H 2 optimal control for a wide class of discrete-time linear stochastic systems

In this article, the problem of H ₂-control of a discrete-time linear system subject to Markovian jumping and independent random perturbations is considered. Different H ₂ performance criteria (often called H ₂-norms) are introduced and characterised via solutions of some suitable linear equations on certain spaces of symmetric matrices. Some aspects specific to the discrete-time framework are revealed. The problem of optimisation of H ₂-norms is solved under the assumption that full state vector is available for measurements. One shows that among all stabilising controllers of higher dimension, the best performance is achieved by a zero-order controller. The corresponding feedback gain of the optimal controller is constructed based on the stabilising solution of a system of discrete-time generalised Riccati equations.     

Robust H 2-control for discrete-time Markovian jump linear systems

This paper deals with the robust H₂-control of discrete-time Markovian jump linear systems. It is assumed that both the state and jump variables are available to the controller. Uncertainties satisfying some norm bounded conditions are considered on the parameters of the system. An upper bound for the H₂-control problem is derived in terms of a linear matrix inequality (LMI) optimization problem. For the case in which there are no uncertainties, we show that the convex formulation is equivalent to the existence of the mean square stabilizing solution for the set of coupled algebraic Riccati equations arising on the quadratic optimal control problem of discrete-time Markovian jump linear systems. Therefore, for the case with no uncertainties, the convex formulation considered in this paper imposes no extra conditions than those in the usual dynamic programming approach. Finally some numerical examples are presented to illustrate the technique.     

Maximal and Stabilizing Hermitian Solutions for Discrete-Time Coupled Algebraic Riccati Equations

Discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control and H _∞-control of Markovian jump linear systems are considered. First, the equations that arise from the quadratic optimal control problem are studied. The matrix cost is only assumed to be hermitian. Conditions for the existence of the maximal hermitian solution are derived in terms of the concept of mean square stabilizability and a convex set not being empty. A connection with convex optimization is established, leading to a numerical algorithm. A necessary and sufficient condition for the existence of a stabilizing solution (in the mean square sense) is derived. Sufficient conditions in terms of the usual observability and detectability tests for linear systems are also obtained. Finally, the coupled algebraic Riccati equations that arise from the H _∞-control of discrete-time Markovian jump linear systems are analyzed. An algorithm for deriving a stabilizing solution, if it exists, is obtained. These results generalize and unify several previous ones presented in the literature of discrete-time coupled Riccati equations of Markovian jump linear systems. Date received: November 14, 1996. Date revised: January 12, 1999.     

Finite horizon quadratic optimal control and a separation principle for Markovian jump linear systems

In this note, we consider the finite-horizon quadratic optimal control problem of discrete-time Markovian jump linear systems driven by a wide sense white noise sequence. We assume that the output variable and the jump parameters are available to the controller. It is desired to design a dynamic Markovian jump controller such that the closed-loop system minimizes the quadratic functional cost of the system over a finite horizon period of time. As in the case with no jumps, we show that an optimal controller can be obtained from two coupled Riccati difference equations, one associated to the optimal control problem when the state variable is available, and the other one associated to the optimal filtering problem. This is a principle of separation for the finite horizon quadratic optimal control problem for discrete-time Markovian jump linear systems. When there is only one mode of operation our results coincide with the traditional separation principle for the linear quadratic Gaussian control of discrete-time linear systems.     

Further results on H∞ control for discrete-time Markovian jump time-delay systems

This paper studies the problem of H_∞ control for a class of discrete-time Markovian jump systems with time delay. The purpose is to improve the existing results on H_∞ controller design for Markovian jump systems. A novel summation inequality is presented and an improved stability criterion for the system is derived by utilising the new inequality, which is proved to be less conservative than most results in the literature. Then the state feedback controller is designed, which guarantees the stochastic stability of the closed-loop system with a given disturbance attenuation. Numerical examples are provided to illustrate the effectiveness and advantages of the proposed techniques.     

Optimal time-weighted H 2 model reduction for Markovian jump systems

This article addresses the optimal time-weighted H ₂ model reduction problem for Markovian jump linear systems. That is, for a given mean square stable Markovian jump system, our aim is to find a mean square stable jump system of lower order such that the time-weighted H ₂ norm of the corresponding error system is minimised. The time-weighted H ₂ norm of the system is first defined, and then a computational method is constructed. The computation requires the solution of two sets of recursive Lyapunov-type linear matrix equations associated with the Markovian jump system. To solve the optimal time-weighted H ₂ model reduction problem, we propose a gradient flow method for its solution. A necessary condition for minimality is derived, and a computational procedure is provided to obtain the minimising reduced-order model. The necessary condition generalises the standard result for systems when Markov jumps and the time-weighting term do not appear. Finally, two numerical examples are given to demonstrate the effectiveness of the proposed approach.     

