Identification and tuning fractional order proportional integral controllers for time delayed systems with a fractional pole

Mechatronics xxx (2013) xxx–xxx

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Identification and tuning fractional order proportional integral controllers for time delayed systems with a fractional pole Hadi Malek a,⇑, Ying Luo b, YangQuan Chen c a

Electrical & Computer Eng. Dept., Utah State University, Logan, UT 84321, United States Department of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China c School of Engineering, University of California, Merced, 5200 North Lake Road, Merced, CA 95343, United States b

a r t i c l e

i n f o

Article history: Available online xxxx Keywords: Fractional order controllers Identification Controller design

a b s t r a c t First order plus time delay model is widely used to model systems with S-shaped reaction curve. Its generalized form is the model with a single fractional pole replacing the integer order pole, which is believed to better characterize the reaction curve. In this paper, using time delayed system model with a fractional pole as the starting point, fractional order controllers design for this class of fractional order systems is investigated. Integer order PID and fractional order PI and [PI] controllers are designed and compared for these class of systems. The simulation comparison between PID controller and fractional order PI and [PI] controllers show the advantages of the properly designed fractional order controllers. Experimental results on a heat flow platform are presented to validate the proposed design method in this paper. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Increasing applications of fractional order calculus in fields of science and engineering have been witnessed recently [1]. Fractional operators are the generalization of integration and differentiation of integer order calculus, which allow us to present more accurate descriptions of real systems which includes a combination of multidisciplinary field of engineerings [2–5]. As a matter of fact, all the real dynamic systems have certain degree of fractionality [6]. But in many cases, this fractionality is not strong enough to affect the behavior of the system and therefore this behavior can be described by an approximated integer order differential equation. In the integer order world, first order plus time delay model is widely used to model systems with S-shaped reaction curve. Its generalized form is the model with a single fractional pole replacing integer order pole [7], which is believed to better characterize the reaction curve. In this paper, the Heat Flow Experiment (HFE) will be modeled by a time delay system with fractional order pole and its reaction curve will be compared with that of first order plus time delay model to show the advantages of the introduced fractional order model. Among all type of controllers, fractional PID and PI controllers are obviously important to the practical world, see [8–11]. As

⇑ Corresponding author. Tel.: +1 435 797 9764. E-mail address: [email protected] (H. Malek).

pointed in [12], for closed-loop control systems, there are four situations: (1) (2) (3) (4)

IO (integer order) plant with IO controller. IO plant with FO (fractional order) controller. FO plant with IO controller. FO plant with FO controller.

There are many methods to tune IO controllers for IO plants. One of the most famous and popular method of IO PID controller tuning is called Ziegler–Nichols method [13,14]. Also [8] and [11] have discussed the FO controllers design for the FO plants and the results with FO controllers are compared with the IO controllers for FO plants. In this paper integer order PID (Eq. (2)) and fractional order PI (Eq. (3)) and [PI] controllers (Eq. (4)) are designed for this class of fractional order systems. The simulation and experimental results with the comparison between integer order PID controller with fractional order PI and [PI] controllers show the advantages of the properly tuned fractional order controllers in terms of time response features. This paper is organized as follows: In Sections 2 and 3 the general form of plant and controllers considered in this paper is presented; we proposed our tuning approaches for PID controller, FO–PI controller and FO–[PI] controller. In Sections 4 and 5, HFE will be identified by a pure time delay model with fractional order pole, the integer order and fractional order controllers are designed for this plant. Sections 6 and 7 present the simulation and experimental results, and the conclusion is given in Section 8.

0957-4158/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechatronics.2013.02.005

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2. General form of plant and controllers considered The general transfer function of the considered fractional order plant with a fractional order pole and constant time delay is,

PðsÞ ¼

where A ¼ 1 þ T xac cosðap=2Þ and B ¼ T xac sinðap=2Þ. According to the first design constraint (5), the phase of considered system with integer order PID controller at the gain cross-over frequency (xc) is,

tan1

K eLs ; Ts þ 1

ð1Þ

a

where T, a 2 (0,2), L and K are constants. Our goal is to design three different controllers, integer order PID, fractional order PI and fractional order [PI], for the fractional order plant (1). The transfer function of the three controllers are given as follows respectively,

Ki C 1 ðsÞ ¼ K p þ þ K d s;  s  Ki C 2 ðsÞ ¼ K p 1 þ k ; s  k Ki C 3 ðsÞ ¼ K p þ ; s

