Design and Implementation of Stable PID Controller for ... - Core

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ScienceDirect Procedia Computer Science 79 (2016) 737 – 746

7th International Conference on Communication, Computing and Virtualization 2016

Design and Implementation of Stable PID Controller for Interacting Level Control System C. B. Kadua* and C.Y.Patilb a

Department of Instrumentation and Control Engineering, Pravara Rural Engineering College, Loni- 413736, Savitribai Phule Pune University, India, [email protected] b Department of Instrumentation and Control Engineering, College of Engineering, Pune -411005 , India, [email protected]

Abstract: In this present study a Stability Region Analysis method for designing PID controller for time delay system is validated with real time experimentation with Interacting process. The higher order system is reduced into first order plus time delay (FOPDT) model. A polyhedral sets in the 3D (three dimensional) search band parameter space

is resulted from set of stable PID controllers for a fixed value of , and it is faster and simple

to identify these regions. The stability region verification is done with dual locus diagram. An approach presented works satisfactorily without sweeping over the parameters, and without any complicated mathematics. While designing optimization method based advanced PID controllers it is beneficial and reliable to determine range of variables. We have presented one simulation example and experimental validation on Interacting Level Control System. Keywords: Stability region analysis, First order plus dead time (FOPDT), Proportional Integral Derivative Controller (PID), and Level control

1.

INTRODUCTION

In the last couple of decades an automatic control theory has been developed significantly; however in industries the use of proportionalintegral-derivative (PID) controllers remains unaffected. In process industries, until the last decade more than 90% of all control loops are Proportional Integral Derivative (PID) [1, 2]. They are being used due to simplicity, effectiveness and better understanding of control action and realization of digital PIDs. For several decades, a variety of PID tuning ways for first-order-plus-delay-time system have been elaborated in the literature. Some of them are Ziegler- Nichols method [2], Cohen-Coon method [2], constant open loop transfer function method [2,4], synthesis method [2,5] internal model controller [6] etc. For high order time delay processes, these tuning methods have some drawbacks and most of the time doesn’t give satisfactory tuning parameters for the PID controllers. C. B. Kadu is working as an Associate Professor in the department of Instrumentation & Control Engineering, Pravara Rural Engineering College, Loni, Ahmednagar 413736, INDIA. Email: [email protected]

1877-0509 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ICCCV 2016 doi:10.1016/j.procs.2016.03.097

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C.B. Kadu and C.Y. Patil / Procedia Computer Science 79 (2016) 737 – 746

The methods based on gain and phase margin (GPM) specifications are used to tune the PID controllers for processes exhibiting time delay and integral with delay time. ([7] - [8] - [9]-[10]-[12]-[13]). For measure of robustness, consideration of damping factor is done which is related to PM. In gain and phase margin the solutions are usually determined by graphically or numerically by trial-and-error. For systems having infinite phase crossover frequencies these methods are not suitable. The major disadvantage of the GPM method is that they are limited to the FOPDT or SOPDT [13]. In this paper, stability region analysis approach is applied to design PID controller for first order plus time-delay system. A polyhedral sets in the 3D (three dimensional) search band parameter space

is resulted from set of stable PID controllers for a fixed value of , and it is faster and simple

to identify these regions. [3]. The stability region verification is done with dual locus diagram. The paper is organized as follows: In Section 2 Model of interacting tank level control system is discussed. In Section 3 design approach of stable PID controller is given. Dual Locus Diagram for Stability Verification is incorporated in section 4. The experimental setup is explained in section 5 with its real time results. Conclusion is summarized in Section 6.

2.

SYSTEM INTRODUCTION AND MODELLING

An Interacting level system is combination of two tanks connected to each other usually also referred as couple tanks. The Setup consist of supply tank with variable speed positive displacement pump for liquid transportation, two acrylic tanks fitted with flow restrictors and transmitter are as shown in Fig. 1. As pump (p) is driven with variable speed it requires VFD with signal of 0 to 5 V. The Transmitter output is 4 to 20 mA for 0 to 100% of liquid level in tank. The process is interfaced with the digital computer through the controller with SCADA.

