Abstract
This paper studies a Non-convex State-dependent Linear Quadratic Regulator (NSLQR) problem, in which the control penalty weighting matrix 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 . 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 , in situations where only the goal discrepancy at the terminal time is of concern, such as in Marathon races and target hitting missions.
Highlights
Introduction1.1 Problem Definition In this paper, we seek an optimal control law u~k(x,t), for which the performance index ðtf
1.1 Problem Definition In this paper, we seek an optimal control law u~k(x,t), for which the performance index ðtfJ(x(t),u(t))~ L(x(t),u(t),t)dtzw(x(tf ),tf ) t0 1 ðtf ~(xT (t)Q(t)x(t)zuT (t)R(x(t))u(t))dt ð1Þ 2 t0 z xTS(tf )x(tf )is minimized along the associated closed-loop system trajectory of the Linear Time-variant (LTV) system x_ (t)~A(t)x(t)zB(t)u(t) x(t0)~x0ð2Þ where u(t)[Rm is the control input, x(t)[Rn is the system state, t0 is the starting time, tf is the terminal time and x0 is the initial value of x(t) at time t0
The main result of this paper is presented: the necessary and sufficient condition of the optimality of the solution to the Non-convex State-dependent Linear Quadratic Regulator (NSLQR) problem defined in Eq (1) and (2)
Summary
1.1 Problem Definition In this paper, we seek an optimal control law u~k(x,t), for which the performance index ðtf. J(x(t),u(t))~ L(x(t),u(t),t)dtzw(x(tf ),tf ) t0 1 ðtf ~. (xT (t)Q(t)x(t)zuT (t)R(x(t))u(t))dt ð1Þ 2 t0 z xT (tf )S(tf )x(tf ). Is minimized along the associated closed-loop system trajectory of the Linear Time-variant (LTV) system x_ (t)~A(t)x(t)zB(t)u(t) x(t0)~x0. Ð2Þ where u(t)[Rm is the control input, x(t)[Rn is the system state, t0 is the starting time, tf is the terminal time and x0 is the initial value of x(t) at time t0. The dependence of variables on t is omitted when no confusion will be introduced in the rest of the paper. It is assumed that A, B, Q are continuous LR in t, R(x) is differentiable with respect to x, and Lx is bounded
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