TY - JOUR

T1 - Polynomial stabilization with bounds on the controller coefficients

AU - Eaton, Julia

AU - Grundel, Sara

AU - Gürbüzbalaban, Mert

AU - Overton, Michael L.

N1 - Publisher Copyright:
© 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

PY - 2015/7/1

Y1 - 2015/7/1

N2 - 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(o) - 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 2deg(x) + 2. We illustrate our results on two examples from the feedback control literature.

AB - 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(o) - 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 2deg(x) + 2. We illustrate our results on two examples from the feedback control literature.

KW - Frequency domain stabilization

KW - Polynomial optimization

KW - Robust control

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U2 - 10.1016/j.ifacol.2015.09.487

DO - 10.1016/j.ifacol.2015.09.487

M3 - Conference article

AN - SCOPUS:84992521845

SN - 2405-8963

VL - 28

SP - 382

EP - 387

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

IS - 14

T2 - 8th IFAC Symposium on Robust Control Design, ROCOND 2015

Y2 - 8 July 2015 through 11 July 2015

ER -