Optimal Decoupling of Dynamical Systems

Systems

A Convex Approach With Aerospace Applications

Ph.D. Dissertation by Tam´ as Ba´ ar

In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

Department of Control for Transportation and Vehicle Systems Budapest University of Technology and Economics

Hungary 2023 Supervisor:

Tam´as Luspay Ph.D.

”It’s a kind of magic” - Queen

Undersigned, Tam´as Ba´ar, 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 con- tents, have been unambiguously marked by a reference to the source.

Nyilatkozat

Alul´ırott, Ba´ar Tam´as kijelentem, hogy ezt a doktori ´ertekez´est magam k´esz´ıtettem ´es abban csak a megadott forr´asokat haszn´altam fel. Minden olyan r´eszt, amelyet sz´o szerint, vagy azonos tartalomban, de ´atfogalmazva m´as forr´asb´ol ´atvettem, egy´ertelm˝uen, a forr´as megad´as´aval meg- jel¨oltem.

Budapest, January 11, 2023

Tam´as Ba´ar

Acknowledgments . . . vi

Part I Introduction 1 Introduction 2 1.1 Motivation and connection to the existing literature . . . 2

1.2 Contributions . . . 5

1.3 Outline . . . 5

2 Preliminaries 7 2.1 Linear Time-Invariant systems . . . 7

2.2 Convex optimization and LMIs . . . 9

2.3 Sensitivity analysis of dynamic MIMO systems . . . 11

2.3.1 Minimum sensitivity . . . 11

2.3.2 Maximum sensitivity . . . 15

2.4 Systems with uncertainties . . . 16

2.4.1 Polytopic modeling . . . 16

2.4.2 Integral Quadratic Constraints . . . 17

2.5 Linear Parameter-Varying systems . . . 23

Part II Subsystem decoupling 3 Optimal decoupling of Linear Time-Invariant systems 28 3.1 Introduction . . . 28

3.2 Problem statement . . . 31

3.3 LTI subsystem decoupling algorithm . . . 32

3.3.1 Input blend calculation . . . 32

3.3.2 Output blend calculation . . . 35

3.3.3 Academic example . . . 39

3.3.4 Batch test . . . 42

3.4 The decoupling of multiple modes . . . 42

3.4.1 Single controller approach . . . 43

3.4.2 Multiple controller approach . . . 44

3.4.3 Examples for the decoupling of multiple modes . . . 45

3.5 Contribution . . . 50

4 Optimal decoupling of uncertain systems 51 4.1 The robust decoupling problem . . . 52

4.2 The polytopic approach . . . 53

4.3 The static IQC multiplier approach . . . 55

4.3.1 Robust analysis over finite frequency . . . 55

4.3.2 Decoupling by static IQC multipliers . . . 57

4.4 The dynamic IQC multiplier approach . . . 60

4.4.1 Robust minimum sensitivity . . . 60

4.4.2 Robust minimum sensitivity over finite frequency . . . 62

4.4.3 Decoupling by dynamic IQC multipliers . . . 63

4.5 Computational complexity . . . 65

4.6 Numerical Examples . . . 68

4.6.1 Robust sensitivity analysis by dynamic IQCs . . . 68

4.6.2 Academic example: subsystem decoupling . . . 69

4.7 Contribution . . . 73

5 Optimal decoupling of Linear Parameter-Varying systems 75 5.1 The parameter-varying subsystem decoupling problem . . . 75

5.2 Polytopic decoupling of parameter-varying subsystems . . . 76

5.3 Grid-based decoupling of parameter-varying subsystems . . . 79

5.4 Computational complexity . . . 84

5.5 Academic example . . . 85

5.6 Contribution . . . 90

Part III Applications 6 Flutter control 92 6.1 Aircraft model . . . 93

6.2 Linear Time-Invariant subsystem decoupling . . . 95

6.3 Linear Parameter-Varying subsystem decoupling . . . 102

6.4 Robust subsystem decoupling by dynamic IQCs . . . 103

6.5 Contribution . . . 106

7 Distributed propulsion aircraft 107 7.1 Problem statement . . . 108

7.2 Aircraft model . . . 108

7.3 Reduced order controller design . . . 110

7.4 Reconfiguration control design . . . 115

7.5 Numerical results . . . 122

7.5.1 Doublet response . . . 122

7.5.2 Path following simulation . . . 123

Conclusion 128 8 Closing remarks and future work 128 Publications of the Author 130 References 138 Appendices 140 A Mathematical background 140 A.1 Norms . . . 140

B Analysis of Linear Time-Invariant systems 143 B.1 Gains of a MIMO system . . . 143

B.2 Proof of Lemma 6 . . . 144

C Uncertain system modeling 146 C.1 On the selection of static IQC multipliers . . . 146 C.2 Dual integral quadratic constraints . . . 147

D Alternating projections 149

First of all I would like to express my gratitude to my supervisor Tam´as Luspay, for his tremen- dous help and endless support. He had a strong vision for the outline of the research topic, and he has been open to explore new directions. Tam´as made me acquire new skills which I never thought I could. I am profoundly grateful for his encouragement and unending patience.

I would like to thank to P´eter Bauer. From the time I have started to work with him as a bachelor student, he has invested a huge effort in my professional development, with continuous support through my masters and Ph.D. studies.

I wish to express my sincere thanks to Roland T´oth for his valuable advice on the dissertation and for drawing my attention to a new area of applications of the developed results.

I also thank my colleagues at SZTAKI, B´alint Patartics, B´ela Takarics, M´arton Farkas and B´alint Vanek for their help, guidance and interesting discussions over the years.

Lastly, I address my special thanks to my family for encouragement, and to Vir´ag Zsig´o for her abundant love and continuous support as a girlfriend and as a wife as well.

List of Acronyms and Abbreviations ASVD Analytic Singular Value Decomposition BMI Bilinear Matrix Inequality

CG Center of Gravity

DEP Distributed Electric Propulsion DLM Doublet Lattice Method

GKYP Generalized Kalman-Yakubovich-Popov IMU Inertial Measurement Unit

IQC Integral Quadratic Constraint LFT Linear Fractional Transformation LMI Linear Matrix Inequality

LPV Linear Parameter-Varying LTI Linear Time-Invariant LTV Linear Time-Varying

MIMO Multiple Input Multiple Output PID Proportional Integral Derivative SISO Single Input Single Output SVD Singular Value Decomposition UAV Unmanned Aerial Vehicle VLM Vortex Lattice Method e.g. for example (exempli gratia) i.e. that is (id est)

R⁺ Set of positive real numbers Rⁿ Set of n dimensional real vectors

R^n×m Set of real matrices with nrows and m columns C Set of complex numbers

C^n×m Set of complex matrices with n rows andm columns

RL_∞ Set of proper, rational functions with real coefficients, without poles on the imaginary axis RH∞ Subset of RL∞, the functions that are analytic in the closed right half complex plane S^m Set of m×msymmetric real matrices

