EE472 Control Principles Semester 2 Project Assignment Student’s Name: Zhiwang Feng Registration Number: 201522397 Date: 24/03/2017 12 Table of Contents 1.INTRODUCTION ................................................................................................................................... - 1 - 2. SYSTEM MODEL CONSTRUCTION IN SIMULINK ........................................................................ - 1 - 2.1 IMPLEMENTATION OF VALVE 1AND VALVE 2 AND THEIR INPUT-OUTPUT CHARACTERISTICS ...................... - 1 - 2.2 IMPLEMENTATION OF TWO TANKS ............................................................................................................. - 4 - 2.3 IMPLEMENTATION OF PI CONTROLLERS AND VALVE LIFT CONSTRAINTS.................................................... - 5 - 2.4 CONSTRUCTION OF WHOLE CLOSED-LOOP SYSTEM. ................................................................................... - 6 - 3.TUNE THE TWO PI CLOSED-LOOP CONTROL SYSTEMS. ........................................................... - 7 - 3.1 TUNE THE PI CONTROLLERS TO GIVE SATISFACTORY STEP RESPONSES. ..................................................... - 7 - 3.2 VALVE LIFT SATURATING ANALYSIS ........................................................................................................ - 10 - 3.3 THE EFFECT OF NOISE ON SYSTEM PERFORMANCE ................................................................................... - 11 - 4. OBJECTIVE FUNCTION MINIMIZATION AND SYSTEM PERFORMANCE IMPROVEMENT. ... - 12 - 4.1 OBJECTIVE FUNCTION IMPLEMENTATION. ................................................................................................ - 12 - 4.2 TUNING PI CONTROLLER TO GET MINIMUM J FOR R=0.1 AND R=1.5. ....................................................... - 13 - 4.3 EFFECT OF WEIGHTING FACTOR R. ........................................................................................................... - 15 - 5. CONCLUSION ..................................................................................................................................... - 16 - APPENDIX ............................................................................................................................................... - 16 - EE472 S2 Project Report –Zhiwang Feng - 1 - 1.Introduction Slug catcher is an important part of oil platform to work as a buffer stage to separate the crude oil from water and gas, which guarantee that the cleaned and pure oil flows into other downstream installation. In this project, a Simulink model is implemented to simulate the slug catcher. As shown in figure 1, the system was made up of two tanks (Slug catcher vessel and Free-water-knock-out vessel). To achieve the slug catcher functions, the tank level has to stay in fixed range and the tank level is also designed to supply continuous and stable outflow to the downstream part. Figure 1. The schematic diagram of modelling system For the desired control process, PI controllers are utilized to act on the valve to form closed control loop to determine the valve lift. The valve lift controls the outflow rate of each tank, which is the key factor to determine the tank level. The requirement of outflow limit and continuous outflow can be maintained by tuning PI controllers. The modelling and control process can be achieved by tuning PI controller to obtain desired system performance in Simulink. 