Modern Control Engineering

694 Chapter 9 / Control Systems Analysis in State Space

The inverse Laplace transforms of these equations give

(9–87)

These nequations make up a state equation.

In terms of the state variables X 1 (s),X 2 (s),p,Xn(s), Equation (9–86) can be written as

The inverse Laplace transform of this last equation is

(9–88)

which is the output equation.

Equation (9–87) can be put in the vector-matrix equation as given by Equation (9–84). Equa-

tion (9–88) can be put in the form of Equation (9–85).

Figure 9–3 shows a block diagram representation of the system defined by Equations (9–84)

and (9–85).

It is noted that if we choose the state variables as

Xˆn(s)=

cn

s+pn

U(s)







Xˆ 2 (s)=

c 2

s+p 2

U(s)

Xˆ 1 (s)=

c 1

s+p 1

U(s)

y=c 1 x 1 +c 2 x 2 +p+cn xn+b 0 u

Y(s)=b 0 U(s)+c 1 X 1 (s)+c 2 X 2 (s)+p+cn Xn(s)

x#n=-pn xn+u







x# 2 =-p 2 x 2 +u

x# 1 =-p 1 x 1 +u

u y

xn

x 2

x 1

(^1) c 2
s+p 2
c 1
b 0
cn
1
s+p 1
... ...
1
s+pn

• 
  


• 
  


• 
  

  ++
  Figure 9–3
  Block diagram
  representation of the
  system defined by
  Equations (9–84)
  and (9–85) (diagonal
  canonical form).
  Openmirrors.com