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

doi:10.1016/j.sysconle.2008.01.006

Systems & Control Letters

Volume 57, Issue 8, August 2008, Pages 680-689

https://doi.org/10.1016/j.sysconle.2008.01.006 Get rights and content

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.

Introduction

In the last decade, research in systems and control have witnessed an effort to compute better and better approximate solutions to robust optimization problems [1]. Take for instance the problem of robust stability analysis of uncertain linear systems [2]. Using Lyapunov stability theory, this problem can be cast as the feasibility of parameter-dependent Linear Matrix Inequalities (LMIs), which are optimization problems of infinite dimension. Much effort went into trying to produce finite dimension tests, through the use of relaxations or other techniques. The following works, among many others, effectively reduced the gap between quadratic stability, which is the simplest parameter-independent form of robust stability test, and robust stability [3], [4], [5], [6], [7], [8], [9], [10]. Related problems, such as robust and gain-scheduled synthesis, filtering, including performance indexes as the $ℋ_{2}$ and the $ℋ_{\infty}$ norms, can also be expressed in terms of parametrized LMIs.

Recent approaches to parameter-dependent LMIs work by transforming the original infinite-dimensional problem into a sequence of finite dimension LMI relaxations, in some cases with guarantees of convergence. Most relaxations are based on finding polynomial parameter-dependent Lyapunov matrix solutions of large enough degree. Indeed, as shown in [11], polynomial solutions completely characterize parameter-dependent LMIs with parameters in compact sets. Within the context of robust stability and $ℋ_{2}$ and $ℋ_{\infty}$ performance analysis, the following are some of the works that follow such polynomial approach [6], [8], [12], [13], [14], [15], [16], [10]. A related strategy is the use of sum-of-squares decompositions [9], [17].

With several convergent LMI relaxations for robust analysis available in the literature, it seems that the challenge now is to improve the quality of the relaxations. For instance, in terms of efficiency, i.e. the ability to obtain the same results with less computational burden; or the easiness to extend a given stability condition to cope with similar problems, such as control and filtering design; and even practical aspects such as the difficulties associated with programming the proposed conditions. Typically, polynomial approaches require a significant amount of preliminary manipulations before the conditions can be numerically solved.

Recently, interesting LMI relaxations have been introduced in [7], [18] that cope with the robust stability analysis of continuous-time time-invariant linear systems in polytopic domains. In these works, robust stability conditions are reformulated as parameter-dependent LMIs by imposing a special parameter-dependent polynomial Lyapunov function. This Lyapunov function depends polynomially on the uncertain system matrices and on a generic polynomial parameter-dependent matrix (Lyapunov matrix) to be determined. Constraining the particular structure of the parameter-dependent Lyapunov matrix, i.e. constant, affine, etc. the associated Lyapunov function can be viewed as a special polynomial whose degree is given as a function of a certain integer $κ$ . For each $κ$ , finite dimension LMI relaxations can be obtained by means of Finsler’s Lemma [19]. Numerical experiments show that conservativeness of such conditions is reduced as $κ$ increases. Extensions to deal with $ℋ_{2}$ performance analysis have also been presented in [20], [21]. In [20], the authors raise the possibility of changing the dynamic matrix in the Lyapunov function by an arbitrary matrix, possibly reducing the conservativeness of the evaluations.

This paper investigates the structure of the Lyapunov function originally proposed in [7], [18] in more detail. One of the main contributions is to show that the Lyapunov matrix can be constrained to be constant, i.e. parameter independent, without loss of generality if $κ$ is allowed to be large enough. This can explain the good practical behavior of the relaxations proposed in [7], [18], [20], [21], where only sufficient conditions for robust stability have been provided. It is also shown how simple yet numerically effective LMI conditions can be obtained in a lifted parameter-space. The use of the proposed Lyapunov function to compute guaranteed $ℋ_{2}$ costs is also discussed and some conditions that ensure convergence to necessity in this case are also given. Numerical aspects are discussed and illustrated by examples.

Section snippets

Problem statement

Consider the linear time-invariant continuous-time uncertain system $\dot{x} = A (α) x + B (α) w, y = C (α) x,$ with $x \in R^{n}$ , $w \in R^{m}$ , $y \in R^{p}$ , $A (α) \in R^{n \times n}$ , $B (α) \in R^{n \times m}$ , and $C (α) \in R^{p \times n}$ . The triplet $(A, B, C) (α)$ is assumed to belong to a convex bounded (polytope-type) uncertain domain $P ≔ {(A, B, C) (α) : (A, B, C) (α) = \sum_{i = 1}^{N} α_{i} (A_{i}, B_{i}, C_{i}); α \in Δ_{N}},$ where $Δ_{N}$ is the unit simplex given by $Δ_{N} = {α \in R^{N} : \sum_{i = 1}^{N} α_{i} = 1, α_{i} \geq 0, i = 1, \dots, N} .$ In other words, any uncertain matrix triplet $(A, B, C) (α) \in P$ can be written as a convex combination $α \in Δ_{N}$ of vertices $(A_{i}, B_{i}, C_{i})$ , $i = 1$

