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

Guillaume Rance (1) , Yacine Bouzidi (2) , Alban Quadrat (2) , Arnaud Quadrat (1)

1 SAFRAN Electronics & Defense
2 NON-A - Non-Asymptotic estimation for online systems

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.

Mots clés: Robust control and H-infinity control; Algebraic Riccati equations; Parametric control theory; Polynomial systems

Domaines: Optimisation et contrôle [math.OC]; Systèmes dynamiques [math.DS]; Calcul formel [cs.SC]; Géométrie algébrique [math.AG]

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⟩