Singular Linear Quadratic Optimal Control Problem for Stochastic Nonregular Descriptor Systems

Abstract

This paper is concerned with the singular linear quadratic (SLQ) optimal control problem for stochastic nonregular descriptor systems with time‐delay. By means of some reasonable assumptions and a series of equivalent transformations, the problem is finally transformed into a positive linear quadratic (LQ) problem for standard stochastic systems. Then dynamic programming principle is used to establish the solvability of the original problem, and the desired explicit presentation of the optimal controller is given in terms of matrix iterative form. The results due to Feng et al. are generalized and improved. As an application, a numerical example is presented to demonstrate the efficiency of the proposed approach.

Indefinite linear quadratic optimal control for discrete time‐varying linear rectangular descriptor systems

“…According to the basic orthogonal property among the Krein space stochastic variables, combining with ( 28), (29), and (31), the following covariances of 𝜙(0) and 𝜁 are obtained…”

“…The LQ problem for linear systems was discussed in detail in [21]. Notice that the definite case of the LQ problem for square and rectangular descriptor systems, where the weight matrices are both semi-positive definite in cost function, attracted the attention of many scholars; see [22][23][24][25][26] and [27][28][29], respectively. Particularly, in [26], the LQ problem in definite case for descriptor Markov jump systems was discussed, where the control variable was constrained to a subspace related to input matrices to guarantee that the system is regular; meanwhile, the limited cost value was derived.…”

Robust global controller design for discrete‐time descriptor systems with multiple time‐varying delays

“…Descriptor systems also commonly called singular, generalized state‐space or differential‐algebraic equation systems, are widely used to describe a variety of practical systems . For instance, constrained robots, flexible‐link manipulators, aerospace applications, biological systems, mechanical systems, electrical circuits, and power systems can be modeled mathematically by descriptor systems.…”

Archives of Control Sciences

“…where ∆ t denotes the derivative operator with respect to t, ξ = ξ (t) denotes state, ω = ω(t) denotes control; E, A ∈ R n×n , with rank(E) < n, B ∈ R n×r ; Q and R are positive definite matrices. It is well known that the problem to be solved in the optimization problem (1) and ( 2) is to find the control-state pairs (ω, ξ ) satisfying the dynamic constraint (2) such that the objective functional (1) is minimized, see [1], [2], [3], [4] and the literatures therein for exhaustively explanation.…”