EE 6225 – Multivariable Control System
Part I – Advanced Process Control
Assignment
(i) Due Date: 7 April 2023
(ii) Select either Q1 or Q2
(iii)Submit softcopy solution to Assignment
folder in NTULearn Course Folder and include
Your Name and Matriculation Number
1
1(a). Direct Synthesis Method (DSM)
• Consider the following block diagram of a feedback control system
2
Gc G
Set-Point R
Gd
Disturbance D
Output Y
Load LControl
UError E +
-
Controller
Process/
Plant
3( )
( )( )
( )
( )( )
0.5
The Second Order Plus Time Delay (SOPTD) process is given by:
2

5 1 2 1
The disturbance to output is

10 1 5 1
Apply the Direct Synthesis Method (DSM) to design a controller us
s
s
d
e
G s
s s
D Y
e
G s
s s
−
−
=
+ +
=
+ +
ing
the following desired closed loop transfer function
1
where
1
1
(i) Assume , design a PID controller for 1 sec and 10 sec.
(Hi
s
d
c
d c
d
c c
Y
Y eR
G
Y R sG
R
G G G


−
  
  
   = = 
+    −   
  
= =
nt: Using first order Taylor series to approximate 1 in the
denominator of ; the in the numerator will be cancelled by identical term in )
(ii) Using MatLab/Simulink or any sim
s
s
c
e s
G e G


−
−
 −
ulation software, plot the output response
to a step input at time 0 sec and a step disturbance input at 60 sec.Y R t D t= =
1(a). Direct Synthesis Method (DSM)
4GR
Gd
D
Y
L
UE +
-
Controller Process
-
Y Y−
G
Y
*
cG
• (a) Block diagram of a feedback control system using IMC
• (b) Block diagram of a feedback control with equivalent standard controller
1(b). 1 Degree of Freedom Internal
Model Control (IMC)
Gc G
Set-Point R
Gd
Disturbance D
Output Y
Load LControl
UError E +
-
Controller
Process/
Plant
5( )
( )
( )( )
( )
( )( )
*
*
The Second Order process with right Half Plane (RHP) zero is given by:
2 1

5 1 2 1
The disturbance to output is

10 1 5 1
An IMC controller is given by
1
where
s
d
c
c
s
G s
s s
D Y
e
G s
s s
G
G f G
G
−
−
− +
=
+ +
=
+ +
= = ( )
*
*
*
, contains time delays and RHP zeros with 0 =1
1
and
1
(i) Assume , design an IMC Controller for 5 sec.
(ii) Derive the equivalent standard controller = an
1
c
c c
c
c
c
G G G G s
f
s
G G G
G
G
G G


+ − + + =
=
+
= =
−
d determine the PID parameters
1
= 1 . Check if you get the same if you use DSM with .
(iii) Using MatLab/Simulink or any simulation software, plot the output res
c c D c
dI
Y
G K s G G f
s R


+
   
+ + =   
  
ponse
to a step input at time 0 sec and a step disturbance input at 60 sec.Y R t D t= =
1(b). 1 Degree of Freedom Internal
Model Control (IMC)
6• (a) Block diagram of a feedback control system using 2 degree of freedom
controller design (the controller is place in feedback path so that setpoint
filter Gs can be designed independently of Gc* from R(s) to Y(s))
1(c). 2 Degree of Freedom Internal
Model Control (IMC)
G
R Gs
D
Y
L
U
+
-
Controller
Process
-
Y Y−
G
Y
*
cG
Gd
( )
( )
( )
( ) ( ) ( )( )
( )
( )
( )
( ) ( )
( ) ( ) ( )( )
( )
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )
*
* *
*
1
,
1 1
Assume ,
and 1
c
s d
c c
s c d
Y s G s Y s G s G s
G s G s
R s D sG s G s G s G s G s G s
G s G s
Y s G s G s R s Y s G s G s D s
−
= =
+ − + −
=
= = −
7( ) ( )
( )
( )
2
*
*
The First Order Plus Time Delay (FOPTD) is given by:
that is, we assume that the disturbance is at the plant input
10 1
An IMC controller is given by
1
where
s
d
c
c
e
G s G s
s
G
G f G G G
G
−
+ −
−
= =
+
= =
( )
*
1
and
1
(i) Assume , design an IMC Controller for 2 sec, 1 and 0.
Set 1. Using MatLab/Simulink or any simulation software, plot the output
response to a step in
r
c
c c
s
s
f
s
G G G r
G
Y


