Modern Control Engineering

666 Chapter 9 / Control Systems Analysis in State Space

Hence,

Noting that we obtain the inverse of the state-transition matrix as follows:

Solution of Nonhomogeneous State Equations. We shall begin by considering

the scalar case

(9–39)

Let us rewrite Equation (9–39) as

Multiplying both sides of this equation by e–at,we obtain

Integrating this equation between 0 and tgives

or

The first term on the right-hand side is the response to the initial condition and the

second term is the response to the input u(t).

Let us now consider the nonhomogeneous state equation described by

(9–40)

where

By writing Equation (9–40) as

and premultiplying both sides of this equation by e–At,we obtain

e-^ AtCx

(t)-Ax(t)D=

d

dt

Ce-^ At x(t)D =e-^ At Bu(t)

x#(t)-Ax(t)=Bu(t)

B=n*r constant matrix

A=n*n constant matrix

u=r-vector

x=n-vector

x# =Ax+Bu

x(t)=eatx(0)+eat

3

t

0

e-atbu(t)dt

e-atx(t)-x(0)=

3

t

0

e-atbu(t)dt

e-atCx#(t)-ax(t)D=

d

dt

Ce-atx(t)D =e-atbu(t)

x# -ax=bu

x# =ax+bu

-^1 (t)=e-^ At= B

2et-e2t

• 2et+2e2t

et-e2t

• et+2e2t

R

-^1 (t)=(-t),

=B

2e-t-e-2t

• 2e-t+2e-2t

e-t-e-2t

• e-t+2e-2t

R

(t)=eAt=l-^1 C(s I-A)-^1 D

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