Design and Implementation of Smith Predictor Based Fractional Order ...

2015 23rd Iranian Conference on Electrical Engineering (ICEE)

Design and Implementation of Smith Predictor Based Fractional Order PID Controller on MIMO Flow-Level Plant Roohallah Azarmi*, Ali Khaki Sedigh*, Mahsan Tavakoli-Kakhki**, and Alireza Fatehi* *Advanced Process Automation & Control (APAC) Research Group, Industrial Control Center of Excellence, Faculty of Electrical Engineering, K. N. Toosi University of Technology, Tehran, Iran, [email protected] , [email protected] , [email protected] **Faculty of Electrical Engineering, K. N. Toosi University of Technology, Tehran, Iran, [email protected] Abstract: The main point of this paper is to present an iterative optimization strategy for tuning the parameters of Smith predictor based fractional order PID (SPFOPID) controller. The control scheme considered in this paper is the standard Smith predictor structure. Also, the internal model is considered to be a First Order Plus Dead Time (FOPDT) transfer function. Finally, the proposed method is implemented on a multi inputmulti output (MIMO) flow-level plant and the obtained results are compared with the results of applying Smith predictor based PID controller (SPPID) in the similar structure.

PI or PID controller has been modified to control unstable and integrative processes [22, 23]. Although, the Smith predictor structure offers potential improvement of the closed-loop performance of the industrial processes with long dead time, it requires an accurate plant model since small modelling errors can lead to poor performance [24]. In this paper, Smith predictor based fractional order PID (SPFOPID) controller is designed through solving a nonlinear optimization problem. In fact, we add a degree of freedom to the Smith predictor based PID (SPPID) by benefiting from fractional order PID (FOPID) controller in the Smith predictor structure. Then, according to the sensitivity functions, the better performance of SPFOPID is shown in comparison with SPPID. Effectiveness and simplicity are key features of the presented method. Finally, the designed SPFOPID controller is implemented on a laboratory multi input-multi output (MIMO) flowlevel plant and the obtained results are compared with the results of applying SPPID controller in a similar structure. This paper is organized as follows. Section 2 devotes to presenting some preliminaries about fractional order calculus, FOPID controllers, and fractional order transfer functions. Design of SPFOPID controller has been described in Section 3. In Section 4, the MIMO flowlevel control plant is introduced and the designed SPFOPID controller has been applied in the control of a MIMO flow-level plant. Also in this section, the experimental results have been compared with the results of applying SPPID controller in a similar structure. Finally, the paper is concluded in Section 5.

Keywords: Smith Predictor, Fractional Order PID Controller, MIMO Flow-Level Plant, Iterative Optimization. 1.

Introduction

The effectiveness of fractional calculus techniques are well recognized in the mathematical modelling, control, and identification of a diverse range of dynamical systems. Also, the issue of fractional order systems and controllers has been addressed by many researchers [1]. Fractional calculus has been used in various fields of engineering; such as control system design [2-4], continuum mechanic [5], signal processing [6], and bioengineering [7]. The extra degrees of freedom in setting the integration and differentiation orders O and P lead to more flexible tuning strategy for achieving control requirements as compared with the classical controllers [8]. Fractional order control has been extended on advanced and simple control methods; such as adaptive control [9], fractional order model predictive control [10], set-point weighted fractional order controller [11, 12], and Smith predictor based fractional order controller [1317]. The long dead time existing in many industrial processes is a difficulty in achieving good performance by using PID controller. In [18], Smith proposed an effective dead time compensator which was able to control processes with long dead times. The Smith predictor and its modifications are a practical approach to delay compensation for the control of processes with long time delay [19-21]. In some cases, Smith predictor based

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2.

