Discrete Time-delay Systems Research Articles

Model order reduction for discrete time-delay systems based on Laguerre function expansion

We investigate model order reduction for discrete time-delay systems via orthogonal polynomial expansion in the frequency domain. The transfer function of systems is expanded in the framework of Laguerre function basis. We show that Laguerre coefficients of the states satisfy a linear system and reduced models generated by projection methods can preserve some Laguerre coefficients of the original systems. We prove that the subspace spanned by Laguerre coefficients is exactly a high order Krylov subspace, thereby leading to an efficient computation of projection matrices and a more accurate coefficient-matching property in the two-sided framework. Further the Laguerre expansion of systems is employed to enable an approximate but fast execution of balanced truncation for discrete time-delay systems. Specifically, Gramians are approximated based on the derived Laguerre coefficients, which circumvents the main bottleneck of balanced truncation methods. Finally, two numerical examples are simulated to demonstrate the feasibility and effectiveness of the proposed methods.

Low-rank balanced truncation of discrete time-delay systems based on Laguerre expansions

This paper introduces a novel model order reduction method based on low-rank Gramian approximations for discrete time-delay systems. Firstly, an efficient algorithm based on Laguerre functions to compute the low-rank decomposition factors of the controllability and observability Gramians for discrete time-delay systems is given, in which the low-rank factors satisfy the iterative recursive formulas of the expansion coefficients of the Laguerre functions. It effectively avoids the direct solutions of Gramians. Then, the reduced-order model of the discrete time-delay system is obtained by combining the low-rank square root method. Additionally, a modified algorithm that combines the dominant subspace projection method is introduced, which alleviates certain drawbacks of the above technique and enhances the stability in certain cases. Finally, two numerical examples are given to verify the accuracy and efficiency of our proposed algorithm.

Nonlinear causal network learning via Granger causality based on extreme support vector regression.

For complex networked systems, based on the consideration of nonlinearity and causality, a novel general method of nonlinear causal network learning, termed extreme support vector regression Granger causality (ESVRGC), is proposed. The nonuniform time-delayed influence of the driving nodes on the target node is particularly considered. Then, the restricted model and the unrestricted model of Granger causality are, respectively, formulated based on extreme support vector regression, which uses the selected time-delayed components of system variables as the inputs of kernel functions. The nonlinear conditional Granger causality index is finally calculated to confirm the strength of a causal interaction. Generally, based on the simulation of a nonlinear vector autoregressive model and nonlinear discrete time-delayed dynamic systems, ESVRGC demonstrates better performance than other popular methods. Also, the validity and robustness of ESVRGC are also verified by the different cases of network types, sample sizes, noise intensities, and coupling strengths. Finally, the superiority of ESVRGC is successful verified by the experimental study on real benchmark datasets.

Delay-variation-dependent summation inequality and its application to stability analysis of discrete-time systems with time-varying delay

This work tends to research the stability of the discrete-time systems with a time-varying delay based on the Lyapunov-Krasovskii functional (LKF) method. In order to acquire a less conservative stability criterion, some techniques in this work are refined and taken into use. Firstly, to reduce the conservatism generated when estimating the forward difference of the LKF, a newly delay-variation-dependent summation inequality is constructed, which includes the existing free-matrix-based and Bessel function-based summation inequalities, using delay-variation-product relaxed matrices to provide more freedom for the estimation results. Secondly, to further show the influence of the introduced variation information in the time-varying delay, we give another selection of the allowable delay set for the delayed discrete-time systems. Thirdly, by taking advantages of the above method and by constructing a delay-product-type LKF, using extended free-weighting-matrices zero equations, and considering different allowable delay sets, two improved linear matrix inequality (LMI)-based delay-variation-dependent criteria for delayed discrete-time systems are formulated. Some classical numerical instances are presented to explain the effectiveness of these proposed stability criteria.

Global exponential stability of nonlinear delayed difference systems with Markovian switching

This work addresses the global exponential stability of nonlinear delayed discrete-time systems with Markovian switching. By a novel approach, some explicit criteria for the exponential stability in mean square of such systems are derived. An application to discrete-time neural networks is presented.

