Optimistic Value Model of Indefinite LQ Optimal Control for Discrete‐Time Uncertain Systems

Abstract

Uncertainty theory is a branch of mathematics which provides a new tool to deal with the human uncertainty. Based on uncertainty theory, this paper proposes an optimistic value model of discrete-time linear quadratic (LQ) optimal control, whereas the state and control weighting matrices in the cost function are indefinite, the system dynamics are disturbed by uncertain noises. With the aid of the Bellman's principle of optimality in dynamic programming, we first present a recurrence equation. Then, a necessary condition for the state feedback control of the indefinite LQ problem is derived by using the recurrence equation. Moreover, a sufficient condition of well-posedness for the indefinite LQ optimal control is given. Finally, a numerical example is presented by using the obtained results.

Uncertain random linear quadratic control with multiplicative and additive noises

“…Remark It is not new for us to use recurrence equations to solve optimal control for multistage systems, such as Yan et al, Chen and Zhu and Sheng et al [12,13,27]. It is valuable to extend this method to uncertain random systems.…”

“…As a counterpart of Hamilton‐Jacobi‐Bellman equation, Zhu [11] presented an equation of optimality. In uncertain environment, Yan [12] tackled with an LQ model with additive noises, Chen [13] studied LQ model with multiplicative noises. And the noises in Yan [12] and Chen [13] are described by linear uncertain variables.…”

“…In uncertain environment, Yan [12] tackled with an LQ model with additive noises, Chen [13] studied LQ model with multiplicative noises. And the noises in Yan [12] and Chen [13] are described by linear uncertain variables. More related works about uncertain control problems with additive noises or multiplicative noises refer to previous studies [13‐16].…”

“…And the noises in Yan [12] and Chen [13] are described by linear uncertain variables. More related works about uncertain control problems with additive noises or multiplicative noises refer to previous studies [13‐16]. Up to now, the LQ model with both additive noises and multiplicative noises has not been considered in uncertain environment.…”

A Single Machine Scheduling with Periodic Maintenance and Uncertain Processing Time

“…To deal with the involved human uncertainty, the uncertainty theory was founded by Liu [12] in 2007 and refined it in 2010 [13]. The uncertainty theory is a branch of axiomatic mathematics for modeling human uncertainty, which has been deeply developed in many fields such as uncertain programming [14][15][16], uncertain scheduling [17][18][19][20][21][22], uncertain risk analysis [23][24][25], uncertain calculus [26][27][28] and uncertain optimal control [29][30][31][32][33][34].…”