MASTER's DISSERTATION No 1200 ROBUST STABILITY OF

MASTER’S DISSERTATION No 1200

ROBUST STABILITY OF POLYTOPIC LPV SYSTEMS: GEOMETRIC ISSUES AND THE COMPUTATIONAL COMPLEXITY

Igor Pereira Vieira

School of Engineering

Graduate Program in Electrical Engineering

Belo Horizonte - MG July 2020 UNIVERSIDADE FEDERAL DE MINAS GERAIS

SCHOOL OF ENGINEERING

GRADUATE PROGRAM IN ELECTRICAL ENGINEERING

ROBUST STABILITY OF POLYTOPIC LPV SYSTEMS: GEOMETRIC ISSUES AND THE COMPUTATIONAL COMPLEXITY

Igor Pereira Vieira

Dissertation presented to the Graduate Program in Electrical Engineering at the Universidade Federal de Minas Gerais in partial fulﬁllment of the requirements for obtaining a M.Sc. degree in Electrical En- gineering.

Concentration area: Signals and Systems Research line: Modeling, Analysis and Control of Nonlinear Systems (MACSNL)

Advisors: Prof. Leonardo Amaral Mozelli, Ph.D. Prof. Fernando de Oliveira Souza, Ph.D.

Belo Horizonte - MG July 2020 UNIVERSIDADE FEDERAL DE MINAS GERAIS

ESCOLA DE ENGENHARIA

PROGRAMA DE PÓS-GRADUAÇÃO EM ENGENHARIA ELÉTRICA

ESTABILIDADE ROBUSTA DE SISTEMAS LPV POLITÓPICOS: ASPECTOS GEOMÉTRICOS E COMPLEXIDADE COMPUTACIONAL

Igor Pereira Vieira

Dissertação apresentada ao Programa de Pós-Graduação em Engenharia Elétrica da Universidade Federal de Minas Gerais como requisito parcial para a obtenção do título de Mestre em Ciências, na área de Engenharia Elétrica.

Área de Concentração: Sinais e Sistemas Linha de Pesquisa: Modelagem, Análise e Controle de Sistemas Não Lineares

Orientadores: Prof. Leonardo Amaral Mozelli, Dr. Prof. Fernando de Oliveira Souza, Dr.

Belo Horizonte - MG Julho de 2020

Vieira, Igor Pereira. V658r Robust stability of polytopic LPV systems: [recurso eletrônico] : geometric issues and the computational complexity / Igor Pereira Vieira. – 2020. 1 recurso online (100 f. : il., color.) : pdf.

Orientador: Leonardo Amaral Mozelli. Coorientador: Fernando de Oliveira Souza.

Dissertação (mestrado) Universidade Federal de Minas Gerais, Escola de Engenharia.

Inclui bibliografia e índice. Exigências do sistema: Adobe Acrobat Reader.

1. Engenharia elétrica - Teses. 2. Liapunov, Funções de - Teses. 3. Complexidade computacional - Teses. 4. Controle robusto - Teses. I. Mozelli, Leonardo Amaral. II. Souza, Fernando de Oliveira. III. Universidade Federal de Minas Gerais. Escola de Engenharia. IV. Título.

CDU: 621.3(043)

Ficha catalográfica: Biblioteca Profº Mário Werneck, Escola de Engenharia da UFMG

"Robust Stability of Polytopic LPV Systems: Geometric Issues and the Computational Complexity"

Igor Pereira Vieira

Dissertação de Mestrado submetida à Banca Examinadora designada pelo Colegiado do Programa de Pós-Graduação em Engenharia Elétrica da Escola de Engenharia da Universidade Federal de Minas Gerais, como requisito para obtenção do grau de Mestre em Engenharia Elétrica.

Aprovada em 28 de julho de 2020.

Por: ______Prof. Dr. Leonardo Amaral Mozelli DELT (UFMG) - Orientador

______Prof. Dr. Fernando de Oliveira Souza DELT (UFMG) - Co-orientador

______Prof. Dr. Leonardo Antônio Borges Tôrres DELT (UFMG)

______Prof. Dr. Víctor Costa da Silva Campos DELT (UFMG)

ACKNOWLEDGEMENTS

My most sincere thanks to my advisors Prof. Leonardo A. Mozelli and Prof. Fernando O. Souza. I am especially grateful for their con- ﬁdence, patience, time and support in all stages of this work. Their dedication and continued commitment to excellence are truly inspiring.

It is a pleasure to express my gratitude to the members of the exa- mination committe, Prof. Víctor C. S. Campos and Prof. Leonardo A. B. Tôrres, for their careful reading and their many insightful comments and suggestions. I also thank Prof. Leonardo Tôrres for the opportunity to participate in his magniﬁcent course on Fundamentals of Nonlin- ear Control.

To all the professors of the Department of Electronics Engineering (DELT/EE-UFMG) and of the Department of Electrical Engineering (DEE/EE-UFMG) who indirectly contributed to the preparation of this work. Special acknowledgments are made to: Prof. Reinaldo M. Pal- hares, Prof. Luis A. Aguirre, Prof. Rodney R. Saldanha, Prof. Eduardo M. A. M. Mendes and Prof. Bruno O. S. Teixeira.

I am grateful to the Universidade Federal de Minas Gerais and the Graduate Program in Electrical Engineering for the opportunity. I also thank the National Council for Scientiﬁc and Technological Develop- ment (CNPq) for the partial support.

I am grateful as well to my family, for the constant support and encouragement. [...]

As mais soberbas pontes e edifícios, o que nas oficinas se elabora, o que pensado foi e logo atinge distância superior ao pensamento, os recursos da terra dominados, e as paixões e os impulsos e os tormentos e tudo que define o ser terrestre ou se prolonga até nos animais e chega às plantas para se embeber no sono rancoroso dos minérios, dá volta ao mundo e torna a se engolfar, na estranha ordem geométrica de tudo, e o absurdo original e seus enigmas, suas verdades altas mais que todos monumentos erguidos à verdade: e a memória dos deuses, e o solene sentimento de morte, que floresce no caule da existência mais gloriosa, tudo se apresentou nesse relance e me chamou para seu reino augusto, afinal submetido à vista humana.

[...]

A Máquina do Mundo Carlos Drummond de Andrade VIEIRA, I.P. Robust Stability of Polytopic LPV Systems: Geometric Issues and the Computational Complexity. Master Dissertation (Electrical Engineering) – School of Engineering, Universidade Federal de Minas Gerais. Belo Horizonte, 2020.

ABSTRACT

This work addresses the problem of computational complexity in robust stability certiﬁcation for continuous-time Linear Parameter Varying (LPV) systems based on Parameter-Dependent Lyapunov Functions (PDLFs). New strategies are presented for the inclusion of the uncertain parameters time derivatives, designed to balance the computational cost, resulting from the reduction of the number of Linear Ma- trix Inequalities (LMIs) to be evaluated, and the conservatism. The methodology is grounded in the exploration of geometric aspects of the LPV polytopic representation. In a ﬁrst approach, the gradual transition between the regular simplex – the convex object with least possible number of vertices in each dimension – and the minimum- hypervolume polytope – with factorial vertex growth as function of the number of time-varying uncertain parameters – is outlined by exchanging elements between these sets. Analytical relations that make possible evaluations about the impact of suppressing one or more simplex vertices, for the case of symmetrical uncertainties parametrically described, are provided, leading to a bendable trade-off solution. It is also shown that conservatism can occasionally be improved through simple procedures involving the rotation of the simplex vertices in the time-derivative parameter space, without any intervention in the geometric structure of the polytope. Subse- quently, procedures that enable the employment of cubic and orthoplectic convex hulls, in detriment of the original one, are proposed. Along with the regular simplexes, already covered in the literature, these polytopic families are the only ones to remain regular as the dimensionality grows. In the numerical analysis conducted, such geometries provided better results, with regard to conservatism, as the system scale was increased. Finally, using well-consolidated tools of the control theory, the previous approaches, conceived within the scope of the stability analysis, are generalized to the context of robust state feedback control synthesis.

Keywords: Linear Parameter-Varying Systems, Parameter-Dependent Lyapunov Func- tions, Linear Matrix Inequalities, Computational Complexity, Robust Control, Poly- topic Geometry, Regular Polytopic Families VIEIRA, I.P. Estabilidade Robusta de Sistemas LPV Politópicos: Aspectos Geométri- cos e Complexidade Computacional. Dissertação de Mestrado (Engenharia Elétrica) – Escola de Engenharia, Universidade Federal de Minas Gerais. Belo Horizonte, 2020.

RESUMO

Este trabalho investiga o problema da complexidade computacional no âmbito da geração de certificados de estabilidade robusta para sistemas lineares a parâmetros variantes (LPV), de tempo contínuo, empregando funções de Lyapunov dependentes de parâmetros (PDLFs). São apresentadas novas estratégias para a inclusão das derivadas temporais dos parâmetros incertos destinadas a balancear o custo computacional, a partir da redução da quantidade de desigualdades matriciais lineares (LMIs) a serem avaliadas, e o conservadorismo. A metodologia é fundamentada na mani- pulação dos aspectos geométricos da representação politópica destes sistemas. Em uma primeira abordagem, é delineada a transição gradual entre o simplex regular – o objeto convexo com o menor número possível de vértices em cada dimensão – e o politopo de menor hipervolume – com crescimento fatorial de vértices em função do número de parâmetros incertos variantes no tempo –, intercambiando elementos entre esses conjuntos. São fornecidas relações analíticas que possibilitam a avaliação acerca do impacto da supressão de um ou mais vértices do simplex para o caso de incertezas simétricas descritas parametricamente, culminando em uma solução de compromisso flexível. Além disso, é mostrado que o conservadorismo pode, ocasionalmente, ser reduzido por procedimentos simples envolvendo a rotação dos vértices do simplex no espaço de parâmetros das derivadas temporais, sem qualquer intervenção na es- trutura geométrica do politopo. Na sequência, propõem-se procedimentos para a uti- lização de fechos convexos cúbicos e ortopléticos, em detrimendo do fecho convexo original. Juntamente com os simplexos regulares, já cobertos na literatura, essas famí- lias politópicas são as únicas a manterem-se regulares à medida que se procede com o crescimento da dimensionalidade. Nas análises numéricas conduzidas, tais geome- trias forneceram melhores resultados, no que se refere ao conservadorismo, conforme a escala do sistema foi aumentada. Por fim, utilizando ferramentas já consolidadas da teoria de controle, as abordagens anteriores, concebidas no escopo da análise de estabilidade, são generalizadas para o contexto da síntese de controle robustos por realimentação de estados.

Palavras-chave: Sistemas LPV, Funções de Lyapunov Dependentes de Parâmetros, Desigualdades Matriciais Lineares, Complexidade Computacional, Controle Robusto, Geometria Politópica, Famílias Politópicas Regulares

CONTENTS

List of Figures xiv

List of Tables xvi

List of Symbols xvii

Acronyms and Abbreviations xix

1 Introduction 21 1.1 Motivation ...... 21 1.2 Literature Review ...... 22 1.3 Text Structure ...... 24

2 Mathematical Background 27 2.1 Convex Analysis and LMIs ...... 27 2.1.1 Introduction ...... 27 2.1.2 Convexity-Preserving Operations ...... 28 2.1.3 Representations ...... 29 2.2 Higher-Dimensional Euclidean Convex Geometry ...... 32 2.2.1 Context ...... 32 2.2.2 Concepts ...... 33 2.3 Overview of Linear Parameter-Varying Systems ...... 35 2.3.1 Introduction ...... 35 2.3.2 Polytopic Representation ...... 36 2.4 Lyapunov Stability for Polytopic Systems ...... 38

3 Methodology 41 3.1 Previous Works ...... 41 3.1.1 Exact Polytope ...... 41 3.1.2 Simplectic Convex Hull Approach ...... 42 3.2 Vertex Elimination and Rotation Procedures ...... 43 3.2.1 Vertex-Rotation Procedure ...... 49 3.3 Regular Polytopic Families for Robust LPV Analysis ...... 51 3.3.1 Hyperplane Mapping via Rotation Matrices ...... 51 3.3.2 The Enclosing Polytope Hypervolume ...... 54 3.3.3 The Enclosing Polytope Vertex-Matrix Composition ...... 63 3.4 An Extension for Robust State Feedback Control ...... 65

4 Results 69 4.1 Model Description ...... 69 4.2 Complexity and Conservatism ...... 70 4.2.1 Vertex Elimination and Rotation Procedures ...... 70 4.2.2 Regular Polytopic Families ...... 72

xi 4.2.3 Control Extension ...... 74

5 Conclusions 77 5.1 Next Steps ...... 80

A A combinatorial interpretation of the vertex growth 81

B Effective number of eliminations 85

C Proof for non-trivial simplex generation in three dimensions via rotation matrix 87

D Dimensionality reduction using Givens rotations 91

Remissive Index 99

xii LIST OF FIGURES

2.1 Convex (a) and nonconvex (b) sets...... 28 2.2 The convex hull for a finite (a) and infinite number of points (b). Also the convex hull of a nonconvex set (c)...... 29 2.3 Generation of a plane in R3...... 30 2.4 Convex sets Q1 and Q2 separated by the hyperplane H (F, q)...... 30 2.5 The supporting hyperplane theorem...... 31 2.6 The five regular polyhedra, as depicted in the work Harmonices Mundi, Libri V [Kepler, 1619], by Johannes Keppler, in a reference to the Timaeus dialogue [Zeyl, 2000]...... 33 2.7 Main Theorem of Polytope Theory...... 34 2.8 Schlegel diagrams for the 2 to 4-simplex...... 34 2.9 Schlegel diagrams for the 2 to 4-cube...... 35 2.10 Schlegel diagrams for the 2 to 4-cross-polytope...... 35

3.1 Original convex hull (black vertices) and 2-simplex (red ones) overlapped, computed via (3.1) and (3.5), respectively, for −1 ≤ θ˙i(t) ≤ 1, i = 1, 2, 3. Conservatism comes from Minkowski difference, P ∩ R Λ2, between the polytopes...... 43 3.2 Some possible scenarios in the scope of simplex vertex suppression. (a), (b), (c) Only one vertex is eliminated, resulting in a total of 4 vertices. (c), (d), (e) Two simplex vertices are eliminated, resulting in a total of 5 vertices...... 44 3.3 Saturation during the vertex elimination process in R4 dimensions. Simplex and original polytope were mapped to R3 subspace using the procedure described in sec 3.3.1. Red vertices are exposed, while the yellow ones are covered...... 48 3.4 Simplex rotation of π rad around normal vector e...... 49 3.5 Mapping Π 7→ Π0...... 52 3.6 Polytope H˘ ...... 54 3.7 Hypervolume minimization procedure...... 55 3.8 Original polytope mapped to Π0 and encapsulated by a 2-cube and by a 2-orthoplex...... 56 3.9 The blue points correspond to (x, | tan (ψc∗) |), where ψc∗ is the new optimal angle in this dimension, and x := r − 2 is the total of rotations performed. a=1.056, b=1.289, e log is in the base e...... 59 3.10 Polytope (3.31)...... 62

xiii 3.11 Cube and cross-polytope encapsulating the original polytope in Π0 and Π...... 64 3.12 Original 4-dimensional polytope mapped to Π0 and encapsulated by the simplex (yellow), cube (blue) and orthoplex (green, identical to the T original polytope). When multiplied by Ξ4 , they return to the original manifold, i.e., Π0 7→ Π...... 65

4.1 The mass-spring-damper system with b = 3...... 69 4.2 Vertex growth in each geometry...... 73

C.1 Triangulation procedure...... √ . . . 88 C.2 Area as a function of the rotation angle. The green line, at A = (9/2) 3, is the simplex area, as calculated in Eq. (C.4)...... 90

xiv LIST OF TABLES

3.1 Vertices growth, p, relative to the number of uncertain time-varying parameters, r, for (3.1)...... 42 3.2 Vertices growth, q, relative to the number of uncertain time-varying parameters, r, and the number of vertices eliminated from simplex, ξ. Content inside the cells is arranged in the following format: total of vertices [simplex contribution + original polytope contribution]. . . . . 46 3.3 Optimal angles, ψ∗ [rads], for the minimization of the enclosing hypercube hypervolume employing the structure (3.23) as a function of the number of uncertain time-varying parameters, r. Step length: 10−5 rads. 57 3.4 Optimal angles, ψ∗ [rads], for the minimization of the enclosing orthoplex hypervolume employing the structure (3.23) as a function of the number of uncertain time-varying parameters, r. Step length: 10−5 rads. 57 3.5 Tangent of the optimal angles, ψ∗ [rads], for the minimization of the enclosing hypercube hypervolume employing the structure (3.23) as a function of the number of uncertain time-varying parameters, r..... 58 3.6 H˘ is the original 8-dimensional polytope, with 70 vertices, mapped to the subspace of dimension 7. Initially, kH˘ k1 = 7.3701...... 62

4.1 Bounds for model parameters. SI units...... 70 4.2 Number of decision variables for each pair r × n in the MSD model. . . 71 4.3 Number of constraints over the decision variables for each geometry. . . 71 4.4 Computational performance of methodologies with respect to the maximum value of γ for which stability can be guaranteed. The element in parentheses corresponds to the number of vertices for the ﬁnal polytope after the rotation of the geometry by an angle of π rad with respect to the hyperplane normal vector...... 72 4.5 Number of constraints over the decision variables for each geometry. . . 73 4.6 Computational performance of methodologies with respect to the maximum value of γ for which stability can be guaranteed for alternative geometries...... 74 4.7 Computational performance of the orthoplectic convex hull, with respect to the maximum value of γ, for r = 8. Optimal adjust compared to that obtained by the heuristic procedure described in sec. 3.3.2. . . . 74

xv 4.8 Computational performance of methodologies. The ﬁrst row of each cell corresponds to the average time required to obtain the gain matrix in that conﬁgu- ration, while the second one indicates the success percentage of the approach. ζi = 1, ∀i ∈ N ∧ i ≤ r...... 75 A.1 Vertices growth, q, relative to the number of vertices eliminated from simplex, ξ, for r = 6 and r = 7. Content inside the cells is arranged in the following format: total of vertices [simplex contribution + original polytope contribution]...... 83

xvi LIST OF SYMBOLS

Q.E.D., completion of proof × Scalar multiplication (often omitted) :=, =: Identity employing new notation ≡ Identity with notation reuse ≈ Approximately ≥, ≤ Greater than or equal to, less than or equal to , Much greater than, much less than 6= Not equal ∑ Summation ∏ Product N Set of the natural numbers Z Set of the integer numbers R Set of the real numbers In Identity matrix of order n On×m Null n × m matrix | • | Absolute value k•k Euclidian norm k•k1 Manhattan norm k•k∞ Maximum norm x˙(t), x¨(t) First and second time derivatives of x(t) A 0 ( 0) Positive definite (semidefinite) matrix A ≺ 0 ( 0) Negative definite (semidefinite) matrix A(i,j) The element contained in the ith row and jth column of matrix A AT Transpose matrix of A A−1 Inverse of A det(•) Determinant ker(•) Kernel space Co(•) Convex hull O (•) Bachmann–Landau notation (Big-O) Q1 ∪ Q2 Union of two sets Q1 ∩ Q2 Intersection of two sets Q1 ⊂ Q2 Q1 is a subset (or equal) to Q2 Q1 \Q2 Set difference Q1 Q2 Minkowski difference Q1 ⊕ Q2 Minkowski sum

xvii word or phrase

∅ Empty set ∈ Belongs to int Q Interior of Q rel int Q Relative interior of Q cl (Q) Closure of Q ∂Q Boundary of Q mod Modulo operation sign(•) Sign function d•e Ceil operator b•c Floor operator [L] Iverson bracket, where L ∈ {0, 1} is a logical operation → Material implication ↔ Material biconditional (denoted as “iff” sometimes) ∧ Logical conjunction ∨ Logical disjunction | Such that ∀ Universal quantifier ∃ Existential quantifier n (k) Binomial coefficient n Trinomial coefficient k, l, m {p, q, ··· , z} Schläfli symbol