Fractional Order PID Controller

Fractional calculus is a generalization of integration and differentiation to non-integer order. A fractional order integral of order J is defined as follows [1]. t 1 J (1) (t  W ) J 1 x(W ) dW , J   0 I t x(t ) *(J ) ³ 0

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2015 23rd Iranian Conference on Electrical Engineering (ICEE)

In (1), *(J ) is the Euler’s Gamma function. It can be shown that L 0 It J x(t ) s J L ^x(t )` , (2)

end, in (6) consider O parameters U is U ª¬k p ki kd P º¼ .

where L{.} denotes the Laplace transform operator [1]. The Caputo definition of the fractional order derivative of order J , where J is a positive non-integer number is given by

The gain and phase margins are two important frequency domain specifications and two stability measures. It is known that the phase margin is related to the damping of the system and therefore can also serve as a performance measure [30]. Thus, the following equations must be established L( jZc ) dB 0 , (8)

^

C 0

DtJ x(t )

`

­ r 1J ­ d r 1 ½ °0 It ® r 1 x(t ) ¾ , r  J  r  1 , r  N ‰ {0} ° ¯ dt ¿ . (3) ® r °d ( ) , J x t r °¯ dt r

`

A fractional order transfer function represented as H (s )

Q (s Ek ) P (s Dk )

H ( s)

is

bm s Em  bm 1s Em 1   b1s E1  b0s E0 , (5) an s Dn  an 1s Dn 1   a1s D1  a0s D0

where ak ( k 0,1,..., n ) and bk ( k 0,1,..., m) are constant coefficients and also, D k ( k 0,..., n ) and Ek ( k 0,..., m) are arbitrary real numbers [1]. Recently, it has been shown that applying fractional order controllers in control systems may lead to an improved performance in comparison with the traditional controllers [2-4]. The FOPID controller has been proposed in [8] as a generalization of the classical PID controller with an integrator of order O and differentiator of order P . The transfer function of this type of controller is k (6) C (s) k p  Oi  kd s P , 0  O  2 , 0  P  2 . s In (6), k p , ki , and kd are respectively the proportional, integral, and derivative gains of the FOPID controller [8]. 3.

(7)

and (9) Arg ( L( jZc )) S  Im , where Zc is the gain crossover frequency and Im is the required phase margin. In (8) and (9), L(s) Gn (s)C (s) where C ( s) is the controller and Gn ( s) is model of the plant without time delay. The sensitivity function must have a small magnitude at low frequencies to achieve good reference tracking and disturbance rejection. Therefore, the following inequality should be established. 1  C ( j Z )G n ( j Z )(1  e  j W Z ) S ( j Z ) dB d B (dB ) 1  C ( j Z )(G n ( j Z )(1  e  j W Z )  P ( j Z )) dB

Also, if the initial conditions are considered as zero, the Laplace transform of the Caputo fractional derivative is given by L C0 DtJ x(t ) sJ L ^x(t )` , r  J d r  1 . (4)

^

1 so the vector of control

n

n

Z d Zs .

(10) In (10), the parameter B is determined by the designer and its value is chosen such that the control requirements on reference tracking and disturbance rejection are satisfied. Also, for reducing high frequency noise, at high frequencies the complementary sensitivity function must have a small magnitude. As a result, the following inequality must be established. C ( j Z )P ( j Z ) d A (dB ) T ( j Z ) dB 1  C ( j Z )(G n ( j Z )(1  e  j W Z )  P ( j Z )) dB n

Z t Zt . (11) In (11), the value of parameter A is chosen according to the intended control objective on noise reduction. In (10) and (11), P( jZ ) is the transfer function of the plant and W n is the estimated time delay of the process. According to (8)-(11), the controller parameters can be tuned through solving a nonlinear optimization problem which can be done iteratively. It is assumed that the process model is accurate. Hence, the following inequalities will result for the design problem: 1  C ( jZ )Gn ( jZ )(1  e jW nZ ) S ( jZ ) dB d B(dB) 1  C ( jZ )Gn ( jZ ) dB

SPFOPID Controller Design

The general schematic of standard Smith predictor control structure is shown in Fig. 1. Different methods have been proposed for tuning the parameters of Smith predictor based fractional order controllers [13-17]. In this paper, an FOPID controller is employed and the internal model is considered to be a First Order Plus Dead Time (FOPDT) transfer function.