ð2Þ ð3Þ ð4Þ

where Kp,Ki, Kd and k are positive real number. In this paper, we consider k 2 (0,1). There are many different approaches for PID tuning. A tuning method for integer order PID and also fractional order PI and [PI] controllers was proposed in [15], which is a special case of this paper. We similarly assume that the gain crossover frequency xc, and phase margin Um, are given. Three tuning constraints for controller design are presented as follows [15]:

    K d x2c  K i B  tan1 j  Lxc ¼ p þ /M ; A x¼xc xc K p

so,

    K d x2c  K i B þ /M þ Lxc  p : ¼ tan tan1 A xc K p

¼ p þ /M ;

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi K 2p þ K d xc  xK ic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : jG1 ðjxc Þj ¼ A2 þ B2 K

According to the second design constraint (6),

K 2p þ K d xc  xK ic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 þ B 2

ð6Þ

3. Constraint for robustness to loop gain variations, which demands that the phase is flat around the gain crossover frequency xc. It means that the derivative of open loop phase around the gain cross-over frequency is zero, i.e.,

dðArg½GðjxÞÞ jx¼xc ¼ 0: dx

2

¼

1 : K

ð13Þ

Based on the third constraint (7), the robustness to loop gain variations can be obtained by forcing phase plot flat around the gain cross-over frequency. The derivative of phase with respect to frequency at the cross-over frequency point should be zero. Then,



aT xca1 A sin

ap

 

 B cos a2p

2

þL

  K p K d x2c þ K i  2 ðK p xc Þ2 þ K d x2c  K i

¼ 0:

where G(jx) is open-loop transfer function of system, C(jx) is controller transfer function and P(jx) is plant transfer function. 2. Gain cross-over frequency constraint

jGðjxc Þ ¼ jCðjxc ÞPðjxc ÞjdB ¼ jCðjxc ÞjdB jPðjxc ÞjdB ¼ 0;

ð12Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi

A2 þ B2

ð5Þ

ð7Þ

ð11Þ

The open-loop gain at the gain cross-over frequency is,

1. Phase margin constraint,

Arg½Gðjxc Þ ¼ Arg½Cðjxc ÞPðjxc Þ ¼ \Cðjxc Þ þ \Pðjxc Þ

ð10Þ

ð14Þ

From Eqs. 11, 13 and 14 the coefficients of PID controller (Kp,Ki and Kd) can be solved,

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u A2 þ B2 ; Kp ¼ t  K 2 1 þ D21

ð15Þ



where D1 ¼ tan tan1 AB þ Lxc þ /m  p ,

Ki ¼

  i 1h E1 K p x2c 1 þ D21  D1 K p xc ; 2

where E1 ¼

Kd ¼

aT xac 1 A2 þB2

ðA sinðap=2Þ  B cosðap=2ÞÞ þ L, and,

K i þ D1 K p xc

xc

ð16Þ

:

ð17Þ

3. Tuning of the controllers

3.2. Fractional order PI controller design

In this section, based on the constraints introduced in Section 2, we present the tuning process of the PID controller (2), fractional PI controller (3) and fractional [PI] controller (4), for the considered plant with constant time delay and fractional order pole (1).

The open-loop transfer function with the fractional order PI (FOPI) controller is,

3.1. Integer order PID controller design The open-loop transfer function with plant (1) and PID controller (2) is,

 G1 ðsÞ ¼ C 1 ðsÞPðsÞ ¼

  Ki K Ls ; e Kp þ þ Kds s Tsa þ 1

ð8Þ

where T, a and L are known and Kp, Ki and Kd should be designed. The phase of the open-loop system at the gain cross-over frequency is,

    K d x2c  K i B  tan1  Lxc ; Arg½G1 ðjxc Þ ¼ tan1 A xc K p

ð9Þ

   Ki K G2 ðsÞ ¼ C 2 ðsÞPðsÞ ¼ K p 1 þ k eLs ; a s Ts þ 1

ð18Þ

where T,a and L are known and Kp,Ki and k should be designed in the controller design process. The FOPI controller can be expressed as:

! !   Ki Ki K i xk ¼ K : C 2 ðsÞ ¼ K p 1 þ k ¼ K p 1 þ 1 þ p k s ðjxÞk j

ð19Þ

As we know j = ep/2, then jk = ekp/2 = cos (kp/2) + j sin (kp/2), therefore

 C 2 ðsÞ ¼ K p 1 þ

 K i xk : cosðkp=2Þ þ j sinðkp=2Þ

ð20Þ

Then open-loop phase at the gain cross-over frequency is,

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" Arg½G2 ðjxc Þ ¼  tan1

kp #   K i xk B c sin 2 kp  tan1 A 1 þ K i xk cos c 2

 Lxc ;