Units of measure

m metre [length] N Newton [force] kg kilogram [mass] s second [time]

Mathematical constants

π ≈ 3.14159 [Pi] e ≈ 2.71828 [Euler’s number]

xviii ACRONYMS AND ABBREVIATIONS app. Appendix ch. Chapter def. Deﬁnition ed. Edition eq. Equation ex. Example ﬁg. Figure no. Number p. (pp.) Page(s) sec. Section tab. Table thm. Theorem vol. Volume apud Lat. apud in the writings of e.g. Lat. exempli gratia for example et al. Lat. et alii, et aliae and others etc. Lat. et cetera and so forth i.e. Lat. id est that is id., ibid. Lat. idem, ibidem the same, in the same place viz. Lat. videlicet namely

SS State-Space DOF Degrees-of-Freedom LMI Linear Matrix Inequality LFT Linear Fractional Transformation LPV Linear Parameter-Varying LTI Linear Time-Invariant LTV Linear Time-Varying MSD Mass-Spring-Damper PDLF Parameter-Dependent Lyapunov Function QS Quadratic Stability SDP Semideﬁnite programming SI Fr. Système Internationale (d’unités) International System of Units

xix

CHAPTER 1

Introduction

1.1 Motivation

Asymptotic stability is the element of primary interest in control systems design [For- tuna and Frasca, 2012]. In practical applications, however, the presence of uncertainties in the models, whether they come from neglected or unknown variables, can make this pursuit unfeasible – or drive the plant to instability –, if not treated properly. Ro- bust control is the branch of control theory that proposes to explicitly consider the presence of uncertainties in the models, offering a framework of mathematical tools to deal with them. Implicitly, the robust treatment of a system implies reducing its sensitivity to parameters, promoting a refinement in the perception of dynamics by the model’s constitutive equations. And while robustness is an indispensable element in many applications, it often leads to performance losses [Tsui, 2003], distancing the point of operation of the system from its optimal region, either, as mentioned above, by reducing sensitivity, or by increasing the complexity required in this new formulation. Thus, it is pertinent to search for alternative mathematical descriptions that culminate in models whose “dialogue” with the control interface take place in the easiest possible way, without incurring, however, significant losses in the quality of the system’s dynamic behavior representation. LPV systems have been an active research field in control and systems engineering since its introduction, by Shamma [1988], fostering contributions in several topics, including: modeling, synthesis, analysis, filtering, etc. To a large extent, this singular acquiescence comes from good practical results combined with a reliable mathematical description, if compared, e.g., to that demanded by non-linear systems, or even by linear time varying systems. Not by chance, processes that demand high-fidelity treatments and/or have emergent properties resulting in complex dynamics, such as aircraft control [Lu et al., 2006, Al-Jiboory et al., 2017], missile autopilot guidance [Shamma and Cloutier, 1993, Pellanda et al., 2002], wind turbines [Bianchi et al., 2005, Shirazi et al., 2012] and satellites [Corti et al., 2012, Jin et al., 2018], are commonly represented in the LPV formalism. As highlighted by Hoffmann and Werner [2014] and references therein, even though LPVs are well-consolidated from a theoretical point of view and have been presented for more than 30 years, they are not widely used in the industrial environment. Tak- ing into account that they are mostly based on the convenient structure of the linear systems theory1, the main reason considered as an obstacle to advances in this di-

1Linear time-invariant (LTI) control theory is widely used in industrial applications. [Tóth, 2010]

21 22 Chapter 1. Introduction rection is the computational burden. In the specialized literature, this is a recurrently overlooked aspect, given the amount of publications devoted purely to methodologies for conservatism reduction. It is noteworthy that, although, at ﬁrst, approaches aimed at mitigating the computational load may raise uncertainties regarding the longevity of the research results, in the sense of practical applications, given the constant escalation of improvements in the ﬁeld of computer architectures and, consequently, of the processing power, the techniques of modeling, analysis and synthesis also keep updating, promoting gradually more accurate and, often, complex representations of physical phenomena.

1.2 Literature Review

The use of Lyapunov’s theory in the context of robust stability analysis of linear systems whose dynamics are somehow subject to uncertain real parameters has been around for more than 60 years – see, e.g., [Parks, 1962, Siljak, 1969, Naredra and Tay- lor, 1973, Horisberger and Belanger, 1976, Leitmann, 1979]. At the forefront of this development, it was already possible to find studies with analyzes based on both the temporal and spectral domains. As we know, these two aspects are forked when it comes to the use of mathematical tools, viz., time-domain studies widely explore state-space (SS) descriptions and, therefore, matrix theory, while those conducted in the frequency-domain predominantly employ descriptions via transfer functions and polynomial algebra. Throughout this work, the focus will be on affine parametric dependence of the state matrices, in the polytopic representation, over the uncertainties; thereby, the present review will concentrate in developments aligned with these features. One of the most significant contributions to the treatment of this class of systems refers to the introduction of the notion of quadratic stability2 (QS) [Barmish, 1983, 1985], where asymptotic stability is ensured by means of a single Lyapunov candidate function, quadratic in states, obtained from the solution of a Riccati-type equation, and which must exist for all linear inequalities arising from the stability conditions imposed on the vertices. This approach, however, culminates in severe complications in regard to the conservatism, since taking into account only the upper and lower bounds for variation in each parameter, it is necessary to guarantee the stability of the system in the worst possible scenario of all uncertainties undergoing abrupt variation. Throughout the 1990s, several studies have emerged aimed at reducing the conservatism emerging from QS, mainly based on the use of linear matrix inequalities (LMIs). In addition, in the same period, the growing interest in the polytopic representation of uncertainties in detriment of the norm-bounded approach is remarkable. An important historical antecedent that contributed to this scenario was the appear- ance of what was later called the Karmarkar’s algorithm [Karmarkar, 1984]: a highly efficient interior-point method, proposed by the Indian mathematician N. Karmarkar, aimed at solving linear programming problems in proven polynomial time. New approaches, henceforth, provided profound advances in knowledge in the field of computational mathematics. Nesterov and Nemirovsky [1988]3, e.g., developed new

2Detailed in sec. 2.4 3apud [Boyd and Vandenberghe, 2004] 1.2. Literature Review 23 techniques in interior points directly applicable to convex optimization problems involving LMIs4. Geromel et al. [1991], in the case of time-invariant uncertainties, used convexity, constraining the uncertain state matrices within a polyhedral domain, so that the treatment could be conducted entirely based on vertex evaluations of that geometry, covering both discrete and continuous time settings. For systems with slow time variations5, the use of parameter-dependent Lyapunov functions (PDLFs) is motivated, above all, by the fact that the effectiveness of the QS will be smaller, the greater the ranges for uncertainties variation [Barmish and Jury, 1994]. Two of the pioneering works in the use of this formalism are due to Barmish and DeMarco [1986], where the affine parameterization of the Lyapunov matrix is introduced, and Mansour and Anderson [1993], where robust stability is proven6 for the special case of bilinear parameterization of this matrix. Dasgupta et al. [1994] extend some results of the stability analysis of uncertain linear time-invariant (LTI) systems for the time varying case using PDLFs and, additionally, in the same work, it is demonstrated that, for a set of polytopic system matrices, the robust stability can be established from the solution of an augmented Lyapunov equation7. In [Gahinet et al., 1996], the multiconvexity of the time-derivative of the Lyapunov function is considered in order to obtain sufficient conditions for the stability of LPV systems, through a methodology that consists of the explicit incorporation of quan- titative information about the rate of parameter variation in the quadratic PDLFs. A myriad of works would follow, based on this concept, trying to gradually reduce the influence of the polytopic treatment, with affine parametric dependence, on conservatism, emerging in several fields within control and systems engineering. Trofino and de Souza [1999], e.g., assume PDLFs with quadratic dependence on uncertain parameters8. Jadbabaie [1999], in the framework of the Takagi-Sugeno fuzzy systems analysis, deals with the term that involves the time-derivative of the Lyapunov matrix from a major that takes into account the contributions of all the 2n hypercube vertices where the system trajectories, in the parameter time derivatives space, lie. Chesi et al. [2004] and Geromel and Colaneri [2006] independently introduced a model able to provide the smallest possible hypervolume polytope9 – and therefore the one that will result in the least conservative analysis –, for the case of politopic LPVs systems, with quadratic PDLFs affine on the parameters, and whose rates of parameter variations as well as their time-derivatives are bounded, when considering geometric connections in the parameters space that manifest themselves from the sets of constraints on the accessible regions to the system trajectories. Despite the problem of conservatism being solved for this class of LPVs, there was still the issue of computational burden, given that, in many cases, even if the set of PDLFs capable of guaranteeing the robust stability of a certain system exists, the algorithms of opti-

4Although the connection between the problem of searching for Lyapunov functions and the semidefined programming algorithms (SDPs) had already been casted years before in [Pyatnitskii and Skorodinskii, 1982]. Ibid., p. 4 5In contrast to systems whose rates of variation in parameters are arbitrary. 6via Kharitonov’s theorem. 7[Dasgupta et al., 1994], thm. 2.1 8Bi-quadratic stability comprises the affine one as a special case. 9Detalhada na sec. 3.1. 24 Chapter 1. Introduction mization did not converge due to the high number of vertices of these polytopes10. Apkarian and Tuan [2000] and Montagner and Peres [2003] are some of the first papers to present strategies to mitigate the computational complexity in this class of LPVs. Mozelli et al. [2009] propose the use of extra variables in order to assist the numerical convergence of the optimization algorithms. In a recent work11, Mozelli and Adriano [2019] present an algorithm that allows a reduction in the number of LMIs constraints from a factorial growth rate – coming from the scenario of less conservatism – to linear, by encapsulating the exact polytope by a simplex. In all cases, however, the efficiency achieved is offset by penalties in terms of conservatism, with the only option remaining the search for trade-off solutions between these two aspects.

1.3 Text Structure

The remaining of the work is organized as follows: • In the next chapter, the mathematical tools will be briefly presented and the notation established. In a first moment, basic notions of topology are exposed, aimed at defining convexity in sets, as well as their connection with the problems of semidefinite programming through LMIs. Next, sec. 2.2 discusses convex polytopic geometry in the Euclidean space, with particular attention to aspects of symmetry and regularity. This section dialogues directly with the methodology described in sec. 3.3. The secs. 2.3 and 2.4 deal with LPV systems, basic def- initions of stability analysis and modeling, and the polytopic representation of uncertainties. • Ch. 3 contains the methodology. Here, all protocols, algorithms and formula- tions that will allow the replication of results will be described. It is divided into three parts. After a brief mathematical presentation of the related literature, in 3.1, the gradual transition between the regular simplex and the smallest possible hypervolume polytope12 is outlined using the vertex elimination and rotation procedures. Then, in 3.3, a proposal is presented for the use of cubic and orthoplectic convex hulls in the context of feasibility analysis. Finally, in 3.4, a possibility of extending the previous results for the state feedback controller synthesis is presented. • Ch.4 contains the contributions regarding the study of polytopic LPV systems proposed in the previous chapter, with results of analysis in sec. 4.2.1 and control in sec. 4.2.2. The “laboratory” used in numerical simulations is the easily scalable mass-spring-damper (MSD) model, described in sec. 4.1. • Ch. 5 is intended for conclusions and discussions about possible routes for the generalization of studies contained in this dissertation. At the end, the ap-

10In this approach, the vertices have factorial growth as a function of the number of time-varying uncertain parameters. 11Detailed in sec. 3.1 12For the case of polytopic LPV systems with quadratic PDLFs depending afﬁnely on the uncertain parameters. 1.3. Text Structure 25

pendices A to D can be found with mathematical proofs of speciﬁc properties, referenced, at the appropriate time, throughout the text. 26 Chapter 1. Introduction CHAPTER 2

Mathematical Background

2.1 Convex Analysis and LMIs

2.1.1 Introduction Due to its properties – to be explored throughout this section –, convexity is a de- sirable feature in the investigation of problems in various branches of mathematics. In particular, it is intended here to present the theoretical basis for the solution of a set of affine matrix constraints, under convex objective functions, as a problem of semidefinite programming (SDP). Following [Dullerud and Paganini, 2013], let points v1, v2 ∈ V, v1 6= v2, where V is a vector space over a field F, so that

l(v1, v2) := [v1, v2] = {v ∈ V | ∃µ [0, 1] . v := µv1 + (1 − µ)v2} (2.1) is the line segment, parametrically deﬁned, with extreme points in v1 and v2. Within the scope of set theory, the notions of convexity and strict convexity can then be established by the assertions: Deﬁnition 1. Let Q ⊂ V, with Q 6= ∅. Such Q is called a convex set iff

l(v1, v2) ⊂ Q, ∀v1, v2 ∈ Q. Deﬁnition 2. Let Q ⊂ V, with Q 6= ∅. Such Q is called strictly convex set iff

l(v1, v2) \ {v1, v2} ⊂ int Q, ∀v1, v2 ∈ Q, where int Q denotes the interior1 of Q.

As a linear combination of the points v1 and v2, v has its range of values arbitrated 2 by the structure ∑i=1 µivi, with µi ∈ F. Nevertheless, the restriction that µ ∈ [0, 1] imposes in their coefficients (specifically for the case of µ1 := µ e µ2 := 1 − µ) two 2 additional conditions: the unit sum, i.e. ∑i=1 µi = 1, characteristic of affine combinations; and µi ≥ 0, which in turn characterizes the conicity. Convex combinations emerge from the simultaneous verification of these two restrictions. An extension 2 k of (2.1) is obtained by taking a set of k extreme points , {vi}i=1, with vi ∈ V, lin- k early combined, such that v ≡ ∑i=1 µivi, with the k coefficients µi ∈ [0, 1] subject to k 2 ∑i=1 µi = 1. This scenario is illustrated in Fig. 2.1, for R . Singletons, line segments, hyperplanes and polyhedrons are some examples of non-empty convex sets.

1See def. 4. 2Convex combinations can also be generalized to include inﬁnite sums, integrals and probability distributions [Boyd and Vandenberghe, 2004].

27 28 Chapter 2. Mathematical Background

(a) (b) Figure 2.1: Convex (a) and nonconvex (b) sets.

2.1.2 Convexity-Preserving Operations Certain algebraic and topological operations do not inﬂuence the convexity of the sets, and can be used in the creation of new sets from original ones whose convexity is a priori guaranteed [Hiriart-Urruty and Lemaréchal, 2012]. Among these operations: T • Intersection: Q ≡ {Qi | i ∈ I} ⊂ V, I ⊂ N \ {0}, will always be convex for a family {Qi}i∈I ⊂ V of convex subsets.

• Affine mappings: (Affine image) If F : V → J is such that F (µv1 + (1 − µ) v2) = µF (v1) + (1 − µ) F (v2), ∀v1, v2 ∈ V ∧ ∀µ ∈ [0, 1], so, the image F (Q) := {F(v) | v ∈ Q}, for Q ⊂ V convex, will be convex in J . (Affine preimage) In addition, for a convex set S ⊂ J , F−1 (v) := {v ∈ V | F (v) ∈ S} will be convex. As immediate consequences: – Opposite set: If Q ∈ V is convex, so is −Q, provided that −Q ⊂ V. – Minkowski sum: For the family of convex subsets in the Euclidean space {Qi}i∈I ⊂ V, I ⊂ N \ {0}, the sum Q1 ⊕ Q2 ⊕ · · · ⊕ Qk := {∑i∈I vi | vi ∈ Qi} is convex.

– Scalling and translation: For the family of convex subsets {Qi}i∈I ⊂ V, I ⊂ N \ {0}, ∑i∈I µiQi := {∑i∈I µivi | vi ∈ Qi} is convex. • Closure and Interior: Definition 3. The closure of a convex set Q ⊂ V is defined as the smallest closed set that contains Q, i.e., cl (Q) = Q ∪ ∂Q, where ∂Q is the boundary of Q. Definition 4. The interior of Q ⊂ V, for Q convex, is defined as the set n o (Q) = ∈ Q ∃ > ∧ k ∈ V ( ) ⊂ Q int : q e 0 {ui}i=1 . Bu q, e , where Bu (q, e) := {a ∈ V | ka − qk < e} (2.2) k is an open e-ball centered at q ∈ V, with {ui}i=1 forming a basis for vector space V. Both topological operations cl (Q) and int (Q) over Q ⊂ V result in convex sets. 2.1. Convex Analysis and LMIs 29

2.1.3 Representations Convex sets are usually described by means of their convex hulls or by the intersection of a ﬁnite number of semispaces. The equivalence between these descriptions is ensured by the Fundamental Theorem of the Polytope Theory3.

Convex Hulls Deﬁnition 5. The convex hull of a set Q ⊂ V, Co (Q), is deﬁned as the intersection of all convex subsets of V that contain Q. Remark 1. Co (Q) is the smallest convex set in V that contains Q.

(a) (b) (c) Figure 2.2: The convex hull for a ﬁnite (a) and inﬁnite number of points (b). Also the convex hull of a nonconvex set (c).

k Taking, e.g., the ﬁnite set S := {vi}i=1 ⊂ V, its convex hull will correspond to the set of all possible convex combinations of its subsets: ( ) k k

Co (S) := v ∈ V v ≡ ∑ µivi, µi ∈ [0, 1] ∧ ∑ µi = 1 , (2.3) i=1 i=1 which, in geometric terms, presents itself as a closed region in V, such that any line segment l(vi, vj), ∀i, j ∈ N \ {0}, it is entirely contained within it. Similarly, for an arbitrary, not necessarily ﬁnite, set Q ⊂ V n o Q = ∈ V ∃ k ∈ k ∈ Q ∈ N Co ( ) : v {vi}i=1 . v Co {vi}i=1 , vi , k . (2.4)

Intersection of Halfspace Let the linear mapping F : V → R be such that, for q ∈ R, F(v) = q is surjective, i.e., always has a solution in the variable v ∈ V. In this case, any element vk of the null space of F can be integrated into the mapping solution set for a ﬁxed variable, vs ∈ V, producing {v ≡ vs + vk | F(vs) = q ∧ vk ∈ ker (F)}. Geometrically, this afﬁne transformation displaces the null space from its origin, giving rise to a linear manifold of dimension dim(V) − 1, i.e., an hyperplane [Coelho and Lourenço, 2001, Dullerud and Paganini, 2013].

