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Table of Contents Table of Contents 1 Abstract 3 Introduction 4 Background of Control Systems 4 Controllers 5 P Controller 5 P+I Controller 6 Feedback Mechanism 7 Positive Feedback 7 Negative Feedback 7 The Experiment 8 Setup 8 Equipment 9 Objectives 9 Methodology 9 Initial Stage 9 Modelling of the Tank 9 Results and Analysis 10 Step Test Experiment 10 Analysis 12 Design of Proportional (P) only Controller 13 Design of P+I Controller 13 Discussion 15 Conclusion 16 References 16 Table of Figures Figure 1: Open Loop System 4 Figure 2: Closed-loop System 5 Figure 3: Block Diagram of P controller 5 Figure 4: Block Diagram of a P+I Controller 6 Figure 5: Experimental Set-up 8 Figure 6: Test Data 11 Figure 7: Test Results for P-only Controller 13 Figure 8: Root Locus Plot 14 Control of a Non-Linear Process by Linear Methods Abstract The purpose of the report is to develop a control system and a method of approach to the developed systems for the purpose of controlling the water level of a V-shaped Perspex water tank. The top side of the height of the tank has parallel sides and V-shaped bottom part. The tank has a constant width throughout its section and is used as a receptacle for water (fig. 5). Evidently, the water tank is non-linear and therefore an experiment was conducted in the lab to investigate systematic control of the water level in the tank. This necessitated the use of Proportional (P) only controllers and Proportional and Integrated (P+I) controllers so as to come up with a model that mathematically represents the non-linearity of the water tank. LabView™ software was used for the purpose of the simulations as it contains the transfer functions and enables the user to edit the parameters to suit their needs. The results collected are analyzed and discussed in the report. A conclusion based on the discussed results is also presented at the end of the report. Keywords: control system, controllers, discharge coefficient, sensors, LabVIEW. Introduction Background of Control Systems Control systems are an integral part of the 21st century society. Various systems around us require a control mechanism. Dorf and Bishop (2001) defined a control system as a branch of engineering that entails the design mechanisms and algorithms that allow for maintaining of process outputs within a desired range. It consists of subsystems and processes grouped together for the purpose of obtaining the desired output with desired performance given specific input (Nise, 2007). There are two major configurations of control systems (1) open loop system and (2) closed loop system. The differentiating feature of the open loop system is the absence of the ability to compensate for possible disturbances i.e. the system cannot correct any disturbances that may occur but are rather simply commanded by the input. Figure 1: Open Loop System A closed loop on the other hand, employs a feedback mechanism to regulate and control disturbances. A sensors is used to measure the output that is then compared to the desired valued. The gap represents the errors that results from possible disturbances. The closed loop system then requires that the input be altered such that the error is reduced to a level that the system becomes optimally efficient. The figure 2 below, illustrates this assertion. Figure 2: Closed-loop System Controllers P Controller P controller is used for the system when a large gain is needed, it is useful when a system needs to improve the steady state error and to make the system stable, when the system is stable the P controller is not need as the system which has large gain is already stabled. P controller has some limitations such as the constant steady state error should be constant and the system must be one energy storage that are 1st order systems. P controller is also suitable when the sensor measures a small amount of error and example of the system which can be controlled by P controller water tank where liquid controlling is need to approximate the level of water surface which is then needed for the plant operations. Figure 3: Block Diagram of P controller P controllers can be merged with Integral (I) controller and with the Derivative (D) control which makes it P+I or P+D controller and the combination of three is known as PID controller. P controller is sometimes used but the most used controller in P+I controller. The System which is to be controlled will produce constant gain that is known by Ks and a proportional controller gain is represented by Kp. P+I Controller It is the combination of P controller with the Integral (I) controller and it is the one of the most used controllers. The P+I controller have high zones which are served by Kp and the low zones by K, when the high zone is complete it can be tuned. It is similar to P controller as the purpose is to control the system, the integral gain in the P+I controller is limited because of the overshoot. P+I controller is used in order to eliminate the error as the P only controller is unable to remove the steady state error because of the algorithm which cannot increase the control signal and the P only controller stays constant. Whereas the P+I controller is able to increase the control signal and the proportional