Diagonal Riccati stability and positive time-delay systems

Mason, Oliver

doi:10.1016/j.sysconle.2011.09.022

Cited by 43 publications

(36 citation statements)

References 18 publications

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“…The above result provides a solution to this problem for the particular cases of linear positive and positively dominated systems with delays. Note that the positive systems case has also been solved in [51,32] using different approaches. These results have since been extended to some other classes of systems in [67].…”

Section: Proposition 13mentioning

confidence: 99%

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Stability and performance analysis of linear positive systems with delays using input–output methods

Briat

2017

International Journal of Control

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It is known that input-output approaches based on scaled small-gain theorems with constant Dscalings and integral linear constraints are non-conservative for the analysis of some classes of linear positive systems interconnected with uncertain linear operators. This dramatically contrasts with the case of general linear systems with delays where input-output approaches provide, in general, sufficient conditions only. Using these results we provide simple alternative proofs for many of the existing results on the stability of linear positive systems with discrete/distributed/neutral time-invariant/-varying delays and linear difference equations. In particular, we give a simple proof for the characterization of diagonal Riccati stability for systems with discrete-delays and generalize this equation to other types of delay systems. The fact that all those results can be reproved in a very simple way demonstrates the importance and the efficiency of the input-output framework for the analysis of linear positive systems. The approach is also used to derive performance results evaluated in terms of the L1-, L2-and L∞-gains. It is also flexible enough to be used for design purposes.

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Section: Proposition 13mentioning

confidence: 99%

“…(iii) A linear positive system with constant discrete delay is stable if and only if the associated Riccati equation has diagonal solutions [32,51].…”

Section: Introductionmentioning

confidence: 99%

Stability and performance analysis of linear positive systems with delays using input–output methods

Briat

2017

International Journal of Control

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“…So, it is of high importance to calculate analytic or numerical approximate for it. In recent decades, some new numerical and analytic approximate methods have been proposed for solving such a difficult problem in the context of delay ordinary differential equations (see for example [31][32][33][34][35][36][37]). To overcome this difficulty, an iterative approach, based on the VIM, will be introduced in the next section.…”

Section: (24)mentioning

confidence: 99%

An iterative method for suboptimal control of linear time-delayed systems

Mirhosseini-Alizamini

Effati

Heydari

2015

Systems & Control Letters

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“…This established the existence of a diagonal Lyapunov functional for asymptotically stable positive linear time-delay systems, providing a natural extension of a fundamental property of positive linear time-invariant (LTI) systems [2]. Furthermore, this fact strengthened the main result of [5] which showed that under the same condition (A + B Hurwitz), there exists a diagonal P ≻ 0 and Q ≻ 0 (not necessarily diagonal) satisfying (1).…”

Section: Background and Introductionmentioning

confidence: 65%

Diagonal Riccati stability and the Hadamard product

Aleksandrov

Mason

Vorob'eva

2017

Linear Algebra and its Applications

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We first present an extension of a recent characterisation of diagonal Riccati stability and, using this, extend a result of Kraaijevanger on diagonal Lyapunov stability to Riccati stability of time-delay systems. We also describe a class of transformations that preserve the property of being diagonally Riccati stable and apply these two results to provide novel stability results for classes of time-delay systems.Keywords: Riccati inequality; diagonal stability; time-delay systems.AMS subject classifications: 15A24; 93D05. Background and IntroductionThe problem of Riccati stability was introduced in [1] and is motivated by the stability theory of linear time-delay systems. Formally, a pair (A, B) is said to be Riccati stable if there exist P = P T ≻ 0, Q = Q T ≻ 0 such thatwhere M ≺ 0 (M ≻ 0) denotes that the matrix M = M T is negative definite (positive definite). Throughout the paper, M 0 (M 0) denotes that M is positive semi-definite (negative semidefinite). When matrices P , Q satisfying (1) exist they define a quadratic Lyapunov-Krasovskii functional establishing stability for the time-delay systeṁwhere τ ≥ 0 can be any fixed nonnegative delay. A ∈ R n×n is Metzler if a ij ≥ 0 for i = j. We denote the spectrum of A by σ(A) and the spectral abscissa of A by µ(A): formally,and say that A is Hurwitz if µ(A) < 0.We denote the standard basis of R n by e 1 , . . . , e n and we use 1 n to denote the vector in R n , all of whose entries are equal to one. For A ∈ R n×n , diag(A) is the vector v in R n with v i = a ii , 1 ≤ i ≤ n. Sym(n, R) denotes the space of n × n symmetric matrices with real entries.For a real number x, sign(x) is given by +1 if x ≥ 0 and −1 for x < 0 respectively.

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Diagonal Riccati stability and positive time-delay systems

Cited by 43 publications

References 18 publications

Stability and performance analysis of linear positive systems with delays using input–output methods

Stability and performance analysis of linear positive systems with delays using input–output methods

An iterative method for suboptimal control of linear time-delayed systems

Diagonal Riccati stability and the Hadamard product

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