H2 control of discrete-time periodic systems with Markovian jumps and multiplicative noise

This paper addresses the problem of optimal and robust H₂ control for discrete-time periodic systems with Markov jump parameters and multiplicative noise. To analyse the system performance in the presence of exogenous random disturbance, an H₂ norm is firstly established on the basis of Gramian matrices. Further, under the condition of exact observability, a necessary and sufficient condition is presented for the solvability of H₂ optimal control problem by means of a generalised Riccati equation. When the transition probabilities of jump parameter are incompletely measurable, an H₂-guaranteed cost norm is exploited and the robust H₂ controller is designed through a linear matrix inequality (LMI) optimisation approach. An example of a networked control system is supplied to illustrate the proposed results.     

Anti-disturbance control for time-varying delay Markovian jump nonlinear systems with multiple disturbances

In this paper, the problem of composite anti-disturbance resilient control is addressed for time-varying delay Markovian jump nonlinear systems with multiple disturbances. The disturbances are assumed to include two parts: the first one in the input channel is described by an external system with perturbations; the second one is supposed to be bounded H₂ norm. By combining disturbance observer and L₂–L_∞ control method, the disturbances are attenuated and rejected, simultaneously, and the desired dynamic performance can be obtained for time-varying delay Markovian jump nonlinear systems. Moreover, the gains of the resilient controller and the observer are acquired by applying linear matrix inequalities (LMIs) technology. Finally, an application example is presented to show the effectiveness of the proposed approach.     

H∞ State Feedback Delay-dependent Control for Discrete Systems with Multi-time-delay

In this paper, H∞ state feedback control with delay information for discrete systems with multi-time-delay is discussed. Making use of linear matrix inequality (LMI) approach, a time-delay-dependent criterion for a discrete system with multi-time-delay to satisfy H∞ performance indices is induced, and then a strategy for H1 state feedback control with delay values for plant with multi-time-delay is obtained. By solving corresponding LMI, a delay-dependent state feedback controller satisfying H∞ performance indices is designed. Finally, a simulation example demonstrates the validity of the proposed approach.     

Stabilisation and H∞ control of neutral stochastic delay Markovian jump systems

This paper investigates the exponential stabilisation and H_∞ control problem of neutral stochastic delay Markovian jump systems. First, a delay feedback controller is designed to stabilise the neutral stochastic delay Markovian jump system in the drift part. Second, sufficient conditions for the existence of feedback controller are proposed to ensure that the resulting closed-loop system is exponentially stable in mean square and satisfies a prescribed H_∞ performance level. Finally, numerical examples are provided to show the effectiveness of the proposed design methods.     

Delay-range-dependent robust stabilisation and H ∞ control for nonlinear uncertain stochastic fuzzy systems with mode-dependent time delays and Markovian jump parameters

This article focuses on the problems of robust stabilisation and H _∞ control for nonlinear uncertain stochastic systems with mode-dependent time delay and Markovian jump parameters represented by the Takagi–Sugeno (T-S) fuzzy model approach. The system under consideration involves parameter uncertainties, Itô-type stochastic disturbances, Markovian jump parameters and unknown nonlinear disturbances. The purpose is to design a state feedback controller such that the closed-loop system is robustly exponentially stable in the mean square and satisfies a prescribed H _∞ performance level. Novel delay-range-dependent conditions in the form of linear matrix inequalities (LMIs) are derived for the solvability of robust stabilisation and H _∞ control problem. A desired fuzzy controller can be constructed by solving a set solutions of LMIs and can be easily calculated by Matlab LMI control toolbox. Finally, a numerical example is presented to illustrate the proposed method.     

Finite-time H∞ control for a class of discrete-time Markovian jump systems with partly unknown time-varying transition probabilities subject to average dwell time switching

An extension of a fixed transition probability (TP) Markovian switching model to combine time-varying TPs has offered another set of useful regime-switching models. This paper is concerned with the problem of finite-time H_∞ control for a class of discrete-time Markovian jump systems with partly unknown time-varying TPs subject to average dwell time switching. The so-called time-varying TPs mean that the TPs are varying but invariant within an interval. The variation of the TPs considered here is subject to a class of slow switching signal. Based on selecting the appropriate Lyapunov–Krasovskii functional, sufficient conditions of finite-time boundedness of Markovian jump systems are derived and the system trajectory stays within a prescribed bound. Finally, an example is given to illustrate the efficiency of the proposed method.     

