Optimal Control for Discrete‐time Descriptor Noncausal Systems

Abstract

In this paper, optimal control problems governed by linear discrete-time descriptor noncausal systems (with quadratic input variables) are investigated in order.A descriptor system assumed to be regular alone is called descriptor noncausal system. According to Bellman's principle of optimality in dynamic programming, a recurrence equation for simplifying the optimal control problems is derived. Then, employing the recurrence equation, a bang-bang optimal control problem subject to a linear descriptor noncausal system and an optimal control problem subject to a descriptor noncausal system with quadratic input variables are both settled, and the optimal solutions are given through exact expressions.A numerical example is presented to illustrate the effectiveness of the results obtained concerning the bang-bang optimal control problem.

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

“…In this example, the optimal control of indefinite LQ problem $$$ {\mathbf{P}}_1 $$$ is shown above. However, the optimal control obtained in [26] and [34] are constrained on a given interval and the fixed subspace based on the input matrix of descriptor Markov jump systems, respectively. Besides, [33] only considered the uniqueness of solution for open‐loop system; thus, the discussion about uniqueness of solution for optimal closed‐loop system in this work is more meaningful compared with [33].…”

“…Remark The exact optimal control can be derived from Theorem 2, which depends on coefficient matrices of original system (). [34] investigated a bang‐bang optimal control problem for the time‐invariant noncausal descriptor system, which was the time‐optimal control problem actually. And the optimal control derived in [34] was restricted on a given interval and bounded.…”

“…In comparison with the previous known results, the contributions of this work are the following. Since the free item exists in state variable and the parameters of time‐varying rectangular descriptor system are all time‐dependent, the structure and properties of time‐varying rectangular descriptor systems are more complex than time‐invariant square descriptor systems in [22–25, 34]. In this paper, not only the existence of solution for optimal closed‐loop systems but also the uniqueness of solution for the difference subsystems of the optimal closed‐loop systems should be considered.…”

“…In addition, the optimal control in [26] was restricted on the fixed subspace which was related to the input matrix for static part of descriptor Markov jump systems, resulting in the limited cost value. Although [34] investigated the indefinite LQ problem, the control input was constrained on a given interval, which leads to the limited optimal control. The accurate optimal control and optimal cost value are derived in this work. Moreover, Krein space method can be directly used to solve the transformed indefinite LQ problem for time‐varying systems without any additional conditions.…”

“…In the last several years, since the weight matrix is required only symmetric, the indefinite LQ problem has growing practical applications in portfolio selection problem and industrial production; see [30][31][32][33][34][35][36][37]40], and the references therein. For instance, the indefinite LQ problem can be applied to solve portfolio optimization, Pareto problem and Nash games [31,[35][36][37].…”