A new condition and equivalence results for robust stability analysis of rationally time-varying uncertain linear systems

Uncertain systems is a fundamental area of automatic control. This paper addresses robust stability of uncertain linear systems with rational dependence on unknown time-varying parameters constrained in a polytope. For this problem, a new sufficient condition based on the search for a common homogeneous polynomial Lyapunov function is proposed through a particular representation of parameter-dependent polynomials and LMIs. Relationships with existing conditions based on the same class of Lyapunov functions are hence investigated, showing that the proposed condition is either equivalent to or less conservative than existing ones. As a matter of fact, the proposed condition turns out to be also necessary for a class of systems. Some numerical examples illustrate the use of the proposed condition and its benefits. © 2011 IEEE.published_or_final_versionThe 2011 IEEE International Symposium on Computer-Aided Control System Design (CACSD), Denver, CO., 28-30 September 2011. In IEEE CACSD International Symposium Proceedings, 2011, p. 234-23

Robust stability of time-varying uncertain systems with rational dependence on the uncertainty

Robust stability of time-varying uncertain systems is a key problem in automatic control. This note considers the case of linear systems with rational dependence on an uncertain time-varying vector constrained in a polytope, which is typically addressed in the literature by using the linear fractional representation (LFR). A novel sufficient condition for robust stability is derived in terms of a linear matrix inequality (LMI) feasibility test by exploiting homogeneous polynomial Lyapunov functions, the square matrix representation and an extended version of Polya's theorem which considers structured matrix polynomials. It is shown that this condition is also necessary for second-order systems, and that this condition is less conservative than existing LMI conditions based on the LFR for any order. © 2010 IEEE.published_or_final_versio

Sufficient and Necessary LMI Conditions for Robust Stability of Rationally Time-Varying Uncertain Systems

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LMI techniques for optimization over polynomials in control: A survey

Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Plya's theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, time-delay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers. © 2006 IEEE.published_or_final_versio

Computing upper-bounds of the minimum dwell time of linear switched systems via homogenous polynomial lyapunov functions

Regular Session - Switched Systems IIThis paper investigates the minimum dwell time for switched linear systems. It is shown that a sequence of upper bounds of the minimum dwell time can be computed by exploiting homogeneous polynomial Lyapunov functions and convex optimization based on LMIs. This sequence is obtained by adopting two possible representations of homogeneous polynomials, one based on Kronecker products, and the other on the square matrix representation. Some examples illustrate the use and the potentialities of the proposed approach.published_or_final_versionThe 2010 American Control Conference (ACC), Baltimore, MD., 30 June-2 July 2010. In Proceedings of the American Control Conference, 2010, p. 2487-249

Toward non-conservative stability conditions for equilibrium points of genetic networks with SUM regulatory functions

An important problem in systems biology consists of establishing whether an equilibrium point of a genetic regulatory network is stable. This paper investigates this problem for genetic networks with SUMregulatory functions. It is shown that a sufficient condition for global asymptotical stability of an equilibrium point of these networks can be derived in terms of convex optimizations with LMI constraints by exploiting polynomial Lyapunov functions and SOS techniques. This condition is interesting because does not introduce approximations of the nonlinearities present in the genetic regulatory network, and the conservatism can be decreased by increasing the degree of the involved polynomials. ©2009 IEEE.published_or_final_versionThe Joint 48th IEEE Conference on Decision and Control and the 28th Chinese Control Conference (CDC/CCC 2009), Shanghai, China, 16-18 December 2009. In Proceedings of the IEEE Conference on Decision and Control, 2009, p. 5631-563

