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 behaviour 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 thelocal algebraic observability problem which is concerned with the existence of a non-trivial Lie subalgebra of model?s symmetries letting the inputs and the outputs be 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]  H. Pohjanpalo.  System identifiability based on the power series expansion of the solution , 1978 .

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

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

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

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

[6]  Alexandre Sedoglavic.  A probabilistic algorithm to test local algebraic observability in polynomial time , 2001, ISSAC '01.

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

[8]  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.

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

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

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

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

[13]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

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

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

[16]  Nicole Ollivier.  Individualisme et mouvements sociaux , 1990 .

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

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

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

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

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

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

[23]  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.

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

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

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

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

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

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

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

[31]  M. Fliess,et al.  Nonlinear observability, identifiability, and persistent trajectories , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

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

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

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

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

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

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

[38]  Michael Clausen,et al.  Algebraic Complexity Theory : With the Collaboration of Thomas Lickteig , 1997 .

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

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

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

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

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