roll steer coefficient Efficiency of forwd./backwd. torque transmission Efficiency of power steering torque transmission Power steering gain Cumulative inertia of steering linkage and tire Total tire trail length Normalizing factor such that δ( ) T d ( ) Wind force factors on slip and yaw-rate C s = Ksc K tb K sc+k tb, k m = Bsc K tb K sc+k tb C a = η s K F R p + η sc G ps K ps K sc+k tb, b = 2η sl β R RN L µ = 1 (nominal).2, 1, 1 (test) Adhesion coefficient in experiment 1,2,3 v = 15 (nominal) 1, 3, 1 m/s (test) Absolute velocity in experiment 1,2,3 The steering system dynamics is considered as unknown and the linearized singletrack model is used for control design. The nominal model is given by β c f+c r mv 1 + crlr c fl f c f mv G n : ψ 2 mv β = ψ (4.18) y nom c rl r c f l f J z crl2 r+c f l 2 f J zv c f l f J z 1 Based on Table 4.2, note that the input is scaled so that T d = δ in the nominal model. 56 T d

61 4.6. CASE STUDY: ACTIVE STEERING FOR VEHICLE STABILITY ENHANCEMENT - LTI CASE r W r T driver W ref ψ ref d + z W d u w u w u W d u = T d G n y = ψ - z 2 W u u K = T c K y K W t z 1 Figure 4.8: Control specification: the closed-loop system with normalizing weighting functions. The nominal value of uncertain adhesion coefficient µ = 1 is built in cornering stiffness parameters (c f,c r ) and the nominal velocity is v = 15 m/s. The neglected steering system dynamics acts on the steering angle input of the nominal model, for this reason a natural choice for the uncertainty structure is input-multiplicative perturbation = T G n 1 that belongs to set S u = { RH }. The effect of the side-wind is modelled by an additive disturbance on[ the output. Thus, ] the system describing the structure of 1 the uncertainty will be U =. G n 1 G n The goal of the control is given by the specification of the closed-loop system in Figure 4.8. Blocks, W, G n and W d constitute the uncertain system. The torque of the driver on the steering wheel T driver drives reference model W ref = 75, which (s+5) 2 represents the required behavior of the vehicle. Block W r normalizes physical signal T driver. Signal ψ ref is to be followed by applying an additional torque T c generated by feedback controller K. Tracking error ψ ref ψ is penalized by low-pass filter W t =.25 s+3 s+.1 in order to achieve good tracking at steady state and at low frequencies. Control signal u K = T c is penalized by high-pass filter W u = 1.4 (s+.4)2 in order to (s+1) 2 avoid high-frequency components. Augmented plant mapping [w u d r u K ] T [z u z 1 z 2 y K ] T is given by W r / 2 1 G = W t G n W t / 2 W t (W ref G n )W r / 2 W t G n W u G n 1/ 2 (W ref G n )W r /, 2 G n where term 2 is incorporated in order to normalize input vector [d r] T. Initial data set E 1 is generated by three open-loop experiments associated with the three systems in S T. In all experiments (open- and closed-loop) the same driver torque T driver, plotted in Figure 4.11(c) (dash-dot), and the same side-wind velocity, plotted in Figure 4.9, will be applied. A sampling time of.1s is used and the frequency domain data is derived by discrete Fourier-transformation (DFT). 57

62 CHAPTER 4. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LTI SYSTEMS 2 Disturbance on the vehicle side wind velocity [m/s] v wind time [s] Figure 4.9: Side-wind velocity in the experiments. 2 Disturbance weighting function W d 1 Perturbation weighting function W 1 1 magnitude, db true sup d DK DKW 1 DKW 2 DKW 3 magnitude, db true sup DK DKW 1 DKW 2 DKW frequency, rad/s frequency, rad/s (a) (b) Figure 4.1: Disturbance (a) and perturbation (b) weighting functions. True uncertainty (o), initial (DK), and subsequent iterations DKW1,2,3. (Data at ω 1 = is plotted at ω 2 1 = rad/s) Given the control specifications, the first task is to chose initial weighting functions for the uncertainty. In this numerical example we can exactly calculate the boundaries of the multiplicative perturbation set as W,true (jω) = max T ST T (jω) G n(jω) 1. Similarly, the maximal magnitude of the true additive disturbance can be calculated. These boundaries are plotted in Figure 4.1 by small o markers. It can be seen from Figure 4.1(b) that the size of the perturbation is greater than -6dB=.512 (minimum at ω = ), i.e. the nominal model is very inaccurate. In the real problems, however, the boundary of the uncertainty set is not known. Assume, the initial weighting functions in the uncertainty model are set according to the following scene: Based on engineering judgement constant 6% is assumed for the neglected dynamics, i.e. W :=.6, which is a quite good estimation - a bit overestimated at low frequencies and a bit underestimated at higher frequencies. In order to start from a correct (consistent) model the following procedure is applied for the determination of the disturbance weighting function. Using the model validation constraints of Section 4.2 the upper-bound of the disturbance weighting function is minimized for each data point in E 1 (open-loop data) min θ lk α lk 58

63 4.6. CASE STUDY: ACTIVE STEERING FOR VEHICLE STABILITY ENHANCEMENT - LTI CASE subject to.6 w u,l(jω k,θ lk ) z ul (jω k ) α lk d,l (jω k,θ lk ) for l = 1,...,N and k = 1,...,n ω. Values α lk are, then, over-bounded by weighting function Wd which is shown in Figure 4.1 by dotted line. The controller designed by Algorithm 1 is initialized by the controller designed by the skew D-K iteration (steps B1 and B3 in Algorithm 1) for the initial model with Wd and W. This controller is denoted by K DK. The evaluation of the uncertainty weighting functions during Algorithm 1 is presented in Figure 4.1. After the first application of D-K-W iteration the size of perturbation decreases significantly to W (Θ 1 ), while that of the disturbance increases to W d (Θ 1 ) (dashed lines in Figure 4.1). This means that the weighting functions got farther from the true weighting functions: the neglected dynamics is modelled partly as an effect of disturbance. The resulting controller is denoted by K DKW,1. Then, the three closed-loop simulations with K DKW,1 are carried out. Step D reveals that the new data invalidates the model. Thus, the iteration continues. After three iterations the analysis shows that data E 4 are consistent with the resulting uncertainty model, therefore, the algorithm stops. The reference-tracking and the control signals of K DK and the last controller K DKW,3 are plotted in Figures 4.11(a)-4.11(c). Among the three plants in S T the one with (µ,v) = (.2,1) is the worst-case showing the largest oscillations 1. This plant is destabilized by initial controller K DK : at about t = 43s the yaw-rate attains the predefined saturation level and u K = T c is set to zero 2, meanwhile the yaw-rate converges to the open-loop case. At t = 53s controller K DK is switched on again and the yaw-rate starts to oscillate. Figure 4.11(b) show the yaw-rate of the final DKW controller for both the worst-case plant with (µ,v) = (.2,1) and for the case with (µ,v) = (1,3). Despite the large uncertainty the final controller achieved good tracking performance for all plants in the set. The L 2 -norm of the tracking error, ψ ref (t) ψ(t) 2, is also computed during the simulations. The norm is plotted in Figure 4.11(d), showing the superiority of the DKW controllers over K DK. Regarding each controller, Table 4.3 shows the achieved robust performance level γ s (K) and the satisfaction of the consistency conditions based on the closed-loop simulation data. Table 4.3: Performance levels of the controllers K DK K DKW,1 K DKW,2 K DKW,3 γ s (K) consistency 1 Only the experiments with parameter setting (µ, v) = (.2, 1) are plotted in Figures 4.11(a)-4.11(c) 2 Applied steering torque T d became equal to driver torque T driver in period t [43s, 53s], see Figure 4.11(c). 59

