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

International audienceIn 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

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

International audienceIn 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

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

International audienceIn 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

Contrôleurs H∞ explicites pour les systèmes d'ordre 1 à 3, à simple entrée et simple sortie, avec paramètres

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Contrôleurs H∞ explicites pour les systèmes d'ordre 1 à 3, à simple entrée et simple sortie, avec paramètres

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Contrôleurs H∞ explicites pour les systèmes d'ordre 4, à une seule entrée et une seule sortie, avec des paramètres et leurs applications au système masse-ressort avec amortissement

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Utilisation de méthodes symboliques pour la résolution d'équations algébriques de Riccati apparaissant en filtrage invariant

International audienceThis paper proposes a new step in the development of invariant observers. In the past, this theory led to impressive simplifications of the error equations encountered in estimation problems, especially those related to navigation. This was used to reduce computation load or derive new theoretical properties. Here, we leverage this advantage to obtain closed-form solutions of the underlying algebraic Riccati equations through advanced symbolic computation methods

Contrôleurs H∞ explicites pour les systèmes d'ordre 4, à une seule entrée et une seule sortie, avec des paramètres et leurs applications au système masse-ressort avec amortissement

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