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fractional order IMC controllers for time delay processes. The tuning algorithm is based on computing the equivalent controller of the IMC structure and imposing ...
International Journal of Mechanical Engineering and Robotics Research Vol. 4, No. 3, July 2015

A Novel Tuning Algorithm for Fractional Order IMC Controllers for Time Delay Processes Cristina I. Muresan, Eva H. Dulf, and Roxana Both Department of Automation, Technical University of Cluj-Napoca, Cluj-Napoca, Romania Email: [email protected]

community, mainly because of the simplicity in the design that is based upon inverting the process model. The IMC approach has been proposed as a method for tuning fractional order PIμDλ based on a bandwidth specification [10], for first order plus time delay process [11] and a class of fractional order systems [12-14]. In this paper, a novel tuning algorithm is proposed for a fractional order IMC (FO-IMC) controller, as compared to the existing design approaches [10-14]. An integer order IMC controller is first tuned to meet settling time requirements. Next, the FO-IMC controller is tuned according to the same performance specification. However, due to the supplementary tuning parameter, the fractional order of the FO-IMC filter, a second performance criterion is imposed to increase the closed loop performance and the robustness of the controller. The simulation results, considering modeling errors, show that the proposed FO-IMC controller offers better closed loop results as compared to its integer order version.

Abstract—This paper presents a novel tuning algorithm for fractional order IMC controllers for time delay processes. The tuning algorithm is based on computing the equivalent controller of the IMC structure and imposing frequency domain specifications for the resulting open loop system. A second order time delay process is used as a case study. An integer order IMC controller is designed, as well as a fractional order IMC controller. The simulation results show that the proposed fractional order IMC controller ensures an increased robustness to modeling uncertainties. Index Terms—fractional order controller, time delay processes, tuning algorithm, robustness

I.

INTRODUCTION

Fractional calculus represents the generalization of the integration and differentiation to an arbitrary order. There is currently a continuously increasing interest in generalizing classical control theories and developing novel control strategies that use fractional calculus. The most commonly used method for controlling a great range of processes is the PID controller, which is in fact a special case of fractional order PIμDλs. The design problem of fractional order controllers has been the interest of many authors, with some valuable works, in which the fractional order controllers have been applied to a variety of processes to enhance the robustness and performance of the control systems [1]-[4]. The choice of fractional order PIμDλ controllers is based on their potential to improve the control performance, due to the supplementary tuning variables involved, μ and λ. Since the fractional controller has more parameters than the conventional controller, more specifications can be fulfilled, improving the overall performance of the system and making it more robust to modeling uncertainties. Apart from the fractional order PIμDλ controller, some extensions and generalizations of advanced control strategies using fractional calculus have been previously proposed, such as fractional optimal control [5], fractional fuzzy adaptive control [6], fractional iterative learning control [7], fractional predictive control [8] and fractional model reference adaptive control [9] to name just a few. The internal model control (IMC) based PID controller has gained widespread acceptance in the control

II.

TUNING ALGORITHM

The basic structure of the IMC is shown in Fig. 1, where Hp(s) is the process transfer function, H m(s) is the model of the process, HFO-IMC(s) is the fractional order IMC controller transfer function and Hc(s) is the equivalent fractional order controller. Considering the second order time delay process, described by the following transfer function: H p (s) 

as  bs  c

e   ms

(1)

where  m is the process time delay.

Figure 1. Basic IMC control structure

The proposed fractional order IMC controller is given by: H FO  IMC (s) 

Manuscript received July 1, 2014; revised June 8, 2015. © 2015 Int. J. Mech. Eng. Rob. Res. doi: 10.18178/ijmerr.4.3.218-221

k 2

218

as 2  bs  c 1  k s  1

(2)

International Journal of Mechanical Engineering and Robotics Research Vol. 4, No. 3, July 2015

with αϵ(0÷1), the fractional order. The equivalent controller in Fig. 1 is computed as: H c (s) 

H FO  IMC (s) as 2  bs  c  1  H FO  IMC (s)H m (s) k s  ms





H ol (s) 

 ms

with e  1   ms . The open loop transfer function with the equivalent controller and the process transfer function is described by: s  ms

e   ms

(4)

