Article
Full entry |
PDF
(1.4 MB)
Feedback
PDF
(1.4 MB)
Feedback
Keywords:
nonlinear output regulation; singularly perturbed equation; gyroscope
nonlinear output regulation; singularly perturbed equation; gyroscope
Summary:
The regulator equation is the fundamental equation whose solution must be found in order to solve the output regulation problem. It is a system of first-order partial differential equations (PDE) combined with an algebraic equation. The classical approach to its solution is to use the Taylor series with undetermined coefficients. In this contribution, another path is followed: the equation is solved using the finite-element method which is, nevertheless, suitable to solve PDE part only. This paper presents two methods to handle the algebraic condition: the first one is based on iterative minimization of a cost functional defined as the integral of the square of the algebraic expression to be equal to zero. The second method converts the algebraic-differential equation into a singularly perturbed system of partial differential equations only. Both methods are compared and the simulation results are presented including on-line control implementation to some practically motivated laboratory models.
References:
[1] C. I. Byrnes, F. Delli Priscoli, A. Isidori, and W. Kang: Structurally stable output regulation of nonlinear systems. Automatica 33 (1997), 369–285. MR 1442555
[2] S. Čelikovský and B. Rehák: Output regulation problem with nonhyperbolic zero dynamics: a FEMLAB-based approach. In: Proc. 2nd IFAC Symposium on System, Structure and Control 2004, Oaxaca 2004, pp. 700–705.
[3] S. Čelikovský and B. Rehák: FEMLAB-based error feedback design for the output regulation problem. In: Proc. Asian Control Conference 2006. Bandung 2006.
[4] S. Devasia: Nonlinear inversion-based output tracking. IEEE Trans. Automat. Control 41 (1996), 930–942. MR 1398777 | Zbl 0859.93006
[5] B. A. Francis and W. M. Wonham: The internal model principle of control theory. Automatica 12 (1976), 457–465. MR 0429257
[6] B. A. Francis: The linear multivariable regulator problem. SIAM J. Control Optim. 15 (1977), 486–505. MR 0446631
[7] J. S. A. Hepburn and W. M. Wonham: Error feedback and internal models on differentiable manifolds. IEEE Trans. Automat. Control 29 (1981), 397–403. MR 0748204
[8] J. Huang: Nonlinear Output Regulation: Theory and Applications. SIAM, New York 2004. MR 2308004 | Zbl 1087.93003
[9] J. Huang: Output regulation of nonlinear systems with nonhyperbolic zero dynamics. IEEE Trans. Automat. Control 40 (1995), 1497–1500. MR 1343825 | Zbl 0841.93022
[10] J. Huang: On the solvability of the regulator equations for a class of nonlinear systems. IEEE Trans. Automat. Control 48 (2003), 880–885. MR 1980601
[11] J. Huang and W. J. Rugh: On a nonlinear multivariable servomechanism problem. Automatica 26 (1990), 963–972. MR 1080983
[12] A. Isidori and C. I. Byrnes: Output regulation of nonlinear systems. IEEE Trans. Automat. Control 35 (1990), 131–140. MR 1038409
[13] A. Isidori: Nonlinear Control Systems. Third edition. Springer, Berlin – Heidelberg – New York 1995. Zbl 0931.93005
[14] H. K. Khalil: Nonlinear Systems. Pearson Education Inc., Upper Saddle River 2000. Zbl 1140.93456
[15] B. Rehák, J. Orozco-Mora, S. Čelikovský, and J. Ruiz-León: FEMLAB-based output regulation of nonhyperbolically nonminimum phase system and its real-time implementation. In: Proc. 16th IFAC World Congress, Prague, IFAC 2005.
[16] B. Rehák, J. Orozco-Mora, S. Čelikovský, and J. Ruiz-León: Real-time error-feedback output regulation of nonhyperbolically nonminimum phase system. In: Proc. American Control Conference 2007, New York 2007.
[17] B. Rehák and S. Čelikovský: Numerical method for the solution of the regulator equation with application to nonlinear tracking. Automatica 44 (2008), 1358–1365. MR 2531803
[18] H.-G. Roos, M. Stynes, and L. Tobiska: Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin 1996. MR 1477665
[19]
J. Ruiz-León, J. L. Orozco-Mora, and D. Henrion: Real-time $H_{2}$ and $H_{\infty }$ control of a gyroscope using Polynomial Toolbox 2.5. In: Latin-American Conference on Automatic Control CLCA 2002, Guadalajara 2002.
[20] A. B. Vasil’eva and B. F. Butuzov: Asymptotic Expansions of Solutions of Singularly Perturbed Equations (in Russian). Nauka, Moscow 1973. MR 0477344
Search
Partner of
