A probabilistic algorithm to test local algebraic observability in polynomial time

The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the input-output behavior of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of the local algebraic observability problem which is concerned with the existence of a non trivial Lie subalgebra of the symmetries of the model letting the inputs and the outputs invariant. We present a probabilistic seminumerical algorithm that proposes a solution to this problem in polynomial time. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the complexity of evaluation of the model and in the number of variables. Furthermore, we show that the size of the integers involved in the computations is polynomial in the number of variables and in the degree of the system. Last, we estimate the probability of success of our algorithm.

[1]  M. Fliess.  Automatique et corps différentiels , 1989 .

[2]  H. T. Kung,et al.  Fast Algorithms for Manipulating Formal Power Series , 1978, JACM.

[3]  Alberto Isidori,et al.  Nonlinear control systems: an introduction (2nd ed.) , 1989 .

[4]  L. Ljung,et al.  PARAMETRIZATION OF NONLINEAR MODEL STRUCTURES AS LINEAR REGRESSIONS , 1991 .

[5]  Alexandre Sedoglavic.  A Probabilistic Algorithm to Test Local Algebraic Observability in Polynomial Time , 2002, J. Symb. Comput..

[6]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[7]  Donald E. Knuth.  The Art of Computer Programming 2 / Seminumerical Algorithms , 1971 .

[8]  F. Ollivier.  Le probleme de l'identifiabilite structurelle globale : approche theorique, methodes effectives et bornes de complexite , 1990 .

[9]  Joseph Johnson,et al.  Kahler Differentials and Differential Algebra , 1969 .

[10]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[11]  Lennart Ljung,et al.  PARAMETRIZATION OF NONLINEAR MODEL STRUCTURES AS LINEAR REGRESSIONS , 1990 .

[12]  K R Godfrey,et al.  Global identifiability of the parameters of nonlinear systems with specified inputs: a comparison of methods. , 1990, Mathematical biosciences.

[13]  Piet Hemker,et al.  Mathematical modelling in blood coagulation ; Simulation and parameter estimation , 1997 .

[14]  A. Raksanyi.  Utilisation du calcul formel pour l'étude des systèmes d'équations polynomiales (applications en modélisation) , 1986 .

[15]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[16]  Giovanni Gallo,et al.  Efficient algorithms and bounds for Wu-Ritt characteristic sets , 1991 .

[17]  R. E. Kalman,et al.  On the general theory of control systems , 1959 .

[18]  S. Vajda,et al.  IDENTIFIABILITY OF POLYNOMIAL SYSTEMS: STRUCTURAL AND NUMERICAL ASPECTS , 1987 .

[19]  V. Pan,et al.  Polynomial and Matrix Computations , 1994, Progress in Theoretical Computer Science.

[20]  C. W. Gear,et al.  The index of general nonlinear DAEs , 1995 .

[21]  Eric Schost.  Sur la resolution des systemes polynomiaux a parametres , 2000 .

[22]  L. Lipshitz,et al.  Power series solutions of algebraic differential equations , 1984 .

[23]  François Boulier,et al.  Efficient computation of regular differential systems by change of rankings using Kähler differentials , 2000 .

[24]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[25]  H. Rabitz,et al.  Similarity transformation approach to identifiability analysis of nonlinear compartmental models. , 1989, Mathematical biosciences.

[26]  Richard Zippel,et al.  Effective polynomial computation , 1993, The Kluwer international series in engineering and computer science.

[27]  H. Pohjanpalo.  System identifiability based on the power series expansion of the solution , 1978 .

[28]  François Boulier,et al.  Representation for the radical of a finitely generated differential ideal , 1995, ISSAC '95.

[29]  Brahim Sadik,et al.  A Bound for the Order of Characteristic Set Elements of an Ordinary Prime Differential Ideal and some Applications , 2000, Applicable Algebra in Engineering, Communication and Computing.

[30]  Kyriakos Kalorkoti.  ALGEBRAIC COMPLEXITY THEORY (Grundlehren der Mathematischen Wissenschaften 315) , 1999 .

[31]  V. Becerra,et al.  Optimal control of nonlinear differential algebraic equation systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[32]  R. Kálmán.  On the general theory of control systems , 1959 .

[33]  Marc Giusti,et al.  A Gröbner Free Alternative for Polynomial System Solving , 2001, J. Complex..

[34]  François Ollivier.  Identifiabilité et identification : du Calcul Formel au Calcul Numérique ? , 2000 .

[35]  Céline Noiret.  Utilisation du calcul formel pour l'identifiabilité de modèles paramètriques et nouveaux algorithmes en estimation de paramètres , 2000 .

[36]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

[37]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[38]  Palaiseau Cedex,et al.  Computing Parametric Geometric Resolutions , 2001 .

[39]  P. Daoutidis,et al.  Feedback control of nonlinear differential-algebraic-equation systems , 1995 .

[40]  Erich Kaltofen.  Computational Differentiation and Algebraic Complexity Theory * , .

[41]  Keith O. Geddes.  Convergence behavior of the Newton iteration for first order differential equations , 1979, EUROSAM.

[42]  Evelyne Hubert,et al.  Factorization-free Decomposition Algorithms in Differential Algebra , 2000, J. Symb. Comput..

[43]  Yves Lecourtier,et al.  Identifiability and distinguishability testing via computer algebra , 1985 .

[44]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[45]  Luis M. Pardo,et al.  Kronecker's and Newton's Approaches to Solving: A First Comparison , 2001, J. Complex..

[46]  Darrell Williamson,et al.  Observation of bilinear systems with application to biological control , 1977, Autom..

[47]  Walter Baur,et al.  The Complexity of Partial Derivatives , 1983, Theor. Comput. Sci..

[48]  W. J. H. Stortelder,et al.  Parameter estimation in chemical engineering ; a case study for resin production , 1996 .

[49]  A. Krener,et al.  Nonlinear controllability and observability , 1977 .

[50]  Lennart Ljung,et al.  On global identifiability for arbitrary model parametrizations , 1994, Autom..

[51]  A. Isidori.  Nonlinear Control Systems , 1985 .

[52]  A. Goldbeter.  A model for circadian oscillations in the Drosophila period protein (PER) , 1995, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[53]  Eric Walter,et al.  Identifiability of State Space Models , 1982 .