Design and Implementation of SISO and MIMO Processes using PID Controller for Coupled Tank Process

Transcription

1 IJIRST International Journal for Innovative Research in Science & Technology Volume 1 Issue 10 March 2015 ISSN (online): Design and Implementation of SISO and MIMO Processes using PID Controller for Coupled Tank Process Yeshudas Muttu PG Student Department of Electronics, Communication & Instrumentation Engineering Goa College of Engineering Abstract The paper deals with the study of SISO and MIMO processes using PID controller for a coupled tank process. The PID controllers have found wide acceptance and applications in the industries. In spite of their simple structures, PID controllers are proven to be sufficient for many practical control problems. This paper presents the PID controller design for controlling liquid level of coupled tank system. These coupled tank systems form a second order system. Simulation study was carried out for this process with more concern given to the heights of the tank & performance evaluating parameters are then determined. Keywords: Coupled tank, MIMO, PID, SISO, Simulink I. INTRODUCTION Characteristics of coupled tanks for SISO and MIMO are of great importance. SISO stands for single input single output & MIMO stands for multiple inputs multiple outputs. This analysis is not possible without the use of proper controller. The controller used is PID controller. This controller is simplest to deal with and used widely. PID stands for Proportional Integral Derivative controller. Proportional-Plus-Integral-Plus Derivative Controller The combination of proportional, integral and derivative control action is called PID control action and the controller is called three action controllers. Mathematically, II. PROCESS DESCRIPTION & MATHEMATICAL MODELLING A. Coupled Tank Process: Mathematical modelling for SISO process is shown in Figure 1. Fig. 1: SISO coupled tank process Let H 1 and H 2 be the fluid level in each tank, measured with respect to the corresponding outlet. Considering a simple mass balance situation, the rate of change of fluid volume in each tank equals the net flow of fluid into the tank. Thus for each of tank 1 and tank 2, the dynamic equation is developed as follows. All rights reserved by 55

2 Where, H 1, H 2 = height of fluid in tank 1 and tank 2 respectively A 1, A 2 =cross sectional area of tank 1 and tank 2 respectively Q o3 =flow rate of fluid between tanks Q i1, Q i2 = pump flow rate into tank 1 and tank 2 respectively Q o1, Q o2 = flow rate of fluid out of tank 1 and tank 2 respectively Bernoulli s equation for a steady, non-viscous, incompressible liquid shows that the outlet flows in each tank is proportional to the square root of the head of water in the tank. Similarly, the flow between the two tanks is proportional to the square root of the head differential. Where α 1, α 2, α 3 are proportional constants which depends on the coefficients of discharge, the cross sectional area of each tank and the gravitational constant. Combining equation (3), (4) and (5) into both equations (1) and (2), a set of nonlinear state equations which describe the system dynamics of the coupled tank are derived as, B. A Linearized Perturbation Model: Suppose that for a set of inflows Qi1 and Qi2, the fluid level in the tanks is at some steady state level H1 and H2. Consider a small variation in each inflow, q1 in Qi1 and q2 in Qi2. Let the resulting perturbation in level be h1 and h2 respectively. From equations (6) and (7), the equation will become: For Tank 1 For Tank 2 i i Subtracting equations (6) and (7) from equation (8) and (9), the equations that will be obtained are, ( ) For small perturbations, ( ) Therefore, Similarly, ( ) And Simplifying equation (10) and (11) with these approximations, we get All rights reserved by 56

3 C. Second Order Single Input Single Output (SISO) plant: This configuration is considered by having the baffle raised slightly. The manipulated variable is the perturbation to tank 1 inflow. Taking Laplace transform of equation (13) and (14) and assuming that initially all variables are at their steady state value s ( ) Rewriting equations (15) and (16) s ( ) Where For the second order configuration that shows on Figure 2, h 2 is the process variable (PV) and q 1 is the manipulated variable (MV). Case will be considered when q 2 is zero. Then, equation (17) and (18) will be expressed into a form that relates between the manipulated variable, q 1 and the process variable, h 2 and the final transfer function can be obtained as, Fig. 2: Block diagram of a second order process Now, the value for each parameters of the equation (25), the equation for the second order configuration needed to be found. Taking the values as,h 1 = 17, H 2 = 5, α 1 = 0.78, α 2 = 11.03, α 3 = 11.03, A 1 = 32 & A 2 = 32 By solving equation (19), (20), (21), (22), (23) and (24 with those values α 1, α 2, α 3, A 1, A 2, H 1 and H 2, the value of T 1, T 2, K 1, K 2, K 12, and K 21 can be determined. So the value of T 1, T 2, K 1, K 2, K 12, and K 21 are T 1 = , T 2 = , K 1 = ,K 2 = , K 12 = & K 21 = By using the value that has been obtained from T 1, T 2, K 1, K 2, K 12, and K 21 and put it in equation (25), the value of transfer functions become: Then, the transfer function will become All rights reserved by 57