64 CHAPTER 4. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LTI SYSTEMS Figure 4.1 show that the weighting functions represent the size and shape of a fictive uncertainty set that serves for a mathematical description (or rather inclusion) of the true system. They values depend on the control specification, not on the true system. It can be seen that uncertainty modelling improved the guaranteed performance. Controller K DKW,1 after the first iteration is the best performing controller, however, it is designed based on an invalid model. In the subsequent iterations the robustness of the controllers is increased. The final controller is consistent with all measurements. yaw rate, rad/s Reference tracking with K= and with K DK, (µ,v)=(.2,1) 1 r ref 1.5 open loop K DK time, s (a) yaw rate, rad/s Reference tracking with the final K DKW r ref K DKW,3, (µ,v)=(.2,1) K DKW,3, (µ,v)=(1,3) time, s (b) Control signals T d with (µ,v)=(.2,1) 5 Tracking performance of the controllers Steering torque, Nm T driver K DK K DKW,3 Norm of the tracking errors K DK K DKW,1 K DKW,2 K DKW, time, s (c) time, s (d) Figure 4.11: Simulation results (a) - yaw-rate reference ψ ref (solid), open-loop response (dash-dot), closed-loop response with K DK (dotted). (b) - yaw-rate with final controller K DKW,3 in two experiments. (c) - control signals: driver torque T driver (dash-dot), T d with K DK (dotted), with K DKW,3 (solid). (d) - evaluation of reference tracking error ψ ref (t) ψ(t) 2 (sum of the three experiments) 6

65 4.7. SUMMARY AND THESIS Summary and Thesis 1 In this chapter, a design algorithm for weighting functions characterizing structured uncertainty is developed. It is based on frequency-domain model validation results by Smith and Doyle [123], but consistency conditions are utilized in an optimization problem as constraints where the objective is to tune the weighting functions in order to minimize a control performance criterion. This weight optimization problem is placed into an iterative experimenting, unfalsification and robust control design framework. No a priori information is assumed on the true system that generates experiment data. It is only assumed that the true system is stable and in every experiment it can be modelled by a given LTI nominal model with LFT interconnection to structured dynamic perturbations and disturbances. The objective is to minimize the robust performance of the true controlled system by designing both the weighting functions and the controller. It is shown that if the model is consistent with all experiment data that can be obtained with a given controller, then an upper-bound of the true performance can be computed based on the model (Lemma 4.1). The lemma suggests the idea of using closed-loop experiments to falsify/unfalsify models, and thus improve the consistency of the model. In fact, one cannot do more than just improving robustness when no a priori information is available on the bounds of uncertainty. Guarantee cannot be given for stability and performance. It is proved, furthermore, in Theorem 4.2 that, when using a control synthesis method developed for minimizing skew µ criterion, then the following statements hold: if a model is not falsified by a new experiment, then the actual performance of the control is, as expected, below the guaranteed level if the realized performance in the new experiment exceeds the guaranteed level, then the model is necessarily falsified Based on these implications a stability uncertain (defined by [13]) iterative scheme can be proposed where instability is detected in time and measurement data is utilized for improving robustness of the controller. The proposed scheme is summarized in Algorithm 1 where details of skew µ-synthesis and weighting function design are elaborated. Finally, a class of uncertainty structures are defined which turns the steps of the algorithm into convex problems which can be solved based on LMIs. The contribution of this chapter can be summarized in the following thesis: Thesis 1 An iterative algorithm has been elaborated for LTI models with the aim of shaping structured uncertainty models and designing robust controllers based on measurement data. The algorithm handles unstable experiments ensuring safe improvement of guaranteed robust performance on the true, unknown system. For additive uncertainty structures, the algorithm is formulated as a series of LMI problems. The steps of skew µ synthesis 61

66 CHAPTER 4. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LTI SYSTEMS has been elaborated as part of the algorithm. It has been proved that performance degradation beyond the guaranteed level implies falsification of the actual model, which forces the continuation of the algorithm. Own publications related to the chapter and thesis: [RB5b, RB6b, RB6a, RGSB8, Röd9, RBar, RG1]. 62

67 Chapter 5 Uncertainty modelling and robust control design for LPV systems The class of LPV systems has become more important in the last two decades due to its modelling power for nonlinear/time-varying systems and the quick development in the related control theory. Many analysis and synthesis results have been successfully inherited from the LTI system s theory, as it is shown e.g. by the unified framework of Chapter 3, and the adapted popular control methods: LQG control and Kalman-filtering [134], [RBL2], fault detection and isolation [11], model predictive control [15], and so on. This line is continued in the chapter: iterative uncertainty modelling and robust control design method is elaborated for LPV systems with LFT scheduling parameter dependence and structured dynamic uncertainty. The design of robust LPV controllers based on intuitive choice of weighting functions is an even more time-consuming and laborious iterative task than in case of LTI controllers. The surplus labour is due to the increased computational burden of controller synthesis and the additional design parameters related to LPV modelling (e.g. choice of scheduling variables and its polytope). The automated and control oriented design of uncertainty weighting functions may drastically decrease the design effort while improving robust performance. In the iterative algorithm presented in Chapter 4, a frequency-domain consistency constraint is applied for the reason of computational advantages. The linearity of nominal model U 23 is not utilized in the uncertainty modelling step of Algorithm 1, even the nominal model does not play a role there. In fact, the nominal model can be of any class of systems. One important property we require of the algorithm is that criterion for uncertainty modelling and control design are the same. This means that the system class of the nominal model must be such that control criterion (robust performance) can be characterized in the frequency-domain. This requirement is fulfilled in case of LPV systems with LFT parametrization of the scheduling variables. The modelling and the analysis are carried out in the frequency-domain and the LPV control design is performed in the time-domain. Analysis and synthesis of LPV systems are carried out via multipliers and IQCs. The relation between µ analysis and IQCs is formulated in Section 5.2. Controller synthesis method for uncertain LPV systems with LFT parameter dependence is discussed in 63

