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Technical Brief

Transient Response Control of Two-Mass System via Polynomial Approach

[+] Author and Article Information
Yue Qiao

University of Michigan–Shanghai Jiao Tong University
Joint Institute,
Shanghai Jiao Tong University,
No. 800 Dongchuan Road,
Shanghai 200240, China
e-mail: chimmy@sjtu.edu.cn

Junyi Cao

School of Mechanical Engineering,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: caojy@mail.xjtu.edu.cn

Chengbin Ma

University of Michigan–Shanghai Jiao Tong University
Joint Institute,
Shanghai Jiao Tong University,
No. 800 Dongchuan Road,
Shanghai 200240, China;
School of Mechanical Engineering,
Shanghai Jiao Tong University,
No. 800 Dongchuan Road,
Shanghai 200240, China
e-mail: chbma@sjtu.edu.cn

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 5, 2013; final manuscript received April 27, 2014; published online August 8, 2014. Assoc. Editor: Qingze Zou.

J. Dyn. Sys., Meas., Control 136(6), 064503 (Aug 08, 2014) (8 pages) Paper No: DS-13-1009; doi: 10.1115/1.4027560 History: Received January 05, 2013; Revised April 27, 2014

This paper discusses the application of polynomial method in the transient response control of a benchmark two-mass system. It is shown that transient responses can be directly addressed by specifying the so-called characteristic ratios and the generalized time constant. The nominal characteristic ratio assignment (CRA) is a good starting point for controller design. And the characteristic ratios with lower indices have a more dominant influence. Two practical low-order control configurations, the integral-proportional (IP) and modified-integral-proportional-derivative (m-IPD) controllers are designed. The primary design strategy of the controllers is to guarantee the lower-index characteristic ratios to be equal to their nominal values, while the higher-index characteristic ratios are determined by the interaction with the generalized time constant and the limits imposed by zeros, a specific control configuration, etc. The demonstrated relationship between the transient responses and the assignments of characteristic ratios and generalized time constant in simulation and experiments explains the effectiveness of the polynomial-method-based controller design.

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References

Kim, Y. C., Keel, L. H., and Bhattacharyya, S. P., 2003, “Transient Response Control Via Characteristic Ratio Assignment,” IEEE Trans. Autom. Control, 48(12), pp. 2238–2244. [CrossRef]
Kim, Y. C., Keel, L. H., and Manabe, S., 2002, “Controller Design For Time Domain Specifications,” Proceedings of the 15th IFAC Triennial World Congress Barcelona, Spain.
Naslin, P., 1969, Essentials of Optimal Control, Boston Technical Publishers, Inc., Cambridge, MA.
Manabe, S., 2003, “Importance of Coefficient Diagram in Polynomial Method,” Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, pp. 3489–3494.
Lee, K., and Blaabjerg, F., 2007, “An Improvement of Speed Control Performances of a Two-Mass System Using a Universal Approximator,” Electr. Eng., 89, pp. 389–396. [CrossRef]
Ma, C., Cao, J., and Qiao, Y., 2013, “Polynomial Method Based Design of Low Order Controllers for Two-Mass System,” IEEE Trans. Ind. Electron., 60(3), pp. 969–978. [CrossRef]
Goodwin, G. C., Woodyatt, A. R., Middleton, R. H., and Shim, J., 1999, “Fundamental Limitations due to jω-Axis Zeros in SISO Systems,” Automatica, 35, pp. 857–863. [CrossRef]

Figures

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Fig. 1

The torsion test bench: (a) experimental setup and (b) two-mass model

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Fig. 2

Bode magnitude plots for the normalized two-mass model

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Fig. 3

Magnitude of the sensitivity functions Sγi(s) for the fifth-order system (all the nominal values of γi are equal to two)

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Fig. 4

Step responses of the fifth-order all-pole closed-loop system with a varying τ from 0.25 to 2 at 0.25 interval

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Fig. 5

Control configurations for the velocity control of the two-mass system: (a) IP controller and (b) m-IPD controller

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Fig. 6

Unity velocity step responses: (a) q = 0.31 (IP control), (b) q = 0.8 (IP control), (c) q = 1/4 (m-IPD control), and (d) q = 0.8 (m-IPD control)

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Fig. 7

The Bode plots of the m-IPD control's loop transfer functions under nominal CRA

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Fig. 8

m-IPD controller parameters versus generalized time constant τ (q = 0.8)

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Fig. 9

Velocity step responses under IP control with γ1 = 2.53 and γ2 = 2: (a) q = 0.29 and (b) q = 0.80

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Fig. 10

Velocity step responses under IP control with q = 0.29, γ1 = 2.53, and τ from 2 to 5 at 1.0 interval: (a) drive velocities and (b) load velocities

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Fig. 11

Velocity step responses under m-IPD control with nominal CRA: (a) q = 0.29 and (b) q = 0.80

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Fig. 12

Velocity step responses under m-IPD control with γ1 = 2.48, γ2 = 2, γ3 = 2 and specified τ's at q = 0.80: (a) τ = 5.5 and (b) τ = 6.5

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