Stabilization of Stochastic Hybrid Linear Systems with Noise

Abstract

This article deals with the class of uncertain stochastic hybrid linear systems with noise. The uncertainties we are considering are of norm bounded type. The stochastic stabilization and robust stabilization problems are treated. Linear matrix inequality (LMI)-based sufficient conditions are developed to design the state feedback controller with constant gain that stochastically (robust stochastically) stabilizes the studied class of systems. Our results are mode independent and require only the complete access to the state vector. Numerical examples are given to show the effectiveness of the proposed results.     

Production set of failure prone flexible manufacturing systems 1

For a flexible manufacturing system, determining the feasible production set at a given time is essential for production management. In this paper, we propose methods for estimating the production set for a general multi–part type failure prone flexible manufacturing system. We also discuss how to get an open loop control that can ensure the system producing at the margin of its expected production set and how to estimate the corresponding variance of the expected production set. Some special flexible manufacturing systems are discussed where we also suggest some simpler approximations. Numerical examples are presented to illustrate our algorithm     

Guaranteed Cost Control for Uncertain Neutral Stochastic Systems via Dynamic Output Feedback Controllers

This paper deals with the problem of guaranteed cost control for uncertain neutral stochastic systems. The parameter uncertainties are assumed to be time-varying but norm-bounded. Dynamic output feedback controllers are designed such that, for all admissible uncertainties, the resulting closed-loop system is mean-square asymptotically stable and an upper bound on the closed-loop value of the cost function is guaranteed. By employing a linear matrix inequality (LMI) approach, a sufficient condition for the solvability of the underlying problem is obtained. A numerical example is provided to demonstrate the potential of the proposed techniques.     

Hedging Point Policy Improvement

This paper deals with the production and the corrective maintenance planning control problem for failure-prone manufacturing systems. The introduction of corrective maintenance increases the availability of the manufacturing system, which guarantees the improvement of the system productivity if the production planning is well done. For the one-machine/one-part system, we show that our planning control problem is more efficient than the one given by Akella and Kumar. We also show that the hedging level is lower than the one obtained by Akella and Kumar.     

On State Feedback Stabilization of Singular Systems with Random Abrupt Changes

This paper deals with the class of continuous-time singular linear systems with random abrupt changes. The state feedback stabilization and its robustness for this class of systems with norm-bounded uncertainties are tackled. Sufficient conditions for designing either a stabilizing controller or a robust stabilizing controller are developed in the LMI setting. The developed sufficient conditions are used to synthesize the state feedback controller that guarantees that either the nominal system or the uncertain system is piecewise regular, impulse free and stochastically stable or robust stochastically stable. The research of this author was supported by NSERC, Grant RGPIN36444-02.     

Output-feedback guaranteed cost control for uncertain time-delay systems with Markov jumps

This paper considers the guaranteed cost control problem ofthe jump linear system with uncertain parameters and constanttime delay. The sufficient and necessary condition for the systemto be mean square quadratically stable (MSQS) is obtained. Usingthe LMI technique, an algorithm to design an output-feedbackguaranteed controller which stabilizes the system in the MSQSsense is developed.     

Piecewise deterministic Markov process model for flexible manufacturing systems with preventive maintenance

In this paper, we consider a maintenance and production model of a flexible manufacturing system. The maintenance activity involves lubrication, routine adjustments, etc., which reduce the machine failure rates and therefore reduce the aging of the machines. The objective of the problem is to choose the rate of maintenance and the rate of production that minimize the overall costs of inventory/shortage, production, and maintenance. It is shown that the value function is locally Lipschitz. Then, the existence of the optimal control policy is shown, and necessary and sufficient conditions for optimality are obtained.This research has been supported by NSERC-Canada, Grant OGP-003644 and FCAR-NC0271F.     

Minimax production planning in failure-prone manufacturing systems

In this paper, we consider a minimax production planning model of a flexible manufacturing system with machines that are subject to random breakdown and repair. The objective is to choose the rate of production that minimizes the related minimax cost of production and inventory/shortage. The value function is shown to be the unique viscosity solution to the associated Hamilton-Jacobi-Isaacs equation. Under certain conditions, it is shown that the value function is continuously differentiable. A verification theorem is given to provide a sufficient condition for optimal control. Finally, two examples are solved explicitly.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grants OGP0036444 and A4169.     

Suboptimal Regulators for Discrete-Time Jump Linear Systems with Time-Multiplied Performance Index

This paper addresses the suboptimal regulator design problem of a discrete-time jump linear system by using a time-multiplied performance index. For a given state feedback control with constant gain matrices, the time-multiplied performance index is derived and a necessary condition for such a control to minimize the time-multiplied cost is provided. A gradient-based algorithm to find a suboptimal control law is developed and some numerical results are given to show the usefulness of the theoretical results.     

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.