Event-triggered control for piecewise affine discrete-time systems

In the present work, we study the problems of stability analysis of piecewise-affine (PWA) discrete-time systems, and trigger-function design for discrete-time event-triggered control systems. We propose a representation for piecewise-affine systems in terms of ramp functions, and we rely on Lyapunov theory for the stability analysis. The proposed implicit piecewise-affine representation prevents the shortcomings of the existing stability analysis approaches of PWA systems. Namely, the need to enumerate regions and allowed transitions of the explicit representations. In this context, we can emphasize two benefits of the proposed approach: first, it makes possible the analysis of uncertainty in the partition and, thus, the transitions. Secondly, it enables the analysis of event-triggered control systems for the class of PWA systems since, for ETC, the transitions cannot be determined as a function of the state variables. The proposed representation, on the other hand, implicitly encodes the partition and the transitions. The stability analysis is performed with Lyapunov theory techniques. We then present conditions for exponential stability. Thanks to the implicit representation, the use of piecewise quadratic Lyapunov functions candidates becomes simple. These conditions can be solved numerically using a linear matrix inequality formulation. The numerical analysis exploits quadratic expressions that describe ramp functions to verify the positiveness of extended quadratic forms. For ETC, a piecewise quadratic trigger function defines the event generator. We find suitable parameters for the trigger function with an optimization procedure. As a result, this function uses the information on the partition to reduce the number of events, achieving better results than the standard quadratic trigger functions found in the literature. We provide numerical examples to illustrate the application of the proposed representation and methods.Ce manuscrit présente des résultats sur l’analyse de stabilité des systèmes affines par morceaux en temps discret et sur le projet de fonctions de déclenchement pour des stratégies de commande par événements. Nous proposons une représentation pour des systèmes affines par morceaux et l’on utilise la théorie de stabilité de Lyapunov pour effectuer l’analyse de stabilité globale de l’origine. La nouvelle représentation implicite que nous proposons rend plus simple l’analyse de stabilité car elle évite l’énumération des régions et des transitions entre régions tel que c’est fait dans le cas des représentations explicites. Dans ce contexte nous pouvons souligner deux avantages principaux, à savoir I) la possibilité de traiter des incertitudes dans la partition qui définit le système et, par conséquent des incertitudes dans les transitions, II) l’analyse des stratégies de commande par événements pour des systèmes affines par morceaux. En effet, dans ces stratégies les transitions ne peuvent pas être définies comme des fonctions des variables d’état. La théorie de stabilité de Lyapunov est utilisée pour établir des conditions pour la stabilité exponentielle de l’origine. Grâce à la représentation implicite des partitions nous utilisons des fonctions de Lyapunov quadratique par morceaux. Ces conditions sont données par des inégalités dont la solution numérique est possible avec une formulation par des inégalités matricielles linéaires. Ces formulations numériques se basent sur des expressions quadratiques décrivant des fonctions rampe. Pour des stratégies par événement, une fonctions quadratique par morceaux est utilisée pour le générateur d’événements. Nous calculons les paramètres de ces fonctions de déclenchement a partir de solutions de problèmes d’optimisation. Cette fonction de déclenchement quadratique par morceaux permet de réduire le nombre de d’événementsen comparaison avec les fonctions quadratiques utilisées dans la littérature. Nous utilisons des exemples numériques pour illustrer les méthodes proposées.No presente trabalho, são estudados os problemas de análise de estabilidade de sistemas afins por partes e o projeto da função de disparo para sistemas de controle baseado em eventos em tempo discreto. É proposta uma representação para sistemas afins por partes em termos de funções rampa, e é utilizada a teoria de Lyapunov para a análise de estabilidade. A representação afim por partes implícita proposta evita algumas das deficiências das abordagens de análise de estabilidade de sistemas afins por partes existentes. Em particular, a necessidade de anumerar regiões e transições admissíveis das representações explícitas. Neste contexto, dois benefícios da abordagem proposta podem ser enfatizados: primeiro, ela torna possível a análise de incertezas na partição, e, assim, nas transições. Segundo, ela permite a análise de sistemas de controle baseado em eventos para a classe de sistemas afins por partes, já que, para o controle baseado em eventos, as transições não podem ser determinadas como uma função das variáveis de estado. A representação proposta, por outro lado, codifica implicitamente a partição e as transições. A análise de estabilidade é realizada com técnicas da teoria de Lyapunov. Condi- ções para a estabilidade exponencial são então apresentadas. Graças à representação implícita, o uso de funções candidatas de Lyapunov se torna simples. Essas condições podem ser resolvidas numéricamente usando uma formulação de desigualdades matriciais lineares. A análise numérica explora expressões quadráticas que descrevem funções de rampa para verificar a postivividade de formas quadráticas extendidas. Para o controle baseado em eventos, uma função de disparo quadrática por partes define o gerador de eventos. Parâmetros adequados para a função de disparo sãoencontrados com um procedimento de otimização. Como resultado, esta função usa informação da partição para reduzir o número de eventos, obtendo resultados melhores do que as funções de disparo quadráticas encontradas na literatura. Exemplos numéricos são fornecidos para ilustrar a aplicação da representação e mé- todos propostos