68 CHAPTER 5. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LPV SYSTEMS Section 5.3. The uncertainty modelling criterion and constraints are presented in Section 5.4. The joint uncertainty modelling and control design algorithm for LPV systems is summarized in Section 5.5. The efficiency of the method is demonstrated on the vehicle steering problem in Section Problem formulation The true plant to be controlled and which generates the data is described by (4.1). Suppose that in finite time experiments its input-output behavior can be described with the help of a fictive disturbance d L n d 2 and by an LFT of a given LPV model and a structured dynamic uncertainty u in the form [ zu ŷ ] [ ] U11 U = 12 U 13 U 21 U 22 U 23 ( s ) }{{} U w u d u, w u = u z u, (5.1) where U 23 ( s ) is an n y n u LPV nominal model with LFT dependence on scheduling parameter s S s, where S s is defined by (3.31). Other blocks U ij, ij 23 belong to RH and describe the interconnection structure of the nominal model and structured dynamic uncertainty, u S u, where S u is defined by (4.3). Normalizing weighting functions, W,W d, are stable LTI systems such that u = W u, u BS u and d = W d d, d BL 2. Suppose that the input-output data of system T in the lth experiment are given in the frequency domain as E l. The data containing N experiments are denoted by E N = N l=1 E l. The consistency definition in Section 4.1 still holds with the modification that transfer function U(jω k ) does not exists, U being an LPV system, but model output ŷ can be transformed to the frequency-domain by DFT. The control objective is to minimize (4.5), where controller K = K( s ) is allowed to depend on the on-line measurable scheduling parameter via LFT. An upper-bound can be calculated for γ(k( s )) based on a consistent model in the same way as described in Chapter 5. To this end the control performance is specified on the model, as well, by augmenting U, as described in Section 4.1, and so, the system configuration plotted in Figure 5.1 is obtained. It is also supposed that assumptions 4.1 and 4.2 hold. The assumptions allow the existence of unfalsified uncertainty models and the establishment of consistency based on generalized plant G. We can summarize the problem statement as follows. Problem 2. Given an unknown system T, a plant model U and control specification defined by G. Provide an iterative uncertainty model unfalsification and robust control design procedure that 1. constructs a consistent model (U,W,W d ) for the true plant, based on closed-loop experiments, and 2. designs an LPV controller K( s ) that minimizes objective (4.5). 64

69 5.2. RELATION BETWEEN SKEW µ ANALYSIS AND IQC APPROACH z u u w u s z s w u W w s z p y K G d u W d u K d r K w c z c c Figure 5.1: Closed-loop system configuration with the LPV model 5.2 Relation between skew µ analysis and IQC approach It is shown in this section that abstract performance characterization with appropriate multipliers leads to the same upper-bounds as in the (skew) µ-analysis. In this way, performance characterization used for LTI systems in Chapter 4 is adopted into the IQC framework, which is the tool of analysis and synthesis of LPV systems. In this section, the scheduling parameter dependence is disregarded temporarily, and the uncertain system in Figure 3.2 [ is considered ] where S u represents a structured Mu M dynamic perturbation and M = up denotes a LTI system. M pu M p It is easy to verify that IQC (3.8) with parametrization Q ω (jω) = γd r (jω) D r (jω), S ω (jω) =, R ω (jω) = γ 1 D l (jω) D l (jω), D r := diag{d 1 I w 1,...,d τ I w τ }, D l := diag{d 1 I z 1,...,d τ I z τ }, d i,d 1 i RH, i = 1,...,τ implies σ( ) γ 1. Quadratic performance specification (3.7) with (3.1) and (3.11) is equivalent to 1 [ ] [ ] w p (jω) I I P 2π M (jω) p w M (jω) p (jω)dω ε w p 2 2 by applying Parseval-theorem and inserting z p = M w p, where M = F U (M, ) is the uncertain closed-loop system. When P p is defined by (3.11) and (3.12), the latter IQC leads to quadratic constraint γ p I + M γ 1 p M <, 65

70 CHAPTER 5. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LPV SYSTEMS by Lemma 2.2. This implies that σ(m ) < γ p, i.e., the L 2 gain of the uncertain system is at most γ p. Condition (3.9) of robust performance reappears as 1 ] 2π 2 2 w z w p z p which is equivalent to I M u M up I M pu M p γd rd r γ 1 D l D l γ p I γ 1 p I γd r D r γ 1 D l D l γ p I γ 1 p I w z w p z p dω ǫ [ w w p I M u M up I M pu M p < for all ω R { }. By simple algebraic manipulation, this quadratic constraint leads to ( )1 ( ) 1 γp 2 σ γ D L MDR 1 γp 2 γ < γ p, I I where D R := diag{d r,i wp } and D L := diag{d l,i zp }. Note that the infimum of this expression in scalings d 1,...,d τ equals the upper-bound of scaled µ ( ) 1 ( )1 γp 2 inf σ γ D L MD 1 γp ( [ 2 γp ]) γ D R = inf σ D L MD 1 γ D R I I I and this proves the following lemma. Lemma 5.1 For all S u with σ( (jω)) γ(ω) 1 loop F U (M, ) is well-posed, internally stable and for all ω: σ (F U (M(jω), (jω))) γ s (ω) if there exist d i (jω), 1 = 1,...,τ, satisfying σ ( D L (jω)m(jω)d R (jω) 1 [ γs(ω) γ(ω) I w I wp ]) < γ s (ω) or equivalently, if there exist d i (jω), 1 = 1,...,τ, satisfying [ ] ] I I Π(γ(ω),γ M(jω) s (ω))[ <, M(jω) where Π(γ(ω), γ s (ω)) = γ(ω)d r (jω) D r (jω) γ s (ω)i γ(ω) 1 D l (jω) D l (jω) γ s (ω) 1 I The derived identity in the lemma provides multipliers for the IQC characterization of structured dynamic uncertainty being in a ball of radius γ(ω) 1 and guaranteed induced L 2 -gain of the system with a given level γ s (ω). Due to this skew µ like characterization the advantageous properties of Lemma 4.1 and Theorem 4.2 can be kept also for LPV systems. 66

