CHAPTER 7 MODELING AND CONTROL OF SPHERICAL TANK LEVEL PROCESS 7.1 INTRODUCTION

Transcription

1 141 CHAPTER 7 MODELING AND CONTROL OF SPHERICAL TANK LEVEL PROCESS 7.1 INTRODUCTION In most of the industrial processes like a water treatment plant, paper making industries, petrochemical industries, nuclear power plants, etc., the control of liquid levels in tanks and the rate of change of flow from a tank to another tank are elementary problems. Most of the liquid is required to be pumped and stored in tanks, from there into another tank and processed by chemical treatment, but the level of liquids must be always controlled during the entire process (Falkus & Damen 1993). The liquid level control in a spherical tank is complex due to the constant change in the cross sectional area. A level which is extremely high may distress the reaction equilibria, damage the equipment, or result in the spillage of the precious or harmful material. If the level is extremely low, it may affect the successive operations adversely. Hence, the level is the acute factor to be controlled here and some form of truthful and best control is critical for this process (Al.-zahrani & NoorWali 1995). Hence, substitute methods based on the use of expert knowledge have been proposed to develop nonlinear empirical models, to be used in the design of control systems.

2 142 This chapter examines the application of a proposed probabilistic fuzzy inference system for modeling and control of a highly nonlinear process using simulation of spherical tank level process and also implemented in the real process. The lab scale experimental set-up and mathematical model of a spherical tank level process are presented. Then, a fuzzy system is applied to develop fuzzy model and design fuzzy logic controller for the same nonlinear spherical tank level process. Probabilistic fuzzy model and probabilistic fuzzy logic controller are designed for the same process. Finally, an effort is also made to compare the accuracy of probabilistic fuzzy model with conventional fuzzy model in terms of RMSE values. Likewise the various performances of probabilistic fuzzy logic controller are compared with conventional fuzzy logic controller and classical PID controller in terms of various performances indices. 7.2 DESCRIPTION OF THE LABORATORY PROCESS SETUP The spherical tanks are widely used in most of the process industries because their shape gives better disposal of solids and liquids. The level control in a spherical tank is a challenging problem due to nonlinearity and uncertainty associated with the change in the shape of the tank. An experimental set-up and the schematic diagram of the hardware set-up for spherical tank level process are shown in Figure 7.1 and Figure 7.2 respectively. The set-up consists of a spherical tank, pump, Rotameter, water reservoir, level sensor, I/P converter (electro pneumatic converter), an interfacing module VMAT (Data Acquisition Module), a pneumatic control valve with positioner and a personal computer (PC). The output of the level sensor is interfaced with the computer by means of VMAT through RS-232 port of the PC. The interface module is used to link the programs written in

3 143 script code using MATLAB software with sampling period of 60 milliseconds. The specifications of the experimental setup of the spherical tank level process are presented in Table 7.1. Figure 7.1 Experimental set-up of the Spherical Tank Level process Figure 7.2 Schematic diagram of the Spherical Tank Level process

4 Table Technical specifications of experimental set-up of Spherical Tank Level process Part name Details Body material SS 316 Sperical tank Diameter 50 cm Capacity 102 literes Storage tank Body material SS 316 Volume 48 literes Make Telien / Equivalent Rotameter Type Variable area Range 0 20 lpm Control valve Type Air to close Input 3 15 psi Pump Centrifugal 0.5 HP Air regulator Size 1/4 BSP Range bar I/P Convertor Input 4-20 ma Output 0.2 1bar V/I Convertor Input 1 5 V Output 4-20 ma I/V Convertor Input 4-20 ma Output 1 5 V Pressure Gauge Range 0 30 psi Range psi The level transmitter senses the water level in the tank which is calibrated from 0 to 50 cm and is converted into an output current of 4 to 20 ma. This current in turn is converted into 0 to 5 v using I/V converter. This voltage is communicated to the interfacing module and hence to the PC. According to the control algorithm in the PC, the control signal is computed which is again 0 to 5 v. This voltage is converted back into 4 to 20 ma by means of V/I converter. This current is then passed to I/P converter, which is

