Discrete Time-delay Systems Research Articles

Non-fragile [formula omitted] filtering for nonlinear discrete-time delay systems with randomly occurring gain variations

In this paper,the problem of H∞ filtering for a class of nonlinear discrete-time delay systems is investigated. The time delay is assumed to be belonging to a given interval, and the designed filter includes additive gain variations which are supposed to be random and satisfy the Bernoulli distribution. By the augmented Lyapunov functional approach, a sufficient condition is developed to ensure that the filtering error system is asymptotically mean-square stable with a prescribed H∞ performance. In addition, an improved result of H∞ filtering for linear system is also derived. The filter parameters are obtained by solving a set of linear matrix inequalities. For nonlinear systems, the applicability of the developed filtering result is confirmed by a longitudinal flight system, and an additional example for linear system is presented to demonstrate the less conservativeness of the proposed design method.

Periodic event-triggered state-feedback control for discrete-time linear systems

This paper investigates the periodic event-triggered control for discrete-time linear systems. The discrete-time plant denotes a combination of the continuous-time controlled plant, the sensors and the actuator. A general packaged component, event machine, is introduced to conduct the detection and judgement of the events. Then the periodic event-triggered control for the discrete-time system including a discrete-time plant, an event machine, a transmitter and a continuous static controller, exhibits some different behaviors from that for the continuous-time system. Especially, in this system, the sampling sequences and the event-verifying sequences may be independent from each other. Furthermore, according to the difference between the two sequences, the periodic event-triggered control system is classified into three cases, and the corresponding models are established for the three cases: a discrete-time perturbed linear system, a discrete time-delay system and a switched system, respectively. Moreover, the stability conditions for the three cases are proposed. Finally, a numerical example is given to illustrate the efficiency and feasibility of the obtained results.

Stability analysis of a general family of nonlinear positive discrete time-delay systems

ABSTRACTIn this paper, we propose a new approach to analyse the stability of a general family of nonlinear positive discrete time-delay systems. First, we introduce a new class of nonlinear positive discrete time-delay systems, which generalises some existing discrete time-delay systems. Second, through a new technique that relies on the comparison and mathematical induction method, we establish explicit criteria for stability and instability of the systems. Three numerical examples are given to illustrate the feasibility of the obtained results.

Output regulation problem for discrete-time linear time-delay systems by output feedback control

In this paper, we study the output regulation problem of discrete linear time-delay systems by output feedback control. We have established some results parallel to those for the output regulation problem of continuous linear time-delay systems.

Robust admissibility and admissibilisation of uncertain discrete singular time-delay systems

abstractThis paper is concerned with the characterisation of robust admissibility and admissibilisation for uncertain discrete-time singular system with interval time-varying delay. Considering the norm-bounded uncertainty and the interval time-varying delay, a new comparison model is introduced to transform the original singular system into two connected subsystems. After this transformation, a singular system without uncertainty and delay can be handled by the Lyapunov–Krasovskii functional method. By virtue of the scaled small gain theorem, an admissibility condition of the original singular system is proposed in terms of linear matrix inequalities. Moreover, the problem of robust admissibilisation of uncertain discrete singular time-varying system is also studied by iterative linear matrix inequality algorithm with initial condition optimisation. Several numerical examples are used to illustrate that the results are less conservative than existing ones.

Linear functional observers with guaranteed ε-convergence for discrete time-delay systems with input/output disturbances

ABSTRACTThe problem of designing linear functional observers for discrete time-delay systems with unknown-but-bounded disturbances in both the plant and the output is considered for the first time in this paper. A novel approach to design a minimum-order observer is proposed to guarantee that the observer error is ε-convergent, which means that the estimate converges robustly within an ε-bound of the true state. Conditions for the existence of this observer are first derived. Then, by utilising an extended Lyapunov–Krasovskii functional and the free-weighting matrix technique, a sufficient condition for ε-convergence of the observer error system is given. This condition is presented in terms of linear matrix inequalities with two parameters needed to be tuned, so that it can be efficiently solved by incorporating a two-dimensional search method into convex optimisation algorithms to obtain the smallest possible value for ε. Three numerical examples, including the well-known single-link flexible joint robotic system, are given to illustrate the feasibility and effectiveness of our results.

Stability analysis and controller design of discrete-time polynomial fuzzy time-varying delay systems

This paper investigates the stability and stabilization problems for discrete-time polynomial systems with time-varying delay via a sum of squares (SOS) approach. Unlike the most existing literature, the delayed state is also considered in the controller design. By constructing a new parameter-dependent Lyapunov–Krasovskii function and introducing some free weighting matrices, some novel delay-dependent stability and stabilization conditions are obtained. Then, the results are extended to discrete time-delay systems with norm-bounded uncertainties. The conditions proposed are all derived in terms of SOS, which are more effective than the LMI method in some cases. All the results can be symbolically and numerically solved via the recently developed SOSTOOLS and a semidefinite-program (SDP) solver, respectively. Three numerical examples are provided to demonstrate the effectiveness and superiority of the proposed analysis and design methods.