3See sec. 2.2. 30 Chapter 2. Mathematical Background

Figure 2.3: Generation of a plane in R3.

Theorem 1 (Separating Hyperplane [Dullerud and Paganini, 2013]). If Q1 ⊂ V and Q2 ⊂ V are two convex subsets such that Q1 ∩ Q2 = ∅, for Q1, Q2 6= ∅, then sets can be separated by a hyperplane H(F, q), q ∈ R, so that

F(v1) ≤ q, ∀v1 ∈ Q1 F(v2) ≥ q, ∀v2 ∈ Q2.

Figure 2.4: Convex sets Q1 and Q2 separated by the hyperplane H (F, q).

Deﬁnition 6. Sets Q1 ⊂ V and Q2 ⊂ V, convex and non-empty, separated by the hyperplane H (F, q), are said to be properly separated if v1, v2 ∈/ H (F, q) , ∀v1 ∈ Q, v2 ∈ Q2.

Deﬁnition 7. Two convex subsets Q1 ⊂ V and Q2 ⊂ V are said to be strictly separated if a scalar e > 0 exists such that

F (v1 + Bu(0, e)) < q ∧ F (v2 + Bu(0, e)) > q, ∀v1 ∈ Q1, v2 ∈ Q2, with q ∈ R. 2.1. Convex Analysis and LMIs 31

Theorem 2 (Strictly Separation by hyperplane). If, in addition to the conditions in thm. 1, cl (Q1) ∩ cl (Q2) = ∅ ∧ ∃∂Q1, then there is a hyperplane H (F, q) that strictly separates them.

Deﬁnition 8. A supporting hyperplane, Hs ⊂ V, of the subset Q ⊂ V, convex and bounded, is the one that Q lies integrally into one of the closed semi-spaces delimited by it. Additionally, at least one element qs ∈ ∂Q, called limit point of Q with respect to Hs, must also belong to the supporting hyperplane. Theorem 3 (Supporting Hyperplane). Q ⊂ V convex, closed and bounded, implies the Hi ⊂ V I ⊂ N \ existence of a family of supporting hyperplanes s i∈I , {0}, associated with it.4

Figure 2.5: The supporting hyperplane theorem.

Thms. 1 and 3 are the cornerstones of several computational optimization methods. Deﬁnition 9 (Linear Matrix Inequality [Dullerud and Paganini, 2013]).

F(X) ≺ Q (2.5) is a Linear Matrix Inequality (LMI) over F if it meets the following conditions simultaneously: • X ∈ V, where V is an real vector space;

• F : V 7→ Hn is linear, where Hn is a real vector space formed by Hermitian matrices; • Q ∈ Hn. Remark 2. LMIs can be strict, as in (2.5), or not (F(X) Q).

Remark 3. A group of LMIs, F(i)(X) ≺ Q(i), ∀i ∈ N \{0} ∧ i ≤ m, can always be expressed (i) m (i) m as a single LMI: diag {F (X)}i=1 ≺ diag {Q }i=1 . m m Let {ui}i=1 ∈ V be a basis for this vector space, then ∃ {xi}i=1 ∀X ∈ V such that m X := ∑ xiui (2.6) i=1

4Conversely, the existence of this family along a contour ∂Q ⊂ V, with int rel (Q) 6= ∅, implies the convexity of Q. [Póczos, 2013] 32 Chapter 2. Mathematical Background

Then, once F is a linear function, (2.6) applied to (2.5) becomes

m ∑ xiF (ui) ≺ Q, (2.7) i=1

n where F (ui) =: Fi ∈ H , so that

m ∑ xiFi ≺ Q, (2.8) i=1

m is convex in the set of variables {xi}i=1. Here, the search for values of xi that verify the inequalities can be implemented via convex optimization algorithms [Boyd and Vandenberghe, 2004].

2.2 Higher-Dimensional Euclidean Convex Geometry

2.2.1 Context In The Elements, a work attributed to the Greek mathematician Euclides of Alexandria, and considered as a fundamental landmark of systematization in the study of the exact sciences, the bases of geometry are laid out from the synthesis of the ideas of the Platonic school about the primordial structures that constitute the Universe. Among the various deﬁnitions contained in Book I5, it is established:

(i) A point is that which has no part;

(ii) A line is breadthless length;

(iii) The ends of a line are points;

(iv) A straight line is a line which lies evenly with the points on itself;

(v) A surface is that which has length and breadth only;

(vi) The edges of a surface are lines;

(vii) A plane surface is a surface which lies evenly with the straight lines on itself.

The particular interest in symmetry that emerged from regular forms was reflected in the encyclopedic aspect assumed in the cataloging and formalizing procedures aimed at obtaining figures limited by lines and plane surfaces with equal internal angles. The infinity of regular polygons existing in two dimensions, however, was in stark contrast to the limited number of regular solids in three-dimensional space: only five polyhedrons (Fig. 2.6), already known at that time. Although the analo- gies between the spaces of dimension one, two and three were already present – as recorded, for example, in the dialogue of Socrates and Glaucon, in Plato’s Republic –, there is no mention about additional dimensions [Banchoff, 1990].

5apud [Barrow et al., 1732] 2.2. Higher-Dimensional Euclidean Convex Geometry 33

Figure 2.6: The ﬁve regular polyhedra, as depicted in the work Harmonices Mundi, Libri V [Kepler, 1619], by Johannes Keppler, in a reference to the Timaeus dialogue [Zeyl, 2000].

The discovery of non-Euclidean geometries, resulting from the studies of C. F. Gauss, N. I. Lobachevsky, B. Riemann and J. Bolyai [Gordon et al., 2012], in the early decades of the 19th century, as well as the recognition of ﬂat and solid geometries as the beginning of a sequence of geometries in higher dimensions and their formal treatment by A. Cayley, H. Grassmann and L. Schläﬂi [Hazewinkel, 2013], shortly afterwards, promoted a fundamental break with the idea, until then in vogue, that the geometry was constituted, only, in the branch of the mathematics that dialogues with the immediate physical reality.

2.2.2 Concepts A polytope is the geometric object corresponding to the generalization of polygons and polyhedra to the dimension Rr, r > 3, and, like them, it can be described as a compact convex set with a ﬁnite number of extreme points6. Therefore, two deﬁni- tions follow immediately:

Deﬁnition 10 (V-polytope). It is called V-polytope the one obtained from the convex hull of a compact and ﬁnite set. Considering V := v1, v2, ··· , vp ⊂ Rr, the V-polytope P corresponds to P := Co (V) .

Definition 11 (H-polytope [Toth et al., 2017]). For a real matrix A ∈ Rr×p and vector b ∈ Rr, a H-polytope is defined by the bounded solution set of a finite system of linear inequalities, i.e., n o p T P ≡ x ∈ R ai x ≤ bi .

Theorem 4 (Main Theorem of Polytope Theory [Ziegler, 2012]). Deﬁnitions (10) and (11) are equivalent.

Only three families of regular polytopes exist for all dimensions Toth et al. [2017], namely:

6See Carathéodory’s Theorem. 34 Chapter 2. Mathematical Background

Figure 2.7: Main Theorem of Polytope Theory.

• The regular simplexes; • The hypercubes (also called “measure polytopes”); • The orthoplexes (also called “cross-polytopes”). The following are the formal deﬁnitions of each one7. The Figs. 2.8, 2.9 and 2.10 show some of these polytopes. Deﬁnition 12 (n-simplex, {3, 3, 3n}). A regular n-simplex in Rn is given by √ ( n ) s 1 2 n 1 − n + 1 k Hn := Co u , u , ··· , u , ∑ u , (2.9) n k=1 where uk denotes the unit vectors coordinates in Rn.

(a) Triangle, {3}. (b) Tetrahedron, {3, 3}. (c) Pentachoron, {3, 3, 3}. Figure 2.8: Schlegel diagrams for the 2 to 4-simplex.

Deﬁnition 13 (n-cube, {4, 3, 3n}). A unit n-cube (or hypercube) in Rn is given by

( n ) c k Hn := Co ∑ αku | αk ∈ {+1, −1} . (2.10) k=1

7 Schläﬂi symbol, {p1, p2, ··· , pn−1}, is a recursive notation that encodes the local structure of the regular polytope. As described in Toth et al. [2017]: “For each i = 1, ··· , n − 1, if F is any (i + 1)-face of P, then pi is the number of i-faces of F that contain a given (i − 2)-face of F”. 2.3. Overview of Linear Parameter-Varying Systems 35

(a) Square, {4}. (b) Cube, {4, 3}. (c) Tesseract, {4, 3, 3}. Figure 2.9: Schlegel diagrams for the 2 to 4-cube.

Deﬁnition 14 (n-cross-polytope, {3, 4, 4n}). A unit n-cross-polytope (or orthoplex) in Rn is given by o n 1 2 no Hn := Co ±u , ±u , ··· , ±u . (2.11)

(a) Square, {4}. (b) Octahedron, {3, 4}. (c) 16-cell, {3, 3, 4}. Figure 2.10: Schlegel diagrams for the 2 to 4-cross-polytope.

2.3 Overview of Linear Parameter-Varying Systems

2.3.1 Introduction Formally, Linear Parameter-Varying (LPV) systems can be defined in terms of their state-space (SS) representation as follows: Definition 15. A dynamical system mathematically described by the generic formulation  x˙(t) = A (θ(t)) x(t) + B (θ(t)) u(t)  y(t) = C(θ(t))x(t) + D(θ(t))u(t)  x(0) = x0, is called a continuous-time Linear Parameter-Varying system. Here, A ∈ Rn×n,B ∈ Rn×m, C ∈ Rq×n and D ∈ Rq×m are matrix functions with continuous dependence on the scheduling 36 Chapter 2. Mathematical Background parameter θ(t) ∈ Rr, t ≥ 0. x(t) ∈ Rn, u(t) ∈ Rm, y(t) ∈ Rq are, respectively, the state, input, and output vectors. Unlike LTI system, the matrix functions in LPV models have temporal dependence, even though implicitly, what makes them time-varying systems. This is also the feature that opposes them to the LTV ones, whose time-dependence occurs in an explicit way. As detailed in Shamma [2012], the distinction between these formalisms is, in principle, quite subtle, and is fundamentally supported by the perspective adopted. When arbitrarily choosing a path for the θ(t) parameter in an LPV model, e.g., one has an LTV. On the other hand, by imposing conditions on the rate variation of the exogenous parameters and, eventually, on their time-derivatives, as is usually done in the scope of stability and performance analysis, a family of LTV systems is typified, through LPV representation. The paradigm of LPV systems has proven to be a suitable approach to handle a wide range of problems, especially in the fields of control and systems engineering. It is possible, for instance, to ease the analytical complexity in the modeling and identification processes by taking a family of linear structures as local approximators for nonlinear dynamics, so that high-fidelity models can be achieved (see, e.g., Lovera et al. [2013]). Moreover, the design of robust state feedback controllers finds in LPVs a straight dialog interface, as originally proposed in Shamma [1988], for the synthesis based on gain scheduling, since time-varying parameters can be taken as local system operating points in a Jacobian linearization procedure [Shamma, 2012, Hoffmann and Werner, 2014]. The literature dedicated to LPV applications is wide, covering from aerospace and automotive projects [Morato et al., 2019, Yang et al., 2019, Al-Jiboory et al., 2017, Fleps-Dezasse and Brembeck, 2016], to wind turbine control [Sloth et al., 2010, Inthamoussou et al., 2016].

2.3.2 Polytopic Representation The mathematical treatment of LPV systems, whether for the purposes of synthesis or analysis, requires the definition of the formalism to be used in the representation of the plant’s dependence, reflected in its SS matrices, on the exogenous parameter. Among the main frameworks, there are those based on gridding [Becker, 1993, Wu et al., 1996, Abbas et al., 2014], affine and polynomial dependence, linear fractional transformations (LTFs) [Lambrechts et al., 1993, Apkarian and Gahinet, 1995, Scherer, 2001], etc. All of them allow the mapping of a problem of infinite dimension, computationally intractable [Zapateiro and Pozo, 2012], due to the continuity of θ(t), in a finite set of LMIs. Naturally, any approach chosen will result in benefits and draw- backs, which are well documented in the references cited above. Definition 16 (Polynomially Parameter Dependent). A parametrically polynomial LPV model is one whose dependence on the exogenous parameter in the system matrices are polynomial functions, i.e.,

A (θ(t)) B (θ(t)) r A B = θαi i i , C (θ(t)) D (θ(t)) ∑ C D i=1 i i

αk i 1 2 r αi r i with α := αi αi ··· αi e θ := ∏k=1 θk . 2.3. Overview of Linear Parameter-Varying Systems 37

Deﬁnition 17 (LFT-based). An LPV/LFT system is one whose parametric dependency is decomposed via linear fractional transformations (LFTs). For

M M M := 11 12 , M21 M22 representing the nominal portion of the system. The parametric portion is delegated to the operator ∆u, for upper-type LFTs, and ∆l, for lower LFTs, so that ( A (θ(t)) B (θ(t)) Fu (M, ∆u) = C (θ(t)) D (θ(t)) Fl (M, ∆l) where −1 Fu (M, ∆u) := M22 + M21 (I − ∆u M11) ∆u M12 is called the upper LFT operator, and

−1 Fl (M, ∆l) := M11 + M12 (I − ∆l M22) ∆l M21, the lower LFT operator. LFT operators, in this case, act to conform the coefﬁcients of the state-space representation matrices into rational functions of the parameter θ, absorbed in the matrices ∆u(l). These, in turn, are unrestricted in terms of both structure and dynamical nature [Marcos et al., 2015], making it a very general approach. The focus of this work, however, is on models with afﬁne dependence, to be treated via polytopic representation. From these assumptions, let be the polytope in Rr

( r ) A ≡ A (θ(t)) = ∑ θi(t)Ai , (2.12) i=1 ¯ ¯ with θi(t) ∈ θi, θi , where θi and θi stands, respectively, for the minimum and maximum values assumed by the ith uncertain parameter, and Ai are constant matrices. In this case, θ (t) belongs to the unit simplex

( r ) r Sr ≡ λ ∈ R : ∑ λi = 1, λi ≥ 0, ∀i = 1, 2, ··· , r , (2.13) i=1 in a barycentric sum arrangement [Bridson and Haeﬂiger, 2013]. In this conformation, the polytope A, in (2.12), and the simplex Sr, in (2.13), are connected by the convex combination ¯ θi(t) = θiλi(t) + [1 − λi(t)] θi, (2.14) and r ! x˙ = ∑ θi(t)Ai x(t), (2.15) i=1 is said to be the polytopic uncertain model for the LPV system, deﬁned in 15, in the absence of an input signal8.

8Similar structures can be built for the other system matrices. 38 Chapter 2. Mathematical Background

2.4 Lyapunov Stability for Polytopic Systems

A common approach to obtain the stability certiﬁcate for polytopic LPV systems is based on Lyapunov’s Direct Method, with the quadratic Lyapunov candidate function, which has become known in the literature as Quadratic Stability (QS) (see, e.g., [Amato, 2006]). Deﬁnition 18 (Quadratic Stability). An LPV system x˙(t) = A (θ(t)) x(t) is said to be quadrically stable if, by taking a Lypunov candidate function in the form

V(x(t)) ≡ xT(t)Px(t), ∀x ∈ Rr \ {0} , (2.16) with P = PT 0; (2.17) it is possible to conclude that h i V˙ (x(t)) = x(t)T A (θ(t))T P + PA (θ(t)) x(t) < 0, ∀x ∈ Rr \ {0} m (2.18) A (θ(t))T P + PA (θ(t)) ≺ 0

The last condition of def. 18 is a problem of inﬁnite dimension. In the polytopic representation (2.15), however, we can eliminate the dependence of A (θ(t)) over this parameter, based on the convexity of A. Since

r T T A (θ(t)) P + PA (θ(t)) = ∑ θi(t) Ai P + PAi (2.19) i=1 a sufﬁcient condition for stability consists on a set of LMIs based only on the vertices. Limiting the search for a single matrix P satisfying the entire set of LMIs, nonetheless, raises the conservatism, taking into account that the worst variation rates scenarios will always be considered for θ(t) trajectories, like jumps and discontinuities [Sename et al., 2013], in such a way that QS is often unable to ensure the stability. On the other hand, having informations about the bounds of the parameters time- ˙ ¯ derivatives, i.e. θi ∈ δi, δi , an interesting alternative is to proceed with the stability analysis turning the P matrix, in Eq. (18), into a function of the uncertainty vector. This approach, originally introduced by Barmish and DeMarco [1986], has been known as Parameter Dependent Lyapunov Functions (PDLFs). The constraint on the derivative of the candidate Lyapunov function is now written as h i V˙ (θ(t), x(t)) = x(t)T A (θ(t))T P (θ(t)) + P (θ(t)) A (θ(t)) + P˙ (θ(t)) x(t) < 0. m (2.20) A (θ(t))T P (θ(t)) + P (θ(t)) A (θ(t)) + P˙ (θ(t)) ≺ 0

This structure have been extensively explored in the treatment of several problems in the LPV theory, including extensions to control [Feron et al., 1996] and discrete- time systems [Daafouz and Bernussou, 2001]. Naturally, the more information about 2.4. Lyapunov Stability for Polytopic Systems 39 the dynamic behavior of the uncertain system the Lyapunov function carries, the greater the possibility of developing less conservative formalisms: Mozelli and Pal- hares [2011], for example, achieved new stability conditions whose reduction in conservatism is obtained from the explicit incorporation of informations regarding high- order time-derivatives of the parameters in (2.4). In the simplest case of PDLF, P exhibits afﬁne dependence on θ, so that

r P = ∑ θi(t)Pi. i=1 This approach is adopted in several seminal studies aimed at assessing the robust stability of LPV system and all of the methodologies in the next chapter will make use of it. 40 Chapter 2. Mathematical Background CHAPTER 3

Methodology

3.1 Previous Works

In this section, two formalisms will be presented aimed at obtaining the polytope in the time-derivative parameter space that will serve as the background for further developments: the ﬁrst one, due to Chesi et al. [2004] and Geromel and Colaneri [2006], consists in generating the polytope from the convex hull resulting from the sum of the r uncertain time-varying parameters, contemplating all possible convex combinations of the extreme values of the time derivatives. This is a standard approach in the literature and, strictly speaking, corresponds to the scenario of least possible conservatism1, which, in contrast, increases the computational complexity during the analysis procedure. In the second formalism, proposed in Mozelli and Adriano [2019], the original polytope, from the aforementioned approach, will be “encapsulated” by a simplex. Two immediate consequences of this procedure are the decrease in the number of vertices growth rate – from exponential to linear – and the consequent improvement in the computational load, as well as the increase in conservatism.

