Uncertain Discrete-time Stochastic Systems Research Articles

Pareto-based guaranteed cost control of discrete-time uncertain stochastic systems

The Pareto-based guaranteed cost control (GCC) of discrete-time uncertain stochastic systems in the infinite horizon is studied. Firstly, the convexity of the weighted sum objective function is proved. Based on this convexity, we demonstrate the relationship between Pareto optimality and minimization of the weighted sum objective function. This relationship confirms that the Pareto-based GCC strategy can be obtained by optimizing the weighted sum objective function. Secondly, to address the Pareto-based GCC issue for discrete-time uncertain stochastic systems, we derive the generalized algebraic Riccati inequality (GARI) specific to these systems. Thirdly, the necessary condition for a Pareto-based guaranteed cost controller is derived using the Karush–Kuhn–Tucker (KKT) condition. Lastly, we introduce a method based on linear matrix inequality (LMI) to determine the Pareto-based GCC strategy. This method reduces computational complexity and establishes the sufficient conditions for a Pareto-based guaranteed cost controller. To validate our findings, we provide a example implemented in MATLAB, confirming the accuracy of the proposed approach.

Robust Finite-Time Stability for Uncertain Discrete-Time Stochastic Nonlinear Systems with Time-Varying Delay.

The main concern of this paper is finite-time stability (FTS) for uncertain discrete-time stochastic nonlinear systems (DSNSs) with time-varying delay (TVD) and multiplicative noise. First, a Lyapunov–Krasovskii function (LKF) is constructed, using the forward difference, and less conservative stability criteria are obtained. By solving a series of linear matrix inequalities (LMIs), some sufficient conditions for FTS of the stochastic system are found. Moreover, FTS is presented for a stochastic nominal system. Lastly, the validity and improvement of the proposed methods are shown with two simulation examples.

Finite-time Stochastic Stability and Stabilization for Uncertain Discrete-time Stochastic Systems with Time-varying Delay

This paper deals with the problems of finite-time stochastic stability and stabilization for discrete-time stochastic systems with parametric uncertainties and time-varying delay. Using the Lyapunov-Krasovskii functional method, some sufficient conditions of finite-time stochastic stability for a class of discrete-time stochastic uncertain systems are established in term of matrix inequalities. Then, a new criterion is proposed to ensure the closed-loop system is finite-time stochastically stable. The controller gain is designed. Finally, two numerical examples are given to illustrate the effectiveness of the proposed results.

Mean square consensus of uncertain discrete-time stochastic multi-agent systems with x-dependent noise

ABSTRACTIn this paper, the mean square consensus control problem is investigated for linear uncertain discrete-time multi-agent systems with x-dependent noise and time-varying delays. Under undirected connected topology, the robust consensus problem of multi-agent systems is converted into the robust stabilisation problem for the discrete-time stochastic systems. By utilising Lyapunov functional and the linear matrix inequality method, some new sufficient conditions are derived to guarantee the consensus of uncertain discrete-time stochastic multi-agent systems. Based on the state feedback controller protocol with time-varying delays, a new consensus criterion is established for the discrete-time stochastic multi-agent systems with parameter uncertainties. Finally, numerical examples are provided to illustrate the effectiveness of the proposed results.

Robust H∞ Control of Uncertain Stochastic Systems with Time-varying Interval Delays

A robust delay-dependent stability criterion for discrete-time uncertain stochastic systems is proposed to achieve asymptotical stability and H∞ performance in this paper. Based on modeling approaches, Linear Parameter Varying (LPV) system with multiplicative noise term is built to represent uncertain stochastic systems. Moreover, state and input delays are considered as two individual time-varying interval cases for general effect. Employing a novel Lyapunov-Krasovskii function, Jensen inequality and transform technology, some relaxed sufficient conditions are derived into Linear Matrix Inequality (LMI) forms to apply convex optimization algorithm. Through solving the derived conditions, Gain-Scheduled (GS) controller can be designed such that robust asymptotical stability and H∞ performance of closed-loop system are achieved in the mean square. At last, two numerical examples are provided to demonstrate applicability and effectiveness of this paper.

