Page 58 - Servo Motors and Industrial Control Theory -

50 3 State Variable Feedback Control Theory

The state variables are not unique and different forms of state variables can be
defined. In modeling high-order system, some arrangements have to be made so
that the state variables are measurable for feedback. Effort must be made to mini-
mize the state variables that are not measurable and have to be predicted from an
observer.

3.2 State Variables

As was mentioned in the introduction, state variables are not unique and they can be
defined in various forms. It will be shown in the proceeding chapters and it is better
to define the state variables right from the beginning when the governing differen-
tial equations for various elements of a control system is derived.
One possible method is to define state variables from the transfer function. The
number of state variables must be equal to the order of the transfer function. Some
books show the derivative with respect to time by a dot above variable, that is, () x .
For clarity, the derivative with respect to time will be shown by the standard symbol
of d/dt .
Without loss of generality, a simple third order transfer function with numerator
as constant value is considered:
y K (3.1)
3
2
u := . 05 s +15 s + sK+
.
This transfer function was studied in the previous chapter. Equation (3.1) may be
written in differential form as
d 3 d 2 d 1
05. y + 15. y + yKy+ : Ku= (3.2)
dt 3 dt 2 dt 1
For the above differential equation, three state variables are defined as
x := y
1
d
x := y (3.3)
2
dt
d
x := x
3 2
dt
Three set of first order differential equations can be defined as
d
1
dt x := x 2
d x := x (3.4)
dt 2 3
d x :=− x 3 − 2 x − 2 Kx + 2 Ku
dt 3 3 2 1