3.1.1 Exact Polytope In the afﬁne PDLF (2.4), it is necessary to consider the term

r T ˙ T ˙ ΦPDLF := x P(θ)x = x ∑ θiPix. i=1 The theoretical framework presented by Chesi et al. [2004] and Geromel and Colaneri [2006] resorts to the geometrical and convexity aspects inherent to this type of modeling: since the parameters time-derivatives are conﬁned to both the hyperrectangle ˙ ˙ r ˙ ¯ θ ∈ R := θ ∈ R : δi ≤ θi ≤ δi, ∀i = 1, 2, ··· , r , and the hyperplane n o θ˙ ∈ π := θ˙ ∈ Rr : eTθ˙ = 0 , T r r ˙ for e ≡ 1 1 ··· 1 ∈ R , and given the restriction ∑i=1 θi = 0, we have n o ˙ 1 2 p j r j ¯ T j θi ∈ π ∩ R ≡ co v , v , ··· , v = v ∈ R : δk ≤ vk ≤ δk, e v = 0 , (3.1)

1For the case of polytopic LPV systems with quadratic PDLFs depending afﬁnely on the uncertain parameters.

41 42 Chapter 3. Methodology

j r×p where vi corresponds to the ith coordinate of the jth vertex. A matrix H ∈ R can be employed in order to group them   j   v1   j     v   H ≡  v1, v2, ··· ,  2  , ··· , vp  . (3.2)   .     .   j vr The resulting polytope, which will be refered to as the original polytope, has the smallest possible hypervolume and, therefore, will provide the least possible conservative analysis, since p r ! T ΦPDLF = x ∑ εj ∑ H(i,j)Pi x, (3.3) j=1 i=1 for ε ∈ Sp. Here, the number of vertices, p, has a factorial growth as function of the number of time-varying uncertain parameters, r. Analytically, this relation is modeled by the swinging factorial (OEIS A056040 [OEIS, 2019]): r! r p(r) = = , (3.4) (br/2c!)2 br/2c, [r mod 2 6= 0] , br/2c where b•c is the ﬂoor operator, [•] is the Iverson bracket, and the right hand-term is the trinomial coefﬁcient.

Table 3.1: Vertices growth, p, relative to the number of uncertain time-varying parameters, r, for (3.1).

r 2 3 4 5 6 7 8 9 10 11 p 2 6 6 30 20 140 70 630 252 2772

3.1.2 Simplectic Convex Hull Approach As an alternative to reduce the computational load, Mozelli and Adriano [2019] proposed, in a recent study, a methodology for vertex selection where the original convex hull (3.1) is replaced by a standard r-simplex, with r vertices, generated by the algorithm ( r j ∆j 2 + δj, if i = j, v¯i = (3.5) δj, otherwise; ¯ ∆j := δj − δj reﬂects the amplitude of the polytopic parametric intervalar representation. Similarly, grouping the vertices in a matrix,   j   v¯1   j     v¯   × H¯ ≡  v¯1, v¯2, ··· ,  2  , ··· , v¯r  ∈ Rr r. (3.6)   .     .   j v¯r 3.2. Vertex Elimination and Rotation Procedures 43

For some η ∈ Sr [Mozelli and Adriano, 2019]:

r r ! T ¯ ΦPDLF < x ∑ ηj ∑ H(i,j)Pi x. (3.7) j=1 i=1

In this case, the number of vertices is drastically reduced, implying, however, an increase in conservatism. Figure 3.1, for the three-dimensional case, and Tab. 3.1 displays a short comparison between these two methodologies.

Figure 3.1: Original convex hull (black vertices) and 2-simplex (red ones) overlapped, computed via (3.1) and (3.5), respectively, for −1 ≤ θ˙i(t) ≤ 1, i = 1, 2, 3. Conservatism comes from Minkowski difference, P ∩ R Λ2, between the polytopes.

3.2 Vertex Elimination and Rotation Procedures

In geometric terms, a possible form of reduction of conservatism consists, essentially in the approximation between the hypervolumes of the simplex polytope and the original one (3.1). For this task, in general, a larger number of vertices provides a better ﬁt, since extra regions of the time-derivative parameter space can be avoided. As illustrated in Fig. 3.2, it is possible, through successive suppression of vertices of the simplex – which is, by deﬁnition, the geometry with the lowest number of them in any dimension – to describe a gradual path between the two formalisms presented in the previous section. Next, we will introduce an analytical procedure for obtaining these hybrid geometries. i p j r Initially, consider the vertex sets v i=1 and v¯ j=1 generated by (3.1) and (3.5), respectively. Take

n oq n op [ n or−ξ v˜k := co vi v¯j , with 0 ≤ ξ ≤ r and 1 ≤ q ≤ p + br/2c − 1, k=1 i=1 j=1 (3.8) 44 Chapter 3. Methodology

(a) a (b) b

(e) e (f) f Figure 3.2: Some possible scenarios in the scope of simplex vertex suppression. (a), (b), (c) Only one vertex is eliminated, resulting in a total of 4 vertices. (c), (d), (e) Two simplex vertices are eliminated, resulting in a total of 5 vertices. 3.2. Vertex Elimination and Rotation Procedures 45 and store these elements into a matrix:

  j   v˜1   j     v˜   × H˜ ≡  v˜1, v˜2, ··· ,  2  , ··· , v˜q  ∈ Rr q. (3.9)   .     .   j v˜r

= ˜ = ¯ i p j r Thus, ξ 0 clearly implies H H, since v i=1 is fully covered by v¯ j=1. However, removing one or more elements from v¯j , i.e., forcing ξ ≥ 1 in (3.8), this will be no longer the case. Ultimately, by setting ξ = r, implies H˜ = H, and the original polytope is recovered. Thus, for some ζ ∈ Sq, q r ! T ˜ x ∑ ζj ∑ H(i,j)Pi x. (3.10) j=1 i=1 This formulation appears as a cost mediator between the previous ones, in the sense that it is up to the user to decide how close ξ and r will be. Tab. 3.2 was generated using Fukuda’s implementation of the double description method of Motzkin et al. [Motzkin et al., 1953, Fukuda and Prodon, 1995], through Polymake’s CDD package [Fukuda, Assarf et al., 2017], for convex hull estimation, and shows the growth dynamics of the vertex matrix (3.9) as ξ simplex vertices are suppressed. Note that the extremal values in each column correspond to the number of vertices of the simplex (ﬁrst row) and of the original polytope (last row), i.e., we move gradually from one geometry to another, as the eliminations are processed (confront with Tab. 3.1, e.g.). It is interesting to note that some of the values of ξ result in well-known sequences for the number of vertices coming from the original polytope, viz., deﬁning ϑ(r) ≡ r − ξ, we have, for ξ = 1 and ξ = 2,

• ξ = 1: ( rB(r) + ϑ(r), if r is even, q(r, 1) = B(r + 1) + ϑ(r), otherwise.

2r Here, B(r) ≡ ( r ) is the rth central binomial coefﬁcient (OEIS A000984 [OEIS, 2019, Rosen, 2017]).

• ξ = 2: ( (3r + 2)C(r) + ϑ(r), if r is even, q(r, 2) = (3r + 1)C(r − 1) + ϑ(r), otherwise;

1 2r 2r where C(r) ≡ r+1 ( r ) = B(r) − (r+1), ∀r ≥ 0, are the Catalan numbers (OEIS A000108 OEIS [2019]). This sequence is known for emerging in triangulation problems of convex n-gons [Stanley, 2015]. 46 Chapter 3. Methodology

Table 3.2: Vertices growth, q, relative to the number of uncertain time-varying parameters, r, and the number of vertices eliminated from simplex, ξ. Content inside the cells is arranged in the following format: total of vertices [simplex contribution + original polytope contribution].

r 4 5 6 7 8 9 10 ξ 0 4 [4+0] 5 [5+0] 6 [6+0] 7 [7+0] 8 [8+0] 9 [9+0] 10 [10+0] 1 6 [3+3] 16 [4+12] 15 [5+10] 66 [6+60] 42 [7+35] 288 [8+280] 135 [9+126] 2 7 [2+5] 24 [3+21] 20 [4+16] 105 [5+100] 61 [6+55] 462 [7+455] 204 [8+196] 3 7 [1+6] 29 [2+27] 22 [3+19] 128 [4+124] 70 [5+65] 561 [6+555] 238 [7+231] 4 6 [0+6] 31 [1+30] 22 [2+20] 139 [3+136] 73 [4+69] 610 [5+605] 252 [6+246] 5 - 30 [0+30] 21 [1+20] 142 [2+140] 73 [3+70] 629 [4+625] 256 [5+251] 6 - - 20 [0+20] 141 [1+140] 72 [2+70] 633 [3+630] 256 [4+252] 7 - - - 140 [0+140] 71 [1+70] 632 [2+630] 255 [3+252] 8 - - - - 70 [0+70] 631 [1+630] 254 [2+252] 9 - - - - - 630 [0+630] 253 [1+252] 10 ------252 [0+252]

For the general case2, the vertex growth is given by r hj r k i q(r, ξ) := r r − U − ξ × b 2 c, [r mod 2 6= 0] , b 2 c 2 nl r m l r m o ϑ(r) × − − 1 [r mod 2 = 0] r + ϑ(r), ∀ξ ∈ N, r ∈ N \ {0} , 2 2 b 2 c (3.11) where d•e stands for the ceiling operator and ( 0, if n < 0, U [n] := 1, otherwise, is the Heaviside step function. Still, Tab. 3.2 also reveals the occurrence of a saturation in the growth rate of the vertices coming from the original polytope, beginning at ξsat = r/2, for ξ even, and at ξsat = dr/2e + 1, for the odd ones. At the (ξsat + 1)th elimination, we have the maximum value for the number of vertices, surpassing even the number of vertices of the original polytope. By inspection, we can infer that the maximum number of eliminations such that the total of vertices is less than the total number of vertices of the original polytope is ξ < ξsat. In Appendix C, we demonstrate that this characteristic holds for higher values of r. This is a limit scenario in the elimination procedure. After this threshold, there is no advantage in using the proposed methodology, once the number of vertices

2See Appendix A for a combinatorial interpretation of this relation. 3.2. Vertex Elimination and Rotation Procedures 47 required is larger than the number of vertices in the original polytope, and there is still the conservatism because of the Minkowski difference between the original polytope and the one generated by H˜ . Below ξsat the elimination procedure can pay-off, since we have a trade-off solution between the original polytope and the simplex. In this situation, the new convex hull has fewer vertices than the original polytope and its volume is less than that of the simplex.

Practical procedure for determining the new convex hull Let H ∈ Rr×p and H¯ ∈ Rr×r be the matrices that contain the vertices of the original polytope and those of the simplex, respectively. For a symmetric uncertainty range, the new vertex matrix, H˜ , can be composed from these vertices by the following procedure:

• Select from the H¯ vertices, arranged along its columns, those to be eliminated. k k ¯ • The vertex v¯ must then be replaced by all vertices of H that meet vi = δj, ∀j = 1, 2, ··· , r.

• Repeated columns will always occur in H˜ for ξ ≥ 4 and must be eliminated.

Example: Consider the case r = 3, with δ ≤ θ˙i ≤ δ¯, ∀i = 1, 2, 3. So

 δ δ δ¯ δ¯ δ + δ¯ δ¯ + δ  H =  δ¯ δ¯ + δ δ δ + δ¯ δ¯ δ  δ + δ¯ δ¯ δ¯ + δ δ δ δ¯ and  3 + δ δ δ  H¯ =  δ 3 + δ δ  . δ δ 3 + δ

Eliminating the ﬁrst simplex vertex, based on the procedure detailed above, we search for all columns of H whose ﬁrst element is δ¯. Thus,

 δ¯ δ¯ δ δ  H˜ =  δ δ + δ¯ 3 + δ δ  , δ¯ + δ δ δ 3 + δ which corresponds to the polytope illustrated in Fig. 3.2a. By the same rule, elimination of the ﬁrst and third simplex vertices produce

 δ¯ δ¯ δ δ δ¯ + δ  H˜ =  δ δ + δ¯ 3 + δ δ¯ + δ δ  , δ¯ + δ δ δ δ¯ δ¯ which, in turn, corresponds to the polytope depicted in Fig. 3.2e. 48 Chapter 3. Methodology

(a) ξ = 0: the simplex is the convex hull. p = 4. (b) ξ = 1: vertex 4 eliminated. p = 6.

(e) ξ = 4: the original polytope. p = 6.

Figure 3.3: Saturation during the vertex elimination process in R4 dimensions. Sim- plex and original polytope were mapped to R3 subspace using the procedure described in sec 3.3.1. Red vertices are exposed, while the yellow ones are covered. 3.2. Vertex Elimination and Rotation Procedures 49

3.2.1 Vertex-Rotation Procedure Consider all possible vertex elimination sets, based on the methodology described in the previous section, for each ξ (number of simplex vertices to be suppressed) and r (number of time-varying uncertain variables) fixed. In this case, the eliminations can be interpreted as rotations around the normal vector eT ≡ 1 1 ··· 1 ∈ Rr, in the manifold which the parameters time derivative trajectories lie. In Fig. 3.2, e.g., with r = 3, all the configurations (a)-(c), for ξ = 1, could be generated, from the first polytope, (a), by successive counterclockwise rotations of 2π/3 rad, as well as (d)-(f) configurations, for ξ = 2, progressively rotating, by this same angle, (d). However, using purely eliminations, it is easy to see that there will be non-accessible configurations: those generated by sequential rotations of π/3 rad. A simple way to integrate them into the process is to pre-multiply the vertex matrix of the simplex, H¯ , by −1, which, geometrically, will promote a rotation of π rads, setting it in the sequence of odd rotations (2m + 1)π/3, m ∈ N. In the three-dimensional space, assuming symmetric uncertainties, rotation possibilities are depleted by considering 6 rotations of π/3 rad for ξ = 2, 6 rotations of π/3 rad for ξ = 1, and one rotation of π/3 rad (or, equivalently, π rad) for ξ = 0 (Fig. 3.4), the full simplex – in Appendix B we prove that, in this scenario, the latter is the only rotation that results in a new simplex.

Figure 3.4: Simplex rotation of π rad around normal vector e.

Although there are no changes in the geometric structure or in the number of 50 Chapter 3. Methodology vertices of the polytope, this methodology allows other regions of the time-derivative parameter space to be evaluated, producing, eventually, less conservative solutions in relation to their non-rotated analogues – other times, however, it can have the opposite effect. The pre-multiplication of the vertex matrix by a factor of −1, as a rotation procedure, is valid for all dimensions, via Givens rotation [Faul, 2018], and results in a simplex distinct from the original, also able to encapsulate the polytope (3.1), generated by the rule:

( r j − ∆j 2 + δj , if i = j, v¯i = −δj, otherwise. 3.3. Regular Polytopic Families for Robust LPV Analysis 51

3.3 Regular Polytopic Families for Robust LPV Analysis

A natural extension of Mozelli and Adriano [2019] consists in reducing the total number of vertices to be computed by replacing the original convex hull with hulls associated with the other families of regular polytopes. Another important factor to consider is the deﬁnition of which of them will survive the increase in dimensionality: as discussed in ch. 2, only hypercubes and cross-polytopes remain. The methodology for the composition of the vertex matrix in these cases will consist of three stages, namely: the generation of the respective encapsulating polytope in the subspace Rr−1, with the last row of the vertex matrix null, the reduction of residual hypervolume, T and its transposition to the hyperplane with normal vector 1 1 ··· 1 in Rr, that contains the system origin, (0, 0, ··· , 0) ∈ Rr, through rotation matrices. For clarity, the procedures will be explained directly by numerical examples.

3.3.1 Hyperplane Mapping via Rotation Matrices The original polytope, obtained from the methodology described in Geromel and Colaneri [2006], is generated in a manifold r − 1 of the Euclidean space, belonging to T the hyperplane, Π, with normal vector e = 1 1 ··· 1 that crosses the system origin. Solving the problem of mapping Π, through a set of rotation matrices, in Π0, T whose normal vector is of the form 0 0 ··· 1 ∈ Rr, and that also contains the origin, automatically produces the solution of the problem of taking a simplex, cube or cross-polytope generated, respectively, by the deﬁnitions 12, 13 and 14, in Π0, and mapping them in Π, just transposing3 these matrices.

Example 1. Mapping of the original polytope from dimension 3 to dimension 2.

Consider, initially, a system with three uncertain variables, such that kθ˙ik ≤ 1, ∀i = 1, 2, 3. The constraints will produce the vertex matrix of the original polytope:

 −1 0 1 1 0 −1  H =  0 −1 −1 0 1 1  (3.12) 1 1 0 −1 −1 0 with trajectories lying in the inclined plane Π. Moreover

T T ! 1 1 0 · 0 1 0 ϕa = arccos √ (3.13a) 2 and T T ! 1 1 1 · 0 0 1 ϕb = arccos √ , (3.13b) 3 T T T are the angles between the vectors 1 1 0 and 0 1 0 , and 1 1 1 T and 0 0 1 , respectively. Taking the rotation matrices around the axes θ˙1 and θ˙3,

3The inverse of a unitary matrix is equal to its conjugated transpose. 52 Chapter 3. Methodology it is possible to perform the mapping Π 7→ Π0 of all vectors vj ∈ Π from

     1 0 0 cos ϕ − sin ϕ 0

H˘ :=  0 cos ϕ − sin ϕ   sin ϕ cos ϕ 0  H . (3.14)   0 sin ϕ cos ϕ 0 0 1 ϕ=ϕb ϕ=ϕa

The matrix (3.12) has its vectors rotated, as shown in Fig. 3.5, preserving the magni- tudes:  √ √ √ √ √ √  −1/√ 2 1/√ 2 2 1/√ 2 −1/√ 2 − 2 H˘ =  − 3/2 − 3/2 0 3/2 3/2 0  . (3.15) 0 0 0 0 0 0

Figure 3.5: Mapping Π 7→ Π0.

An analogous effect can be obtained for higher dimensions using the so-called Givens rotations. In this case, the entire coordinate system in dimension r will be rotated along a two-dimensional plane repeatedly, i.e., the transformation will consist of a sequence of rotations, in appropriate order and angles, to be performed in the planes θ˙rθ˙r−1 → θ˙r−1θ˙r → · · · → θ˙2θ˙1.

Deﬁnition 19 (Givens Rotation Matrix [Faul, 2018]). Ω[k,l,ϕ] ∈ Rr×r is called a Givens Rotation Matrix if det Ω[k,l,ϕ] = 1 and if all its elements are such as those of an identity matrix, I ∈ Rr×r, with the exception of four elements, which must assume the following 3.3. Regular Polytopic Families for Robust LPV Analysis 53 values:   cos (ϕ) , if i = j = k ∨ i = j = l; [k,l,ϕ]  Ωi,j := sin (ϕ) , if i = k ∧ j = l; (3.16)  − sin (ϕ) , if i = l ∧ j = k; for ϕ ∈ [−π, π].

Example 2. Mapping of the original polytope from dimension 4 to dimension 3.