5 145 finally converted into a pressure signal of range, 3 to 15 psi. This pressure signal changes the stem position of the pneumatic control valve to produce the necessary level in a tank. 7.3 MODELING OF THE SPHERICAL TANK LEVEL PROCESS Mathematical Modeling of Spherical Tank Level Process According to the law of conservation of mass, the first order differential equation describing the nonlinear dynamics of a spherical tank system as shown in Figure 7.3 is given by the equation (7.1). Figure 7.3 Spherical Tank dynamics Application of the Chain Rule and a little algebra produces an ordinary differential equation which defines the effect of system inputs (inflow and outflow ) to the system output (level). The model is nonlinear in nature since the system inputs, outputs, and states are not in a linear combination. = (7.1)

6 146 = (7.2) = (2 ) (7.3) where is the inlet flow rate; is the outlet flow rate, ; is the tank volume, h is the level of the tank (head), is the radius of the tank, is the outlet flow capacity coefficient and A is the area of the tank (Rajinikanth & Latha 2012) Black box modeling For the tuning of PID controller, black box modeling is applied to obtain the model. The spherical tank is allowed to fill with water from 0 to 50 cm by keeping the inlet and outlet flow rates at a fixed value. At every sample time, the level of the tank is collected from the level sensor and fed to the PC and thereby the data is scaled up in terms of level in cm. Two point method is employed to determine the parameters of the transfer function, FOPTD (First Order Plus Dead Time), by allowing the response of the model and the actual system to meet at two points that describe the parameters and. From the step response curve, the proposed times (time to reach 35.3% of response) and (time to reach 85.3% of response) are estimated (Nithya et al 2008). The time constant and time delay are computed as = 0.67( ) = (7.4) where - Time constant and - delay time (or dead time).

7 147 The spherical tank process is allowed to reach the steady state by keeping the inlet flow rate and the output flow rate as fixed. Then, a step increment is given in the input flow rate and the various values of process variable were noted down until the process reaches the steady state again and approximated to be a FOPDT model as () = (7.5) The above model is used in the implementation of PID controller, fuzzy logic controller and the probabilistic fuzzy logic controller Fuzzy Modeling of Spherical Tank Level Process The spherical tank level process is simulated using the nonlinear first principle model equation (7.3) in all simulation runs. A collection of N input-output data training pairs; ) ) ), ( ) ) ),, ( () () ), where x is the vector input and y is the scalar output. Simulation is based on 500 samples, the first 250 data samples are applied for training and the remaining samples are applied for testing. Three inputs (antecedents) are used for estimating the model of spherical tank level process; the inflow rate ), 1), and liquid level 1). The training pair is represented as 1), ), 1), )]. Three fuzzy sets are used for each of the antecedent, so the number of rules is 3 3 = 27. The fuzzy model of spherical tank level process is designed from 250 data samples generated using simulink model. That is extracting the set of rules from the data which models the process using fuzzy c-means (FCM) clustering algorithm as discussed in the section Each rule is characterized by six antecedent membership function parameters (mean and standard deviation of each fuzzy set of every antecedent) and one consequent

8 148 parameter. The initial locations of antecedents MFs are based on the mean and standard deviation of the first 250 samples. Here, Sugeno fuzzy model with Gaussian MFs, product implication and t-norm, and weighted average are considered. The designed fuzzy model is validated with the last 250 samples of test data ( ), which is presented in Figure 7.4. The RMSE for fuzzy model is computed as defined by equation (5.12). Data is corrupted by adding the Gaussian noise with different SNR and in each case the value of RMSE is calculated and presented in Table 7.2. Figure 7.4 Validation of Fuzzy model of Spherical Tank Level process with test data Probabilistic Fuzzy Modeling of Spherical Tank Level Process To design, probabilistic fuzzy model for a spherical tank level process, the same three antecedents, 1), ), 1) and 500 data samples are used to estimate the liquid level in a tank). Three fuzzy sets are used for each of the antecedent, so the number of rules is 3 3 = 27. The Sugeno fuzzy model is considered with Gaussian primary MFs with an uncertain mean (but constant standard deviation) and interval secondary membership functions (shown in Figure 4.5) are used for antecedents, product

