A polynomial approach to a nonlinear model matching problem; pp. 330–344Full article in PDF format | https://doi.org/10.3176/proc.2016.4.02
The paper studies the model matching problem for nonlinear systems, described by a higher order input–output differential equation, not necessarily realizable in the state–space form. Only the feedforward solution is looked for. The probleem statement and solution rely on the recently introduced concept of a generalized transfer function for nonlinear systems. We require that the transfer functions of the compensated system and that of the prespecified model be equal, like in the linear case. However, in the nonlinear transfer function formalism one does not work with equations but with differential one-forms, and the existence of the compensator is restricted by integrability of the one-form corresponding to the compensator. Necessary and sufficient but nonconstructive solvability conditions are given. The second theorem lists a number of different (constructive) conditions under which the one-form is integrable. Additional freedom is sometimes obtained by forcing the conditions of the second theorem to hold via introducing assumptions suggested by the Euclidean division algorithm.
1. Belikov, J., Halás, M., Kotta, Ü., and Moog, C. H. Model matching problem for discrete-time nonlinear systems. Proc. Estonian Acad. Sci., 2015, 64, 457–472.
2. Conte, G., Moog, C. H., and Perdon, A. M. Algebraic Methods for Nonlinear Control Systems. Springer-Verlag, London, 2007.
3. Di Benedetto, M. D. Nonlinear strong model matching. IEEE Trans. Autom. Control, 1990, 35, 1351–1355.
4. Di Benedetto, M. D. and Isidori, A. The matching of nonlinear models via dynamic state feedback. In Proceedings of the 23rd IEEE Conference on Decision and Control, Las Vegas, NV, USA, December 1984, 416–420.
5. Fliess, M. Une interprétation algébrique de la transformation de laplace et des matrices de transfert. Linear Algebra Appl., 1994, 203, 429–442.
6. Fliess, M., Lévine, J., Martin, P., and Rouchon, P. Flatness and defect of non-linear systems: introductory theory and examples. Int. J. Contr., 1995, 61, 1327–1361. 344 Proceedings of the Estonian Academy of Sciences, 2016, 65, 4, 330–344
7. Glad, S. T. Nonlinear regulators and Ritt’s remainder algorithm. In Analysis of Controlled Dynamical Systems (Gauthier, J. P., Kupka, I., Bournard, B., and Bride, B., eds), Birkh¨auser, Boston, 1991, 224–232.
8. Halás, M. An algebraic framework generalizing the concept of transfer functions to nonlinear systems. Automatica, 2008, 44, 1181–1190.
9. Halás,M. and Kotta, Ü. Realization problem of SISO nonlinear systems: a transfer function approach. In The 7th International Conference on Control and Automation, Christchurch, New Zealand, December 2009, 546–551.
10. Halás, M. and Kotta, Ü. A transfer function approach to the realisation problem of nonlinear control systems. Int. J. Contr., 2012, 85(3), 320–331.
11. Halás, M., Kotta, Ü., and Moog, C. H. Transfer function approach to the model matching problem of nonlinear systems. In The 17th IFAC World Congress, Seoul, Korea, July 2008, 15197–15202.
12. Huijberts, H. J. C. A nonregular solution of the nonlinear dynamic disturbance decoupling problem with an application to a complete solution of the nonlinear model matching problem. SIAM J. Control Optim., 1992, 30(2), 350–366.
13. Johnson J. Kähler differentials and differential algebra. Ann. Math., 1969, 89(1), 92–98.
14. Kotta, Ü. Comments on a structural approach to the nonlinear model matching problem. SIAM J. Control Optim., 1994, 32(6), 1555–1558.
15. Kučera, V. and Toledo, E. C. A review of stable exact model matching by state feedback. In The 22nd Mediterranean Conference on Control and Automation, Palermo, Italy, June 2014, 85–90.
16. Li, Z., Ondera, M., and Wang, H. Simplifying skew fractions modulo differential and difference relations. In International Symposium on Symbolic and Algebraic Computation, Linz, Austria, 2008.
17. Marinescu, B. Model-matching topics for linear time-varying systems: computation rules in an algebraic approach. In American Control Conference, July 2007, 4357–4362.
18. Marinescu, B. and Bourlès, H. The exact model-matching problem for linear time-varying systems: an algebraic approach. IEEE Trans. Autom. Control, 2003, 48(1), 166–169.
19. Moog, C. H., Perdon, A., and Conte, G. Model matching and factorization for nonlinear systems: a structural approach. SIAM J. Control Optim., 1991, 29(4), 769–785.
20. Ore, O. Linear equations in non-commutative fields. Ann. Math., 1931, 32(3), 463–477.
21. Ore, O. Theory of non-commutative polynomials. Ann. Math., 1933, 34, 480–508.
22. Perdon, A. M., Moog, C. H., and Conte, G. The pole-zero structure of nonlinear control systems. In 7th IFAC Symposium NOLCOS, Pretoria, South Africa, August 2007, Vol. 7, 717–720.
23. Pommaret, J. F. Géométrie differentielle algébrique et théorie du contrôle. CR Acad. Sci. Paris, 1986, 302, 547–550.
24. Rudolph, J. Viewing input-output system equivalence from differential algebra. JMSEC, 1994, 4(3), 353–383.
25. Yamamoto, Y. Simple model matching control of nonlinear discrete time systems. In International Conference on Control, Automation and Systems, 14–17 October 2008, Seoul. IEEE, 2008, 2325–2329.
26. Zheng, Y. and Cao, L. Transfer function description for nonlinear systems. Journal East China Normal University, 1995, 2, 15–26.
27. Zheng, Y., Willems, J. C., and Zhang, C. A polynomial approach to nonlinear system controllability. IEEE Trans. Autom. Control, 2001, 46(11), 1782–1788.
Back to Issue