2. System model construction in Simulink 2.1 Implementation of valve 1and valve 2 and their input-output characteristics The implementation of valve 1: Valve 1 is a proportional value with the expression： 12 = 1((1 − 1)1 + 1)√ ℎ1 ℎ1 − 1 ℎ 1 = 0.02, 1 = 1, ℎ1 = 0.4-------(1) EE472 S2 Project Report –Zhiwang Feng - 2 - To plot the input-output characteristics(q vs L) for minimum, maximum and half of tank level h, the following models can be built in Simulink: Figure 2. Simulink model for valve 1. Setting the valve lift L1 with a Ramp signal with its slope set in 1, the characteristic of valve 1 output 12 can be obtained: Figure 3. The input-output characteristic of valve 1 The valve expression (1) can be expressed: 12 = 1 ∗ ((1 − 0.02)1 + 0.02)√ ℎ1 0.04 − 1, the input signal L is a ramp signal with slope in 1. For ℎ1 = ℎ1, 12 = 0, which corresponds to the purple line in figure 3. For ℎ1 = ℎ1 or ℎ1 = ℎ1, the expression of valve 1 can be simplified : 12 = 1 + , EE472 S2 Project Report –Zhiwang Feng - 3 - The input signal L is a ramp signal with slope in 1, the output outflow rate 12 is linear with the input L as indicted in figure 2 and it is evident that the valve 1 is a proportional valve from the blue and orange line in figure 3. The implementation of valve 2: Valve 2 is an equal percentage valve with the expression: 2 = 2 2−1√ ℎ2 ℎ2 − 1 ℎ 2 = 0.7, ℎ2 = 0.2, = 25 ------------(2) Setting ℎ2 in minimum, maximum and half of tank level and the input L with a ramp signal with slope 1, the modelling of valve 2 can be built in Simulink: Figure 4. Simulink model for valve 2. The simulation results can be obtained from scope: Figure 5. The input-output characteristic of valve 2. EE472 S2 Project Report –Zhiwang Feng - 4 - For ℎ2 = ℎ2, 2 = 0, which is corresponding to the purple line. For ℎ2 = ℎ2ℎ ℎ2 , the expression (2) can be simplified in 2 = ×25 2−1，A is constant. It is evident from blue and yellow lines that the output 2 is increasing in exponential form with the variation of time for the input Ramp signal varies linearly with time, which prove that the valve 2 is an equal percentage valve. 2.2 Implementation of two tanks For tank 1: 1 ℎ1 = 1 − 12 ------------------------(3) The expression (3) can be simplified into ℎ1 = 1 1 ∫(1 − 12) ℎ 1 = 1.324. -------------(4) The tank 1 model can be built: Figure6. Model for subsystem tank 1. The input 1 is in expression: 1() = (0.5 + 0.1 sin ( 2 50 ) + ()) Where () is a band-limited white noise with noise power 0.02 and sampling time 10 seconds. 1 can be constructed as flowing: Figure 7. Model for subsystem 1 For tank 2: 2 ℎ1 = 12 − 2 -------------------------(5) The expression (5) can be simplified into EE472 S2 Project Report –Zhiwang Feng - 5 - ℎ2 = 1 2 ∫(12 − 2) ℎ 2 = 2.02. -----------(6) The tank 2 model can be constructed: Figure 8. Model of subsystem tank 2. 2.3 Implementation of PI controllers and valve lift constraints. PI controllers with expression: () = ( + 1 ) (ℎ() − ℎ()), = 1,2 ---------------(7) According to expression (7), the subsystems for PI controllers can be built: Figure 9. Model of subsystem PI controller 1. Figure 10. Model of subsystem PI controller 2. The reference tank level for two tanks are listed in following table: Tank 1 Tank 2 Time(s) 0-200 200-800 800-1000 0-400 400-600 600-1000 ℎ() 0.8 0.96 0.8 0.5 0.45 0.5 Table 1. Reference height for two tanks. Implement the reference tank level with Stair generator with following settings: EE472 S2 Project Report –Zhiwang Feng - 6 - Figure 11. Reference tank level generation The output of PI controller is the valve lift , which is subject to the following opening and varying speed constraints. 0 ≤ ≤ 1 − 0.1 ≤ ≤ 0.1 = 1,2 The valve constraints can be implemented by Rate limiter and saturation with settings: Figure 12. Valve lift constraints setting Figure 13. Model of subsystem valve constraints 1 and 2 2.4 Construction of whole closed-loop system. For tank 1, the tank level ℎ1is determined by the inflow rate 1and the outflow rate 12 which is the output of valve 1 and is directly controlled by the tank level ℎ1 and the valve lift 1 that is controlled by tank level ℎ1via a PI controller and constraint. The ℎ1 − outflow rate 12 from valve 1 can work on the tank 1to determine tank level ℎ1, forming the level control closed loop for the whole tank 1 system. So the closed-loop system for tank 