EEE3001 – EEE8013
State Space Analysis and Controller Design
This lecture will be recorded and
you will be able to download it
Dr Damian Giaouris
http://www.staff.ncl.ac.uk/damian.giaouris/
Goals/Aims of Chapter 4
 Controller design in state space




Controllability/Observability
State feedback control (pole placement)
Linear Quadratic Regulator (LQR)
Estimator design in state space
 Open loop estimator
 Closed loop estimator
 Summary
Feedback Controlled Systems
Open Loop Transfer Function (OLTF):
U
Y
G(s)
Closed Loop Transfer Function (CLTF): OLTF + Feedback of the output
R
RRR
GcK(s)
1
Gc(s)
RR11
U
G(s)
U
GK
c(s)
Y
G(s)
G(s)
YY
State Feedback State Space Models
Open Loop Transfer Function (OLTF):
U
Y
G(s)
u
Dx
B
dt
x
y
C
A
Closed Loop Transfer Function (CLTF): OLTF + Feedback of the State
R
Gc(s)
U
G(s)
Y
r
Controller
x
u
B
Dx
  dt
A
x
C
y
State Feedback State Space Models
R
K1
U
R1
G(s)
Y
K
r
+
K1
+
u
B
Bu +
x
  dt
+
Ax
Plant
-K
A
x
y
C
State Feedback - Equations
r
+
K1
+
u
B
Bu +
x
  dt
x
The task of the controller is to produce the
appropriate control signal u that will insure
that y=r.
y
C
+
Ax
A
u  t   K1 r  t   Kx  t 
Plant
-K
u  t   K1 r  t   Kx  t  


x  t   Ax  t   Bu  t  


  
 
 
 

WE MUST
CHECK
IF
THE
SYSTEM


 
 

x t  Ax t  B K1 r t  Kx t
 x t  A  BK x t  BK1r t
y t  Cx t  Du t  C  DK x t  DK1 r t
x  t   ACL x  t   BCL r  t  


y  t   CCL x  t   DCLu  t  

IS CONTROLLABLE.
K? : A  faster/stable.
ACL  A  BK 

BCL  BK1


CCL  C  DK 
DCL  DK1 
CL
This method is called pole placement.
RL Circut
L
R
di 1
di
R 1
 V  iR     i  V
dt L
dt
L L
I
x  Ax  Bu
A=-R/L, x=i, B=1/L and u=V
V
Obviously the eigenvalue is –R/L (assume here 0.5 and u=0)
1
0.8
x(t)
0.6
0.4
0.2
0
0
2
4
6
time, s
8
10
RL Circut
L
R
di 1
di
R 1
 V  iR     i  V
dt L
dt
L L
I
x  Ax  Bu
A=-R/L, x=i, B=1/L and u=-Kx=-Ki
V
ACL  A  BK 
R K R  K
 
L
L
L
Choose to place the eigenvalue at -6R/L
0.5
Open loop
Closed Loop
0.4
x
0.3
0.2
0.1
0
0
2
4
6
Time, s
8
10
12
RK
R
 6  K  5R
L
L
u=V=-5Ri
Unstable System
x  3  k x
u  kx
Hence the eigenvalue of the CL system is 3-k
k=13
Hence the eigenvalue of the CL system is -10 (stable)
2
Open loop
Closed Loop
1.5
x
x  3x  u
1
0.5
0
0
0.2
0.4
0.6
Time, s
0.8
1
LQR
1 2 
1 
1 0 
x(t )  
x
(
t
)

u
t
,
y
(
t
)




0 
0 1  x(t )
3
4


 


2
4
x1
2
1
|x |
x2
1
2
0
0
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
time, s
1
x,x
I x    x1  x2 dt
u   Kx, K  26 72 
2
2
|x |
-1
-2
0
0
0.4
0.6
0.8
0.4
time, s
1
time, s
1
0.9383
0.8
0.6
x
0.2
0.2
I
-3
0
1
0.4
0.2
0
0
0.2
0.4
0.6
time, s
0.8
1
LQR
1 2 
1 
1 0 
x(t )  
x
(
t
)

u
t
,
y
(
t
)




0 
0 1  x(t )
3
4


 


u   Kx, K  26 72 
2
2
x1
2
1
1
0
0
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
time, s
1
x,x
x2
x2
1
I x    x12  x2 2  dt
-1
2
x2
2
-2
0
0
0.2
0.4
0.6
0.8
0.2
0.4
time, s
1
time, s
1.5
Ix=1.403
x
1
I
-3
0
1
0.5
0
0
0.2
0.4
0.6
time, s
0.8
1
LQR
Poles at -10, -11
2
1.5
x1
Ix=1.403
x2
1
1
1
I
x
x,x
2
0
-1
0.5
-2
0
0
-3
0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
1
time, s
1
time, s
Poles at -15, -20
1.5
1
5
Ix=1.271
0.2
0.4
0.6
0.8
1
x
-5
0
1
I
x
0
time, s
0.5
x
2
2
1
0
0
0.2
0.4
0.6
time, s
0.8
1
0
0
0.2
0.4
0.6
time, s
0.8
1
LQR
50
u=-Kx
0
case 1
case 2
-50
-100
-150
-200
0
0.2
0.4
0.6
time, s
0.8
1
LQR
I u   u 2 dt
600
500
I
u
400
Case 1
Case 2
540
300
200
230.5
100
0
0
0.2
0.4
0.6
time, s
0.8
1
LQR
I   xT  t  Qx  t   u T  t  Ru  t dt
Reduced Riccati Equation
AT P  PA  PBR 1 BT P  Q  0
K  R1 BT P
Estimating techniques
u
Bu +
B
x
  dt
x
y
C
+
A
Ax
e  t   x t   x t   e t   x t   x t  
Ax  t   Bu  t   A  t  x  t   Bu  t   e  t   Ae t 
Plant
B
Bu +
+
x
Ax
  dt
C
x
A
Estimator
y
Estimating techniques
u
B
Bu +
x
  dt
x
y
C
+
Ax
y +
A
Plant
- y
G
B
Bu +
-
x
  dt
x
y
C
+
Ax
A
Estimator
e  t    A  GC  x  t    A  GC  x  t    A  GC  e  t 
But the system must be observable
Estimating techniques
u
B
x
Bu +
  dt
x
x(t )  Ax(t )  Bu (t )
y
C
y (t )  Cx(t )  Du (t )
+
y +
A
Ax
Plant
u  t    Kx  t 
- y
x(t )  Ax(t )  BKx  t  
G
B
Bu +
-
x
  dt
x
y
C
+
Ax
 A  BK  x(t )  BK  x(t )  x  t  
 A  BK  x(t )  BKe  t 
e  t    A  GC  e
A
Estimator
-K
 x  t   A  BK


e
t



  0
BK   x  t 


A  GC   e  t  
sI  ( A  BK )
 BK
0
sI   A  GC 
 0  sI  ( A  BK ) sI   A  GC   0
This means that the pole placement design and
the estimator design are independent of each
other. Then they can be designed separately and
combined to form the observed-state feedback
control system.
Estimation and Tracking
rss
+
K1
+
u
B
Bu +
x
  dt
x
y
C
+
Ax
y +
A
Plant
- y
G
B
Bu +
+ x
  dt
+
Ax
A
Estimator
-K
1
 N x   A B  0 
 N   C 0  1 
  
 u 
K1  N u  KN x
u  K1 rss  Kx
e  t    A  GC  e
x
y
C