Question

PID controller: KG (5) - K(s’+1.55 +0.5) Plant: G (s)=- 1 (20s +1)(10s +1)(0.5s +1) For this control system, do the following tasks. Advice: Before you start, it is advised that you convert G,(s) to a standard form so that the leading coefficients of the denominator polynomials become 1 (without altering the overall transfer function). (1) Plot the root locus for 0 <K<using a computer tool, e.g., Matlab tools such as rlocus(), rltool() or any appropriate computer tools. Also, read out the values of K at which the system is marginally stable and the values of sand K for the breakaway point. (See Page 285 text for more details about Matlab root locus tools.) (2) By analyzing the open-loop transfer function G (5)G,(s), determine the following quantities of the root locus: (a) Real-axis segment (b) Breakaway point (You may use the computer to find the roots of a 3rd order polynomial.) (c) Angles of asymptotes (e.). (d) Point of convergence of asymptotes, i.e., the centroid of poles and zeros (o,). (3) Verify that the quantities you have found in part(2) by analysis are consistent with the plot you have obtained in part(1). (4) Using Routh array/table, determine the ranges of K for which the system is stable. Check to see if they are consistent with the values of K you have obtained in part(1). (5) Plot the unit step response for k=300 and find the % overshoot and 2% settling time (i.e.,the earliest time after which the response remains within +2% of the target value). (6) Add the "precompensator” G (s) to the system as in Fig.P2B and plot the unit step responses of the system for k=300 and of the precompensator (alone) on the same graph. precomp

Answer 1

matlab code

controller=tf([1 1.5 0.5],[1 0])
plant=zpk([],[-0.05 -0.1 -2],0.01)
open_loop=series(controller,plant)
figure
rlocus(open_loop)
grid on
closed_loop=feedback(300*open_loop,1)
figure
step(closed_loop)
grid on
stepinfo(closed_loop)

output:

controller =

s^2 + 1.5 s + 0.5
-----------------
s

Continuous-time transfer function.

plant =

0.01
----------------------
(s+0.05) (s+0.1) (s+2)

Continuous-time zero/pole/gain model.

open_loop =

0.01 (s+1) (s+0.5)
------------------------
s (s+0.05) (s+0.1) (s+2)

Continuous-time zero/pole/gain model.

closed_loop =

3 (s+0.5) (s+1)
-------------------------------------------
(s+1.505) (s+0.4439) (s^2 + 0.201s + 2.245)

Continuous-time zero/pole/gain model.

ans =

struct with fields:

RiseTime: 0.6466
SettlingTime: 40.0134
SettlingMin: 0.2265
SettlingMax: 1.9152
Overshoot: 91.5246
Undershoot: 0
Peak: 1.9152
PeakTime: 1.9582

Root Locus 0,3 0.16 0.105. 0.065 0.03 10.44 Imaginary Axis (seconds) ...6 .... -0.44 ..... W.... 0.105 0.065 0.03 -2.5 -1.5 -0.5 Real Axis (seconds)

Root Locus 0.03. ..0.012..............0.006......... 0.058 System: open,oop Gain: 136 Pole: 0.000296 + 0.9771 Damping: -0.000303 Overshoot (%): 100 Frequency (rad/s): 0.977 0.09 0.18......... Imaginary Axis (seconds) 0.18 ...0.09 : N 0.058 **.......0.04.... -0.15 -0.1 -0.012..........0.006............ -0.05 Real Axis (seconds) 0.05

. . . . Step Response 30 Time (seconds) - - .- .- - .- CO apni duy