Ricardo C. L. F. Oliveira

Parameter-Dependent LMIs in Robust Analysis: Characterization of Homogeneous Polynomially Parameter-Dependent Solutions Via LMI Relaxations

This note investigates the robust stability of uncertain linear time-invariant systems in polytopic domains by means of parameter-dependent linear matrix inequality (PD-LMI) conditions, exploiting some algebraic properties provided by the uncertainty representation. A systematic procedure to construct a family of finite-dimensional LMI relaxations is provided. The robust stability is assessed by means of the existence of a Lyapunov function, more specifically, a homogeneous polynomially parameter-dependent Lyapunov (HPPDL) function of arbitrary degree. For a given degree , if an HPPDL solution exists, a sequence of relaxations based on real algebraic properties provides sufficient LMI conditions of increasing precision and constant number of decision variables for the existence of an HPPDL function which tend to the necessity. Alternatively, if an HPPDL solution of degree exists, a sequence of relaxations which increases the number of variables and the number of LMIs will provide an HPPDL solution of larger degree. The method proposed can be applied to determine homogeneous parameter-dependent matrix solutions to a wide variety of PD-LMIs by transforming the infinite-dimensional LMI problem described in terms of uncertain parameters belonging to the unit simplex in a sequence of finite-dimensional LMI conditions which converges to the necessary conditions for the existence of a homogeneous polynomially parameter-dependent solution of arbitrary degree. Illustrative examples show the efficacy of the proposed conditions when compared with other methods from the literature.

LMI conditions for robust stability analysis based on polynomially parameter-dependent Lyapunov functions

Abstract The robust stability of uncertain linear systems in polytopic domains is investigated in this paper. The main contribution is to provide a systematic procedure for generating sufficient robust stability linear matrix inequality conditions based on homogeneous polynomially parameter-dependent Lyapunov matrix functions of arbitrary degree on the uncertain parameters. The conditions exploit the positivity of the uncertain parameters, being constructed in such a way that: as the degree of the polynomial increases, the number of linear matrix inequalities and free variables increases and the test becomes less conservative; if a feasible solution exists for a certain degree, the conditions will also be verified for larger degrees. For any given degree, the feasibility of a set of linear matrix inequalities defined at the vertices of the polytope assures the robust stability. Both continuous and discrete-time uncertain systems are addressed, as illustrated by numerical examples.

Convergent LMI Relaxations for Quadratic Stabilizability and

This paper investigates the quadratic stabilizability of Takagi-Sugeno (T-S) fuzzy systems by means of parallel distributed state feedback compensators. Using Finslers lemma, a new design condition assuring the existence of such a controller is formulated as a parameter-dependent linear matrix inequality (LMI) with extra matrix variables and parameters in the unit simplex. Algebraic properties of the system parameters and recent results of positive polynomials are used to construct LMI relaxations that, differently from most relaxations in the literature, provide certificates of convergence to solve the control design problem. Due to the degrees of freedom obtained with the extra variables, the conditions presented in this paper are an improvement over earlier results based only on Polyas theorem and can be viewed as an alternative to the use of techniques based on the relaxation of quadratic forms. An extension to cope with guaranteed H infin attenuation levels is also given, with proof of asymptotic convergence to the global optimal controller under quadratic stability. The efficiency of the proposed approach in terms of precision and computational effort is demonstrated by means of numerical comparisons with other methods from the literature.

{{\mathscr H}}_{\infty}

This paper presents new convex optimization procedures for full order robust H 2 and H ∞ filter design for continuous and discrete-time uncertain linear systems. The time-invariant uncertain parameters are supposed to belong to a polytope with known vertices. Thanks to the use of a larger number of slack variables, linear matrix inequalities for the design of robust filters can be derived from the proposed conditions, outperforming the existing methods. The superiority and efficiency of the proposed method for filter design are illustrated by means of numerical comparisons in benchmark examples from the literature.