Hⁿ The set of n×nHermitian matrices, A∈Hⁿ ifA^∗ =A A^T Transpose of matrix A

A^∗ Conjugate transpose A⁻¹ Inverse of matrix A

A≻0 Positive definite matrix, i.e. x^TAx >0, ∀x̸= 0, x∈Rⁿ A⪰0 Positive semi-definite matrix, i.e. x^TAx≥0, ∀x̸= 0, x∈Rⁿ In n dimensional identity matrix

s Laplace variable j Imaginary unit

¯

z Complex conjugate of z∈C

M^∼ Para-Hermitian conjugate, M ∈RL^m×n_∞ and M^∼(s) :=M(−¯s)^T L2 Lebesgue-2 space, i.e., y∈ L2 if||y||²2 =R^∞

0 |y(t)|²dt <∞

⋆ Symmetric term in an expression, i.e., ⋆=A^T inA+⋆=A+A^T

∆ Uncertainty block, collecting all uncertainties δ Scalar parametric uncertainty

∆_i Dynamic uncertainty, a sub-block of ∆ End of proof

∀ for all

A large scale complex dynamic system is generally composed of several subsystems. These serve as building blocks, describing various parts of the underlying dynamics. The control of the system as a whole might be a challenging task, however, splitting up the control problem to distinct subproblems may ease the control design process. A straightforward idea is the control of the individual building blocks (subsystems), where on this local level the complexity of the design is significantly reduced. The thesis explores the question of how to interact with a specific subsystem, without interfering the others. To be more specific, static input and output transformation vectors are introduced, which are ensuring that, a given controller will only interact with the targeted subsystem.

For the calculation of each input- and output transformation, the subsystems are rendered into two groups. One consists of those to be controlled, while others belong to the ones to be decoupled. The transformation vectors are designed such that they are maximizing the transfer through the subsystems to be controlled, while they are minimizing it through the other dynam- ics (to be decoupled). These transformation vectors are squaring down the problem for each subsystem to be a Single Input Single Output (SISO) one. At the input side the transformation assures that the excitation of the controlled subsystem is maximized, while interactions with the decoupled one are minimized. Similarly the output transformation maximizes the information regarding the controlled subsystem, and at the same time suppresses the effects of the remaining dynamics.

The thesis covers the decoupling of various system classes, involving Linear Time-Invariant (LTI), uncertain, and Linear Parameter-Varying (LPV) systems as well. The proposed synthesis methods are heavily building on the minimum and maximum sensitivity characterizations of these systems. The maximum or peak gain of a system is a widely discussed topic in the control systems literature, however, the question of minimum sensitivity has received less attention.

When the minimum sensitivity analysis conditions are not yet developed for a certain system class, its extension is presented in detail. The synthesis methods for the transformations are derived from the analysis conditions corresponding to the minimum and maximum sensitivity characterizations.

The analysis and synthesis methods are expressed as optimization problems with LMI con- straints, because this formulation guarantees an efficient way to solve the underlying problems.

System uncertainties are handled by the most widely accepted methods, including polytopic modeling and integral quadratic constraint (static and dynamic as well) based uncertainty de- scriptions. The thesis handles plant nonlinearities in two different ways. If the nonlinearity is induced by a continuously measurable parameter, then a linear parameter-varying framework is applied. Otherwise the previously mentioned uncertainty handling methodologies are followed.

The developed decoupling methods open up new design strategies in structured observer and controller synthesis theory. These allow specific designs for the individual targeted subsystems, potentially resulting in smaller observer or controller dimensions. These results are discussed in a separate chapter, along with corresponding examples.

Throughout the thesis, easily reproducible examples are provided after each developed anal- ysis or synthesis condition. These serve as introductory examples, allowing the reader to focus on the core of the concepts without overcomplicating the discussions. Special, real life examples, originating from the aerospace control field are presented in a distinct chapter. These examples necessitates more heavy computations, with larger system dimensions.

A nagym´eret˝u komplex rendszerek a legt¨obbsz¨or t¨obb, kisebb alrendszer egy¨uttesek´ent mod- ellezhet˝oek, melyek ´ep´ıt˝oelemekk´ent a dinamika k¨ul¨onb¨oz˝o r´eszeit ´ırj´ak le. Ir´any´ıt´as szem- pontj´ab´ol a teljes rendszernek az egyk´ent val´o kezel´ese sokszor bonyolult feladat, azonban a probl´ema r´eszfeladatokra bont´as´aval az ir´any´ıt´astervez´es k¨onny´ıthet˝o. Ebben az esetben egy k´ezenfekv˝o megk¨ozel´ıt´es a rendszert alkot´o ´ep´ıt˝oelemek (alrendszerek) egym´ast´ol f¨uggetlen ir´any´ıt´asa, ahol ezeken a lok´alis szinteken az ir´any´ıt´asi feladat komplexit´asa cs¨okken. A dissz- ert´aci´o azt a k´erd´est j´arja k¨or¨ul, hogy hogyan lehet egy kiv´alasztott alrendszerrel ´ugy kapcsolatba l´epni, hogy az nem befoly´asolja a rendszerdinamika m´as alegys´egeit. Pontosabban megfogal- mazva, olyan statikus be- ´es kimeneti transzform´aci´ok ker¨ulnek bevezet´esre, amelyek biztos´ıtj´ak, hogy egy k´es˝obbiekben megtervezett szab´alyz´o csak a c´elzott alrendszerrel l´epjen kapcsolatba.

A t´argyalt tervez´esi elj´ar´asban, a statikus be- ´es kimeneti transzform´aci´ok meghat´aroz´as´ahoz az alrendszereket k´et csoportba sz¨uks´eges sorolni. Az els˝oh¨oz tartoznak az ir´any´ıtand´oak, m´ıg a m´asodikhoz a lecsatoland´oak (amelyekkel nem szeretn´enk kapcsolatba l´epni). A tran- szform´aci´okat alkot´o vektorok ´ugy ker¨ulnek meghat´aroz´asra, hogy azok az ir´any´ıtand´o alrend- szereken kereszt¨ul az ´atvitelt n¨ovelj´ek, m´ıg a lecsatoland´oakon ´at cs¨okkents´ek. Bel´athat´o, hogy ezen transzform´aci´ok alkalmaz´asa az egyes alrendszerekhez tartoz´o ir´any´ıt´asi feladatot egy, egy bemenet˝u ´es egy kimenet˝u rendszer ir´any´ıt´asi feladat´av´a reduk´alja. A bemeneti oldalon a transz- form´aci´o biztos´ıtja, hogy az ir´any´ıtand´o alrendszer gerjeszt´ese maxim´alis, m´ıg a lecsatoland´oak´e minim´alis legyen. Hasonl´oan, a kimeneti oldalon az ir´any´ıtand´o alrendszerr˝ol kinyerhet˝o in- form´aci´o maxim´alis, m´ıg a lecsatoland´oak hat´as´at elnyomja az alkalmazott transzform´aci´o.

