Certified Algorithms for proving the structural stability of two dimensional systems possibly with parameters

International audienceIn [1], a new method for testing the structural stability of multidimensional systems has been presented. The key idea of this method is to reduce the problem of testing the structural stability to that of deciding if an algebraic set has real points. Following the same idea, we consider in this work the specific case of two-dimensional systems and focus on the practical efficiency aspect. For such systems, the problem of testing the stability is reduced to that of deciding if a bivariate algebraic system with finitely many solutions has real ones. Our first contribution is an algorithm that answers this question while achieving practical efficiency. Our second contribution concerns the stability of two dimensional systems with parameters. More precisely, given a two-dimensional system depending on a set of parameters, we present a new algorithm that computes regions of the parameter space in which the considered system is structurally stable

Improved algorithm for computing separating linear forms for bivariate systems

We address the problem of computing a linear separating form of a system of two bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and this is the bottleneck of these algorithms in terms of worst-case bit complexity. We present for this problem a new algorithm of worst-case bit complexity \sOB(d^7+d^6\tau) where $d$ and $\tau$ denote respectively the maximum degree and bitsize of the input (and where \sO refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity). This algorithm simplifies and decreases by a factor $d$ the worst-case bit complexity presented for this problem by Bouzidi et al. \cite{bouzidiJSC2014a}. This algorithm also yields, for this problem, a probabilistic Las-Vegas algorithm of expected bit complexity \sOB(d^5+d^4\tau).Comment: ISSAC - 39th International Symposium on Symbolic and Algebraic Computation (2014

Tests certifiés pour la stabilité structurelle de systèmes multidimensionnels

In this paper, we present new computer algebra based methods for testing the structural stability of n-D discrete linear systems (with n >= 2). More precisely, we show that the standard characterization of the structural stability of a multivariate rational transfer function (namely, the denominator of the transfer function does not have solutions in the unit polydisc of \C^n) is equivalent to the fact that a certain system of polynomials does not have real solutions. We then use state-of-the-art computer algebra algorithms to check this last condition, and thus the structural stability of multidimensional systems.Nous présentons dans cet article de nouvelles méthodes, basées sur des techniques de calcul formel, pour tester la stabilité structurelle de systèmes n-D linéaires et discrets (avec n > 2). Plus précisément, nous montrons dans un premier temps que la condition classique de stabilité structurelle d'une fonction de transfert rationnelle multivariée (à savoir que le dénominateur de celle-ci n'a pas de zéros à l'intérieur du polydisque unité de \C^n) est équivalente au fait que des systèmes d'équations polynomiales, obtenus via certaines transformations, n'ont pas de zéros réels. Nous utilisons ensuite des algorithmes de résolutions de systèmes algébriques pour vérifier cette dernière condition et ainsi la stabilité structurelle de systèmes multidimensionnels

Computation of the L∞\mathcal{L} \infty -norm of finite-dimensional linear systems

International audienceIn this paper, we study the problem of computing the $\mathcal{L}\infty$- norm of finite-dimensional linear time-invariant systems. This problemis first reduced to the computation of the maximal x-projection of the real solutions $(x, y)$ of a bivariate polynomial system $\sum=\{P,{\frac{\partial{P}}{\partial{y}}}\}$, with ${P} \in \mathbb{Z}[x, y]$. Then, we use standard computer algebra methods to solve the problem. In this paper, we alternatively study a method based on rational univariate representations, a method based on root separation, and finally a method first based on the sign variation of the leading coefficients of the signed subresultant sequence and then based on the identification of an isolating interval for the maximal $x$-projection of the real solutions of $\sum$