71 5.3. OUTPUT-FEEDBACK CONTROL OF UNCERTAIN LPV SYSTEMS 5.3 Output-feedback control of uncertain LPV systems The analysis and synthesis of LPV systems with general or affine parameter dependence, lead to parameter dependent LMIs which cannot be transformed to frequency-domain inequalities. However, an equivalent frequency-domain formulation of robust performance would be required for tuning models of dynamic uncertainty. Fortunately, LPV systems with LFT parameter dependence can be rewritten in forms appropriate for frequency-domain analysis. The following robust LPV control synthesis method is similar to the LPV/µ iterative result of Ref. [125], except that in [125] function c is fixed to c ( s ) = s, which introduces some conservatism as it involves more constraints. z u u w u s z s w s z p G w p y K K u K w c c z c Figure 5.2: Uncertain closed-loop LPV system We consider the LPV LFT system in Fig. 5.2 where generalized plant G is given in the state-space by ẋ z u z s z p y = A B u B s B p B C u D u D us D up E u C s D su D s D sp E s C p D pu D ps D p E p C F u F s F p The uncertainty and the scheduling parameters enter the system via channels w u z u and w s z s, respectively, as [ ] [ ] [ ] wu u zu =, w s s z s where u S u is a structured LTI uncertainty where set S u is defined by (4.3). x w u w s w p u 67

72 CHAPTER 5. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LPV SYSTEMS z c z u u z s s c ( s ) w u w s w c G a z p G w p y K u K w c K z c Figure 5.3: Uncertain closed-loop LPV system redrawn to -G-K configuration s S s is a time-varying on-line measurable scheduling matrix contained in starshaped convex hull S s defined by (3.31). We assume that the controller has also an LFT dependence on measurable scheduling parameter s, although through a possible nonlinear matrix function c (.), as follows ẋ c [ ] x u Ac B c = c y, C z c D c c w c w c (t) = c ( s (t))z c (t) Analysis The goal of the analysis is to determine whether the closed-loop system is stable and the quadratic performance specification defined by (3.7), (3.1) with multiplier (3.11) is satisfied on channel w p z p for all admissible parameter curves in S s and uncertainties u BS u. The solution is a combination of robust control and LPV LFT control synthesis presented in Sections 3.3 and 3.4. The closed-loop system is reformulated in the general -G-K form as shown in Figure 5.3 where a new augmented plant G a is perturbed by generalized uncertainty which are described by 68

73 5.3. OUTPUT-FEEDBACK CONTROL OF UNCERTAIN LPV SYSTEMS ẋ z u z s z c z p y w c = A B u B s B p B C u D u D us D up E u C s D su D s D sp E s I C p D pu D ps D p E p C F u F s F p I } {{ } G a x w u w s w c w p u z c w u w s w c = u s } c ( s ) {{ } z u z s z c According to Theorem 3.2 the analysis starts with the description of the IQCs characterizing structured perturbation. Let the multipliers for u be parameterized as in Section 5.2 [ ] [ Qω S Π u (jω) := ω Dr (jω) = ] D r (jω) D l (jω) (5.2) D l (jω) S ω R ω where D r = diag{d 1 I wu,1,...,d τ I wu,τ }, D l = diag{d 1 I zu,1,...,d τ I zu,τ }, d i,d 1 i RH, i = 1,...,τ. Note that Π u can be factorized as Π u (jω) = Ψ(jω) P u Ψ(jω) [ Dr (jω) Ψ(jω) = D l (jω) ] [ I, P u = I In section 3.4, the LTV scheduling parameter matrix a = diag{ s, c ( s )} [ is characterized by augmented multiplier P a defined by (3.32) where sub-matrix P s = ] Qs S s belongs to set S P. Multipliers Π u and P a are combined to satisfy (3.8) for extended perturbation 1 2π z u z s z c u I a I Q ω Q a S a S T a R ω R a u I a I ] z u z s z c S T s R s dω which is, according to Theorem 2.2, satisfied when u I a I Q ω Q a S a S T a R ω R a u I a I > (5.3) 69

74 CHAPTER 5. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LPV SYSTEMS holds for all ω R { }. Note, the quadratic form of the left hand side of (5.3) is a block-diagonal matrix, so the condition comes apart to [ ] [ ] u (jω) u (jω) Π I u (jω) >, for all ω R { }, I implying σ( u (jω)) 1, and (3.33). Before establishing the robust quadratic performance analysis results for uncertain LPV systems, several notations are introduced. Define D L = diag{d l,i} and D R = diag{d r,i} with which G D := D L GDR 1 is introduced as the scaled nominal system. Its state-space realization is marked by tilde as G D = à Bu Bs Bp B C u Du Dus Dup Ẽ u C s Dsu Ds Dsp Ẽ s C p Dpu Dps Dp Ẽ p C Fu Fs Fp The scaled augmented nominal system is consequently denoted by G ad := D L G a D 1 R. Let the scaled closed-loop system be denoted by M D = F L (G ad,k) and the unscaled closed-loop system by M := F L (G a,k). The state-space matrices of M D are affine functions of the controller. M D := = A B u B s B c B p C u D u D us D uc D up C s D su D s D sc D sp C c D cu D cs D c D cp C p D pu D ps D pc D p à Bu Bs Bp C u Du Dus Dup C s Dsu Ds Dsp C p Dpu Dps Dp + B I Ẽ u Ẽ s I Ẽ p [ ] Ac B c C c D c I C Fu Fs Fp I The following theorem states the conditions of robust quadratic performance of the uncertain closed-loop LPV system. Theorem 5.1 (Analysis inequalities) The closed-loop system in Figure 5.3 is robust stable and robust quadratic performance defined by (3.7), (3.1) and (3.11) is satisfied 7 if there exist a Lyapunov matrix X = X T > and multipliers Π u (jω) with (5.2) and P a with (3.32), (3.33) satisfying [ ] T [ ] I I P M M <, (5.4) D M D

