Polynomial Stabilization with Bounds on the Controller Coefficients

Let b/a be a strictly proper reduced rational transfer function, with a monic. 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) of ax + 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 of ax + by, that is the largest of the moduli of the roots of ax + by, subject to lower and upper bounds on the coefficients of x and y, 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 controller y/x minimizing the root radius of ax + by subject to given bounds on the coefficients of x and y with root activity (the number of roots of ax+ by whose modulus equals its radius) and bound activity (the number of coefficients of x and y that are on their lower or upper bound) summing to at least 2 deg(x) + 2. We illustrate our results on two examples from the feedback control literature.