Exact Stability Analysis of 2-D Systems Using LMIs

Abstract

In this note, we propose necessary and sufficient conditions for the asymptotic stability analysis of two-dimensional (2-D) systems in terms linear matrix inequalities (LMIs). By introducing a guardian map for the set of Schur stable complex matrices, we first reduce the stability analysis problems into nonsingularity analysis problems of parameter-dependent complex matrices. Then, by means of the discrete-time positive real lemma and the generalized-procedure, we derive LMI-based conditions that enable us to analyze the asymptotic stability in an exact (i.e., nonconservative) fashion. It turns out that, by employing the generalized-procedure, we can derive smaller size of LMIs so that the computational burden can be reduced.

Delay-independent stability analysis of linear time-delay systems based on frequency discretization

“…This property is attractive because it is useful for instability test. Conditions with a similar property can also be found in Chesi (2013) and Ebihara, Ito, and Hagiwara (2006), which deal with Schur stability with a modulus-bounded uncertain complex parameter and (Schur or Hurwitz) stability with a bounded uncertain real parameter of the polynomial form, respectively. However, the stability conditions presented in this paper do not have this property, because there is no a priori knowledge on what the discretized frequency intervals should be, which is a drawback of the proposed method.…”

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

“…It has to be mentioned that most of the proposed LMI stability conditions, when tractable, are only sufficient. Recently, some preliminary results on formulating necessary and sufficient stability condition have been proposed in Ebihara et al (2006), Fu et al (2006). However, these conditions are only formulated for 2-D discrete system case and cannot be easily extended to controller design case.…”

Time-relevant stability of 2D systems

“…LMI tests for checking various properties of 2D systems, among which stability, have been proposed in [1,6], building on the pioneering work done in [17] for the study of parameter-dependent inequalities. Other relevant results in this area aimed at eliminating the parameterdependency in stability tests for 2D systems have been reported in [5]; applications to discrete linear repetitive processes have been reported in [9].…”