Problem 1. This problem considers the closed-loop behavior of an integrating plant K p(s) = 5...

 

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Problem 1. This problem considers the closed-loop behavior of an integrating plant K p(s) = 5 (1) using two different types of classical feedback controllers. a) Consider first the use of Proportional-Only control: ci(s) = K (2) Obtain, p, and for this control system. Write these transfer functions in standard (i.e., gain/time constant) form and use this information to generate detailed sketches of the responses to a unit step setpoint change and a unit ramp disturbance change. Sketch each change separately; show both the y and u responses. You may assume in your sketches that Ke is selected such that the responses are internally stable. Please answer the following questions based on the results of your responses: Which case(s) leads to offset? How do these results differ from those shown in lecture for the classical first-order system subject to proportional-only control? b) Consider instead a Proportional-Integral (PI) controller tuned using the model-based tuning rule K = 2 KX T 2A, A>0 (3) Obtain n, p, and for this control system as before; write these in standard form. Sketch the responses for the controlled variable (only) to step setpoint and ramp disturbance changes (note: use the information on pole/zero locations to be accurate regarding the shape of the response in this case). Use the Final Value Theorem to determine if no offset conditions are met by this control system for both step setpoints and ramp disturbances. c) Use the command ltiview (part of the Control System Toolbox in MATLAB) as a means to verify the sketches asked for in items a) and b). Use K, K=A=1 for simplicity. For example, the transfer function = h(s) 28+1 (8+1)2 is defined on the MATLAB command line by the system object (4) Problem 1. This problem considers the closed-loop behavior of an integrating plant K p(s) = 5 (1) using two different types of classical feedback controllers. a) Consider first the use of Proportional-Only control: ci(s) = K (2) Obtain, p, and for this control system. Write these transfer functions in standard (i.e., gain/time constant) form and use this information to generate detailed sketches of the responses to a unit step setpoint change and a unit ramp disturbance change. Sketch each change separately; show both the y and u responses. You may assume in your sketches that Ke is selected such that the responses are internally stable. Please answer the following questions based on the results of your responses: Which case(s) leads to offset? How do these results differ from those shown in lecture for the classical first-order system subject to proportional-only control? b) Consider instead a Proportional-Integral (PI) controller tuned using the model-based tuning rule K = 2 KX T 2A, A>0 (3) Obtain n, p, and for this control system as before; write these in standard form. Sketch the responses for the controlled variable (only) to step setpoint and ramp disturbance changes (note: use the information on pole/zero locations to be accurate regarding the shape of the response in this case). Use the Final Value Theorem to determine if no offset conditions are met by this control system for both step setpoints and ramp disturbances. c) Use the command ltiview (part of the Control System Toolbox in MATLAB) as a means to verify the sketches asked for in items a) and b). Use K, K=A=1 for simplicity. For example, the transfer function = h(s) 28+1 (8+1)2 is defined on the MATLAB command line by the system object (4)