On equivalence of quadratic loss functions†

Abstract

When a linear, time.invariant plant is optimized with respect to the performance index where x is the state vector and u the control, the optimal control can be expressed as a feedback law u = –Kx Two pairs of matrices [Q, R] and [Q _e, R _e], yielding the same control law are equivalent. A necessary and sufficient condition is derived, in the single–input case, for a symmetric non–negative definite Q to be equivalent to a diagonal matrix Q*. This condition is satisfied by a plant described by equations in phase–variable canonical form, and a formula for Q* in terms of Q is given. It is shown that an equivalent Q _e can be parameterized by exactly n non–negative parameters. For the multi–input case, Q _e and R _e must contain at least nr parameters, where n and r are the dimensions of x and u, respectively.