 
+
=
+
= = = =
=
( )( ) ( ) ( )
( )
*
put at time 0 sec and a step disturbance input at 20 sec.
(ii) Next design another IMC Controller with 2 sec, 2 and is determined by setting
1
1 0 at the pole of 1
c
c d
R t D t
r
s
G s G G s G s
G s
 

−
= =
= =
+
− =  −
( )
1 3, 0.2, 2
1
0
1
1
Choose and judiciously select . Using MatLab/Simulink or any simulation software,
1
plot the output response to a step input at time 0 sec an
c
r
c
s r
s s
s
s
G
Y R t




=− = =
 
= 
 + 
=
+
= d a step disturbance input
at 20 sec.
(iii) Compare the responses obtained in (i) and (ii).
D
t =
1(c). 2 Degree of Freedom Internal
Model Control (IMC)
1(d). Cascade Structure and Feedforward
Control
• IMC Cascade Control Structure
8
gd(s)
d1
9( ) ( ) ( )
( ) ( )
( )
( )( )
( )
1 2
20 4
1 2
30
2 2
The Second Order Plus Time Delay (SOPTD) is given by:

where by
2
and
15 1 1
and

10 1 0.3 1
(i) Design the inner controller for using the 1 Degree
s s
s
d
p s p s p s
e e
p s p s
s s
e
g s
s s
q p s
− −
−
=
= =
+ +
=
+ +
( )
( )
1
2
of Freedom IMC design method
1
as in Part 1(b) with . Next design the outer controller for using the
4 1
1
1 Degree of Freedom IMC design method as in Part 1(b) with . Using
18 1

f q p s
s
f
s
=
+
=
+
( )
2 MatLab/Simulink or any simulation software, if =0, plot the output response to a step
input at time 0 sec and a step disturbance input to at 40 sec.d
d y
r t g s t= =
1(d). Cascade Structure and Feedforward
Control
10
( )
( )
( ) ( )( ) ( ) ( )
( ) ( )
_
2 2 2
2 2 1 2 2
2
1
1
1
(ii) Design the inner controller for with and is determined by setting
4 1
1 1
1 0 at the pole of 1 0
4 1
Use the same outer controller fo
s
s
q p s f
s
s
p s q s p s p s
p s s
q



=−
+
=
+
 +
 − =  − =
 + 
( )
( )
2
r designed in Part 1(d)(i). Using MatLab/Simulink
or any simulation software, if =0, plot the output response to a step input at
time 0 sec and a step disturbance input to at d
p s
d y r
t g s t=
2
2
40 sec.
(iii) Design a feedforward controller against the distance . Together with the designed IMC
controllers designed in Part 1 (d)(ii), if =0, plot the output response to a step input
d
d y r
=
( )
at
time 0 sec and a step disturbance input to at 40 sec.
(iv) Compare the responses obtained in (i), (ii) and (iii).
dt g s t= =
1(d). Cascade Structure and Feedforward
Control
2. Linear Quadratic Regulator (LQR)
11
Under stabilizability condition on (A,B) and detectability condition on (A,G), the
LQR control results in a closed-loop system that is asymptotically stable.
(Algebraic Ricatti Equation)
(assuming all states x are available for feedback)
( ) ( ) ( ) ( )LQR
0

