Switched Discrete-Time Delay Systems: Delay-Dependent Analysis and Synthesis

A delay-dependent analysis and synthesis approach is established for a class of linear discrete-time switched systems with time-varying delays using switched Lyapunov–Krasovskii functionals (SLKFs). The problem of water-quality control is cast as the problem of delay-dependent ℒ2 gain analysis and synthesis. New delay-dependent asymptotic stability criteria are developed under arbitrary switching based on appropriately constructed SLKFs. Delay-dependent switched control feedback is then designed, based on state- and output-measurements, to render the corresponding switched closed-loop system delay-dependent asymptotically stable with a prescribed ℒ2 gain measure. The developed results are cast as linear matrix inequalities and tested by Matlab simulation on a representative water-quality example.

This paper studies the problem of designing observer-based controllers for a class of delayed neural networks with nonlinear observation. The system under consideration is subject to nonlinear observation and an interval time-varying delay. The nonlinear observation output is any nonlinear Lipschitzian function and the time-varying delay is not required to be differentiable nor its lower bound be zero. By constructing a set of appropriate Lyapunov---Krasovskii functionals and utilizing the Newton---Leibniz formula, some delay-dependent stabilizability conditions which are expressed in terms of Linear Matrix Inequalities (LMIs) are derived. The derived conditions allow simultaneous computation of two bounds that characterize the exponential stability rate of the closed-loop system. The unknown observer gain and the state feedback observer-based controller are directly obtained upon the feasibility of the derived LMIs stabilizability conditions. A simulation example is presented to verify the effectiveness of the proposed result.

New filter design for linear time-delay systems

This study investigated the problem of robust finite-time stability (FTS) in uncertain genetic regulatory networks (GRNs) affected by delays that vary over time. The proposed work introduced new criteria for evaluating robust FTS, which accounted for parametric uncertainties, discrete interval time-varying delays, and distributed time-varying delays. These criteria were developed using the approach of selecting the appropriate Lyapunov–Krasovskii functionals (LKFs). The analysis considered cases where the time-varying delays were continuous and nondifferentiable. To enhance the robustness of the results, integral inequalities based on auxiliary functions were employed, along with convex and reciprocally convex techniques. The findings provided the FTS criteria for GRNs with delays that vary over time under uncertainties in the form of linear matrix inequalities (LMIs). Numerical examples illustrated the effectiveness of the proposed framework, validated the theoretical results, and enabled comparison with existing approaches. The results thus demonstrated the advantages of the developed criteria over previous methods.

Guaranteed cost control problems for discrete-time systems with uncertainties and time delays are addressed in this paper. The delays may simultaneously appear not only in the state, but in control input and measurement output. Also, parametric uncertainties may exist in all system matrices of the systems considered in this paper. First, a new linear matrix inequality (LMI)-based approach is developed to construct dynamic guaranteed cost controllers for delay systems without uncertainties. Then this result is extended to discrete-time delay systems with uncertainties in linear fractional form in all system matrices. The solvability of these robust control problems for uncertain delay systems can be reduced to that for certain auxiliary systems without uncertainty. Numerical examples are provided to show the effectiveness of our approach.

In this technical note, the problem of delay-independent minimum dwell time for exponential stability of uncertain switched delay systems is considered. Piecewise time-varying Lyapunov functionals/functions which are decreasing at switching times by construction are introduced to investigate exponential stability of switched delay systems with constant or time-varying delays. This type of delicately constructed Lyapunov functionals/functions can efficiently eliminate the “jump” phenomena of adjacent Lyapunov functionals/functions at switching times without imposing any restriction on the sizes of time-delays. By applying this type of Lyapunov functionals/functions, it is shown that if each subsystem is delay-independently exponentially stable, then under some conditions there exists a delay-independent minimum dwell time in the sense that the switched delay system with such minimum dwell time is exponentially stable irrespective of the sizes of the time-delays. Two numerical examples are provided to demonstrate the efficiency of the proposed approach.

This thesis is concerned with the dissipative control and filtering problems of singular systems. Four classes of singular systems are considered: delay-free singular systems, singular systems with constant time-delay, uncertain singular systems with time-varying delay and sensor failures, and singular Markovian jump systems with actuator failures.
\nFor delay-free singular systems, the system augmentation approach is employed to study the dissipative control and filtering problems. First, the approach is used to solve the dissipative control problem by static output-feedback for standard state-space systems which are the special cases of singular systems. For a continuous-time standard state-space system, the closed-loop system is represented in an augmented system form. Based on the augmented system, a necessary and sufficient dissipativity condition is proposed, which decouples the Lyapunov matrix and controller matrix. To further separate the Lyapunov matrix and the system matrices, an equivalent condition is obtained by introducing some slack matrices. Then, a necessary and sufficient condition for the existence of a static output-feedback controller is proposed, and an iterative algorithm is given to solve the condition. For discrete-time singular systems, by giving an equivalent representation of the solution set, a necessary and sufficient dissipativity condition is proposed in terms of strict linear matrix inequality (LMI) which can be easily solved by standard commercial software. Then a state-feedback controller design method is given based on the augmentation system approach. The method is extended to the static output-feedback control problem and the reduced-order dissipative filtering problem.
\nFor continuous-time singular time-delay systems, the problem of state-feedback dissipative control is considered. An improved delay-dependent dissipativity condition in terms of LMIs is established by employing the delay-partitioning technique, which guarantees a singular system to be admissible and dissipative. Based on this, a delay-dependent sufficient condition for the existence of a state-feedback controller is proposed to guarantee the admissibility and dissipativity of the closed-loop system. In addition to delay-dependence, the obtained results are also dependent on the level of dissipativity. Moreover, the results obtained unify existing results on H∞ performance analysis and passivity analysis for singular systems.
\nFor discrete-time singular systems with polytopic uncertainties, time-varying delay and sensor failures, the problem of robust reliable dissipative filtering is considered. The filter is designed by the reciprocally convex approach such that the filtering error singular system is admissible and strictly (Q, S, R)-dissipative. For singular systems with time-varying delay and sensor failures, a sufficient condition of reliable dissipative analysis is obtained in terms of LMIs. Then the result is extended to the uncertain case by introducing some variables to decouple the Lyapunov matrices and the filtering error system matrices. Moreover, a desired filter for uncertain singular systems with time-varying delay and sensor failures is obtained by solving a set of LMIs.
\nFor continuous-time singular Markovian jump systems with actuator failures, the problem of reliable dissipative control is addressed. Attention is focused on the state-feedback controller design method such that the closed-loop system is admissible and strictly (Q, S, R)-dissipative. A sufficient condition is obtained in terms of strict LMIs. Moreover, the results obtained unify existing results on H∞control and passive control on singular Markovian jump systems.