Robust Finite-Time <i>H</i><sub>∞</sub> Filtering for Discrete-Time Markov Jump Stochastic Systems

This study is concerned with the problem of finite-time H∞ filter design for uncertain discrete-time Markov Jump stochastic systems. Our attention is focused on the design of mode-dependent H∞ filter to ensure the finite-time stability of the filtering error system and preserve a prescribed H∞ performance level for all admissible uncertainties. Sufficient conditions of filtering design for the system under consideration are developed and the corresponding filter parameters can be achieved in terms of linear matrix inequalities (LMI). Finally, a numerical example is provided to illustrate the validity of the proposed method.

Simultaneous fault detection and control for uncertain discrete-time stochastic systems with limited communication

In the paper, the simultaneous fault detection and control (SFDC) problem is concerned for uncertain discrete-time stochastic systems with limited communication. An integrated module with a filter and a dynamic output feedback controller is designed to achieve the desired fault detection (FD) and control objectives, simultaneously. The event-triggered technology is employed to save the network communication resources, while two event detectors are utilized to decrease the amount of data transmitted from the sensor to the SFDC module and from the SFDC module to the plant. A novel method is proposed to ensure that the obtained closed-loop model is robustly stochastically stable and satisfies the desired detection and control performances. Sufficient conditions are derived to obtain the parameters of filter, controller and event detectors. Finally, the validity of the design strategy is verified by a simulation case.

Delay-Dependent Robust Control for Discrete-Time Uncertain Stochastic Systems With Time-Varying Delays

This paper investigates a delay-dependent robust control problem of discrete-time uncertain stochastic systems with delays. The uncertainty considered in this paper is time-varying but norm-bounded, and the delays are considered as interval time-varying case for both state and input. According to the considerations of uncertainty, stochastic behavior, and time delays, the problem considered in this paper is more general than the existing works for uncertain stochastic systems. Via the proposed Lyapunov–Krasovskii function, some sufficient conditions are derived into the extended linear matrix inequality form. Moreover, Jensen inequality and free matrix equation are employed to reduce conservatism of those conditions. Through using the proposed design method, a gain-scheduled controller is designed to guarantee asymptotical stability of uncertain stochastic systems in the sense of mean square. Finally, two numerical examples are provided to demonstrate applicability and effectiveness of the proposed design method.

Robust non-fragile stabilisation and H∞ control for uncertain discrete-time stochastic systems with Markovian jump parameters and time-varying delay

In this paper, the robust non-fragile stabilisation and H∞ control problem is investigated for a class of uncertain discrete-time stochastic systems with Markovian jumping parameters and time-varying delay. The parameter uncertainties are supposed to be time-varying as well as norm-bounded. The aim of the robust non-fragile stabilisation problem is to design a non-fragile state feedback controller which guarantees the robust stability of the closed-loop system for all admissible uncertainties. At the same time, in addition to the robust stability requirement, a prescribed H∞ performance level is required to be achieved for the robust H∞ control problem. By Lyapunov stability theory, delay-segment-dependent conditions for the solvability of these problems are formulated in terms of linear matrix inequality technique. Finally, numerical examples are shown to demonstrate the usefulness and applicability of the proposed design method.

Robust Stability and Stabilization of a Class of Uncertain Nonlinear Discrete-Time Stochastic Systems with Interval Time-Varying Delays

This paper deals with the problems of the robust stochastic stability and stabilization for a class of uncertain discrete-time stochastic systems with interval time-varying delays and nonlinear disturbances. By utilizing a new Lyapunov-Krasovskii functional and some well-known inequalities, some new delay-dependent criteria are developed to guarantee the robust stochastic stability of a class of uncertain discrete-time stochastic systems in terms of the linear matrix inequality (LMI). Then based on the state feedback controller, the delay-dependent sufficient conditions of robust stochastic stabilization for a class of uncertain discrete-time stochastic systems with interval time-varying delays are established. The controller gain is designed to ensure the robust stochastic stability of the closed-loop system. Finally, illustrative examples are given to demonstrate the effectiveness of the proposed method.

