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LOOP SHAPING
Steps Involved in Loop Shaping
140530L Ravinath, W. D. A. P.
This is submitted as a partial fulfillment for the module
EN3143: Electronic Control Systems
Department of Electronics and Telecommunication Engineering
University of Moratuwa
22nd of May, 2017
1
Steps involved in Loop Shaping of a Simple Plant
Amila Pasan,
Undergraduate, Electronics and Telecommunication Department, University of Moratuwa,
Katubedda, Sri Lanka
[email protected]
Abstract—In this report we have shown the steps required to
follow in order to adjust a plant with respect to bandwidth, phase
margin and steady state error (SSE). Furthermore we have done
a simulation in order to check the tracking capability of the
adjusted system and compared that with the original plant.
I. I NTRODUCTION
Loop shaping is one of the vital things of control systems.
This may be needed as an optimum solution to repair a
plant without replacing the plant which no longer gives its
original specifications. This comprehensive report deals with
adjustment of bandwidth, correction of phase margin and
steady state error (SSE).
II. M ETHOD
The given system:
3(s + 3)
2
s + 3s + 20
The adjustments we have to make to the plant are as follows:
Band Width = 15[rad/s]
Phase Margin = 400
Steady State Error = 0.02
G(s) =
A. Step 1: Draw Gain and Phase Plots
16.0022 = 22.99780 , the additional margin required. This is
clearly depicted in Fig. 1. As Cle should compensate for this,
∠Cle (jω)|ω=15[rad/s] = 22.99780 . After numerically solving
for this we can find that,
s + 9.742
Cle (s) =
s + 23.1
D. Step 4: Adjusting lead compensator gain to unity (Using
a gain Kle )
Here we adjust lead compensator gain at ω = 15[rad/s]
using the gain Kle s.t.,
Kle |Cle |ω=15[rad/s] = 1
which is 1.5398.
E. Step 5: Steady state error ( ess ) adjustment (Using a lag
compensator)
First we have to figure out ess of G2 .
Using Final Value Theorem,
ess = lim e(t) = lim sE(s)
t→∞
First we need to draw the gain and phase plots of original
plant in order to identify it. They are shown in Fig.1.
E(s) = R(s) − Y (s)
= R(s) − E(s)G2 (s)
1
E(s) =
R(s)
1 + G2 (s)
B. Step 2: Bandwidth adjustment (Using a forward gain K)
First thing we have to adjust is the bandwidth to
BW. For that we introduce K s.t.
K|G(jω)|ω=15[rad/s] = 1
s→0
Here e is the error of the system (actually this is when r(t) is
unit step), i.e.,
So ess becomes,
ess = lim
s→0
We get K = 70.999010 . The bode diagrams of gain and
phase are shown in Fig. 1.
C. Step 3: Phase margin adjustment (Using a lead compensator)
We want our system to have phase magnitude 400 more
than 1800 within our adjusted bandwidth of 15 [rad/s]. To do
that we introduce a lead compensator Cle to the plant,
s + zle
Cle =
s + ple
We get the unadjusted phase margin, ∠G(jω)|ω=15 +
1800 = −163.99780 + 1800 = 16.00220 which is 400 −
1
1 + G2 (s)
ess of G2 happens to be 0.1616 which is way over than
required, 0.02. So we need to adjust ess of the system using
a lag compensator.
F. Designing a lag compensator
By introducing a lag compensator,
s + zla
Cla =
s + pla
to the system, ess becomes,
ess = lim
s→0
1
1 + Cla (s)G2 (s)
2
By plugging in ess = 0.02 we get the ratio,
zla
= 9.4462
pla
Bode Diagram of Lead Compensator
0
−1
−2
Magnitude (dB)
We want our lagging compensator to incur less interference
to the system as possible with respect to other parameters, so
we make both zla and pla to be small as possible, i.e. closer
to the origin. By choosing pla to be 0.1 we end up zla being
0.9446. So our lad compensator Cla is s.t.
−3
−4
−5
−6
−7
−8
25
s + 0.9446
Cla (s) =
s + 0.1
Phase (deg)
20
G. Step 6: Simulation
15
10
5
Reference signals of 5 rad/s and 20 rad/s signals are input
to the signal in order check the tracking performance of the
system. The block diagram of simulink model is shown in Fig.
5.
0
−1
0
10
1
10
2
10
Frequency (rad/s)
3
10
10
Fig. 2. Bode Diagram of Lead Compensator
.
III. R ESULTS
How loop shaping works is summarized in Fig. 1. The
simulation result is shown in Fig. 4.
Bode Diagram of Lag Compensator
20
18
IV. C OMMENTS
16
Magnitude (dB)
14
As Fig. 4 shows the system is quite capable of tracking the
reference compared to the plant without any adjustments (Fig.
6) of which the result of the simulation is shown in Fig. 7
12
10
8
6
4
2
0
Bode Diagram
0
Phase (deg)
Magnitude (dB)
50
0
−30
−50
−60
−3
10
−2
−1
10
0
10
1
10
2
10
10
Frequency (rad/s)
−100
0
Fig. 3. Bode Diagram of Lag Compensator
Phase (deg)
−45
.
