GáborRödönyi ITERATIVEDESIGNOFSTRUCTUREDUNCERTAINTYMODELSANDROBUSTCONTROLLERSFORLINEARSYSTEMSLINEÁRISRENDSZEREKSTRUKTURÁLTBIZONYTALANSÁGIMODELLJÉNEKÉSROBUSZTUSSZABÁLYOZÁSÁNAKITERATÍVTERVEZÉSE

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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

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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 meg- jelöltem.

Budapest, 2010. 05. 22.

Rödönyi Gábor

Az értekezésről készült bírálatok és a jegyzőkönyv a későbbiekben a Budapesti Műszaki és Gaz- daságtudományi 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.

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Édesanyám emlékére Anna Mártának

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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 op- timizing 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.

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CONTENTS

Notations

R,C real and complex fields

L Lebesgue-measurable functions

L² finite energy Lebesgue-measurable functions

RH∞ space of proper and real-rational stable transfer matrices C⁰ imaginary axis of the complex plane

A^T, A^∗ transpose and conjugate transpose of matrix A nx length of vector x

I, I₂, Ix identity matrices possibly with indication of the size (Ix:nx×nx) w1, w2, ... 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 L2-gain systems S^P multiplier set for parametric, polytopic perturbations

B unit ball of a set, e.g., BS∆={∆∈ S∆| k∆k ≤1}

>, <,≤,≥ for matrices it denotes definiteness or semi-definiteness co{A1, ..., An}convex hull of n matrices

F^U,F^L upper- and lower linear fractional transformation P() integral quadratic form specifying perturbation P

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

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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 asrobust control anditerative 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 onvehicle safety enhancement in a lateral control problem and is introduced in Section 1.4.

1.1 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

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bounded set of stable systems. If the uncertainty set isunstructured (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 asubset 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 compu- tational 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.

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1.2 Trends in system identification

Before the 1990s 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 [40, 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 1980s 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 1990s.

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 ofH∞/µmethods. 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 [110] on real parametric uncertainty, [112, 70]

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, 50, 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

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uration 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 [60].

The above formulation motivated the iterative identification and control design methods in both the probabilistic [20, 21, 75, 15, 90] 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., [101, 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, 60] 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 [10, 91, 68] assume deterministic additive noise and additive unstructured perturbation ∆. Based on time-domain model validation results of [108], 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, 102, 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.

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Thewindsurfer 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, model- free 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., [80, 68] where the criterion is the worst- case performance of the system. In [130] 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 [130] 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 aH∞ 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. [Röd09]. 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 calledgeneric 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

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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 exam- ples 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 [130]: 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 pur- poses.

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

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(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 ex- amples 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 [109, 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 as- pects, 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

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ordered, e.g. [120].

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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.

LetLⁿ denote the set of all signals x: [0,∞)7→ Rⁿ that are Lebesgue-measurable.

All piece-wise continuous signals are contained in Lⁿ. A linear subspace of Lⁿ is the set of finite energy signals defined by

Lⁿ2 :={x∈ Lⁿ:kxk2 <∞}, where kxk2 = qR∞

0 kx(t)k²dt denotes L2-norm of signal x. This norm is called the energy of the signal. With the help of the truncation operator

PT :Lⁿ7→ Lⁿ, PTx(t) =

x(t) for t∈[0, T] 0 for t∈(T,∞) an extended signal space Lⁿ2e is defined as

Lⁿ2e={x∈ Lⁿ: PTx∈ Lⁿ2 for all T ≥0}

Any signal that is bounded over a finite interval belongs to this signal space.

A dynamical system M is a mapping M :Lⁿ2e^u 7→ Lⁿ2e^y. The system is causal if its output depends only on past inputs, i.e. PTM(u) = PTM(PTu) for all T ≥ 0, and all inputs u ∈ Lⁿ2e^u. A stable system maps any signal in L2 into a signal that is also contained in L2.

Definition 2.1 (L₂-gain) TheL2-gain of the system M :Lⁿ2e^u 7→ Lⁿ2e^y is defined as kMk2 := sup{kPTM(u)k2

kP_Tuk2 |u∈ Lⁿ2e^u, T ≥0, kP_Tuk2 6= 0}

= inf{γ ∈R| ∀u∈ Lⁿ2e^u, T ≥0 : kPTM(u)k2 ≤γkPTuk2}

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If M is causal thenkMk2 = sup_u∈L₂_,kuk₂_>0^kM(u)k_kuk ²

2 . In the dissertation we restrict our focus on linear, causal systems.