Z d Zs , and Fig. 1. Schematic of Smith predictor structure [19]

T ( jZ ) dB

In [25], a method has been proposed for tuning FOPID controller. This method has been also used for tuning robust FOPI and FOPID controllers in [26-29]. In this paper, the method presented in [25] is used for tuning FOPID controller in the Smith predictor structure. To this

Z t Zt .

(12)

C ( jZ )Gn ( jZ ) d A(dB) 1  C ( jZ )Gn ( jZ ) dB

(13) The specification in (8) is considered as the main function to be minimized and the rest of specifications

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2015 23rd Iranian Conference on Electrical Engineering (ICEE)

where l and f are the level and the flow outputs. Also, ul and u f are respectively the position valve actuators. Relative gain array (RGA) is used for input-output pairing of the decentralized controller [35, 36]. The RGA matrix in this process is obtained as follows. /(GFlow Level ) GFlow Level (0). GFlow Level T (0)

(9), (12), and (13) are considered as the constraints for the nonlinear minimization problem defined based on (8). 4.

Experimental Results

In this section, the results obtained through practical implementation of SPFOPID controller on a MIMO flowlevel process are given. The schematic of interconnection between RT522 process trainer flow and RT512 process trainer level have been illustrated in Fig. 2. The motor No. 1 pumps water into the pipes. Valves No. 7 and No. 8 are closed. Valve No. 3 is used to interconnect the two systems. We set it in 40 degrees opening position. Valve No. 2 in the level system and valve No. 6 in the flow system are the inputs of the MIMO system. The level input valve and the flow valve can be adjusted from 0 to 100 percent of their opening position and -100 to 100 opening and closing rate respectively. Valve No. 4 is used to produce disturbance for the level system. This valve is set on 30 degrees of opening position during our experiments. There are two outputs for the system, the level in tank No. 5 and the flow passing through the flow system measured by flow-meter No. 10 [31, 32].

(16) ª 1.039 0.039 º « 0.039 1.039 » . ¬ ¼ The schematic of the decentralized control structure of this process is shown in Fig. 3.

Fig. 3. Decentralized control of MIMO flow-level plant

According to relation (15), the single input-single output (SISO) transfer functions of the level and the flow subsystems are respectively obtained as Wn s 3.481  6.4 s , (17) G11- Level ( s) G n1 ( s) e 1 e 1  37.8 s and Wn s 0.6828  4.4 s . (18) G22- Flow ( s) G n 2 ( s) e 2 e 1  8.4 s Now, for transfer function (17) consider rad rad Z c1 0.13( ), I m1 95o , Z t 1 6( ), sec sec rad Z s 1 0.008( ), A 1 20(dB ), B 1 20(dB ) . (19) sec Implementing the method described in Section 3, a SPFOPID controller can be designed for the control of the level output as follows 0.0479 CFOPID _ level (s) 1.361   1.0883 s 0.7899 . (20) s Similarly, by considering rad rad Z c 2 0.22( ), I m 2 90o , Z t 2 10( ), sec sec rad Z s 2 0.008( ), A2 20(dB), B 2 20(dB), (21) sec for transfer function (18), a SPFOPID controller can be obtained for the control of the flow output as follows

Fig. 2. Diagram of the interconnection of MIMO flow-level plant [31-33]

The approximation method proposed in [34] is used for implementation of the SPFOPID controller. The order and the frequency range of the approximation filter are considered as N 5 and Z [0.001,100](rad / sec) respectively and also, Ts 0.4(Sec) is the selected as the sampling time. The process is identified by applying step inputs and the resulted transfer function matrix is ª ul º ªl º (14) « f » GFlow Level «u » , ¬ ¼ ¬ f¼

GFlow  Level ( s)

ª 3.481 6.4 s «1  37.8 s e « « 5.2524 5.2 s « 1  1.8 s e ¬

0.017 13.2 s º e » 1  32.4 s », 0.6828 4.4 s » e » 1  8.4 s ¼

(15)

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2015 23rd Iranian Conference on Electrical Engineering (ICEE)