T T
J z t Qz t u t Ru t dt

= +
0 TN G QH= =
1, , , T T Tu Kx K RR B P RR H QH I QQ G QG−= − = = + =
( )1 0T TA P PA QQ PB RR B P−+ + − =
12
2. LQR Loop Transfer Function L(s)
(assuming all states x are available for feedback)
13
2. LQR Robustness to Model Uncertainties
2. LQR Sensitivity and Complementary
Sensitivity Functions
14
2(a). Roll Dynamics of Aircraft
15
Consider the roll angle dynamics of an aircraft:
 
Defining the state variable as :
We have
and controlled output
with
0 1 0 0
0 0.875 20 , 0
0 0 50 50
T
x
x
x Ax Bu z Gx Hu
A B
  =
= + = +
   
   = − − =
   
   −   
2(a). LQR Loop Transfer Function
16
 
2 2
The controlled output is chosen to be which corresponds to
1 0 0 0
,
0 0 0
The LQR controllers are to be designed with 1, and different values of and .
(Reference Slide 11)
T
x
z
G H
R Q I
 

 
=
   
= =   
   
= =
(i) Design 3 LQR controllers for a fixed =0.01 and =0.1,10,100.
Using MatLab/Simulink or any other software, plot on the same graph the Bode frequency
responses (magnitude and phase) of the 3 corresp
 
( )
1
onding loop transfer functions
(refer to Slides 17 and 18 for hints).
(ii) Design 3 LQR controllers for a fixed =0.01 and =0.01,0.1,1.
Using MatLab/Simulink or any other software, plot on the
K sI A B
 
−
−
( )
1
same graph the Bode frequency
responses (magnitude and phase) of the 3 corresponding loop transfer functions .
(iii) Determine the phase margins for the LQR controllers designed in Part 2(a)(i
K sI A B
−
−
)and (ii).
How does and affect the phase margins. Estimate the slope (in terms of dB/decade) of the
loop transfer functions at high frequencies.
 
2(a). MatLab Commands
17
For the Loop Transfer
Function L(s) = K(sI-A)-1B,
use sys_ltf = ss(A,B,K,0)
2(a). MatLab Commands
18
and x Ax Bu z Gx Hu= + = +
2(b) Nyquist Plot
19
2 2(i) Select , 1, =0.1, =1. Plot the Nyquist plot of the
loop transfer function (refer to Slide 17 for hints)
(ii) Under state feedback , the closed-loop dynamics are given by

xQ I R
u Kx
x A
 = =
= −
= ( ) and the closed-loop poles are the eigenvalues of .
Using the MatLab function (implemented via Bass Gura
or Ackerman's formula), you can compute a state feedback gain matrix
BK x A BK
place
− −

that assigns these poles to any locations. For the same and in Part 2(a),
choose the closed loop poles 1, 2 3 and then use ( )
MatLab function to compute th
, ,
K
A B
K place A B pp = − − − =
( )
1
e gain matrix . Plot the Nyquist plot of the
loop transfer function .
(iii) Compare the Nyquist plots obtained in (i) and (ii) in terms of whether these
plots enter within a circl
K
K sI A B
−
−
e of radius 1 centred at -1. What can you conclude about
the robustness of the 2 design methods against multiplicative uncertainty?
2(c) Step Response
(i) Plot on the same graph, the step responses to roll angle
setpoint command of 1 rad for the 3 LQR state feedback
controllers designed in Part 2(a) (i) for
(ii) Plot on the same graph, the step responses to roll angle
setpoint command of 1 rad for the 3 LQR state feedback
controllers designed in Part 2(a) (ii) for
(iii) Compare on the effect of the step responses (speed and
overshoot) for variations in
Hint: Specify the closed loop system:
20
0.1,10,100. =
0.01,0.1,1. =
and . 
   
( )
  ( )
, 1 0 0 , 1 0 0 , 0
sys_cl=ss , , ,
, , step sys_cl
T
cl cl cl cl
cl cl cl cl
A A BK B C D
A B C D
y t x
= − = = =
=