1. 3.Closed-Loop Transfer Functions
3.1 Block Diagram Algebra (Interconnection rules)
DEU-MEE 5017 Advanced Automatic Control

3. From the block diagram:
3.2 Closed-Loop Transfer Function with Unity Feedback
a) Block diagram
From the block diagram:

4. Forward paths: Gc(s)Gp(s) Mason formula :
3.3 Closed-Loop Transfer Function with Unity Feedback
b) Mason formula
R(s) : Reference
N(s): Disturbance ) s ( R
Forward paths
Loops
Forward paths: Gc(s)Gp(s)
Mason formula :
Loops: Gc(s)Gp(s)(-1)
Closed loop transfer function from R(s) to C(s) :
Closed loop transfer function from N(s) to C(s) :
When N(s)0, the Laplace transform of output C(s) :

5. R(s) : Reference N(s): Disturbance ) s ( R
3.4 Closed-Loop Transfer Function with Sensor Dynamics
R(s) : Reference
N(s): Disturbance ) s ( R
indicated output
actual output
Frequently, the sensor has its own dynamics and might be modeled as a first–order system.

6. PID control: Fd=0 Fd=-mgsin
Example 3.1: Study the control of cruise control (Example1.1)
a) Without disturbance (Fd=0) :
Fp: Pushing force (from engine), Fd: Distrubance
Fd=0
Fd=-mgsin
PID control: Ks
Gc(s) Ka
G1(s) + - Ke

7. MATLAB code for closed-loop response : Fd=0
clc,clear,%close all
kp=12;ki=0;kd=0; % kp=1,4,12
v0=72; % km/h
m=1000;b=60;ke=55.56;ka=5;kkm=3.6;ks=0.1;
vref=v0/kkm*ks
nc=[kd,kp,ki];dc=[1,0];
n1=[1];d1=[m,b];
nh=ka*ke*ks*kkm*conv(nc,n1);
dh=polyadd(conv(dc,d1),nh);
h=tf(nh,dh);
p=roots(dh),%return
p0=p(2);cksi_v2;%return
t=0:dt:ts;
vs=vref*step(h,t);
hold on;plot(t,vs,'k');
return
vsan= *exp(-0.33*t).*cos(0.3018*t )+10;
plot(t,vsan,'or')
Settling time decreases
Steady-sate error decreases

8. Steady-sate error increases
b) Effect of disturbance :
MATLAB code for closed-loop response:
clc,clear;%close all
kp=12;ki=0;kd=0; % kp=1,4,12
fi=5;g=9.81;v0=72; % km/h
m=1000;b=60;ke=55.56;ka=5;kkm=3.6;ks=0.1;
vref=v0/kkm*ks
nc=[kd,kp,ki];dc=[1,0];
n1=[1];d1=[m b];
nh=ka*ke*ks*conv(nc,n1);
dh=polyadd(conv(dc,d1),nh);
nhn=ks*conv(dc,n1);
fd=-m*g*sind(fi);
p=roots(dh),%return
p0=p(2);cksi_v2
t=0:dt:ts;
h=tf(nh,dh);hn=tf(nhn,dh)
vs=vref*step(h,t);vsn=fd*step(hn,t);
hold on;plot(t,vs,'b');plot(t,vsn,'g')
plot(t,vs+vsn, 'k')
Steady-sate error increases

9. Control action: PID control
Example 3.2 : Find the closed-loop response of 2-DOF mechanical system (Example2.2)
Control action: PID control
Closed loop transfer functions:
When F2(s)0, the Laplace transform of output X2(s) :

10. a) Without disturbance (F2=0) :
kp=45
a) Without disturbance (F2=0) :
Steady-sate error
MATLAB code for closed-loop response:
clc,clear all
kp=45;ki=0;kd=0;
%kp=45;ki=150;kd=0;
%kp=45;ki=150;kd=2;
m1=1.2;m2=1;k1=350;k2=300;c1=4;c2=3;
nc=[kd,kp,ki];dc=[1,0];
n21=15*[1 100];d21=[ ];
n22=[ ];d22=d21;
nh=conv(nc,n21);dh=polyadd(conv(dc,d21),nh);
p =roots(dh)%,p0=p(2);cksi_v2;return
dt=0.001;ts=25;
t=0:dt:ts;
h=tf(nh,dh);x2r=0.1*step(h,t);
hold on;plot(t,x2r,'b','LineWidth',1.5)
kp=45,ki=150
Eliminates steady-sate error
kp=45,ki=150, kd=2
Decrease settling time

11. Study effect of disturbance, develop MATLAB program, find x2(t)
b) With disturbance (F2 unit step input) :
Study effect of disturbance, develop MATLAB program, find x2(t)
MATLAB code for closed-loop response:

12. c) Simulation with Simulink (F2 unit step input) :
Comparison of Matlab code and Simulink results