0.3922 (22)  1.1051 s 0.2995 . s In [20], a method for tuning the parameters of PID controllers based on the standard Smith predictor structure is given. In this section, we compare the efficiency of the designed SPFOPID controllers with the SPPID controller designed based on the method described in [20]. Applying this method, the following SPPID controllers are respectively obtained for the control of the level output and the flow output. 0.0345 (23) CPID _ level ( s) 1.4152   4.1753 s . s 0.256 (24) CPID _ flow ( s) 2.7132   4.7302 s . s The magnitudes of the sensitivity and complementary sensitivity functions of the level and the flow output for the nominal plant are shown in Figs. 4-7. As can be seen in Fig. 4 and Fig. 6, the magnitudes of complementary sensitivity functions by applying the SPFOPID controllers (20) and (22) are smaller than those of obtained by applying the SPPID controllers (23) and (24) at high frequencies. Therefore, reduction of high frequency noise by SPFOPID controllers is significantly better done in comparison with the SPPID controllers. Also, the magnitudes of sensitivity functions by applying the SPFOPID controllers are smaller than those of obtained by applying the SPPID controllers at low frequencies. Therefore, disturbance rejection of the SPFOPID controllers is better than the SPPID controllers (See Fig. 5 and Fig. 7). CFOPID _ flow (s)

2.0807 

Fig. 6. Magnitude of Tflow ( jZ ) by applying two controllers (22, 24)

Fig. 7. Magnitude of S flow ( jZ ) by applying two controllers (22, 24)

The obtained experimental results by applying SPFOPID controllers (20) and (22) and SPPID controllers (23) and (24) are shown in Figs. 8-13.

Fig. 4. Magnitude of Tlevel ( jZ ) by applying two controllers (20, 23)

Fig. 8. Level output by applying the SPFOPID controller (20)

Fig. 5. Magnitude of Slevel ( jZ ) by applying two controllers (20, 23)

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2015 23rd Iranian Conference on Electrical Engineering (ICEE)

Fig. 9. Flow output by applying the SPFOPID controller (22)

Fig. 13. The comparison of two methods in the flow output (22, 24)

The results by applying the designed SPFOPID controllers (20) and (22) and SPPID controllers (23) and (24) are summarized in Table I and Table II. As it can be seen in the obtained results for the control of level output in Table I, the set-point tracking, reduction of high frequency noise, and disturbance rejection of the proposed method are better than the SPPID controller. On the other hand, as it can be seen in Table II for the obtained results of the flow output, the set-point tracking, reduction of high frequency noise, and disturbance rejection of the proposed method are better than the SPPID controller. Table I. The experimental results of level output

Fig. 10. Level output by applying the SPPID controller (23)

Set-point tracking and Comparison of the designed controllers

reduction of high frequency noise

Disturbance rejection

Transient time

Steady state

(MSE)

(MSE)

SPFOPID (20)

0.7025

0.0035

0.0061

SPPID (23)

0.7244

0.0036

0.0064

(MSE)

Table II. The experimental results of flow output Set-point tracking and Comparison of the designed controllers

Fig. 11. Flow output by applying the SPPID controller (24)

reduction of high frequency noise

Disturbance rejection

Transient time

Steady state

(MSE)

(MSE)

SPFOPID (22)

47.3047

1.8174

12.8216

SPPID (24)

47.3508

1.9565

13.1868

5.

(MSE)

Conclusions

In this paper, a method for designing SPFOPID controller was presented by which compromise between set-point tracking, rejecting of disturbance, and reduction of high frequency noise can be achieved. The efficiency of the designed SPFOPID controllers was verified in the control of an experimental MIMO flow-level plant and the obtained results were compared with those obtained

Fig. 12. The comparison of two methods in the level output (20, 23)

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2015 23rd Iranian Conference on Electrical Engineering (ICEE)

by applying SPPID controllers in a similar structure. The presented method would be a simple and effective control strategy to design a Smith predictor controller for the processes with long dead time.

[18] [19]

Acknowledgement The authors would like to thank Dr. Siavash Fakhimi Derakhshan and Mohammad Jahvani for their effective comments and suggestions to improve the quality of the paper and Farzad Ravosh for helping to put the method into practice.

[20]

[21]

[22]

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