A disszert´aci´o sz´amos, az irodalomban sz´eles k¨orben alkalmazott rendszeroszt´aly sz´etcsatol´asi probl´em´aj´at t´argyalja, ide ´ertve a line´aris id˝oinvari´ans, a bizonytalan ´es a line´aris v´altoz´o param´eter˝u rendszereket. A bemutatott tervez´esi elj´ar´asok a dinamikus rendszerek legkisebb ´es legnagyobb ´erz´ekenys´eg´en (er˝os´ıt´es´en) alapulnak. A rendszer legnagyobb er˝os´ıt´ese a robusztus ir´any´ıt´asok t´emak¨or´eben j´ol ismert ´es sz´eles k¨orben alkalmazott. Ezzel ellent´etben a legkisebb er˝os´ıt´es kevesebb figyelmet kapott a kapcsol´od´o szakirodalomban. Azokban az esetekben amikor a minim´alis ´erz´ekenys´egre vonatkoz´o anal´ızis felt´etelek m´eg nem ker¨ultek kidolgoz´asra egy adott rendszeroszt´alyra, a kiterjeszt´es¨uk r´eszletes bemutat´asra ker¨ul. A sz´etcsatol´ast biztos´ıt´o transz- form´aci´ok tervez´ese a rendszer ´erz´ekenys´egeket le´ır´o anal´ızis felt´etelek m´odos´ıt´as´an alapul.

Az anal´ızis ´es szint´ezis felt´etelek fel´ır´asa egyar´ant line´aris m´atrix egyenl˝otlens´egek form´aj´aban t¨ort´enik. Amellett, hogy ez a megk¨ozel´ıt´es a probl´em´anak egy hat´ekony megold´as´at teszi lehet˝ov´e, egyben j´ol is illeszkedik az eddig kialakult szakirodalomba. A rendszermodellekben fellelhet˝o bizonytalan komponensek hat´asainak le´ır´asa vagy polit´opikus modellez´esi elj´ar´assal, vagy kvadratikus integr´al felt´etelek seg´ıts´eg´evel t¨ort´enik. A rendszerben fellelhet˝o k¨ul¨onb¨oz˝o nemlinearit´asok kezel´es´ere k´et elj´ar´ast t´argyal a disszert´aci´o. Amennyiben a nemlinearit´ast egy folytonosan v´altoz´o, m´erhet˝o param´eter okozza, akkor a line´arisan v´altoz´o param´eterekre vonatkoz´o megk¨ozel´ıt´es ker¨ul alkalmaz´asra. M´ask¨ul¨onben a m´ar eml´ıtett, bizonytalan rendsz- ereket le´ır´o technik´ak sz¨uks´egesek.

A kidolgozott sz´etcsatol´asi elj´ar´asok ´uj lehet˝os´egeket nyithatnak meg a struktur´alt szab´alyz´o tervez´es elm´elet´eben. Az alkalmaz´asuk c´elzott tervez´est tesz lehet˝ov´e k¨ul¨onb¨oz˝o alrendszerekhez tartoz´oan, amelyek v´arhat´oan egyszer˝ubb, kisebb ´allapotdimenzi´oj´u szab´alyz´okat eredm´enyeznek.

A disszert´aci´o nagy hangs´ulyt helyez a k¨onnyen reproduk´alhat´o p´eld´akra, melyek az ´ujonnan bevezetett anal´ızis ´es szint´ezis felt´etelek k¨onnyebb meg´ert´es´et szolg´alj´ak. Ezen bevezet˝o p´eld´ak lehet˝ov´e teszik az olvas´o sz´am´ara, hogy a r´eszletek t´ulbonyol´ıt´asa n´elk¨ul a l´enyegre f´okusz´aljon.

Val´os rendszereken alapul´o alkalmaz´asi p´eld´ak t´argyal´asa egy k¨ul¨on fejezetben t¨ort´enik.

Introduction

Introduction

1.1 Motivation and connection to the existing literature

In the control of multivariable complex systems it is often desirable to ease the complexity of the underlying analysis or synthesis problem. In the vast field of large-scale dynamical systems many approaches have been developed for this purpose in the past decades. These methods can be categorized into three main groups [10]. Decentralization aims for separate control design processes and their independent implementation. Decomposition aims for reducing the computational complexity by breaking the system into subsystems. Model reduction seeks for an approximate dynamical description, with lowered complexity.

The thesis focuses on the decoupling (or decomposition) of dynamical systems, where the general aim is to control a certain fraction of the system, without affecting other parts. Synthesis methods are presented for designing input and output transformation vectors in order to assure decoupled control of individual subsystems with simple SISO controllers. The approach is based on theH−index and theH∞norm of dynamical systems. TheH−index is a sensitivity measure mainly used in fault detection, based on the smallest singular value of a transfer function matrix over a given frequency range [74]. By its maximization between given inputs and outputs the system’s sensitivity can be increased. Oppositely, the H∞ norm defines the maximal singular value of a transfer function matrix and it is mainly used in robust analysis and synthesis problems [116]. By minimizing the H∞norm, the maximum sensitivity of the transfer function matrix is reduced. The present approach seeks input and output blend vectors which are maximizing the sensitivity for a given subsystem, while minimizing it for another one. This way, decoupling can be achieved and consequently a suitably designed control law will affect one subsystem, while leaving unattained the other one(s)^*. After laying down the basics of the algorithm, discussion is not limited to linear time-invariant systems, and investigations are extended to uncertain linear and parameter-varying systems as well. System uncertainties are dealt with the most widely accepted methods, including polytopic modeling and integral quadratic constraint (static and dynamic as well) based uncertainty descriptions. In the thesis plant nonlinearities are managed in two different ways. If the nonlinearity is induced by a continuously measurable parameter, than a linear parameter-varying framework is applied. Otherwise the previously mentioned uncertainty handling methodologies are followed.

The method is closely related to various well established fields in the control theory. The most important ones are the decoupled control, structured control, squaring down, and modal control.

*Note that the thesis focuses on the design of input- and output transformations, and not the controller itself.

From a certain point of view these transformation vectors are part of the controller, but we rather look at them as interfaces which assure that the control designer will only interact with the targeted dynamics. After applying the transformation vectors, the controllers can be designed by any suitable design approaches in the literature for this reduced problem.

This decoupling objective is in line with the recent trends of systems- and control engineer- ing aiming for the design of structured controllers [1], where each block of the controller may correspond to a specific subsystem. A conventional decoupling methodology in the control sys- tem literature is the so called input-output decoupling, which is a frequently used approach to simplify the control system design by enforcing a diagonal controller structure. This is a well established research field [75], and most approaches trace back to the application of a suitable method which converts the system into a diagonally dominant one (such as: decoupling by static and dynamic pre- and post-compensators [116, 90], decoupling by state-feedback [133], etc.). A common feature is that the outputs are defined to be the controlled variables. These methods limit the interaction between certain loops, and consequently open up the possibility to design a feedback controller with diagonal structure, where each diagonal block in the controller’s trans- fer function matrix is responsible for controlling one output. However, it has been noted [133]

that these methods are very sensitive to modeling errors and plant uncertainties.

