Research Papers

Simultaneous Compensation of the Gain, Phase, and Phase-Slope

[+] Author and Article Information
Vahid Badri

College of Engineering,
School of Electrical and Computer Engineering,
University of Tehran,
Tehran 1439957131, Iran
e-mail: badri.v@ut.ac.ir

Mohammad Saleh Tavazoei

Electrical Engineering Department,
Sharif University of Technology,
P.O. Box 11155-4363,
Tehran 1458889694, Iran
e-mail: tavazoei@sharif.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received March 6, 2015; final manuscript received June 21, 2016; published online August 11, 2016. Assoc. Editor: YangQuan Chen.

J. Dyn. Sys., Meas., Control 138(12), 121002 (Aug 11, 2016) (7 pages) Paper No: DS-15-1091; doi: 10.1115/1.4034073 History: Received March 06, 2015; Revised June 21, 2016

This paper deals with the problem of simultaneous compensation of the gain, phase, and phase-slope at an arbitrary frequency by using a fractional-order lead/lag compensator. The necessary and sufficient conditions for feasibility of the problem are derived. Also, the number of existing solutions (i.e., the number of distinct fractional-order lead/lag compensators satisfying the considered compensation requirements) is analytically found. Moreover, as a sample application, it is shown that the obtained results for the considered compensation problem are helpful in tuning fractional-order lead/lag compensators for simultaneously achieving desired phase margin, desired gain cross frequency, and flatness of the Bode phase plot of the loop transfer function at this frequency.

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Monje, C. A. , Chen, Y. Q. , Vinagre, B. M. , Xue, D. , and Feliu, V. , 2010, Fractional-Order Systems and Controls—Fundamentals and Applications, Springer, Heidelberg, Germany.
Caponetto, R. , Dongola, G. , Fortuna, L. , and Petras, I. , 2010, Fractional Order Systems: Modeling and Control Applications, World Scientific, Singapore.
Luo, Y. , and Chen, Y. Q. , 2012, Fractional Order Motion Controls, Wiley, New York.
Efe, M. Ö. , 2011, “ Fractional Order Systems in Industrial Automation—A Survey,” IEEE Trans. Ind. Inf., 7(4), pp. 582–591. [CrossRef]
Tavazoei, M. S. , 2012, “ From Traditional to Fractional PI Control: A Key for Generalization,” IEEE Ind. Electron. Mag., 6(3), pp. 41–51. [CrossRef]
Tavazoei, M. S. , 2014, “ Time Response Analysis of Fractional-Order Control Systems: A Survey on Recent Results,” Fractional Calculus Appl. Anal., 17(2), pp. 440–461.
Wang, D. J. , and Gao, X. L. , 2012, “ H Design With Fractional-Order PDμ Controllers,” Automatica, 48(5), pp. 974–977. [CrossRef]
Badri, V. , and Tavazoei, M. S. , 2013, “ On Tuning Fractional Order [Proportional–Derivative] Controllers for a Class of Fractional Order Systems,” Automatica, 49(7), pp. 2297–2301. [CrossRef]
Djennoune, S. , and Bettayeb, M. , 2013, “ Optimal Synergetic Control for Fractional-Order Systems,” Automatica, 49(7), pp. 2243–2249. [CrossRef]
Badri, V. , and Tavazoei, M. S. , 2013, “ On Tuning FO[PI] Controllers for FOPDT Processes,” Electron. Lett., 49(21), pp. 1326–1328. [CrossRef]
Padula, F. , and Visioli, A. , 2014, Advances in Robust Fractional Control, Springer, Heidelberg, Germany.
Podlubny, I. , 1999, “ Fractional-Order Systems and PIλDμ-Controllers,” IEEE Trans. Autom. Control, 44(1), pp. 208–214. [CrossRef]
Tavazoei, M. S. , and Tavakoli-Kakhki, M. , 2014, “ Compensation by Fractional-Order Phase-Lead/Lag Compensators,” IET Control Theory Appl., 8(5), pp. 319–329. [CrossRef]
Bonnet, C. , and Partington, J. R. , 2000, “ Coprime Factorizations and Stability of Fractional Differential Systems,” Syst. Control Lett., 41(3), pp. 167–174. [CrossRef]
Anton, H. , 1999, Calculus: A New Horizon, 6th ed., Wiley, New York.
Ogata, K. , 2002, Modern Control Engineering, 4th ed., Prentice-Hall, Upper Saddle River, NJ.
Chen, Y. Q. , and Kevin, L. M. , 2005, “ Relay Feedback Tuning of Robust PID Controllers With Iso-Damping,” Trans. Syst. Man Cybern., Part B, 35(1), pp. 23–31. [CrossRef]
Luo, Y. , and Chen, Y. Q. , 2009, “ Fractional Order [Proportional Derivative] Controller for a Class of Fractional Order Systems,” Automatica, 45(10), pp. 2446–2450. [CrossRef]
Luo, Y. , and Chen, Y. Q. , 2012, “ Stabilizing and Robust Fractional Order PI Controller Synthesis for First Order Plus Time Delay Systems,” Automatica, 48(9), pp. 2159–2167. [CrossRef]
Wang, Q. G. , Ye, Z. , and Hang, C. C. , 2006, “ Tuning of Phase-Lead Compensators for Exact Gain and Phase Margins,” Automatica, 42(2), pp. 349–352. [CrossRef]
Zhang, R. , Xue, A. , and Gao, F. , 2014, “ Temperature Control of Industrial Coke Furnace Using Novel State Space Model Predictive Control,” IEEE Trans. Ind. Inf., 10(4), pp. 2084–2092. [CrossRef]


Grahic Jump Location
Fig. 1

A unity negative feedback control system

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Fig. 2

The achievable region of compensator (1) in the p–c plane for different values of ρωc (the dark gray region specifies the area in which there are two different compensators in the form (1) for ensuring requirements (i)–(iii), whereas in the light gray area there is a unique compensator in the form (1) for satisfying these requirements): (a) ρωc=0.3, (b) ρωc=0.896, (c) ρωc=1.5, and (d) ρωc=−1

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Fig. 3

The exact solution of Eq. (22) for different values of k3=2/πk1∈(0,1) and its approximation which is given by Eq.(28)

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Fig. 4

Bode diagrams of the loop transfer functions in Example 1 (case kv=2.3469)

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Fig. 5

Bode diagram of the loop transfer function in Example 1 (case kv=2.3571)

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Fig. 6

Step response of the closed-loop system in Example 1 (case kv=2.3571)

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Fig. 7

Bode diagram of the loop transfer function in Example 2

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Fig. 8

Step response of the closed-loop system in Example 2




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