$κ$ -Robust stability

Robust stability and $ℋ_{2}$ performance analysis of the uncertain system (1) is verified using the particular polynomial parameter-dependent quadratic Lyapunov function $V_{κ} (α, x) ≔ x^{T} X_{κ} (α, P_{κ} (α)) x,$ where $κ \geq 1$ is an integer, $X_{κ} (α, P_{κ} (α)) \in R^{n \times n}$ is such that $X_{κ} (α, P_{κ} (α)) ≔ A_{κ} {(α)}^{T} P_{κ} (α) A_{κ} (α) > 0,$ the matrix $A_{κ} (α) \in R^{κ n \times n}$ is defined by $A_{κ} (α) ≔ [\begin{matrix} A {(α)}^{0} \\ ⋮ \\ A {(α)}^{κ - 1} \end{matrix}] = [\begin{matrix} I_{n} \\ ⋮ \\ A {(α)}^{κ - 1} \end{matrix}],$ and the matrix $P_{κ} (α) = P_{κ} {(α)}^{T} \in R^{κ n \times κ n}$ , a polynomial in $α$ , is partitioned as $P_{κ} (α) ≔ [\begin{matrix} P_{11} (α) & \dots & P_{1 κ} (α) \\ ⋮ & ⋱ & ⋮ \\ P_{1 κ} {(α)}^{T} & \dots & P_{κ κ} (α) \end{matrix}],$ where each block $P_{j k} (α) \in R^{n \times n}$ , for all

$κ$ -robust $ℋ_{2}$ performance

In this section the previous discussion is extended to the question of robust $ℋ_{2}$ performance. Parameter-dependent LMIs which are equivalent to the inequalities of Lemma 1 can also be obtained by using the concept of $κ$ -robust stability. The following Lemma provides the condition of $κ$ -robust $ℋ_{2}$ performance analysis.

Lemma 9

Matrix $A (α)$ is $κ$ -robustly stable and the inequality ${‖ H_{w y} (s, α) ‖}_{2}^{2} < γ$ holds for all $α \in Δ_{N}$ if and only if there exists a polynomial parameter-dependent matrix $X_{κ} (α, P_{κ} (α)) > 0$ in the form

Numerical complexity

The numerical complexity associated with an optimization problem based on LMIs can be estimated by the number $K$ of scalar variables and the number $L$ of LMI rows involved. Table 1 shows the values of $K$ and $L$ for Theorem 8 (T8) on $κ$ -robust stability and Theorem 11 (T11) on $κ$ -robust $ℋ_{2}$ performance.

Note that Theorem 11 considers $P_{κ} (α)$ affine on $α$ . If $P_{κ}$ is made constant, $(N - 1) κ n (κ n + 1) / 2$ can be subtracted from $K$ . Moreover, if it is known a priori that matrix $A (α)$ is Hurwitz in $A$ , the constraints

Numerical experiments

All numerical tests have been performed in a Pentium IV 2.6 GHz, 512 MB RAM, using the SeDuMi [30] and Yalmip [31] within Matlab. The source code can be downloaded from http://www.dt.fee.unicamp.br/~ricfow/robust.htm.

In all robust stability examples, Theorem 8 provided a positive answer for $κ \leq 3$ . The worst case $ℋ_{2}$ norm has been reached by the LMI conditions of Theorem 11 for $κ = 3$ (with $P_{κ} (α)$ affine on $α$ ) and $κ = 4$ (with $P_{κ} (α) = P_{κ}$ ).

Example 1

Theorem 8 is used to check robust stability of 1000 stable polytopes

Conclusion

A new approach to produce convergent LMI relaxations to cope with the robust stability analysis of continuous-time uncertain linear systems in polytopic domains has been presented. A special polynomial parameter-dependent Lyapunov matrix is used to derive equivalent parameter-dependent LMIs which can be lifted into a larger dimensional space. In this space, efficient LMI relaxations can be obtained. An extension to cope with guaranteed $ℋ_{2}$ cost computation is also provided. The results are

Acknowledgments

Supported by CNPq and FAPESP. The authors thank Pierre-Alexandre Bliman for many fruitful discussions and also the anonymous reviewers.

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^☆: This research has been supported in part by grants from “Fundação de Amparo à Pesquisa do Estado de São Paulo” — FAPESP and from “Conselho Nacional de Desenvolvimento Científico e Tecnológico” — CNPq, Brazil.

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

Abstract

Introduction

Section snippets

Problem statement

κ-Robust stability

κ-robust ℋ2 performance

Numerical complexity

Numerical experiments

Conclusion

Acknowledgments

Linear Algebra and Its Applications

Systems & Control Letters

Systems & Control Letters

Linear Algebra and Its Applications

Convex Optimization

Affine parameter-dependent Lyapunov functions and real parametric uncertainty

IEEE Transactions on Automatic Control

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

IEEE Transactions on Automatic Control

Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: An LMI approach

IEEE Transactions on Automatic Control

Matrix sum-of-squares relaxations for robust semi-definite programs

Mathematical Programming Series B

Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations

IEEE Transactions on Automatic Control

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$κ$ -Robust stability

$κ$ -robust $ℋ_{2}$ performance