On the Asymptotic Stability of a Class of Saturating Sampled-Data Systems

Pearson, J.; Gibson, J. E.

doi:10.1109/tai.1964.5407770

Cited by 34 publications

(11 citation statements)

References 6 publications

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“…The function is inspired by the continuoustime framework and can be considered as a candidate function due to the time invariance of nonlinearity ϕ(·), which guarantees the positiveness of the integral. To our knowledge, the function v was first used in the discretetime domain, in [18], for stability of input-saturating sampled-data systems. Because of the time invariance of ϕ(·), the following integral term appears in the Lyapunov difference I = y k+1 y k ϕ(s)ds.…”

Section: Motivationmentioning

confidence: 99%

“…Because of the time invariance of ϕ(·), the following integral term appears in the Lyapunov difference I = y k+1 y k ϕ(s)ds. An extensive literature (see [18,21,12,13,20,14,6]) is forced to make additional assumptions about the slope of the nonlinearity to define an upper bound of I and to conclude on the stability. The spirit of these assumptions is to upper and/or lower bound the slope of the nonlinearity to allow a bound of the integral I via a trapezoidal rule.…”

Section: Motivationmentioning

confidence: 99%

“…By replacing u k given by (18) and (19) in (16), the closedloop model is described, with A cl = A + BK and F cl = F + BΓ as…”

Section: Discrete Time Lur'e Systems Subject To Input Saturationmentioning

confidence: 99%

“…′ is an element of S( K − J, ρ), then the nonlinearity Ψ(u k ), with u k is defined in Equation (18), satisfies the following sector condition, with U ∈ R m×m diagonal and positive definite…”

Section: Discrete Time Lur'e Systems Subject To Input Saturationmentioning

confidence: 99%

“…Let us define this problem Problem 2 (Local Stability Analysis) For a given set of gains K ∈ R m×n and Γ ∈ R m×p of the control law (18), determine a region, in the state space, as large as possible included in the basin of attraction B 0 of the system (16)- (17), under Assumption 1.…”

Section: Discrete Time Lur'e Systems Subject To Input Saturationmentioning

confidence: 99%

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Stability analysis of discrete-time Lur’e systems

2012

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A class of Lyapunov functions is proposed for discrete-time linear systems interconnected with a cone bounded nonlinearity. Using these functions, we propose sufficient conditions for the global stability analysis, in terms of linear matrix inequalities (LMI), only taking the bounded sector condition into account. Unlike frameworks based on the Lur'e-type function, the additional assumptions about the derivative or discrete variation of the nonlinearity are not necessary. Hence, a wider range of cone bounded nonlinearities can be covered. We also show that there is a link between global stability LMI conditions based on this new Lyapunov function and a transfer function of an auxiliary system being strictly positive real. In addition, the novel function is considered in the local stability analysis problem of discrete-time Lur'e systems subject to a saturating feedback. A convex optimization problem based on sufficient LMI conditions is formulated to maximize an estimate of the basin of attraction. Another specificity of this new Lyapunov function is the fact that the estimate is composed of disconnected sets. Numerical examples reveal the effectiveness of this new Lyapunov function in providing a less conservative estimate with respect to the quadratic function.

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Section: Motivationmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

“…By replacing u k given by (18) and (19) in (16), the closedloop model is described, with A cl = A + BK and F cl = F + BΓ as…”

Section: Discrete Time Lur'e Systems Subject To Input Saturationmentioning

confidence: 99%

Section: Discrete Time Lur'e Systems Subject To Input Saturationmentioning

confidence: 99%

Section: Discrete Time Lur'e Systems Subject To Input Saturationmentioning

confidence: 99%

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Stability analysis of discrete-time Lur’e systems

2012

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Advanced Lyapunov Functions for Lur'e Systems

Gonzaga¹,

Jungers²,

Daafouz³

2013

Hybrid Systems With Constraints

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Explicit construction of quadratic lyapunov functions for the small gain, positivity, circle, and popov theorems and their application to robust stability. part II: Discrete‐time theory

Haddad

Bernstein

1994

Intl J Robust & Nonlinear

103

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In a companion paper (‘Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle, and Popov theorems and their application to robust stability. Part I: Continuous‐time theory’), Lyapunov functions were constructed in a unified framework to prove sufficiency in the small gain, positivity, circle, and Popov theorems. In this Part II, analogous results are developed for the discrete‐time case. As in the continuous‐time case, each result is based upon a suitable Riccati‐like matrix equation that is used to explicitly construct a Lyapunov function that guarantees asymptotic stability of the feedback interconnection of a linear time‐invariant system and a memoryless nonlinearity. Multivariable versions of the discrete‐time circle and Popov criteria are obtained as extensions of known results. Each result is specialized to the linear uncertainty case and connections with robust stability for state‐space systems is explored.

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On the Asymptotic Stability of a Class of Saturating Sampled-Data Systems

Cited by 34 publications

References 6 publications

Stability analysis of discrete-time Lur’e systems

Stability analysis of discrete-time Lur’e systems

Advanced Lyapunov Functions for Lur'e Systems

Explicit construction of quadratic lyapunov functions for the small gain, positivity, circle, and popov theorems and their application to robust stability. part II: Discrete‐time theory

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