Time-varying Discrete-time Systems Research Articles

Distributed Exponential State Estimation for Discrete-Time Linear Systems Over Jointly Connected Switching Networks

The existing results on distributed state estimation problems for discrete-time linear systems were obtained over connected static networks, or every-time connected switching networks by a two-time-scale design. This paper further studies the same problem over jointly connected switching networks which can be disconnected at every time instant by a single-time-scale distributed design. Since the existing distributed designs critically rely on the connectedness assumption on the networks and thus may not apply to jointly connected switching networks, we have to modify the local observers and develop a novel approach which makes use of the uniform complete observability property for a discrete-time time-varying system. Additionally, we establish two exponential stability results for two classes of discrete-time switched systems, which may be of independent interest.

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.

An Optimization Framework for Resilient Batch Estimation in Cyber-Physical Systems

This article proposes a class of resilient state estimators for linear time-varying discrete-time systems. The dynamic equation of the system is assumed to be affected by a bounded process noise. As to the available measurements, they are potentially corrupted by a noise of both dense and impulsive natures. The latter, in addition to being arbitrary in its form, need not be strictly bounded. In this setting, we construct the estimator as the set-valued map, which associates with the measurements the minimizing set of some appropriate performance functions. We consider a family of such performance functions, each of which yielding a specific instance of the proposed general estimation framework. It is then shown that the proposed class of estimators enjoys the property of resilience, i.e., it induces an estimation error, which, under certain conditions, is independent of the extreme values of the (impulsive) measurement noise. Hence, the estimation error may be bounded, while the measurement noise is virtually unbounded. Moreover, we provide several error bounds (in different configurations), whose expressions depend explicitly on the degree of observability of the system being observed and on the considered performance function. Finally, a few simulation results are provided to illustrate the resilience property.

Input–Output Finite-Time Stabilization of Linear Time-Varying Discrete-Time Systems

This article deals with the input–output finite-time stability (IO-FTS) of linear discrete-time systems. In current control science, discrete-time systems play a very important role in many engineering contexts. Moreover, many discrete-time control problems are defined over a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">finite</i> -interval of time; therefore, the development and application of finite-time control methodologies is of particular relevance. The first contribution of this article is a pair of necessary and sufficient conditions for IO-FTS. The former involves the solution of a set of generalized difference Lyapunov equations; the latter allows one to establish the feasibility of an optimization problem by solving a set of difference linear matrix inequalities (DLMIs). The second contribution of this article is a theorem for IO finite-time stabilization via state feedback, followed by a more general one for stabilization via output feedback. Both conditions are necessary and sufficient, and lead to optimization problems cast in the form of DLMIs. The applicability of the devised results is illustrated through a numerical example.

Input-to-State Stability Criteria of Discrete-Time Time-Varying Impulsive Switched Delayed Systems With Applications to Multi-Agent Systems

This paper considers weakly and strongly input-to-state stability (ISS) of a discrete-time time-varying delayed system with synchronous impulses and switches. By employing the uniformly asymptotically stable function and mode-dependent dwell time conditions, ISS criteria are derived using the Krasovskii and Razumikhin techniques. The time differences of the Krasovskii functionals or the Razumikhin functions can be mode-dependent and time-varying, which release constraints of some existing results. As subsequent results, we apply the derived ISS criteria to the multi-agent systems (MASs). Finally, two examples considering the quasi-consensus of MASs are given to verify the established results.

Input-to-state stability of discrete-time time-varying impulsive delay systems

This paper investigates the input-to-state stability (ISS) of nonlinear discrete-time time-varying impulsive delay systems with the coexistence of stabilising and destabilising impulses. Some ISS criteria are established for discrete-time impulsive delay systems by employing the Lyapunov functional where the coefficient of the estimated upper bound of the time difference of the Lyapunov functional is allowed to be a time-varying function which can take both negative and positive values. In addition, the ISS of a class of discrete-time linear time-varying impulsive delay systems is discussed by utilising a non-quadratic Lyapunov functional. Numerical examples are presented to verify the effectiveness of the main results.

