Example Problems and Solutions 849
where
Referring to Equation (10–51), the error equation for the system is given by
where
and the control signal is given by Equation (10–41):
where
Using MATLAB, determine the state feedback gain matrix such that the following
performance index Jis minimized:
J=
3
q0(e* Qe+u* Ru)dtKˆ
x=D
x 1
x 2
x 3
x 4T = D
u
ux
xT
e=B
xe
jeR =B
x(t)-x(q)
j(t)-j(q)R
Kˆ =CK-kID=Ck 1 k 2 k 3 k 4 -kID
ue=-Kˆe
Aˆ =B
A
- C
0
0
R = E
0
20.601
0
- 0.4905
0
1
0
0
0
0
0
0
0
0
- 1
0
0
1
0
0
0
0
0
0
0
U, Bˆ = B
B
0
R= E
0
- 1
0
0.5
0
U
e# =Aˆe+Bˆue
A= D
0
20.601
0
- 0.4905
1
0
0
0
0
0
0
0
0
0
1
0
T , B=D
0
- 1
0
0.5
T, C=[ 0 0 1 0 ]