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

New bivariate system solver and topology of algebraic curves

International audienceWe present a new approach for solving polynomial systems of two bivariate polynomials with rational coefficients. We first use González-Vega and Necula approach [3] based on sub-resultant sequences for decomposing a system into subsystems according to the number of roots (counted with multiplicities) in vertical lines. We then show how the resulting triangular subsystems can be efficiently solved by computing lexicographic Gröbner basis and Rational Univariate Representations (RURs) of these systems. We also show how this approach can be performed using modular arithmetic, while remaining deterministic. Finally we apply our solver to the problem of computing the topology of algebraic curves using the algorithm Isotop [2]. We show that our approach yields a substantial gain of a factor between 1 to 10 on curves of degree up to 28 compared to directly computing a Gröbner basis and RUR of the input system, and how it leads to a very competitive algorithm compared to the other state-of-the-art implementations

Separating linear forms and Rational Univariate Representations of bivariate systems

International audienceWe address the problem of solving systems of bivariate polynomials with integer coefficients. We first present an algorithm for computing a separating linear form of such systems, that is a linear combination of the variables that takes different values when evaluated at distinct (complex) solutions of the system. In other words, a separating linear form defines a shear of the coordinate system that sends the algebraic system in generic position, in the sense that no two distinct solutions are vertically aligned. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and, moreover, the computation of a separating linear form is the bottleneck of these algorithms, in terms of worst-case bit complexity. Given two bivariate polynomials of total degree at most $d$ with integer coefficients of bitsize at most~$\tau$, our algorithm computes a separating linear form {of bitsize $O(\log d)$} in \comp\ bit operations in the worst case, which decreases by a factor $d^2$ the best known complexity for this problem (where \sO refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity). We then present simple polynomial formulas for the Rational Univariate Representations (RURs) of such systems. {This yields that, given a separating linear form of bitsize $O(\log d)$, the corresponding RUR can be computed in worst-case bit complexity \sOB(d^7+d^6\tau) and that its coefficients have bitsize \sO(d^2+d\tau).} We show in addition that isolating boxes of the solutions of the system can be computed from the RUR with \sOB(d^{8}+d^7\tau) bit operations in the worst case. Finally, we show how a RUR can be used to evaluate the sign of a bivariate polynomial (of degree at most $d$ and bitsize at most $\tau$) at one real solution of the system in \sOB(d^{8}+d^7\tau) bit operations and at all the $\Theta(d^2)$ real solutions in only $O(d)$ times that for one solution

Separating Linear Forms for Bivariate Systems

International audienceWe present an algorithm for computing a separating linear form of a system of bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at distinct (complex) solutions of the system. In other words, a separating linear form defines a shear of the coordinate system that sends the algebraic system in generic position, in the sense that no two distinct solutions are vertically aligned. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and, moreover, the computation of a separating linear form is the bottleneck of these algorithms, in terms of worst-case bit complexity. Given two bivariate polynomials of total degree at most $d$ with integer coefficients of bitsize at most $\tau$, our algorithm computes a separating linear form in \sOB(d^8+d^7\tau+d^5\tau^2) bit operations in the worst case, where the previously known best bit complexity for this problem was \sOB(d^{10}+d^9\tau) (where \sO refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity)

Improved algorithms for solving bivariate systems via Rational Univariate Representations

Given two coprime polynomials $P$ and $Q$ in $\Z[x,y]$ of degree bounded by $d$ and bitsize bounded by $\tau$, we address the problem of solving the system $\{P,Q\}$. We are interested in certified numerical approximations or,more precisely, isolating boxes of the solutions. We are also interested in computing, as intermediate symbolic objects, rational parameterizations of he solutions, and in particular Rational Univariate Representations (RURs), which can easily turn many queries on the system into queries on univariate polynomials. Such representations require the computation of a separating form for the system, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system. We present new algorithms for computing linear separating forms, RUR decompositions and isolating boxes of the solutions. We show that these three algorithms have worst-case bit complexity $\widetilde{O}_B(d^6+d^5\tau)$, where $\widetilde{O}$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to thebit complexity. We also present probabilistic Las-Vegas variants of our two first algorithms, which have expected bit complecity $\widetilde{O}_B(d^5+d^4\tau)$. A key ingredient of our proofs of complexity is an amortized analysis of the triangular decomposition algorithm via subresultants, which is of independent interest