Page 59 - Servo Motors and Industrial Control Theory

3.2 State Variables 51

It can be seen that by defining three state variables, the transfer function is con-
verted to three first order differential equations. Various methods can now be used
to solve the above set of first order differential equations. The above process can
be repeated for any transfer function regardless of its order. The number of state
variables equals the order of the transfer function. The above set of equation can be
written in matrix form as

x 0 1 0x 0
1
d x:= 1 x⋅ 1 + 0 ⋅
dt 2 0 0 2 u
x − 2K − 2 − 3 x 2K (3.5)
3 3
x 1
y : (1 0 0) x= ⋅ 2
x 3

It should be noted that the matrices are printed without the usual bracket form.
In the above transfer function, there was one input variable ( u) and one output
variable ( y). The matrix equation of (3.5) may be written in compact form of
d
dt X := AXBU+ (3.6)
y := CX
In the above equation, X is the vector containing n state variables and its dimension
is n × 1. A is called the system matrix and its dimension is n × n. B is called the input
matrix and its dimension is n × m, where m is the number of input variables. In the
above example, there was only one input variable. In servo control systems, there
are generally two input variables, one is the command signal and the other one is
external torque or force applied to the system. The second equation is the output
equation where Y is the output variables and its dimension is l × 1 and C is the output
matrix and its dimension is l × n. In servo control systems, there is generally one
output variable of output position.
If the numerator of the transfer function is a polynomial in s, then the transforma-
tion of the transfer function to state equation is slightly different. Without loss of
generality, the procedure will be explained with one example. Consider the follow-
ing transfer function,
2
y := (5 s + 2 s + 2 ) (3.7)
3
2
u s + 6 s + 9 s + 3
For real systems, usually the order of the numerator is smaller than the order of
denominator. First, consider the following transfer function which includes on the
denominator
w := 1 (3.8)
2
u s + 6 s + 9 s + 3
3