4 D. MIMO System Design: Fig. 3: The coupled tank MIMO process Mathematical modelling is same as for SISO. Here, equilibrium conditions are used and the equations become: in ( ) in ( ) Where, A i = Cross sectional area of tank, a i = Cross sectional area of outlet hole, h i = Water level in tank i. E. Equilibrium Point Calculation: We can calculate the equilibrium points from above equations by equating them to zero: After linearizing and taking Laplace transforms, we get ( )( ) ( )( ) Table - 1 Coupled tank system Modelling parameters System Parameters Value Cross sectional area of coupled tank reservoir (A) m 2 Cross sectional area of the outlet (a i ) Range of the input signal (u i) 0 5 Volts Maximum allowable height in tank (h i ) 0.3 m Constant relating control voltage with the water flow from the pump (n) m/v-sec Substituting the parameters specified in the above table results in the following plant transfer function: 2 [ ] From the above transfer function, we can easily derive the transfer function for a coupled tank system without interaction as follows: [ ] All rights reserved by 58

5 III. SIMULINK DESIGN The equation for coupled tank system will refer the equation (3.4.1). Figure 3 shows the coupled tank system simulink model without controller. Fig. 4: Coupled tank without controller A. Coupled Tank with PID Controller: Fig. 5: Block Diagram of PID Controller combines with plant Fig. 6: Block Diagram of inside Model Plant System Fig. 7: Block Diagram of PID Controller combines with plant for MIMO All rights reserved by 59

6 IV. SIMULATION RESULTS A. Simulation result without Controller for any process: Fig. 8: Plot of Liquid Level at the Coupled Tank 1 From Figure 7, it is seen that the liquid will constantly overflow. This situation arises because this system is running without a controller to control the Pump 1 speed, so the Pump 1 will continuously pump the liquid into the tank until it overflows. A PID controller must be added as it controller element so that the liquid will not overflow and will indicate as required. V. SIMULATION RESULT WITH PID CONTROLLER FOR SISO PROCESS This section will show the simulation result with the PID Controller. The configuration of the MATLAB simulink model for PID Controller combines with coupled tank is shown as in Figure 9. The performance result for each parameter are also been discussed based on overshoot, rise time and steady state error. Fig. 9: Plot Performance of PID Controller (SISO) Here, the set point is set equal to 3. The proportional gain is set equal to 12, integral gain is set equal to 4 and derivative gain is set equal to 7 to provide the desired response. The plot shows that the output voltage achieves the set point voltage at time equal to 10 second. The output voltage have slightly overshoot before stabilize at time equal to 20 second. VI. SIMULATION RESULT WITH PID FOR MIMO PROCESS As per the transfer function, PID is tuned and the output response was observed on scope as shown below (Figure 9). All rights reserved by 60

7 Fig. 10 (a): Plot Performance of PID Controller (MIMO) with Set-point = 1 Fig. 10 (b): Plot Performance of PID Controller (MIMO) with Set-point = 10 We have given the set point and observed the influence of one on the other. Study of rise time, peak overshoot and settling time was done. Table 2 Study of Rise Time, Peak Overshoot and Settling Time System Rise Time (sec) Overshoot Settling Time (sec) SISO (set point = 3) % 150 MIMO(set point = 1) 35 5% 180 MIMO(set point = 10) 40 5% 220 VII. CONCLUSION & FUTURE WORK The coupled tank process with respect to SISO and MIMO was studied and the rise time, peak overshoot, and settling time was calculated for respective set points and the waveforms obtained. I had considered rectangular tanks. The project may be applied for various non-linear tanks like spherical tanks. The study can not only be applied for two tanks, but is applicable for three tanks & even quadruple tank system. All rights reserved by 61

8 REFERENCES [1] J. Gireesh Kumar &Veena Sharma, Model Predictive Controller Design for Performance Study of a Coupled Tank Process, ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE), ISSN (PRINT), , Volume -1, Issue -3, [2] MohdIzzat, BDzolkafle, Implementation of PID Controller for Controlling the Liquid Level of the Coupled Tank System, unpublished. [3] M.Senthilkumar, Dr.S.Abraham Lincon, Design of Stabilizing PI Controller for Coupled Tank MIMO Process, International Journal of Engineering Research and Development, Volume 3, Issue 10 (September 2012), PP [4] Pawan Kumar Kushwaha and Vinod Kumar Giri, Control Strategies for Water Level Control of Two Tank System, IJBSTR Research paper Vol. 1,Issue 8, August [5] M. Senthilkumar a and S.Abraham Lincon, Design of Multiloop Controller for Multivariable System using coefficient diagram method, IJARET, Volume 4, Issue 4, May June 2013, pp [6] Truong Nguyen Luan Vu, Jietae Lee, and Moonyong Lee, Design of Multi-loop PID Controllers Based on the Generalized IMC-PID Method with Mp Criterion, International Journal of Control, Automation, and Systems, vol. 5, no., pp. -217, April [7] D. Dinesh Kumar, C. Dinesh & S.Gautham, Design And Implementation of Skogestad PID Controller For Interacting Spherical Tank System, International Journal of Advanced Electrical and Electronics Engineering, (IJAEEE), ISSN (Print): , Volume-2, Issue-4, [8] Qamar Saeed, Vali Uddin and Reza Katebi, Multivariable Predictive PID Control for Quadruple Tank, World Academy of Science, Engineering and Technology All rights reserved by 62