-100 4

Phase (deg)

0

x 10

System: H Phase Margin (deg): 52.9 Delay Margin (sec): 15.2 At frequency (rad/sec): 0.0609 Closed Loop Stable? Yes

-1.152 -2.304 -3.456 -4.608 -5.76 -2 10

-1

0

10

1

10

2

10

10

Frequency (rad/sec)

Figure 2. Bode diagram of the open loop transfer function with integer order IMC controller

(5) 10

8



6





      sin   tan     mgc cos    2   2 

(6)

following relation was used         jgc    The gain   jsin   . gc  cos  2   2    crossover frequency condition is written in a mathematical form as:



(11)

eq. (6) eq. (8)

m1gc 

where

e   ms

-50

which leads to: 

 s  2  m s

Bode Diagram

To tune the FO-IMC controller, a new tuning technique is proposed, that allows the computation of the time constant λ and the fractional order α based on two imposed performance specifications, a specified gain crossover frequency, ωgc, and a phase margin, γ. The specified ωgc is given in order to ensure a certain closed loop settling time, while an increased phase margin will ensure increased stability of the closed loop system. The phase margin condition is written in a mathematical form as: Hol ( jgc )    

1 2 2

0

Magnitude (dB)

1 

(10)

with the time constant λ=3.2. The open loop transfer function is computed similarly to (4) as:

(3)

where a series approximation for the time delay was used,

Hol (s)  Hc (s)H p (s) 

s 2  2s  10 1 5 3.2s  12

H IMC (s) 

the

4



Hol ( jgc )  1

2

0 0.8

0.85



(7)

0.9

0.95

Figure 3. Graphical solution for the tuning algorithm

which leads to:

To compute the values for λ and α, a graphical approach is used in which for different values of α, relations (6) and (8) are used to compute λ and the resulting values are plotted as a function of α. The intersection point yields the final values for the fractional order and the time constant.

e 10s

-40 -60

Phase (deg)

System: Hd Phase Margin (deg): 61 Delay Margin (sec): 17.9 At frequency (rad/sec): 0.0595 Closed Loop Stable? Yes

-2.3593 -4.7186 -7.0779 -4 10

-2

10

0

10

2

10

4

10

Frequency (rad/sec)

Figure 4. Bode diagram of the open loop transfer function with fractional order IMC controller

(9)

s 2  2s  10 For comparison purposes, an integer order IMC controller is designed: © 2015 Int. J. Mech. Eng. Rob. Res.

0 -20

7

To exemplify the tuning procedure described in the previous section, the following second order process is used as a case study: 5

20

-80 x 10 0

III. CASE STUDY: SECOND ORDER TIME DELAY PROCESS

H p (s) 

Bode Diagram

40

(8) Magnitude (dB)

   2 2 1 2gc   2m gc  2 m   1 gc sin  2 

The Bode diagram of (11) is given in Fig. 2, yielding a gain crossover frequency ωgc=0.06 and a phase margin, 219

International Journal of Mechanical Engineering and Robotics Research Vol. 4, No. 3, July 2015

γ=53o. To tune the FO-IMC controller, based on (6) and (8), the same gain crossover frequency is imposed of 0.06 rad/sec, while the phase margin is increased for improved performance γ=60o. Fig. 3 presents the graphical representation of the solutions of (6) and (8) for α ranging from 0 to 1. The solution is found at the intersection point: α=0.88 and λ=4.87. Fig. 4 shows the Bode diagram of the open loop transfer function with the resulting FO-IMC controller. As noted, the obtained gain crossover frequency and the phase margin meet the design requirements previously mentioned. 1.4 FO-IMC controller Integer order IMC controller

1.2

Fig. 5 presents the simulation results considering both the integer order, as well as the proposed FO-IMC controllers. Under nominal conditions, assuming Hp(s)=Hm(s), the two controllers achieve similar results, with the settling time equal to 30 seconds, and no overshoot. To test the robustness of the designed FO-IMC controller, two case scenarios are considered. Fig. 6 presents the simulations results considering a +30% variation of the time delay, while Fig. 7 shows the simulation results considering a+50% variation of the process gain. The comparative results in terms of settling time and overshoot are given in Table I, showing the increased robustness of the proposed FO-IMC controller.

