Baozhu Du

Stability and Stabilization of Delayed T--S Fuzzy Systems: A Delay Partitioning Approach

This paper proposes a new approach, namely, the delay partitioning approach, to solving the problems of stability analysis and stabilization for continuous time-delay Takagi-Sugeno fuzzy systems. Based on the idea of delay fractioning, a new method is proposed for the delay-dependent stability analysis of fuzzy time-delay systems. Due to the instrumental idea of delay partitioning, the proposed stability condition is much less conservative than most of the existing results. The conservatism reduction becomes more obvious with the partitioning getting thinner. Based on this, the problem of stabilization via the so-called parallel distributed compensation scheme is also solved. Both the stability and stabilization results are further extended to time-delay fuzzy systems with time-varying parameter uncertainties. All the results are formulated in the form of linear matrix inequalities (LMIs), which can be readily solved via standard numerical software. The advantage of the results proposed in this paper lies in their reduced conservatism, as shown via detailed illustrative examples. The idea of delay partitioning is well demonstrated to be efficient for conservatism reduction and could be extended to solving other problems related to fuzzy delay systems.

Technical communique: A delay-partitioning approach to the stability analysis of discrete-time systems

This paper revisits the problem of stability analysis for linear discrete-time systems with time-varying delay in the state. By utilizing the delay partitioning idea, new stability criteria are proposed in terms of linear matrix inequalities (LMIs). These conditions are developed based on a novel Lyapunov functional. In addition to delay dependence, the obtained conditions are also dependent on the partitioning size. We have also established that the conservatism of the conditions is a non-increasing function of the number of partitions. Numerical examples are given to illustrate the effectiveness and advantage of the proposed methods.

Positive Observers and Dynamic Output-Feedback Controllers for Interval Positive Linear Systems

This paper is concerned with the design of observers and dynamic output-feedback controllers for positive linear systems with interval uncertainties. The continuous-time case and the discrete-time case are both treated in a unified linear matrix inequality (LMI) framework. Necessary and sufficient conditions for the existence of positive observers with general structure are established, and the desired observer matrices can be constructed easily through the solutions of LMIs. An optimization algorithm to the error dynamics is also given. Furthermore, the problem of positive stabilization by dynamic output-feedback controllers is investigated. It is revealed that an unstable positive system cannot be positively stabilized by a certain dynamic output-feedback controller without taking the positivity of the error signals into account. When the positivity of the error signals is considered, an LMI-based synthesis approach is provided to design the stabilizing controllers. Unlike other conditions which may require structural decomposition of positive matrices, all proposed conditions in this paper are expressed in terms of the system matrices, and can be verified easily by effective algorithms. Two illustrative examples are provided to show the effectiveness and applicability of the theoretical results.

Further Results on Exponential Estimates of Markovian Jump Systems With Mode-Dependent Time-Varying Delays

This technical note studies the problem of exponential estimates for Markovian jump systems with mode-dependent interval time-varying delays. A novel Lyapunov-Krasovskii functional (LKF) is constructed with the idea of delay partitioning, and a less conservative exponential estimate criterion is obtained based on the new LKF. Illustrative examples are provided to show the effectiveness of the proposed results.

Neural networks letter: Stability analysis of static recurrent neural networks using delay-partitioning and projection

This paper introduces an effective approach to studying the stability of recurrent neural networks with a time-invariant delay. By employing a new Lyapunov-Krasovskii functional form based on delay partitioning, novel delay-dependent stability criteria are established to guarantee the global asymptotic stability of static neural networks. These conditions are expressed in the framework of linear matrix inequalities, which can be verified easily by means of standard software. It is shown, by comparing with existing approaches, that the delay-partitioning projection approach can largely reduce the conservatism of the stability results. Finally, two examples are given to show the effectiveness of the theoretical results.

Brief paper: Stabilization for state/input delay systems via static and integral output feedback

This paper addresses the static and integral output feedback stabilization problems of continuous-time linear systems with an unknown state/input delay. By combining an augmentation approach and the delay partitioning technique, criteria for static and integral output feedback stabilizability are proposed in terms of nonlinear matrix inequalities with a free parameter matrix introduced. These new characterizations possess a special structure, which leads to linearized iterative computation. The effectiveness and merits of the proposed approach are shown through numerical examples.

Stability and Stabilization for Markovian Jump Time-Delay Systems With Partially Unknown Transition Rates

This paper focuses on the stability analysis and controller synthesis of continuous-time Markovian jump time-delay systems with incomplete transition rate descriptions. A general stability criterion is formulated first for state- and input-delay Markovian jump time-delay systems with fully known transition rates. On the basis of the proposed condition, an equivalent condition is given under the assumption of partly known/unknown transition rates. A new design technique based on a projection inequality has been applied to design both state feedback and static output feedback controllers. All conditions can be readily verified by efficient algorithms. Finally, illustrative examples are provided to show the effectiveness of the proposed approach.

A Delay-Partitioning Projection Approach to Stability Analysis of Neutral Systems

Abstract This paper introduces a new effective approach to study the stability of neutral systems. By employing a special Lyapunov-Krasovskii functional form based on delay partitioning, delay-dependent stability criteria are established for the nominal and the uncertain case (polytopic type) in terms of linear matrix inequalities (LMI). Numerical examples are employed to illustrate that the delay-partitioning projection approach can be applied to estimate the upper bounds for the delays for the system to maintain stability. Judging from these numerical results, the stability criteria obtained are less conservative than those of existing methods.

On reachable sets for positive linear systems under constrained exogenous inputs

This paper focuses on positive linear time-invariant systems with constant coefficients and specific exogenous disturbance. The problem of finding a hyper-pyramid to bound the set of the states that are reachable from the origin in the Euclidean space is addressed, subject to inputs whose ( 1 , 1 ) -norm or ( ∞ , 1 ) -norm is bounded by a prescribed constant. The Lyapunov approach is applied and a bounding hyper-pyramid is obtained by solving a set of inequalities. Iterative procedures (with an adjustable parameter) for reducing the hyper-volume of the bounding hyper-pyramid for the reachable set are proposed.

Improved stability and stabilization results for discrete singular delay systems via delay partitioning

In this paper, the problems of delay-dependent robust stability analysis and robust stabilization are investigated for uncertain discrete-time singular systems with state delay. First, by making use of the delay partitioning technique, a new delay-dependent criterion is given to ensure the nominal system to be regular, causal and stable. This new criterion is further extended to singular systems with both delay and parameter uncertainties. Moreover, without the assumption that the considered systems being regular and causal, robust controllers are designed for discrete-time singular time-delay systems such that the closed-loop systems have the characteristics of regularity, causality and asymptotic stability. These results are illustrated, via a few numerical examples, to be much less conservative than most of the existing results in the literature.