Abstract
Shaping the response of a control system has long been a central problem in the analysis and design of feedback systems. The widespread use of both frequency domain techniques and state-space methods is at least in part due to the relative ease and intuitive content of these methods in addressing problems such as asymptotic tracking and disturbance attenuation for linear systems. Recently, a combination of methods drawn from geometric nonlinear control theory and from nonlinear dynamics was developed to give an admittedly unanticipated local solution to the nonlinear regulator problem, yielding necessary and sufficient conditions for nonlinear regulation for the class of detectable and stabilizable nonlinear systems ([1], [2]). In section 2, we state the basic nonlinear regulator problem and give conditions for solvability of the problem in terms of the solvability of a system of nonlinear partial differential equations. In the linear case, these “regulator equations” coincide with the linear equations derived by Francis [3] in his rather complete treatment of the linear multivariable regulator problem. The derivation of the nonlinear regulator equations and the consequent design of a nonlinear controller repose on two essential problems: feedback stabilization for nonlinear systems, a research area currently enjoying intense activity and success, and an analysis of the “steady-state response” of a nonlinear system to a driving input. In section 3, we sketch our solution to the problem of existence of such a steady-state response using center manifold methods, from which the regulator equations can be derived mutatis mutandis.
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References
C.I. Byrnes and A. Isidori, Règulation asymptotique des systèmes nonlinedres, C.R. Acad. Sci. Paris 309 (1989), 527–530.
A. Isidori and C.I. Byrnes, Output Regulation of Nonlinear Systems, IEEE Trans. Aut. Contr. AC-35 (1990), 131–140.
B.A. Francis, The Linear Multivariable Regulator Problems, SIAM J. Contr. Optimiz. 115 (1977), 486–505.
C.I. Byrnes, “Some Partial Differential Equations Arising in Nonlinear Control, Computation and Control, II,” (K. Bowers, J. Lund, eds.),Birkhaiiser-Boston, to appear..
E.J. Davison, The Output Control of Linear Time-Invariant Multi-Variable System with Unmeasurable Arbitrary Disturbances, IEEE Trans. Aut Contr AC-17 (1972), 621–630.
B.A. Francis and W.M. Wonham, The Internal Model Principle for Linear Multivariable Regulators, J. Appl. Math. Optimiz. 2 (1975), 170–194.
C.I. Byrnes and A. Isidori, Steady State Response, Separation Principle and the Output Regulation of Nonlinear Systems, Proceedings of the 28th IEEE Conference on Decision and Control, Tampa (1989), 2247–2251.
J.E. Marsden and M. McCracken, “The Hopf Bifurcation and Its Applications,” Springer-Verlag, New York, Heidelberg, Berlin, 1976.
M. Hautus, Linear Matrix Equations with Application to the Regulator Problem,Outils and Modeles Mathematique pour l’Automatiquechrw(133)(I.D.Landau ed.),C.N.R.S. (1983), 399–412.
C.I. Byrnes and A. Isidori, A Frequency Domain Philosophy for Nonlinear Systems with Applications to Stabilization and Adaptive Control, Proc. of 23rd IEEE Conf. on Dec. and Control, Las Vegas, NV (1984).
A.J. Krener, A. Isidori, Nonlinear Zero Distributions, Proc. of the 19th IEEE Conf. on Dec. and Control, Albuquerque (1980).
C.I. Byrnes and A. Isidori, “Heuristics for Nonlinear Control,” in Modelling and Adaptive Control, (Proc. of the IIASA Conf. Sopron, July 1986, C.I. Byrnes, A. Kurzhansky, eds.), Springer-Verlag, Berlin, 1988, pp. 48–70.
A. Isidori and C. Moog, On the Nonlinear Equivalent of the Notion of Transmission Zeros, in Modelling and Adaptive Control, (Proc. of the IIASA Conf. Sopron, July 1986, C.I. Byrnes, A. Kurzhansky, eds.), Springer-Verlag, Berlin.
C.I. Byrnes and A. Isidori, Local Stabilization of Minimum-phase Nonlinear Systems, Systems and Control Letters 11 (1988), 9–17.
J.P. Aubin, C.I. Byrnes and A. Isidori, Viability Kernels, Controlled Invariance and Zero Dynamics for Nonlinear Systems in Analysis and Optimization of Systems, (Proc. of 9th Int’l Conf., Antibes, June 1990, A. Bensoussan and J.L. Lions, eds) (1990), 821–832. Springer-Verlag, Berlin.
J.P. Aubin and H. Frankowska, Viability Kernel of Control Systems, Nonlinear Synthesis (C. I. Byrnes and A. Kurzhansky, eds ). Birkhâuser-Boston, 1991.
X.M. Hu, Robust Stabilization of Nonlinear Control Systems, Ph.D. dissertation (1989). Arizona State University.
A. Ben-Artzi and J.W. Helton, A Riccati Partial Differential Equation for Factoring Nonlinear Systems,preprint.
A.P. Willemstein, Optimal Regulation of Nonlinear Systems on a Finite Interval, SIAM J. Control Opt. 15 (1977), 1050–1069.
D.L. Lukes, Optimal Regulation of Nonlienar Dynamical Systems, SIAM J. Control and Opt. 7 (1969), 75–100.
E.G. Al’brekht, On the optimal stabilization of nonlinear systems, J. Appl. Math. Mech. 25 (1962), 1254–1266.
E.G. Al’brekht, Optimal stabilization of nonlinear systems,Mathematical Notes, vol. 4, no. 2, The Ural Mathematical Society, The Ural State University of A. M. Gor’kii, Sverdiovsk (1963). In Russian.
P. Brunovsky, On optimal stabilization of nonlinear systems, Mathematical Theory of Control, A. V. Balakrishnan and Lucien W. Neustadt, eds., Academic Press, New York and London (1967).
M.K. Sain (Ed), Applications of tensors to modelling and control, Control Systems Technical Report #38, Dept. of Elec. Eng., Notre Dame University (1985).
T. Yoshida and K.A. Loparo, Quadratic Regulatory Theory for Analytic Non-linear Systems with Additive Control, Automatics 25 (1989), 531–544.
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Byrnes, C.I., Isidori, A. (1991). New Methods for Shaping the Response of a Nonlinear System. In: Byrnes, C.I., Kurzhansky, A.B. (eds) Nonlinear Synthesis. Progress in Systems and Control Theory, vol 9. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-2135-5_3
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DOI: https://doi.org/10.1007/978-1-4757-2135-5_3
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