1 can be built by connecting all the sub-systems constructed in section 2.1 to 2.3 and is shown in figure 14. For tank 2, the tank level ℎ2 is determined by the outflow rate 12 from valve 1 and the outflow rate 2 of tank 2. The outflow rate 2 is controlled directly by the tank level ℎ2 and the valve lift 2 which is controlled by the tank level ℎ2 via a PI controller. The ℎ2 − outflow rate 2 from valve 2 can work on the tank 2 to determine tank level ℎ2 , forming a level control closed loop for the whole tank 2 system. So the closed-loop system for tank 2 can be constructed by connecting all the sub- systems constructed in section 2.1 to 2.3 and is shown in the following figure: EE472 S2 Project Report –Zhiwang Feng - 7 - Figure 14. The model for whole closed-loop system 3.Tune the two PI closed-loop control systems. 3.1 Tune the PI controllers to give satisfactory step responses. In section 2.3, the reference signal for tank levels have been implemented by setting the stair generator signal. The parameters of PI controllers are written in a Matlab script to tune the PI controllers. For PI controller 1, with expression: 1() = (1 + 1 1 ) (ℎ1() − ℎ1()). The proportional gain results in the change in the output for a given error. Small value of proportional gain will make the system less sensitive to system disturbances. With the increment of proportional gain P(1), the system can response faster but the steady-state error decrease significantly and the overshoot will increase, which have a huge influence on reference tracking performance. So proportional gain is supposed to be controlled in a proper value. And with the increasing of integral gain I, the system steady-state error can be eliminated, however the overshoot will increase and the system become more instable with more oscillations, which shapes both the reference tracking performance and disturbance rejection performance. At the beginning, set 1 = 1, 1 1 = 1, and initial setting in 0.8. EE472 S2 Project Report –Zhiwang Feng - 8 - Figure 15. The step response of tank 1. It is evident that the step response represents large overshoot and the system has significant oscillations. The disturbance peak values are relatively large and the reference tracking performance as well as the disturbance rejection performance are not as participated. To recede the oscillations and overshoot to make the step response more stable , tune the PI controller by increasing the proportional gain 1and decreasing integral gain 1 1 to degrade the disturbance peak value and maintain stable oscillation. When 1 increases to 4 and 1 1 decreses to 0.8, the tank 1 system response is shown in figure16, the disturbance peak values have been receded significatly to desired level and the system’s response has relatively small overshoot, the reference tracking performance and disturbance rejection performance are much better as shown in this satisfactory response. Figure 16. The step response of tank 1 after PI controller tuning. EE472 S2 Project Report –Zhiwang Feng - 9 - For PI controller 2, with expression: 2() = (2 + 1 2 ) (ℎ2() − ℎ2()). Use the same way to tune PI controller 2, setting 2 = 1 and 1 2 = 1, the system response: Figure 17. The step response of tank 1. It is evident that this response has relatively disturbance peak value and the system oscillation is relatively stable. To get more satisfactory response with small disturbance peak value, tune the PI controller by increasing the proportional gain 2and decreasing integral gain 1 2 . When 2 = 4, 1 2 = 0.8, the system response represent much smaller disturbance peak values which are acceptable, this system operate with satisfactory reference tracking performance and disturbance rejection performance. Figure 18. The step response of tank 1 after PI controller tuning EE472 S2 Project Report –Zhiwang Feng - 10 - 3.2 Valve lift saturating analysis Setting 