The vertex matrix of the original polytope in the hyperplane Π is

 −1 −1 −1 1 1 1   −1 1 1 −1 −1 1  H =   .  1 −1 1 −1 1 −1  1 1 −1 1 −1 −1

Three Givens rotation matrices will be necessary in order to reach Π 7→ Π0, namely:

 1 0 0 0  √  0 1 0 0  [3,4,arccos(1/ 4)]  h √ i h √ i  Ω =    0 0 cos arccos 1/ 4 − sin arccos 1/ 4   h √ i h √ i  0 0 sin arccos 1/ 4 cos arccos 1/ 4

 1 0 0 0  h √ i h √ i √  0 cos arccos 1/ 3 − sin arccos 1/ 3 0  [2,3,arccos(1/ 3)]   Ω =  h √ i h √ i     0 sin arccos 1/ 3 cos arccos 1/ 3 0  0 0 0 1

 h √ i h √ i  cos arccos 1/ 2 − sin arccos 1/ 2 0 0 √  h √ i h √ i  [1,2,arccos(1/ 2)]   Ω =  sin arccos 1/ 2 cos arccos 1/ 2 0 0     0 0 1 0  0 0 0 1

Thus, √ √ √ H˘ ≡ Ω[3,4,arccos(1/ 4)]Ω[2,3,arccos(1/ 3)]Ω[1,2,arccos(1/ 2)]H  √ √ √ √  0 −2/ 2 −2/ 2 2/ 2 2/ 2 0  −1.6330 0.8165 −0.8165 0.8165 −0.8165 1.6330  =   . (3.17)  −1.1547 −1.1547 1.1547 −1.1547 1.1547 1.1547  0 0 0 0 0 0

The polytope described by the vertices of the matrix (3.17) is shown in Fig. 3.6. 54 Chapter 3. Methodology

(b) Plane θ˙1θ˙2.

(a) Π 7→ Π0. (d) Plane θ˙3θ˙1.

Figure 3.6: Polytope H˘ .

In the general case, this type of mapping will require a set of r − 1 Givens rotation matrices: √ √ √ H˘ ≡ Ω[r−1,r,arccos(1/ r)]Ω[r−2,r−1,arccos(1/ r−1)] ··· Ω[1,2,arccos(1/ 2)]H

=: Ξr H, ∀r ≥ 3, r ∈ N, (3.18) i.e., −1 √ [−l,1−l,arccos(1/ 1−l)] Ξr = ∏ Ω , ∀r ≥ 3, r ∈ N. (3.19) l=−r+1 In Appendix D there is a formal proof about the reduction of dimensionality using (3.19). Besides that, in the next section a simpler way to perform this mapping will be presented.

3.3.2 The Enclosing Polytope Hypervolume Once in the Π0 hyperplane, a new rotation procedure is required in order to determine the smallest hypervolume required for the encapsulating polytope. In addition to the rotation angles themselves, two other elements directly affect this minimization: the geometry of the encapsulating polytope (cubic or orthoplectic) and the and the order assumed in the rotation matrix multiplication. In the case of cubic convex hulls, it is demanded that the rotations minimize c ˘ max |Ψr H| ( ) , (3.20) 1≤i≤r i,j 1≤j≤p 3.3. Regular Polytopic Families for Robust LPV Analysis 55

c where Ψr is a new set of Givens rotation matrices. For orthoplectic convex hulls, rotations must promote the minimization of the 1-norm4, i.e., the function to be min- imized in this case is given by o ˘ kΨr Hk1 (3.21) with r ˘ ˘ kHk1 := max |H(i,j)|, (3.22) ≤j≤p ∑ 1 i=1 o c where Ψr is another set of rotations, not necessarily equal to Ψr. In the three- dimensional case, reported in example 1, when proceeding with the rotation of the original polytope in the Π0 hyperplane, the values of (3.20) and (3.22) vary as shown in Fig. 3.7, so that, when encapsulating the original polytope by a 2-cube without rotating it, the residual area will be greater than that obtained if the 2-cube is previously rotated by an angle of π/4 rads5. A similar analysis can be conducted for orthoplex.

1.415 2

1.41 1.99 1.405

1.4 1.98

1.395 1.97 1.39 1.96 1.385 Eq. (3.17) [a. u.] Eq. (3.18) [a. u.] 1.38 1.95

1.375 1.94 1.37

1.365 1.93 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Angle [rads] Angle [rad] (a) Cube. (b) Cross-Polytope. Figure 3.7: Hypervolume minimization procedure.

Note that the set of rotations of the polytope in the subspace Rr−1, to which it has been mapped, will allow for an increasing variety of angle combinations as the number of dimensions grows (since that the only restriction imposed is that the last row of the vertex matrix remains null). In addition, as already mentioned, given the non-commutativity of the matrix multiplication, the order adopted for rotations will determine the conﬁgurations accessible to this polytope and, therefore, subordinate the minimization of the hypervolume to a speciﬁc matrix structure. In view of the standardization of the analysis, the following structure of the rotation matrix will be

4Manhattan distance: the r-cross-polytope in the taxicab geometry corresponds to a r-hyperesphere in the Euclidian one. 5In the three-dimensional case, the minimum values have a periodicity of π/6 rad. For higher dimensions, however, this behavior is not kept. 56 Chapter 3. Methodology

(a) 2-cube, without rotation. 2-ortoplex, ro- (b) 2-cube, rotated by π/4 rad. 2-ortoplex, tated by π/4 rad. without rotation.

Figure 3.8: Original polytope mapped to Π0 and encapsulated by a 2-cube and by a 2-orthoplex. assumed here6 for the minimization process: • ( ) [1,2,ψ1] [2,3,ψ2] [r−2,r−1,ψr−2] Ψr := Ω Ω ··· Ω (3.23) Through a linear search algorithm, the set of optimal angles was determined, n (•)∗or−2 ψ , that will minimize the residual hypervolumes of the encapsulating poly- i i=1 topes for the two proposed geometries, as provided in the Tab. 3.3. It should be noted that the set of optimal angles was determined from rotations of the original polytope through the structure (3.23), so that, when applying this structure, with the (•)T proper rotation angles, in the encapsulating polytope, its transpose, Ψr , must be employed. Although it is possible to obtain the optimal angles through a linear search or spe- ciﬁc optimization algorithms, this procedure creates a new problem, whose computational cost is high. Below are two approaches aimed at minimizing the hypervolumes of the encapsulating polytopes using the same structure (3.23), but with negligible computational complexity7.

An Heuristic Approach for Minimizing the Hypervolume of the Enclosing Hyper- cube In the Tab. 3.5 the tangent of the optimal angles for the hypercube, presented in Tab. 3.3, is evaluated. A ﬁrst point to highlight is that, taking the module of each element, a pattern is formed: the same values of the tangents for the optimal angles of the dimension r = k occur in the dimension r = k + 1, with the introduction of a new c∗ c∗ tangent angle between the elements tan ψr−1 and tan (ψr ).

6The matrix Ω[r−1,r,ϕr−1] is not allowed to be incorporated into Ψr because this Givens matrix would modify the null line of the polytope in Π0. 7Assuming a computer with standard hardware speciﬁcations. 3.3. Regular Polytopic Families for Robust LPV Analysis 57

Table 3.3: Optimal angles, ψ∗ [rads], for the minimization of the enclosing hypercube hypervolume employing the structure (3.23) as a function of the number of uncertain time-varying parameters, r. Step length: 10−5 rads.

r 3 4 5 6 7 8 {ψc∗} c∗ ψ1 2.8798 2.3562 2.3562 2.3562 2.3562 2.3562 c∗ ψ2 - 2.8798 0.9600 2.1816 0.9600 2.1816 c∗ ψ3 - - 2.8798 1.0472 2.0944 2.0944 c∗ ψ4 - - - 2.8798 1.1006 2.0410 c∗ ψ5 - - - - 2.8798 1.1372 c∗ ψ6 - - - - - 2.8798 c ˘ max Ψr H (i,j) 1.3660 1.3494 1.3171 1.2948 1.2888 1.2823

Table 3.4: Optimal angles, ψ∗ [rads], for the minimization of the enclosing orthoplex hypervolume employing the structure (3.23) as a function of the number of uncertain time-varying parameters, r. Step length: 10−5 rads.

r 3 4 5 6 7 8 {ψo∗} o∗ ψ1 0.0000 2.3562 2.3562 3.0756 0.0000 0.0000 o∗ ψ2 - 0.9553 2.1863 0.9553 0.9553 2.5260 o∗ ψ3 - - 2.4825 0.0000 0.0000 0.0000 o∗ ψ4 - - - 2.4569 2.4905 0.7000 o∗ ψ5 - - - - 1.0600 2.9290 o∗ ψ6 - - - - - 2.4279 o ˘ kΨr Hk1 1.9319 2.0000 3.4175 4.2365 5.1487 5.8749

In Fig. 3.9 a logarithm fit with data from Tab. 3.5 is performed, through the equation y = a log(bx). The regression model adjusted produced a coefficient of determi- nation8 R2 = 1. Here a=1.056, b=1.289, and the logarithm base is e. y corresponds to the modulus of the tangent of the new angle in each dimension. x is the total number of rotations to be performed in the minimization procedure, i.e., x=r-2. Therefore, only the definition of which angles correspond to that absolute value of the tangent is required. For symmetry reasons, only two angles will correspond to this tangent value, and a test on the vertex matrix is required in order to find out which one will reduce the element with the greatest absolute value. Although it lacks formal proof about this behavior, the numerical results suggest that this pattern will remain for higher dimensions.

8The other ﬁt-statistic measures are: the sum of squares due to error (SSE) = 4.3895 × 10−9, degree of freedom in the error (DFE) = 4, adjusted R-square = 1.0000, and root mean squared error (RMSE) = 3.3127 × 10−5. 58 Chapter 3. Methodology

Table 3.5: Tangent of the optimal angles, ψ∗ [rads], for the minimization of the enclosing hypercube hypervolume employing the structure (3.23) as a function of the number of uncertain time-varying parameters, r.

r 3 4 5 6 7 8 {ψc∗} c∗ tan(ψ1 ) -0.2679 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 c∗ tan(ψ2 ) - -0.2679 1.4282 -1.4282 1.4282 -1.4282 c∗ tan(ψ3 ) - - -0.2679 1.7321 -1.7321 -1.7321 c∗ tan(ψ4 ) - - - -0.2679 1.9677 -1.9677 c∗ tan(ψ5 ) - - - - -0.2679 2.1602 c∗ tan(ψ6 ) ------0.2679

An Heuristic Approach for Minimizing the Hypervolume of the Enclosing Ortho- plex

The set of optimal angles intended to minimize the 1-norm, shown in Tab. 3.4, appar- ently does not follow a pattern. The methodology in this case will be based on increasing the sparsity of the vertex matrix of the original polytope, H˘ , when it is mapped to the subspace Rr−1. Despite several studies are available in the literature aimed at minimizing the 1-norm by numerical optimization methods, these procedures tend to be computationally costly. In addition, the minimization in the proposed approach must be performed through rotation matrices. The same mapping (3.18) can be generated more easily using the original vertex matrix rows. Let H ∈ Rr×p be the vertex matrix of the original polytope in r dimensions, lying in the hyperplane Π. It is easy to show that the matrix resulting from the mapping Π 7→ Π0 described in the previous section, is given by

 √  √1/ 1 × 2 (h1 − h2)  × ( + − )   1/ 2 3 h1 h2 2h3   .   .     p × + k −  H˘ = Ξr H =  1/ k (k 1) ∑i=1 hi khk+1  , (3.24)    .   .   p −   1/ (r − 2) × (r − 1) r 1 h − rh   ∑i=1 i r  0 where hk denotes the kth row of H. The sparsity increase of this matrix will also be accomplished by row manipulations. Here, however, the rows of H˘ will assist in determining the best projection angles, in order to make the mapped polytope vertices and the Cartesian axes coincident (as much as possible). Clearly, not all vertices will line up. It is intended to determine the projection angle argument, τˆk, as well as the signal 3.3. Regular Polytopic Families for Robust LPV Analysis 59

2.2

1.8

1.6

1.4

y 1.2 y = a*log(b*x) 1 sse: 4.3895e-9 rsquare: 1.0000 dfe: 4 0.8 adjrsquare: 1.000 rmse: 3.3127e-05 0.6

0.4

0.2 1 2 3 4 5 6 x

Figure 3.9: The blue points correspond to (x, | tan (ψc∗) |), where ψc∗ is the new optimal angle in this dimension, and x := r − 2 is the total of rotations performed. a=1.056, b=1.289, e log is in the base e. in 1 ψk = arccos ±√ . (3.25) τˆk This argument must be taken in such a way that it minimizes, at each step, the value of the 1-norm of H˘ when multiplied by the Givens rotation matrix of the kth iteration. Therefore, one must evaluate

 ˘ 2 ˘ 2  hk+1 + hk τ := (3.26) k  ˘ 2  hk i

The subscript i indicates that the operation in (3.26) must be carried out element by element of these rows. See that τk will result in a row vector. τˆk, in turn, will correspond to the most incident element at τk. Next, the minimization process will be applied to the case of the original 4-dimensional polytope.

Example 3. Minimization of 1-norm of the original 4-dimensional polytope mapped to 3 dimensions. 60 Chapter 3. Methodology

The vertex matrix of the original polytope in the hyperplane Π is

 −1 −1 −1 1 1 1   −1 1 1 −1 −1 1  H =   .  1 −1 1 −1 1 −1  1 1 −1 1 −1 −1 Starting from the alternative formulation (3.24) for mapping, it is obtained, directly from row manipulations, √ √ √ √ √ √   0/√2 −2/√2 −2/√2 2/√2 2/√2 0/√2  −4/ 6 2/ 6 −2/ 6 2/ 6 −2/ 6 4/ 6  H˘ =  √ √ √ √ √ √  , (3.27)  −4/ 12 −4/ 12 4/ 12 −4/ 12 4/ 12 4/ 12  0 0 0 0 0 0 which is exactly the same as the matrix (3.17). Once the mapping is complete, the procedures to increase sparsity begin. In view of the structure (3.23), only two Givens matrices will be used, namely:  √ √  cos arccos 1/ τˆ1 − sin arccos 1/ τˆ1 0 0 √ √ √ [1,2,arccos(1/ τˆ )]  sin arccos 1/ τˆ1 cos arccos 1/ τˆ1 0 0  Ω 1 =    0 0 1 0  0 0 0 1 and  1 0 0 0  √ √ √ [2,3,arccos(1/ τˆ )]  0 cos arccos 1/ τˆ2 − sin arccos 1/ τˆ2 0  Ω 2 =  √ √  .  0 sin arccos 1/ τˆ2 cos arccos 1/ τˆ2 0  0 0 0 1

The procedure should be conducted from the (r − 1)th row, in this case, i = 3, up to the ﬁrst one, taken two by two 9.

• Obtaining the ψ2 angle:  h √ √ i √  (−4/ 6)2 + (−4/ 12)2 /(−4/ 6)2  h √ √ i √     2 2 2  3/2  (2/ 6) + (−4/ 12) /(2/ 6)   h √ √ i √   3   2 2 2     (−2/ 6) + (4/ 12) /(−2/ 6)   3  τT =  h √ √ i √  =   (3.28) 2  2 2 2   3   (2/ 6) + (−4/ 12) /(2/ 6)     h √ √ i √   3   2 2 2   (−2/ 6) + (4/ 12) /(−2/ 6)  3/2  h √ √ i √  (4/ 6)2 + (4/ 12)2 /(4/ 6)2

Since the 3 element took place 4 times (against 2 occurrences of 3/2), τˆ2 = 3 is the value that allows the best projection. Another important point to consider

9Based on numerical simulations, minimization from (r − 1)th row of the matrix towards the ﬁrst showed better results than other combinations. 3.3. Regular Polytopic Families for Robust LPV Analysis 61

is the sign of the arccos function argument 10, which can be deﬁned based on the number of occurrences of ±1 in a signal function applied to the division of the two “active” rows. The sign of the argument will now be the one with the lowest incidence. For h i sign h˘ 3/h˘ 2 = 1 −1 −1 −1 −1 1 , i

√ √ √ √  0 −2/ 2 −2/ 2 2/ 2 2/ 2 0  √ √ √ √ √ [2,3,arccos(1/ τˆ )]  0 2/ 2 −2/ 2 2/ 2 −2/ 2 0  Ω 2 | = H˘ =   . τˆ2 3  −2 0 0 0 0 2  0 0 0 0 0 0 (3.29) As it is a heuristic procedure, a check on the 1-norm of (3.29) must be conducted at each new rotation, in order to verify if there was, in fact, a reduction in relation to the previous conﬁguration. If the norm has not decreased, the rotation is discarded. In particular, (3.29) shows a reduction in the 1-norm compared to (3.27), from 3.35454 to 2.8284 and, therefore, this rotation will be maintained.

• Obtaining the ψ1 angle: Applying the same procedure, now to the ﬁrst and second rows of (3.29),

 (0)2 + (0)2 /(0)2  h √ √ i √    2 2 2  undeﬁned  (−2/ 2) + (2/ 2) /(−2/ 2)   h √ √ i √   2   2 2 2     (−2/ 2) + (−2/ 2) /(−2/ 2)   2  τT =  h √ √ i √  =   . (3.30) 1  2 2 2   2   (2/ 2) + (2/ 2) /(2/ 2)     h √ √ i √   2   2 2 2   (2/ 2) + (−2/ 2) /(2/ 2)  undeﬁned (0)2 + (0)2 /(0)2

Ergo, τˆ1 = 2 (4 occurrences in the τ1 vector) and h i sign h˘ 2/h˘ 1 = −1 −1 1 1 −1 −1 , i √ resulting in the projection angle arccos 1/ 2 . Thus,

√ √ Hˆ := Ω[1,2,arccos(1/ τˆ1)]Ω[2,3,arccos(1/ τˆ2)] H˘ | {zo } Ψ4  0 −2 0 0 2 0   0 0 −2 2 0 0  =   , (3.31)  −2 0 0 0 0 2  0 0 0 0 0 0

10| tan [arccos (x)] | = | tan [arccos (−x)] | 62 Chapter 3. Methodology

which is the lowest 1-norm conﬁguration for this dimension: in relation to the matrix (3.29), there is a reduction of the 1-norm from 2.8284 to 2. Although it coincides with the optimal angles for r = 4, as a rule, this will not be the case. The polytope (3.31) is shown in Fig. (3.10).

Tab. 3.6 displays the results of this procedure for r = 8. Using optimal angles, as shown in Tab. 3.4 the norm-1 of the mapped polytope is 5.8749, while the procedure leads to 6.0388. Whereas, without any intervention on H˘ , kH˘ k = 7.3701.

