Question

This problem explores the effect of closed-loop zeros on the system response. Suppose ?(?) = ?? (?)?? (?) where ?? (?) = ? a proportional controller. The plant ??(?) takes on two different systems in Problem 1 and in Problem 2

R(S) + OE(S) This problem explores the effect of closed-loop zeros on the system response. Suppose G(s) = G(s)Gp(s) where Gc(s) = K a proportional controller. The plant Gp(s) takes on two different systems in Problem 1 and in Problem 2. John Wiley & Soni Aligher 1. Suppose Gpi(s) = S(52+65+45) a. Find the open-loop poles and the closed-loop poles when K = 40 b. Find and plot the closed-loop unit step response, that is, find c(t), t > 0 when r(t) = u(t) 3 s2+05+40 2. Repeat for G 2. Repeat for Gp2(S) = 40 s(52+35+5) - Compare the two answers relative to the location of closed-loop poles, and the effect of the closed-loop zeros Hint: explore using the Matlab function ilaplace Type help ilaplace See file MATLAB Examples in BB Modules, Example #3

1. Suppose ??1 (?) = 1 ?(? 2+6?+45) a. Find the open-loop poles and the closed-loop poles when ? = 40 b. Find and plot the closed-loop unit step response, that is, find ?(?), ? > 0 when ?(?) = ??(?)

2. Repeat for ??2 (?) = 3 40 ? 2+ 40 3 ?+ 40 3 ?(? 2+3?+5) Compare the two answers relative to the location of closed-loop poles, and the effect of the closed-loop zeros Hint: explore using the Matlab function ilaplace Type help ilaplace

The question didn't state the K for problem 2.

Answer 1

The open loop teansfer function is Kapil) = 40 0 s(5²+65+45) to any system shown below, the state loop Transfer function is closed Ris) o al Cis) (1) - Ç - RIS) I+GH " The closed loop transfer function is Cs - 40 s(5²+ 65+45) 40 ☺ It 1x 40 8+68+ 455740 s(s+ 65+45) The open loop poles one Rosts of ☺ Piz o, -3765 The closed loop poles are roots of ③ i. - P. =-1, -2.5 15815 There are neither open loop zeros nol Closed loop zeros. GALD = 3 ( $ +40 + ) s(8+35+5) Anm .

step response using MATLAB:

The - RIS) kapsy - yox (s'+ 405 +40) 40 8 5 (8735+5) = (3.5+ 405+40) -0 5 (8+35+5) Closed loop Teansfer function is C(s) - 35² + 405+40 - ② S 3 +65 +455+40 The open loop poles and zeos one co) 212 = -1.0889, - 12.2444 Piz3 = 0,-1.51.65835' The closed loop poles and seros one (①) 22 = -1.0889, -12.2444 Pizz = -1, -2-575.815

clc;

clear all;

close all;

s=tf('s');

k=40; % P-gain

gp1=1/(s*(s^2+6*s+45));

gp2=(3*s^2+40*s+40)/(s*(s^2+3*s+5));

step(feedback(k*gp1,1),feedback(k*gp2,1));grid % step response

legend(' plant 1 ', ' plant 2')

The system response is faster with the plant with zeros added to it

File Edit View x? Figure 1 x 1 Insert Tools Debug Desktop Window Help Ha a my . EI E O | ID E F D) Step Response plant 1 plant 2 Amplitude Time (seconds) Property Editor - Waveform