LUT Controller Design with Piecewise Bilinear Systems Using Estimation of Bounds for Approximation Errors
Tadanari Taniguchi*, Luka Eciolaza**, and Michio Sugeno**
*Tokai University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan
**European Center for Soft Computing, c/ Gonzalo Gutierrez Quieros, s/n, 33600 Mieres, Asturias, Spain
We propose the stabilization of nonlinear control systems approximated by Piecewise Bilinear (PB) models. The approximated model is fully parametric and a Look-Up-Table (LUT) represents its controller. Input-Output (I/O) feedback linearization is applied to stabilize PB control systems. We further propose PB modeling combined with conventional feedback linearization as a very powerful tool for analyzing and synthesizing nonlinear control systems. We also propose a method for designing robust stabilization controllers taking modeling error into consideration. Examples confirm the feasibility of our proposals.
-  J. Imura and A. van der Schaft, “Characterization of well-posedness of piecewise-linear systems,” IEEE Trans. Autom. Control, Vol.45, pp. 1600-1619, 2000.
-  M. Johansson and A. Rantzer, “Computation of piecewise quadratic lyapunov functions of hybrid systems,” IEEE Trans. Autom. Control, Vol.43, Issue 4, pp. 555-559, 1998.
-  E. D. Sontag, “Nonlinear regulation: the piecewise linear approach,” IEEE Trans. Autom. Control, Vol.26, pp. 346-357, 1981.
-  D. A. Babayev, “Piece-wise linear approximation of functions of two variable,” J. of Heuristics, Vol.2, pp. 313-320, 1997.
-  H. Grandin, Jr., “Fundamentals of the Finite Element Method,” Macmillan, 1986.
-  J. S. Shamma and M. Athans, “Analysis of gain scheduled control for nonlinear plants,” IEEE Trans. Autom. Control, pp. 898-907, 1990.
-  T. Takagi and M. Sugeno, “Fuzzy identification of systems and its application to medelling and control,” IEEE Trans. Systems, Man and Cybernetics, SMC-15:116-132, 1985.
-  K. Tanaka and H. O. Wang, “Fuzzy control systems design and analysis: A linear matrix inequality approach,” John Wiley & Sons, 2001.
-  M. Sugeno, “On stability of fuzzy systems expressed by fuzzy rules with singleton consequents,” IEEE Trans. Fuzzy Syst., Vol.7, Issue 2, pp. 201-224, 1999.
-  M. Sugeno and T. Taniguchi, “On improvement of stability conditions for continuous mamdani-like fuzzy systems,” IEEE Tran. Systems, Man, and Cybernetics, Part B, Vol.34, Issue 1, pp. 120-131, 2004.
-  T. Taniguchi and M. Sugeno, “Stabilization of nonlinear systems based on piecewise lyapunov functions,” FUZZ-IEEE 2004, pp. 1607-1612, 2004.
-  K.-C. Goh, M. G. Safonov, and G. P. Papavassilopoulos, “A global optimization approach for the BMI problem,” Proc. the 33rd IEEE CDC, Vol.3, pp. 2009-2014, 1994.
-  T. Taniguchi and M. Sugeno, “Piecewise bilinear system control based on full-state feedback linearization,” SCIS & ISIS 2010, pp. 1591-1596, 2010.
-  T. Taniguchi and M. Sugeno, “Stabilization of nonlinear systems with piecewise bilinear models derived from fuzzy if-then rules with singletons,” FUZZ-IEEE 2010, pp. 2926-2931, 2010.
-  T. Taniguchi and M. Sugeno, “Design of LUT-controllers for nonlinear systems with PB models based on I/O linearization,” FUZZIEEE 2012, pp. 997-1022, 2012.
-  G. O. Guarabassi and S. M. Savaresi, “Approximate linearization via feedback – an overview,” Automatica, Vol.37, pp. 1-15, 2001.
-  A. J. Krener, “Approximate linearization by state feedback and coordinate change,” Systems & Control Letters, Vol.5, pp. 181-185, 1984.
-  C. Reboulet and C. Champetier, “A new method for linearizing nonlinear systems: the pseudolinearization,” Int. J. of Control, Vol.40, pp. 631-638, 1984.
-  W. J. Rugh, “Design of nonlinear compensators for nonlinear systems by an extended linearization technique,” 23rd control and decision conf., No.69-73, 1984.
-  G. O. Guardabassi, A. Righettini, and I. Ruffoni, “Output linearizable models and nonlinear control of a distillation column,” System Science, 12, 1986.
-  J. Hauser, “Nonlinear control via uniform system approximation,” Systems & Control Letters, Vol.17, pp. 145-154, 1991.
-  C. A. Desoer and Y. T. Wang, “Foundations of feedback theory for nonlinear dynamical systems,” IEEE Trans. on Circuit and Systems, Vol.27, No.2, pp. 104-123, 1980.
-  A. J. Stack and F. J. Doyle III, “A measure for control relevant nonlinearity,” American control conf., No.2200-2204, 1995.
-  F. Allgöwer, A. Rehm, and E. D. Gilles, “An engineering perspective on nonlinear H∞ control,” 33rd control and decision conf., pp. 2537-2542, 1994.
-  R. Isermann, S. Ernst, and O. Nelles, “Identification with dynamic neural networks – architectures, comparisons, applications,” IFAC symp. on system identification, pp. 997-1022, 1997.
-  T. Taniguchi and M. Sugeno, “Robust stabilization of nonlinear systems modeled with piecewise bilinear systems based on feedback linearization,” Advances on Computational Intelligence, Vol.297 of Communications in Computer and Information Science, pp. 111-120, Springer, 2012.
-  H. K. Khalil, “Nonlinear systems,” (Third Ed.), Prentice hall, 2002.
-  A. Isidori, “The matching of a prescribed linear input-output behavior in a nonlinear system,” IEEE Trans. Autom. Control, Vol.30, Issue 3, pp. 258-265, 1985.
-  S. Sastry, “Nonlinear Systems,” Springer, 1999.
This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.