MODERN CONTROL THEORY Lecturer:Qilian Bao 鲍其莲 1
MODERN CONTROL THEORYLecturer:Qilian Bao鲍其莲
1
Chapter 7 State Feedback and State
Estimator
Objectives:
• State feedback and design of feedback matrix
• Design of state estimator
• Feedback control with estimated states
2Chapter 7
7.1 State Feedback
• Output feedback control system
3Chapter 7
Output feedback system
4
let HyVu
DuCxy
BuAxx
VDDHIBHBxCDHIBHAHyVBAxx ])([])([)( 11
DVDHICxDHIy11 )()(
Chapter 7
State feedback system• A linear system:
• Then
5
DuCxy
BuAxx
Let: KxVu
DVxDKCy )(
BVxBKAKxVBAxx )()(
Kx - State feedback
V – input of system
u – output of controller/input of plant
Chapter 7
7.2 Controllability of Feedback system
• Theorem: The system (A-BK, B) is controllable IFF system
(A,B) is controllable.
• Note: The observability may be changed.
6
Linear system:
Cxy
BuAxx
State feedback
n
i
iin xkrxkkr-Kxru1
1
Cxy
BrBK)x(AxFeedback system
Chapter 7
• Example:
• Let
• Feedback system
• Controllability matrix controllable
• Observability matrix un-observable
•
7
xy
uxx
21
1
0
13
21
xru 13
xy
uxx
21
1
0
00
21
01
20fC
21
21fO
Chapter 7
7.3 Pole placement and design of
feedback matrix
• State feedback can be used to place eigenvalues(pole)of
A in any position if the system is controllable.
• Pole placement: choosing state feedback gains to place
poles in desired positions.
• Theorem: If the n-dimensional linear system is
controllable, then the eigenvalues of A-BK can arbitrarily
assigned by state feedback u=r-Kx, where k is a 1n real
constant vector.
8Chapter 7
9Chapter 7
10Chapter 7
Example:
• Consider a linear system
11
uxx
0
1
13
31
2
1 2( ) ( 1) 9 ( 4)( 2), 4, 2s s s s s s
rxkk
uxkk
x
0
1
13
31
0
1
0013
31 2121
)kk(s)k(s)s( 832 211
2
State feedback
n
i
iin xkrxkkV-Kxru1
1
New s1 s2 is determined by values of k1 and k2.
Chapter 7
• If linear system is controllable, then it can be transfer to
canonical controllable form by linear transformation:
• Transfer function:
12
u
aaa n
1
0
0
1000
0100
0010
110
xx
x110 nβββy
xPx1
α(s)
β(s)
asasas
βsβsβsβ
b]A[sICbA]C[sIg(s)
n
n-
n
n
n-
n
n-
01
1
1
01
2
2
1
1
11
Chapter 7
• Let state feedback
• K is unknown gains.
• Then
13
xKrxKPrKxru 1
110
1
nkkk KPK
)k(a)k(a)k(a
kkk
aaa
KbA
nn
n
n
111100
110
110
10
010
1
0
0
10
010
)k(a)sk(a)sk(as)]KbA([sIdet(s)Δn
nn
n
K 0011
1
11
Chapter 7
• For feedback system
• Assume that desired eigenvalues are s1, s2, …, sn
Feedback gains after transformation
Feedback gains or feedback matrix
14
)k(a)sk(a)sk(as)]KbA([sIdet(s)Δn
nn
n
K 0011
1
11
*
0
*
1
1*
1
1
* )()(Δ asasassss n
n
nn
i
iK
1
*
11
*
10
*
0 nn aaaaaa K
110 nkkk PKK
Chapter 7
Example: Consider a linear system
15
uxx
0
1
13
31
242491 21
2 s,s),s)(s()s()s(
rxkk
uxkk
x
0
1
13
31
0
1
0013
31 2121
)kk(s)k(s)s( 832 211
2
State feedback
n
i
iin vkrxkkV-Kxru1
1
New s1 s2 are determined by values of k1 and k2.
