Guaranteed Cost Control for Uncertain Neutral Systems with a Minimal Order Observer

Stability and state-feedback stabilization of linear systems with uncertain coefficients and uncertain time-varying delays are considered. The system under consideration may be unstable without delay, but it becomes asymptotically stable for positive values of the delay. A new descriptor discretized Lyapunov-Krasovskii functional (LKF) method is introduced, which combines the application of the complete LKF and the discretization method of K. Gu with the descriptor model transformation. For the first time, the new method allows to apply the discretized LKF method to synthesis problems. Moreover, the analysis of systems with polytopic time-invariant uncertainties is less restrictive by the new discretized method. Sufficient conditions for robust stability and stabilization of uncertain neutral type systems are derived in terms of linear matrix inequalities (LMIs) via input-output approach to stability. Numerical examples illustrate the efficiency of the new method.

Eigenstructure assignment-based robust stability conditions for uncertain systems with multiple time-varying delays

This paper focuses on the robust stability and the memory feedback stabilization problems for a class of uncertain switched nonlinear systems with multiple time-varying delays. Especially, the considered time delays depend on the subsystem number. Based on a novel common Lyapunov functional, the aggregation techniques, and the Borne and Gentina criterion, new sufficient robust stability and stabilization conditions under arbitrary switching are established. Compared with existing results, the proposed criteria are explicit, simple to use, and obtained without finding a common Lyapunov function for all subsystems through linear matrix inequalities, considered very difficult in this situation. Moreover, compared with the memoryless one, the developed controller guarantees the robust stability of the corresponding closed-loop system with more performance by minimizing the effect of the delays in the system dynamics. Finally, two numerical simulation examples are shown to prove the practical utility and the effectiveness of the proposed theories.

Robust stability problem of discrete-time fractional-order systems (DTFOSs) with interval uncertainties is investigated in this paper. Firstly, a new theorem for matrix root-clustering in union-region is established. Based on this theorem, the stability regions of DTFOSs are described as the union-region of closed sub-regions, and sufficient conditions for stability of DTFOSs are presented. Then, new sufficient conditions for robust stability of DTFOSs with interval uncertainties are derived. All the results are obtained in terms of linear matrix inequalities (LMIs) which are more tractable than the existing ones. Finally, numerical examples are given to show that our results are valid and less conservative than the existing ones.

Stabilization of fractional-order singular uncertain systems

Robust stability analysis of polytopic linear time-invariant (LTI) systems is a basic problem in engineering. This paper analyzes the complexity of three fundamental methods used to address this problem, all of them providing a sufficient and necessary condition (with finite and known sizes) for robust stability. The first method is based on the use of a polynomially parameter-dependent Lyapunov function. The second method is a simplified version of the Routh-Hurwitz stability criterion. Lastly, the third method is based on eigenvalue combinations. It is explained that the robust stability conditions provided by these three methods require to establish positive definiteness of symmetric matrix forms (SMFs) over the simplex. Also, it is explained that a sufficient condition for the latter problem can be given in terms of a linear matrix inequality (LMI) feasibility test. Hence, the complexity of the three methods is analyzed and compared by deriving the number of scalar variables in the LMI feasibility tests. A numerical example is also presented to investigate the computational time of these tests.

This paper describes the synthesis of robust decentralized controllers for uncertain large-scale discrete-time systems with time delays in the subsystem interconnections. Based on the Lyapunov method, a sufficient condition for robust stability is derived in terms of a linear matrix inequality (LMI). The solutions of the LMIs can be obtained easily using efficient convex optimization techniques. A numerical example is given to illustrate the proposed method.

In this note, a new result that extends the scaled small gain theorem is presented. As is well known, the scaled small gain theorem gives necessary and sufficient conditions for robust stability of a nominal linear time-invariant system in feedback with structured operators of norm less than or equal to unity. We propose alternative linear matrix inequality conditions that give necessary and sufficient conditions for robust stability against the class of structured unitary operators (invertible operators of exactly unit norm). It is shown that this result, besides being a less conservative version of the scaled small gain theorem, has connections to several recent results on the control of spatially interconnected systems and serves to unify and quantify the conservatism of those results.

The robust stabilization problem of uncertain discrete system with delay and saturating actuators is investigated. A polytopic representation is used to describe the saturation behavior. The sufficient condition of robust stabilization is obtained by using parameter dependent Lyapunov function. Furthermore, the attractive domain is estimated. Finally, numerical example shows the effectiveness of the results.

The robust H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> stabilization problem of uncertain discrete systems with delays is investigated. By means of descriptor system transformation, a new delay-dependent sufficient condition of robust stabilization is given. Then, a new memoryless H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> feedback-controller is designed based on the new stability criterion, which can be constructed by solving a set of linear matrix inequalities(LMIs). Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

On the robustness of uncertain time-delay systems with structured uncertainties

In this paper, the problem of the robust stabilization for a class of uncertain linear dynamical systems with time-varying delay is considered. By making use of an algebraic Riccati equation, we derive some sufficient conditions for robust stability of time-varying delay dynamical systems with unstructured or structured uncertainties. In our approach, the only restriction on the delay functionh(t) is the knowledge of its upper boundh −. Some analytical methods are employed to investigate these stability conditions. Since these conditions are independent of the delay, our results are also applicable to systems with perturbed time delay. Finally, a numerical example is given to illustrate the use of the sufficient conditions developed in this paper.

In this paper, we consider generalized delay-dependent stability conditions of Takagi-Sugeno fuzzy systems with discrete and distributed time-delays. For fuzzy systems with discrete time-delays, both delay-independent stability conditions and delay-dependent stability conditions have already been obtained. However, no stability condition for fuzzy systems with distributed time-delays has appeared in the literature. This is also true in case of the robust stability for uncertain fuzzy systems with distributed time-delays. With the aid of the stability theorems of linear systems with distributed time-delays, we employ a similar Lyapunov functional for the stability of fuzzy time-delay systems. Then we introduce many free matrices to such a Lyapunov functional, in order to obtain delay-dependent stability conditions. These techniques lead to generalized and less conservative stability conditions. In fact, delay-dependent stability conditions thus obtained are given in linear matrix inequalities(LMIs) and guarantee a wide stability region. We also consider the robust stability of fuzzy time-delay systems with uncertain parameters. Applying the same techniques made on the stability conditions, we obtain delay-dependent sufficient conditions for the robust stability of uncertain fuzzy systems with discrete and distributed time-delays. Moreover, we consider the state feedback stabilization. Based on stability and robust stability conditions, we obtain conditions for the state feedback controller to stabilize the fuzzy time-delay systems. Finally, we take some examples to illustrate our results.

This article concerns the finite-time robust guaranteed cost control problem for a class of linear continuous-time singular systems with norm-bounded uncertainties. In this study, the problem is to design a state feedback controller such that the closed-loop system is finite-time stable, and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. By constructing an appropriate Lyapunov function, a sufficient condition for the finite-time robust stability of the system based on linear matrix inequality (LMI) is established. Furthermore, the sufficient condition for the existence of the guaranteed cost controller is formulated in terms of LMIs, which can make the closed-loop uncertain singular system finite-time robust stable. Finally, two numerical examples are given for illustration of the proposed theoretical results.

ABSTRACTA new design approach to delay‐dependent robust stabilization and robust H∞ control for a class of uncertain time‐delay systems is provided in this paper. The sufficient conditions for delay‐dependent robust stabilization and robust H∞ control are derived based on a new state transformation and given in terms of linear matrix inequalities (LMI). Numerical examples are presented to show that the proposed results can be less conservative and can be used to deal with not only small but also large delay systems.