Accordingly, attempts for robust decoupling have also gained attention in the control com- munity. A Linear Matrix Inequality (LMI) based robust state-feedback design approach for uncertain systems is presented in [133], which results in a diagonally dominant closed-loop plant. The method relies on the LMI description of the minimum and maximum gains of a dy- namical system; the designed state-feedback maximizes the transfer through the diagonal term of the closed-loop transfer function matrix, and minimizes the maximum sensitivities over the off diagonal terms. Due to the presence of uncertainties exact decoupling is not possible, hence [133] labels this approach as near decoupling. However, several subsystems may contribute to a decoupled input-output pair, while the approach presented in this thesis assures that only the targeted subsystems are excited.

Another main direction in decoupled control design is the approach of controlling selected subsystems by specific control laws. Here, the frequency-wise separation of the different subsys- tems is one well-known tool. Traditionally notch-, or roll-off filters are introduced to suppress certain frequency ranges, limiting the controller’s interaction with various parts of the dynamics [26, 95]. In this framework, an H∞ closed-loop shaping approach has been presented in [125], involving band-stop weighting functions to achieve decoupled interaction between the plant and the controller. The concept of dynamic filters often leads to satisfactory results with good robust- ness properties against input and output multiplicative uncertainties, however, the dimension of the controller is increased by each applied weighting filter.

The proposed approach fits more closely into a recently developed trend in the decoupled control design; which issubsystem decoupling by means of suitable input and output transforma- tions. The advantage of these approaches is the unaltered dimension of the underlying control problem: the static transformations convert the design into a SISO one. The method of ’Modal Isolation and Damping for Adaptive Aeroelastic Suppression’ (MIDAAS) [32] is a constrained least-squares optimization based algorithm, which designs controller for specifically damping the undesired dynamical components in the system, without affecting the remaining dynamics.

For this purpose a special combination of the available input- and output signals are used. It has been implemented on an aerospace system to achieve flutter suppression without interact- ing with the aircraft’s rigid body dynamics. Another static decoupling approach is presented in [98] and [100] which relies on a joint H2 norm based input- and output-blend calculation method to maximize the controllability and observability of the selected modes and accordingly to facilitate their decoupled control. The method has been successfully applied in a gust load alleviation system on an experimental flexible wing [101], and in a structured controller design of a flexible wing flutter demonstrator aircraft for suppressing unstable wing oscillations [102].

Despite their successful applications, these approaches do not have any extensions for uncertain or parameter-varying systems.

During the discussion of subsystem decoupling methods by input-, output transformation vectors, one cannot avoid mentioning the so called squaring down technique. Its roots were founded in the 1970’s -80’s by [66, 106], and [107] among others, while it recently gained attention

in [143] and [142]. It employs static or dynamic pre- and postcompensators, and used for inverse- based control law design. When a system has a different amount of inputs and outputs (with possible open loop right half plane zeros), then by squaring down their dimensions are reduced to an equal number, while the resulting system is invertible. The squaring down process has a significant effect on the zero structure of the transformed system. The squared down system always has an additional set of finite zeros apart from the zeros inherited from the original nonsquare system. Right half plane zeros has adverse and prohibitive effects on the control system design [116]. The additional zeros that are due to the squaring down process may be placed at desired locations in the open left-half complex plane by choosing appropriate pre- and postcompensators. However, this is not always possible with static compensators.

Before proceeding it is important to take one further note. An LTI system can be brought to a subsystem form if its state matrix is diagonalizable by a corresponding so called modal transformation. However, even in this specific modal form, the subsystems are still coupled through the input and output matrices. This means a certain input might excite all subsystems, and a given measurement may contain information from all subsystems as well. By implying input and output transformations, which decouple a certain subsystem from the remaining dynamics, the proposed approach assures that these couplings are minimized. This way the applied method can be viewed as an extension of the modal transformation, which points in the direction of a diagonal transfer function matrix, having the subsystems (or modes) on its diagonal. This property indicates a strong connection to the literature of modal control, where the design aim is to influence certain modes of a system.

Traditionally system modes have a great importance in structural dynamics [44, 43] and in aerospace control. In case of the latter one, they are widely used to understand and describe the lateral, longitudinal motions [28, 118] and flexible deformations [111, 125] of an aircraft. Modal control has an exhaustive literature [43, 83], with dozens of applications to aerospace examples [117, 98]. Furthermore, note that simple intuitive input combinations are already used in the aerospace control for reducing the design complexity, and to assuring interaction with a certain part of the dynamics. Such an example is the joint use of the left and right ailerons to interact with the lateral dynamics of the aircraft. In this case, the virtual aileron command is given as u_ail=u_L,ail−u_R,ail, with a corresponding input combinationk^T_u =

1 −1

. Similarly a blend of k^T_u =

1 1

is frequently used to combine elevator deflections to excite solely the longitudinal dynamics, by a symmetric actuation on both sides. However, in case of a large number of inputs and outputs, the targeted interaction with a specific subsystem (mode) is a challenging task, which can be eased by the application of the decoupling tools developed in the dissertation.

The theoretical research has originally been motivated by the aerospace industry, with special attention on the control of the flexible structure of an aircraft. It is shown that, the introduced methods are capable to find complex, less intuitive input and output combinations to isolate the flexible dynamics, and so to facilitate noninteracting^„ flutter suppression control law synthesis.

Furthermore, as Chapter 7 shows the developed approach is also valuable for the lateral control augmentation of a distributed propulsion aircraft.

Nevertheless the proposed methods are general tools which may be applied to a broader set of control problems, where a targeted interaction with a specific over-actuated subsystem is needed.

In order to emphasize this generality, beside the aerospace applications some connections are also drawn to subsystem identification and to the control of wafer scanners (complex machines used in the semiconductor industry [97]). The dissertation is centered around the decoupling of three generic fundamental system classes, involving linear time-invariant, uncertain, and linear parameter-varying systems. The next section discusses these contributions in detail.

„The flutter controller is not affecting the rigid body dynamics and the baseline controller.

1.2 Contributions

The novel contributions of this dissertation are listed here. The majority of them has appeared previously in conference or journal papers, and they are referenced accordingly. These contri- butions are related to the minimum sensitivity calculations for various system classes, to the development of a subsystem decoupling algorithm, and to corresponding aerospace applications.

These results strongly build on each other. To emphasize these connections, their relation is depicted in Figure 1.1.

1, Subsystem decoupling algorithm for linear time-invariant systems. In Chapter 3, Section 3.3 a subsystem decoupling algorithm is presented for continuous time linear time- invariant systems. The method yields static input and output transformations which isolate the targeted subsystem from the remaining dynamics. This facilitates controller synthesis which only interacts with the targeted dynamics. The corresponding thesis is Thesis 1, and the related publications are [TB6, TB14, TB3] and [TB7].

2, The robust Minimum Gain Lemma. In Chapter 4, Sections 4.3 and 4.4 linear matrix inequality based analysis conditions are developed for characterizing the minimum gain of uncertain systems relying on integral quadratic constraints. These formulations are extensions of the widely accepted minimum sensitivity measures for LTI systems, and can be calculated over finite and infinite frequency ranges as well. This is summarized in Thesis 2, with a corresponding publication in [TB9].

3, Subsystem decoupling algorithm for uncertain linear systems. The subsystem de- coupling algorithm developed for LTI systems is extended to the class of uncertain systems.