−90
Original System
Bandwidth Adjusted
Phase Margin Adjusted
SSE Corrected
−135
−180
−3
10
−2
10
−1
10
0
10
Frequency (rad/s)
1
10
2
10
3
10
2.5
yout
r
Fig. 1. Bode plots of original, bandwidth adjusted, phase margin adjusted
and SSE corrected systems. Note that the phase bode plots of both original
and bandwidth adjusted systems overlap each other since those differ only by
a factor
.
Tracking Response (y_{out}) and Reference Input (r)
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
0
0.5
1
1.5
Time (s)
2
2.5
Fig. 4. This is how the adjusted system tracks the reference r.
3
3
r
sin(5t)
Reference
Input
yout
Tracking
Response
sin(20t+pi/2)
tf([1 0.9446],[1 0.1])
Lag Compensator
ess = 0.02
1.5398
tf([1 9.742],[1 23.1])
Gain
Lead Compensator
||=1 @ 15 [rad/s] PM Adjustment 40 [deg]
70.999
BW Adjustment
15 [rad/s]
tf([3 9],[1 7 32 80])
Plant
Fig. 5. Simulation diagram adjusted system to check its tracking capability. Here, we have used 5 rad/s and 20 rad/s sinusoidals as reference signals.
.
r
sin(5t)
Reference
Input
sin(20t+pi/2)
yout
Tracking
Response
tf([3 9],[1 7 32 80])
Plant
Fig. 6. Simulation diagram of original plant
.
2
yout
r
Tracking Response (y_{out}) and Reference Input (r)
1.5
1
0.5
0
−0.5
−1
−1.5
−2
0
0.5
1
1.5
Time (s)
2
2.5
Fig. 7. This is how the original plant G tracks the reference r.
3
Appendix: MATLAB Code
clear,close all;clc
% requirements
BW = 15;
PM = 40;
SSE = 0.02;
% bandwidth
% required phase margin
% required steady state error
numG = 3*[1 3];
denG = conv([1 4],[1 3 20]);
G = tf(numG, denG);
leg_G = ’Original System’;
bode(G); grid on;hold on
[gainG, phaseG] = bode(G,BW);
K = 1/gainG;
G1 = K*G;
leg_G1 = ’Bandwidth Adjusted’;
bode(G1);
PM_G = 180+phaseG;
PR = PM−PM_G;
% bandwidth adjusted system
% required additional margin
% getting the numerical solution to z_le
syms z_le
eqn = PR/180*pi == atan(BW/z_le)−atan(z_le/BW);
sol_z_le = solve(eqn,z_le);
p=eval(sol_z_le);
clear z_le
z_le = p(1);
p_le = BW^2/z_le;
C_le = tf([1 z_le],[1 p_le]);
% lead compensator
K_le = 1/bode(C_le,BW);
G2 = K_le*C_le*G1;
leg_G2 = ’Phase Margin Adjusted’;
bode(G2);hold on;
% phase margin adjusted system
E = 1/(tf([0 1],[0 1])+G2);
SSE_G2 = evalfr(E,0);
% steady state error of G2
if (SSE_G2<SSE)
C_la = 1;
else
ratio = (1/SSE−1)/evalfr(G2,0);
p_la = 0.1;
z_la = p_la*ratio;
C_la = tf([1 z_la],[1 p_la]);
end
G3 = C_la*G2;
% SSE adjusted ssytem
bode(G3);
leg_G3 = ’SSE Corrected’;
leg = legend(leg_G, leg_G1,leg_G2,leg_G3,’Location’,’SouthWest’);
set(leg,’FontSize’,12)
print −depsc html\controlsystem_04.eps
hold off
1
figure
bode(C_le);grid on
title(’Bode Diagram of Lead Compensator’)
print −depsc html\controlsystem_05.eps
figure
bode(C_la);grid on
title(’Bode Diagram of Lag Compensator’)
print −depsc html\controlsystem_06.eps
%%
% loop shaped system simulation
clear, close all;clc
tStart = 0;
tFinal = 3;
Simulation_loopFilt
print −s −dsvg html\diagram
sim(’Simulation_loopFilt’,[tStart tFinal]);
figure
plot(yout)
hold on
plot(r,’−−’)
grid on
xlabel(’Time (s)’)
ylabel(’Tracking Response (y_{out}) and Reference Input (r)’)
legend(’y_{out}’,’r’)
print −depsc html\tracker
%%
% original system simulation
clear, close all;clc
tStart = 0;
tFinal = 3;
Simulation_loopFilt_plant
print −s −dsvg html\diagram_plant
sim(’Simulation_loopFilt_plant’,[tStart tFinal]);
figure
plot(yout)
hold on
plot(r,’−−’)
grid on
xlabel(’Time (s)’)
ylabel(’Tracking Response (y_{out}) and Reference Input (r)’)
legend(’y_{out}’,’r’)
print −depsc html\tracker_plant
2

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