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Research Papers

Tracking Control of Linear Time-Invariant Nonminimum Phase Systems Using Filtered Basis Functions

[+] Author and Article Information
Keval S. Ramani

G. G. Brown Laboratory,
Department of Mechanical Engineering,
University of Michigan,
2350 Hayward,
Ann Arbor, MI 48109
e-mail: ksramani@umich.edu

Molong Duan

G. G. Brown Laboratory,
Department of Mechanical Engineering,
University of Michigan,
2350 Hayward,
Ann Arbor, MI 48109
e-mail: molong@umich.edu

Chinedum E. Okwudire

Mem. ASME
G. G. Brown Laboratory,
Department of Mechanical Engineering,
University of Michigan,
2350 Hayward,
Ann Arbor, MI 48109
e-mail: okwudire@umich.edu

A. Galip Ulsoy

Fellow ASME
G. G. Brown Laboratory,
Department of Mechanical Engineering,
University of Michigan,
2350 Hayward,
Ann Arbor, MI 48109
e-mail: ulsoy@umich.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received October 3, 2015; final manuscript received July 22, 2016; published online September 8, 2016. Assoc. Editor: Zongxuan Sun.

J. Dyn. Sys., Meas., Control 139(1), 011001 (Sep 08, 2016) (11 pages) Paper No: DS-15-1479; doi: 10.1115/1.4034367 History: Received October 03, 2015; Revised July 22, 2016

An approach for minimizing tracking errors in linear time-invariant (LTI) single-input single-output (SISO) discrete-time systems with nonminimum phase (NMP) zeros using filtered basis functions (FBF) is studied. In the FBF method, the control input to the system is expressed as a linear combination of basis functions. The basis functions are forward filtered using the dynamics of the NMP system, and their coefficients are selected to minimize the error in tracking a given desired trajectory. Unlike comparable methods in the literature, the FBF method is shown to be effective in tracking any desired trajectory, irrespective of the location of NMP zeros in the z-plane. The stability of the method and boundedness of the control input and system output are discussed. The control designer is free to choose any suitable set of basis functions that satisfy the criteria discussed in this paper. However, two rudimentary basis functions, one in time domain and the other in frequency domain, are specifically highlighted. The effectiveness of the FBF method is illustrated and analyzed in comparison with the truncated series (TS) approximation method.

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References

Figures

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Fig. 1

Block diagram for tracking control

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Fig. 2

Flowchart of filtered basis functions approach for tracking NMP systems. The controller can be implemented offline since the desired trajectory is assumed to be entirely known.

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Fig. 3

Variation of 3 dB bandwidth with number of basis functions for BPF and DCT

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Fig. 4

Desired trajectory (yd(k)) and its first and second derivatives

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Fig. 5

Comparison of tracking error for FBF (using BPF and DCT) and TS, n = n1 = 50: (a) a = 2, (b) a = 1.001, and (c) a = −1

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Fig. 6

Comparison of tracking error for FBF (BPF and DCT) and TS, n = n1 = 100: (a) a = 2, (b) a = 1.001, and (c) a = −1

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Fig. 7

Effect of n on error and control matrix norms: (a) a = 2, (b) a = 1.001, and (c) a = −1

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Fig. 8

Effect of n on L for DCT basis functions: (a) n = 25, (b) n = 50, (c) n = 75, and (d) n = 100

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