Table 3.6: H˘ is the original 8-dimensional polytope, with 70 vertices, mapped to the subspace of dimension 7. Initially, kH˘ k1 = 7.3701. Plane ψo kΨo H˘ k √ 8 1 θ˙6 × θ˙7 arccos 1/ 2.3333 6.7895 θ˙ × θ˙ 0 6.7895 5 6 √ θ˙4 × θ˙5 arccos −1/ 1.6667 6.4326 θ˙ × θ˙ 0 6.4326 3 4 √ θ˙2 × θ˙3 arccos −1/ 1.5 6.0388

θ˙1 × θ˙2 0 6.0388

(b) Plano θ˙1θ˙2.

(a) Π 7→ Π0. (d) Plano θ˙3θ˙1

Figure 3.10: Polytope (3.31). 3.3. Regular Polytopic Families for Robust LPV Analysis 63

3.3.3 The Enclosing Polytope Vertex-Matrix Composition

The Ξr transformation matrix allows elements belonging to the Π hyperplane to be mapped to Π0, making the last row of the vertex matrix null. In this section, polytopes will be generated in the subspace Rr−1 following deﬁnitions 13 (hypercube) and 14 T (cross-polytope), and by means of Ξr , these will be taken to the conﬁguration of the original polytope, after having their dimensions adjusted. However, before mapping Π0 7→ Π, i.e., to the original region where the stability analysis will take place, the (•)T encapsulating polytope will be rotated through Ψr in order to reduce the residual hypervolume. This entire process will result in a new convex, cubic or orthoplectic, hull. c | c ˘ | In possession of Ψr and max Ψr H (i,j), obtained in the previous sections, the r-hypercube in the original manifold, that contains in its interior H, is given by    H¯ c  c = T | c ˘ c| ( c)T r−1 H : Ξr max Ψr Hr (i,j) Ψr . (3.32) 1≤i≤r 01×2r−1  1≤j≤p 

To generate r−cross-polytope in the original manifold, covering H, it is required that o o ˘ Ψr and kΨr Hk1 have already been obtained in the procedure described in the previous sections, so H¯ o o T o ˘ o T r−1 H := Ξr kΨr Hk1 (Ψr ) , (3.33) 01×2(r−1) where H¯ c and H¯ o are generated according to defs. 13 and 14, respectively.

Example 4: Generating the cubic convex hull in three dimensions.

 0.7071 0.4082 0.5774  Hc =  −0.7071 0.4082 0.5774  × 0 −0.8165 0.5774 | {z } T Ξ3              √  −0.9659 0.2588 0   -1 1 -1 1   1 + 3  ×  −0.2588 0.9659 0   -1 -1 1 1  .  2   | {z } 0 0 1 0 0 0 0   c ˘ c  max1≤i≤3[Ψ3 H3]( ) | {z } | {z }  i,j T " c #   1≤j≤6 Ψc H   ( 3) 2    01×2r−1 64 Chapter 3. Methodology

(a) (b)

Figure 3.11: Cube and cross-polytope encapsulating the original polytope in Π0 and Π.

Example 5: Generating the orthoplectic convex hull in three dimensions.

 0.7071 0.4082 0.5774  Ho =  −0.7071 0.4082 0.5774  × 0 −0.8165 0.5774 | {z } T Ξ3              √ !  1 0 0   1 0 -1 0  √ 3 + 3  × 2 √  0 1 0   0 1 0 -1  .  2 3 0 0 1 0 0 0 0  | {z }   o | {z } | {z }  kΨ H˘ k1 T     3 Ψo o   ( 3) H2       01×2(3−1)  3.4. An Extension for Robust State Feedback Control 65

(b) Plane θ˙1θ˙2.

(a) Π 7→ Π0. (d) Plane θ˙3θ˙1

Figure 3.12: Original 4-dimensional polytope mapped to Π0 and encapsulated by the simplex (yellow), cube (blue) and orthoplex (green, identical to the original polytope). T 0 When multiplied by Ξ4 , they return to the original manifold, i.e., Π 7→ Π.

3.4 An Extension for Robust State Feedback Control

In this section, the results Chesi et al. [2004], Geromel and Colaneri [2006] and Mozelli and Adriano [2019], both within the scope of robust stability analysis, will be ex- tended to the state feedback control synthesis, based on the development presented in Montagner and Peres [2006]. Thus, let

u(t) ≡ K(θ(t))x(t) (3.34) the control law, where K(θ(t)) ∈ Rm×n is a gain matrix.

Deﬁnition 20. Based on Lemma 18, the LPV system (2.15) is said state feedback (SF) stabi- lizable if, besides the existence of P(θ(t)) = P(θ(t))T 0, the inequality

V˙ = x(t)T He{P(θ(t)) (A(θ(t) + B(θ(t))K(θ(t)))} + P˙ (θ(t)) x(t) < 0 (3.35) holds, where He {A} := A + AT.

Through variable substitution W (θ(t)) := P(θ)−1 and Z(θ(t)) := K (θ(t)) W(θ(t)), and after some algebra, it is possible to derive, from Eq. (3.35), the following condition

He{A(θ(t)W(θ(t)) + B(θ(t))Z(θ(t))} − W˙ (θ(t)) ≺ 0. (3.36) 66 Chapter 3. Methodology

Three distinct formalisms will be employed, which will differ basically in how the term corresponding to the time derivative of P (θ(t)) will be treated. In all of them, the scheduled gain will be a function of the time-varying parameters and of the matrices A(θ(t)) and B(θ(t)) evaluated at the polytope vertices. The ﬁrst one is based on the following theorem (proved in Montagner and Peres [2006]). Here, the time-varying parameters boundaries must be symmetrical, which ¯ implies to make δi =: −ζi and δi = ζi.

Theorem 5. For a given set of scalar real parameters ζi ≥ 0, i ∈ N ∧ i ≤ r − 1, if there exist n×n m×n symmetric positive deﬁnite matrices Wi ∈ R and matrices Zi ∈ R , i ∈ N ∧ i ≤ r, such that11 r−1 He {AiWi + BiZi} + ∑ ±ζi (Wi − Wr) ≺ 0 (3.37) i=1 r−1 He {AiWk + AkWi + BiZk + BkZi} + 2 ∑ ±ζi(Wi − Wr) ≺ 0, with k = i + 1, ··· , r, i=1 (3.38) then the stability of the closed loop LPV system (2.15) can be ensured by SF control law u(t) = −1 r K(θ(t))x(t), with K(θ(t)) = Z(θ(t))W(θ(t)) , for Z(θ(t)) = ∑i=1 θi(t)Zi,W(θ(t)) = r r ∑i=1 θi(t)Wi and ∑i=1 θi = 1, θi ≥ 0, ∀i ∈ N ∧ i ≤ r. With minor changes in the previous approach – especially in the inequality (3.36) –, the existence of the gain matrix K(θ(t)) able to stabilize the system relying on the theoretical framework presented in Chesi et al. [2004], Geromel and Colaneri [2006] is guaranteed by the next theorem.

Theorem 6. For a given set of scalar real parameters ζi ≥ 0, i ∈ N ∧ i ≤ r and εj, j ∈ n×n N ∧ j ≤ p, εj ∈ Sp, if there exist symmetric positive deﬁnite matrices Wi ∈ R and m×n matrices Zi ∈ R , i ∈ N ∧ i ≤ r, such that

p r He {AiWi + BiZi} − ∑ εj ∑ H(i,j)Wi ≺ 0, i = 1, ..., r (3.39) j=1 i=1

p r He {AiWk + AkWi + BiZk + BkZi} − 2 ∑ εj ∑ H(i,j)Wi ≺ 0, i = 1, ..., r − 1; k = i + 1, ..., r, j=1 i=1 (3.40) 1 2 p r×p with H = [ v v ··· v ] ∈ R and |θ˙i| ≤ ζi, ∀i ∈ N ∧ i ≤ r, then the stability of the closed loop LPV system (2.15) can be ensured by SF control law u(t) = K(θ(t))x(t), with −1 r r K(θ(t)) = Z(θ(t))W(θ(t)) , for Z(θ(t)) = ∑i=1 θi(t)Zi,W(θ(t)) = ∑i=1 θi(t)Wi and r ∑i=1 θi = 1, θi ≥ 0, ∀i ∈ N ∧ i ≤ r. In this section the term involving the derivative of time-varying parameters will be considered according to the algorithm proposed in Mozelli and Adriano [2019]. As in the feasibility analysis frameworks, this procedure is expected to reduce the order of computational complexity from exponential, proper to both previously presented methods, to linear.

11All possible combinations of ± should be considered in the composition of the LMIs. 3.4. An Extension for Robust State Feedback Control 67

Theorem 7. For a given set of scalar real parameters ζi ≥ 0, i ∈ N ∧ i ≤ r and ηj, n×n j ∈ N ∧ j ≤ r, ηj ∈ Sr, if there exist symmetric positive deﬁnite matrices Wi ∈ R and m×n matrices Zi ∈ R , i ∈ N ∧ i ≤ r, such that

r r ¯ He {AiWi + BiZi} − ∑ ηj ∑ H(i,j)Wi ≺ 0, i = 1, ..., r (3.41) j=1 i=1

r r ¯ He {AiWk + AkWi + BiZk + BkZi} − 2 ∑ ηj ∑ H(i,j)Wi ≺ 0, i = 1, ..., r − 1; k = i + 1, ..., r, j=1 i=1 (3.42) 1 2 r r×r with H¯ = [ v¯ v¯ ··· v¯ ] ∈ R and |θ˙i| ≤ ζi, ∀i ∈ N ∧ i ≤ r, then the stability of the closed loop LPV system (2.15) can be ensured by SF control law u(t) = K(θ(t))x(t), −1 r r with K(θ(t)) = Z(θ(t))W(θ(t)) , for Z(θ(t)) = ∑i=1 θi(t)Zi,W(θ(t)) = ∑i=1 θi(t)Wi r and ∑i=1 θi = 1, θi ≥ 0, ∀i ∈ N ∧ i ≤ r. Remark 4. Theorems 6 and 7 only contemplate the polytopes treated in sec. 3.1. All other polytopic geometries presented secs. 3.2 and 3.3 can, nonetheless, be easily integrated into this formalism by simply reformulating the negative portion of (3.41) e (3.42), referring to the summations, replacing it with the condition corresponding to the polytope of interest. 68 Chapter 3. Methodology CHAPTER 4

Results

4.1 Model Description

For comparison purposes, we will borrow the same damped harmonic oscillator model from Mozelli and Adriano [2019], which consists of b masses interconnected by a spring-damping system, such that the masses at the beginning and at the end of the chain of interconnected masses m1 and mb are attached to ﬁxed structures, as in Fig. 4.1. A total of 3b + 2 parameters are required for its full description. The system dynamics, with n = 2b degrees of freedom (DOF), obey

Figure 4.1: The mass-spring-damper system with b = 3.

Mx¨(t) + Cx˙(t) + Kx(t) = 0, (4.1) = b ∈ Rb×b for M : diag {mi}i=1 , the mass matrix,   c1 + c2 −c2 0 ··· 0  −c c + c −c ··· 0   2 2 3 3   0 −c c + c ··· 0  C :=  3 3 4  ,  . .. .   . . .  0 0 0 ··· cb + cb+1 the damping coefﬁcients matrix, and   k1 + k2 −k2 0 ··· 0  −k k + k −k ··· 0   2 2 3 3   0 −k k + c ··· 0  K ≡  3 3 4  ,  . .. .   . . .  0 0 0 ··· kb + kb+1

69 70 Chapter 4. Results the stiffness matrix. From (4.1) the state-space representation is given by x˙ 0 × I x = n n n (4.2) x¨ −M−1K −M−1C x˙

The variation range allowed for the parameters is listed in Tab. 4.1.

Table 4.1: Bounds for model parameters. SI units.

Parameter Lower Upper Stiffness, ki 100 200 Damping coefﬁcient, ci 4.00 8.00 Mass, mi 5.00 5.50

The aspects related to conservatism and computational complexity will be evaluated by gradual increases in dimensionality, n, and number of uncertain time-varying parameters, r. Uncertainties will fall on l elastic coefﬁcients, so that the total number of vertices for the system will be r = 2l, with l ≤ b + 1. The remaining parameters are considered to be exact. Simulations were conducted in MATLABr R2015a on Linux Ubuntur 18.04, with solver SeDuMi 1.3 [Strum, 2001] and parser YALMIP R20190425 [Lofberg, 2004], while the convex hulls for the rotated polytopes were lined off in the current version 3.0 of polymake[Assarf et al., 2017] via GMP’s exact rational arithmetic of the cddlib package [Fukuda, Toth et al., 2017].

4.2 Complexity and Conservatism

Let’s take k¯ := (kmax + kmin) /2 = 150 N/m as the nominal value of the elasticity constant. It is required to ﬁnd the highest possible value of γ such that

ki = k¯ ± γ, ∀i = 1, 2, ··· , l, with |k˙ i| ≤ 1, ∀i = 1, 2, ··· , l, results in a feasible configuration, assuming that all masses have a fixed value, set at 5.25 Kg, as well as all damping coefficients, at 6.00 Ns/m. The number of decision variables for each pair r × n is shown in Tab. 4.2.

4.2.1 Vertex Elimination and Rotation Procedures In terms of complexity, Tab. 4.3 presents the number of constraints for each geometry in the elimination procedure. It should be emphasized that from Eq. (2.4) there T is a baseline number of constraints because of the term Ai Pi + Pi Ai. Additional constraints arise from the numerical treatment of the term ΦPDLF, according to the options in Eq. (3.10). 4.2. Complexity and Conservatism 71

Table 4.2: Number of decision variables for each pair r × n in the MSD model.

n 4 6 8 10 12 r 2 52 114 200 310 444 4 72 156 272 420 600 8 112 240 416 640 912

The procedure for eliminating simplex vertices was performed sequentially, from the ﬁrst to the fourth column of H¯ . Note that the effect of the constraints, other than those related to ΦPDLF, causes, in the ﬁrst elimination, the hybrid geometry and the original one to equate at 28, for r = 4. As we proceed, taking ξ = 2, the saturation region begins, and the number of constraints is increased to 32, which ends only in ξ = 4, i.e., in the original system. The range of options, however, expands as the scale of the system increases: the interval ξ ∈ [0, 4], for r = 8, covers from the simplex to the saturated hybrid geometry. In this case, there are three options for mitigating the computational load: the simplex (ξ = 0), which reduces the original total of constraints by approximately 90%; ξ = 1, with a reduction of approximately 40%; and ξ = 2, with 6% reduction. Naturally, the choice to reduce complexity should be taken in view of the impacts produced on conservatism.

Table 4.3: Number of constraints over the decision variables for each geometry.

ξ 0 1 2 3 4 H r 2 6 6 6 NA NA 6 4 20 28 32 32 28 28 8 72 344 496 568 592 568

Tab. 4.4 presents the results of conservatism of the MSD system, both for geometries coming directly from the elimination process and for those rotated from π rads. Starting from the non-rotated case, in connection with the aforementioned complexity data, a ﬁrst point to be highlighted is that the same number of vertices between the hybrid geometry and the original polytope does not necessarily imply that they will exhibit the same results for conservatism: ξ = 1, for r = 4, e.g., although, from a computational point of view, it requires the solution of the same number of LMIs as ξ = 4 (original polytope), it incurs a slightly more conservative solution. Furthermore, although there is no change in the total number of constraints, due to saturation, as in ξ = 2 and ξ = 3, for r = 4, both with 32, conservatism will suffer a reduction, since the residual hypervolume is reduced for every elimination. In the case of r = 8, it is noted that with only one elimination, in the worst case, n = 10, conservatism already reaches a fraction of 94% of the exact result, saving 224 constraints (approximately 40% of the total). Evidently, the eliminations will be so much more opportune, the 72 Chapter 4. Results greater the scale of the system, considering that, in these cases, the large computational load comes from the term ΦPDLF.

Table 4.4: Computational performance of methodologies with respect to the maximum value of γ for which stability can be guaranteed. The element in parentheses corresponds to the number of vertices for the ﬁnal polytope after the rotation of the geometry by an angle of π rad with respect to the hyperplane normal vector.

n 4 6 8 10 r 2 153 (153) 118 (118) 99 (99) 88 (88) H¯ 4 108 (105) 89 (89) 73 (72) 63 (62) 8 64 (62) 63 (62) 55 (55) 46 (46) 2 153 (153) 118 (118) 99 (99) 88 (88) ξ = 1 4 108 (110) 89 (89) 74 (74) 65 (65) 8 67 (66) 65 (65) 57 (57) 49 (49) 2 153 (153) 118 (118) 99 (99) 88 (88) ξ = 2 4 109 (110) 90 (89) 75 (74) 66 (65) 8 67 (67) 66 (66) 58 (58) 50 (49) 2 NA (NA) NA (NA) NA (NA) NA (NA) ξ = 3 4 110 (110) 90 (90) 75 (75) 66 (66) 8 68 (68) 66 (67) 58 (58) 50 (50) 2 NA (NA) NA (NA) NA (NA) NA (NA) ξ = 4 4 110 (NA) 90 (NA) 75 (NA) 67 (NA) 8 68 (68) 67 (67) 59 (58) 51 (50) 2 153 (NA) 118 (NA) 99 (NA) 88 (NA) H 4 110 (NA) 90 (NA) 75 (NA) 67 (NA) 8 69 (NA) 68 (NA) 59 (NA) 52 (NA)

Another interesting aspect from the point of view of conservatism is the rotation by π rad. This strategy allows, eventually, to improve such results at a negligible computational cost1 (viz., the multiplication of H¯ by −1), by allowing LMIs to be evaluated in another region of the time-derivative parameter space. There is also the possibility of using it in conjunction with the elimination processes: for ξ = 1 and n = 4, in r = 4, e.g., though subject to the same number of constraints as the original polytope, the non-rotated hybrid geometry has a conservatism approximately 2% lower, its rotated version, however, is less conservative, and matches the exact value.

4.2.2 Regular Polytopic Families Tab. 4.5 shows the number of LMI constraints in each geometry (as in Tab. 4.3, for the elimination of vertices, there is also the baseline contribution due to the term 1It must be considered, however, that, although the rotation procedure itself is not computationally costly, the lack of a rule to ensure that rotation will reduce the conservatism imply, from a practical point of view, that the two conﬁgurations, rotated and non-rotated, demand evaluation. 4.2. Complexity and Conservatism 73

T Ai Pi + Pi Ai). Orthoplex, like simplex, has linear growth, viz., 2(r − 1), which makes it an attractive option for ease the complexity. Cubic geometry, in this case, is always the one with the greatest number of restrictions. This occurs, as can be seen more easily in Fig. 4.2, as a consequence of the number of dimensions in the MSD model, in the conﬁguration used, which is always even (since r = 2l). Mathematically,

r! r! < (r−1) < 2 2 2 , r h r−1 i 2 ! r even ! 2 r odd where the ﬁrst term of the inequality is the growth in the number of vertices in the original polytope when r is even, while the last one refers to the odd case. This characteristic of the swinging factorial variation conditions the use of the hypercube, in the sense of computational load attenuation, to the number of time-varying uncertain parameters, whether even or odd.

