Linear Matrix Inequality Conditions For Stability Research Articles

Exact LMI Conditions for Stability and $\mathcal {L}_2$ Gain Analysis of 2-D Mixed Continuous–Discrete Time Systems via Quadratically Frequency-Dependent Lyapunov Functions

This article addresses the problems of establishing structural stability and the <inline-formula><tex-math notation="LaTeX">$\mathcal {L}_2$</tex-math></inline-formula> gain of 2-D mixed continuous&#x2013;discrete time systems. The first contribution is to show that Lyapunov functions quadratically dependent on the frequency are exact for establishing structural stability. This is particularly important since the existing works that exploit Lyapunov functions provide a much larger upper bound on the dependence on the frequency or other parameters. The second contribution is to propose a novel linear matrix inequality (LMI) necessary and sufficient condition for establishing the existence of such Lyapunov functions. It is shown, analytically and through several examples, for both best and worst cases, that the numerical complexity of this novel condition is significantly smaller than that of the existing methods. The third contribution is to show that the proposed methodology can be used to establish upper bounds on the <inline-formula><tex-math notation="LaTeX">$\mathcal {L}_2$</tex-math></inline-formula> gain, in particular, deriving a novel necessary and sufficient LMI condition based on Lyapunov functions quadratically dependent on the frequency. Finally, the article presents the generalization of the proposed methodology to nonmixed 2-D systems.

Computational method for estimating the domain of attraction of discrete-time uncertain rational systems

Abstract Using linear matrix inequality (LMI) conditions, we propose a computational method to generate Lyapunov functions and to estimate the domain of attraction (DOA) of uncertain nonlinear (rational) discrete-time systems. The presented method is a discrete-time extension of the approach first presented in [39], where the authors used Finsler’s lemma and affine annihilators to give sufficient LMI conditions for stability. The system representation required for DOA computation is generated systematically by using the linear fractional transformation (LFT). Then a model simplification step not affecting the computed Lyapunov function (LF) is executed on the obtained linear fractional representation (LFR). The LF is computed in a general quadratic form of a state and parameter dependent vector of rational functions, which are generated from the obtained LFR model. The proposed method is compared to the numeric n-dimensional order reduction technique proposed in [11]. Finally, additional tuning knobs are proposed to obtain more degrees of freedom in the LMI conditions. The method is illustrated on two benchmark examples.

Lyapunov method for stability of descriptor second-order and high-order systems

In this study, the stability problem of descriptor second-order systems is considered. Lyapunov equations for stability of second-order systemsare established by using Lyapunov method. The existence of solutions for Lyapunov equations are discussed and linear matrixinequality condition for stability of second-order systems aregiven. Then, based on the feasible solutions of the linear matrixinequality, all parametric solutions of Lyapunov equations are derived.Furthermore, the results of Lyapunov equations and linear matrixinequality condition for stability of second-ordersystems are extended to high-order systems. Finally, illustratingexamples are provided to show the effectiveness of the proposed method.

Necessary and Sufficient LMI Conditions for Stability and Performance Analysis of 2-D Mixed Continuous-Discrete-Time Systems

This paper proposes necessary and sufficient conditions for stability and performance analysis of 2-D mixed continuous-discrete-time systems that can be checked with convex optimization, in particular linear matrix inequalities (LMIs). Specifically, the first contribution of the paper is a condition for exponential stability based on the introduction of a complex Lyapunov function depending polynomially on a parameter and on the use of the Gram matrix method. It is shown that this condition is sufficient for any chosen degree of the complex Lyapunov function, and necessary for an a priori known degree. The second contribution is a non-Lyapunov condition for exponential stability based on eigenvalue products. This condition is necessary and sufficient, and has the advantage of requiring a significantly smaller computational burden for achieving necessity. Lastly, the third contribution is to show how upper bounds on the H∞ norm of 2-D mixed continuous-discrete-time systems can be obtained through a semidefinite program based on complex Lyapunov functions. A necessary and sufficient condition is provided for establishing the tightness of the found upper bounds.

LMI conditions for stability of impulsive stochastic Cohen–Grossberg neural networks with mixed delays

National Natural Science foundation of China [10871120]; Natural Science of Shandong Province [Y2007A10]

LMI conditions for stability of stochastic recurrent neural networks with distributed delays

Abstract In this paper, the global asymptotic stability of stochastic recurrent neural networks with discrete and distributed delays is analyzed by utilizing the Lyapunov–Krasovskii functional and combining with the linear matrix inequality (LMI) approach. A new sufficient condition ensuring the global asymptotic stability for delayed recurrent neural networks is obtained in the stochastic sense using the powerful MATLAB LMI Toolbox. In addition, an example is also provided to illustrate the applicability of the result.

LMI conditions for stability of neural networks with distributed delays

Abstract This paper presents new sufficient conditions for global asymptotic stability of neural networks with discrete and distributed delays. By using appropriate Lyapunov–Krasovskii functionals, we derive stability conditions in terms of linear matrix inequalities (LMIs). This is convenient for numerically checking the system stability using the powerful MATLAB LMI Toolbox. Moreover, existing conditions are mostly based on certain diagonal dominance or M matrix conditions on weight matrices of the neural networks, which only make use of absolute values of the weights and ignore their sign, and hence are somewhat conservative. The LMI-based results obtained here can get rid of this disadvantage and give less conservative stability conditions. We illustrate this with numerical examples.