title: The optimal solution of a non-convex state-dependent LQR problem and its applications.
abstract: This paper studies a Non-convex State-dependent Linear Quadratic Regulator (NSLQR) problem, in which the control penalty weighting matrix [Formula: see text] in the performance index is state-dependent. A necessary and sufficient condition for the optimal solution is established with a rigorous proof by Euler-Lagrange Equation. It is found that the optimal solution of the NSLQR problem can be obtained by solving a Pseudo-Differential-Riccati-Equation (PDRE) simultaneously with the closed-loop system equation. A Comparison Theorem for the PDRE is given to facilitate solution methods for the PDRE. A linear time-variant system is employed as an example in simulation to verify the proposed optimal solution. As a non-trivial application, a goal pursuit process in psychology is modeled as a NSLQR problem and two typical goal pursuit behaviors found in human and animals are reproduced using different control weighting [Formula: see text]. It is found that these two behaviors save control energy and cause less stress over Conventional Control Behavior typified by the LQR control with a constant control weighting [Formula: see text], in situations where only the goal discrepancy at the terminal time is of concern, such as in Marathon races and target hitting missions.
authors.affiliation: School of Automation Science and Electrical Engineering, Beihang University, Beijing, China; School of Electrical Engineering and Computer Science, Ohio University, Athens, Ohio, United States of America.
authors.forename: Xudan
authors.initials: X
authors.lastname: Xu
authors.affiliation: School of Electrical Engineering and Computer Science, Ohio University, Athens, Ohio, United States of America.
authors.forename: J Jim
authors.initials: JJ
authors.lastname: Zhu
authors.affiliation: School of Automation Science and Electrical Engineering, Beihang University, Beijing, China.
authors.forename: Ping
authors.initials: P
authors.lastname: Zhang
mesh_terms: D006040:Goals; D006801:Humans; D016014:Linear Models; D008960:Models, Psychological; D008962:Models, Theoretical; D013223:Statistics as Topic
references.citation: J Appl Psychol. 2010 Nov;95(6):985-1008
references.pmid: 20804232
references.citation: Psychol Sci. 2011 May;22(5):607-12
references.pmid: 21474842