Abstract
Based on state space reachable sets we formulate a two-player differential game for stability. The role of one player (the bounded disturbance) is to remove as much of the system’s stability as possible, while the second player (the control) tries to maintain as much of the system’s stability as possible. To obtain explicit computable relationships we limit the control selection to setting basic parameters of the system in their stability range and the disturbance is a bounded scalar function. This stability game provides for an explicit quantification of uncertainty in control systems and the value in the game manifests itself as the L ∞-gain of the dynamic input/output disturbance system as a function of the control parameters. It has been discovered recently that the L ∞-gain can be expressed as an explicit parametric formula for linear second order games, and design charts here provide quantitative information in third order linear games. A model predictive scheme will be given for the higher order linear and nonlinear stability games.
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References
Friedman A. (1971) Differential Games, Wiley, New York.
Ryan E. P. (1980) On the Sensitivity of a Time-Optimal Switching Function, IEEE Trans. on Automatic Control, 25, 275–277.
Doyle J. C., Francis B. A. et al. (1992) Feedback Control Theory, Macmillan, New York.
Franklin G., Powell J. D. et al. (1994) Feedback Control of Dynamic Systems, 3rd edn, Addison-Wesley.
Koivuniemi A. J. (1966) Parameter Optimization in Systems Subject to Worst (Bounded) Disturbance, IEEE Trans. on Automatic Control, 11, 427–433.
Polanski A. (2000) Destability Strategies for Uncertain Linear Systems, IEEE Trans. on Automatic Control, 45, 2378–2382.
Wang H. H., Krstic M. (2000) Extreme Seeking for Limit Cycle Minimization, IEEE Trans. on Automatic Control, 45, 2432–2437.
Kirk D. E. (1970) Optimal Control theory-An Introduction, Prentice-Hall, New Jersey.
Lee E. B., Markus L. (1967) Foundations of Optimal Control Theory, Wiley, New York.
Lee E. B., Luo J. C. (2000) On Evaluating the Bounded Input Bounded Output Stability Integral for Second Order Systems, IEEE Trans. on Automatic Control, 45, 311–312.
Luo J. C., Lee E. B. (2000) A Closed Analytic Form for the Time Maximum Disturbance Isochrones of Second Order Linear Systems, System & Control Letters, 40, 229–245.
You K. H., Lee E. B. (2000) Time Maximum Disturbance Switch Curve and Isochrones of Linear Second Order Systems with Numerator Dynamics, Journal of The Franklin Institute, 337, 725–742.
You K. H., Lee E. B. (2001) Time Maximum Disturbance Design for Stable Linear Systems; A Model Predictive Scheme, IEEE Trans. on Automatic Control, 46, 1327–1332.
Meadows E. S., Rawlings J. B. (1995) Topics in Model Predictive Control in Methods of Model Based Process Control, Kluwer, New York.
Rawlings J. B. (1999) Tutorial: Model Predictive Control Technology, Proc. of 1999 ACC., 662–676.
Martin G. (1999) Nonlinear Model Predictive Control, Proc. of 1999 ACC., 677–678.
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You, KH., Lee, E.B. (2002). The Stability Game. In: Pasik-Duncan, B. (eds) Stochastic Theory and Control. Lecture Notes in Control and Information Sciences, vol 280. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48022-6_35
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DOI: https://doi.org/10.1007/3-540-48022-6_35
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