JZUS - Journal of Zhejiang University SCIENCE

Journal of Zhejiang University SCIENCE C 2012 Vol.13 No.11 P.850-858

http://doi.org/10.1631/jzus.C1200096

Uniform modeling of parameter dependent nonlinear systems

Author(s): Najmeh Eghbal, Naser Pariz, Ali Karimpour
Affiliation(s): Department of Electrical Engineering, Ferdowsi University of Mashhad, P. O. Box 91775-1111, Iran
Corresponding email(s): najmeh.eghbal@gmail.com
Key Words: Parameter dependent nonlinear systems, Approximation method, Parameter dependent piecewise affine systems, Modeling

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Abstract: This paper addresses the problem of approximating parameter dependent nonlinear systems in a unified framework. This modeling has been presented for the first time in the form of parameter dependent piecewise affine systems. In this model, the matrices and vectors defining piecewise affine systems are affine functions of parameters. modeling of the system is done based on distinct spaces of state and parameter, and the operating regions are partitioned into the sections that we call ‘multiplied simplices’. It is proven that this method of partitioning leads to less complexity of the approximated model compared with the few existing methods for modeling of parameter dependent nonlinear systems. It is also proven that the approximation is continuous for continuous functions and can be arbitrarily close to the original one. Next, the approximation error is calculated for a special class of parameter dependent nonlinear systems. For this class of systems, by solving an optimization problem, the operating regions can be partitioned into the minimum number of hyper-rectangles such that the modeling error does not exceed a specified value. This modeling method can be the first step towards analyzing the parameter dependent nonlinear systems with a uniform method.

Reference

[1]Bemporad, A., 2004. Efficient conversion of mixed logical dynamical systems into an equivalent piecewise affine form. IEEE Trans. Autom. Control, 49(5):832-838.

[2]Bergami, M., Bizzarri, F., Carlevaro, A., Storace, M., 2006. Structurally stable PWL approximation of nonlinear dynamical systems admitting limit cycles: an example. IEICE Trans. Fund. Electron. Commun. Comput. Sci., 89(10):2759-2766.

[3]Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V., 1994. Linear Matrix Inequalities in Systems and Control Theory. SIAM, Philadelphia, PA.

[4]Breiman, L., 1993. Hinging hyper-planes for regression, classification, and function approximation. IEEE Trans. Inf. Theory, 39(3):999-1013.

[5]Chien, M., Kuh, E., 1977. Solving nonlinear resistive networks using piecewise-linear analysis and simplicial subdivision. IEEE Trans. Circ. Syst., 24(6):305-317.

[6]de Feo, O., Maggio, G., Kennedy, M., 2000. The Colpitts oscillator: families of periodic solutions and their bifurcations. Int. J. Bifurc. Chaos, 10(5):935-958.

[7]di Federico, M., Julián, P., Poggi, T., Storace, M., 2010. Integrated circuit implementation of multi-dimensional piecewise-linear functions. Dig. Signal Process., 20(6):1723-1732.

[8]Gao, Y., Chen, Z.H., 2009. Robust H_∞ control for constrained discrete-time piecewise affine systems with time-varying parametric uncertainties. IET Control Theory Appl., 3(8):1132-1144.

[9]Heemels, W.P.M.H., de Schutter, B., Bemporad, A., 2001. Equivalence of hybrid dynamical models. Automatica, 37(7):1085-1091.

[10]Julian, P., Desages, A., Agamennoni, O., 1999. High-level canonical piecewise linear representation using a simplicial partition. IEEE Trans. Circ. Syst. I, 46(4):463-480.

[11]Julian, P., Desages, A., D′Amico, B., 2000. Orthonormal high level canonical PWL functions with applications to model reduction. IEEE Trans. Circ. Syst. I, 47(5):702-712.

[12]Lin, H., Antsaklis, P., 2003a. Robust Regulation of Polytopic Uncertain Linear Hybrid Systems with Networked Control System Applications. In: Liu, D., Antsaklis, P.J., (Eds.), Stability and Control of Dynamical Systems with Applications: a Tribute to Anthony N. Michel. Birkhauser, Boston, p.83-108.

[13]Lin, H., Antsaklis, P., 2003b. Robust Invariant Control Synthesis for Discrete Time Polytopic Uncertain Linear Hybrid Systems. American Control Conf., p.5221-5226.

[14]Lin, J.N., Unbehauen, R., 1992. Canonical piecewise-linear approximations. IEEE Trans. Circ. Syst. I, 39(8):697-699.

[15]Maggio, G., de Feo, O., Kennedy, M., 1999. Nonlinear analysis of the Colpitts oscillator and applications to design. IEEE Trans. Circ. Syst. I, 46(9):1118-1130.

[16]Poggi, T., 2010. Circuit Implementation of Piecewise-Affine Functions and Applications. PhD Thesis, University of Genoa, Italy.

[17]Storace, M., Bizzarri, F., 2007. Towards accurate PWL approximations of parameter-dependent nonlinear dynamical systems with equilibria and limit cycles. IEEE Trans. Circ. Syst. I, 54(3):620-631.

[18]Storace, M., de Feo, O., 2005a. PWL approximation of nonlinear dynamical systems. Part I: structural stability. J. Phys., 22:208-221.

[19]Storace, M., de Feo, O., 2005b. Piecewise-linear approximation of nonlinear dynamical systems. IEEE Trans. Circ. Syst. I, 51(4):830-842.

[20]Storace, M., Julián, P., Parodi, M., 2002. Synthesis of nonlinear multiport resistors: a PWL approach. IEEE Trans. Circ. Syst. I, 49(8):1138-1149.

[21]Storace, M., Repetto, L., Parodi, M., 2003. A method for the approximate synthesis of cellular nonlinear networks. Part 1: circuit definition. Int. J. Circ. Theor. Appl., 31(3):277-297.

[22]Tarela, J.M., Martinez, M.V., 1993. Region configurations for realizability of lattice piecewise-linear models. Math. Comput. Model., 30(11-12):17-27.

[23]Thomas, J., Olaru, S., Buisson, J., Dumur, D., 2009. Attainability and Set Analysis for Uncertain PWA Systems with Parameter Variations and Bounded Disturbance. 21st Chinese Control and Decision Conf., p.4238-4244.

[24]Ulbig, A., Olaru, S., Dumur, D., Boucher, P., 2010. Explicit nonlinear predictive control for a magnetic levitation system. Asian J. Control, 12(3):434-442.

[25]Zhai, G., Lin, H., Antsaklis, P.J., 2003. Quadratic stabilizability of switched linear systems with polytopic uncertainties. Int. J. Control, 76(7):747-753.

[26]Zhang, L., Wang, C., Chen, L., 2009. Stability and stabilization of a class of multimode linear discrete-time systems with polytopic uncertainties. IEEE Trans. Ind. Electron., 56(9):3684-3692.

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