Ki ¼

" tan

2E2 x2k c

ð29Þ

;

ð21Þ

where A ¼ 1 þ T xac cosðap=2Þ and B ¼ T xac sinðap=2Þ. Based on the first design constraint (5), 1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 2  F 22  4E22 x2k c

kp #   K i xk B c sin 2 kp  tan1  Lxc ¼ p þ /M ; A cos 1 þ K i xk c 2

ð22Þ

k1 where F 2 ¼ 2E2 xk sinðkp=2Þ. c cosðkp=2Þ  kxc Obviously, based on Eqs. 24, 27 and 28, k,Ki and Kp can be determined by different approaches like fminsearch in MATLAB or by crossing Eqs. (24) and (28) graphically, and from the intersection point as in Fig. 1, Ki and k can be obtained and then Kp can be determined by (27).

or 3.3. Fractional order [PI] controller design

kp     K i xk B c sin 2 kp ¼ tan tan1 þ Lxc þ /M : k A 1 þ K i xc cos 2

ð23Þ

Then, relationship between Ki and k can be established as,

Ki ¼

D kp 2 kp ; k xk c sin 2 þ xc cos 2

ð24Þ

where D2 = tan[tan1(B/A) + /M + L]. Open-loop gain using FOPI controller at the crossover frequency is,

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  kp2  kp2 KK p 1 þ K i xk þ K i xk c cos 2 c sin 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : jG2 ðjxc Þj ¼ A2 þ B2

ð25Þ

According to the second constraint (6),

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2   2 KK p 1 þ K i xk cos k2p þ K i xk sin k2p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1; A2 þ B2

ð26Þ

ð27Þ

K i xk1 sin

kp

x þ 2K i xk cos

k2p

aT xa1

2

þ K 2i

jx¼xc  E2 ¼ 0;

ð28Þ

where E2 ¼ A2 þBc 2 ðA sinðap=2Þ  B cosðap=2ÞÞ þ L. Then, we can establish an equation for Ki versus k as follows,



Kp þ

Ki s

k 

 K Ls : e Tsa þ 1

ð30Þ

Open-loop phase at the gain cross-over frequency is,

Arg½G3 ðjxc Þ ¼ k tan1



Ki

K p xc



 tan1

  B  Lxc : A

ð31Þ

According to the design rules, the open-loop phase satisfies the following equation.



Ki

K p xc



 tan1

  B j  Lxc ¼ p þ /M ; A x¼xc

ð32Þ

so,

Ki ¼ D3 : K p xc

According to the third constraint (7),

2k

G3 ðjxÞ ¼ C 3 ðsÞPðSÞ ¼

k tan1

or

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u A2 þ B2 Kp ¼ t 2 kp2  kp2 : K 1 þ K i xk þ K i xk c cos 2 c sin 2

The open-loop transfer function with the fractional order [PI] controller is,

ð33Þ

where D3 = tan[(p  /m  tan1(B(xc)/A(xc))  Lxc)/k], Open-loop gain at the gain cross-over frequency should satisfy the second constraint (6),

  2 2k K K 2p þ xK ic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1: A2 þ B2

ð34Þ

Based on the third constraint (7), for the robustness to loop gain variations, we get,

Fig. 1. Finding Ki and k graphically.

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H. Malek et al. / Mechatronics xxx (2013) xxx–xxx

ters of each structure. The first one is the time delayed system with an integer order pole,

PIO ¼

K eLs ; Ts þ 1

ð39Þ

and second one is a time delayed system with a fractional order pole.

PFO ¼ Fig. 2. Heat flow experiment.

kK i K p ðK p xc Þ2 þ K 2i

¼ E3 ;

ð35Þ

    aT xa1  where E3 ¼ A2 þBc 2 A sin a2p  B cos a2p þ L. From Eqs. 32, 34 and 35, we can get,

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 E3 3 Ki ¼ x D3 ðA2 ðxc Þ þ B2 ðxc ÞÞk ; k sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1 E x 3 c 2 2 K i ¼ xc ðA ðxc Þ þ B ðxc ÞÞk 1  ; kD3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 E3 xc ðA2 ðxc Þ þ B2 ðxc ÞÞk : Kp ¼ kD3

K eLs : Tsa þ 1

ð40Þ

Now we try to identify the unknown parameters K,T,L in both model and additional parameter a, in the second model. we use fminsearch in MATLAB to find these parameters. This is generally referred to unconstrained nonlinear optimization. In this case we define an error signal and try to minimize this error by fminsearch command in MATLAB. The error signal is,

err ¼

tf X jyP ðtÞ  yM ðtÞj;

