Upper bound estimation of the spectral abscissa for switched linear systems via coordinate transformations

summary:In this paper, we develop computational procedures to approximate the spectral abscissa of the switched linear system via square coordinate transformations. First, we design iterative algorithms to obtain a sequence of the least $\mu_1$ measure. Second, it is shown that this sequence is convergent and its limit can be used to estimate the spectral abscissa. Moreover, the stopping condition of Algorithm 1 is also presented. Finally, an example is carried out to illustrate the effectiveness of the proposed method

Bounds and Invariant Sets for a Class of Switching Systems with Delayed-state-dependent Perturbations

We present a novel method to compute componentwise transient bounds, ultimate bounds, and invariant regions for a class of switching continuous-time linear systems with perturbation bounds that may depend nonlinearly on a delayed state. The main advantage of the method is its componentwise nature, i.e. the fact that it allows each component of the perturbation vector to have an independent bound and that the bounds and sets obtained are also given componentwise. This componentwise method does not employ a norm for bounding either the perturbation or state vectors, avoids the need for scaling the different state vector components in order to obtain useful results, and may also reduce conservativeness in some cases. We give conditions for the derived bounds to be of local or semi-global nature. In addition, we deal with the case of perturbation bounds whose dependence on a delayed state is of affine form as a particular case of nonlinear dependence for which the bounds derived are shown to be globally valid. A sufficient condition for practical stability is also provided. The present paper builds upon and extends to switching systems with delayed-state-dependent perturbations previous results by the authors. In this sense, the contribution is three-fold: the derivation of the aforementioned extension; the elucidation of the precise relationship between the class of switching linear systems to which the proposed method can be applied and those that admit a common quadratic Lyapunov function (a question that was left open in our previous work); and the derivation of a technique to compute a common quadratic Lyapunov function for switching linear systems with perturbations bounded componentwise by affine functions of the absolute value of the state vector components.Comment: Submitted to Automatic

Periodic Adaptive Stabilization of Rapidly Time-Varying Linear Systems

This is a post-peer-review, pre-copyedit version of an article published in Mathematics of Control, Signals, and Systems. The final authenticated version is available online at: http://dx.doi.org/https://doi.org/10.1007/s00498-019-0236-6Adaptive control deals with systems that have unknown and/or time-varying parameters. Most techniques are proven for the case in which any time variation is slow, with results for systems with fast time variations limited to those for which the time variation is of a known form or for which the plant has stable zero dynamics. In this paper, a new adaptive controller design methodology is proposed in which the time variation can be rapid and the plant may have unstable zero dynamics. Under the structural assumptions that the plant is relative degree one and that the plant uncertainty is a single scalar variable, as well as some mild regularity assumptions, it is proven that the closed-loop system is exponentially stable under fast parameter variations with persistent jumps. The proposed controller is nonlinear and periodic, and in each period the parameter is estimated and an appropriate stabilizing control signal is applied.C. Nielsen and D. E. Miller: Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada

Nonlinear Periodic Adaptive Control for Linear Time-Varying Plants

In adaptive control the goal is to deal with systems that have unknown and/or time-varying parameters. Adaptive control techniques have been developed since 1950’s and most results were proven in the cases when the time-variations were non-existent or slow. However the results pertaining to systems with fast time-variations are still limited, in particular, when it comes to plants with unstable zero dynamics. In this work we adopt the controller design technique from the area of gain scheduling, where the time-varying parameter is assumed to be measurable. We propose the design of a nonlinear periodic controller, where in each period the state and parameter values are estimated and an appropriate stabilizing control signal is applied. It is shown that the closed loop system is stable under fast parameter variations with persistent jumps: the trajectory of the closed loop state in response to the initial condition is bounded by a decaying exponential plus a gain times the size of the noise. Our approach imposes several constraints on the plant; however, we show that there exists at least one interesting class of systems, which includes plants with unstable zero dynamics, that can be stabilized by our controller.1 yea

Estabilidade e controle de sistemas lineares variantes no tempo e de sistemas chaveados lineares

Orientador: Pedro Luis Dias PeresTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de ComputaçãoResumo: Esta tese apresenta contribui»coes para a solucao de problemas de anaalise de estabilidade e de sintese de controladores para sistemas lineares com parametros variantes no tempo pertencentes a um politopo e para sistemas chaveados lineares com funcoes de chaveamento arbitrarias atraves de condicoes na forma de desigualdades matriciais lineares baseadas em funcoes de Lyapunov. Para sistemas lineares variantes no tempo (caso continuo), sao fornecidas condicoes de verificacao de estabilidade e de computo de custos garantidos H1 quando os parametros pertencentes a um politopo sao supostos incertos e com taxas de variacao limitadas. Para o problema de sintese, supondo que os parametros sao conhecidos em tempo real, sao fornecidas condicoes de projeto de ganhos de realimentacao de estados que variam de forma nao-linear com os parametros e que asseguram a estabilidade com um certo custo garantido H1 para o sistema em malha fechada sujeito a taxas de variacoes parametricas limitadas. No caso de taxas de variacoes parametricas arbitrarias, sao fornecidas condicoes de c^omputo de ganhos que variam de forma linear (caso continuo) ou de forma nao-linear (caso discreto) com os par^ametros, assegurando a estabilidade com requisitos de desempenho H1 para o sistema em malha fechada. Para sistemas chaveados lineares (casos continuo e discreto), sao fornecidas condicoes para computar ganhos chaveados de realimentacao de estados que resolvem os problemas de estabiliza cao e de controle H1, incluindo especi¯cacoes de alocacao de piolos, permitindo melhorar o desempenho do sistema em malha fechada sujeito a funcoes de chaveamento arbitrarias disponiveis em tempo real. Exemplos numericos incluindo problemas de controle com restri»cao de estrutura, de controle sob falhas de atuadores e uma aplicacao em circuitos eletricos chaveados ilustram como as condi»coes propostas reduzem o conservadorismo nos problemas de analise e de sintese das classes de sistemas dinamicos sob investigacaoAbstract: This thesis presents contributions to the solution of problems of stability analysis and control synthesis applied to linear systems with time-varying parameters belonging to a polytope and to switched linear systems subject to arbitrary switching functions using linear matrix inequality conditions based on Lyapunov functions. Concerning linear time-varying systems (continuous-time case), the proposed conditions assess the problems of stability analysis and computation of H1 guaranteed costs when the parameters belonging to a polytope are supposed to be uncertain with bounded rates of variation. For the problem of synthesis, assuming that the parameters are available in real time, the thesis provides conditions to design state feedback gains which depend nonlinearly on the parameters and assure stability with a given H1 guaranteed cost to the closed-loop system for the case of bounded rates of parametric variations. When the rates of parametric variations are assumed to be arbitrary, the given conditions can determine gains that depend linearly (continuous-time case) or nonlinearly (discrete-time case) on the parameters, assuring stability with H1 performance to the closed-loop system. In the context of switched linear systems (continuous and discrete-time cases), the proposed conditions are suitable to determine switched state feedback gains that solve the problems of stabilization and H1 control including pole location speci¯cations, allowing to improve the performance of the closed-loop system subject to arbitrary switching functions available in real time. Numerical examples including problems of structurally constrained control, robustness against actuator failures and an application on switched electrical circuits illustrate how the proposed conditions reduce the conservatism of the problems of analysis and synthesis for the classes of dynamic systems under investigationDoutoradoAutomaçãoDoutor em Engenharia Elétric