Suppose desired eigenvalue is -1, -2. Then
2321 2 ss)s)(s()s(*
283
32
21
1
kk
k
5
5
2
1
k
k
Chapter 7
16
A cart with inverted pendulum:
;0
1
000
1000
000
0010
1
1
4
3
2
1
)(
4
3
2
1
u
x
x
x
x
x
x
x
x
Ml
M
Ml
gmM
M
mg
4
3
2
1
0001
x
x
x
x
y
Chapter 7
• Exercise: for a cart with inverted pendulum system
• Assume the desired eigenvalues
• Try to find the feedback gain matrix K.
17
;u
x
x
x
x
x
x
x
x
2
0
1
1
0500
1000
0100
0010
4
3
2
1
4
3
2
1
4
3
2
1
0001
x
x
x
x
y
1 2 3 411, 12, 13, 14s s s s
Chapter 7
• Exercise:
18
;u
x
x
x
x
x
x
x
x
2
0
1
1
0500
1000
0100
0010
4
3
2
1
4
3
2
1
4
3
2
1
0001
x
x
x
x
y
00505 23422 ssss)s(s)s(
)s)(s)(s)(s()s(* 14131211
Chapter 7
• Note: State feedback can shift the poles of a plant but has
no effect on the zeros. This can be used to explain why a
state feedback may alter the observability property of a
state equation.
19Chapter 7
Control system design using state feedback
A rough guide for system design
• Place all eigenvalues inside the
region denoted by G in the right
figure
• Better to place all eigenvalues
evenly around a circle with radius r
inside the sector as shown
• A final selection may involve
compromises among many
conflicting requirements
20Chapter 7
7.4 State estimator
• Why do we need state estimator or state observer?
– State feedback requires real-time values of state variables
– State variables may not be accessible for direct connection
in practice
– Sensing devices or transducers may be unavailable or very
expensive
21Chapter 7
• Why we can estimate state variable?
State vector x(t) may be estimated from u and y over any
time interval [t, t+t1].
22Chapter 7
23
Open-loop estimator
Closed-loop estimator
Chapter 7
How to estimate state variables?
• If the states of the linear system can not be measured
directly, alternative approach is to construct an equivalent
system and measure the states of equivalent system
instead.
24Chapter 7
25
closed-loop estimator
Chapter 7
• The original system
• Design a system with same parameters
• Then
• Let
• Then
26
)x()x(t,Cxy
BuAxx0
0
xCy
BuxAx
ˆˆ
ˆ
)ˆ(ˆ
)ˆ(ˆ
xxCyy
xxAxx
lyBuxlC)(A
)xlC(xBuxA)yl(yBuxAx
)xlC)(x-(Aly]BuxlC)[(ABuAxx-xe
Chapter 7
27
State
estimator
State
estimator
Chapter 7
)yl(y
a) Can be
rewrote as b)
• If (A-lC) is stable, i.e. all the eigenvalues of (A-lC) have
negative real parts, then
• So the states of equivalent system can be used for state
feedback of original system.
• Theorem: All eigenvalues of (A-lC) can be assigned
arbitrary by selecting a real constant vector l if and only if
(A, C) is observable.
28Chapter 7
0)ˆ(lim
xxt
lC)e(A
)xlC)(x-(Aly]BuxlC)[(ABuAxx-xe
• Example: Design a state estimator with desired
eigenvalues as -3,-4,-5 for the following system.
Solution:
(1) observability of the system
It is observable, then the state estimator can be assigned
eigenvalues.
