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

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 form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V=∫min{t}max{∞}[q(x)+h(u)]dτ</tex> are obtained. It is shown that if the value of the optimum V 0 is required to be a quadratic form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Vᵒ = 1/2x'Mx</tex> of the current state <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> , and if certain state variables cannot be measured, then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> cannot 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 performance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V⁰(x)</tex> .