### 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 language | English (US) |
---|---|

Pages (from-to) | 382-387 |

Number of pages | 6 |

Journal | Unknown Journal |

Volume | 28 |

Issue number | 14 |

DOIs | |

State | Published - Jul 1 2015 |

### Fingerprint

### Keywords

- Frequency domain stabilization
- Polynomial optimization
- Robust control

### ASJC Scopus subject areas

- Control and Systems Engineering

### Cite this

*Unknown Journal*,

*28*(14), 382-387. https://doi.org/10.1016/j.ifacol.2015.09.487

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

Research output: Contribution to journal › Article

*Unknown Journal*, vol. 28, no. 14, pp. 382-387. https://doi.org/10.1016/j.ifacol.2015.09.487

}

TY - JOUR

T1 - Polynomial stabilization with bounds on the controller coefficients

AU - Eaton, Julia

AU - Grundel, Sara

AU - Gürbüzbalaban, Mert

AU - Overton, Michael

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

UR - http://www.scopus.com/inward/record.url?scp=84992521845&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84992521845&partnerID=8YFLogxK

U2 - 10.1016/j.ifacol.2015.09.487

DO - 10.1016/j.ifacol.2015.09.487

M3 - Article

AN - SCOPUS:84992521845

VL - 28

SP - 382

EP - 387

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 14

ER -