Robust stabilisation control for discrete-time networked control systems

We consider the analysis and synthesis of discrete-time networked control systems (NCSs), where the plant has additive uncertainty and the controller is updated with the sensor information at stochastic time intervals. It is shown that the problem is linked to H_∞-control of linear discrete-time stochastic systems and a new small gain theorem is established. Based on this result, sufficient conditions are given for ensuring mean square stability of the NCS, and the genetic algorithm is utilised to design the controller of the NCS based on a linear matrix inequality technique. An illustrative example is given to demonstrate the effectiveness of our proposed method.     

H<Subscript>2</Subscript>-optimal tuning algorithms for fixed structure controllers

An H₂-method of optimal tuning is proposed for a fixed order controller. The SISO plant model is considered in state space. The H₂-method of tuning parameter design is based on the minimization of a transient process closeness criterion for appropriate open-loop and closed-loop control systems and their reference models. The controller tuning algorithms use the plant parameter estimations obtained during the plant parameter identification. The analytical expressions are obtained for the square of H₂-norm of a stable dynamic system. The following theorem is proven: the minimum necessary conditions for the functionals of transfer function H₂-norm of open-loop and closed-loop systems are the same as the minimum necessary conditions for the Frobenius norm of the controller parameter tuning polynomial.     

H∞ filtering for piecewise homogeneous Markovian jump nonlinear systems

This paper concerns the problem of H_∞ filtering for piecewise homogeneous Markovian jump nonlinear systems. Different from the existing studies in the literatures, the existence of variations in transition rates for Markovian jump nonlinear systems is considered. The purpose of the paper is to design mode-dependent and mode-independent filters, such that the dynamics of the filtering errors are stochastic integral input-to-state stable with H_∞ performance index. Using the linear matrix inequality method and the Lyapunov functional method, sufficient conditions for the solution to the H_∞ filtering problem are derived. Finally, three examples are proposed to illustrate the effectiveness of the given theoretical results.     

Output-feedback control for sampled-data systems with variable sampling rate

In this paper, we propose design method of controller for sampled-data systems with variable sampling rate. First, we give design method for both H₂ and H_∞ controller. For H₂ control, performance of the system is introduced according to a standard sampled-data setting. A discrete-time H₂ control problem is employed for solving the original problem. Its solvability condition is then established as a parameter-dependent linear matrix inequality. A probabilistic approach is taken for coping with the parameter-dependency. H_∞ controller is designed by almost the same manner. Applying both results, we have design method for multi-objective control.     

Reduced-order filtering for linear systems with Markovian jump parameters

This paper addresses the reduced-order H_∞ filtering problem for continuous-time Makovian jump linear systems, where the jump parameters are modelled by a discrete-time Markov process. Sufficient conditions for the existence of the reduced-order H_∞ filter are proposed in terms of linear matrix inequalities (LMIs) and a coupling non-convex matrix rank constraint. In particular, the sufficient conditions for the existence of the zero-order H_∞ filter can be expressed in terms of a set of strict LMIs. The explicit parameterization of the desired filter is also given. Finally, a numerical example is given to illustrate the proposed approach.     

Observer-based H ∞-control for discrete-time T–S fuzzy systems

This article further studies the observer-based H _∞-control problem for discrete-time Takagi–Sugeno (T–S) fuzzy systems. By using fuzzy Lyapunov functions and introducing slack variables, a sufficient condition, which can guarantee observer-based H _∞-control performance for T–S fuzzy systems, is proposed in terms of a set of bilinear matrix inequalities. Moreover, in the so-called two-step procedure, results of the first step are allowed to select in order to reduce the conservatism of previous approaches. In comparison with the existing literature, the proposed approach not only provides more relaxed H _∞-control conditions but also ensures better H _∞-control performance. Finally, the validity and applicability of the proposed approach are successfully demonstrated through two numerical examples.     

Robust H∞ estimation of stationary discrete-time linear processes with stochastic uncertainties

The problem of H_∞ filtering of stationary discrete-time linear systems with stochastic uncertainties in the state space matrices is addressed, where the uncertainties are modeled as white noise. The relevant cost function is the expected value, with respect to the uncertain parameters, of the standard H_∞ performance. A previously developed stochastic bounded real lemma is applied that results in a modified Riccati inequality. This inequality is expressed in a linear matrix inequality form whose solution provides the filter parameters. The method proposed is applied also to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂/H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a simple tracking example.