H∞Gain-Scheduled Control for LPV Stochastic Systems

A robust control problem for discrete-time uncertain stochastic systems is discussed via gain-scheduled control scheme subject toH∞attenuation performance. Applying Linear Parameter Varying (LPV) modeling approach and stochastic difference equation, the uncertain stochastic systems can be described by combining time-varying weighting function and linear systems with multiplicative noise terms. Due to the consideration of stochastic behavior, the stability in the sense of mean square is applied for the system. Furthermore, two kinds of Lyapunov functions are employed to derive their corresponding sufficient conditions to solve the stabilization problems of this paper. In order to use convex optimization algorithm, the derived conditions are converted into Linear Matrix Inequality (LMI) form. Via solving those conditions, the gain-scheduled controller can be established such that the robust asymptotical stability andH∞performance of the disturbed uncertain stochastic system can be achieved in the sense of mean square. Finally, two numerical examples are applied to demonstrate the effectiveness and applicability of the proposed design method.

A class of robust bounded controllers tracking a nonlinear discrete-time stochastic system: Attractive ellipsoid technique application

This paper deals with designing of robust tracking controllers for a class of nonlinear uncertain discrete-time stochastic systems which are affine with respect to a control action on a bounded energy level. The considered nonlinear dynamics is admitted to be a priori unknown but belonging to the class of the so-called quasi-Lipschitz vector-field. The current states may be unavailable on-line, but the corresponding output vector is assumed to be measured during the process. Both the state dynamics and the output measurements are disturbed by external additive noises which are also supposed to be immeasurable. In this situation any suitable controller can only provide the boundedness of the tracking-error trajectories within a bounded zone with probability one. In this paper we suggest designing of both the control and observer “optimal” gain-matrices minimizing the “size” of the attractive ellipsoid containing all tracking-error trajectories in the vicinity of the origin. It is shown that this design problem of an output bounded control may be converted into the corresponding attractive ellipsoid “minimization” under some constraints of BMI's (bilinear matrix inequalities) type. The application of an adequate coordinate changing transforms these BMI's into a set of LMI's (linear matrix inequalities) that permits to use directly the standard MATLAB-toolbox. Two illustrative examples are considered: the first one concerns a two state—single output stochastic model, and the second one deals with a discrete-time model of electric-magnetic-tape-drive containing the four states-positions and two measured outputs.

Fuzzy control with robust and passive properties for discrete-time Takagi–Sugeno fuzzy systems with multiplicative noises

This paper deals with the problem of robust fuzzy control for uncertain passive discrete-time non-linear stochastic systems with multiplicative noise. In this paper the non-linear stochastic system is represented by a Takagi–Sugeno fuzzy model. Using that approach the concept of parallel distributed compensation is applied to construct the fuzzy controller. Sufficient stability conditions are derived in terms of a linear matrix inequality by applying the Lyapunov theory. In addition, the passivity theory and robust technique are used to guarantee strict input passivity and robust asymptotical stability of the closed-loop system with admissible parameter uncertainties. Finally, a numerical example concerning the control of a truck–trailer system is provided to demonstrate the validity of the proposed fuzzy controller design method.

Robust state estimation for uncertain discrete-time stochastic systems with missing measurements

In this paper, results of robust estimation of Zhou (2010a) are extended to state estimation with missing measurements. A new procedure is derived which inherits the main properties of that of Zhou (2010a). In this extension, a covariance matrix used in the recursions is replaced by its estimate which makes its asymptotic property investigation mathematically difficult. Though introducing a monotonic function and using the so-called squeeze rule, this new robust estimator is proved to converge to a stable system. Numerical simulation results indicate that the proposed estimator may have an estimation accuracy better than the estimator of Wang, Yang, Daniel, and Liu (2005).

Guaranteed cost PID control for uncertain discrete-time stochastic systems with additive gain perturbations

This paper investigates a novel design method for robust nonfragile proportional-integral-derivative (PID) control that is based on the guaranteed cost control (GCC) problem for a class of uncertain discrete-time stochastic systems with additive gain perturbations. On the basis of linear matrix inequality (LMI), a class of fixed PID controller parameters is obtained, and some sufficient conditions for the existence of the GCC are derived. Although the additive gain perturbations are included in the feedback systems, both the stability of closed-loop systems and adequate cost bound are attained. As a sequel, decentralized GCC PID for a class of discrete-time uncertain large-scale stochastic systems is also considered. Finally, the numerical results demonstrate the efficiency of the proposed controller synthesis. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Further Results on Robust Variance-Constrained Filtering for Uncertain Stochastic Systems with Missing Measurements

