Modern Control Engineering

Section 9–4 / Solving the Time-Invariant State Equation 667

Integrating the preceding equation between 0 and tgives

or

(9–41)

Equation (9–41) can also be written as

(9–42)

where Equation (9–41) or (9–42) is the solution of Equation (9–40). The

solutionx(t)is clearly the sum of a term consisting of the transition of the initial state

and a term arising from the input vector.

Laplace Transform Approach to the Solution of Nonhomogeneous State

Equations. The solution of the nonhomogeneous state equation

can also be obtained by the Laplace transform approach. The Laplace transform of this

last equation yields

sX(s)-x(0)=AX(s)+BU(s)

or

(sI-A)X(s)=x(0)+BU(s)

Premultiplying both sides of this last equation by(sI-A)–1,we obtain

X(s)=(sI-A)–1x(0)+(sI-A)–1BU(s)

Using the relationship given by Equation (9–36) gives

X(s)=lCeAtDx(0)+lCeAtDBU(s)

The inverse Laplace transform of this last equation can be obtained by use of the

convolution integral as follows:

Solution in Terms of Thus far we have assumed the initial time to be zero.

If, however, the initial time is given by t 0 instead of 0,then the solution to Equation

(9–40) must be modified to

x(t)=eAAt-t^0 B xAt 0 B+ (9–43)

3

t

t 0

eA(t-t) Bu(t)dt

xAt 0 B.

x(t)=eAt x(0)+

3

t

0

eA(t-t) Bu(t)dt

x# =Ax+Bu

(t)=eAt.

x(t)=(t) x(0)+

3

t

0

(t-t) Bu(t)dt

x(t)=eAt x(0)+

3

t

0

eA(t-t) Bu(t)dt

e-^ At x(t)-x(0)=

3

t

0

e-^ At Bu(t)dt