1 = 4, 1 1 =0.8, 2 = 4, 1 2 =0.8, the unconstrained and constrained control signal of valve lift can be plotted: Figure 19. The unconstrained and constrained control signal of valve 1. Figure 20. The unconstrained and constrained control signal of valve 2. It is clear form figure 19 and figure 20, the evident valve lift saturations happen at the time when the reference tank level changes. Due to the changes of reference tank level, an obvious difference between the measured tank level and reference tank level will represent, generating a disturbance in the operating system. The valve lift will be tripped to degrade the disturbance to maintain the stability. The lift has limited opening magnitude constraint and varying speed constrain, the lift saturation will occur when the disturbance is beyond the limitation. Apart from the point when reference tank level EE472 S2 Project Report –Zhiwang Feng - 11 - changes, there still exist some slight lift saturations for the existence of disturbance signals resulted from the noise, which are also beyond the lift constraints. 3.3 The effect of noise on system performance The input 1 is expressed in 1() = (0.5 + 0.1 sin ( 2 50 ) + ()), to investigate the effect of noise on system performance, firstly remove the noise signal from inflow 1 : Figure 21. The step response of tank 1 and tank 2. Figure 22. The unconstrained and constrained control signal of valve 1 & 2. Then increase the power of noise to 0.25: Figure 23. The step response of tank 1 and tank 2. EE472 S2 Project Report –Zhiwang Feng - 12 - Figure 24. The unconstrained and constrained control signal of valve 1 & 2. By comparing the system performance and valve saturation levels under the condition that no noise inflows and stronger noise signal inflows, it is evident that when system with noise inflows, the disturbance peak value is much large than that no noise inflows, and the system has much more valve lift saturations. With the increment of power of noise, the disturbance peak value become much larger and more valve lift saturations occur, the control system stability decrease significantly. The noise in the inflow weakens both the reference tracking performance and disturbance rejection performance. 4. Objective function minimization and system performance improvement. 4.1 Objective function implementation. The objective function is shown below: J = √∫ [(ℎ1() − ℎ1()) 2 + (ℎ2() − ℎ2()) 2 ] 0 + ∫ [| 1() | + | 2() |] 0 ℎ = 1000 According to the expression of objective function, the Simulink sub-system model can de designed: Figure 25. Model of the sub-system of objective function J. EE472 S2 Project Report –Zhiwang Feng - 13 - Connect the sub-system to the closed loop system constructed in section 2. And use display to show the value of J. Figure 26. The model for whole closed-loop system 4.2 Tuning PI controller to get minimum J for r=0.1 and r=1.5. Tune the parameters of PI controller to minimize J by following steps: 1) Fix three parameters and tune one parameter to record the value of J shown in Display. 2) When J commence to increase, narrow the range of tuning parameter and tune again until get local minimization of J. 3) Fix the tuned parameter and repeat the first step again to tune other parameters. For r=0.1, the tuning process is shown below: Setting 1 = 1, 1 1 = 1, 2 = 1, 1 2 = 1 Tuning 1 with 1 1 = 1, 2 = 1, 1 2 = 1 1: 0.5 1.0 1.5 2.0 2.5 2.4 2.3 2.2 2.1 J 2.8643 2.6880 2.6576 2.6342 2.6645 2.6513 2.6334 2.6354 2.6336 Table 2. The value of J when tuning the value of 1 The local minimization of J is 2.6334 when 1 = 2.3. Tuning 1 1 with 1 = 2.3, 2 = 1, 1 2 = 1 EE472 S2 Project Report –Zhiwang Feng - 14 - 1 1 0.5 0.7 0.8 1.0 1.2 0.6 0.73 0.75 J 2.6448 2.6234 2.6229 2.6334 2.6941 2.6308 2.6201 2.6178 Table 3. The value of J when tuning the value of 1 1 The local minimization of J is 2.6178 when 1 1 = 0.75. Tuning 2 with 1 = 2.3, 1 1 = 0.75, , 1 2 = 1 2 0.5 1.0 1.5 2.0 2.5 3.0 2.8 2.2 J 2.6243 2.6178 2.5972 2.5956 2.5915 2.6123 2.6007 2.5926 Table 4. The value of J when tuning the value of 2 The local minimization of J is 2.5915 when 2 = 2.5. Tuning 1 2 with 1 = 2.3, 1 1 = 0.75 2 = 2.5 1 2 1.5 1.2 1.0 0.9 0.8 0.7 0.85 0.95 J 2.6087 2.6089 2.5915 2.5895 2.5913 2.5976 2.6021 2.5888 Table 5. The value of J when tuning the value of 1 2 The local minimization of J is 2.5888 when 1 2 = 0.95. For the first tuning, the local minimization of J is 2.5888, when 1 = 2.3, 1 1 = 0.75, 2 = 2.5, 1 2 = 0.95. 