On cost design in applications of optimal control

TL;DR

This paper introduces a simplified feedback control design method using control Lyapunov functions derived from optimal control problems, avoiding complex PDE solutions and enabling easier cost design.

Contribution

It presents a novel approach that replaces the Hamilton-Jacobi-Bellman PDE with an algebraic relationship, streamlining optimal control and $ ext{H}_ ext{infty}$ control applications.

Findings

Simplifies control design by replacing PDE with algebraic equations.

Demonstrates effectiveness in $ ext{H}_ ext{infty}$ control and oscillator systems.

Reduces computational complexity in optimal control problems.

Abstract

A new approach to feedback control design based on optimal control is proposed. Instead of expensive computations of the value function for different penalties on the states and inputs, we use a control Lyapunov function that amounts to be a value function of an optimal control problem with suitable cost design and then study combinations of input and state penalty that are compatible with this value function. This drastically simplifies the role of the Hamilton-Jacobi-Bellman equation, since it is no longer a partial differential equation to be solved, but an algebraic relationship between different terms of the cost. The paper illustrates this idea in different examples, including $\mathcal{H}_{\infty}$ control and optimal control of coupled oscillators.