On the Kalman–Yakubovich–Popov lemma and the multidimensional models

Bachelier, Olivier; Paszke, Wojciech; Mehdi, Driss

doi:10.1007/s11045-008-0055-2

Cited by 31 publications

(18 citation statements)

References 43 publications

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“…It's worth noting that, the result in Lemma 5 and the corresponding result in Li et al (2015) both can be rewritten as special cases of Theorem 2 in Bachelier et al (2008) for 2-D continuous-discrete systems and 2-D discrete systems, respectively.…”

Section: Lemmamentioning

confidence: 99%

Stability and robust stabilization of 2-D continuous–discrete systems in Roesser model based on KYP lemma

Wang

Gao

et al. 2015

Multidim Syst Sign Process

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This paper investigates the problem of stability and robust stabilization of two-dimensional (2-D) continuous-discrete systems in Roesser model. Based on KalmanYakubovich-Popov (KYP) lemma, sufficient conditions of stability of the systems are proposed in terms of linear matrix inequalities (LMIs). Moreover, combining generalized KYP lemma with frequency-partitioning idea, the conservativeness of the stability conditions may be further reduced. Robust stabilization using state feedback is studied as well and stabilizing feedback gain matrices are constructed based on the solutions of certain LMIs. In addition, the stability analysis of a class of differential repetitive processes, a special case of 2-D continuous-discrete systems in Roesser model, can adopt the results in this manuscript. Numerical examples are presented to verify the effectiveness of the method.

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Section: Lemmamentioning

confidence: 99%

Stability and robust stabilization of 2-D continuous–discrete systems in Roesser model based on KYP lemma

Wang

Gao

et al. 2015

Multidim Syst Sign Process

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“…The KYP lemma has been extended in other directions as well, including FDIs with frequencydependent weights (Graham and de Oliveira 2010), internally positive systems (Tanaka and Langbort 2011), full rank polynomials (Ebihara et al 2008), real multipliers (Pipeleers and Vandenberghe 2011), a more general class of FDIs (Gusev 2009), multidimensional systems (Bachelier et al 2008), negative imaginary systems (Xiong et al 2012), symmetric formulations for robust stability analysis (Tanaka and Langbort 2013), and multiple frequency intervals (Pipeleers et al 2013). Extensions of the KYP lemma and related S-procedures are thoroughly reviewed in Gusev and Likhtarnikov (2006).…”

Section: Summary and Further Directionsmentioning

confidence: 99%

(Generalized) KYP lemma and applications

Iwasaki¹

2013

Encyclopedia of Systems and Control

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Various properties of dynamical systems can be characterized in terms of inequality conditions on their frequency responses. The Kalman-Yakubovich-Popov (KYP) lemma shows equivalence of such frequency domain inequality (FDI) and a linear matrix inequality (LMI). The fundamental result has been a basis for robust and optimal control theories in the past several decades. The KYP lemma has recently been generalized to the case where an FDI on a possibly improper transfer function is required to hold in a (semi)finite frequency range. The generalized KYP lemma allows us to directly deal with practical situations where design parameters are sought to satisfy FDIs in multiple (semi)finite frequency ranges. Various design problems, including FIR filter and PID controller, reduce to LMI problems which can be solved via semidefinite programming.

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“…It is also possible 1 The S-procedure we refer to is the one formulated in [19]; there are many other formulations but this one is particularly elegant and well designed for our purpose to define the continuous or hybrid versions of Roesser model to describe -D systems such as the following form [2] …”

Section: A Hybrid Roesser Modelsmentioning

confidence: 99%

“…Lemma 1: Consider a multidimensional system represented by (1). Then, such a system is asymptotically stable if and only if…”

Section: B Polynomial-based Stability Conditionmentioning

confidence: 99%