E. K. Boukas

Analysis and Synthesis of Markov Jump Linear Systems With Time-Varying Delays and Partially Known Transition Probabilities

In this note, the stability analysis and stabilization problems for a class of discrete-time Markov jump linear systems with partially known transition probabilities and time-varying delays are investigated. The time-delay is considered to be time-varying and has a lower and upper bounds. The transition probabilities of the mode jumps are considered to be partially known, which relax the traditional assumption in Markov jump systems that all of them must be completely known a priori. Following the recent study on the class of systems, a monotonicity is further observed in concern of the conservatism of obtaining the maximal delay range due to the unknown elements in the transition probability matrix. Sufficient conditions for stochastic stability of the underlying systems are derived via the linear matrix inequality (LMI) formulation, and the design of the stabilizing controller is further given. A numerical example is used to illustrate the developed theory.

Manufacturing flow control and preventing maintenance: a stochastic control approach

A stochastic control model for simultaneously planning production and maintenance in a flexible manufacturing system (FMS) is proposed, and an efficient technique for computing the optimal control policy is developed. The model extends previous formulations by including an age-dependent machine failure rate and by allowing the control to influence some jump rates (namely the preventive maintenance activities). By using an adaptation to the case of piecewise-deterministic systems of the approximation technique initially proposed by H.J. Kushner (1977) in the realm of the optimal control of diffusions, one shows how it is possible to computer the optimal control for a two-machine system. >

H ∞ -control for Markovian jumping linear systems with parametric uncertainty

This paper studies the problem of H∞-control for linear systems with Markovian jumping parameters. The jumping parameters considered here are two separable continuous-time, discrete-state Markov processes, one appearing in the system matrices and one appearing in the control variable. Our attention is focused on the design of linear state feedback controllers such that both stochastic stability and a prescribed H∞-performance are achieved. We also deal with the robust H∞-control problem for linear systems with both Markovian jumping parameters and parameter uncertainties. The parameter uncertainties are assumed to be real, time-varying, norm-bounded, appearing in the state matrix. Both the finite-horizon and infinite-horizon cases are analyzed. We show that the control problems for linear Markovian jumping systems with and without parameter uncertainties can be solved in terms of the solutions to a set of coupled differential Riccati equations for the finite-horizon case or algebraic Riccati equations for the infinite-horizon case. Particularly, robust H∞-controllers are also designed when the jumping rates have parameter uncertainties.

Robust H/sub /spl infin// control of discrete-time Markovian jump linear systems with mode-dependent time-delays

Considers the class of discrete-time Markovian jump linear system with norm-bounded uncertainties and time-delay, which is dependent on the system mode. Linear matrix inequality (LMI)-based sufficient conditions for the stability, stabilization and H/sub /spl infin// control are developed. A numerical example is worked out to show the usefulness of the theoretical results.

Robust H∞ control of descriptor discrete-time markovian jump systems

This paper considers the stochastic stability and the robust control of descriptor discrete-time systems with Markovian jumping parameters. In terms of linear matrix inequalities, a necessary and sufficient condition is proposed, which ensures a discrete-time descriptor Markovian jump system to be regular, causal and stochastically stable. A robust admissibility condition and a robust bounded real lemma are also developed. Based on these, a sufficient condition on the existence of a state-feedback controller which guarantees the robust admissibility and the performance is also given by employing the linear matrix inequality technique. A robustly stabilizing state feedback controller can be constructed through the numerical solutions of linear matrix inequalities. Finally, an example is provided to demonstrate the effectiveness of the proposed approach.

H ∞ model reduction for discrete-time Markov jump linear systems with partially known transition probabilities

In this article, the fault detection (FD) problem for a class of discrete-time Markov jump linear system (MJLS) with partially known transition probabilities is investigated. The proposed systems are more general, which relax the traditional assumption in Markov jump systems that all the transition probabilities must be completely known. A residual generator is constructed and the corresponding FD is formulated as an H ∞ filtering problem by which the error between residual and fault are minimised in the H ∞ sense. The linear matrix inequality-based sufficient conditions for the existence of FD filter are derived. A numerical example on a multiplier–accelerator model economic system is given to illustrate the potential of the developed theoretical results.

Stochastic stability and guaranteed cost control of discrete-time uncertain systems with Markovian jumping parameters

In this paper, we first study the problems of robust quadratic mean-square stability and stabilization for a class of uncertain discrete-time linear systems with both Markovian jumping parameters and Frobenius norm-bounded parametric uncertainities. Necessary and sufficient conditions for the above problems are proposed, which are in terms of positive-definite solutions of a set of coupled algebraic Riccati inequalities. Then, the problem of robust quadratic guaranteed cost control for the underlying systems is investigated. A guaranteed cost control is designed to ensure the cost function is within a certain bound, irrespective of all admissible uncertainities.

Delay-dependent robust stability and H/sub /spl infin// control of jump linear systems with time-delay

Considers the class of continuous-time jump linear system with time-delay and polytopic uncertain parameters. When the time-delay is a known constant, by using the linear matrix inequality (LMI) technique, we first establish a delay dependent sufficient condition for robust stability and robust H/sub /spl infin// control of the class of systems under study. The problem of determining the maximum time-delay under which the system will remain stable is cast into a generalized eigenvalue problem and thus solved by LMI techniques. When the control input contains time-delay, an algorithm to design a state feedback controller with constant gain matrix is developed.

On stabilization of uncertain linear systems with jump parameters

In this paper, we study the problem of robust stabilizability of the class of uncertain linear systems with Markovian jumping parameters. Under the assumption of complete access to the continuous state, the stochastic stabilizability of the nominal system and the boundedness of the systems uncertainties, sufficient conditions which guarantee the robust stability of the uncertain systems are presented, which are in terms of a set of coupled algebraic Riccati equations. A numerical example is given to illustrate the potential of the proposed technique.

Robust production and maintenance planning in stochastic manufacturing systems

In this paper the authors consider a preventive maintenance and production model of a flexible manufacturing system with machines that are subject to breakdown and repair. The preventive maintenance can be used to reduce the machine failure rates and improve the productivity of the system. The control variables are the rate of maintenance and the rate of production; the objective is to choose a control process that optimizes a robust cost of inventory/shortage, production, and maintenance. The value function is shown to be locally Lipschitz and to satisfy a Hamilton-Jacobi-Isaacs equation. A sufficient condition for optimal control is obtained. Finally, an algorithm is given for solving the optimal control problem numerically. >