1

Output

TABLE I. 0.8

COMPARATIVE CLOSED LOOP PERFORMANCE RESULTS UNDER PARAMETER VARIATIONS

0.6 0.4

Integer order IMC Fractional order IMC

0.2 0 0

10

20

30 40 Time (sec)

50

60

Figure 5. Closed loop simulation results using the two controllers under nominal conditions

Integer order IMC Fractional order IMC

1.4 1.2

+30% variation of the time delay Settling time Overshoot 73 seconds 20% 51 seconds

12%

+50% variation of the process gain Settling time Overshoot 64 seconds 37% 42.4 seconds

21%

1

Output

IV. CONCLUSIONS 0.8

The paper presents a novel approach for tuning FOIMC controllers in the frequency domain that is based on improving the closed loop performance and robustness of a classical integer order IMC controller. The simulation results considering a second order time delay system show that the proposed FO-IMC controller provides improved closed loop performance, despite modeling uncertainties, if compared to the integer order IMC.

0.6 0.4 0.2 0 0

FO-IMC controller Integer order IMC controller 20

40 Time (sec)

60

80

Figure 6. Closed loop simulation results using the two controllers under +30% variation of the process time delay

ACKNOWLEDGMENT This work was supported by a grant of the Romanian National Authority for Scientific Research, CNDI– UEFISCDI, project number PNII-RU-TE-2012-3-0307.

1.4 1.2

Output

1

REFERENCES

0.8

[1]

0.6 0.4

[2]

0.2 0 0

FO-IMC controller Integer order IMC controller 20

40 Time (sec)

60

[3] 80

[4]

Figure 7. Closed loop simulation results using the two controllers under +50% variation of the process gain

© 2015 Int. J. Mech. Eng. Rob. Res.

220

C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. Feliu: Fractional-Order Systems and Controls: Fundamentals and Applications, London: Springer, 2010 D. Xue, C. Zhao, and Y. Q. Chen, “Fractional order PID control of a DC-Motor with elastic Shaft: A case study,” in Proc. American Control Conference, Minnesota, USA, 2006, pp. 3182-3187 C. I. Pop (Muresan), C. Ionescu, R. De Keyser, and E. H. Dulf, “Robustness evaluation of fractional order control for varying time delay processes,” Signal, Image and Video Processing vol. 6, pp. 453-461, 2012 A. Oustaloup, La Commande CRONE: Commande Robust d’ordre non Entire, Paris: Hermes, 1991.

International Journal of Mechanical Engineering and Robotics Research Vol. 4, No. 3, July 2015

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[7]

[8]

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[13]

[14]

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Cristina I. Muresan received the degree in Control Systems in 2007, and the Ph.D. in 2011 from Technical University of ClujNapoca, Romania. She is currently lecturer within the Automation Department of the same university. She has published over 50 papers and book chapters. Her research interests include modern control strategies, such as predictive algorithms, fractional order control, time delay compensation methods and multivariable systems. Eva-Henrietta Dulf received her B. Eng. degree in Automation from Technical University of Cluj-Napoca, Romania, in 1997, and her M.Sc. and Ph.D. degrees from the same university in 1998 and 2006, respectively. In November 1998, she joined Technical University of Cluj-Napoca as teaching assistant and she is now Associate Professor. Current research interests include various areas in process control, including mathematical modeling, controller design, optimization methods and practical implementation. She’s published more than 100 technical journal papers and conference talks, and she is a member of IEEE. Roxana Both received her B.Eng. degree in Control Systems in 2008 and the Ph.D. in 2011 from Technical University of ClujNapoca, Romania. She is currently lecturer at the same university at the Automation Department. Her current research interests include process modelling and simulation, process control strategies design and implementation: robust control, model predictive control, fractional control. She has published more than 30 papers in journal and conferences.