Page 66 - Servo Motors and Industrial Control Theory -

58 3 State Variable Feedback Control Theory

It can be seen that the open loop system is stable but the complex roots are un-
derdamped. The gain vector must be selected such that to move all roots to de-
sired position on s-plane. For this, the practical limitations such saturations, the
limit of power amplifier, and nonlinearity of the system must be considered. Some
practically achievable locations must be assumed for the eigenvalues. This can be
achieved by trial and error strategy. A feedback strategy such as,

x 1
u:= ( 1 k 2 k 3 ) x⋅ 2
k
x 3

Substituting the above feedback to the original system, assuming that u = 0 gives
i
x 0 1 0 x 0 x
1
1
d x:= 0 0 1 ⋅ x +⋅ K K 1
0
dt 2 2 (K 1 2 3 ) x⋅ 2 (3.20)
x 3 − 1 − 3 − 2x 3 1 x 3
which gives the system matrix as
0 1 0
0 0 1 (3.21)
−+ 1 −+ 2 −+ 3
1 K
3 K
2 K
Suppose it is required that the eigenvalues must be changed from the original under-
damped case to an overdamped values of − 2, − 3, − 4, respectively. Therefore, the
required characteristic equation becomes
2
(s + 2 )(s + 3 )(s + 4 ) : s= 3 + 9 s + 26 s + 24 (3.22)
Forming the dynamic matrix of the state variable feedback system gives
−s 1 0
0 −s 1 (3.23)
−+ K1 −+ K3 −+ K2 − s
1 2 3
Expanding the determinant of the matrix (3.23) gives

−− −+ss( 2 K 3 − s) − −+ K( 3 2 ) −− + K(11 1 ) : = s 3 +− + K s( 2 3 ) 2 + −+ K sK( 3 2 ) + 1 −1
Equating the above equation to zero gives the characteristic equation. Equating the
coefficients of s in the above equation with Eq. (3.22) gives
2 − K : = 9 K : =− 7
3 3
3 − K : = 26 K : =− 23
2 2
1− K : = 24 K : =− 23
1 1
Therefore, the gain can be calculated with the above method. The gains become
negative which shows that similar to the feedback control theory the state variables