Foundations of feedback theory for nonlinear dynamical systems

We study the fundamental properties of feedback for nonlinear, time-varying, multi-input muldt-output, distributed systems. The classical Black formula is generalized to the nonlinear case. Achievable advantages and limitations of feedback in nonlinear dynamical systems are classified and studied in five categories: desensitization, disturbance attenuation, linearizing effect, asymptotic tracking and disturbance rejection, stabilization. Conditions under which feedback is beneficial for nonlinear dynamical systems are derived. Our results show that if the appropriate linearized inverse return difference operator is small, then the nonlinear feedback system has advantages over the open-loop system. Several examples are proyided to illustrate the results.

[1]  H. S. Black,et al.  Stabilized feedback amplifiers , 1934 .

[2]  H. W. Bode,et al.  Network analysis and feedback amplifier design , 1945 .

[3]  E. Hille.  Analytic Function Theory , 1961 .

[4]  I. Horowitz.  Synthesis of feedback systems , 1963 .

[5]  G. Zames.  Functional Analysis Applied to Nonlinear Feedback Systems , 1963 .

[6]  J. Cruz,et al.  A new approach to the sensitivity problem in multivariable feedback system design , 1964 .

[7]  I. Sandberg.  Some results on the theory of physical systems governed by nonlinear functional equations , 1965 .

[8]  Austin Blaquière,et al.  Nonlinear System Analysis , 1966 .

[9]  G. Zames.  On the input-output stability of time-varying nonlinear feedback systems--Part II: Conditions involving circles in the frequency plane and sector nonlinearities , 1966 .

[10]  G. Zames.  On the input-output stability of time-varying nonlinear feedback systems Part one: Conditions derived using concepts of loop gain, conicity, and positivity , 1966 .

[11]  W. Wonham.  On pole assignment in multi-input controllable linear systems , 1967 .

[12]  J. Dieudonne.  Foundations of Modern Analysis , 1969 .

[13]  J. Pearson,et al.  Pole placement using dynamic compensators , 1970 .

[14]  O. Mayr.  The Origins of Feedback Control , 1970 .

[15]  S. Mitter,et al.  Controllability, observability, pole allocation, and state reconstruction , 1971 .

[16]  Jacob Katzenelson,et al.  A generalized Nyquist-type stability criterion for multivariable feedback systems† , 1974 .

[17]  C. Desoer,et al.  The feedback interconnection of lumped linear time-invariant systems☆ , 1975 .

[18]  D.L. Elliott,et al.  Feedback systems: Input-output properties , 1976, Proceedings of the IEEE.

[19]  H. Kwakernaak.  Asymptotic root loci of multivariable linear optimal regulators , 1976 .

[20]  Dante C. Youla,et al.  Modern Wiener-Hopf Design of Optimal Controllers. Part I , 1976 .

[21]  E. Davison.  The robust control of a servomechanism problem for linear time-invariant multivariable systems , 1976 .

[22]  H. S. Black,et al.  Inventing the negative feedback amplifier: Six years of persistent search helped the author conceive the idea “in a flash” aboard the old Lackawanna Ferry , 1977, IEEE Spectrum.

[23]  R. Curtain,et al.  Functional Analysis in Modern Applied Mathematics , 1977 .

[24]  W. Kwon,et al.  A modified quadratic cost problem and feedback stabilization of a linear system , 1977 .

[25]  I. Postlethwaite,et al.  The generalized Nyquist stability criterion and multivariable root loci , 1977 .

[26]  Bruce A. Francis,et al.  The multivariable servomechanism problem from the input-output viewpoint , 1977 .

[27]  Edward W. Kamen,et al.  An operator theory of linear functional differential equations , 1978 .

[28]  E. Cherry.  A new result in negative-feedback theory, and its application to audio power amplifiers , 1978 .

[29]  W. Kwon,et al.  On feedback stabilization of time-varying discrete linear systems , 1978 .

[30]  J. Doyle.  Robustness of multiloop linear feedback systems , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[31]  Charles A. Desoer,et al.  Pertubation in the I/O Map of a Non-linear Feedback System Caused by Large Plant Perturbation☆ , 1978 .

[32]  C. Desoer,et al.  The robust nonlinear servomechanism problem , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[33]  C. Desoer,et al.  On the generalized Nyquist stability criterion , 1979, 1979 18th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[34]  V. Cheng.  A direct way to stabilize continuous-time and discrete-time linear time-varying systems , 1979 .

[35]  C. Desoer,et al.  Linear Time-Invariant Robust Servomechanism Problem: A Self-Contained Exposition , 1980 .