Observer-Based Piecewise Multi-Linear Controller Designs for Nonlinear Systems Using Feedback and Observer Linearizations
Tadanari Taniguchi* and Michio Sugeno**
4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan
**Tokyo Institute of Technology
4259 Nagatsuta-cho, Midori-ku, Yokohama, Kanagawa 226-8503, Japan
This paper proposes observer-based piecewise multi-linear controllers for nonlinear systems using feedback and observer linearizations. The piecewise model is a nonlinear approximation and fully parametric. Feedback linearizations are applied to stabilize the piecewise multi-linear control system. Furthermore, observer linearizations are more conservative in modeling errors compared with feedback linearizations. In this paper, we propose robust observer designs for piecewise multi-linear systems. Moreover, we design piecewise multi-linear controllers that combine the robust observer with various performance such as a regulator and tracking controller. These design methods realize a separation principle that allows an observer and a regulator to be designed separately. Examples are demonstrated through computer simulation to confirm the feasibility of our proposals.
-  E. D. Sontag, “Nonlinear regulation: the piecewise linear approach,” IEEE Trans. Autom. Control, Vol.26, No.2, pp. 346-358, doi: 10.1109/TAC.1981.1102596, 1981.
-  M. Johansson and A. Rantzer, “Computation of piecewise quadratic Lyapunov functions for hybrid systems,” IEEE Trans. Autom. Control, Vol.43, No.4, pp. 555-559, doi: 10.1109/9.664157, 1998.
-  J. Imura and A. van der Schaft, “Characterization of well-posedness of piecewise-linear systems,” IEEE Trans. Autom. Control, Vol.45, No.9, pp. 1600-1619, doi: 10.1109/9.880612, 2000.
-  G. Feng, G. P. Lu, and S. S. Zhou, “An approach to H∞ controller synthesis of piecewise linear systems,” Communications in Information and Systems, Vol.2, No.3, pp. 245-254, doi: 10.4310/CIS.2002.v2.n3.a2, 2002.
-  M. Sugeno, “On stability of fuzzy systems expressed by fuzzy rules with singleton consequents,” IEEE Trans. Fuzzy Syst., Vol.7, No.2, pp. 201-224, doi: 10.1109/91.755401, 1999.
-  M. Sugeno and T. Taniguchi, “On improvement of stability conditions for continuous Mamdani-like fuzzy systems,” IEEE Trans. Systems, Man, and Cybernetics, Part B, Vol.34, No.1, pp. 120-131, doi: 10.1109/TSMCB.2003.809226, 2004.
-  K.-C. Goh, M. G. Safonov, and G. P. Papavassilopoulos, “A global optimization approach for the BMI problem,” Proc. of the 33rd IEEE Conf. on Decision and Control, Vol.3, pp. 2009-2014, doi: 10.1109/CDC.1994.411445, 1994.
-  T. Taniguchi and M. Sugeno, “Stabilization of Nonlinear Systems with Piecewise Bilinear Models derived from Fuzzy If-Then Rules with Singletons,” Proc. of Int. Conf. on Fuzzy Systems, pp. 2926-2931, doi: 10.1109/FUZZY.2010.5584807, 2010.
-  T. Taniguchi, L. Eciolaza, and M. Sugeno, “Full-Order State Observer Design for Nonlinear Systems Based on Piecewise Bilinear Models,” Int. J. of Modeling and Optimization, Vol.4, No.2, pp. 120-125, doi: 10.7763/IJMO.2014.V4.358, 2014.
-  T. Taniguchi, L. Eciolaza, and M. Sugeno, “Designs of Minimal-Order State Observer and Servo Controller for a Robot Arm Using Piecewise Bilinear Models,” Proc. of The 2014 IAENG Int. MultiConf. of Engineers and Computer Scientists, 2014.
-  T. Taniguchi and M. Sugeno, “Robust Observer-Based Piecewise Multi-Linear Controller Design for Nonlinear Systems,” Proc. of 2018 Joint 10th Int. Conf. on Soft Computing and Intelligent Systems (SCIS) and 19th Int. Symp. on Advanced Intelligent Systems (ISIS), pp. 902-907, doi: 10.1109/SCIS-ISIS.2018.00149, 2018.
-  T. Taniguchi, L. Eciolaza, and M. Sugeno, “LUT Controller Design with Piecewise Bilinear Systems Using Estimation of Bounds for Approximation Errors,” J. Adv. Comput. Intell. Intell. Inform., Vol.17, No.6, pp. 828-840, doi: 10.20965/jaciii.2013.p0828, 2013.
-  A. Isidori, “The matching of a prescribed linear input-output behavior in a nonlinear system,” IEEE Trans. Autom. Control, Vol.30, No.3, pp. 258-265, doi: 10.1109/TAC.1985.1103939, 1985.
-  S. Sastry, “Nonlinear Systems,” Springer, 1999.
-  T. Taniguchi and M. Sugeno, “Robust Lookup Table Controller Based on Piecewise Multi-Linear Model for Nonlinear Systems with Parametric Uncertainty,” Proc. of 17th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems, pp. 727-738, doi: 10.1007/978-3-319-91479-4_60, 2018.
-  H. Guillard and H. Bourlès, “Robust feedback linearization,” Proc. of 14th Int. Symp. Mathematical Theory of Networks and Systems, pp. 1-6, 2000.
-  T. Taniguchi, L. Eciolaza, and M. Sugeno, “Look-Up-Table Controller Design for Nonlinear Servo Systems with Piecewise Bilinear Models,” Proc. of 2013 IEEE Int. Conf. on Fuzzy Systems (FUZZ-IEEE), doi: 10.1109/FUZZ-IEEE.2013.6622512, 2013.
-  R. Sepulchre, M. Jankovic, and P. V. Kokotovic, “Constructive Nonlinear Control,” Springer, 1997.