Fig.1. Interacting Level System Portrayal

In order to tune the PID controller the process model is determined by operating the pump (p)with 100% step, valve (V1) is fully open the valve (V2) is adjusted such that the level in tank (T2) is reached to the 93% of the full scale range. From this open loop experimentation the process model is obtained from process reaction as

#,(3

(1)

The above model is validated by matching the actual process reaction curve with step response of the model and responses are found to be comparable with each other and hence model is validate.

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3.

STABLE PID CONTROLLER DESIGN APPROACH

Here we consider standard unity feedback (SISO) control system with FOPDT process * and ideal PID controller *. The problem is to find out the proper values of PID controller parameter i.e. set of values. * *#,(

#,(

#,(

* 2* *

Then the closed-loop transfer function is given by * * *2 * *

(4)

Substitute Eq.(2) into Eq.(4) and denote * **, the closed-loop characteristic polynomial can be obtained as -* ** * * *#,(

(5)

Then, by putting * %/ and decomposing the numerator and denominator polynomials of * into real and imaginary part, there exists %/ / %/ / 2 / %/ / Dropping /for simplicity and using Euler’s identity for #,(

(6)

"'*+ / 3 %*$&+ /into Eq.(5), characteristic polynomial can

be expressed as -%/ / % /

(7)

Where / 3 / 3 / "'*+ / / / /

3 / *$&+ / 3 /

3 / "'*+ / / 3 / *$&+ / /

The characteristic equation for Eq. (5) then becomes / /

(9)

From Eq. (8), we obtain / / / / / / / /

(10)

Where / / *$&+ / 3 /

"'*+ /

(8)

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/ "'*+ / /

*$&+ /

/ 3/ "'*+ /

*$&+ /

/ /

(11)

/ / "'*+ / / / /

*$&+ /

"'*+ / 3 *$&+ /

/ / *$&+ / 3 /

"'*+ /

/ 3/

(12)

It is difficult to get a unique solution for the parameters , and as a function of / from Eq. (10) because number of equations is less than that of required one. In classical tuning, usually varies with and/or .Hence, we can assume 2&

(13)

Approximately industrial engineering value of n(proportion factor) is4 . Substituting Eq. (13) into Eq. (10), and removing / for simplification. Thus it can be expressed as

3 3 3 3 3 3 & 3 3

& 3 & 3 3

(14)

Generally, the frequency range /0 / is consider for the stability analysis due to operability of controllers, where, the crossover frequency of plant is given by/ . Since the phase of %/at / / is3., from Eq. (2) and Eq. (6), one can obtain +!&

3 + / 3.

3 +!&

(15)

Taking the tangent for the first two terms, and noticing that the tangent of other two terms +!&+ / 3 . +!&+ /let /

(16)

To determine / , find the minimum positive value of / at which two plot intersects with each other from the plot of /and +!&+ /versus /, and note that / +!&+ /.The minimum positive value of / in Eq. (14) which make will be denoted as / . The stability boundary of PID controller parameters and in Eq. (14) can be easily determined by comparing the / to / , the smaller one will be considered as stability frequency / . The special case as per assumption in Eq. (13), when , the feedforward path transfer function * * has the same nature as like plant *. So, in this case / / / .

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4. SIMULATION EXAMPLES In this section simulation of one example is done in MATLAB 7.11.0 to illustrate how effective is the method. The PID controller designed by the stability region analysis method is compared with numerical optimization approach for PID design by [14] Example Consider the higher order and moderately oscillatory process given in [11] 1

#,(3

(17)

The FOPDT model is obtained by using frequency response method

#,(3

(18)