75 5.3. OUTPUT-FEEDBACK CONTROL OF UNCERTAIN LPV SYSTEMS where P M = X I Q s Q 12 S s S 12 Q 21 Q 22 S 21 S 22 Q p S p X I Ss T S21 T R s R 12 S12 T S22 T R 21 R 22 Sp T R p (or equivalently) if there exist multipliers Π u (jω) with (5.2) and P a with (3.32), (3.33) satisfying [ I M(jω) ] P e (jω)[ I M(jω) ] <, for all ω R { }, (5.5) where P e (jω) = Q ω Q s Q 12 S s S 12 Q 21 Q 22 S 21 S 22 Q p S p R ω Ss T S21 T R s R 12 S12 T S22 T R 21 R 22 Sp T R p Proof. The theorem is a straightforward combination of the analysis results for uncertain and nominal LPV systems. The one issue that might require explanation is that (5.5) and (5.4) are equivalent. [ ] [ w I I P M e M w u w s w c = w p z u z s z c z p ] w = Dr D r Q s Q 12 S s S 12 Q 21 Q 22 S 21 S 22 Q p S p Dl D l Ss T S21 T R s R 12 S12 T S22 T R 21 R 22 Sp T R p w u w s w c w p z u z s z c z p 71

76 CHAPTER 5. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LPV SYSTEMS u u D 1 l z u u w u D r D l z u D r w u D l zu w u D r D l z u M D D r w u D l z u w u D 1 r z s z c z p M w s w c w p z s z c z p M w s w c w p Figure 5.4: Handling dynamic multipliers in performance analysis = D r w u w s w c w p D l z u z s z c z p I Q s Q 12 S s S 12 Q 21 Q 22 S 21 S 22 Q p S p I Ss T S21 T R s R 12 S12 T S22 T R 21 R 22 Sp T R p D r w u w s w c w p D l z u z s z c z p, Scaled w u and z u signals of the last expression can be generated by two alternative formulations of the closed-loop uncertain system: Figure 5.4 clarifies that (5.5) is equivalent to [ I M D ] I Q s Q 12 S s S 12 Q 21 Q 22 S 21 S 22 Q p S p I Ss T S21 T R s R 12 S12 T S22 T R 21 R 22 Sp T R p [ I M D ], (5.6) By commutativity of u and the scalings, u = D 1 l u D r holds, so the same uncertainty, u S u, affects the scaled system. Finally, apply Lemma 2.3 to show that (5.6) is equivalent to (5.5) Synthesis The goal of the synthesis is to construct an LPV controller such that for all admissible parameter curves in S s and uncertainties u BS u the closed-loop system is stable and the quadratic performance specification defined by (3.7), (3.1) with multiplier (3.11) is satisfied on channel w p z p. 72

77 5.3. OUTPUT-FEEDBACK CONTROL OF UNCERTAIN LPV SYSTEMS Time-domain analysis condition (5.4) depends nonlinearly on the controller parameters and Lyapunov matrix X. Using a nonlinear transformation of the variables and elimination of the transformed controller parameters, we arrive at the equivalent synthesis conditions presented in Theorem 5.2 which can be proved based on Lemma 5.1 and Theorem 3.5. Let Υ denote the basis matrix of Ker[C F u F s F [ p ] and Φ a ] basis matrix of Ker[B T Eu T Es T Ep T]. Let the inverse of multiplier Qp S p Sp T be denoted by R [ ] p Qp Sp P 1 p := S T p R p. Theorem 5.2 (Synthesis inequalities) There exists a controller K, a scheduling function c ( s ), a Lyapunov matrix X = X T > and multiplier P a with (3.32)-(3.33) satisfying (5.4) if and only if there exist X, Y and multipliers P s S P and P s S P that satisfy the LMIs [ ] Y I >,, Γ T I X XP X Γ X <, Γ T Y P Y Γ Y >, (5.7) where P X = X I Q s S s Q p S p X I Ss T R s Sp T R p Γ X = P Y = I I I I Ã Bu Bs Bp C u Du Dus Dup C s Dsu Ds Dsp C p Dpu Dps Dp Υ Y I Qs Ss Qp Sp Y I ST s Rs ST p Rp 73

78 CHAPTER 5. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LPV SYSTEMS Γ Y = ÃT C u T C s T C p T B u T D u T D su T D pu T B s T D us T D s T D ps T B p T D up T D sp T D p T I I I I Φ Since uncertainty is present, the output-feedback controller synthesis for robust performance problem is a non-convex optimization problem, as discussed in Section 3.3. One approach is to solve the optimization by iteration using both (5.7) and (5.5). In one step, the scalings related to the dynamic uncertainty are designed in the frequency-domain: point-wise search over a frequency grid followed by fitting the magnitude of a stable minimum-phase transfer function. In a second step, the fitted scalings are connected to system G to build system G ad for which a nominal LPV controller is designed as described in Section 3.4. The algorithm is extended by the design of uncertainty weighting functions, and the complete procedure is placed in the iterative uncertainty modelling and control design algorithm as discussed in the next sections. 5.4 Optimizing the uncertainty model The goal of this section is to extend the conditions of robust performance of uncertain LPV systems by model consistency conditions in order to simultaneously test consistency and performance. When a scalar performance level is defined, then an optimization problem can be formulated in the uncertainty model parameters. By this means an unfalsified uncertainty model can be tuned in order to tighten the upper-bound of robust performance. According to Theorem 5.1 robust quadratic performance is satisfied if (5.5) holds. In the proof of the theorem, it is shown that (5.5) is equivalent to the frequency domain inequality (5.6). In order to connect consistency and control performance conditions, inequality (5.6) has to be reformulated as a function of uncertainty model parameters v ik, i = 1,...,τ +n d defined by (4.9)-(4.1), which are parameters of consistency constraints (4.12)-(4.13) as well. We derive results for the special case of additive uncertainty, discussed in Section 4.5, because this leads to an LMI problem. The general case follows from this result by simply replacing the LMI associated with the consistency constraints by general constraints (4.12)-(4.13). By pulling out the weighting functions related to the uncertainty model, scaled unweighted closed-loop nominal system M D is defined by M D = M D W with W = diag{w,i ws+w c,w d,i nr } As in Section 4.5, define uncertainty weighting function magnitude parameters v ik, i = 1,...,τ + n d, constituting the diagonal matrix V k as in (4.14) and (4.16) and define 74

