Polynomial stabilization with bounds on the controller coefficients

Julia Eaton, Sara Grundel, Mert Gürbüzbalaban, Michael L. Overton

Research output: Contribution to journalConference article


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.

Original languageEnglish (US)
Pages (from-to)382-387
Number of pages6
Issue number14
StatePublished - Jul 1 2015
Event8th IFAC Symposium on Robust Control Design, ROCOND 2015 - Bratislava, Slovakia
Duration: Jul 8 2015Jul 11 2015



  • Frequency domain stabilization
  • Polynomial optimization
  • Robust control

ASJC Scopus subject areas

  • Control and Systems Engineering

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