9 149 implication and t-norm, center-of-sets order-reduction and defuzzification employing the centroid of the order reduced set are considered for the design of the probabilistic fuzzy model. There are nine parameters associated with all antecedent membership functions i.e. lower and upper limits on the mean and standard deviation (i.e. 3*3=9). And two parameters, i.e. the left-end and right-end points associated with the centroid of the consequent probabilistic fuzzy set. Every rule of the probabilistic fuzzy model is described by these antecedent and consequent parameters. The design steps discussed in the section 4.7 are applied to build the probabilistic fuzzy model of spherical tank level process. To prepare this, first best possible fuzzy model is obtained by tuning all of its parameters and then these parameters are applied to initialize the parameters of the probabilistic fuzzy model of the process. Hence, the initial values of the parameters of the probabilistic fuzzy model are taken from the parameters of the fuzzy model, i.e. obtained from the data previously in the section These initial values of the parameters are considered as the parameters of the antecedent membership functions, and, where = 1,, 4and = 1,, 16. The initial shapes of these MFs are chosen randomly as shown in Figure 4(b) by distorting the fuzzy sets of the built fuzzy model shown in Figure 4(a). Similarly, the centroids of the membership functions of the consequent sets are taken from consequents of the same fuzzy model. For each training pair by employing the product t-norm, the total firing interval for each rule, i.e., values of and are determined. The values of the left-end point and the right end points are computed. The average of these two points is taken as the defuzzified value ). If the deviation of this defuzzified value is more from the desired value then the parameters of the FIS are retuned. Next, the active branches of the upper and lower antecedent MFs for () are determined. After determining the active branches, their

10 150 antecedent and consequent MF parameters are tuned using the steepest descent algorithm according to the error function. The developed probabilistic fuzzy model is validated with the test data, which is presented in Figure 7.5. The data is corrupted by adding the Gaussian noise with different SNR and in each case the value of RMSE is calculated and presented in Table 7.2. Figure 7.5 Validation of Probabilistic Fuzzy model of Spherical Tank Level process with test data The simulation comparison of approximation error and RMSE values of fuzzy model with a probabilistic fuzzy model for the spherical tank level process are shown in Figure 7.6 and Figure 7.7 respectively. From the simulation results, it is concluded from the Table 7.2 and Figure 7.7 that the RMSE values of the probabilistic fuzzy model have been found to be considerably less than that of conventional fuzzy model in each case.

11 Table RMSE values of Fuzzy and Probabilistic Fuzzy models for noise with various SNR Noise with various SNR db RMSE Fuzzy model Probabilistic Fuzzy model Figure 7.6 Comparison of approximation error for Fuzzy and Probabilistic Fuzzy models Fuzzy model Probabilistic Fuzzy model Figure 7.7 RMSE values of Fuzzy and Probabilistic Fuzzy models for noise with various SNR

12 DESIGN AND IMPLEMENTATION OF CONTROLLERS FOR SPHERICAL TANK LEVEL PROCESS In this section, design of different controllers such as PID controller, fuzzy logic controller and probabilistic fuzzy logic controller to control the liquid level, h by controlling the inflow rate,, using simulation of spherical tank level process is presented. These controllers are also designed and implemented on real process. The performances of these controllers for spherical tank level process in both simulation and real time implementation are examined and compared in terms of their various performance indices and specifications Design of PID Controller for Spherical Tank Level Process The modified Internal Model Control (IMC) structure based PID tuning procedure proposed by Skogestad (2003) is used for the design of PID controller. The Skogestad s IMC tuning procedure is derived analytically. It is simple and works well on various processes. A prominent feature is that it is intended for PI/PID controllers. Hence, a first- or second-order process model for the process must be attained. He has suggested a modest technique based on a half-rule to acquire an approximate process model. The result is an approximate FOPDT or SOPDT process model. When these models are achieved, the PI/PID controller gains are calculated to correct the closed-loop response to a first- or second order model reference response (Nithya et al 2008). This technique provides a set of modest analytical PI/PID tuning rules Design of Fuzzy Logic Controller for Spherical Tank Level Process The conventional FLC is designed with two inputs, which are the difference between the required level and the actual level of the tank, error