ð41Þ

0

ð36Þ ð37Þ

ð38Þ

We can find Ki and k graphically from (36) and (37) and then we can find Kp from (38) or alternatively, fminsearch command of MATLAB can be used to find Kp, Ki and k of FO[PI] controller. 4. System identification In this section, a Heat Flow Equipment (HFE) as shown in Fig. 2 is used as our benchmark for the experiment in this paper. Applying an input to the heater terminal of HFE, three different sensors detect the heated air flow in different locations of channel. Fig. 3 shows the sensor outputs of HFE to a step input. Obviously, further sensor is with more delay and less temperature. we consider two structures for this plant and try to identify the parame-

where yP(t) is the practical output data which is obtained by applying a step input to the HFE (we consider the response of the third sensor) and yM(t) is the step response of the considered models to the same input. Also tf is the final time for simulation and experimental results. In this experiment, we have investigated three different plants:  Integer order plant.  Fractional order plant 1 with unknown a.  Fractional order plant 2 with a = 0.5. Fig. 4 shows the simulation step responses of these three plants and compares them with the experimental step response. Obviously, step responses of the estimated fractional order plants can fit the experimental data better than that of the integer order plant. This fact can be seen clearly from the last column in Table 1 which shows the errors between the responses of estimated systems and experimental data. Then we can conclude that the considered fractional order model can characterize the real behavior of HFE better than the integer order model.

Fig. 3. Heat flow experiment outputs.

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Fig. 4. Simulation step responses of three identified systems.

Table 1 Estimated parameters.

IO plant FO plant 1 FO plant 2

a

K

T

L

Error

1.00 0.61 0.50

38.86 50.34 66.16

23.43 16.91 12.72

3  1011 1.31 1.93

771.324 139.629 199.47

Since the equation which describe the relation between the heat flux and temperature at the desired point, (t, k), follows a parabolic partial differential equation [16], the considered transfer function plant for this heat flow experiment should be a fractional order plant with a = 0.5,

PðsÞ ¼

K Ts0:5 þ 1

eLs :

ð42Þ

where K = 66.16,T = 12.72 and L = 1.93, according to Table 1. 5. Controller design procedures 5.1. Integer order PID (IOPID) design In this section, we present a practical example to verify the performance of the introduced design approach for S-shape response plant.

Given K,T,L,xc = 0.08 rad/s and /m = 50°, Kp,Ki and Kd can be obtained by solving the set of Eqs. (15)–(17): Kp = 0.005, Kd = 0.7942 and Ki = 0.01.

Fig. 5. Frequency responses of open-loop plant with designed IO-PID.

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H. Malek et al. / Mechatronics xxx (2013) xxx–xxx

Gain of open-loop system by FOPI 30

Gain (dB)

20 10 0 -10 -20 -30 10-2

10-1

100

Frequency (rad/sec) Phase of open-loop system by FOPI -120

Phase (Deg)

-130 X: 0.08 Y: -130.1

-140 -150 -160 -170 -180 10-2

10-1

100

Frequency (rad/sec) Fig. 6. Frequency responses of open-loop plant with designed FOPI.

5.3. Fractional order [PI] (FO[PI]) design Given K,T,L,xc = 0.08 rad/s and /m = 50°, plot Ki with respect to k according to (36), then plot Ki with respect to k according to (37). Next step is finding the intersection between these two curves which can gives us Ki = 0.2927 and k = 1.097. Last step is calculating Kp from (27), Kp = 0.7820. The open-loop Bode plot with the considered plant (42) and this designed FO[PI] controller is shown in Fig. 7. We can see that all the three constraints are satisfied precisely.

6. Simulation results

Fig. 7. Frequency responses of open-loop plant with designed FO[PI].

For our considered plant (42), the open-loop Bode plot with the designed IOPID is shown in Fig. 5. We can see all the three constraints are satisfied precisely. 5.2. Fractional order PI (FOPI) design Given K,T,L,xc = 0.08 rad/s and /m = 50°, plot Ki with respect to k according to (24), then plot Ki with respect to k according to (29). Next step is finding the intersection between these two curves which gives us Ki = 0.3153 and k = 1.082. Last step is calculating Kp from (27), Kp = 0.8812. The open-loop Bode plot with the considered plant (42) and this designed FOPI controller is shown in Fig. 6. We can see that all the three constraints are satisfied precisely.