Definition 2.2 (Linear time-invariant (LTI) systems)

˙

x(t) = Ax(t) +Bu(t), y(t) = Cx(t) +Du(t).

Throughout the dissertation M =

A B 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)⁻¹B+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 calli- graphic symbol M represents the block matrix containing the state-space matrices of M

M=

A B C D

The space L^∞ is a Banach space of matrix valued functions that are essentially bounded onjR, with the norm

kMk∞=esssup

ω∈R

¯

σ[M(jω)]

The spaceH∞is a closed subspace ofL∞with functions that are analytic and bounded on the open right-half plane. The H∞ norm defined as

kMk∞= sup

ω∈R

¯

σ[M(jω)]

is equal to the L2-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)

˙

x(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 setS^ρ⊂Rⁿ^ρ.

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2.2. LINEAR MATRIX INEQUALITIES (LMIS) Definition 2.4 (Affine LPV systems) The LPV system is called affine if the state- space matrices depend affinely on the scheduling parameter which belongs to a convex set S^ρ.

M(ρ(t)) :=

A(ρ(t)) B(ρ(t)) C(ρ(t)) D(ρ(t))

=M⁰+ρ1M¹+...+ρnρMⁿ^ρ, ρ∈ S^ρ Definition 2.5 (Linear fractional transformation (LFT)) The LFT is a feedback interconnection of two systems M and ∆. Let M :=

M₁₁ M₁₂ M₂₁ M₂₂

be appropriately partitioned. The lower LFT is defined by

F^L(M,∆) :=M₁₁+M₁₂∆(I−M₂₂∆)⁻¹M₂₁ provided that the inverse exists. The upper LFT is defined by

F^U(M,∆) :=M₂₂+M₂₁∆(I −M₁₁∆)⁻¹M₁₂ provided that the inverse exists.

Definition 2.6 (LFT LPV systems) LFT LPV system FU(M,∆_s) is defined by a LTI system

M :=





A B₁ B₂ C₁ D₁₁ D₁₂ C₂ D₂₁ D₂₂





and time-varying scheduling matrix∆_s(t)which belongs to a convex setS∆s: ∆_s ∈ S∆s. Alternative notion can be that∆s(ρ(t)), ρ∈ S^ρ.

TheL²-gain of the LPV system M(ρ(t)) :=

A(ρ(t)) B(ρ(t)) C(ρ(t)) D(ρ(t))

is defined by

kM(ρ)k2 := sup

ρ∈Sρ

sup

u∈L2,kuk2>0

kM(ρ)uk2

kuk2

2.2 Linear matrix inequalities (LMIs)

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 xi ∈ R is an ex- pression of the form

F(x) :=F₀+x₁F₁+. . .+xmFm >0 (2.1) where F₀, . . . , F_m are real symmetric matrices, F_i =F_i^T ∈R^n×n,i= 1, ..., m.

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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 solutionx 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 λsubject toA(x)< λB(x),B(x)>0andC(x)<0, where A, B and C are affine matrix functions ofx.

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 F(x) =

Q(x) S(x) S(x)^T R(x)

If R(x)>0, then the nonlinear matrix inequality

Q(x)−S(x)R(x)⁻¹S(x)^T >0, is equivalent toF(x)>0. If Q(x)>0, then

R(x)−S(x)^TQ(x)⁻¹S(x)>0 is equivalent toF(x)>0.

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 formΣ(x)≥0, where Σ(x) is an integral quadratic function that can be defined in the frequency-domain as

Σ(x) = 1 2π

Z _∞

−∞

x(jω)^∗Π(jω)x(jω)dω

via a measurable Hermitian bounded mapping calledmultiplier: Π :jω∈ C⁰ →Π(jω) ∈ Cⁿ^x^×n^x,kΠ(jω)k ≤c, wherecis a constant, andx∈ L2. The time-domain counterpart

Σ(x) = Z _∞

0

z(t)^∗P z(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.

