摘要：AbstractLetb/abe a strictly proper reduced rational transfer function, withamonic. Consider the problem of designing a controller y/x, with deg(y) < deg(x) < deg(a) – 1 and x monic, subject to lower and upper bounds on the coefficients of y and x, so that the poles of the closed loop transfer function, that is the roots (zeros) ofax+by, are, if possible, strictly inside the unit disk. One way to formulate this design problem is as the following optimization problem: minimize the root radius ofax+by, that is the largest of the moduli of the roots ofax+by, subject to lower and upper bounds on the coefficients ofxandy, as the stabilization problem is solvable if and only if the optimal root radius subject to these constraints is less than one. The root radius of a polynomial is a non-convex, non-locally-Lipschitz function of its coefficients, but we show that the following remarkable property holds: there always exists an optimal controllery/xminimizing the root radius ofax+bysubject to given bounds on the coefficients of x and y withroot activity(the number of roots ofax+bywhose modulus equals its radius) andbound activity(the number of coefficients of x and y that are on their lower or upper bound) summing to at least 2deg(x) + 2. We illustrate our results on two examples from the feedback control literature.
