Julia Eaton, Sara Grundel, Mert Gürbüzbalaban, Michael L Overton
Polynomial Stabilization with Bounds on the Controller Coefficients∗ J Eaton, S Grundel, M Gürbüzbalaban, ML Overton - IFAC-PapersOnLine, 2015
Publication year: 2015

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 coecients 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 coecients 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.