By applying integral quadratic constraints, the proposed method can handle various sources of uncertainties. In Chapter 4, Section 4.3.2 discusses the decoupling approach building on static IQCs, while in Section 4.4.3 dynamic IQCs are applied. The related thesis is Thesis 3, and the results have been appeared in [TB8].

4, The decoupling of parameter-varying subsystems. Chapter 5, Section 5.3 discusses the minimum sensitivity of parameter-varying systems, when the input signal is constrained to have a finite frequency spectra. This allows the extension of minimum sensitivity of parameter-varying systems to a limited frequency range, which has crucial importance for the decoupling design. A corresponding LMI based analysis condition is provided. In Section 5.3 the subsystem decoupling algorithm is discussed for linear parameter-varying systems.

This is solved over the resulting infinite set of LMIs by applying a gridding method. These are discussed in Thesis 4, along with a corresponding publication [TB13].

5, Aerospace application for flutter suppression control. In Chapter 6, the previously developed subsystem decoupling algorithms (Thesis 1, Thesis 3 and Thesis 4) are applied to the model of a flexible wing aircraft. These techniques assure that a suitably designed flutter controller will solely interact with the targeted flexible dynamics. The decoupling performance is evaluated in the frequency and time domain as well. The results are summarized in Thesis 5. For providing a highlight of corresponding control law synthesis, Section 6.2 collects simple control design examples for LTI systems. The corresponding publications are [TB6, TB14, TB8, TB13].

1.3 Outline

The thesis covers the topic of subsystem decoupling by static input and output transformations for various system classes. The dissertation consists of three parts.

Linear Time-Invariant (LTI) systems

Analysis H−: [74]

H^∞: [109]

Decoupling Thesis 1.

Uncertain systems

Analysis H−: Thesis 2.

H^∞: [94]

Decoupling Thesis 3.

Linear Parameter-Varying (LPV) systems

Analysis H−: [72]

H^∞: [136]

Decoupling Thesis 4.

Aerospace applications

Control of a distributed propulsion aircraft

Thesis 5.

Flutter suppression control

Figure 1.1: Thesis overview, where H− and H∞ denote a system’s minimum and maximum sensitivities

Part I collects the necessary prerequisites in Chapter 2. The discussion starts with important definitions and properties of linear time-invariant systems (Section 2.1), a short introduction to convex optimization and its connection to state-space theory (Section 2.2). Next linear matrix inequality based sensitivity analysis conditions are given for LTI systems (Section 2.3). In case one’s knowledge of the underlying plant dynamics is uncertain, then special modeling, analysis and synthesis techniques have to applied. These are reviewed in Section 2.4. Last, but not least the class of linear parameter-varying systems (being continuous functions of time-varying parameters) are overviewed in Section 2.5.

Part II contains the main results. The proposed subsystem decoupling approach is presented in Chapter 3 for linear time-invariant systems. Based on a simple academic example, the impor- tant properties of the decoupling transformations are discussed in detail. Chapter 4 discusses the extension of the subsystem decoupling approach for uncertain systems. For this a necessary prerequisite is the minimum sensitivity analysis condition for uncertain systems with various sources of uncertainties. These are discussed in Sections 4.3 and 4.4 for static and dynamic integral quadratic constraints respectively. The case of linear parameter-varying systems is ad- dressed in Chapter 5, where the special case of minimum sensitivity over a finite frequency range is discussed as well. Simple examples are provided to evaluate the proposed approaches.

Part III is devoted to aerospace applications. This part explores the various application possibilities of the proposed decoupling transformations. Chapter 6 evaluates the benefits of the decoupling transformations in flutter control, based on the model of a flexible wing aircraft.

Section 6.2 discusses their use cases in static state- and output-feedback, and then for dynamic output-feedback design. Sections 6.3 and 6.4 provide parameter-varying and robust subsystem decoupling examples targeting the flexible flutter modes. Chapter 7 highlights the benefits of reducing the input dimension by applying a static input transformation to a distributed propulsion aircraft. The chapter aims to design a control law for roll control augmentation, which uses a suitable combination of various propulsion inputs.

Preliminaries

The work in this dissertation covers various system classes involving Linear Time-Invariant (LTI), uncertain and Linear Parameter-Varying (LPV) systems. This chapter collects the nec- essary prerequisites for the reader. Section 2.1 covers the general LTI systems, focusing on those properties which will exhaustively used in later chapters. Section 2.2 discusses convex optimiza- tion and corresponding Linear Matrix Inequality (LMI) techniques. These are used in Section 2.3 to formalize sensitivity analysis conditions for LTI MIMO systems. In Sections 2.4 and 2.5 these analysis inequalities are extended to uncertain and parameter-varying systems as well.

2.1 Linear Time-Invariant systems

Linear time-invariant systems are defined in the control system literature as follows [116]. The input-output responses of an LTI system are governed by ordinary differential equations, with constant coefficients. These constant coefficients may be grouped together to matrices to provide the well known time domain description of LTI systems, termed as state-space form

˙

x(t) =Ax(t) +Bu(t),

y(t) =Cx(t) +Du(t). (2.1)

Here x(t)∈R^nx represents the states of the system with ˙x(t) = ^dx_dt,u(t)∈R^nu is the input and y(t) ∈R^ny is the output signal. The system is time invariant since the A,B, C, D coefficient matrices are independent of time. Whenever it is necessary to emphasize the input, output dimensions of the system, (2.1) will be denoted by Gny×nu. Furthermore, A will be referenced as the state matrix of the system, whileB andC as the input and output matrices respectively.

The frequency dependent description of the underlying system is described by theG(s) transfer function matrix, defined as

G(s) =C(sI−A)⁻¹B+D, (2.2)

where sis the Laplace variable.

Throughout the dissertation the following definitions are used.

Definition 1.: The definition of proper system [116].

ˆ A systemG(jω) is strictly proper ifG(jω)→0 asω → ∞.

ˆ A systemG(jω) is semi-proper ifG(jω)→D̸= 0 asω→ ∞.

ˆ A systemG(jω) which is strictly proper or semi-proper is proper.

ˆ A systemG(jω) is improper ifG(jω)→ ∞ as∞ → ∞.

Furthermore, one can introduce the tall, square and wide notations for the Gny×nu system [71]. It is called tall when the ny > nu, and oppositely it is called wide when nu > ny. The system is square if n_u =n_y.

Dual representation

Assume that the system Gny×nu is given in state-space form by (2.1) . According to [67], the state-space matrices of the dual system ˜Gnu×ny are

A˜=A^T, B˜ =C^T, C˜ =B^T, D˜ =D^T. (2.3) This dual representation has a favorable property, which will be used throughout the dissertation.

Namely, the input-output norm of the system is preserved (see Chapter 3), while the input and output dimensions are interchanged. It follows immediately that ifGny×nu is wide, then ˜Gnu×ny

is tall.

Modal representation

The discussed methods in the thesis are closely related to the theory of modal control. The so called modal form is a natural subsystem form of an LTI system, and it is always achievable by suitable similarity transformations, if the underlying plant is diagonalizable (by applying the similarity trasformations the system’s state matrix will have a blockdiagonal structure).