As per the design presented in section 3, the plant Gp s is controlled by a PID controller. By Eq. (13), we can Substitute n 4, from Eq. (14) Kp sin 1012 −cos 1012 Ki 4 sin 1012cos 1012 24 Kd sin 1012cos 1012 24 Using Eq. (16), it is easy to find the crossover frequency

as well as

. The stability frequency

shown in Fig.3. A spatial curve is plotted in space coordinates (Kp, Ki, Kd) by substituting

f = 2.1200 is obtained as

from 0 to 2.1200 into Eq. (19). The closed

Stable spatial dimension is shown in Fig.4. For a specific time-delay system to obtained search band for optimum results in this stable spatial dimension without sweeping over all possible regions is very effective while designing an optimal PID controller. . If it gives Gc(s) = 10.19+ 8.69/s+2.17s, by using dual locus diagram it is easy to determine stability of the time-delay system as in fig. 5. * * * * First of all, determine the minimum value of = j into Eq. (20), at

c

at which the locus of H(s) intersects with the unity circle, denoted by

=1.5550. Hence, the phase angles 1 of H(s) and 2 for −exp(t0s) at

1 !)"+!&

!)"+!&

/* . Obviously,

1

is larger than

2.

(20)

s

c.

Substituting s

are given by below equations respectively

= 5.2231 (21)

As per the stability condition of dual-locus diagram, the locus of −exp (t0s) reaches later at the

intersection than that of H(s). Hence from this observation we conclude that designed PID controller will make the time-delay system stable. We have simulated the system by using designed Stable PID controller and results are compared with the controller designed using Numerical optimization approach given by [15] is as shown in Fig.2. For unit step set point change at time t = 0. Thus the Stable PID controller gives better response with n

4 (i.e. Kd = 2.80).

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Fig. 2 Unit step response of system with Stable & NOA PIDcontroller

Fig. 3 Graphical computation of

and

for higher order and moderately oscillatory system

Fig. 4 3D Region of higher order and moderately oscillatory system


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Fig. 5 Dual-locus diagram of higher order and moderately oscillatory system Table 1 Performance specifications (overshoot in % (OS), settling time in seconds (ST), ITAE: integral time absolute Error, NOA: Numerical Optimization Approach) Ex.

5.

Method Stable PID NOA PID

Kp 10.19 4.54

Controller gain Ki Kd 8.69 2.80 4.63 1.11

OS 0.00 0.00

Parameter ST 4.40 7.91

ITAE 12.50 37.30

EXPERIMENTAL VALIDATION AND RESULTS

To validate the stable PID controller, we have carried out a real-time experimentation on level control loop. The prototype experimental setup of the level system is as shown in Fig.6.

Fig.6. Laboratory Interacting Level System

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A. Simulation Result: The Stable PID algorithm is implemented in MATLAB for the process model given by Eq. (1), the as shown in Fig. 3. By plotting and

2

Fig.7. Graphical Computation of

and

is find out graphically

versus . The PID controller obtained by using this Stable PID algorithm is (22)

and

Fig.8. Dual Locus Diagram

The stability of the PID controller given in Eq. (22) with plant Eq. (1) is verified by plotting Dual Locus Diagram as shown in Fig. 4.

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B. Experimental Result: The PID Controller in Auto Mode with settings in Eq. (22) the step response of the interacting tank level control system for different set point input and disturbance input is obtained as shown in Fig. 5. The summary of performance specifications of experimental validation are given in table 2. Table 2 Controller

Specifications

Plant

Interacting Level System

6.

76.2566

0.7996

0.1999

OS

ST

(Overshoot in %)

(Settling Time in Sec.)

0.6000

313

Fig.9. Step response of the Interacting Level System with Stable PID Controller for

Fig.10. Step response of the Interacting Level System with Stable PID

S.P.tracking

Controller for disturbance rejection

CONCLUSION

In this work, we have presented easy and reliable tuning method of PID controller for reduced FOPDT model. The stability regions of time-delay systems with PID controllers shows the effectiveness of the stable PID tuning method. The validity of the method is determine by dual-locus diagram. An experimental validation of the method is done by applying this method to simple Interacting level control loop. The controller designed with the given method provides good set point tracking and also offers better disturbance rejection. The stability region analysis method is applicable only for the FOPDT model of process and it is simplest method used for tuning of the PID controller. Acknowledgments

!

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[2]

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