Denis Arzelier

A new robust D-stability condition for real convex polytopic uncertainty

Abstract The problem of robust D -stability analysis with respect to real convex polytopic uncertainties is tackled. A new LMI -based sufficient condition for the existence of parameter-dependent Lyapunov functions is proposed. This condition generalises previously published conditions. Numerical comparisons with quadratic stability results as well as previous results based on parameter-dependent Lyapunov functions illustrate the relevance of this new condition. Finally, this result appears to be promising for robust multi-objective performance analysis and control synthesis purposes.

Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation

Topological separation is investigated in the case of an uncertain time-invariant matrix interconnected with an implicit linear transformation. A quadratic separator independent of the uncertainty is shown to prove losslessly the closed-loop well-posedness. Several applications for LTI descriptor system analysis are then given. First, some known results for stability and pole location of descriptor systems are demonstrated in a new way. Second, contributions to robust stability analysis and time-delay systems stability analysis are exposed. These prove to be new even when compared to results for usual LTI systems (not in descriptor form). All results are formulated as linear matrix inequalities (LMIs).

Robust stabilization of discrete-time linear systems with norm-bounded time-varying uncertainty

Abstract This paper presents an algorithm for the stabilization of a class of discrete-time uncertain systems suffering from uncertainty of the norm bounded time varying type. The stabilizing control gain matrix is obtained by solving a certain discrete Riccati equation. The solution of this Riccati equation is also a Lyapunov matrix which is used to establish the stability of closed-loop system. The results presented in this paper generalize those obtained in previous works for continuous-time uncertain systems.

Robust performance analysis with LMI-based methods for real parametric uncertainty via parameter-dependent Lyapunov functions

Robust performance analysis for linear time-invariant systems with linear fractional transformation real parametric uncertainty is considered. New conditions of robust stability/performance based on parameter-dependent Lyapunov functions are proposed. The robust stability/performance measures are: robust pole location, robust H/sub /spl infin// performance and robust H/sub 2/ performance. Linear matrix inequality (LMI)-based sufficient conditions for the existence of parameter-dependent Lyapunov functions are derived by using the quadratic separation concept. The performances of the proposed conditions are compared with existing tests.

Robust H2 performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs

A particular class of uncertain linear discrete-time periodic systems is considered. The problem of robust stabilization of real polytopic linear discrete-time periodic systems via a periodic state-feedback control law is tackled here, along with H2 performance optimization. Using additional slack variables and the periodic Lyapunov lemma, an extended sufficient condition of robust H2 stabilization is proposed. Based on periodic parameter-dependent Lyapunov functions, this last condition is shown to be always less conservative than the more classic one based on the quadratic stability framework. This is illustrated on a numerical example from the literature.

Brief Positive polynomial matrices and improved LMI robustness conditions

Recently, several new LMI conditions for stability of linear systems have been proposed, introducing additional slack variables to reduce the gap between conservative convex quadratic stability conditions and intractable non-convex robust stability conditions. In this paper, we show that these improved LMI conditions can be derived with the help of some basic results on positive polynomial matrices. The approach allows us to derive in a unifying way results in the state-space and polynomial frameworks. Applications to robust stability analysis and robust stabilization of systems with multi-linear parametric uncertainty are fully described.

Pole assignment of linear uncertain systems in a sector via a Lyapunov-type approach

The problem of designing robust control laws, in performance and in stability, for uncertain linear systems is considered. Performances are taken into account using root clustering of the closed-loop dynamic matrix in a sector of the complex plane. A synthesis procedure, based on a sufficient condition for quadratic stabilization and root clustering, such as stabilizability, is given, using an auxiliary convex problem. The results are illustrated by a significant example from the literature. >

On parameter-dependent Lyapunov functions for robust stability of linear systems

For a linear system affected by real parametric uncertainty, this paper focuses on robust stability analysis via quadratic-in-the-state Lyapunov functions polynomially dependent on the parameters. The contribution is twofold. First, if n denotes the system order and m the number of parameters, it is shown that it is enough to seek a parameter-dependent Lyapunov function of given degree 2nm in the parameters. Second, it is shown that robust stability can be assessed by globally minimizing a multivariate scalar polynomial related with this Lyapunov matrix. A hierarchy of LMI relaxations is proposed to solve this problem numerically, yielding simultaneously upper and lower bounds on the global minimum with guarantee of asymptotic convergence.

Brief An LMI condition for robust stability of polynomial matrix polytopes

A sufficient LMI condition is proposed for checking robust stability of a polytope of polynomial matrices. It hinges upon two recent results: a new approach to polynomial matrix stability analysis and a new robust stability condition for convex polytopic uncertainty. Numerical experiments illustrate that the condition narrows significantly the unavoidable gap between conservative tractable quadratic stability results and exact NP-hard robust stability results.

Robust Static Output Feedback Stabilization for Polytopic Uncertain Systems: Improving the Guaranteed Performance Bound

Abstract A new sufficient condition of robust stabilizability via static output feedback is proposed for polytopic uncertain systems. It is based on a new parameterization of all static output feedback stabilizing gains and uses parameter-dependent Lyapunov functions to systematically reduce conservatism of the usual quadratic stability approach. These results are then extended to deal with the worst-case H 2 guaranteed synthesis problem. A coordinate descent-type algorithm is defined to solve this nonlinear non convex optimization problem. Two numerical examples are provided to illustrate the relevance of the new condition.