Dynamic programming in stochastic control of systems with delay

We consider optimal control problems for systems described by stochastic differential equations with delay (SDDE). We prove a version of Bellman's principle of optimality (the dynamic programming principle) for a general class of such problems. That the class in general means that both the dynamics and the cost depends on the past in a general way. As an application, we study systems where the value function depends on the past only through some weighted average. For such systems we obtain a Hamilton-Jacobi-Bellman partial differential equation that the value function must solve if it is smooth enough. The weak uniqueness of the SDDEs we consider is our main tool in proving the result. Notions of strong and weak uniqueness for SDDEs are introduced, and we prove that strong uniqueness implies weak uniqueness, just as for ordinary stochastic differential equations.     

An online identification algorithm of unknown time-varying delay and internal multimodel control for discrete non-linear systems

In this paper, an online algorithm is proposed for the identification of unknown time-varying input delay in the case of discrete non-linear systems described by decoupled multimodel. This method relies on the minimization of a performance index based on the error between the real system and the partial internal models outputs. In addition, a decoupled internal multimodel control is proposed for the compensation of discrete non-linear systems with time-varying delay. This control scheme incorporates partial internal model controls. Each partial controller is associated to a specified operating zone of the non-linear system. The switching between these controllers is ensured by a supervisor that contains a set of local predictors. A simulation example is carried out to illustrate the significance of the proposed time-varying delay identification algorithm and the proposed internal multimodel control scheme.     

Observer-based robust control for uncertain systems with time-varying delay

This paper is concerned with the function observer-based stabilizationfor time-varying delay systems with parameter uncertainties.The uncertain systems tackled in this paper involve uncertaintyin quadratic constrained form which includes the well-knownnorm-bounded time-varying uncertainty as a special case. Interms of Razumikhin-type stability theorem, we show that thefeasibility of several matrix inequalities guarantees the solvabilityof the addressed problem. Furthermore, we propose a parametricapproach for function observer-based robust stabilizer design.     

Dynamic output guaranteed cost controller for neutral systems with input delay

In this paper, we consider a design problem of dynamic output guaranteed cost controller (GCC) of a class of neutral systems with input delay. A quadratic cost function is considered as a performance measure for the closed-loop system. Based on the Lyapunov second method, two stability criteria for existence of the controller are derived in terms of matrix inequalities. The solutions of the matrix inequalities can be easily obtained using existing efficient convex optimization techniques. A numerical example is given to illustrate the proposed design method.     

Optimal control of switched systems with time delay

In this note, we develop a computational method for solving an optimal control problem which is governed by a switched dynamical system with time delay. Our approach is to parameterize the switching instants as a new parameter vector to be optimized. Then, we derive the required gradient of the cost function which is obtained via solving a number of delay differential equations forward in time. On this basis, the optimal control problem can be solved as a mathematical programming problem.     

Singular control of delay systems

A necessary condition, equivalent to Jacobson's condition, for optimality of a singular control is derived for systems represented by difference-differential equations.     

Decentralized robust control design for uncertain delay systems

We consider a class of large-scale uncertain delay systems. The uncertain parameter vector in the system is possibly fast time-varying. It may be nonlinear in the system dynamics. No statistical or fuzzy information of the uncertainty is known. Based on only the possible bound of the uncertain parameter, a decentralized linear robust control is proposed, which renders the system asymptotically stable.     

Global stabilization of switched control systems with time delay

In this paper, the stabilization problem of switched control systems with time delay is investigated for both linear and nonlinear cases. First, a new global stabilizability concept with respect to state feedback and switching law is given. Then, based on multiple Lyapunov functions and delay inequalities, the state feedback controller and the switching law are devised to make sure that the resulting closed-loop switched control systems with time delay are globally asymptotically stable and exponentially stable.     

Stabilization of linear distributed control systems with unbounded delay

In this paper we study the asymptotic stabilization of linear distributed parameter control systems with unbounded delay. Assuming that the semigroup of operators associated with the uncontrolled and nondelayed equation is compact and that the phase space is a uniform fading memory space, we characterize those systems that can be stabilized using a feedback control. As consequence we conclude that every system of this type is stabilizable with an appropriated finite dimensional control.     

Optimal guaranteed control of delay systems

A linear problem of optimal guaranteed control of a delay system is considered in which geometric constraints on control actions and terminal constraints on states are present. A new concept of a state of the problem that represents a finite-dimensional vector is introduced. Three kinds of optimal feedback are defined. We describe methods for implementing open-loop and closable optimal feedbacks. They are based on a fast dual method for the correction of optimal programs. The results are illustrated by examples.     

Finite-time L1 control for positive switched linear systems with time-varying delay

This paper is concerned with the problem of finite-time L₁ control for a class of positive switched linear systems with time-varying delay. Firstly, by using the average dwell time approach, sufficient conditions which can guarantee the L₁ finite-time boundedness of the underlying system are given. Then, in virtue of the results obtained, a state feedback controller is designed to ensure that the resulting closed-loop system is finite-time bounded with L₁-gain performance. All the obtained results are formulated in terms of linear matrix inequalities (LMIs), which can be solved conveniently. Finally, an example is given to illustrate the efficiency of the proposed method.     

Impulsive control for differential systems with delay

This paper deals with the impulsive control for a class of differential systems with delay. Using Lyapunov functions and the comparison principle, we present some sufficient conditions for the asymptotic stability and exponential stability of impulsive control systems with delay. Moreover, we give an estimate of the upper bound of impulse interval. The results in this paper extend and improve the earlier publications. Copyright © 2012 John Wiley & Sons, Ltd.     

A method of solving boundary-value problems for a non-linear control system allowing for delay in the control action

An algorithm is developed for constructing a control function that guarantees that a wide class of non-linear control systems, described by ordinary differential equations, is transferred from an initial state into a specified final state, allowing for constraints on the system and a delay in the control signal. A constructive criterion is found for the choice of the final states and the value of the delay in the control signal for which the above transfer is possible. An interorbital flight problem is studied as an example and is modelled numerically.     

Relative controllability of stochastic nonlinear systems with delay in control

The existence of solutions to systems is a natural premise to carry our study about controllability. Under the basic and readily verified conditions to guarantee the existence of the solutions to a system, in this paper, we prove the relative controllability (approximate controllability ) of the stochastic differential systems with delay in control. Sufficient conditions are given firstly for the relative controllability and relative approximate controllability in finite dimensional spaces, and these results are then generalized to infinite-dimensional Hilbert spaces. Finally, examples are given to illustrate the effectiveness of the proposed methods.