H∞ Stabilization of Discrete Time-Delayed Systems with Anti-windup Approach

H∞ Stabilization of Discrete Time-Delayed Systems with Anti-windup Approach

Regulator Problem for Linear Discrete-Time Delay Systems with Symmetrical Constraints

Regulator Problem for Linear Discrete-Time Delay Systems with Symmetrical Constraints

Criteria for finite-time stability of singular large-scale discrete-time delay systems

This paper concerns a problem of finite-time stability for a class of linear singular large-scale systems with delays. Based on matrix transformations, Lyapunov function method combined with new estimation techniques, we derive sufficient conditions for solving the finite-time stability of the system. A numerical example is given to illustrate the validity and effectiveness of the theoretical results.

Improved Stability and Passivity Results for Discrete Time-Delayed Systems with Saturation Nonlinearities and External Disturbances

Improved Stability and Passivity Results for Discrete Time-Delayed Systems with Saturation Nonlinearities and External Disturbances

Conditions of stability and guaranteed convergence rate for linear time-varying discrete-time delay systems

This work introduces novel results on the stability of linear time-varying discrete-time delay systems, addressing both internal stability and input-to-state stability. We start from the special case of positive delay systems and provide conditions of global exponential stability, both delay-dependent and delay-independent. Moreover, we formulate an explicit bound on the exponential decay rate of the state evolution, as a function of the maximum delay. We then show that the aforementioned stability conditions also prove input-to-state stability. The stability conditions consist of simple linear inequalities, albeit infinite-many due to the time-varying nature of the system. To overcome the computational burden, we also introduce a simple relaxation that allows to check the stability with a finite test, at the expense of conservatism. Finally, leveraging comparison systems and results from the theory of non-negative matrices, we generalize all the aforementioned results to systems with no sign constraint, for which we still provide stability conditions by means of linear inequalities and guaranteed convergence rate. Some consequences and a comparative example are provided for the well investigated special case of systems with constant matrices and time-varying delays.

Information Fusion Tracking Control for a Class of Discrete-Time Systems With Delays

In this article, the optimal tracking control problem for a class of discrete time-delay systems with disturbance is studied using the information fusion control methodology. First, the disturbance estimation is regarded as a minimization problem, and an information fusion disturbance observer (IFDO) is designed. Second, combining the soft constraint requirement of the output tracking error contained in the performance index function, a fusion filter is constructed to estimate the system co-state and its information weight under system state and input delays. Finally, the obtained co-state and its information weight, the soft constraint requirement of the control input contained in the performance function, as well as the system delay dynamics, are all transformed into the information equations about the control input, and then the information fusion delay controller (IFDC) is directly obtained by fusing these information equations. As a result, the negative effects caused by disturbance and time delays can be effectively compensated at the same time. The convergence of the tracking errors of the closed-loop system is guaranteed by the Lyapunov stability theory. Numerical simulation results have been provided to demonstrate the effectiveness and robustness of the proposed control algorithm.

An anti-windup control strategy for discrete time-delay system based on triple Lyapunov functional

This paper addresses the design of a dynamic output feedback-based anti-windup compensator to mitigate the windup phenomenon for a discrete time-varying delayed system with input saturation to establish the asymptotic stability. Linear matrix inequalities–based stability conditions are derived locally and globally. The controller for the closed-loop system is designed to reduce the effect of external bounded disturbances by utilizing the linear matrix inequality approach. The novel triple Lyapunov–Krasovskii functional along with reciprocal convex inequality is used to solve the expressions contained in the forward difference of the functional and to maximize the basin of attraction. Finally, industrial examples are simulated to prove the effectiveness of the proposed criterion.

Model reduction of discrete time-delay systems based on Charlier polynomials and high-order Krylov subspaces

In this paper, we present an efficient model reduction method for discrete time-delay systems based on the expansions of systems under Charlier polynomials. Making full use of the properties of Charlier polynomials and the structure of discrete time-delay systems, the projection space built by state variables is embedded in a high-order Krylov subspace. Further, a high-order Krylov subspace method is developed to generate discrete time-delay reduced systems. The proposed method is independent of the choice of inputs since the high-order Krylov subspace sequence does not involve the expansion coefficients of inputs. Besides, theoretical analysis shows that the resulting discrete time-delay reduced system characterizes the property of invariable coefficients. Finally, two numerical examples demonstrate the effectiveness of the proposed method.

