Page 65 - Servo Motors and Industrial Control Theory -

3.4 State Variable Feedback Control Theory 57

d X := AXBKX Bu
+
+
dt i (3.19)
d X :(= A BK)X Bu
+
+
dt i
In Eq. (3.19), A is the system matrix, B is the input vector with dimension n × 1
which is a column vector and K is the gain vector with dimension 1 × n and X is
the state variables. In Eq. (3.19) it should be possible to choose the gain so that the
roots of characteristic equation are moved to the desired location on the s-plane. To
do this first it is assumed that u = 0 and the eigenvalue problem as discussed earlier
i
will be studied. Without loss of generality, a third order transfer function will be
considered. Suppose,
y := 1
2
3
u s + 2 s + 3 s +1
The state variables are defined as
x := y
1
d
x := x
2
dt 1
d
x := x 2
3
dt
It should be noted that the operator s and derivative can be used interchangeably
assuming that the initial conditions are zero.
The state equation with some algebraic manipulation becomes
x 0 1 0 x 0
1
1
d x:= 0 0 1 ⋅ x +⋅
0 u
dt 2 2
x 3 − 1 − 3 − 2x 3 1
x 1
y : (1 0 0) x= ⋅ 2
x 3

Now the object is to choose a gain vector and all the state variables are fed back to
the input summation junction. It would be interesting to the eigenvalues (roots of
characteristic equation) of the open loop state equation. MathCad program software
is used for this purpose as follows:
S:= eigenvals A( )

−043.
S = −0 785. +1 307. i
−0 785. −1 307. i