Control of Takagi–Sugeno Fuzzy Systems

Abstract This paper investigates the problems of checking robust stability and evaluating robust ℋ 2 performance of uncertain continuous-time linear systems with time-invariant parameters lying in polytopic domains. The novelty is the ability to check robust stability by constructing a particular parameter-dependent Lyapunov function which is a polynomial function of the uncertain system matrices, as opposed to a general polynomial function of the uncertain parameter. The degree of the polynomial is tied to a certain integer κ . The existence of such Lyapunov function can be proved by solving parameter-dependent Linear Matrix Inequalities (LMIs), which are guaranteed to be solvable for a sufficiently large yet finite value of κ whenever the system is robustly stable. Extensions to guaranteed ℋ 2 cost computation are also provided. Numerical aspects concerning the programming and the evaluations of the proposed tests are discussed and illustrated by examples.

Robust H 2 and H ∞ filter design for uncertain linear systems via LMIs and polynomial matrices

This paper presents new results concerning the stability analysis and design of state-feedback controllers for continuous-time Takagi-Sugeno (T-S) fuzzy systems via fuzzy Lyapunov functions. The membership functions of the T-S fuzzy systems are modeled in a space that is defined by the Cartesian product of simplexes called a multisimplex. If the time derivatives of the membership functions are bounded, the bounds are used to construct a polytope that models the space of the time derivatives of the membership functions. Linear matrix inequality (LMI) relaxations that are based on polynomial matrices are provided for stability analysis and controller design. Extensions for the design of control laws that minimize upper bounds to H2 and H∞ norms are also given. The main novelty of this method is that it allows one to synthesize control gains, which depends only on some premise variables that are selected by the designer. Numerical experiments illustrate the flexibility and advantages of the proposed method.

Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions

This paper presents some general results concerning the existence of homogeneous polynomial solutions to parameter-dependent linear matrix inequalities whose coefficients are continuous functions of parameters lying in the unit simplex. These results are useful in the context of robust analysis and synthesis of parameter-dependent feedback gains (gain-scheduling) for uncertain linear systems in polytopic domains. A result showing the generality of the class of static gains with homogeneous polynomial dependence and a result dealing with the solutions of parameter-dependent linear matrix inequalities with slowly time-varying parameters are also given

Selective

This paper provides a brief survey on the subject of LMI (Linear Matrix Inequality) methods for robust state feedback control design. The focus is on continuous-time linear systems with time-invariant uncertain parameters belonging to a polytope. Several LMI conditions from the literature are reviewed and discussed. The relationship between quadratic stabilizability (i.e. constant Lyapunov matrix) and LMI conditions based on parameter-dependent Lyapunov functions is highlighted. As a contribution, a generalization of a family of parameter-dependent conditions is proposed. Discussions, possible extensions and interpretations are provided along the presentation. Finally, the numerical efficacy of the LMI conditions in finding robust controllers when one stabilizing gain is known to exist is investigated. The methods have been tested against a set of hard uncertain systems that are guaranteed to be stabilized by some robust state feedback controller, including a large subset of problems which are known to be stabilized by some robust controller but not to be quadratically stabilizable by any controller.

\hbox{\scr H}_2

This technical note is concerned with the problem of reduced order robust H∞ dynamic output feedback control design for uncertain continuous-time linear systems. The uncertain time-invariant parameters belong to a polytopic domain and affect all the system matrices. The search for a reduced-order controller is converted in a problem of static output feedback control design for an augmented system. To solve the problem, a two-stage linear matrix inequality (LMI) procedure is proposed. At the first step, a stabilizing state feedback scheduled controller with polynomial or rational dependence on the parameters is determined. This parameter-dependent state feedback controller is used at the second stage, which synthesizes the robust (parameter-independent) output feedback H∞ dynamic controller. A homogeneous polynomially parameter-dependent Lyapunov function of arbitrary degree is used to assess closed-loop stability with a prescribed H∞ attenuation level. As illustrated by numerical examples, the proposed method provides better results than other LMI based conditions from the literature.

and

This paper presents new results concerning the existence of solutions for robust (parameter-dependent) LMIs with parameters lying in a Cartesian product of simplexes, called multi-simplex. These results allow to derive convergent procedures based on LMI relaxations to check the positivity of polynomial matrices with parameters in multi-simplexes. As an application, the robust stability analysis of uncertain linear systems is investigated. As an immediate advantage of this flexible representation, polynomially parameter-dependent Lyapunov functions can be constructed to handle simultaneously time-invariant, arbitrarily time-varying and bounded time-varying parameters in an appropriate way. Numerical experiments illustrate the advantages of the method.