1 1 1 2 1 2 J 1 1 1 2 1 2 J 2.3 0.75 2.5 0.95 2.5888 1.8 0.6 2.5 0.95 2.5898 2.5 0.75 2.5 0.95 2.6000 1.8 0.75 2.7 0.95 2.5718 2.1 0.75 2.5 0.95 2.5828 1.8 0.75 2.3 0.95 2.5681 1.9 0.75 2.5 0.95 2.5717 1.8 0.75 2.1 0.95 2.5743 1.7 0.75 2.5 0.95 2.5772 1.8 0.75 2.2 0.95 2.5703 1.8 0.75 2.5 0.95 2.5696 1.8 0.75 2.3 1.0 2.5690 1.8 0.8 2.5 0.95 2.5748 1.8 0.75 2.3 0.85 2.5702 1.8 0.9 2.5 0.95 2.5806 1.8 0.75 2.3 0.75 2.5746 1.8 0.7 2.5 0.95 2.5722 1.8 0.75 2.3 0.95 2.5681 Table 6. The value of J when tuning the value of 1, 1 1 , 2, 1 2 As shown in table 6. Tuning the parameter of PI controller again by the above steps. After further tuning, the minimization of J is 2.5681, when 1 = 1.8, 1 1 = 0.75, 2 = 2.3, 1 2 = 0.95. EE472 S2 Project Report –Zhiwang Feng - 15 - Setting r=1.5, repeat above steps to tune PI controller parameters to find minimization of J. 1 1 1 2 1 2 J 1 1 1 2 1 2 J 0.6 1 1 1 32.99 1.2 0.1 0.8 1 26.1469 0.8 1 1 1 32.1365 1.2 0.1 1.2 1 26.0824 1.0 1 1 1 31.7723 1.2 0.1 1.4 1 26.1179 1.2 1 1 1 31.5147 1.2 0.1 1.3 1 26.0368 1.4 1 1 1 31.6345 1.2 0.1 1.35 1 26.0175 1.3 1 1 1 31.6307 1.2 0.1 1.38 1 26.0235 1.1 1 1 1 31.607 1.2 0.1 1.35 1.2 26.2246 1.2 1.2 1 1 32.7628 1.2 0.1 1.35 0.8 25.7876 1.2 1.4 1 1 34.2355 1.2 0.1 1.35 0.6 25.5063 1.2 0.8 1 1 30.6026 1.2 0.1 1.35 0.3 24.6438 1.2 0.6 1 1 29.8118 1.2 0.1 1.35 0.1 23.4330 1.2 0.4 1 1 28.8735 1.3 0.1 1.35 0.1 23.6704 1.2 0.2 1 1 27.3908 1.1 0.1 1.35 0.1 24.1023 1.2 0.1 1 1 26.1368 1.2 0.1 1.4 0.1 23.5022 1.2 0.1 0.6 1 26.3584 1.2 0.1 1.3 0.1 23.6082 Table 7. The value of J when tuning the value of 1, 1 1 , 2, 1 2 = 23.4330 when 1 = 1.2, 1 1 = 0.1, 2 = 1.35, 1 2 = 0.1 . 4.3 Effect of weighting factor r. Figure 27．The system response for weighting factor r=0.1 and r=1.5. EE472 S2 Project Report –Zhiwang Feng - 16 - Figure 28. The constrained and unconstrained valve lift for weighting factor r=0.1 and r=1.5. It is evident from figure 27 and 28 that with the increasing of weighting factor, the system response slower and has larger disturbance peak value, the disturbance rejection performance and the reference tracking performance are both regraded. The valve lift saturation level decreases with the increment of r. 5. Conclusion In this project, a closed loop control model with two tanks, two valves and two PI controllers was implemented in Simulink to simulate the control process of slug catcher in section 2. By tuning the PI controller parameter, the tank level can be constrained into certain range and guarantee continuous and constant outflow to downstream sections in this module. In section 3, the reference tracking performance and disturbance rejection performance were tuned to satisfactory level by tuning the PI controller parameters. The constrained and unconstrained valve lift was plotted to insect the saturations of valve lift. At the end of this part, by adjusting the noise inflowing the system, the effects of noise were represented in the graphs and analysed. In section 4, using the trail-and-error method to tune the parameters of two PI controllers to minimize the objective function for different weighting factor (r). by comparing the two designs in section 4.2, the effects of weighting factor on control performance were analysed in section 4.3. Appendix
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