29
u
1
0
1
200
120
001
xx
x011y
441
121
011
OQ 3OQrank
Chapter 7
(2) Design of state estimator
Let
Then for desired eigenvalues
The state estimator
The vector is
30
2
1
0
l
l
l
L
604712)5)(4)(3()(Δ 23* sssssssG
)()()(
detΔ
4248345 210210
2
10
3
lllslllslls
LCAsIG
210
103
120
2
1
0
l
l
l
L
60424
47834
125
210
210
10
lll
lll
ll
Chapter 7
7.5 Reduced-dimensional state estimator
(optional)• Theorem: If (A, C) is observable, and rank(C)=m<n, then
the minimal dimension of state estimator is (n-m).
• m states can be observed from outputs, so we only
estimate the rest (n-m) states.
31
21 CCC
mCrank
Chapter 7
• If the following system is observable
• Let transformation matrix
• After transform
• Only need to be estimated. The estimator is called
reduced-dimensional state estimator.
32
Pxx
1 PAPA
PBB 1CPC
Cxy
BuAxx
21
0
CC
IP
2
2
1
2
1
2
1
2221
1211
2
1
0 xx
xIy
uB
B
x
x
AA
AA
x
x
1x
Chapter 7
• Rewrite the above system as
• Let
• Subsystem
• Design the state estimator as
33
2
2
1
2
1
2
1
2221
1211
2
1
0 xx
xIy
uB
B
x
x
AA
AA
x
x
2221212
11211112121111
uByAxAyx
uByAxAuBxAxAx
xAy
u)ByA(xAx
121
1121111
yGyAGAuBGBxAGA
yGuByAxAGAx
122112211121111
11121211111
)()(ˆ)(
)(ˆ)(ˆ
121222 xAuByAyy
Chapter 7
• Let
• Then
• So the estimate of is .
• and
• Where
34
yGxz 11
ˆ
yGxz 11ˆ
yGzx 11
ˆ
yGzx 11ˆ
yAGAGAGAuBGBzAGAz ])[()()( 2211212111121121111
2
1
x
xx 0
0
ˆlimlim 111
xx
y
yGzx
tt
y
yGz 1x
y
yGzQQxQxPx
1
21
1
])[(~
221121211111 AGAGAGAG
Chapter 7
35Chapter 7
7.6 State feedback from estimated states
36
State
estimator
Chapter 7
• Linear system
• State estimator
• Choose state feedback control
• Then the system equations are
37
Cxy
buAxx
xK ˆVu
bVxbK)GC(AlCxx VbxbKAxx ˆ
Cxy
Vb
b
x
x
bKlCAlC
bKA
x
x
x
xC
ˆ0y
Chapter 7
State
estimator
GybuxlC)(Ax
• Equivalent transformation
• Then
• The poles of system are
• Eigenvalues of (A-bK) and (A-lC) can be assigned
independently.
38
II
IP
0
II
IP
01
Vb
x~x
lCA
bKbKA
Vb
b
II
I
x~x
II
I
bKlCAlC
bKA
II
I
x~x
00
000
x
xC
x
x
II
IC ~0
ˆ
00y
lC)A(sIdetbK)A(sIdetlCAsI
bKbKAsIdet
0
Chapter 7
Transfer function
Note:
• The transfer function of the above system equals the transfer
function of the original state feedback system without using a
state estimator.
39
bbKAsICb
lCAsI
bKbKAsIC(s)gK
1
1
000
Chapter 7
Remark:
1. The eigenvalues are the union of those of A-bk and A-lc.
2. Inserting the state estimator does not affect the
eigenvalues of the original state feedback; nor are the
eigenvalues of the state estimator affected by the
connection.
3. The design of state feedback and the design of state
estimator can be carried out independently. This is
called the separation property.
4. Better to choose bigger-magnitude eigenvalues of (A-
lC) than those of (A-bK) to make the convergences of
state estimator faster than the state feedback control
system.
40Chapter 7
MATLA B
Homework:
• Learn and practice functions related to :
1. Pole placement
2. Computation of state feedback matrix
3. State estimator
41Chapter 7
Summary
• State feedback
• Determination of feedback gains
• State estimator
• Design of state estimators
• State feedback with state estimator
42Chapter 7