Table 4.5: Number of constraints over the decision variables for each geometry.

r Hs Hc Ho H 2 6 6 6 6 4 20 36 28 28 8 72 1032 120 568

3000

2500

2000 Original Simplex Cube 1500 Cross-Polytope

1000 Number of vertices

500

0 2 3 4 5 6 7 8 9 10 11 Dimension

Figure 4.2: Vertex growth in each geometry.

With regard to conservatism, both the orthoplex and the hypercube begin to show better results when the scale of the system increases. In r = 4, e.g., The results of the 74 Chapter 4. Results hypercube will gradually approach those presented by the simplex for large n: in n = 4, while the simplex reaches approximately γ = 108, the hypercube get only γ = 103, however, in n = 12, they are already practically identical in terms of conservatism; for r = 8, the conservatism of the simplex remains inferior to that of the hypercube in n = 4. Logically, although this scenario is a priori interesting, the computational cost of the hypercube is much higher, not justifying its choice. Orthoplex, in turn, exhibits results of conservatism very close to that of the hypercube: for n = 10 and r = 8, e.g., both provide γ = 48 (approximately 92.3% of the maximum value), against γ = 46 (approximately 88.46% of the maximum value) of the simplex, with the fundamental difference that while the hypercube imposes 1032 constraints, the simplex and the orthoplex, with linear complexity growth, demand 72 and 120 LMIs, which implies a reduction of 496 and 448 LMIs in relation to the original polytope, respectively.

Table 4.6: Computational performance of methodologies with respect to the maximum value of γ for which stability can be guaranteed for alternative geometries.

n 4 6 8 10 12 r 2 153 118 99 88 81 Hs 4 108 89 73 63 56 8 64 63 55 46 40 2 153 118 99 88 81 Hc 4 103 86 72 62 55 8 62 64 56 48 42 2 153 118 99 88 81 Ho 4 110 90 75 67 61 8 60 61 55 48 41 2 153 118 99 88 81 H 4 110 90 75 67 61 8 69 68 59 52 46

4.2.3 Control Extension This analysis, unlike the previous ones, will focus on aspects of complexity. It is known that conservatism will tend to increase the greater the residual hypervolume produced in the encapsulation process, however, it is possible that, when subjected to

Table 4.7: Computational performance of the orthoplectic convex hull, with respect to the maximum value of γ, for r = 8. Optimal adjust compared to that obtained by the heuristic procedure described in sec. 3.3.2. n 4 6 8 10 12 o o H (Ψ8) 60 61 55 48 41 o o∗ H (Ψ8 ) 62 62 56 48 41 4.2. Complexity and Conservatism 75

performance indexes – H2 or H∞ –, some of the approaches studied result in more efﬁcient compensators, making the speculations in this line to sound hasty. The conservatism analysis will be restricted to evaluating the capacity of the methods in the generation of gain matrices. In addition, as another perspective, the complexity analysis here will be conducted based on the computational time. The simulations were conducted in the same machine previously described, but using LMI Labr package. In order to exemplify the characteristics related to computational burden of the approaches described in sec. 3.4, 300 conﬁgurations, randomly generated, were computed for each pair n × r shown in Table 4.82. For simplicity, the matrix B(θ(t)) was taken parameter-independent in all of them. Besides that, A(θ(t)) was required to have at least one positive eigenvalue and verifying the rank condition for controllabil- ity in all its vertices, i.e.,

2 n−1 = ∀ ≤ rank BAiBAi B ··· Ai B n, i p.

Table 4.8: Computational performance of methodologies. The ﬁrst row of each cell corresponds to the average time required to obtain the gain matrix in that conﬁguration, while the second one indicates the success percentage of the approach. ζi = 1, ∀i ∈ N ∧ i ≤ r.

Theorem 5 Theorem 6 Theorem 7 r=2 0.17 ± 0.03 s 0.0279 ± 0.005 s 0.026 ± 0.003 s 1 1 1 r=3 25 ± 4 s 0.3 ± 0.2 s 0.18 ± 0.09 s n=4 0.2 0.96 0.96 r=4 − 2 ± 1 s 1.8 ± 0.9 s − 0.9 0.9 r=2 0.3 ± 0.1 s 0.06 ± 0.02 s 0.06 ± 0.02 s 1 1 1 r=3 144 ± 80 s 1.3 ± 0.6 s 0.6 ± 0.2 s n=6 0.4 1 1 r=4 − 15 ± 5 s 8 ± 2 s − 1 1 r=2 0.9 ± 0.2 s 0.19 ± 0.06 s 0.19 ± 0.06 s 1 1 1 r=3 886 ± 384 s 5 ± 2 s 2 ± 1 s n=8 0.4 1 1 r=4 − 63 ± 25 s 40 ± 16 s − 1 1

Theorem 5 is especially sensitive to the number of vertices, signiﬁcantly increasing the mean time compared to the others when r = 3 and not being able to produce solutions for r = 4; besides, it exhibits reasonable conservatism due to the process of implementing the information concerning the parameter θr(t) variation rates in

2Except for the approach I with r = 3, where only 15 conﬁgurations were computed. 76 Chapter 4. Results

LMIs. Both approaches, theorems 6 and 7, were able to ensure the existence of the gain matrix K(θ(t)) for all tested conﬁgurations. Theorem 7, however, did so with considerably shorter times. As originally conceived, it is an interesting strategy for investigating systems with many vertices in the sense of promoting substantial reduction in the computational costs. CHAPTER 5

Conclusions

In this work, a set of new approaches for the treatment of polytopic linear parameter- varying (LPV) systems with quadratic parameter-dependent Lyapunov functions , affine on the uncertain parameters, all aimed at balancing aspects of computational complexity and conservatism.The strategy employed consists only of modifying the problem geometry from the parameter time derivatives space, based on the condition r ˙ ∑i=1 θi = 0, where r is the total number of uncertain parameters. Essentially, in algebraic terms, one group of linear inequalities is replaced by another, containing a smaller number of them, capable of maintaining the characteristics of convexity; from a geometrical point of view, a new polytope, with a lower number of vertices, is defined so as to contain the exact polytope within it. Although there is a gain in terms of computational burden, the robust stability analysis tends to become more conservative in these cases, since it starts to consider extra regions in the parameter space, which are never visited by the system trajectories. A myriad of studies on stability analysis and control synthesis in polytopic LPV systems, using PDLFs, are available in the literature. To a large extent, investigations focus on the development of methodologies for reducing conservatism, and are often relapsing in terms of the computational demand aspects of their approaches, making, in certain cases, their use in practical applications prohibitive, especially those that in- volve relatively complex models and require faster response times. Thence comes the option adopted throughout this dissertation in the concentration of efforts for the development of analytical methodologies or of low computational cost in the definition of new geometries. In terms of general contributions, this work can be identified as a “bridge” between the studies developed by Chesi et al. [2004], Geromel and Colaneri [2006] and Mozelli and Adriano [2019] from two perspectives: the graduation one, through the strategy of vertex elimination, and the generalization, through the introduction of new families of polytopes that, together with the simplex, exhaust the possibilities of regular geometries for higher dimensions. A new way of eventually reducing conservatism is also presented, based on rotations of the polytopes in the derivative parameters space, i.e., without any intervention on the number of LMIs/vertices involved in the problem. In addition, the formalism proposed by Montagner and Peres [2006], within the framework of robust state feedback control,is used in conjunction with the geometries of the aforementioned works in order to obtain the gain matrices with a lower computational cost. Below, the conclusions will be presented in detail about each of the developed approaches, highlighting the specific contributions as well as the points that still need improvement.

77 78 Chapter 5. Conclusions

• The vertex elimination methodology, explained in sec. 3.2, consists of the idea that, starting from a simplectic convex hull, in the derivative parameters space, gradually move towards the exact convex hull, through successive eliminations of elements. Convexity is maintained due to the fact that the simplex contains the original polytope in its configuration, with the lowest possible residual hypervolume configuration, as presented by Mozelli and Adriano [2019]. With this, by eliminating one or more vertices of the simplex, some of the vertices of the original polytope are discovered, becoming part of the convex hull. Each elimination, therefore, tends to increase the number of LMIs to be evaluated, avoiding, however, that regions whose trajectories of the system do not access, are taken into account in the analysis. The dynamics of vertex growth as the eliminations are processed has been studied and an analytical relation has been established, which makes possible to determine a priori how many vertices will come from the parcel involving the temporal derivative of the Lyapunov matrix based on the total number of time-varying uncertain parameters and the number of vertices of the simplex to be eliminated. This relation can be understood in the light of the combinatorics as detailed in the Appendix A. It was found that some of the numerical sequences generated for certain number of eliminations are related to well-known sequences within the scope of number theory, such as the central binomial coefficients, which occur in the elimination of a simplex vertex, and the Catalan numbers, for two vertices. In Appendix B, an analytical proof is presented that the method is no longer effective after a certain number of eliminations, in the sense that the number of vertices thereafter can even sup- plant the number of vertices of the exact polytope, disagreeing with the original purpose of the methodology. From a purely algebraic point of view, this behavior can be explained by the presence of Newton’s binomials in the relation that provides the dynamics of vertex growth. Throughout the dissertation, the effect of this behavior is discussed in terms of the problem geometry. It is also proposed a practical procedure, with easy computational implementation, which allows the vertex matrix for these hybrid geometries to be obtained from the vertex matrices of the original polytope and simplex. Another way to interpret a set of eliminations of the same number of vertices in different positions of the simplex is by means of rotations of the polytope: it is shown that without any change in the geometric structure, one can eventually reduce conservatism, simply considering other regions in the parameter space. One way to get these geometries rotated by a fixed value is stated. In this line, there are still some interesting points to be explored, in particular: (i) the development of a formal demonstration that there are only two possible simplexes, in the configuration of the lowest residual hypervolume, that contain the original polytope for any dimension (as was done in Appendix C, for the three-dimensional case) and (ii) the formulation of a test that allows to define a priori which of these simplexes will promote the reduction in conservatism in the robust stability analysis via LMIs (or a mathematical proof of its nonexistence). 79

• In sec. 3.3 some proposals were presented for the application of the other two families of regular polytopes – hypercubes and orthoplexes – for the robust analysis of stability. Of course, this approach can also be identified as an extension of the use of the simplex to reduce computational complexity in the sense of investigating the performance of other regular geometries. As the first contribution of this part, the Givens rotation matrices are introduced with the purpose of reducing the dimensionality of the original polytope, since the trajectories of the systems, in the Chesi et al. [2004], Geromel and Colaneri [2006] formulation lie in a Rn−1 manifold in the parameter space. Appendix D presents formal proof that these polytopes can always be mapped to a subspace Rn−1 through these rotation matrices, regardless of their dimensionality. It is possible that this formalism assists in the conception of new results with less conservative than those currently available in the literature for norm-bounded uncertainties. A practical rule for mapping the polytopes is described based purely on manipulating the their vertex matrix rows: an immediate benefit of this treatment is the mitiga- tion of numerical errors, since the need for repeated matrix multiplications is eliminated. Although less important, but interesting from a conceptual point of view, it is the possibility of graphical visualization of the polytopes generated in dimension 4 when mapped to the three-dimensional space. The conservative results show that these alternative families of polytopes – cubic and orthoplectic – fit well with the original polytope, especially as the scale of the system is increased. The most promising result is carried by the orthoplectic family, since it delivers one of the smallest conservatisms among the families in addition to a linear order growth for the number of vertices. The heuristic for determining the angles that minimize the hypervolume in the case of the orthoplex, minimizing the 1-norm through specific projections that increase the matrix sparsity, can eventually find potential applications in other fields of study, such as signal processing and big data. The angles of rotation that minimize the residual hypervolume for these polytopes are defined by means of heuristic processes and, despite the good performance, it would be interesting to develop exact methods for the determination of these hulls, similar to what is done in Mozelli and Adriano [2019] for the simplex. The logarithm fit obtained in Fig. 3.9 is a strong indication of the existence of an exact methodology for the hypercube family.

• In sec. 3.4, the formalism presented by Montagner and Peres [2006] was used to extend the previous results of stability analysis to the scope of state feedback control. Although it constitutes only a first step in this direction, the results obtained significantly reduced the time for calculating the feedback gain matrices. The exact geometry and the simplex were able to generate the matrices for most of the tested configurations, presenting the same percentage of failure. A deeper investigation still needs to be conducted in order to discriminate about the quality of this control, by specifying, for example, performance indexes for the controllers. 80 Chapter 5. Conclusions

5.1 Next Steps

A non-exhaustive list of ramiﬁcations from the present study includes:

• Control synthesis and performance indexes (e.g., H∞ and H2); • Time-delayed LPV systems;

• LPV systems with strong nonlinearities (e.g., saturation and hysteresis);

• Norm bounded approach;

• Reproduction of the results in the spectral domain;

• Exact methods of minimizing the residual hypervolume of hypercubes and orthoplexes;

• Investigation of number sequences produced in the light of Number Theory. APPENDIX A

A combinatorial interpretation of the vertex growth

The dynamics of vertex growth as the elimination procedure follows can be interpreted in the light of combinatorics. Initially, based on this formalism, the equations that model the even and odd sequences of the swinging factorial (Eq. (3.4)) will be deduced, as well as the linear growth shown in the case of the simplex. Then, based on numerical examples, an interpretation of Eq. (3.11) – regarding the growth of the number of vertices in the elimination process – will be developed. We will restrict ourselves to the case of symmetric uncertainties1. Note that the structure of the vertex matrix H is closely related to the nature of r being odd or even: in the ﬁrst case,

j n j j o Vodd := v | vi ∈ O, ∀i = 1, 2, ··· , r , ∀j = 1, 2, ··· , p, (A.1) with O := δ, δ¯ ; while in the latter,

j n j j o Veven := v | vi ∈ E, ∀i = 1, 2, ··· , r , ∀j = 1, 2, ··· , p, (A.2) j ¯ j with E := O ∪ {0}; both the cases are subject to the restriction n(V(•), δ) − n(V(•), δ) = 0, where n(S, s) stands for the cardinality of the set S with respect to the element s. • Even: Here we want all possible combinations of the elements of E, included repeti- tions, so that results in k1 times the ﬁrst element, δ¯, and k2 times the second one, δ, totalizing r elements, i.e., r = k1 + k2. The total of subsets, p, is given by the generalized permutation r! Peven(r; k1, k2) := . (A.3) k1!k2! Based on the cardinality constraint relative to the elements δ¯ and δ, it is necessary that we have k1 = k2. Moreover k1 + k2 must total r elements. Consequently, k1 = r/2 = k2. So, (A.3) becomes r! Peven(r; r/2, r/2) = , (A.4) [(r/2)!]2 which corresponds to the even portion of the swinging factorial (OEIS A000984 OEIS [2019]).

1There is no loss of generality since the uncertain system can always be written in this form.

81 82 Appendix A. A combinatorial interpretation of the vertex growth

• Odd: This case repeats the previous one with the fundamental difference that each vector vj must contain one, and only one, null element, so that the cardinality condition over δ¯ and δ is met. So, in addition to the k1 elements δ¯ and to the k2 elements δ, there will be an 0 element, so that each position assumed by him along the vector vj will result in a new set of permutations for this new conﬁguration2. Thus, we arrive at the relation r − 1 r − 1 (r − 1)! r! ( ) = − = = Podd r, k1, k2 rPeven r 1; , r 2 2 (A.5) 2 2 h r−1 i h r−1 i 2 ! 2 !

where the r pre-multiplying Peven corresponds to the number of positions that the zero element can assume in the vector. There remains, for the generalized permutation, r − 1 slots to be ﬁlled equally by elements of E. This behavior is easily seen from Tab. 3.1 for the odd portion of the swinging factorial (OEIS A002457 OEIS [2019]).

The linear growth in the number of vertices as a function of r in the case of the simplex also allows an interpretation in this formalism. If δ¯ = −δ in the algorithm (3.5) [Mozelli and Adriano, 2019], then ∆j = 2δj and the vertex matrix composition rule can be rewritten as ( j (r − 1) δ¯, if i = j, v¯i = (A.6) δj, otherwise; The ﬁrst point to be remarked is the non-occurrence of zeros in the vertex matrix, implying an equal growth rate for the even and odd sequences. Here, obligatorily, there will be an element δ¯(r − 1) occupying one of the slots3, making necessary (r − 1) elements δ arranged along the other positions of the vector so that the condition r ˙ ∑i=1 θi = 0 is satisﬁed. In terms of generalized permutation, we will have, a total of r slots, with the element (r − 1) δ¯ repeating k1 = 1 time, and δ, repeating k2 = r − 1 times. Thus, a total of r! R(r; 1, r − 1) := = r (A.7) 1!(r − 1)! column vectors are produced for H¯ . Finally, to study the growth of the number of vertices in the elimination process in odd and even dimensions, a numerical example will be used for a sake of simplicity. Tab. A.1 shows the growth of the vertices, q, for r = 6 and r = 7 as ξ vertex elimination proceed. Next, it will be described some of the elimination possibilities for r = 6, since the generalization for the other cases is direct. • ξ = 0: If no vertex of the simplex is eliminated, the total number of vertices is given by the relation (A.7), i.e., 6! = 6 vertices. (A.8) 1!5! 2Which explains why the odd sequence grows faster than even one. 3or (r − 1)δ for the simplex rotated by π rad. 83

Table A.1: Vertices growth, q, relative to the number of vertices eliminated from simplex, ξ, for r = 6 and r = 7. Content inside the cells is arranged in the following format: total of vertices [simplex contribution + original polytope contribution].

