Nonlinear Synthesis pp 34-52 | Cite as

# New Methods for Shaping the Response of a Nonlinear System

## 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.

## Keywords

Center Manifold Output Regulation Stable Limit Cycle Linear Quadratic Regulator Regulator Equation## Preview

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## References

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