Optimality of Nonlinear Design Techniques: A Converse HJB Approach

The issue of optimality in nonlinear controller design is confronted by using the converse HJB approach [1] to classify dynamics under which certain design schemes are optimal. In particular, the techniques of Jacobian linearization, pseudo-Jacobian linearization, and feedback linearization are analyzed. Finally, the conditions for optimality are applied to the 2-D nonlinear oscillator, where simple, nontrivial examples are produced in which the various design techniques are optimal. 1 Introduction Determination of the optimal feedback law for nonlinear optimal control problems requires the solution of the Hamilton-Jacobi--Bellman (HJB) partial differential equation. Difficulties in solving the HJB equation for high dimensional systems have precluded their use except in specific areas, and have motivated the study of alternative control techniques. Many of such alternative techniques attempt to "approximately" solve the HJB equation by using some simplified scheme (Jacobian linearizat..