Section 2–5 / State-Space Representation of Scalar Differential Equation Systems 35

Note that

(Refer to Appendix C for the inverse of the 2  2 matrix.)

Thus, we have

which is the transfer function of the system. The same transfer function can be obtained from

Equation (2–16).

Transfer Matrix. Next, consider a multiple-input, multiple-output system. Assume

that there are rinputs and moutputs Define

The transfer matrix G(s)relates the output Y(s)to the input U(s),or

whereG(s)is given by

[The derivation for this equation is the same as that for Equation (2–29).] Since the

input vector uisrdimensional and the output vector yismdimensional, the transfer ma-

trixG(s) is an m*rmatrix.

2–5 STATE-SPACE REPRESENTATION OF SCALAR

DIFFERENTIAL EQUATION SYSTEMS

A dynamic system consisting of a finite number of lumped elements may be described

by ordinary differential equations in which time is the independent variable. By use of

vector-matrix notation, an nth-order differential equation may be expressed by a first-

order vector-matrix differential equation. If nelements of the vector are a set of state

variables, then the vector-matrix differential equation is a stateequation. In this section

we shall present methods for obtaining state-space representations of continuous-time

systems.

G(s)=C(s I-A)-^1 B+D

Y(s)=G(s )U(s )

y= F

y 1

y 2







ym

V, u=F

u 1

u 2







ur

V

u 1 ,u 2 ,p,ur , y 1 ,y 2 ,p,ym.

=

1

ms^2 +bs+k

G(s)=[1 0]

1

s^2 +

b

m

s+

k

m

D

s+

b

m

• 
  

k

m

1

s

T

C

0

1

m

S

C

s

k

m

- 1

s+

b

m

S

• 
  

  1

  
  

1

s^2 +

b

m

s+

k

m

D

s+

b

m

• 
  

k

m

1

s

T