Polynomial stabilization with bounds on the controller coefficients

Julia Eaton, Sara Grundel, Mert Gürbüzbalaban, Michael Overton

Research output: Contribution to journalArticle

Abstract

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
JournalUnknown Journal
Volume28
Issue number14
DOIs
StatePublished - Jul 1 2015

Fingerprint

Transfer functions
Stabilization
Polynomials
Roots
Controller
Controllers
Rational functions
Polynomial
Coefficient
Radius
Feedback control
Poles
Monic
Transfer Function
Upper and Lower Bounds
Modulus
Strictly
Lipschitz Function
Rational function
Feedback Control

Keywords

  • Frequency domain stabilization
  • Polynomial optimization
  • Robust control

ASJC Scopus subject areas

  • Control and Systems Engineering

Cite this

Polynomial stabilization with bounds on the controller coefficients. / Eaton, Julia; Grundel, Sara; Gürbüzbalaban, Mert; Overton, Michael.

In: Unknown Journal, Vol. 28, No. 14, 01.07.2015, p. 382-387.

Research output: Contribution to journalArticle

Eaton, Julia ; Grundel, Sara ; Gürbüzbalaban, Mert ; Overton, Michael. / Polynomial stabilization with bounds on the controller coefficients. In: Unknown Journal. 2015 ; Vol. 28, No. 14. pp. 382-387.
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