This paper is concerned with the problem of state estimation for a class of discrete-time coupled uncertain stochastic complex networks with missing measurements and time-varying delay. The parameter uncertainties are assumed to be norm-bounded and enter into both the network state and the network output. The stochastic Brownian motions affect not only the coupling term of the network but also the overall network dynamics. The nonlinear terms that satisfy the usual Lipschitz conditions exist in both the state and measurement equations. Through available output measurements described by a binary switching sequence that obeys a conditional probability distribution, we aim to design a state estimator to estimate the network states such that, for all admissible parameter uncertainties and time-varying delays, the dynamics of the estimation error is guaranteed to be globally exponentially stable in the mean square. By employing the Lyapunov functional method combined with the stochastic analysis approach, several delay-dependent criteria are established that ensure the existence of the desired estimator gains, and then the explicit expression of such estimator gains is characterized in terms of the solution to certain linear matrix inequalities (LMIs). Two numerical examples are exploited to illustrate the effectiveness of the proposed estimator design schemes.

This paper addressed the delay-dependent design problem for BAM neural networks with time-varying delays. By employing the integral inequality and constructing Lyapunov-Krasovskii functional, the delay-dependent linear matrix inequality (LMI) conditions are obtained to estimate the neuron states through available output measurements such that the dynamics of the estimation error is globally asymptotically stable. These criteria can be easily verified by utilizing the recently developed algorithms solving LMIs. A numerical example is provided to demonstrate the effectiveness of the proposed method.

This paper gives a novel delay-dependent admissibility condition of discrete-time singular systems with time-varying delays. For convenience, the time-varying delay is assumed to be the sum of delay lower bound and the integral multiples of a constant delay. Specially, if the constant delay is of unit length, the delay is an interval-like time-varying delay. The proposed admissibility condition is presented and expressed in terms of linear matrix inequality (LMI) by Lyapunov approach. Generally, the uncertainty of time-varying delay would lead to conservatism. In this paper, this critical issue is tackled by accurately estimating the time-varying delay. Consequently, the proposed admissibility condition is less conservative than the existing results, which is demonstrated by a numerical example.

In this paper, the synchronization stability problem for a class of general complex dynamical networks with interval time-varying coupling delay and delay in the dynamical node is investigated. By dividing the delay interval into two variable subintervals, slightly different Lyapunov---Krasovskii functionals are constructed on these two subintervals. Then several less conservative delay-dependent synchronization stability criteria are derived in terms of linear matrix inequality via reciprocally convex approach, which can be easily solved by using the standard numerical software. Numerical examples are given to illustrate the effectiveness and less conservatism of the proposed method.

The problem of exponentially synchronizing class of delayed neural networks is studied. Both constant and time-varying delays are considered, to obtain the delay-dependent state feedback synchronization gain matrix. By means of the method of Lyapunov–Krasovskii functional, combined with linear matrix inequalities, exponential synchronization of the master–slave structure of neural networks is achieved. The delay interval is decomposed into multiple nonequidistant subintervals, on which Lyapunov–Krasovskii functionals are constructed. On the basis of these functionals, a new exponential synchronization condition, one that is time-delay dependent, is proposed in terms of linear matrix inequalities. A numerical example showing the effectiveness of the proposed method is presented.

Load frequency control (LFC) requires transmission of remote measurements to the control center and of control signals from the control center to the plant. Constant delays exist in the dedicated communication channel and an open communication network introduces time-varying delays. Those delays would degrade the dynamic performance of LFC and even cause instability. This paper investigates the delay-dependent stability of the LFC scheme by adopting a delay-dependent criterion and linear matrix inequalities (LMIs). The maximal delay time which allows a power system with a LFC scheme embedded to retain stable is defined as the delay margin for stability analysis. A proportional-integral (PI) controller using the area control error (ACE) as input signal is employed in the LFC scheme. Special attention is paid to the relationship between the gains of PI controller and the delay margin. Case studies are carried out based on one-area and two-area LFC schemes to demonstrate the effectiveness of the criterion. Both constant and time-varying delays are considered. Moreover, the accuracy of the criterion used is also verified by simulation studies.

A robust control approach to stabilization of networked control systems with time-varying delays

This paper considers the problem on formation tracking control of second-order multi-agent systems (MASs) with communication time-varying delay. Sufficient conditions on the directed interaction topology and existence of the feedback gains to ensure the desired control are presented. Through choosing two augmented Lyapunov–Krasovskii (L–K) functionals and using some novel Wirtinger-based integral inequalities, the previously ignored information can be reconsidered and the application area of derived results can be greatly extended. Moreover, a novel constructive technique is given to compute out the controller gain by resorting to solving the achieved linear matrix inequalities (LMIs). Finally, a numerical example with comparisons and simulations is provided to illustrate the obtained results.