Robust stability and passivity analysis for discrete-time neural networks with mixed time-varying delays via a new summation inequality

<abstract><p>The summation inequality is essential in creating delay-dependent criteria for discrete-time systems with time-varying delays and developing other delay-dependent standards. This paper uses our rebuilt summation inequality to investigate the robust stability analysis issue for discrete-time neural networks that incorporate interval time-varying leakage and discrete and distributed delays. It is a novelty of this study to consider a new inequality, which makes it less conservative than the well-known Jensen inequality, and use it in the context of discrete-time delay systems. Further stability and passivity criteria are obtained in terms of linear matrix inequalities (LMIs) using the Lyapunov-Krasovskii stability theory, coefficient matrix decomposition technique, mobilization of zero equation, mixed model transformation, and reciprocally convex combination. With the assistance of the LMI Control toolbox in Matlab, numerical examples are provided to demonstrate the validity and efficiency of the theoretical findings of this research.</p></abstract>

Deep-learning based design of cascade observers for discrete-time nonlinear systems with output delay

Looking for a proper Lyapunov-based stability analysis for nonlinear discrete-time delay systems is non-trivial, even for constant delay. This paper proposes a learning approach that relies on neural networks to compute the state transformation map involved in the design of Kazantzis-Kravaris-Luenberger (KKL) based observers. Based on this transformation, we propose a chain of observers in cascade with exponential stability guarantees through the inverse transformation mapping in the original coordinates. Finally, we provide a simulation example that includes a sensitivity analysis of the delay parameter to illustrate the performance and demonstrate the effectiveness of the proposed learning-based design of chain observers for discrete-time nonlinear systems with output measurement delay.

Non-fragile $$H_{\infty }$$ Filtering for Discrete Time-Delay Systems with Quantization, TOD Protocol and Deception Attacks

Non-fragile $$H_{\infty }$$ Filtering for Discrete Time-Delay Systems with Quantization, TOD Protocol and Deception Attacks

Event-triggered control for discrete-time delay systems

This study focuses on event-triggered control of nonlinear discrete-time systems with time delays. Based on a Lyapunov–Krasovskii type input-to-state stability result, we propose a novel event-triggered control algorithm that works as follows. The control inputs are updated only when a certain measurement error surpasses a dynamical threshold depending on both the system states and the evolution time. Sufficient conditions are established to ensure that the closed-loop system maintains its asymptotic stability. It is shown that the time-dependent portion in the dynamical threshold is essential to derive the lower bound of the times between two consecutive control updates. As a special case of our results, we demonstrate the performance of the designed event-triggering algorithm for a class of linear control systems with time delays. Numerical simulations are provided to demonstrate the effectiveness of our algorithm and theoretical results.

Discrete-time delay systems: part 2. Sub-fully actuated case

Discrete-time delay systems: part 2. Sub-fully actuated case

Balanced truncation for discrete time-delay systems via the interpretation of system energy

In this paper, the balanced truncation method is investigated for discrete time-delay systems. We show that the energy associated with the system controllability and observability can be characterized via the delay Lyapunov matrices, similar to the case of continuous time-delay systems. Then, we balance the system via a coordinate transformation in order to retain the delay structure of systems naturally. In this way, the balanced truncation method is conducted to obtain structure-preserving reduced models. Further, we provide an efficient process to compute a low-rank approximation to delay Lyapunov matrices based on the equivalent expression of discrete time-delay systems, which enables an approximate but fast execution of the proposed method. The stability of reduced models is also discussed in the paper. Finally, numerical examples are simulated to verify the feasibility and efficiency of the proposed method.

Discrete-time delay systems: part 1. Global fully actuated case

Discrete-time delay systems: part 1. Global fully actuated case