Suppose that A diagonalizable, continuous time LTI dynamics is given by (2.1). After applying the linear diagonalizing Tmtransformation, the resulting state-space system

˙

x_m(t) =T_mAT_m⁻¹x_m(t) +T_mBu(t),

y(t) =CT_m⁻¹xm(t) +Du(t). (2.4) is labeled as the modal form of (2.1) [61], where T_m is constructed from the eigenvectors of A.

The transformed state-space matrices, for two modes (subsystems) has the form

A_m=T_mAT_m⁻¹ =

A_c 0 0 A_d

, B_m=T_mB = B_c

B_d

, Cm=CT_m⁻¹ =

Cc C_d

, D= D

.

(2.5)

In an A_m ∈ R^nx×nx blockdiagonal matrix each block corresponds to a dynamical mode of the system. These dynamical modes can be represented by either real (R) or complex (with imaginary part I) eigenvalues λ, which determine the structure of the corresponding block of theA_m= diag (A₁, ..., A_n) matrix as

A_i =





λ_i if I(λ_i) = 0

R(λ_i) I(λ_i)

−I(λi) R(λi)

if I(λi)̸= 0. (2.6)

In addition, the transfer function matrix representation in modal form for (2.5) is given by G(s) = X

i∈{c,d}

C_i(sI−A_i)⁻¹B_i +D=Gc(s) +Gd(s) +D, (2.7)

where Gc(s) andGd(s) are the transfer functions of the subsystems, with the standard notation of sbeing the Laplace variable and I being the identity matrix.

Remark 1. Equation (2.5) assumes that the system’sA_nx×nx state matrix is similar to anA_m block diagonal matrix, that is there exists an invertible T such that T⁻¹AT = Am. If Anx×nx

hasn_x linearly independent eigenvectors, thenA is diagonalizable and T is nonsingular.

However, on the other hand if T is singular, then A is deffective, nondiagonalizable and it does not have nx linearly independent eigenvectors. In this case there exists an invertible P such that P⁻¹AP = J, with J having Jordan blocks on its diagonal corresponding to those eigenvalues whose eigenvectors are linearly dependent [67].

2.2 Convex optimization and LMIs

In the thesis and in the control system literature as well, several control problems are formalized as convex optimization problems. The strongly appealing property of convex optimization is that, it provably has only one (global) optimum [85], and a wide variety of efficient solvers are available to find it.

Definition 2 .: The convex optimization problem [18]. The convex optimization problem is given in the form

minimize f₀(x)

subject to fi(x)≤bi, i= 1, ...m, (2.8) where the functionsf₀, ..., f_m : R² →Rare convex, i.e., satisfy

f_i(αx+βy)≤αf_i(x) +βf_i(y) (2.9) for all x, y∈Rⁿ and all α, β∈Rwith α+β = 1 α≥0,β ≥0.

In the control system literature Linear Matrix Inequalities (LMIs) are of great importance, because they describe a convex constraint. This is discussed in detail in [19]. In general various linear inequalities, convex quadratic inequalities, matrix norm inequalities can be formulated as LMIs. In the control system engineering field Lyapunov and Riccati inequalities, as well as analysis and synthesis conditions can be turned into LMIs.

Definition 3.: Linear Matrix Inequality [19]. A linear matrix inequality has the form F(x) =F₀+

Xm i=1

x_iF i≻0, (2.10)

where x∈R^m is the variable, and F_i ∈Sⁿ denotes fix matrices withi= 0, ..., m.

The ≻denotes that the F(x) matrix is positive definite.

Definition 4.: Bilinear Matrix Inequality [129]. The bilinear matrix inequality has the form

F(x, y) =F₀+ Xm i=1

x_iF_i+ Xn j=1

y_jG_j+ Xm i=1

Xn j=1

x_iy_jH_ij ≻0,

where x∈R^m,y ∈Rⁿ are the variables, and the symmetricF_i,G_j H_ij matrices are fixed.

Multiple LMIs F⁽¹⁾(x) ≻ 0, ..., F^(p)(x) ≻ 0 can be expressed as a single one, by stacking them in a blockdiagonal way to give diag F⁽¹⁾(x), ..., F^(p)(x)

≻0. When the F_i matrices are

diagonal, the LMI F(x) > 0 is just a set of linear inequalities. A deep introduction to LMIs, and their use in control system theory can be found in [19], [37].

Next, two important lemmas are discussed, which are strongly bonded to LMIs. The first is the Schur complement lemma. By its aid, nonlinear (convex) inequalities are converted to LMI form.

Lemma 1.: The Schur complement lemma [37]. Suppose that Q,M, and R are ma- trices and thatM and Q are self-adjoint. Then the following are equivalent.

(a) The matrix inequalities Q≻0 and

M−RQ⁻¹R^∗ ≻0 both hold.

(b) The matrix inequality

M R

R^∗ Q

≻0 is satisfied.

Proof. The proof is available in [37].

Note that an identical result holds in the negative definite case, by replacing all ”≻”” by ”≺”.

The second important lemma is the Kalman-Yakubovich-Popov (KYP) lemma, which pro- vides a connection between the frequency-wise description of a transfer function matrix, and its corresponding state-space form. As a consequence frequency dependent matrix inequalities can often be solved by first replacing the frequency dependence with a new matrix valued decision variable, using the KYP lemma, and then solving the inequality with corresponding effective numerical tools.

Lemma 2 .: The KYP lemma [103]. Given A ∈ R^nx×nx, B ∈ R^nx×nu, P ∈ S^nx+nu, with det(jωI−A)̸= 0 forω ∈R and (A, B) controllable, the following two statements are equivalent.

(a)

(jωI−A)⁻¹B I

_∗ M

(jωI−A)⁻¹B I

⪯0, ∀ω∈R∪ {∞}. (2.11) (b) There exists a matrix X∈S^nx such that

M+

A^TX+XA XB B^TX 0

⪯0. (2.12)

The corresponding equivalence for strict inequalities holds even if ((A, B) is not controllable.

Proof. The detailed proof can be found in [103].

Note that M in the above lemma is often selected as M =

C DT

Π C D

, where Π is a symmetric multiplier. The Generalized KYP lemma allows the inequality given in (2.11) to hold over a finite frequency interval. It involves the solution of an LMI with additional optimization variables.

low middle high Ω |ω| ≤ω¯l

¯ω≤ω≤ω¯ |ω| ≥ωh

Φ

−1 0 0 ω_l²

−1 −jω˜ jω˜ −¯ωω¯

1 0

0 −ω_h²

Ξ

−Y X X ω²_lY

−Y X−jωY˜ X+jωY˜ −¯ωωY¯

Y X

X −ω_h²Y

Table 2.1: The Ω frequency ranges and the corresponding Φ and Ξ multipliers

Lemma 3.: The Generalized KYP lemma [57, 131]. Consider a system (2.1) with transfer function matrix (2.2), and let a symmetric matrix multiplier Π∈S^nx+ny be given.

