Please solve part b, will upvote ! Problem 2In this problem, we are going to investigate how the feedback controller can havedifferent impact on the closed-loop performance depending on the plant transfer function. For eachcombination of plant transfer function and feedback controller, we need to first write down the errorequation and ascertain the conditions for stability. Finally, we will investigate the steady-state trackingerror and evaluate the closed-loop poles.a) Consider the following second-order plant:1s²+1with a proportional-derivative (P-D) feedback controllerG(s) Gp + GD8G(s). We refer to L(s) = G(s)K(s) as the loop gain. The closed-loop system is stable if all theroots of the equation 1+L(s) = 0 are in the left half-plane. Recall that the closed-loop trackingerror is given by:K(s) =+0+8E(s) =where R(s) is the reference signal. In the following, we will investigate the impact of the presenceof integrators, i.e., terms of the form 1/s, in the feedback controller G(s) on the steady-statetracking error given. Recall that the steady-state tracking error can be computed using the FinalValue Theorem:esse(x) = lim s E(s) (only when the closed-loop is system is stable!)11+L(s) R(s)s* =Show that the closed-loop (CL) tracking error is:s²+1E(s) = ² +Gps +1+Gp▪ Apply Routh's test to show that the CL system is stable if Gp> 0 and Gp> -1.. Under the above conditions, show that the steady-state tracking error is 1/(1+ Gp) for aunit step reference R(s) = 1/s.▪ Show that there are a complex conjugate pair of CL poles atGD2for small enough GD given by 0

The plant transfer function is shown below..The feedback controller is shown below..

b) Now we repeat the above exercise for a different plant K(s)= with a proportional-derivative (P-1) feedback controller G(s)=Gp+5 s²+1 3 (note the s in the numerator) GI • Show that the closed-loop (CL) tracking error is: E(s) = -R(s) . Can you comment on the equivalence of the roles played by Gp and G₁ in this case, with the roles played by Gp and Gp (respectively) in the previous case? s²+1 s2+Gps+1+G₁

b) Now we repeat the above exercise for a different plant K(s)= with a proportional-derivative (P-1) feedback controller G(s)=Gp+5 s²+1 3 (note the s in the numerator) GI • Show that the closed-loop (CL) tracking error is: E(s) = -R(s) . Can you comment on the equivalence of the roles played by Gp and G₁ in this case, with the roles played by Gp and Gp (respectively) in the previous case? s²+1 s2+Gps+1+G₁

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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