part will be added with the integral part in order to stabilize the system. Figure 4: Block Diagram of a P+I Controller The P+I controller will tend to increase the control signal and the controlled variable and error will be act as zero. P+I controller is not able to increase the speed of the response and this problem can be solved by adding the derivative mode but in this experiment only P controller and P+I controlled is illustrated. P+I controller works when there is no internal gain and the Kp is increased to get the satisfactory value of response and when the desired value of Kp is measured then the integral can be added to remove the steady state error. Feedback Mechanism Positive Feedback Refers to the sort of incremental feedback that add to the reference input and the output feedback. Transfer function can be illustrated as; Whereby; - total Gain - gain on the open loop - gain of feedback path Negative Feedback The distinguishing feature from the positive feedback is the fact that negative feedback reduces the error between the reference inputs and the system output. It still has a similar transfer function (eqn. 1 above) as the positive feedback. Equation 1 above indicates that the total system gain can be represented as and by first principles, the overall gain can be derived subject to values. The sensitivity of the system to adjustments is also dependent on the values of the derivative and can be derived as follows; Partial differentiation on both sides of equation 3 w.r.t G; Simplifying; Therefore; Negative feedback results when the values of are less than 1 whereas the reverse is true for positive feedback. A system whose output is well controlled is considered stable and is most desirable as it guarantees an expected output. Equation 1 also indicates that an output of indicates an unstable system. These important aspects were put into consideration when carrying out the test to ensure that the most appropriate feedback was applied. The Experiment Setup The experiment was set up as shown in figure 5 below; Figure 5: Experimental Set-up Equipment The set-up above was composed several task specific equipment that included; V-shaped water tank, Centrifugal Pump, Rectangular Tank, DC motor, Amplifier, Voltage Signal control, Pressure transducer, Analog to Digital Convertor, Personal Computer and LabVIEW (Software) Objectives The experiment looked to investigate the control of a non-linear process by linear methods. The following objectives provides a roadmap to achieve the aforementioned; i. To investigate the control of water level in a lab scale non-linear water tank which is fed by a centrifugal pump and discharges to a sump tank through a valve, ii. To understand the dynamics of the water tank by modelling it from first principles and by applying step tests to identify the system model at various operating points, iii. To design a proportional + integral controller for a specific operating point, implement it in software (as an analogue s-domain system, i.e. very fast sampling). iv. To allow scope for further investigations. Methodology Initial Stage The Sump Tank was first filled with the water and the DC motor powered Pump then pumped the water into the V shaped tank at the top. Ordinarily, the water started filling from the bottom of the V-shaped Tank. The motor was powered by an amplifier with a proportional voltage signal to the head (h) in the tank was measure by pressure transducer. The signals produced by the system was supplied in the PC through the signal convertor and the software LabVIEW was the medium to support all the experiment from which the results were obtained. Modelling of the Tank The most important aspect in modelling the tank’s structure is by considering the top section of the tank with both sides and then whole tank can be considered. It was assumed that the tank had a rectangular cross-section with known values of inflow and outflow. The principle of mass conservation was considered in that the inflow was taken to be equal to the outflow. It therefore means that; But as per the principle of mass conservation; Therefore; The V shape of the tank is considered by altering the cross-sectional area based on the height of the tank. Due to the V shape, a change is h however small, will result in a change in the cross-sectional area, A. Results and Analysis Step Test Experiment The data obtained from the test experiment was represented in the graph as shown in fig. 6. It is evident that the data gets noisy at the beginning of the experiment but gets better as time flow by. The noise can be attributed to the v shape of the tank that resulted in sharp changes in cross-sectional area with change in height. Other causes of the noise could be the set up not being warmed up enough before the experiment began. Figure 6: Test Data As the experiment progressed, the input voltage was noted using a ruler and when the entire measurements are then recorded once the output voltage values became steady. It is from the results in figure 6 above that the time constant data was derived for each step and recorded in table 1 below. This helped in designing a P only controller with closed loop dynamics that represent a third of an open loop process. Table 1: Steady State and time constant values h (cm) 