79 5.4. OPTIMIZING THE UNCERTAINTY MODEL V Ik := W(jω k ), k = 1,...,n ω. Then, W(jω k ) can be factorized as W(jω k ) = V Ik Ω(jω k ), where Ω(jω k ) = diag{ω (jω k ),I ws+w c,ω d (jω k ),I nr } is a unity amplitude diagonal phase function with Ω(jω k ) = Ω(jω k ) 1. Introduce the following real matrices for simplifying notation of the multiplier in (5.6): Q D = diag{ I,Q a,q p } S D = diag{,s a,s p } R D = diag{i,r a,r p } At frequency points ω = ω k, (5.6) is equivalent to [ I M D (jω k )V Ik Ω(jω k ) ] [ QD ][ S D I M D (jω k )V Ik Ω(jω k ) S T D R D ] < (5.8) Thus, the required connection between performance and consistency inequalities is, in principle, established through amplitude parameters v ik. Unfortunately, inequality (5.8) is practically useless, since changing the amplitude parameters while keeping the phase functions implies a strong constraint at every frequency point ω k when creating new tight over-bounding function W. We need low order weighting functions, for that reason we must get rid of term Ω in (5.8). This, however, can be accomplished at a price of loss of generality in quadratic performance specifications: congruent transformation by Ω(jω k ) on inequality (5.8) reveals (dropping (jω k )) ΩQ D Ω + ΩS D M D V Ik + V T Ik M D ST D Ω + V T Ik M D R DM D V Ik < which is equivalent to Q D + S D M D V Ik + V Ik M D ST D + V IkM D R DM D V Ik < (5.9) if and only if performance multiplier P p has the following structure: Q p,s p,r p are blockdiagonal matrices, such that the n d n d left upper block of Q p and R p are diagonal and that of S p is zero, i.e., Q p = diag{q p1,q p2 } with Q p1 R n d n d diagonal R p = diag{r p1,r p2 } with R p1 R n d n d diagonal (5.1) S p = diag{ nd,s p2 } Recall that R D, so by using Schur complement, inequality (5.9) can be written as [ ] Q D + S D M D V Ik + V Ik MD ST D V IkMD R 1 2 D < R 1 2 D M D V Ik I [ ] VIk Applying congruent transformation by and noticing that V I Ik S D = S D, the following lemma is proved. It provides conditions of robust quadratic performance for an LPV model at frequency grid ω k, k = 1,...,n ω. 75

80 CHAPTER 5. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LPV SYSTEMS Lemma 5.2 Suppose that quadratic performance is specified by multiplier P p with structure (5.1). Then, inequality (5.5) is equivalent to [ Q D VIk 2 DM D VIk 2 Ik 2 D ST D V Ik 2 D R 1 2 D R 1 2 D M D VIk 2 I at frequency points ω k, k = 1,...,n ω. ] < (5.11) Conditions (5.11) and (4.15) form a system of LMIs in the variables v 2 ik and Θ lk, i = 1,...,τ + n d, l = 1,...,N, k = 1,...,n ω. 5.5 Iterative design of controller and uncertainty model Iterative uncertainty modelling and robust control design algorithm can be formulated for uncertain LPV systems with LFT scheduling parameter and perturbation dependencies. Algorithm 2. A. Initialization B. D-K-W iteration 1) LPV controller synthesis(k) 2) Optimization of uncertainty model(w) 3) Skew µ analysis by finding scalings(d) C. Closed-loop experiments D. Update of the uncertainty model Differences between Algorithm 1 and 2 appear in Step B, which is detailed as follows. Step B. D-K-W iteration In this step, induced L 2 gain of the uncertain [ LPV system is minimized, ] i.e., robust γp I nd +n quadratic performance is specified by P p = r γp 1. Accordingly, we I nzp are looking for LPV controller (K, c ()), uncertainty weighting functions (W,W d ) and dynamic scalings (d i, i = 1,...,τ) for the system in Figure 5.1 which guarantee sup s S s,r BL 2 z p 2 < γ p i.e. the L 2 -norm of the performance output of the model for the worst-case scheduling trajectory and input r is less than γ p. When the model is consistent with all experiment data obtained from system T then γ p is a guaranteed upper-bound for z p 2, the performance of the true system. Lemma 4.1 holds for LPV systems as well (with γ s (K,W) = γ p ), and can be proved in the same way. The following three steps summarize the results of this chapter. The steps are iterated until the improvement of performance level γ p is less than a given number in 76

81 5.6. CASE STUDY: ACTIVE STEERING FOR VEHICLE STABILITY ENHANCEMENT - LPV CASE the last iteration steps of a given number. During the iteration only the stable controller with improved performance is stored. For notational brevity index N is omitted. Recall that normalized dynamic uncertainty u S u is characterized by dynamic multiplier Π u in (5.2), and set of scheduling parameters S s by multiplier P s S P defined in (3.6). Step B1. Controller synthesis Compute generalized plant G ad including scalings and weighting functions. Then, a nominal LPV controller is designed according to Section Criterion γ p is minimized subject to synthesis inequalities (5.7). Step B2. Optimization of the uncertainty model This step is introduced in Section 5.4. Conditions (5.11) can be rewritten as [ Q D V 2 Ik + S DM D V 2 Ik + V 2 Ik M D ST D V 2 Ik M D R 1 2 D T TR 1 2 D M D V 2 Ik T 2 ] <, (5.12) by congruent transformation with diag{i,t }, where T = diag{i nzu +n zs +n zc,γ 1 2 p I zp }, in order to obtain an LMI in γ p as well. For each frequency point ω k, minimize γ p = γ k subject to (5.12) and (4.15) in the variables γ k, v 2 ik and Θ lk, i = 1,...,τ + n d, l = 1,...,N. Then, point-wise bounds v ik are over-bounded by stable minimal-phase transfer function. The orders of the transfer functions are increased until the criterion with the fitted weighting functions is below εγ k (ε > 1). This is a similar procedure to that in Step B2 in Section 4.4. Step B3. Skew µ analysis by finding the scalings By applying Schur complement on the term with γp 1, analysis inequality (5.5) can be transformed to an LMI in variables d 2 ik, i = 1,...,τ, and γ p. For each frequency point γ p = γ k can be minimized. Then, point-wise scalings d ik are fitted in magnitude by stable, stable invertible transfer functions where the selection of order and the weighting of the fitting criterion are similar to those in the previous step. Remark. 5.1 The implications of Theorem 4.2 hold also for the presented method, since the applied controller synthesis procedure corresponds to skew µ synthesis, as it has been shown in Section Case study: Active steering for vehicle stability enhancement - LPV case In order to illustrate Algorithm 2, the example of Section 4.6 is continued by designing an LPV (velocity-scheduled) controller for the steering problem. 77