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2.3. INTEGRAL QUADRATIC CONSTRAINTS (IQCS) Lemma 2.2 (Frequency-domain inequality (FDI)) SupposeΠis a measurable bounded Hermitian valued mapping on C⁰. Then the following statements are equivalent

• There exists an ǫ >0 such that 1

2π Z _∞

−∞

x(jω)^∗Π(jω)x(jω)dω≤ − ǫ 2π

Z _∞

−∞

x(jω)^∗x(jω)dω

• There exists an ǫ >0 such that

Π(jω)≤ −ǫI for all ω ∈R

•

Π(jω)<0 for all ω∈R∪ {∞}

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

˙

x=Ax+Bu, y =Cx+Du, is controllable. Let M :=

A B

C D

. Let P be a real symmetric matrix. Then the following statements are equivalent

• There exists a nonnegative function V :Rⁿ^x 7→R such that V(x(t₀)) +

Z t1

t0

s(u(t), y(t))dt≥V(x(t₁)), (2.3) s(u, y) =−

u y

T

P u

y

for all t₀ ≤t₁ and all trajectories (u, x, y) of system M.

• For all ω∈Rwithdet(jωI−A)6= 0 there holds I

M(jω) ∗

P I

M(jω)

≤0

• There exists X =X^T ≥0such that A^TX+XA XB

B^TX 0

+

0 I C D

T

P

0 I C D

≤0

The system with the so-named quadratic supply functionsand satisfying (2.3) is called dissipative. All non-strict inequalities can be replaced by strict inequalities and the system is called strictly dissipative.

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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 3.2-3.4 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 [120], [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^∆:={∆ :Lⁿ2e^z 7→ Lⁿ2e^w, ∆is a causal and linear system with finiteL²-gain} and nominal modelM :Lⁿ2e^w 7→ Lⁿ2e^z which is a causal, linear system with finiteL2-gain.

∆

M +

-+

6

-

?

z₀ z

−

w w₀ +

Figure 3.1: The general uncertain feedback configuration for analyzing robust stability

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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 setS^∆u ⊂ S^∆, where

S∆u :={∆_u ∈ RH∞|∆_u = diag{∆₁, ...,∆_τ}, ∆_i(jω)∈Cⁿ^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 setS∆s ⊂ S∆, defined by

S∆s :={∆s(t) :R7→Rⁿ^w^×n^z, ∆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∈[0,1].

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

w₀ z₀

=LM(∆) w

z

, where LM(∆) =

I −∆ M −I

.

Definition 3.1 ((Uniform) robust stability (RS)) The interconnection in Figure 3.1 is uniform robust stable if

• LM(∆) has a causal inverse (well-posedness)

• LM(∆)⁻¹ has finite L2-gain for all∆∈ S∆ (robust stability)

• there exists a common bound on kLM(∆)⁻¹k² for all∆∈ S^∆ (uniformity) The following theorem is the combination of Theorem 3.7 and Theorem 3.8 of reference [120] 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 (L2-gain) perturbations.

Theorem 3.1 (An abstract stability characterization) Let Σ : Lⁿ2^w⁺ⁿ^z 7→ R be an integral quadratic function. Suppose that all ∆∈ S∆ satisfy

Σ

∆z z

≥0 for all z∈ Lⁿ2^z. (3.1) The interconnection in Figure 3.1 is uniform robust stable for all ∆∈ S^∆ if and only if there exists an ǫ >0 with

Σ

w M w

≤ −ǫkwk²2 for all w∈ Lⁿ2^w. (3.2)

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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

Z ∞

−∞

w(jω) z(jω)

∗

Π(jω)

w(jω) z(jω)

dω withw= ∆z is parameterized through multiplier Π(jω) =

Qω(jω) Sω(jω) S_ω(jω)^∗ R_ω(jω)

by specifying all Qω, Sω and Rω 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, 120] several uncertainty types are characterized via IQCs and summarized in the following list. (When multiplier Π is restricted to be a real symmetric matrix, it is denoted by P =

Q S S^T R

throughout the dissertation in accordance with the notation in (2.2)).

• Structured linear causal mappings ∆ = diag{∆1, ...,∆τ} with k∆k^∞ ≤ 1 fulfill (3.1) for the class of multipliers

R= diag{d₁I, ..., dκI}, S= 0, Q=−R, (3.3) where di>0,i= 1, ..., κ, are real scalars.

