A Discussion on Nonlinear Quadratic Control and Sontag's Formula

TL;DR

This paper discusses nonlinear quadratic control, focusing on Sontag's formula, its relation to inverse optimal control, and the challenges of solving the Hamilton-Jacobi-Bellman equation for nonlinear systems.

Contribution

It analyzes Sontag's formula within inverse optimal control for nonlinear systems and explores its connection to quadratic cost functions, providing insights and discussion points.

Findings

Sontag's formula can be adapted for nonlinear systems with quadratic costs.

The minimized cost function can be explicitly derived in certain cases.

Remarks are provided on the applicability and limitations of the approach.

Abstract

The quadratic optimal state feedback (LQR) is one of the most popular designs for linear systems and succeeds via the solution of the algebraic Riccati equation. The situation is different in the case of non-linear systems: the Riccati equation is then replaced by the Hamilton Jacobi Bellman equation (HJB), the solution of which is generally difficult. A compromise can be the so-called Inverse Optimal Control, a form of which is Sontag's formula [1]; here the minimized cost function follows from the feedback law chosen, not vice versa. Using Sontag's formula in the variant according to Freeman and Primbs [2, 9], the actually minimized cost function is given in the following sections, including cases when it reduces to the quadratic cost. Also some remarks and thoughts are presented for discussion.