"Experiment: Design of a Feedback Control system\r\n\r\nInstructions: \r\n- This is a design experiment. \r\n- Keep a record of all measurements taken, designs and screenshots.\r\n\r\nLearning outcomes \r\n- Be able to build a control system using MATLAB Simulink.\r\n- Be able to investigate the control of a process using a proportional (P) and proportional- integral (PI) controllers.\r\n\r\n917_Figure.jpg\r\n\r\nObjectives:\r\n\r\nIt is required to design and build a P and PI controllers to control a process.\r\n\r\nExperimental work:\r\n\r\nThe following transfer function represents a model of a certain process (plant):\r\n\r\nG(s) = K/(Ts + 1)2\r\n\r\nChoose values for the plant parameters using your data of birth such as:\r\n\r\nK= day of birth (17)\r\nT= month of birth (01)\r\n\r\nPart 1: First-order Plus Time Delay Approximation\r\n\r\nUsing MATLAB Simulink, simulate the process with a unit-step change input (open-loop). Approximate this process with a first-order plus time delay model (see appendix A) and simulate with a unit-step input as well compare both responses and find the steady-state error, %OS (percentage overshoot). Setting time and the rise time for both cases.\r\n\r\nPart 2: the P Controller Design \r\nDesign a P controller for the process and simulate the closed-loop response to a step input. Investigate the effect of changing (increase and decrease) the controller gain on the controlled response in term of the steady-state error, %OS. Setting time and rise time.\r\n\r\nPart 3: the PI controller Design\r\nDesign a PI controller for the process and simulate the closed-loop response to a step input. Compare the response with the previously designed P controller in terms of the steady-state error, %OS, setting time and rising time. Investigate the effect of changing the integral action time on the controlled response.\r\n\r\nPart 4: the effect of disturbance \r\nInclude a step disturbance at the input of the process (figure.1). obtain the unit-step responses for using the P and PI controllers. Measure and compare the maximum output responses, setting time and steady-state error in both cases. Which controller can remove the effect of the disturbance in steady-state? Why?\r\n\r\n973_Figure1.jpg\r\n\r\nFigure .1\r\n\r\nPart 5: Extra work\r\n\r\nDesign a PID controller to control the process (using Ziegler-Nichols method or otherwise). Find the parameters of the controller and test your system against disturbance at the input of the process.\r\n\r\nReport marking:\r\n\r\nInclude a separate section for each of the main parts above. Each section should include method used and explanation and comments, appropriate diagrams (SIMULINK diagrams and simulation results) and a discussion of the results.\r\nDesign of a Feedback Control SystemThe original system is of the formThe parameters choosen are K = 17 and T = 1.Part 1The first-order plus delay model is defined aswhere K is the gain, Ta is the time constant and t is the delay. The delay has been chosen to t =1. Ina order to use this model to approximate the original system, the parameters need to be estimated, whichcan be done from the relationsy = Kss a -1 y(T )=(1-e )Ka a The parameter T can therefore be found from the step response plot of the original system. This givesa K = K and T = 2.3. The delay factor can be approximated by the first-order Padé approximationa a The resulting transfer function isand the step responses are shown in Figure 1. Figure 1. Step responses for original system and approximation.For the model, we have the following characteristicsCharacteristic ApproximationOvershoot 0.0Steady-state error 0.0Settling time - system 4.9Rise time - system 3.3Settling time - approx. 11.0Rise time - approx. 7.4The approximation is fairly good in approximating the open-loop system. However, the settling timeand rise time is much longer. Part 2To design a P-controller with gain K for the system, we can derive the transfer functionp The block diagram of the closed-loop response to a step input is shown in Figure 2.Figure 2. Closed-loop response of a P-controller.The step response for different choices ofare shown in Figure 3. The controller becomes increasingly unstable as K increases. pFigure 3. Step response for a P-controller with different gains. The characteristics of the P-controller are shown in Figure 4. Figure 4. Characteristics for the P-controller.For the controller, the steady-state error decreases with gain K , but it remains non-zero for finite gains.p The overshoot depends on the gain too. The settling times and rise times remain fairly constant.Part 3To design a PI-controller with gains K and K for the system, we can derive the transfer function,p iThe block diagram of the closed-loop response to a step input is shown in Figure 5.Figure 5. Closed-loop response of a PI-controller.The step response for different choices ofwith K = 0.3 are shown in Figure 6, andp its characteristics in Figure 7. Figure 6. Step response for a PI-controller with different gains.Figure 7. Characteristics for the PI-controller.For the controller, all characteristics except the overshoot are fairly constant with the gain K . Thei overshoot, however, increases linearly with the gain.Part 4The diagrams for the the P- and PI-controlled system subject to a step disturbance are shown in Figure8 and 9. A comparison of the step responses is shown in Figure 10.Figure 8. Diagram for the step response for a P-controller with added step disturbance.Figure 9. Diagram for the step response for a PI-controller with added step disturbance. Figure 10. Comparison of step responses of the P and PI-controllersin a system with a step disturbance.For the two controlled systems, we have the following characteristicsCharacteristic P-controller PI-controllerOvershoot 16.4 0.0Steady-state error 2.8 0.0Settling time 2.8 3.8Rise time 1.6 2.4The PI-controller can remove the disturbance by "looking backwards", integrating the signal up to thecurrent point in time. In the P-controller, the gain needs to be larger than zero to have effect, but at thesame time, the steady-state error is proportional to the gain. The settling time and rise time, however,are longer for the PI-controller. Part 5A PID-controller is designed using the Ziegler-Nichols method. The transfer function is The approximate parameters are K =0.42, K =1.00 and K =0.25. p i d The diagrams for the the PID-controlled system subject to a step disturbance is shown in Figure 11, andits response is shown in Figure 12. Figure 11. Diagram for the step response for a PID-controller with added step disturbance. Figure 12. Step response of the PID-controller in a system with a step disturbance.The characteristics of the PID-controller are shown below.Characteristic PID-controllerOvershoot 1.6Steady-state error 0.0Settling time 5.8Rise time 4.7The controller has nearly zero steady-state error and low overshoot. This, however, comes at theexpense of longer settling time and rise time. "