• Structured linear time-varying uncertainties

∆(t) = diag{∆₁, ...,∆κ} withk∆(t)k∞≤1

form a special class of linear causal mappings and fulfill not only (3.1) but quadratic constraint

∆(t) I

T

P

∆(t) I

≥0 (3.4)

for any P in form (3.3). The quadratic constraint still holds ifP is time-varying.

• Repeated structured time-varying uncertainties

∆(t) = diag{δ₁(t)I, ..., δκ(t)I} withkδ(t)k∞≤1

fulfill the quadratic constraint (3.4) for the class of multipliers parameterized as R= diag{R₁, ..., Rκ}>0, Q=−R, S= diag{S₁, ..., Sκ}, S+S^T = 0, (3.5) where Ri, Si, i = 1, ..., κ, are real matrices. Again, we can generalize to time- varying multipliers.

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∆

M -

z_∆ w_∆

z_p 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

S^P ={P =

Q S S^T R

| Q <0, ∆j

I T

P ∆j

I

>0, 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 k∆k∞ ≤1satisfies IQCs (3.1) with parametrization

Rω(jω) = diag{d₁(ω)I, ..., dκ(ω)I}, Sω(jω) = 0, Qω(jω) =−Rω(jω), with0< di(ω)∈R.

Robust performance can be analyzed on the setup depicted in Figure 3.2 whereM is partitioned as M =

Mu Mup

Mpu Mp

. Performance specifications can be characterized as follows: there exists an ǫ >0 with

Σp

wp

zp

≤ −ǫkwpk²2 for all wp ∈ L2, (3.7) where Σp is an arbitrary mapping Σp :L2 7→R satisfying Σp

0 z_p

≥0. The conditions of meeting the general performance specification (3.7) are given in the following theorem, [120, Theorem 3.16].

Theorem 3.2 (An abstract performance characterization) Suppose that all∆∈ S∆ satisfy the IQC

Σ

w_∆ z∆

≥0 (3.8)

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3.1. ABSTRACT STABILITY AND PERFORMANCE CHARACTERIZATIONS

Suppose there exists an ǫ >0 such that Σ

w_∆ z∆

+ Σp

w_p zp

≤ −ǫ kw∆k²2+kwpk²2

(3.9)

for allw_∆, wp ∈ L2. ThenI−Mu∆has a causal inverse whoseL2-gain is bounded uni- formly for ∆∈ S∆ (uniform robust stability) and uncertain system FU(M,∆) satisfies the performance criterion (3.7).

For example,quadratic performance is defined by the choice Σp

wp

zp

= Z _∞

0

wp

zp

T

Pp

wp

zp

dt, (3.10)

where performance index P_p is a fixed symmetric matrix that satisfies P_p =

Qp Sp

S_p^T Rp

, R_p≥0 (3.11)

As a special case, strict passivity Z _∞

0

zp(t)^Twp(t)dt≤ −ǫ Z _∞

0

wp(t)^Twp(t)dt for all wp∈ L2,

can be specified by Pp =

0 ¹₂I

1 2I 0

. Another case is theinduced L2-norm of a system being less than a given number γ which can be expressed with

Qp=−γpI, Sp = 0, Rp =γ_p⁻¹I, (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{Π₁,Π₂, ...}. Similarly, multiple performance channels (in mixed performance problems) are specified by IQCs whose performance indices Pp,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 inducedL2-norm performance criterion leads to the classical µ-synthesis, which is summarized in the following section.

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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 byµ. It is known that the satisfaction of robust performance (kF^U(M,∆)k∞≤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 ∈ RHⁿ^∞^wp^×n^zp : zp 7−→ wp, k∆pk^∞ ≤ 1. See also [36, 37, 137, 120, 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)) = 0}

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 withk∆k∞≤ β¹ loop F^U(M,∆) is well-posed, internally stable and kF^U(M,∆)k∞≤β if and only ifµ_∆_a(M)< β .

The theory ofµ-analysis has been developed for mixedfull 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 := {di | di, d⁻¹_i ∈ RH∞, i = 1, ..., τ} and repeated diagonal matrices DL:= diag{d₁Iz_∆,1, ..., dτIz_∆,τ, Izp}andDR:= diag{d₁Iw_∆,1, ..., dτIw_∆,τ, Iwp}. 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_Dσ(D¯ _L(jω)M(jω)D_R⁻¹(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.

