Indefinite LQ optimal control with cross term for discrete‐time uncertain systems

Abstract

Recently, there has been an increasing interest in the study on uncertain optimal control problems. In this paper, a linear quadratic (LQ) optimal control with cross term for discrete‐time uncertain systems is considered, whereas the weighting matrices in the cost function are allowed to be indefinite. Firstly, a recurrence equation for the problem is presented based on Bellman's principle of optimality in dynamic programming. Then, a necessary condition for the existence of an optimal linear state feedback control of the indefinite LQ problem is given by the recurrence equation. Moreover, a sufficient condition of well‐posedness for the indefinite LQ problem is presented by introducing a linear matrix inequality (LMI) condition. Furthermore, it is shown that the well‐posedness of the indefinite LQ problem, the solvability of the indefinite LQ problem, the LMI condition, and the solvability of the constrained difference equation are equivalent to each other. Finally, an example is presented to illustrate the results obtained.

A systematic review of uncertainty theory with the use of scientometrical method

“…In addition, uncertain optimal control has attracted scholars’ attention in recent years, and is also a hot topic in this sub-field. The research objects of optimal control are uncertain explosion control (Xu & Zhu, 2012 ), multi-dimensional case control (Deng & Zhu, 2012 ; Chen & Li, 2015 ; Deng et al, 2018 ), control problems with jumps (Deng & Zhu, 2013 ; Deng & Chen, 2016 ; Deng & Shen, 2019 ), linear quadratic equation optimal control problems (Chen & Zhu, 2016 ; Li & Zhu, 2017 ; Sheng et al, 2018 ; Li & Zhu, 2018 ; Chen et al, 2019 ), discrete-time control problems (Chen & Zhu, 2018a , b ; Chen et al, 2019 ), nonlinear uncertain control (Ding et al, 2017 ), switched systems (Yan & Zhu, 2015 ; Yan et al, 2017 ), singular systems (Shu & Zhu, 2017a , b ) and uncertain dynamic systems (Deng & Chen, 2017 ). Scholars have derived optimal equations, recursive equations, optimal principles, analytical expressions, sufficient and necessary conditions and well-posedness for these research objects, and have founded a continuous-time model, bang-bang optimal control model, optimistic value model, and optimal value model, so as to make use of uncertainty theory to settle practical problems.…”

“…Owing to the fact that the literature growth rate for the uncertain processes topic is faster than that for the uncertain risk analysis topic, we predict that the former will become a mainstream sub-field of uncertainty theory more quickly and be promoted to the “Hard Core” quadrant. In the last two years, more attention has been paid to the uncertain renewal reward process (Yao & Zhou, 2018a ; Yao, 2019 ), the optimal control of the discrete-time uncertain system (Chen & Zhu, 2018a , b ; Chen et al, 2019 ) and uncertain linear quadratic optimal control with jump (Deng & Shen, 2019 ; Chen et al, 2019 , 2019 ). As one of the latest sub-fields to emerge, with the shortest half-life, uncertain risk analysis is a relatively new research direction of uncertainty theory, and the amount of published literature on the topic is increasing year by year.…”

Optimal Control for a Multistage Uncertain Random System

“…Yang and Gao [12] considered an optimal control problem of the uncertain linear quadratic differential game model. Currently, there are many research achievements on uncertain optimal control [13], [14], [15], [16], [17], [18].…”

“…This paper makes the following contributions. First, many works focus on the stochastic optimal control concerning randomness [3] or uncertain optimal control involving uncertainty [14]. This paper further develops optimal control theory and considers an uncertain random optimal control including uncertainty and randomness.…”

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].…”