Section 10–4 / Design of Servo Systems 745
Define
Then Equation (10–37) can be written as
(10–38)
where
(10–39)
Define a new (n+1)th-order error vector e(t)by
-vector
Then Equation (10–38) becomes
(10–40)
where
and Equation (10–39) becomes
(10–41)
where
The state error equation can be obtained by substituting Equation (10–41) into
Equation (10–40):
(10–42)
If the desired eigenvalues of matrix (that is, the desired closed-loop poles) are
specified as m 1 ,m 2 ,p,mn+1,then the state-feedback gain matrix Kand the integral
gain constant kIcan be determined by the pole-placement technique presented in Section
10–2, provided that the system defined by Equation (10–40) is completely state
controllable. Note that if the matrix
has rank n+1,then the system defined by Equation (10–40) is completely state
controllable. (See Problem A–10–12.)
B
A
- C
B
0
R
Aˆ -BˆKˆ
e# =AAˆ -BˆKˆBe
Kˆ =CK-kID
ue=-Kˆe
Aˆ = B
A
- C
0
0
R, Bˆ = B
B
0
R
e# =Aˆe+Bˆue
e(t)= B
xe(t)
je(t)
R =(n+1)
ue(t)=-Kxe(t)+kI je(t)
B
x
e(t)
j
e(t)
R = B
A
- C
0
0
RB
xe(t)
je(t)
R + B
B
0
Rue(t)