Question

ME 3281- Spring 2024 Homework Assignment 11 (HW11) The purpose of this assignment is for you to gain some experience designing and analyzing controllers for linear system models, to build understanding of feedback controls, and to gain intuition around controller gains and design specifications. Reading Assignment 1. Textbook Chapter 13.4 (Block Diagrams for Controls) 2. Textbook Chapter 15 (Feedback Controls) Problem 1. Consider a plant with the transfer function (a) Find the time constant Tp for the plant. 2 Gp = 4s + 1 A closed-loop feed back controller is placed around the plant with the controller being a proportional gain K (a "P control"). The system input is r(t) and output is y(t). r(t). + e(t) K u(t) y(t) Gp y(t) As a control system designer, you have the ability to choose the control gain K. The control objective is for the output y to track the input r. (b) Show that the closed-loop transfer function (See Section 13.2 in the book) is 2K Gcl = 4s + 1 + 2K (c) From this transfer function, find the time constant Tcl of the closed-loop system. (d) Find the DC gain of the transfer function. (e) Find the value of control gain K which results in the closed-loop transfer function having a time constant that is four times faster than the time constant for the open-loop plant. (f) Find the value of the control gain K which results in the closed-loop transfer function having a steady- state error of less than or equal to 10%. Note that the steady state error for a unit step is (1 – D.C. gain). (g) Use MATLAB to create a single plot with 3 step response traces: one for the open-loop plant and one for each of the closed loop systems from (e) and (f). Hint: Use the command step(sys1, sys2, sys3) to get 3 step responses on one plot. If you want the plot to have a different final time, use the command step(sys1,sys2, sys3,tfinal). Problem 2. Given the same plant as Problem 1, you now want to completely eliminate the steady-state error of the closed-loop system with a PI controller. r(t) + e(t) u(t) y(t) Ge Gp y(t) Gc(s) = kp + k¡ _ kps + ki = Gp(s) = S S 2 4s + 1 (a) Show that the closed-loop transfer function is Gel(s) 2(k₁s + k₁) 4s² + (1 + 2kp)s + 2ki (b) From this transfer function, find equations for the natural frequency w and damping ratio 3 in terms of the controller gains. (c) Find the DC gain of the transfer function. Comment on the steady-state error of the closed-loop system. (d) Find the gains kp and ki for the closed loop transfer function to have w₁ = 1 and 3 = 1 (critical damping). (e) Find the poles of the closed-loop system with the controller gains from (d). (f) Use MATLAB to create a single plot with 2 step response traces: one for the open loop plant and one for the closed-loop system designed in (d).

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