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TECHNICAL PAPERS

Observer and Controller Design Using Time-Varying Gains: Duality and Distinctions

[+] Author and Article Information
Vladimir Polotski

 Frontline Robotics, 1920 Research Road, U62, NRC Uplands Campus Ottawa, ON K1V984, Canada Fax: (613) 739–2687vpolotski@frontline-robotics.ca

J. Dyn. Sys., Meas., Control 127(2), 267-274 (Jul 14, 2004) (8 pages) doi:10.1115/1.1898230 History: Received July 05, 2002; Revised July 14, 2004

Stabilization of linear systems by state feedback is an important problem of the controller design. The design of observers with appropriate error dynamics is a dual problem. This duality leads, at first glance, to the equivalence of the responses in the synthesized systems. This is true for the time-invariant case, but may not hold for time-varying systems. We limit ourselves in this work by the situation when the system itself is time invariant, and only the gains are time varying. The possibility of assigning a rapidly decaying response without peaking is analyzed. The solution of this problem for observers using time-varying gains is presented. Then we show that this result cannot be obtained for state feedback controllers. We also analyze the conditions under which the observer error dynamics and the response of the closed loop time-varying controllers are equivalent. Finally we compare our results to recently proposed observer converging in finite time and Riccati-based continuous observer with limited overshoots.

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Copyright © 2005 by American Society of Mechanical Engineers
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Figures

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Figure 1

Continuous matrix commuting with matricant and not commuting with intregral

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Figure 2

Phase portraits of time-varying vs time-invariant observers

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Figure 3

Transition processes

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Figure 4

Finite-time observer dynamics

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Figure 5

Continuous (Riccati-based) vs piecewise continuous observer

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Figure 6

Fundamental solutions: continuous (Riccati-based) vs piecewise continuous case

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