Time varying optimal control of a non-linear system

M.J. Grimble, P. Martin

Research output: Contribution to conferencePaper

1 Citation (Scopus)

Abstract

The solution is given to a time-varying optimal state feedback problem with stochastic disturbances. The system is composed of a plant and disturbance model represented by polynomials in the delay operator, z(-1), leading to a solution involving spectral factorisation of operator equations and Diophantine operator equations. The cost function is over infinite time and the assumption is made that the system is time-varying for T steps into the future from the current sample and time-invariant thereafter. For a time-invariant system over infinite time, the optimal controller is a constant state-feedback matrix gain. Thus, with the assumption of time-invariance from T to, the feedback gain may be calculated using constant system polynomials. The solution of the spectral factors and Diophantine equations may then be computed recursively, for a scalar plant, working from T steps ahead to the current time. The controller calculated for the current time is then applied to the system. If the input non-linearity of a plant is represented in time-varying form, the time-varying ideas may be used to produce a nonlinear controller for the system. The example in this paper is for a smooth saturation non-linearity represented by a tanh function. Simulation results are given and it is clear that performance gains over a time-invariant controller are possible.
LanguageEnglish
Pages3495-3500
Number of pages6
DOIs
Publication statusPublished - Dec 2003
Event42nd IEEE Conference on Decision and Control 2003 - Maui, United States
Duration: 9 Dec 200312 Dec 2003

Conference

Conference42nd IEEE Conference on Decision and Control 2003
CountryUnited States
CityMaui
Period9/12/0312/12/03

Fingerprint

Nonlinear systems
Mathematical operators
Controllers
State feedback
Polynomials
Time varying systems
Invariance
Factorization
Cost functions
Feedback

Keywords

  • time varying
  • optimal control
  • non-linear system
  • control systems
  • time varying systems
  • stochastic processes
  • state feedback
  • polynomials
  • nonlinear equations
  • nonlinear control systems
  • delay
  • cost function

Cite this

Grimble, M. J., & Martin, P. (2003). Time varying optimal control of a non-linear system. 3495-3500. Paper presented at 42nd IEEE Conference on Decision and Control 2003, Maui, United States. https://doi.org/10.1109/CDC.2003.1271689
Grimble, M.J. ; Martin, P. / Time varying optimal control of a non-linear system. Paper presented at 42nd IEEE Conference on Decision and Control 2003, Maui, United States.6 p.
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Grimble, MJ & Martin, P 2003, 'Time varying optimal control of a non-linear system' Paper presented at 42nd IEEE Conference on Decision and Control 2003, Maui, United States, 9/12/03 - 12/12/03, pp. 3495-3500. https://doi.org/10.1109/CDC.2003.1271689

Time varying optimal control of a non-linear system. / Grimble, M.J.; Martin, P.

2003. 3495-3500 Paper presented at 42nd IEEE Conference on Decision and Control 2003, Maui, United States.

Research output: Contribution to conferencePaper

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AB - The solution is given to a time-varying optimal state feedback problem with stochastic disturbances. The system is composed of a plant and disturbance model represented by polynomials in the delay operator, z(-1), leading to a solution involving spectral factorisation of operator equations and Diophantine operator equations. The cost function is over infinite time and the assumption is made that the system is time-varying for T steps into the future from the current sample and time-invariant thereafter. For a time-invariant system over infinite time, the optimal controller is a constant state-feedback matrix gain. Thus, with the assumption of time-invariance from T to, the feedback gain may be calculated using constant system polynomials. The solution of the spectral factors and Diophantine equations may then be computed recursively, for a scalar plant, working from T steps ahead to the current time. The controller calculated for the current time is then applied to the system. If the input non-linearity of a plant is represented in time-varying form, the time-varying ideas may be used to produce a nonlinear controller for the system. The example in this paper is for a smooth saturation non-linearity represented by a tanh function. Simulation results are given and it is clear that performance gains over a time-invariant controller are possible.

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Grimble MJ, Martin P. Time varying optimal control of a non-linear system. 2003. Paper presented at 42nd IEEE Conference on Decision and Control 2003, Maui, United States. https://doi.org/10.1109/CDC.2003.1271689