In this section, using Numerical Inverse Laplace Transform (NILT), results of applying the designed controllers to the identified model of HFE are illustrated. IOPID, FOPI and FO[PI] controllers are designed for (42) to satisfy /m = 50°, gain cross-over frequency, xc = 0.08 rad/s, and flat phase constraint. In Fig. 8, step responses of the designed IOPID controller (43) for HFE model (42), with ±20% open-loop gain variations are presented. The amplitude of input step is 60 °C. According to the design scheme in this paper, the FOPI and FO[PI] controllers are designed as well,

C 1 ðsÞ ¼ 0:005 þ 0:01=s þ 0:7942s;

ð43Þ

  0:3153 C 2 ðsÞ ¼ 0:8815 1 þ 1:082 ; s

ð44Þ

 1:097 0:2927 C 3 ðsÞ ¼ 0:7820 þ s

ð45Þ

As we can see in Figs. 9 and 10, step responses for identified HFE model, using the designed FOPI and FO[PI] with ±20% open-loop gain variations are presented respectively.

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H. Malek et al. / Mechatronics xxx (2013) xxx–xxx

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Fig. 8. IO-PID responses against gain variations.

Fig. 9. FO-PI responses against gain variations.

It can be seen that, comparing with the designed IOPID, the designed FOPI and FO[PI] work more effectively for the single fractional pole plant with constant time delay (42). From Fig. 11, the overshoots of step responses using the designed FO[PI] controller and FOPI controller are smaller than that of using the designed IOPID. Also, this overshoot remains almost constant under loop gain variations when fractional order controllers are applied, which means the system with the designed fractional order controllers is more robust against loop gain changes over the designed IOPID.

7. Experimental results As shown in Fig. 2, HFE system consists of a duct equipped with a heater and a blower at one end and three temperature sensors located along the duct. The power delivered to the heater is controllable. For the control analog signal generation and sensor temperature measurement, we use Quanser MQ3 analog input/output libraries via MATLAB/Simulink/RTW. The fan speed of HFE is also controllable. Fan speed is constant for this experiment. The HFE system also includes built-in power module, analog signals power, an onboard and fast settling platinum temperature transducers (3 sensors along

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Fig. 10. FO-[PI] responses against gain variations.

the duct) for the temperature measurement. The heater voltage is applied in the range of 0–5 V and the temperature measurement from the third sensor is recorded. The heating operation takes effect only when the heater voltage is greater than 3 V. In this paper, we identified the HFE experimental system, and it is shown that fractional order model behaves more accurate and is closer to the behavior of real system. IOPID, FOPI and FO[PI] controllers are designed by the proposed approach in this paper. While in the simulation results (Fig. 11) and also experimental data, the response of FOPI and FO[PI] are similar for our considered plant, then in the following we have just considered FOPI controller. In order to apply FOPI controller in the HFE system, fractional order operator, sk, of controller is implemented by the impulse response invariant discretization methods in time domain [17]. For the implementation of the fractional order plant (1), the impulse response invariant discretization method in time domain [17] for the fractional order operator sa is also used. Fig. 11. Step responses of IOPID, FOPI and FO[PI].

60

PID FOPI

55

Temp. (oC)

50

45

40

35

30

0

10

20

30

40

50

60

70

80

90

100

Time (s) Fig. 12. Comparison of output signals in the experimental results.

Please cite this article in press as: Malek H et al. Identification and tuning fractional order proportional integral controllers for time delayed systems with a fractional pole. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.005

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30 PID FOPI

20

9

From the simulation and the experimental results, the two designed fractional order controllers work more effectively than the designed integer order PID controller. References

o

Error ( C)

10

0

−10

−20

−30 0

10

20

30

40

50

60

70

80

90

Time (s) Fig. 13. Comparison of error signal in the experimental results.

Fig. 12 shows the step responses using the designed IOPID and FOPI. Obviously, applying FOPI results less settling time and rise time comparing IOPID as we expected from the simulation results. Fig. 13 shows the error signals for different controllers in the experiment. 8. Conclusion In this paper, an integer order PID and two fractional order proportional integral controllers are designed for a class of single fractional order pole systems with constant time delay. For fair comparison, the same set of design specifications are used to design the IOPID controller, FOPI controller and FO[PI] controller, respectively. The method used for designing the controllers can guarantee the robustness of the designed controllers to the loop gain variations. In this paper, we identified a Heat Flow Equipment (HFE) by a single fractional order pole system with constant time delay and it is shown that this model can fit the HFE step response better than the corresponding integer order system. The HFE system is controlled by the designed IOPID, FOPI and FO[PI] controllers.

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Please cite this article in press as: Malek H et al. Identification and tuning fractional order proportional integral controllers for time delayed systems with a fractional pole. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.005