On the inverse optimum control problem for a class of nonlinear autonomous systems

Abstract

Some aspects of the inverse optimum control problem are considered for a class of nonlinear autonomous systems. A closed-loop system with a known control law is given; the problem is to determine performance criteria for which the given control law is optimum. Algebraic conditions that must be satisfied by a class of scalar performance criteria of the formV=\int\min{t}\max{\infty}[q(x)+h(u)]d_{\tau}are obtained. It is shown that if the value of the optimum V⁰is required to be a quadratic formV^{o} = \frac{1}{2}x'Mxof the current statex, and if certain state variables cannot be measured, thenMcannot be positive definite. The inverse optimum control problem corresponding to the problem of Lur'e is considered. Examples are given to illustrate the techniques and to compare the properties of a linear and nonlinear system having the same optimum performanceV^{0}(x).