13 153 ) and the rate of change in the level, i.e. change in error). The controller utilizes these inputs to manipulate the inflow rate, and their universes of discourses are -30 to + 30 cm and -4 to +4 cm. The controller output ) is current whose universe of discourse is 4 to 20 ma which is normalized between 0 and 1 by means of suitable scaling factors; taking into account the operating restrictions. The fuzzy logic controller is designed with Gaussian MFs for each of the two inputs and also for the output. Each of the input variable and the output variable are divided into seven fuzzy sets such as NB, NM, NS, ZO, PS, PM, PB and are presented in Figure 7.8. The fuzzy rules derived from the expert s knowledge considered for the design of an FLC are presented in Table 7.3. The Mamdani type fuzzy inference system is used with minimum implication, maximum aggregation and centroid defuzzification. The control surface of an FLC is presented in Figure 7.9. Figure 7.8(a) Membership functions for )- FLC Figure 7.8(b) Membership functions for )

14 154 Figure 7.8(c) Membership functions for ) - FLC Figure 7.9 Control surface of an FLC for Spherical Tank Level process Table 7.3 Control rules (FAM table) for FLC and PFLC e NB NM NS ZO PS PM PB NB NB NB NB NB NM NS ZO NM NB NB NB NM NS ZO PS NS NB NB NM NS ZO PS PM ZO NB NM NS ZO PS PM PB PS NM NS ZO PS PM PB PB PM NS ZO PS PM PB PB PB PB ZO PS PM PB PB PB PB

15 Design of Probabilistic Fuzzy Logic Controller for Spherical Tank Level Process The proposed probabilistic fuzzy logic controller is designed with the same two input variables, error ) and change in error) and also with the same universes of discourses as in the design of fuzzy logic controller. The same controller output ) with the same universe of discourse which is normalized between 0 and 1 by means of suitable scaling factors; taking into account the operating restrictions. This probabilistic fuzzy logic controller is designed by employing Gaussian primary membership functions with uncertain mean and constant standard deviation, and interval secondary membership functions (shown in Figure 4.5) both antecedent and consequent membership functions. Seven MFs are used for each of the two inputs, error and change in error, and also for the controller output. They are NB, NM, NS, ZO, PS, PM and PB, which are presented in Figure The same control rules presented in Table 7.3 are used. The Mamdani fuzzy inference system is used with minimum implication, maximum aggregation and center of sets method of order reduction. The very similar procedure is followed to design PFLC using the PFIS as that of the probabilistic fuzzy model as discussed in section From the parameters of an FLC designed above, choose the initial values of the parameters of the antecedent membership functions, and. The initial shapes of these MFs are chosen randomly as mentioned in the case of probabilistic fuzzy modeling. Similarly, the centroids of the membership functions of the consequent sets are assumed from consequents of FLC. For each training pair by employing the product t-norm, find the total firing interval for each rule, i.e., values of and. Compute the left-end point and the right end points. The average of these two points is the defuzzified value. If the deviation is more in this defuzzified value of the desired

16 156 value then the parameters of FIS are tuned again in the same manner as discussed in the section The control surface of a PFLC of the spherical tank level process is presented in Figure Figure 7.10(a) Membership functions for ) - PFLC Figure 7.10(b) Membership functions for ) -PFLC Figure 7.10(c) Membership functions for ) - PFLC

17 157 Figure 7.11 Control surface of PFLC of the Spherical Tank Level process Various Performances of the Designed Controllers for Spherical Tank Level Process This section emphasizes the various performances of the designed controllers for spherical tank level process both in the simulation studies and real time implementation. The performances of these controllers are compared based on certain performance measures. The tool used for simulation of both modeling and controller is a Matlab ( ) R2012a Servo performance of the designed controllers The setpoint variations as shown in Figure 7.12(a) and Figure 7.12(c) has been introduced for assessing the tracking capability of (i) PID controller (ii) conventional FLC and (iii) PFLC. It can be inferred from the response that all the controllers are able to maintain the liquid level at the setpoint. The response of PFLC settles a little bit faster with small overshoot than the other controllers. The corresponding variation in the controller outputs is presented in Figure 7.12(b) and Figure 7.12(d). The ISE, IAE and ITAE values of PID, FLC and PFLC are reported in Table 7.4 and