H∞ filtering for uncertain systems with time-delay and randomly occurred sensor nonlinearities

This paper investigates the H∞ filtering problem for discrete time-delay systems with quantization and stochastic sensor nonlinearity. The problem of quantization considered in this paper is logarithmic quantization and the quantization error is processed into an uncertain term. The sensor nonlinearity is supposed to occur randomly on the basis of a stochastic variable obeying the Bernoulli distribution and the nonlinear decomposed technique is introduced to deal with the nonlinear term. The time-varying delay of systems is processed by using the Scaled Small Gain (SSG) theorem. The LMI-based sufficient conditions in view of the SSG and the Lyapunov–Krasovskii functional (LKF) approach are presented to guarantee the error system mean square stable and with the prescribed H∞ performance index γ. An example is given to illustrate the effectiveness of the proposed method.

Finite-Time Stability and H ∞ Control of Linear Discrete-Time Delay Systems with Norm-Bounded Disturbances

This paper deals with the finite-time stability and H∞ control of linear discrete-time delay systems. The system under consideration is subject to interval time-varying delay and norm-bounded disturbances. Linear matrix inequality approach is used to solve the finite-time stability problem. First, new sufficient conditions are established for robust finite-time stability of the linear discrete-time delay system with norm-bounded disturbances, then the state feedback controller is designed to robustly finite-time stabilize the system and guarantee an adequate level of system performance. The delay-dependent sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed results.

Reliable finite-time H∞ filtering for discrete time-delay systems with Markovian jump and randomly occurring nonlinearities

This paper is concerned with the problem of reliable finite-time H∞ filtering for discrete time-varying delay systems with Markovian jump and randomly occurring nonlinearities (RONs). RONs are introduced to model a class of sector-like nonlinearities that occur in a probabilistic way according to a two-dimensional discrete random variables joint distribution. The failures of sensors are quantified by a variable varying in a given interval. The time-varying delay is unknown with given lower and upper bounds. Firstly, the filtering errr dynamic system is constructed based on an H∞ filter, sufficient criteria are provided to guarantee that the resulting filtering error system is stochastic finite-time boundedness and stochastic finite-time H∞ filtering in both normal and fault cases. The gain matrices of the controller and filter are achieved by solving a feasibility problem in terms of linear matrix inequalities with a fixed parameter, respectively. Finally, numerical examples are given to demonstrate the effectiveness of the proposed design approach.

Abel lemma-based finite-sum inequality and its application to stability analysis for linear discrete time-delay systems

This paper is concerned with stability of linear discrete time-delay systems. Note that a tighter estimation on a finite-sum term appearing in the forward difference of some Lyapunov functional leads to a less conservative delay-dependent stability criterion. By using Abel lemma, a novel finite-sum inequality is established, which can provide a tighter estimation than the ones in the literature for the finite-sum term. Applying this Abel lemma-based finite-sum inequality, a stability criterion for linear discrete time-delay systems is derived. It is shown through numerical examples that the stability criterion can provide a larger admissible maximum upper bound than stability criteria using a Jensen-type inequality approach and a free-weighting matrix approach.

New Results for Robust Stability of Discrete Bilinear Uncertain Time-Delay Systems

In this paper, new results for the problem of the robust stability of discrete homogeneous bilinear time-delay systems subjected to nonlinear or parametric uncertainties are addressed. Applying a concise upper bound to the Lyapunov equation approach and then associating some linear algebraic techniques, several delay-independent conditions are presented to assure the robust stability of the aforementioned systems. One of the features of the present criteria is that they are independent of any Lyapunov equation, although the Lyapunov equation approach is adopted. Comparing to an existing work, it is shown that the obtained results are sharper. Finally, all obtained results are applied to solve the same problem of discrete bilinear uncertain systems and discrete time-delay systems with/without uncertainties. Concise criteria for the robust stability of the mentioned systems are developed, and comparisons between the obtained results and those appeared in the literature are also made. It is shown that all obtained results in this work are either tighter or more concise.

Further results on stability analysis of discrete-time systems with time-varying delays via the use of novel convex combination coefficients

This paper focuses on deriving an improved stability criterion for discrete time-delay systems via a Lyapunov–Krasovskii functional approach that takes advantage of triple summation terms. To this end, novel convex combination coefficients coupled with second-order time-varying delay terms are introduced, and the corresponding reciprocally convex approach is established with the use of the minimal number of slack matrix variables. Finally, three numerical examples are provided to illustrate the effectiveness of the proposed method, with particular regard to robustness and H∞ performance.

Stability of Discrete-Time Systems With Time-Varying Delays via a Novel Summation Inequality

This technical note is concerned with the stability analysis of discrete linear systems with time-varying delays. The novelty of the technical note comes from the consideration of a new inequality which is less conservative than the celebrated Jensen inequality employed in the context of discrete-time delay systems. This inequality is a discrete-time counterpart of the Wirtinger-based integral inequality that was recently employed for the improved analysis of continuous-tine systems with delays. However, differently from the continuous-time case, the proof of the new inequality is not based on the Wirtinger inequality. The method is also combined with an efficient representation of the improved reciprocally convex combination inequality in order to reduce the conservatism induced by the LMIs optimization setup. The effectiveness of the proposed result is illustrated by some classical examples from the literature.