Then the following statements are equivalent.

(a) The finite frequency inequality G(jω)

I _∗

Π

G(jω) I

≺0, ∀ω ∈Ω, (2.13)

where Ω is defined in Table 2.1.

(b) There exists an n_x×n_x Hermitian X andY withY ≻0 such that A B

I 0 T

Ξ

A B I 0

+

C D 0 I

T

Π

C D 0 I

≺0, (2.14)

where Ξ is selected from Table 2.1 corresponding to Ω, with

¯ω, ¯ωdenoting the minimum and maximum frequencies in the frequency interval, and ˜ω = ¯ω+¯ω

2 .

Proof. The detailed proof is available in [57].

2.3 Sensitivity analysis of dynamic MIMO systems

Sensitivity analysis describes LMI based, efficient computational algorithms to calculate the lower and upper bounds of the gain of an LTI G(s) dynamical system. The section is divided into two parts. First LMI techniques to achieve the lower bounds of the system gain (referred as the minimum sensitivity) are presented. Next algorithms for calculating the upper bound of the system gain are discussed.

2.3.1 Minimum sensitivity

Throughout the thesis minimum sensitivity analysis techniques are based on two similar, but separate analysis conditions. They start from a similar definition and through different proofs they arrive at similar LMI conditions for evaluating the minimum sensitivity of a system. They are referred as theH− index and the minimum gain.

The H− index

The first one is labeled as the H− index^* of a system G(s). It is mainly used in the Fault Detection Filtering (FDI) literature to characterize the minimum sensitivity of a system (see i.e., [132] and [46]). The state-space system given by (2.1) has a minimum sensitivity β if

||uinf||2̸=0

||y||2

||u||2

> β (2.15)

holds, where u is the input andy is the output signal and β is a positive scalar. The H− index in the frequency domain is defined as

||G(s)||−:= inf

ω∈[0,∞]¯σ G(jω)

, (2.16)

with σ denoting the minimum singular value (for details see (B.7) in Appendix B.1). The computation of theH− index over an infinite frequency range can be written as a semi-definite problem.

Lemma 4.: TheH−index over infinite frequency range [74]. Letβ >0 be a positive constant scalar. Then ||G(s)||^[0,∞]₋ > β, if and only if there exists an X such that X ∈S^nx

and

A^TX+XA+C^TC XB+C^TD B^TX+D^TC D^TD−β²I

≻0. (2.17)

Proof. The proof can be found in [74].

Note that (2.17) is a linear matrix inequality, therefore Lemma 4 can be seen either as a feasibility test (fixed value ofβ) or as a semi-definite optimization problem (βis a variable to be maximized) subject to LMI constraints.

It is obvious, that for strictly proper systems (i.e.,D= 0) the above definition and formula- tion yields zero. In order to overcome this problem and compute the minimum sensitivity over a limited frequency interval, [74] proposed the use of specific frequency filters to augment the plant. Then an estimation can be given on the frequency limitedH−index of the strictly proper system. In order to avoid the introduction of frequency filters, a finite frequency extension from the work of [131] is used throughout the thesis. This formulation is based on the Generalized Kalman-Yakubovich-Popov (GKYP) lemma and restricts the analysis condition in Lemma 4 to a finite frequency range.

Lemma 5 .: The H− index over a finite frequency range [55, 131]. Consider the system given in (2.1) with transfer function matrix (2.2). Let Π =

−I 0 0 β²I

∈ R^{(nx+ny)×(nx+ny)}. Then ||G(s)||^[Ω]₋ > β if and only if there exists HermitianX and Y, with Y ≻0 satisfying

A B I 0

T

Ξ

A B I 0

+

C D 0 I

T

Π

C D 0 I

≺0, (2.18)

where Ξ is selected from Table 2.1 with respect to the Ω frequency range.

*It is important to emphasize that despite theH−index is not a norm, i.e. the norm property||e||= 0↔e= 0 does not hold for strictly proper systems, it provides a useful lower bound on the system’s gain.

Proof. The Π =

−I 0 0 β²I

multiplier selection in the Ω frequency range induces that G(jω)^∗G(jω)≻β²I, ∀ω∈Ω,

which is the same as

¯σ(G(jω))> β, ∀ω∈Ω.

Then applying Lemma 3 completes the proof. More detailed discussion and derivation can be found in [55].

Note that (2.17) is a special case of (2.18), and with the selection ofY = 0, (2.18) reduces to (2.17). Since theH−index denotes the smallest singular value, the above mentioned formulations are also applicable to unstable systems without any modifications. However, there is a restriction for the application of the concept ofH− index. It has been discussed in [71] that this index can only be calculated for tall and square systems. Nevertheless, involving the dual representation the following lemma can be stated.

Lemma 6.: Calculation constraints of the minimum sensitivity index. The H−

index with the LMI formulations given in (2.17) and (2.18) can only be calculated for tall and square systems. For wide systems the dual representation provides the appropriateH−

index value.

Proof. No formal proof of this property was found, however, it is believed that this is useful and helpful for understanding the developed results. Due to the length of the proof, it can be found in Appendix B.2.

The minimum gain

The authors in [20] proposed an alternative, yet similar definition for the minimum sensitivity of a system, defined as follows.

Definition 5.: The minimum gain [20]. A causal systemG:L2e→ L2e, has minimum gain 0≤β≤ ∞if there existsν, depending only on the initial conditions, such that

||y||2T −β||u||2T ≥ν, ∀u∈ L2e, ∀T ∈R⁺. (2.19) Here L2e denotes the extended L2 signal space, where||y||²2T = ^∞R

0 |y_T(t)|²dt <∞,T ∈R⁺, with y_T(t) =y(t) for 0≤t≤T andy_T(t) = 0 for t≥T.

For LTI systems, an LMI-based computation has also been derived in [20], which is referred as the Minimum Gain Lemma.

Lemma 7 .: The Minimum Gain Lemma [20]. The LTI system given in (2.1) has minimum gain 0≤β ≤ ∞if there existsX∈S^nx ⪰0 such that

XA+A^TX−C^TC XB−C^TD (XB−C^TD)^T β²I−D^TD

⪯0. (2.20)

Proof. The following sketch of proof follows the steps given in [20]^a. Denote (2.20) by M(X, β) and note thaty=Gu. Then one can write

||y||²2T −β²||u||²2T = ZT

0

|Cx+Du|²−β²|u|²− d

dtx^TXx+ d dtx^TXx

dt=

= ZT

0

x u

T

C^TC C^TD D^TC D^TD

−

0 0 0 β²I

−

XA+A^TX XB B^TX 0

x u

dt+

ZT 0

x u

T

XA+A^TX XB B^TX 0

x u

dt.