18 158 presented in Figure The various performance specifications of these controllers are presented in Table 7.5. Figure 7.12(a) Servo response of Spherical Tank Level process with PID, FLC and PFLC - Process output Figure 7.12(b) Servo response of Spherical Tank Level process with PID, FLC and PFLC - Controllers outputs

19 159 Figure 7.12(c) Servo response of Spherical Tank Level process with FLC and PFLC - Process output Figure 7.12(d) Servo response of Spherical Tank Level process with FLC and PFLC - Controllers outputs

20 Table ISE, IAE and ITAE Values of PID, FLC and PFLC for setpoint tracking - Simulation Time interval ISE IAE ITAE 0 t t t t 4000 PID 1.689e e e e04 FLC 1.352e e e e04 PFLC 1.131e e e e04 PID FLC PFLC PID 3.188e e e e06 FLC 2.624e e e e06 PFLC 2.101e e e e E E E E E+04 PID FLC PFLC 0.00E+00 0 t t t 3001 t Figure 7.13(a) ISE values of PID, FLC and PFLC for setpoint tracking - Simulation

21 t t t t 4000 PID FLC PFLC Figure 7.13(b) IAE values of PID, FLC and PFLC for setpoint tracking - Simulation 1.00E E E E E+06 PID FLC PFLC 0.00E+00 0 t t t 3001 t Figure 7.13(c) ITAE values of PID, FLC and PFLC for setpoint tracking Simulation Table 7.5 Performance specifications of PID, FLC and PFLC for setpoint tracking- Simulation Time interval 0<t< <t< <t< <t<4000 (%) PID (sec) (sec) FLC (sec) (sec) PFLC (sec) (sec)

22 162 It can be inferred from Table 7.4, Table 7.5 and Figure 7.13 that the ISE, IAE, ITAE and other performance indices of PFLC are found to be considerably less than PID and FLC. However, the performance of the PFLC controller at all operating points is found to be better than conventional PID and FLC and settles to the setpoint little bit faster. The real time servo responses of the spherical tank level process with PID, FLC and PFLC are shown in Figure 7.14 and it shows that PFLC performs better than the others in real-time too. The values of ISE for PID, FLC and PFLC are presented in Table 7.6 and in Figure The various performance indices are reported in Table 7.7. It may be concluded from Table 7.6, Table 7.7 and Figure 7.15 that the values of various performance indices are found to be considerably less for PFLC than other controllers in real time implementation also. Figure 7.14 Real-time servo response of Spherical Tank Level process with PID, FLC and PFLC - Process output

23 Table ISE values of PID, FLC and PFLC for setpoint tracking in real-time implementation ISE Time interval 0 t t 2000 PID 1.877e e04 FLC 1.462e e04 PFLC 1.251e e E E E E+03 PID FLC PFLC 0.00E+00 0 t t 2000 Figure 7.15 ISE values of PID, FLC and PFLC for setpoint tracking in real-time implementation Table 7.7 Performance specifications of PID, FLC and PFLC for setpoint tracking in real-time implementation Time interval 0 < t < < t < 2000 (%) PID (sec) (sec) FLC (sec) (sec) PFLC (sec) (sec)

24 Servo - regulatory performance of the designed controllers The disturbance rejection capability of the PID controller, FLC and PFLC has been demonstrated both in simulation studies and real time and also compared. A step signal producing 20% of load disturbance is given at t = 1500 seconds. These results are shown in Figure 7.16(a). The controller output is presented in Figure 7.16(b). The ISE, IAE and ITAE values of PID, FLC and PFLC are reported in Table 7.8 and presented in Figure The following observations can be made from this part of the simulation study and real time implementation. It is concluded that from Figure 7.16(a), the PFLC is able to quickly discard the disturbance and brings the level to the desired setpoint than other two controllers. The values of ISE, IAE and ITAE for the PFLC controller are found to be considerably less from Table 7.8 and Figure 7.17 than other controllers. Table 7.8 ISE, IAE and ITAE values of PID, FLC and PFLC in the presence of load change Time interval 0 t t 3000 PID 2.651e e04 ISE FLC PFLC 3.496e e05 PID 1.989e e04 IAE FLC PFLC 3.176e e05 PID 1.681e e04 ITAE FLC PFLC 2.411e e05

