State-Variable Representation - Control of Color Imaging Systems: Analysis and Design - page 181

The output equation is given by

2
4

3
5

x 1 (t)
x 2 (t)
x 3 (t)

y(t) ¼ x 1 (t) ¼ ½ 100 (4 : 24)

Note that the states of a system are not unique. Any linear combinations of a set
of states can be used as a new set of states for a given system. This means that if the
N-dimensional vector x is a state of a dynamic system with state and output
equations given by

dx(t)
dt ¼ Ax(t)

þ Bu(t)

(4 : 25)

y(t) ¼ Cx(t) þ Du(t)

then the N-dimensional vector z is de ned as

z ¼ Px (4 : 26)

where P is a nonsingular N N transformation, which is another state vector for
the same system. The new state and output equations are given by

dz(t)
dt ¼ PAP 1 z(t)

þ PBu(t)

(4 : 27)

y(t) ¼ CPz(t) þ Du(t)

4.3.4 S TATE -S PACE E QUATIONS OF M ECHANICAL S YSTEMS
The state-space modeling of mechanical systems can be derived from the

rst
principles similar to the electrical circuits. The main equation in this case is Newton s

'

law of motion.

Example 4.3

As the first example, consider the ideal spring - mass system shown in Figure 4.4.

is law, F ¼ ma, where m is the mass, a is the acceleration,
and F is the overall forcing function with the assumption of linear spring, we have

Applying Newton

'

d 2 x(t)
dt 2

m ¼ u(t) kx(t) (4 : 28)

x = 0

x

k

u ( t )

m

FIGURE 4.4 Spring - mass system.