State-Variable Representation - Control of Color Imaging Systems: Analysis and Design - page 179

Differentiating both sides of Equation 4.9 yields

dv C (t)
dt ¼

i L (t)
C ¼

1
C x 2 (t)

x 1 (t) ¼ (4 : 11)

Similarly, we differentiae both sides of Equation 4.10:

di L (t)
dt ¼

v L (t)
L ¼

1
L

1
L

x 2 (t) ¼ [u(t) v R (t) v C (t)] ¼ [u(t) Rx 2 (t) x 1 (t)]

u(t)
L

R
L x 2 (t)

1
L x 1 (t)

¼ (4 : 12)

The output equation is given by

y(t) ¼ x 1 (t) (4 : 13)

The above equations can be written in matrix form as

" # x 1 (t)

x 2 (t)

¼ þ

u(t)

1
C

x 1 (t)
x 2 (t)

0 0

1
L

1
L

R
L

(4 : 14)

x 1 (t)
x 2 (t)

y(t) ¼ ½ 10

Therefore, the A, B, C, and D matrices of the system are

, B ¼ , C ¼

1
C

0 L

A ¼ ½ 10 , and D ¼ 0 (4 : 15)

1
L

R
L

As it can be seen from this simple example, the number of states needed to
model an electric circuit in state-space form is equal to the total number of
independent energy-storage elements in the circuit. In the circuit shown in Figure
4.2, there are two elements that can store energy, the capacitor and the inductor;
hence, two states are needed. In Example 4.2, we consider an electric circuit with
three independent energy-storage elements.

Example 4.2

Consider the circuit shown in Figure 4.3.
The three states of the circuit are de ned as follows:

x 1 (t) ¼ v C 2 (t)

x 2 (t) ¼ v C 1 (t) (4 : 16)

x 3 (t) ¼ i L (t)