Lebesgue piecewise affine approximation of nonlinear systems

This paper addresses a piecewise affine (PWA) approximation problem, i.e., a problem of finding a PWA system model which approximates a given nonlinear system. First, we propose a new class of PWA systems, called the Lebesgue PWA approximation systems, as a model to approximate nonlinear systems. Next, we derive an error bound of the PWA approximation model, and provide a technique for constructing the approximation model with specified accuracy. Finally, the proposed method is applied to a gene regulatory network with nonlinear dynamics, which shows that the method is a useful approximation tool.

[1]  Jun-ichi Imura,et al.  Controllability Analysis of Biosystems Based on Piecewise-Affine Systems Approach , 2008, IEEE Transactions on Automatic Control.

[2]  Hassan K. Khalil,et al.  Nonlinear Systems Third Edition , 2008 .

[3]  Eduardo Sontag.  Nonlinear regulation: The piecewise linear approach , 1981 .

[4]  Antoine Girard,et al.  Reachability Analysis of Nonlinear Systems Using Conservative Approximation , 2003, HSCC.

[5]  K. Åström,et al.  Comparison of Riemann and Lebesgue sampling for first order stochastic systems , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[6]  Jun-ichi Imura,et al.  Characterization of well-posedness of piecewise-linear systems , 2000, IEEE Trans. Autom. Control..

[7]  Ian M. Mitchell,et al.  Safety verification of conflict resolution manoeuvres , 2001, IEEE Trans. Intell. Transp. Syst..

[8]  Manfred Morari,et al.  A clustering technique for the identification of piecewise affine systems , 2001, Autom..

[9]  Jun-ichi Imura,et al.  Polynomial-Time Probabilistic Observability Analysis of Sampled-Data Piecewise Affine Systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[10]  R. Tempo,et al.  Probabilistic robustness analysis: explicit bounds for the minimum number of samples , 1997 .

[11]  Jun-ichi Imura,et al.  Polynomial-Time Probabilistic Controllability Analysis of Discrete-Time Piecewise Affine Systems , 2007, IEEE Transactions on Automatic Control.

[12]  Alberto Bemporad,et al.  Control of systems integrating logic, dynamics, and constraints , 1999, Autom..

[13]  A. Mochizuki.  An analytical study of the number of steady states in gene regulatory networks. , 2005, Journal of theoretical biology.

[14]  Ashish Tiwari,et al.  Automated Symbolic Reachability Analysis; with Application to Delta-Notch Signaling Automata , 2003, HSCC.

[15]  Alberto Bemporad,et al.  Hybrid model predictive control based on wireless sensor feedback : An experimental study , 2010 .

[16]  Alberto Bemporad,et al.  Hybrid model predictive control based on wireless sensor feedback: An experimental study , 2007, 2007 46th IEEE Conference on Decision and Control.

[17]  O. Stursberg,et al.  Approximating switched continuous systems by rectangular automata , 1999, 1999 European Control Conference (ECC).

[18]  T. Sugie,et al.  Lebesgue Piecewise Affine Approximation of Nonlinear Systems and Its Application to Hybrid System Modeling of Biosystems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[19]  Antoine Girard,et al.  Approximate solutions of ODEs using piecewise linear vector fields , 2002 .

[20]  T. Henzinger,et al.  Algorithmic Analysis of Nonlinear Hybrid Systems , 1998, CAV.

[21]  Alberto Bemporad,et al.  Observability and controllability of piecewise affine and hybrid systems , 2000, IEEE Trans. Autom. Control..

[22]  Pravin Varaiya,et al.  Epsilon-Approximation of Differential Inclusions , 1996, Hybrid Systems.