2.6 23.35 12.5 4.1 0.09 3.5 8.82 14 4.67 0.23 5.0 8.07 48 16.00 0.25 5.1 2.77 47 15.67 0.72 6.6 5.04 95 31.67 0.40 8.9 3.74 49 16.33 0.53 10.6 3.32 13 4.33 0.60 12.5 2.87 43 14.33 0.70 16.3 4.49 26 8.67 0.45 21.1 8.55 39 13.00 0.23 24.0 8.95 50 16.67 0.22 26.0 5.30 47 15.67 0.38 Analysis The equation below was used to derive the time constant and the steady state gains of the output voltage; The value represents the steady state gain; Table 1 is used to calculate the value in equation 11 and used in tandem with figure 6 above to develop the time constant, Assumptions are made regarding the value of the rise needed. To perform the P Test, a value of input, is keyed into LabVIEW. The values of steady state gain and time constants already calculated are used together with the new information to determine the value of. The following steps are take; Design of Proportional (P) only Controller The value of steady state gain, calculated from table 1 above was maintained in the design of the P controller. However, when the value was used in design, the P-only controller developed was ineffective since the errors obtained were outside the effective range. The figure below illustrates this assertion. Figure 7: Test Results for P-only Controller Upon close examination, it was determined that the P only controller could not meet the set height demand of 26cm. It was a huge difference from the original value and therefore rendering the P only controller ineffective. By measuring the input and output values, the error value could be determined with ease. It is evident that a more effective controller (e.g. P+I Controller) with minimal errors is required to analyze the non-linearity of the tank. Design of P+I Controller As with any design process, the design of P+I controllers required one to obtain certain variables. For instance, a Root Locus plot would assist in developing feedback parameters used in control as long as specific set criteria is in place. It was necessary to derive the values of steady state input and rise time. The illustration in the figure below could was used to obtain these values; Figure 8: Root Locus Plot The value of that met the system performance requirement was chosen from table 1. It meant that during the root locus plot, it produced the correct s-plane poles. The sensitivity of the system given the chosen k value was also put into consideration and the values; and were chosen. Applying the second order transfer function; Therefore frequency, was determined as follows; The rads/sec can be calculated as follows; Therefore; Replacing the values calculated, the value; But Equation 13 can be used to derive Discussion The results showed different values for the two controllers with the P controllers having the highest error and instability. In theory, Proportional only with a first order process was expected to give a poor result. The results are dependent on the variables loaded on the system, and the P-only controller would be very effective with rapid decrease in error. The experiment showed that P controller has many limitations making it ineffective for the water tank experiment. On the other hand, the outcome of the P+I controller design proved that it was effective and could be used to control the system. The calculations corresponded to the actual test observations and the error was low. A P+I controller was designed perfectly and had the ability to reduce constant steady state error. Further tests on the P+I controller was carried out by dumping extra tests into the tank during testing and then the system reaction was observed. The controller was able to compensate and react to this large disturbance by adjusting the height to the correct position. The extra water adding is in close relation to a situation where aircraft are sometimes faced with a gust of wind or turbulence, it shows how the controls linked to the autopilot would react and adjust rapidly. Conclusion The experiment of the non-linear V-shaped tank was a success. The objectives were met but the step tests gave the error in output and the system was unstable, after calculating the time constants and other parameters to derive the Proportional controller it was seen that the system start behaving in the normal manner. Steady state gain was calculated which concluded that the error in the system still arises after the P controller. For future improvement on controlling the water, adding a Derivative would provide even more stability and reliability. Other form of control could be tested such as a digital controller. In digital control systems, the controller operation is implemented digitally. References Control System | Closed Loop Open Loop Control System | Electrical4u. Electrical4u.com. N.p., 2017. Web. 2 May 2017. Dorf, Richard C, and Robert H Bishop. Modern Control Systems. 1st ed. Upper Saddle River (New Jersey): Prentice-Hall, 2001. Print Dr Grant Covic. (1998). Proportional & Integral Controllers. Systems and Control . 1 (2). Nise, N.S., 2007. CONTROL SYSTEMS ENGINEERING, (With CD). John Wiley & Sons. Tarba, Larisa, and Pavel Mach. "Control Tools For Process Improvement In Electrical Engineering". Proceedings of the 2011 34th International Spring Seminar on Electronics Technology (ISSE) (2011): n. pag. Web. 5 May 2017 Read More