(2.21)

By realizing that −M(X, β) appears in the first integral, (2.21) can be written as

||y||²2T−β²||u||²2T =− ZT

0

x u

T

M(X, β) x

u

dt+ ZT 0

x u

T

XA+A^TX XB B^TX 0

x u

dt. (2.22) Evaluating the second integral (with x(T) = 0) gives

||y||²2T −β²||u||²2T =− ZT

0

x u

T

M(X, β) x

u

dt+x^T(T)Xx(T)

| {z }

0

−x^T₀Xx0. (2.23)

Since the integral term with the negative sign describes a positive quantity (due to (2.20)), it can be neglected to yield the inequality in (2.19) as

||y||²2T −β²||u||²2T ≥ −x^T₀Xx₀= ˜ν. (2.24) The detailed proof can be found in [20] and hence omitted here.

aNote that in order to be consistent with the previous sections, the meaning ofβandνhas been changed compared to [20].

The minimum gain condition similarly to the H− index one, can be restricted to a finite frequency interval by the aid of the GKYP lemma. Nevertheless, a few remarks have to be given regarding the connection of the H− index and the Minimum Gain Lemma. First, Lemma 4 is restricted to stable plants, while the definition and the computation of the minimum gain extends to unstable systems as well. Second, the resulting LMI constraints from the two sensitivity characterizations are structurally similar and connected. In order to show this, borrow the argument presented in [141]. In particular, [141] is using an auxiliary description for unstable (sub)systems, which is defined by ˆG = (−A,−B, C, D). The time-domain interpretation of the auxiliary system is given by reversing the time variable t. For this t=τ is introduced and the signals are rewritten: ˆx(τ) =x(−t). For the computation of the unstable Gramians in [141], it is then showed that they are the solution of a minimal energy problem for the corresponding auxiliary system. What is interesting for the present case is that theH− index for an unstable system can be computed by using the same arguments and the auxiliary description. Namely:

following the same train of thoughts substituting ˆGinto (2.17) yields (2.20) for unstable systems.

2.3.2 Maximum sensitivity

The well-known H∞ norm is used in the thesis for characterizing the maximum sensitivity of theG(s) transfer function matrix given in (2.2). In the frequency domain it is defined as

||G(s)||∞:= sup

ω

¯ σ

G(jω)

, (2.25)

where ¯σdenotes the maximum singular value. The time domain definition of the induced 2-norm of the G state-space system given in (2.1) is defined by

||G||i2 := sup

u∈L2, u̸=0

||Gu||2

||u||2

= sup

u∈L2, u̸=0

||y||2

||u||2

. (2.26)

Using Parseval’s theorem it can be shown that ||G(s)||∞ =||G||i2. This relationship has partic- ular importance at system classes which do not have transfer function forms, such as nonlinear and parameter-varying systems. For LTI systems an LMI based computation of the H∞ norm over the [0,∞) frequency range is given by the Bounded Real Lemma, which is summarized in Lemma 8.

Lemma 8.: The Bounded Real Lemma [109]. Letγ ≥0 be a positive constant scalar.

Then ||G(s)||^[0,∞^∞) < γ if and only if there exists a positive definite symmetricX∈S^nx ≻0,

such that

A^TX+XA+C^TC XB+C^TD B^TX+D^TC D^TD−γ²I

⪯0. (2.27)

Proof. The proof can be found in most of the robust control textbooks, see e.g. [109].

The H∞ norm is defined only for stable systems (i.e., poles having negative real part). At the same time, unstable systems that have no poles on the imaginary axis have an L∞ norm (also known as the peak gain). This peak gain can be computed using (2.27) after mirroring the unstable poles over the imaginary axis [140].

For completeness of the LTI sensitivity analysis techniques, note that, finite frequency ex- tension of the Bounded Real Lemma is also available, relying on the Generalized Kalman- Yakubovich-Popov Lemma (introduced by [57] and [55]).

Lemma 9.: The Bounded Real Lemma over a finite frequency range [131]. Con- sider the system given in (2.1) with transfer function matrix (2.2). Let Π =

I 0 0 −γ²I

∈ R^{(nx+ny)×(nx+ny)}. Then||G(s)||^[Ω]∞ < γ if and only if there exists Hermitian X and Y, with X ≻0,Y ≻0 satisfying

A B I 0

T

Ξ

A B I 0

+

C D 0 I

T

Π

C D 0 I

≺0, (2.28)

where Ξ is selected from Table 2.1 with respect to the Ω frequency range.

Proof. The proof is similar to the proof of Lemma 5. More details can be found in [55].

2.4 Systems with uncertainties

So far exact knowledge of the plant model had been assumed. This section discusses available modeling and analysis techniques for systems with uncertain or varying parameters and other sources of uncertainties (e.g., unmodeled dynamics), or nonlinearities (such as saturation, delay, etc.). These uncertainties may be constant or vary with time, but they are restricted to belong to a certain ∆ set with known bounds on their magnitude. In Section 2.5 the case when the time-varying parameters are measurable (linear parameter-varying systems) is discussed.

2.4.1 Polytopic modeling

A relatively simple, but still effective modeling approach is the polytopic modeling of an uncer- tain system. The following discussion is based on [109]. Consider a linear system which depends on uncertain time-varying parameters collected in ∆(t)

˙

x(t) =A(∆)x(t) +B(∆)u(t),

y(t) =C(∆)x(t) +D(∆)u(t), (2.29) with x(t)∈ R^nx,u(t) ∈R^nu and y(t)∈ R^ny being the state, the input and the output signals, respectively. Note that the time dependence in ∆ has been omitted to ease the notation. Assume that ∆ belongs to a set ∆, which constraints its values to certain intervals. Denote the system matrix as

G(∆) =

A(∆) B(∆) C(∆) D(∆)

, (2.30)

and suppose that for any t ∈ R time instant it can be written as a convex combination of np

system matrices G1, ...,Gnp. This means that there exists functions ξ_i: R→[0,1] such that for any t∈R

G(∆) =

np

X

i=1

ξ_i(t)Gi, withGi=

Ai Bi

C_i D_i

, (2.31)

and Pnp

i=1ξ_i(t) = 1 and i = 1, ..., n_p. The Gi denotes a sequence of constant matrices with equal dimension. This implies that the G(∆) system matrices belong to a convex hull of G1, ...,Gnp denoted by G(∆) ∈ co G1, ...,Gnp

for all t ∈ R. Such models are labeled as poly- topic models, and the corresponding convex hull is called a polytope and will be denoted by Θ (Θ = co G1, ...,Gnp

). The vertex points of the polytope are the G1, ...,Gnp system models (see subfigure a, in Figure 2.1).

The model description given in (2.31) states that, the uncertain system in every instant can be represented as the weighted sum of carefully selected models. These are given at the vertex points of a corresponding polytope shown in subfigure a, in Figure 2.1. Then (2.31) provides a description of the uncertain system if it stays inside this polytope. Various approaches are available to select these vertex points models. Probably the most straightforward is analyz- ing the underlying uncertainty structure (see e.g. Section 4.6.2). Furthermore, an effective computational method for vertex point selection is discussed in [11, 123].

Polytopic analysis conditions

A great advantage of the polytopic technique is that analysis and synthesis conditions are easily extendable from LTI system models, due to the Relaxation Theorem [19]. Simply speaking it means that the LTI analysis and synthesis conditions have to be solved for all vertex point models parallel. This leads to the following lemmas for the sensitivity analysis of uncertain polytopic systems.