• Initialization: DL:=I,DR:=I

• NominalH∞-controller designvia bisection algorithm: a scalarγ¯is minimized in the controller parameters subject to

¯

σ(DLM D_R⁻¹)<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.

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3.3. ROBUST CONTROL OF UNCERTAIN LTI SYSTEMS USING IQCS

• µ analysis. Scalars γk are minimized over a frequency grid ωk,k= 1, ..., nω, in scaling matricesD_L,kandD_R,k, which are parameterized by setD_k={d_i,k|d_i,k∈ R, i= 1, ..., τ} as before.

infD_kσ(D¯ L,kM(jωk)D_R,k⁻¹)< γk

This is an LMI problem. Then, stable, stable invertible transfer functions (di ∈ RH∞) are fitted in magnitude to the positive real scalarsd_i,k,i= 1, ..., τ, which results in updated values of DL and DR.

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_γ¹₃ while the L2-gain of the closed-loop system (performance) by ^γ_γ³₂.

Lemma 3.2 For all∆∈ S∆u withk∆k∞≤ ^γγ¹3 loopF^U(M,∆)is well-posed, internally stable and kF^U(M,∆)k∞≤ ^γγ³2 if and only if µ_∆_a(Mdiag{γ₁I, γ₂I})< γ₃.

A special case of scalings with γ₁ =γ₃ = γs and γ₂ = 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 FU(M,∆) is well-posed, internally stable and for all ω: σ¯(FU(M(jω),∆(jω)))≤γ_s(jω) if and only if

µ_∆_a(M(jω)diag{γs(jω)Iw∆, Iwp})< γs(jω) (3.13) It is shown in Chapter 4 how the skew µanalysis results can be used for controller synthesis. Details of an iterative method, similar to the above D-K iteration, are elab- orated 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. [120], 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.

3.3.1 Problem formulation

The closed-loop system in consideration is plotted in Figure 3.3. Augmented plantGis an LTI system







˙ x z_∆

z_p yK





 =







A Bu Bp B Cu Du Dup Eu

C_p D_pu D_p E_p C Fu Fp 0











 x w_∆

w_p uK







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G

∆

K

wp

u_K z∆

zp

- yK

w∆ -

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 L2-gain and described indirectly by IQC (3.8) in the form

1 2π

Z _∞

−∞

w∆(jω) z_∆(jω)

∗

Π(jω)

w∆(jω) z_∆(jω)

dω≥0, (3.14)

Π(jω) =

Q_ω(jω) S_ω(jω) Sω(jω)^∗ Rω(jω)

(3.15) The controller K is an LTI system,uK =KyK where

K=

Ac Bc

Cc Dc

Robust quadratic performance of the controlled system is specified by (3.7) with (3.10) 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∆.

3.3.2 Analysis

Let the closed-loop nominal system be denoted by M = FL(G, K) and partitioned according to

z_∆ zp

=

Mu Mup

Mpu Mp

| {z }

M

w_∆ wp

(3.16)

GáborRödönyi ITERATIVEDESIGNOFSTRUCTUREDUNCERTAINTYMODELSANDROBUSTCONTROLLERSFORLINEARSYSTEMSLINEÁRISRENDSZEREKSTRUKTURÁLTBIZONYTALANSÁGIMODELLJÉNEKÉSROBUSZTUSSZABÁLYOZÁSÁNAKITERATÍVTERVEZÉSE

ITERATIVE DESIGN OF STRUCTURED UNCERTAINTY MODELS AND ROBUST CONTROLLERS FOR LINEAR

SYSTEMS

LINEÁRIS RENDSZEREK STRUKTURÁLT

BIZONYTALANSÁGI MODELLJÉNEK ÉS ROBUSZTUS SZABÁLYOZÁSÁNAK ITERATÍV TERVEZÉSE

Nyilatkozat

Contents

Notations

Chapter 1

Introduction

1.1 Robust control background and motivation

1.2 Trends in system identification

1.3 Iterative identification and control background

1.4 Vehicle steering problems

1.5 Layout of the thesis

Chapter 2

Basic notions

2.1 Signal and system classes

2.2 Linear matrix inequalities (LMIs)

2.3 Integral quadratic constraints (IQCs)

Chapter 3

Analysis and synthesis of LTI and LPV systems

3.1 Abstract stability and performance characterizations

3.2 Analysis and synthesis in H

/µ framework

3.3 Robust control of uncertain LTI systems using IQCs