Nonlinear control systems theory in Romania and the influence of the contributions of V. A. Yakubovich

Theory of automatic control started in Romania as an essentially linear theory based on transfer functions (input/output representations). The introduction of the nonlinear theory started in the late 50ies of the past century by some research papers due to A. Halanay and V. M. Popov -with special reference to absolute stability. But only after the discovery by V.M. Popov of the famous frequency domain absolute stability inequality it became possible to connect his method to the classical ones which relied on Liapunov functions: this led to the assimilation of the results of V.A. Yakubovich on matrix inequalities which grew in the Yakubovich Kalman Popov lemma and the positiveness theory. The positiveness theory at its turn is strongly connected with the hyperstability theory and thus, with dissipativeness and passivity. The next decades witnessed applications to forced oscillations, generalization of the positiveness theory to the indefinite sign case, the start of the rigorous theory for adaptive systems -with some roots in Romania also -and the extension to discrete time and time delay systems. Consideration of the periodic discrete time systems witnessed a return to the book of Yakubovich and Starzhinskii on systems with periodic coefficients and its first extensions to the discrete time case, including some results on discrete time parametric resonance. Worth mentioning that the influence of V.A. Yakubovich generated the opening to the representatives of his school and, in general, to the Sankt Petesburg University school of differential equations and qualitative control theory.

Auxiliary Function-based Summation Inequalities for Quadratic Functions and their Application to Discrete-time Delay Systems

Jensen inequality has become a powerful tool of supporting summation inequalities for quadratic functions in order to obtain stability criteria for time-delayed systems since it achieves remarkable performance with a small number of decision variables. This paper suggests a new summation inequality for quadratic functions based on an auxiliary function, which is superior to the Jensen inequality. To demonstrate the superiority of the new inequality, its application to stability analysis for discrete-time delay system is provided with a simple numerical example.

On finite time delay dependent stability of linear discrete delay systems: Numerical solution approach

In this paper, a possible solution of the basic nonlinear quadratic matrix equation was proposed. The solution is crucial in the formulation of the particular criteria for the delay-dependent finite time stability of discrete time delay systems represented as x(k+1)=A0(k)+A1x(k-h). The time delay-dependent criteria have been derived. In addition, the significance of the nonlinear discrete polynomial matrix equation is explained. With the use of the mathematical formalism based on the Traub and Bernoulli's algorithms, it was concluded that the computation of the dominant solvent of the matrix polynomial equation does not guarantee a necessary convergence in all cases, unlike in the traditional numerical procedures. In this paper, we presented one particular and one general solution valid in the case when the discrete matrix equation was presented in its factorial form. The numerical computations are performed to illustrate the suggested results.

Observer-Based Bounded Control for Discrete Time-Delay Uncertain Nonlinear Systems

A bounded controller is proposed for a class of uncertain discrete time-delay systems with nonlinearity and disturbance based on state estimator and disturbance observer technique. A state estimator is developed to estimate the unmeasured system state vector. Suppose that the disturbance is generated by an exogenous system; a disturbance observer is designed to estimate the unknown disturbance. The parameters of the state estimator and the disturbance observer are calculated by solving linear matrix inequalities (LMIs). By applying the outputs of the state estimator and the disturbance observer, the sufficient condition for the existence of the bounded controller is derived based on an appropriate Lyapunov function candidate. Under the developed bounded controller, the stability of the closed-loop system can be guaranteed. Simulation examples are provided to show the effectiveness of the proposed bounded control scheme.

Robust H∞ filtering for discrete nonlinear delayed stochastic systems with missing measurements and randomly occurring nonlinearities

In this paper, we are concerned with the robust filtering problem for a class of nonlinear discrete time-delay stochastic systems. The system under consideration involves parameter uncertainties, stochastic disturbances, time-varying delays and sector nonlinearities. Both missing measurements and randomly occurring nonlinearities are described via the binary switching sequences satisfying a conditional probability distribution, and the nonlinearities are assumed to be sector bounded. The problem addressed is the design of a full-order filter such that, for all admissible uncertainties, nonlinearities and time-delays, the dynamics of the filtering error is constrained to be robustly exponentially stable in the mean square, and a prescribed disturbance rejection attenuation level is also guaranteed. By using the Lyapunov stability theory and some new techniques, sufficient conditions are first established to ensure the existence of the desired filtering parameters. Then, the explicit expression of the desired filter gains is described in terms of the solution to a linear matrix inequality. Finally, a numerical example is exploited to show the usefulness of the results derived.

STABILITY ANALYSIS OF SWITCHED DISCRETE TIME-DELAY SYSTEMS WITH CONVEX POLYTOPIC UNCERTAINTIES

This article is concerned with robust stability of switched discrete time-delay systems with convex polytopic uncertainties. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for robust stability for switched system with convex polytopic uncertainties is designed via linear matrix inequalities. AMS Subject Classification: 47N10, 93C55, 93D20, 94C10