# A symbolic-numeric method for the parametric H$\infty$ loop-shaping design problem

2 NON-A - Non-Asymptotic estimation for online systems
CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189, Inria Lille - Nord Europe
Abstract : In this paper, we present a symbolic-numeric method for solving the H_infinity loop-shaping design problem for low order single-input single-output systems with parameters. Due to the system parameters, no purely numerical algorithm can indeed solve the problem. Using Gröbner basis techniques and the Rational Univariate Representation of zero-dimensional algebraic varieties, we first give a parametrization of all the solutions of the two Algebraic Riccati Equations associated with the H_infinity-control problem. Then, following the works H. Anai, S. Hara, M. Kanno, K. Yokoyama, Parametric polynomial factorization using the sum of roots and its application to a control design problem, J. Symb. Comp., 44 (2009), 703-725, and M. Kanno, S. Hara, Symbolic-numeric hybrid optimization for plant/controller integrated design in H_infinity loop-shaping design, Journal of Math-for-Industry, 4 (2012), 135-140, on the spectral factorization problem, a certified symbolic-numeric algorithm is obtained for the computation of the positive definite solutions of these two Algebraic Riccati Equations. Finally, we present a certified symbolic-numeric algorithm which solves the H_infinity loop-shaping design problem for the above class of systems. This algorithm is illustrated with a standard example.
Keywords :
Document type :
Conference papers
Domain :

https://hal.inria.fr/hal-01415294
Submitted on : Friday, December 30, 2016 - 10:57:27 AM
Last modification on : Friday, March 22, 2019 - 1:35:16 AM
Long-term archiving on: Monday, March 20, 2017 - 11:33:33 PM

### File

Files produced by the author(s)

### Identifiers

• HAL Id : hal-01415294, version 1

### Citation

Guillaume Rance, Yacine Bouzidi, Alban Quadrat, Arnaud Quadrat. A symbolic-numeric method for the parametric H$\infty$ loop-shaping design problem. 22nd International Symposium on Mathematical Theory of Networks and Systems (MTNS) , Jul 2016, Minneapolis, United States. pp.8. ⟨hal-01415294⟩

Record views