Sufficient conditions for assignability of nonuniform dichotomy spectrum of discrete time-varying linear systems

We consider a version of the pole placement problem for tempered one-sided linear discrete-time time-varying linear systems. We prove a sufficient condition for assignability of the nonuniform dichotomy spectrum by linear feedback. The main result is that the nonuniform dichotomy spectrum is assignable if the system is completely controllable and certain lower asymptotic bound for the controllability Gramian holds.

Global Stability of Positive Discrete-time Time-varying Nonlinear Feedback Systems

The global stability of positive discrete-time time-varying nonlinear systems with time-varying scalar feedbacks is investigated. Sufficient conditions for the asymptotic stability of discrete-time positive time-varying linear systems are given. The new conditions are applied to discrete-time positive time-varying nonlinear systems with time-varying feedbacks. Sufficient conditions are established for the global stability of the discrete-time positive time-varying nonlinear systems with feedbacks.

On Sufficient Optimality Conditions for a Guaranteed Control in the Speed Problem for a Linear Time-Varying Discrete-Time System with Bounded Control

On Sufficient Optimality Conditions for a Guaranteed Control in the Speed Problem for a Linear Time-Varying Discrete-Time System with Bounded Control

The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

Multirate control strategies for avoiding sample losses. Application to UGV path tracking

When in a digital control strategy there are samples lost due to limitations, different multirate (MR) control options can be used for solving the problem: Dual-rate inferential control (IC) and model-based dual-rate control (MBDR). The objective of this contribution is to analyze, compare, and to assess their behavior under different perspectives. Is a dual-rate inferential control better than a model-based dual-rate control? Both options lead to a periodically time-varying discrete-time system and for this reason a lifted modeling is considered. An efficient algorithm is used for computing a MR system’s frequency response for these control structures. The robust performance and disturbance effects are studied in detail under sample losses and process uncertainty, and some considerations are reported. A new QFT (quantitative feedback theory) procedure for dual-rate systems analysis is also described. Analysis and simulation examples and experimental results for UGV path tracking are introduced in this work, revealing that MBDR outperforms IC when the model contains important uncertainties.

Equivalent conditions of finite-time time-varying output-feedback control for discrete-time positive time-varying linear systems

This paper studies the issue of finite-time time-varying output-feedback control for positive time-varying discrete-time linear systems. Finite-time stability for the systems with positivity is defined. To make sure that the system can be finite-time stable with positivity, an analysis condition is established first, and then two conditions for solving the static output-feedback controller are derived in this work where all the obtained results are necessary and sufficient conditions. An iterative algorithm for solving the controller is developed and finally, an example is given in the simulation to verify the effectiveness of our results and algorithm.

QCs characterization of robust stability with simultaneous uncertainties in plant and controller

Within the frame work of nest algebra, the robust stability of feedback interconnections of time-varying discrete-time systems with combined uncertainties in the plant and controller is derived via quadratic constraint (QC) approach, where the plant and controller are taken from two path connected uncertainty sets with respect to the topology induced by the gap metric. Some sufficient conditions for the existence of complementary QCs for the components of stable uncertain feedback systems are derived. The path-connectedness is established for some uncertainty sets with particular properties. The fundamental robust stability result represents a generalization of the input–output operator robust stability theorem of uncertainties appearing in the plant alone, to include the case of simultaneous uncertainties appearing in the controller.

Optimal control using adaptive resonance theory and Q-learning

Motivated by recent advancement in neurocognitive in brain modeling research, a multiple model-based Q-learning structure is proposed for optimal tracking control problem of time-varying discrete-time systems. This is achieved by utilizing a multiple-model scheme combined with adaptive resonance theory (ART). The ART algorithm generates sub-models based on the match-based clustering method. A responsibility signal governs the likelihood of contribution of each sub-model to the Q-function. The Q-function is learned using the batch least-square algorithm. Simulation results are added to show the performance and the effectiveness of the overall proposed control method.

Adaptive dynamic programming for finite-horizon optimal control of linear time-varying discrete-time systems

This paper studies data-driven learning-based methods for the finite-horizon optimal control of linear time-varying discrete-time systems. First, a novel finite-horizon Policy Iteration (PI) method for linear time-varying discrete-time systems is presented. Its connections with existing infinite-horizon PI methods are discussed. Then, both data-driven off-policy PI and Value Iteration (VI) algorithms are derived to find approximate optimal controllers when the system dynamics is completely unknown. Under mild conditions, the proposed data-driven off-policy algorithms converge to the optimal solution. Finally, the effectiveness and feasibility of the developed methods are validated by a practical example of spacecraft attitude control.