82 CHAPTER 5. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LPV SYSTEMS r Wr T driver W ref ψ ref u = T + d z u w u w u u W s G n Ḡ n + d W d d + + y = ψ - + z 2 W u u K = T c K y K W t z 1 c Figure 5.5: Specification of the closed-loop. z 1 and z 2 are performance outputs, F U (G n, s ) is the LPV nominal model, and Ḡn is an LTI nominal model. The true vehicle is the same as in Section 4.6. The nominal model is described by (4.18), however, it is renamed to G n (v) expressing the velocity-dependence of the linear model. The control specification is plotted in Figure 5.5. It differs from Figure 4.8 in the uncertainty model. Since the iterative uncertainty modelling and control design algorithm applies for LTI (speed independent) uncertainty models, the input-multiplicative perturbation u cannot excite nominal LPV model G n (v). Instead, it excites a fixed-speed single-track model Ḡn = G n (v ), for example at an average speed v. The LPV nominal model is reformulated by an LFT, in Figure 5.5, as G n (v) = F U (G n, s ). Blocks u, s, W, G n, Ḡ n and W d constitute the uncertain LPV system. The LPV feedback controller is F L (K, c ). By redrawing the system of Figure 5.5 we can arrive to Figure 5.1. The input-output data set for the uncertainty modeling consists of u = δ and y = ψ measurements of three experiments, where the adhesion coefficient is 1,.7 and.4. The side-wind and vehicle velocities are plotted in Figure 5.6. The yaw-rate measurement for some T driver input with T c = is plotted in Figure 5.7 by dash-dot lines. At this point, all input parameters of the algorithm have been constructed. The initial uncertainty model is set according to the following scene. 1% for the neglected dynamics is assumed: W :=.1. The minimal necessary disturbance weighting functions are designed as in Section 4.6 for satisfying the consistency constraints for the initial data set. The controller designed by the iterative algorithm is compared to the controller designed for the initial model which is denoted by K. After the first D-K-W iteration the size of the perturbation W increased significantly, specially at the higher frequency range while that of the disturbance decreased. Then, the three closed-loop experiments are carried out by the controller. The new data invalidated the model and the iteration was continued with the extended data set resulting the controller K 2. After this second iteration the algorithm finished. The reference-tracking of D-K controller K and D-K-W controllers K 1 and K 2 78

83 5.6. CASE STUDY: ACTIVE STEERING FOR VEHICLE STABILITY ENHANCEMENT - LPV CASE 3 Side wind speed and vehicle velocity 25 velocity [m/s] v 5 wind v time [s] Figure 5.6: Side-wind speed (solid) and forward velocity (dashed). are illustrated in Figure 5.7. It can be seen that both disturbance attenuation and steady state tracking performance (see also Figures ) have been improved due to the algorithm. With the initial controller the achieved robust performance level is γ =.16, after the first D-K-W iteration it is γ =.544 due to the shaping of W, and then, with additional data, the final controller achieved γ =.95. This implies that the robust performance has been improved. Remark. 5.2 When comparing the results with the LTI case (Figures 4.11(b,d) and Figures 5.7, 5.8), it can be seen that LPV controllers achieved weaker performance. The reason is that the true uncertainty is very large in this example and its embedding by the uncertainty model in the LTI case can easily describe the effects of speed variation as well. The additional constraints imposed by the LPV design allow weaker performance..8 Reference tracking, µ= yaw rate [rad/s] reference.6 K= DK controller DKW final time [s] Figure 5.7: Simulation results with µ =.4. Yaw-rate reference ψ ref (solid), yaw-rate measurement ψ without controller (dash-dot), with initial controller K (dotted), with final controller K 2 (bold dashed). 79

84 CHAPTER 5. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LPV SYSTEMS 3 Tracking performance 2.5 tracking error norm DK DKW 1 DKW time [s] Figure 5.8: Evaluation of the norm of the reference tracking error, ψ ref (t) ψ(t) 2, during the simulation with µ =.4. The norm with K (dotted), with K 1 (dashed), with K 2 (solid). 1 Actuator performance Control energy DK.1 DKW 1 DKW time [s] Figure 5.9: Evaluation of the norm of control input, u K 2, during the simulation with µ =.4. The norm with K (dotted), with K 1 (dashed), with K 2 (solid). 8

85 5.7. SUMMARY AND THESIS Summary and Thesis 2 In this chapter, an iterative uncertainty modelling and robust control design algorithm (Algorithm 2) is elaborated for LPV systems. Since LFT scheduling parameter dependence enables robust quadratic performance analysis to be performed in both time- and frequency-domain, LPV LFT is the appropriate system class for which a common criterion can be defined for both the time-domain control design and the frequency-domain uncertainty modelling methods. By using IQCs a more general uncertainty and system class can be handled as compared to (skew) µ analysis tools. In Section 5.2 the relation between the two approaches is exhibited. The derived identity in Lemma 5.1 provides multipliers for the IQC characterization of structured dynamic uncertainty in a unit ball and guaranteed induced L 2 -gain of systems with a given level γ s. In Section 5.3 analysis and synthesis inequalities are derived for uncertain LPV systems with LFT dependence on scheduling parameters and dynamic uncertainty. Due to the particular structure of dynamic multiplier, it can be factorized and the factors are utilized for scaling the nominal augmented LPV plant. Thus, a two step LMI-based iterative algorithm, similar to D-K iteration in (skew) µ synthesis, can be applied for robust LPV controller synthesis. An optimization problem based on LMIs is derived for tuning additive, structured uncertainty models. Robust quadratic performance criterion of special structure, (5.1), is formulated as an LMI optimization problem in terms of the parameters of the unfalsified uncertainty models, see Lemma 5.2. Due to the skew µ like characterization of the performance problem, the advantageous properties presented in Lemma 4.1 and Theorem 4.2 hold also for Algorithm 2: If the LPV model is consistent with the data of all experiments performed in closed-loop with the LPV controller, then an upper-bound of the performance is guaranteed, even if the true system and disturbance bounds are not known. The upper-bound is calculated based on the LPV model. Performance degradation beyond the guaranteed level in a new experiment implies the falsification of the model, which enforces the update and re-tuning of both the uncertainty model and the controller. The contribution of this chapter can be summarized in the following thesis: Thesis 2 An iterative uncertainty modelling and robust control design algorithm is elaborated for LPV models with LFT dependence on scheduling parameters, structured dynamic uncertainty and disturbances. Based on time- and frequency-domain IQCs, guaranteed robust quadratic performance level is minimized by searching for an unfalsified uncertainty model and a robust LPV controller. For additive uncertainty structures, the algorithm is formulated as a series of LMI problems. Due to the advantageous properties of the 81