ξ 1 2 3 4 5 6 7 15 20 22 22 21 20 q(r = 6) - [5+10] [4+16] [3+19] [2+20] [1+20] [0+20] 66 105 128 139 142 141 140 q(r = 7) [6+60] [5+100] [4+124] [3+136] [2+140] [1+140] [0+140]

• ξ = 1: For reasons of symmetry, we can choose to eliminate any of the vertices T of the simplex. Eliminating 5δ¯ δ δ δ δ δ , we must replace it with all vectors of the original polytope whose first component is δ¯, i.e., by vectors in the form T δ¯ [•][•][•][•][•] . From the cardinality condition, one of these slots must be filled in by an δ element, while the remaining 4 slots should be equally filled in with elements δ¯ and δ. So there will be 5 slots available to be filled, necessarily, with the element δ¯ repeating 3 times and the element δ, 2. So

5! = 10. (A.9) 3!2! In total, removing one vertex from the simplex will result in

6 − 1 +10 = 15 vertices. (A.10) | {z } r−ξ

• ξ = 2: Now, we are looking for vectors in the form

T δ¯ [•][•][•][•][•]

and T [•] δ¯ [•][•][•][•] , which are essentially the same as in the previous case. So there will be, in principle, 5! 2 × = 20 (A.11) 3!2! vectors. However, vectors of the type

T δ¯ δ¯ [•][•][•][•]

were counted by both previous conﬁgurations. Thus, it is necessary to subtract from the 20 vectors counted those that are repeated. For that, it is enough to 84 Appendix A. A combinatorial interpretation of the vertex growth

compute the number of vectors that have δ¯ simultaneously as the ﬁrst and second elements. Here vacant slots must contain two elements δ, and the remainder should be equally ﬁlled by δ¯ and δ. Therefore,

4! = 4. (A.12) 3!1! The total of vectors in the elimination of two vertices of the simplex is

6 − 2 +20 − 4 = 20. (A.13) | {z } r−ξ

• ξ = 3: In the same vein as the previous ones, eliminating the repeated vectors, the counting process will produce

5 4 3 6 − 3 + 3 − 3 + = 22 vertices. (A.14) 2 1 0

In the odd case, each position assumed by zero along the vector will generate a new set of permutations, so that all the arguments applied to obtain Eq. (A.5) remain valid. Over-counting problems during the vertex elimination process can be solved from a dual interpretation4 of counting. For example, as the uncertainties are symmetric, remove from the count all vectors whose first element is not δ¯ is the same as requiring, in counting, that the first element be δ. Although not very intuitive, this approach results in less expensive calculations. For example, consider the case of ξ = 3 in dimension 7: 6! 4! 7 − 3 + 7 − 4 = 4 + 140 − 16 = 128 vertices. (A.15) 3!3! 3!1! Here, the blue term corresponds to the simplex contribution, i.e., 4 vertices, since 3 were eliminated. The green term is Eq. (A.5): all vertices of the original polytope are included, even those that do not fit the valid configurations; the 7 that pre-multiplies this term refers to each of the positions that the zero element can assume in the vector. In red, vectors from the original polytope that do not meet the specifications of ξ = 3, namely, those with structures

T 0 δ δ [•][•][•][•] ,

T δ 0 δ [•][•][•][•] , T δ δ 0 [•][•][•][•] , T δ δ δ [•][•][•][•] . There are 4 possible vectors for each of these structures, totaling 16 to be eliminated.

4Used in Eq. (3.11). APPENDIX B

Effective number of eliminations

Proof. We want ξ such that r q(r, ξ) < r r b 2 c, [r mod 2 6= 0] , b 2 c or hj r k i nl r m l r m o ϑ(r) − U − ξ − − 1 [r mod 2 = 0] r + ϑ(r) < 0. 2 2 2 b 2 c Even and odd sequences will be treated separately. However, in both cases we will employ Stirling’s formula for the factorial terms: √ nn 1 n! = 2πn 1 + O . (B.1) e n • Even: ϑ ϑ! − + ϑ < 0 ⇒ − + ϑ < 0. (B.2) r r 2ϑ−r 2 2 ! 2 ! Applying (B.1), − √ ϑ ϑ q r r/2 2ϑ − r ϑ r/2 2πϑ > πϑ r (2ϑ − r) . e 2e 2e After some algebra, we get s − 2 (2ϑ − r)r 1 2ϑ ϑ > 1. πϑrr+1 2ϑ − r Once the binomial coefﬁcient is deﬁned only for 0 ≤ r/2 < ϑ, it is reasonable to take 2θ − r ≈ 2θ. This approach tends to become better when ξ r/2. We − + obtain 2rϑr 2 & πrr 1. Since ϑ = r − ξ, ( − ) r+1 π1/ r 2 r r−2 ξ . r − . 2r/(r−2) In terms of the asymptotic behavior for large values of r,

( − ) r+1 π1/ r 2 r r−2 r ≈ , 2r/(r−2) 2 therefore r ξ . (B.3) . 2 85 86 Appendix B. Effective number of eliminations

• Odd: In this case,

l r m ϑ l r m ϑ! − r + ϑ < 0 ⇒ − r r + ϑ < 0. (B.4) 2 2 2 2 ! ϑ − 2 ! Following the same steps of the previous development and ignoring the ﬂoor and ceiling operators1, we obtain

s − (2ϑ − r)r 1 2ϑ ϑ > 1. 2πϑrr−1 2ϑ − r For 2ϑ − r ≈ 2ϑ, in terms of ξ, the relation can be simpliﬁed to

( − ) r−1 π1/ r 2 r r−2 ξ r − . . 2 Once again, employing the asymptotic analysis of each term, for large r, we can take ( − ) r−1 π1/ r 2 r r−2 r ≈ . 2 2 So, r ξ . (B.5) . 2

1The outcome of these operators will not take effect when we proceed with the asymptotic analysis right below. APPENDIX C

Proof for non-trivial simplex generation in three dimensions via rotation matrix

Here it will be proved that the only simplex rotation performed around the normal T vector e = 1 1 1 which results in a distinct simplex from the original is that of ϕ = π rad. Proof. Let initially be the three-dimensional simplex given by

 2 −1 −1  H¯ =  −1 2 −1  (C.1) −1 −1 2 and the original polytope, also in three dimensions,

 1 1 0 0 −1 −1  H =  0 0 −1 1 1 −1  . (C.2) 1 −1 1 −1 0 0 Rodrigues’ formula [Rodrigues, 1840] states that a ϕ rotation around the versor T u ≡ ux uy uz is given by

2 Rϕ := I + W sin ϕ + W (1 − cos ϕ) , (C.3) where   0 −uz uy W :=  uz 0 −ux  −uy ux 0 √ is an array of cross products. Taking u = 1/ 3 e and multiplying (C.1) by (C.3), we have

2 Rϕ H¯ = χ Υχ Υ χ for     2 cos√ϕ 0 0 1 χ :=  −cosϕ + √3 sin ϕ  , and Υ :=  1 0 0  , −cosϕ − 3 sin ϕ 0 1 0

87 Appendix C. Proof for non-trivial simplex generation in three dimensions via 88 rotation matrix a cyclic permutation matrix. We want to know how the vertices of the original polytope (C.2) behave as the ϕ angle varies in the range [0, 2π). To do so, we will use a triangulation approach. From Stokes theorem [Riley et al., 2006], the simplex area is

1 I Z A4 ≡ s × ds = da , (C.4) 2 ∂Ω Ω where ∂Ω corresponds to the simplex contour described by the si lines connecting its vertices: T T s1(t) ≡ 2 −1 −1 + t −3 3 0 , s2(t) ≡ Υs1(t), s3(t) ≡ Υs2(t), with t ∈ [0, 1]. Then,

T ds1 = −3 3 0 dt, ds2 = Υds1, ds3 = Υds2.

Replacing in (C.4),

3 Z 1 √ 1 ˆ ˆ ˆ 9 A4 ≡ ∑ 3 i + j + k dt = 3, (C.5) 2 i=1 0 2

T Let’s take now, beyond the three vertices of simplex, a point P ≡ px py pz . As illustrated in Fig. C.1, this point will be within the simplex, i.e., it will belong to the convex hull described by the columns of H¯ if and only if the total area, resulting from the sum of the triangle areas 4v1v2P, 4v2v3P and 4v3v1P, coincides with the simplex area.

Figure C.1: Triangulation procedure.

For three dimensions, the areas of three triangular regions must be computed. Once the corners corresponding to each region are parameterized, the integrands in (C.4) are obtained from following cross products: 89

• Region I: 4v1v2P: s11 × ds11 = 3 iˆ+ jˆ+ kˆ ,

h i h i ˆ ˆ s12 × ds12 = i pzχ(1) − pyχ(2) + s13 × ds13 = i pyχ(3) − pzχ(2) + h i h i ˆ ˆ + j pxχ(2) − pzχ(3) + + j pzχ(1) − pxχ(3) + h i h i ˆ ˆ + k pyχ(3) − pxχ(1) , + k pxχ(2) − pyχ(1) .

• Region II: 4v2v3P: s21 × ds21 = 3 iˆ+ jˆ+ kˆ ,

h i h i ˆ ˆ s22 × ds22 = i pzχ(3) − pyχ(1) + s23 × ds23 = i pyχ(2) − pzχ(1) + h i h i ˆ ˆ + j pxχ(1) − pzχ(2) + + j pzχ(3) − pxχ(2) + h i h i ˆ ˆ + k pyχ(2) − pxχ(3) , + k pxχ(1) − pyχ(3) .

• Region III: 4v3v1P: s31 × ds31 = 3 iˆ+ jˆ+ kˆ ,

h i h i ˆ ˆ s32 × ds32 = i pzχ(2) − pyχ(3) + s33 × ds33 = i pyχ(1) − pzχ(3) + h i h i ˆ ˆ + j pxχ(2) − pzχ(1) + + j pzχ(2) − pxχ(1) + h i h i ˆ ˆ + k pyχ(1) − pxχ(2) , + k pxχ(3) − pyχ(2) . where sij denotes the parameterization of the j-th edge of the i-th triangle, and χ(k) is the k-th element of χ. Finally, the total surface area can be obtained from

3 I 3 3 Z 1 1 1 A(ϕ, P) ≡ ∑ si × dsi = ∑ ∑ sij × dsijdt 2 i=1 ∂Ω 2 i=1 j=1 0

Computing A for ϕ ∈ [0, 2π ) and taking P as the columns of (C.2), we see, according to Fig. C.2 that the values of ϕ for which A(ϕ, P) − A4 = 0, considering all the vertices simultaneously, are those integers multiples of the simplex dihedral angle, i.e., ϕ = kπ/3, k ∈ N. We know that a same simplex will be generated for k = 2m, m ∈ N, which will correspond to the original simplex. The only other nontrivial possibility is the simplex produced by the rotations in k = 2m + 1. Appendix C. Proof for non-trivial simplex generation in three dimensions via 90 rotation matrix

Vertex 1: [-1, 0, 1] Vertex 2: [1, 0, -1] 9 9

8.5 8.5

8 8

7.5 7.5 0 π/2 π 3 π/2 2 π 0 π/2 π 3 π/2 2 π

Vertex 3: [0, -1, 1] Vertex 4: [0, 1, -1] 9 9

8.5 8.5

8 8

7.5 7.5 0 π/2 π 3 π/2 2 π 0 π/2 π 3 π/2 2 π

Vertices 5 and 6: [-1, 1, 0] and [1, -1, 0], respectively 9

8.5

8 Area (D) [au]

7.5 0 π/2 π 3 π/2 2 π Angle (ϕ) [rad]

√ Figure C.2: Area as a function of the rotation angle. The green line, at A = (9/2) 3, is the simplex area, as calculated in Eq. (C.4). APPENDIX D

Dimensionality reduction using Givens rotations

Proof. Consider the mapping matrices for dimensions r = 5, 6 and 7, respectively: √ √ √ √ [4,5,arccos(1/ 5)] [3,4,arccos(1/ 4)] [2,3,arccos(1/ 3)] [1,2,arccos(1/ 2)] Ξ5 = Ω Ω Ω Ω   κ2 −ς2 0 0 0  ς κ κ κ −ς 0 0   2 3 2 3 3  =  ς ς κ κ ς κ κ κ −ς 0  ,  2 3 4 2 3 4 3 4 4   ς2ς3ς4κ5 κ2ς3ς4κ5 κ3ς4κ5 κ4κ5 −ς5  ς2ς3ς4ς5 κ2ς3ς4ς5 κ3ς4ς5 κ4ς5 κ5

√ √ √ √ [5,6,arccos(1/ 6)] [4,5,arccos(1/ 5)] [3,4,arccos(1/ 4)] [2,3,arccos(1/ 3)] Ξ6 = Ω Ω Ω Ω × √ × Ω[1,2,arccos(1/ 2)]   κ2 −ς2 0 0 0 0  ς κ κ κ −ς 0 0 0   2 3 2 3 3   ς ς κ κ ς κ κ κ −ς 0 0  =  2 3 4 2 3 4 3 4 4  ,  ς ς ς κ κ ς ς κ κ ς κ κ κ −ς 0   2 3 4 5 2 3 4 5 3 4 5 4 5 5   ς2ς3ς4ς5κ6 κ2ς3ς4ς5κ6 κ3ς4ς5κ6 κ4ς5κ6 κ5κ6 −ς6  ς2ς3ς4ς5ς6 κ2ς3ς4ς5ς6 κ3ς4ς5ς6 κ4ς5ς6 κ5ς6 κ6

√ √ √ √ [6,7,arccos(1/ 7)] [5,6,arccos(1/ 6)] [4,5,arccos(1/ 5)] [3,4,arccos(1/ 4)] Ξ7 = Ω Ω Ω Ω × √ √ × Ω[2,3,arccos(1/ 3)]Ω[1,2,arccos(1/ 2)]   κ2 −ς2 0 0 0 0 0  ς κ κ κ −ς 0 0 0 0   2 3 2 3 3   ς ς κ κ ς κ κ κ −ς 0 0 0   2 3 4 2 3 4 3 4 4  =  ς ς ς κ κ ς ς κ κ ς κ κ κ −ς 0 0  ,  2 3 4 5 2 3 4 5 3 4 5 4 5 5   ς ς ς ς κ κ ς ς ς κ κ ς ς κ κ ς κ κ κ −ς 0   2 3 4 5 6 2 3 4 5 6 3 4 5 6 4 5 6 5 6 6   ς2ς3ς4ς5ς6κ7 κ2ς3ς4ς5ς6κ7 κ3ς4ς5ς6κ7 κ4ς5ς6κ7 κ5ς6κ7 κ6κ7 κ7  ς2ς3ς4ς5ς6ς7 κ2ς3ς4ς5ς6ς7 κ3ς4ς5ς6ς7 κ4ς5ς6ς7 κ5ς6ς7 κ6ς7 κ7 with 1 1 κk := cos arccos √ and ςk := sin arccos √ . k k

91 92 Appendix D. Dimensionality reduction using Givens rotations

Ergo, it is possible to infer that √ √ √ √ [r−1,r,arccos(1/ r)] [r−2,r−1,arccos(1/ r−1)] [2,3,arccos(1/ 3)] [1,2,arccos(1/ 2)] Ξr = Ω Ω ··· Ω Ω

  κ2 −ς2 0 ··· 0 ··· 0 0  κ3ς2 κ2κ3 −ς3 ··· 0 ··· 0 0     .. . . .   κ4 (ς2ς3) κ2κ4ς3 κ3κ4 . . . .     . . . ..   . . . . −ςj 0 0  =    i i i .. . .   κi+1 ∏k=2 ςk κ2κi+1 ∏k=3 ςk κ3κi+1 ∏k=4 ςk ··· κjκi+1 . . .     ......   . . . . . −ςr−1 0   r−1 r−1 r−1 r−1   κr ∏k=2 ςk κ2κr ∏k=3 ςk κ3κr ∏k=4 ςk ··· κjκr ∏k=j+1 ςk ··· κr−1κr −ςr  r r r r ∏k=2 ςk κ2 ∏k=3 ςk κ3 ∏k=4 ςk ··· κj ∏k=j+1 ςk ··· κr−1ςr κr

1 By multiplying the last row of Ξr by each column in the matrix H, Eq. (3.2) ,

r r ! r ! j + j + j + ··· + j−1 + j v1 ∏ ςk v2 κ2 ∏ ςk v3 κ3 ∏ ςk vr (κr−1ςr) vrκr, k=2 k=3 k=4 ∀j ∈ N \ {0} , j ≤ p. (D.1)

r j r−1 As ∑k=1 vk = 0, the mapping of H to the subspace R occurs if the terms in the last row of Ξr are equal. The product that appears in these terms can be rewritten as s s r r r 1 2 j(j + 1)(j + 2) ··· (r − 1) j ς = 1 − √ = = . (D.2) ∏ k ∏ ( + )( + ) ··· k=j+1 k=j+1 k j 1 j 2 r r

For terms at the extremes, i.e., Ξr(1, r) and Ξr(r, r), r r 1 ∏ ςk = = κr, (D.3) k=2 r once s √ 1 2 κk = 1/ k and ςk = 1 − √ . k For terms in the interval j ∈ [2, r − 1], it is required that r r r r 1 j 1 j + 1 1 1 κj ∏ ςk = κj+1 ∏ ςk ⇒ p = p ⇒ √ = √ = κr. (D.4) k=j+1 k=j+2 j r j + 1 r r r Thus, Eq. (D.1) yields to:

j j j κr v1 + v2 + ··· + vr = 0. (D.5)

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Algorithm, 23, 24, 42, 66 hybrid, 71, 72 Karmarkar, 22 taxicab, 55 linear search, 56 optimization, 24, 32, 56 LMI, 22–24, 31, 36, 38, 66, 71, 72, 74, 76 semideﬁned programming, 23 Matrix, 22, 23, 35, 36, 38, 42, 45, 52, 53, Analysis 55, 58, 60, 62, 69, 75 stability, 23 cyclic permutation, 88 damping coefﬁcients, 69 Complexity, 21, 70, 71, 73–75 gain, 65, 66, 75, 76 analytical, 36 Givens rotation, 56, 59 computational, 24, 41, 56, 66, 70, 77 identity, 52 Control, 21, 23, 24, 36, 38, 65 Lyapunov, 23 robust, 21 real, 33 state-feedback, 66, 67 rotation, 54–56 synthesis, 65 stiffness, 70 Convex hull, 29, 33, 41, 45, 47, 70, 88 transformation, 63 cubic, 24, 54, 63 unitary, 51 original, 42, 51 vertex, 45, 47, 49–51, 53, 55, 57, 58, orthoplectic, 24, 55, 64 60, 63

Equation, 57 Parameter, 23, 36–38, 41, 50, 66, 69, 70, constitutive, 21 72, 75 Lyapunov, 23 exogenous, 36 Riccati-type, 22 scheduling, 36 time-varying, 36, 66 Formalism, 23, 36, 41, 43, 66, 67 time-varying uncertain, 24, 41–43, LPV, 21 70, 73 Function, 24, 38, 42, 55, 57, 58, 61, 66, uncertain, 24, 37, 41, 77 90 PDLF, 23, 39, 77 convex objective, 27 afﬁne, 41 Heaviside step, 46 quadratic, 23, 24, 41 Lyapunov, 23 Lyapunov candidate, 22, 38 Representation, 21, 22, 36 matrix, 35, 36 intervalar, 42 parameter-dependent Lyapunov, polytopic, 22, 24, 37 23, 77 state-space, 35, 70 rational, 37 signal, 61 Sensitivity, 21 transfer, 22 Stability, 22, 23, 38, 66, 67, 72, 74 analysis, 24, 38, 63, 65 Geometry, 43, 45, 54, 70, 72 asymptotic, 21, 22 cubic, 73 bi-quadratic, 23

99 100 Remissive Index

quadratic, 22 non-stationary, 36 robust, 22, 23 polytopic, 23 System, 21–23, 36, 37, 51, 66, 69–73, 76 Systems, 36 control, 21 coordinate, 52 Theory control, 21 discrete-time, 38 linear systems, 21 dynamical, 35 linear time-invariant control, 21 engineering, 23 LPV, 38 ﬁnite, 33 Lyapunov, 22 fuzzy, 23 matrix, 22 linear, 21, 22 set, 27 LPV, 21, 23, 24, 35–38, 41, 65–67, 77, 80 Uncertainty, 21, 22, 24, 49 LPV/LFT, 37 range, 23 LTI, 23, 36 symmetric, 47 MSD, 69, 71 time-invariant, 23 non-linear, 21 vector, 38