Model structure simplification of Nonlinear Systems via immersion

This paper concerns the simplification of model structures of nonlinear systems while preserving their input-output maps. The basic technique is an immersion of a system, which is a mapping of the initial state from the original system to another system of an identical input-output map. The necessary and sufficient conditions are presented for the immersibility of a given system into a state-space representation of such particular structures as rational functions or polynomials with respect to the state. The conditions are so mild that many types of nonlinear systems can be represented by rational or polynomial model structures. Moreover, it is also shown that a polynomial state-space representation can always be constructed to be at most quadratic with respect to the state.

[1]  Claude E. Shannon,et al.  Mathematical Theory of the Differential Analyzer , 1941 .

[2]  R. Pearson.  Nonlinear Input/Output Modeling , 1994 .

[3]  J. Lévine,et al.  Nonlinear system immersion, observers and finite-dimensional filters , 1986 .

[4]  R. Pearson.  Nonlinear Input/Output Modeling , 1994 .

[5]  Eduardo Sontag.  Polynomial Response Maps , 1979 .

[6]  S. Sastry.  Nonlinear Systems: Analysis, Stability, and Control , 1999 .

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

[8]  E. H. Kerner.  Universal formats for nonlinear ordinary differential systems , 1981 .

[9]  A. Levin.  An analytical method of estimating the domain of attraction for polynomial differential equations , 1994, IEEE Trans. Autom. Control..

[10]  H. Sussmann.  Semigroup Representations, Bilinear Approximation of Input-Output Maps, and Generalized Inputs , 1976 .

[11]  Lee A. Rubel,et al.  A survey of transcendentally transcendental functions , 1989 .

[12]  John Milnor,et al.  Introduction to Algebraic K-Theory. (AM-72), Volume 72 , 1972 .

[13]  Arthur J. Krener,et al.  Bilinear and Nonlinear Realizations of Input-Output Maps , 1975 .

[14]  J. Lo.  Global Bilinearization of Systems with Control Appearing Linearly , 1975 .

[15]  W. Steeb,et al.  Nonlinear dynamical systems and Carleman linearization , 1991 .

[16]  Jie Huang,et al.  Global robust stabilization of cascaded polynomial systems , 2002, Syst. Control. Lett..

[17]  A. Seidenberg,et al.  SOME BASIC THEOREMS IN DIFFERENTIAL ALGEBRA (CHARACTERISTIC p, ARBITRARY) , 1952 .

[18]  Alberto Tesi,et al.  On the stability domain estimation via a quadratic Lyapunov function: convexity and optimality properties for polynomial systems , 1996, IEEE Trans. Autom. Control..

[19]  Alekseĭ Ivanovich Kostrikin,et al.  Introduction to algebra , 1982 .

[20]  M. Fliess,et al.  Flatness and defect of non-linear systems: introductory theory and examples , 1995 .

[21]  L. Rubel,et al.  A differentially algebraic replacement theorem, and analog computability , 1987 .

[22]  J. Seo,et al.  Immersion of Non-Linear Systems into Linear Systems up to Output Injection: Characteristic Equation Approach , 2004 .

[23]  W Wu,et al.  A MECHANIZATION METHOD OF GEOMETRY AND ITS APPLICATIONS——I.DISTANCES,AREAS AND VOLUMES , 1986 .

[24]  Heinz Unbehauen,et al.  Structure identification of nonlinear dynamic systems - A survey on input/output approaches , 1990, Autom..

[25]  J. Vickers.  MANIFOLDS, TENSOR ANALYSIS AND APPLICATIONS (2nd edition) (Applied Mathematical Sciences 75) , 1990 .

[26]  S. Monaco,et al.  On the immersion of a discrete-time polynomial analytic system into a polynomial affine one , 1983 .

[27]  Lennart Ljung,et al.  Nonlinear black-box modeling in system identification: a unified overview , 1995, Autom..

[28]  Arjan van der Schaft,et al.  Non-linear dynamical control systems , 1990 .

[29]  Krister Forsman.  On Rational State Space Realizations , 1992 .

[30]  Alessandro Astolfi,et al.  Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems , 2001, IEEE Trans. Autom. Control..

[31]  M. Fliess,et al.  A Finiteness Criterion for Nonlinear Input–Output Differential Systems , 1983 .

[32]  P. Lindskog.  Methods, Algorithms and Tools for System Identification Based on Prior Knowledge , 1996 .

[33]  Michel Fliess,et al.  Un Outil Algebrique: Les Series Formelles Non Commutatives , 1976 .

[34]  T. Glad,et al.  An Algebraic Approach to Linear and Nonlinear Control , 1993 .

[35]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[36]  Michael F. Singer,et al.  A differentially algebraic elimination theorem with application to analog computability in the calculus of variations , 1985 .

[37]  Alberto Isidori,et al.  Nonlinear Control Systems, Third Edition , 1995, Communications and Control Engineering.

[38]  Philippe Jouan,et al.  Immersion of nonlinear systems into linear systems modulo output injection , 2002, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[39]  M. B. Pour-El,et al.  Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers) , 1974 .