This paper revisits the problem of robust filtering for uncertain discrete-time stochastic systems with missing measurements. The measurements of the system may be unavailable at any sample time. Our aim is to design a new filter such that the error state of the filtering process is mean-square bounded. Furthermore, the steady-state variance of the estimation error of each state does not exceed the individual prescribed upper bound subject to all admissible uncertainties and all possible incomplete observations. It is shown that the design of a robust filter can be carried out by directly solving a set of linear matrix inequalities. The nonsingular assumption on the system matrix A and the inequality which is used to handle the uncertainties are not necessary in the derivation process of our results. Thus, it is expected that a less conservative condition can be obtained. The advantage of the new method is demonstrated via an illustrative example.

On designing robust controllers under randomly varying sensor delay with variance constraints

In this paper, we deal with a new stochastic control problem for a class of uncertain discrete-time stochastic systems with randomly varying sensor delay. The parameter uncertainties are allowed to be time-varying and norm-bounded, and appear in the system state matrix. The system measurements are subject to randomly varying sensor delays, which often occur in information transmission through network. The aim of this paper is to design an output feedback controller such that, for all admissible parameter uncertainties and all probabilistic sensor delays, the closed-loop system is Bounded-Input Bounded-Output (BIBO) stable in the mean square (i.e. the system state is mean square bounded), and the steady-state variance of each state is not more than the individual prescribed upper bound. It is shown that such a controller design problem is feasible if there are positive definite solutions to a couple of algebraic Riccati-like inequalities or linear matrix inequalities. The explicit expression of the desired robust controllers is derived in terms of some free parameters. An illustrative numerical example is exploited to demonstrate the usefulness and flexibility of the proposed design approach.

Robust H∞ Filtering for Uncertain Discrete Stochastic Systems with Time Delays

This paper considers the problem of robust H∞ filtering for uncertain discrete-time stochastic systems with time-varying delays. The parameter uncertainties are assumed to be real time-varying norm-bounded in both the state and measurement equations. The problem to be addressed is the design of a stable filter that guarantees stochastic stability and a prescribed H∞ performance level of the filtering error system for all admissible uncertainties and time delays. A sufficient condition for the existence of such filters is obtained in terms of a linear matrix inequality (LMI). For the case when this LMI is feasible, an explicit expression for a desired filter is given. An illustrative example is also provided to demonstrate the effectiveness and applicability of the proposed method.

Variance-Constrained Control for Uncertain Stochastic Systems With Missing Measurements

In this paper, we are concerned with a new control problem for uncertain discrete-time stochastic systems with missing measurements. The parameter uncertainties are allowed to be norm-bounded and enter into the state matrix. The system measurements may be unavailable (i.e., missing data) at any sample time, and the probability of the occurrence of missing data is assumed to be known. The purpose of this problem is to design an output feedback controller such that, for all admissible parameter uncertainties and all possible incomplete observations, the system state of the closed-loop system is mean square bounded, and the steady-state variance of each state is not more than the individual prescribed upper bound. We show that the addressed problem can be solved by means of algebraic matrix inequalities. The explicit expression of the desired robust controllers is derived in terms of some free parameters, which may be exploited to achieve further performance requirements. An illustrative numerical example is provided to demonstrate the usefulness and flexibility of the proposed design approach.

RobustH? control for uncertain discrete-time stochastic bilinear systems with Markovian switching

This paper is concerned with the problems of robust stochastic stabilization and robust H∞ control for uncertain discrete-time stochastic bilinear systems with Markovian switching. The parameter uncertainties are time-varying norm-bounded. For the robust stochastic stabilization problem, the purpose is the design of a state feedback controller which ensures the robust stochastic stability of the closed-loop system irrespective of all admissible parameter uncertainties; while for the robust H∞ control problem, in addition to the robust stochastic stability requirement, a prescribed level of disturbance attenuation is required to be achieved. Sufficient conditions for the solvability of these problems are obtained in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, explicit expressions of the desired state feedback controllers are also given. An illustrative example is provided to show the effectiveness of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.