Interval State Estimator Design Using the Observability Matrix for Multiple Input Multiple Output Linear Time-Varying Discrete-Time Systems

In this paper, a novel numerical scheme to set-membership interval state estimator design is proposed for the multiple-input-multiple-output (MIMO) linear time-varying (LTV) discrete-time systems using systems observability matrix and its past input/output values. The proposed method is more simple and efficient. First, an interval state estimator is designed that will generate a tight interval vector for the real state vector in a guaranteed way by employing interval analysis and consistency techniques for the single-input-single-output (SISO) systems. The proposed interval state estimator technique is then extended easily to the MIMO systems. Secondly, the estimation errors dynamics bounds are computed a-priori to measurements for the unknown but bounded uncertainties. Finally, the convergence of the width of the interval state vector towards a known value in finite time is provided to prove the boundedness of the interval state vector and estimation error that further quantify the accuracy of the developed technique. The performance and comparison with already existing techniques are highlighted through numerical examples.

Remarks on Generalized Stability for Difference Equations in Banach Spaces

This note focuses on the problem of generalized exponential stability for linear time-varying discrete-time systems in Banach spaces. Characterizations for this concept are presented and connections with the classical concept of exponential stability existent in the literature is pointed out. An illustrative example clarifies the implications between these concepts.

MTN Optimal Tracking Control of SISO Nonlinear Time-Varying Discrete-Time Systems without Mechanism Models

Nonlinear time-varying systems without mechanism models are common in application. They cannot be controlled directly by the traditional control methods based on precise mathematical models. Intelligent control is unsuitable for real-time control due to its computation complexity. For that sake, a multidimensional Taylor network (MTN) based output tracking control scheme, which consists of two MTNs, one as an identifier and the other as a controller, is proposed for SISO nonlinear time-varying discrete-time systems with no mechanism models. A MTN identifier is constructed to build the offline model of the system, and a set of initial parameters for online learning of the identifier is obtained. Then, an ideal output signal is selected relative to the given reference signal. Based on the system identification model, Pontryagin minimum principle is introduced to obtain the numerical solution of the optimal control law for the system relative to the given ideal output signal, with the corresponding optimal output taken as the desired output signal. A MTN controller is generated automatically to fit the numerical solution of the optimal control law using the conjugate gradient (CG) method, and a set of initial parameters for online learning of the controller is obtained. An adaptive back propagation (BP) algorithm is developed to adjust the parameters of the identifier and controller in real time, and the convergence for the proposed learning algorithm is verified. Simulation results show that the proposed scheme is valid.

Output feedback preview tracking control for discrete-time polytopic time-varying systems

ABSTRACTThis paper focuses on the problem of static output feedback preview tracking control for discrete-time systems with time-varying parameters subject to a previewable reference signal. We develop a design method of a robust controller with integral and preview actions achieving robust tracking performance. First, an augmented error system including future information on reference signal is constructed by introducing two new related auxiliary variables to the original system state and input. This leads to a tracking problem being transformed into a regulator problem. Then, a previewable reference signal is fully utilised through reformulation of the output equation for the augmented error system while considering a static output feedback. Meanwhile, the static output preview control gains are solved explicitly by the proposed conditions. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.

Improved Razumikhin and Krasovskii approaches for discrete-time time-varying time-delay systems

This paper studies stability of discrete-time time-varying time-delay systems. The existing Razumikhin and Krasovskii stability approaches for this class of systems are improved in the sense that the time-shifts of the Razumikhin functions and Krasovskii functionals are allowed to take both negative and positive values. Three kinds of stability concepts, say, uniform stability, uniformly asymptotic stability and uniformly exponential stability, are considered. The improvements of the Razumikhin and Krasovskii approaches are achieved by using the concept of uniformly asymptotically stable (UAS) function, the notion of overshoot associated with the UAS function and an improved comparison lemma. Both delay-dependent and delay-independent stability theorems are obtained for a class of discrete-time linear time-delay systems by using the improved Razumikhin and Krasovskii stability approaches. Numerical examples demonstrate the effectiveness of the proposed methods.