Pedro L. D. Peres

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.

Brief Robust filtering with guaranteed energy-to-peak performance - an LMI approach

The problem of robust energy-to-peak filtering for linear systems with convex bounded uncertainties is investigated in this paper. The main purpose is to design a full order stable linear filter that minimizes the worst-case peak value of the filtering error output signal with respect to all bounded energy inputs, in such a way that the filtering error system remains quadratically stable. Necessary and sufficient conditions are formulated in terms of linear Matrix Inequalities - LMIs, for both continuous- and discrete-time cases.

Robust H∞ filter design for uncertain linear systems with multiple time-varying state delays

The problem of robust H∞ filtering for continuous-time uncertain linear systems with multiple time-varying delays in the state variables is investigated in this paper. The uncertain parameters are supposed to belong to a given convex bounded polyhedral domain. The aim is to design a stable linear filter assuring asymptotic stability and a prescribed H∞ performance level for the filtering error system, irrespective of the uncertainties and the time delays. Sufficient conditions for the existence of such a filter are established in terms of linear matrix inequalities, which can be efficiently solved by means of powerful convex programming tools with global convergence assured. An example illustrates the proposed methodology.

An LMI condition for the robust stability of uncertain continuous-time linear systems

A new sufficient condition for the robust stability of continuous-time uncertain linear systems with convex bounded uncertainties is proposed in this note. The results are based on linear matrix inequalities (LMIs) formulated at the vertices of the uncertainty polytope, which provide a parameter dependent Lyapunov function that assures the stability of any matrix inside the uncertainty domain. With the aid of numerical procedures based on unidimensional search and the LMIs feasibility tests, a simple and constructive way to compute robust stability domains can be established.

A less conservative LMI condition for the robust stability of discrete-time uncertain systems

Abstract In this paper, a less conservative condition for the robust stability of uncertain discrete-time linear systems is proposed. The uncertain parameters, assumed to be time-invariant, are supposed to belong to convex bounded domains (polytope type uncertainty). The stability condition is formulated in terms of a set of linear matrix inequalities involving only the vertices of the polytope domain. A simple and numerically efficient feasibility test provides a set of Lyapunov matrices whose convex combination can be used to assess the stability of any dynamic matrix inside the uncertainty domain. Examples illustrate the results.

Decentralized control through parameter space optimization

The decentralized control problem for linear dynamic systems is revisited using a parameter space approach which enables the definition of the decentralized feedbacks from the existence of non-empty parameter convex sets. The convexity property enables the derivation of efficient numerical algorithms based on standard approaches in convex programming. The continuous-time and discrete-time cases are investigated and the decentralized control design is also treated to meet other important assignments such as: optimal H2 performance index, absolute stability, H∞ prescribed attenuation and robustness against actuator failures. Some numerical experiments illustrate the potential of this new control design.

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 addresses the problem of optimal H/sub 2/ control by output feedback. Necessary and sufficient conditions on the existence of a linear stabilizing output feedback gain are provided in terms of the intersection of a convex set and a set defined by a nonlinear real valued function. The results can be easily extended to deal with linear uncertain systems, where uncertainties are supposed to belong to convex bounded domains providing an H/sub 2/-guaranteed cost output feedback control. Thanks to the properties of the above-mentioned function, we show that under certain conditions, convex programming tools can be used for numerical purposes. Examples illustrate the theoretical results.

Control of Takagi–Sugeno Fuzzy Systems

Abstract This paper proposes an LMls characterization of guaranteed ℋ∞ and ℋ2 norms costs for linear systems with convex bounded parameter uncertainties. Both continuous-time and discrete-time systems are addressed.