25 165 Figure 7.16(a) Servo and regulatory response of Spherical Tank Level process with PID, FLC and PFLC- Process output Figure 7.16(b) Servo and regulatory response of Spherical Tank Level process with PID, FLC and PFLC - Controller output

26 E E E E E E E+00 t t 3000 PID FLC PFLC Figure 7.17(a) ISE values of PID, FLC and PFLC in the presence of load change t t 3000 PID FLC PFLC Figure 7.17(b) IAE values of PID, FLC and PFLC in the presence of load change 1.20E E E E E E E+00 t t 3000 PID FLC PFLC Figure 7.17(c) ITAE values of PID, FLC and PFLC in the presence of load change

27 167 The real time servo and regulatory responses of the spherical tank process with PID, FLC and PFLC are shown in Figure 7.18 for the same load disturbance as given in the simulation, which shows that the PFLC discards quickly the disturbance and brings to the desired level than others. The value of ISE for PID, FLC and PFLC are presented in Table 7.9 and Figure These values are found to be considerably less for PFLC than other controllers. Figure 7.18 Real-time servo and regulatory response of Spherical Tank Level process with PID, FLC and PFLC- Process output 2.500E E E E E E+00 ISE PID FLC PFLC Figure 7.19 ISE values of PID, FLC and PFLC in the presence of load change in real-time implementation

28 Table ISE values of PID, FLC and PFLC in the presence of load change in real-time implementation Controllers ISE PID 2.205e04 FLC 1.918e04 PFLC 1.527e Performance of FLC and PFLC in the presence of measurement noises Gaussian noise of zero mean and various standard deviations (7 cm and 10 cm) has been added to the true value of the process variable, i.e. liquid level in the tank. The performances of the FLC and proposed PFLC in the presence of measurement noise of various levels are shown in the Figure 7.20(a), Figure 7.21(a) and Figure 7.22(a) and their corresponding controller outputs are shown in the Figure 7.20(b), Figure 7.21(b) and Figure 7.22(b). The mean and the standard deviation of the measured variable for noise signals with various values of the standard deviations are reported in Table The following observations can be made from this part of the simulation study. From Figure 7.20(a) it seems that the FLC performs satisfactorily if the noise level is small in magnitude. From Figure 7.21(a) it is proved that the PFLC outperforms irrespective of magnitude of noise level. From Figure 7.22(a) it is proved that the PFLC produces robust control action than FLC in the presence of measurement noise. The standard deviation of the controlled variable has been found to be less in the case of PFLC than FLC from Table 7.10.

29 169 Figure 7.20 (a) Performance of FLC in the presence of measurement noise of various levels - Process output Figure 7.20(b) Performance of FLC in the presence of measurement noise of various levels - Controller outputs Figure 7.21(a) Performance of PFLC in the presence of measurement noise of various levels - Process output

30 170 Figure 7.21(b) Performance of PFLC in the presence of measurement noise of various levels - Controller outputs Figure 7.22(a) Performance of FLC and PFLC in the presence of measurement noise of standard deviation of 10 cm - Process output Figure 7.22(b) Performance of FLC and PFLC in the presence of measurement noise of standard deviation of 10 cm - Controllers output

31 Table Mean and standard deviation of the true value of the PV for noise with various standard deviations Standard deviation of noise signal (cm) Time interval Mean FLC Standard deviation of process variable Mean PFLC Standard deviation of process variable 0 t t t t DISCUSSIONS In this chapter, proposed probabilistic fuzzy inference system has been applied to handle complexity, nonlinearity and random uncertainties associated in the modeling and control problem with the spherical tank level process. From the widespread simulation studies, it is concluded that the probabilistic fuzzy model of the spherical tank level process brings better modeling accuracy than the conventional fuzzy model even in the presence of measurement noises of appreciable magnitude. Similarly, the proposed probabilistic fuzzy logic controller has good setpoint tracking capabilities, disturbance rejection capabilities and robustness properties in both simulation studies and real time implementation. The performance of the PFLC has been compared with conventional PID and FLC, whose performance is proven to be better than the others in both the cases. It has also been proved that the PFLC captures the stochastic uncertainties effectively and outperforms particularly in the presence of measurement noises than other methods proposed in the literature.