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Technical Briefs

Achieving Minimum Settling Time Subject to Undershoot Constraint in Systems With One or Two Real Right Half Plane Zeros

[+] Author and Article Information
Shen Zhao

National Superconducting Cyclotron Laboratory,
Michigan State University,
East Lansing, MI 48824
e-mail: zhaos@nscl.msu.edu

Wenchao Xue

Key Laboratory of Systems and Control,
Academy of Mathematics and Systems Science,
and National Center for Mathematics and
Interdisciplinary Sciences,
Chinese Academy of Sciences,
Beijing 100190, China
e-mail: wenchaoxue@amss.ac.cn

Zhiqiang Gao

Center for Advanced Control Technologies,
Department of Electrical and Computer Engineering,
Cleveland State University,
Cleveland, OH 44115
e-mail: z.gao@csuohio.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received November 4, 2011; final manuscript received November 25, 2012; published online March 28, 2013. Assoc. Editor: Rama K. Yedavalli.

J. Dyn. Sys., Meas., Control 135(3), 034505 (Mar 28, 2013) (5 pages) Paper No: DS-11-1346; doi: 10.1115/1.4023211 History: Received November 04, 2011; Revised November 25, 2012

This note concerns with the problem of achieving minimum settling time in linear systems with one or two real right half plane (RHP) zeros subject to the condition that undershoot does not exceed a given threshold. Such problem is of great practical significance, but it has not been formally addressed, to our knowledge. Time optimal control solutions for such systems are readily available based on the well known optimal control theory, but it does not address the practical consideration that the “wrong way response,” i.e., undershoot, must be limited. To be sure, the relationship between the minimum settling time and the undershoot constraint for systems with one or two real RHP zeros has already been given in the literature. How to find the control signal that achieves the minimum settling time, however, is still an open question. In this paper, such control signal is obtained constructively and, combined with feedback, is shown to be rather effective in controlling the system in the presence of model uncertainties and external disturbances, as shown in simulation.

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References

Hoagg, J. B., and Bernstein, D. S., 2007, “Nonminimum-Phase Zeros: Much to Do About Nothing,” IEEE Control Syst. Mag., 27(3), pp. 45–57. [CrossRef]
Skogestad, S., and Postlethwaite, I., 2005, Multivariable Feedback Control: Analysis and Design, 2nd ed., John Wiley, Hoboken, NJ, pp. 173–180.
Rigney, B. P., Pao, L. Y., and Lawrence, D. A., 2009, “Nonminimum Phase Dynamic Inversion for Settle Time Applications,” IEEE Trans. Control Syst. Technol., 17(5), pp. 989–1005. [CrossRef]
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Lau, K., Middleton, R. H., and Braslavsky, J. H., 2003, “Undershoot and Settling Time Tradeoffs for Nonminimum Phase Systems,” IEEE Trans. Autom. Control, 48(8), pp. 1389–1393. [CrossRef]
Widder, D. V., 1934, “The Inversion of the Laplace Integral and the Related Moment Problem,” Trans. Am. Math. Soc., 36(1), pp. 107–200. [CrossRef]
Åström, K. J., and Hägglund, T., 1995, PID Controllers: Theory, Design and Tuning, 2nd ed., Instrument Society of America, Research Triangle Park, NC, pp. 284–287.
Zhang, Y., 2009, “Load Frequency Control of Multiple Area Power Systems,” M.S. thesis, Cleveland State University, Cleveland, OH.

Figures

Grahic Jump Location
Fig. 1

Input and output signals in example 1

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Fig. 2

Control signal, its derivative and the output response in example 2

Grahic Jump Location
Fig. 3

The combined feedforward and feedback control system

Grahic Jump Location
Fig. 4

Control signals and output responses in example 3

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