An inverse optimality method to solve a class of third order optimal control problems

Abstract

The main contribution of this paper is to analytically solve the Hamilton-Jacobi-Bellman equation for a class of third order nonlinear optimal control problems for which the dynamics are affine and the cost is quadratic in the input. The proposed solution method is based on the notion of inverse optimality with a variable part of the cost to be determined in the solution. One special advantage of the proposed method is that the solution is directly obtained for the control input without the computation of a cost function first. The cost can however also be obtained based on the control input. Furthermore, a Lyapunov function can be constructed for a subclass of optimal control problems, yielding a proof certificate for stability. Experimental results of a path following problem of a unicycle are also presented.

An inverse optimal approach to design of feedback control: Exploring analytical solutions for the Hamilton‐Jacobi‐Bellman equation

“…Several researchers have exploited this notion to establish a framework for design of optimal feedback control 8‐10 . Within this framework, other researchers developed optimal feedback laws for some practical problems, for instance, stabilization of rigid spacecrafts, 11 control of Euler‐Lagrange systems, 12 stabilization of inverted pendulums, 13 and optimal feedback control of a certain class of second‐ and third‐order nonlinear systems 14,15 . Despite this significant body of prior work, a systematic characterization of the class of cost functionals with analytical HJB solution has not been proposed yet.…”