86 CHAPTER 5. UNCERTAINTY MODELLING AND ROBUST CONTROL DESIGN FOR LPV SYSTEMS algorithm, unstable experiments are handled and improvement of guaranteed robust performance on the true, unknown system is ensured. Own publications related to the chapter and thesis: [RLB7, MKD + 9, RGB9]. 82

87 Chapter 6 Vehicle safety enhancement by steer-by-brake control The topic is motivated by the research into emergency situations of heavy trucks when the driver becomes incapable of controlling the vehicle due to e.g. lipothymy or drowsiness. The need for keeping the vehicle on the road and drive it to a safe position emerges due to increased traffic density. In the most common vehicles where no electronic steering system is available but the braking is controlled by onboard computers, the only way to automate or assist steering is the use of the electronic brake system, the application of individual or unilateral wheel brakes. There are many papers concerning different approaches and aims that improve steering by the braking systems. In [4] active steering and individual wheel braking are discussed from a yaw and roll control point of view. The advantage of individual wheel braking is that it is implementable with less hardware effort since the actuators and wheel speed sensors are available in the existing anti-lock braking systems. A method for unilateral braking for rollover prevention can be found in [51], for preventing unintended lane departure in [71, 1, 14]. Figure 6.1: Block scheme of the control concept. The topic of this paper covers the diagram with the solid lines. In this section a yaw-rate trajectory-tracking problem is solved by using the electronic brake system. The concept of the control algorithm is presented in Figure 6.1. It is assumed that the vehicle is equipped with sensors, such as a camera, GPS or radar, providing information about the environment and the position of the vehicle. It 83

88 CHAPTER 6. VEHICLE SAFETY ENHANCEMENT BY STEER-BY-BRAKE CONTROL is assumed that there is a monitoring and path planning system (MPPS) that detects the emergency situation. A decision can be made based on for example detecting lane avoidance without turn signal. When this occurs, MPPS defines a path along the road and switches on the emergency controller that steers, slows up and halts the vehicle unless the controller is switched off by some driver action. Such monitoring systems are treated in many papers, e.g. [19, 66, 71, 72, 84, 93, 114]. The desired path can be specified in terms of the curvature of the trajectory, thus, by taking the vehicle speed into account, a yaw-rate reference signal can be defined for a trajectory-tracking controller. Based on the yaw-rate and steering wheel angle measurements, the proposed controller affects the front wheel brakes by small pressures in order to turn the steering system which turns the vehicle. In case of heavy trucks this approach has an advantage over unilateral braking 1 : in frequent cases of uneven load distribution between front and rear axles, the rear brake has practically no effect on the lateral dynamics. The performance aims are specified in terms of yaw-rate tracking error and control energy. Both must be bounded for all occurrences of the model uncertainty, such as neglected and unknown dynamics of the vehicle, unknown adhesion characteristics and the lateral slope of the road. Because of strict safety demands on vehicle control, the H /µ design is chosen, which guarantees robust performance against bounded uncertainty of the nominal model. This problem setup suits well the D-K-W iteration algorithm. The advantage of this method is that uncertainty is modelled automatically: choose a simple uncertainty structure, then consider the set of all models in this structure that are not invalidated by measurement data; and this set is parameterized and tuned in order to achieve robust performance specifications. The resulting joint algorithm simplifies the design, reduces the number of design parameters and tailors modelling criterion to the final goal, i.e. closed-loop robust performance. In Section 6.1 the experimental environment is presented. The model used for control design is derived in Section 6.2. The concept of trajectory generation is discussed in Section 6.3. Section 6.4 presents details about the control design and uncertainty modelling by using D-K-W iteration. 6.1 Experimental conditions The modelling and control synthesis are based on measurement data which are generated by the following experimental conditions. A High-Fidelity Vehicle Simulator (HFVS) program was implemented under Matlab/Simulink, which is a relatively good approximation of the MAN truck shown in Figure 6.2a. Some physical parameters (lengths, masses, inertias) of the HFVS match the parameters of the MAN truck, while other parameters (cumulative damping and spring coefficients) are results of identification based on real-life experiments. The HFVS model contains all important features that may be relevant in the yaw motion control problem: the 17-degree of freedom (DoF) nonlinear Matlab/Simulink model contains the dynamics of the suspension system with vertical wheel center motion (4DoF), the roll, pitch, heave motions of the sprung mass (3DoF), 1 unilateral braking means the braking of both front and rear wheels on one side 84

89 6.2. MODELLING FOR STEER-BY-BRAKE CONTROL (a) (b) Figure 6.2: (a) The experimental MAN truck was approximated by a HFVS model. (b) Part of the suspension model of the HFVS. The 17DoF HFVS Matlab/Simulink model serves as the real vehicle in the laboratorial experiments. the yaw, lateral and longitudinal motion of the whole vehicle (3DoF), the steering system (3DoF), the rotational dynamics of the wheels (4DoF) and additionally the brake actuator dynamics and a road model, [3, 26, 65]. The road model pretends effects of a lateral road slope as well. The number of state-variables is 31. Figure 6.2b illustrates one part of the suspension model. The detailed description of the HFVS s dynamics is found in [Röd7]. The advantage of using HFVS instead of real-life experiments on the MAN truck is that D-K-W iteration can be tested and the controller can be tuned before applying it on the real system in cheap and safe laboratorial conditions. On the other hand, as shown below, the algorithm does not exploit any details of the HFVS model, so there is no principal difference in the application of the proposed algorithm on the simulator or on the real system. This will be clear in the next section where a simplified (three state-variables) linear model is derived for controller design purposes. 6.2 Modelling for steer-by-brake control HFVS programs are able to imitate the real vehicle behavior to acceptable accuracy for controller testing (e.g., CarSim/TruckSim [117, 118]). Unfortunately, they are inadequate to be the base of control design, therefore, much simpler models must be identified. A model in the modern robust control concept consists of a nominal model and an uncertainty description. This description defines a bounded set of possible models and a bounded set of possible outer (disturbing) signals. The available robust design methods guarantee a certain level of robust performance for the whole model set, but cannot handle too sophisticated descriptions of model uncertainty. When building a 85