664 Chapter 9 / Control Systems Analysis in State Space(The inverse Laplace transform of a matrix is the matrix consisting of the inverse Laplace
transforms of all elements.) From Equations (9–35) and (9–36), the solution of Equation
(9–34) is obtained as
The importance of Equation (9–36) lies in the fact that it provides a convenient
means for finding the closed solution for the matrix exponential.
State-Transition Matrix. We can write the solution of the homogeneous state
equation
(9–37)
as
(9–38)
where is an n*nmatrix and is the unique solution of
To verify this, note that
and
We thus confirm that Equation (9–38) is the solution of Equation (9–37).
From Equations (9–31), (9–35), and (9–38), we obtain
Note that
From Equation (9–38), we see that the solution of Equation (9–37) is simply a
transformation of the initial condition. Hence, the unique matrix is called the state-
transition matrix. The state-transition matrix contains all the information about the free
motions of the system defined by Equation (9–37).
If the eigenvalues l 1 ,l 2 ,p, lnof the matrix Aare distinct, than will contain
thenexponentials
In particular, if the matrix Ais diagonal, then
(t)=eAt= F
el^1 t
0
el^2 t
0
eln^ t
V (A: diagonal)
el^1 t, el^2 t,p,eln^ t
(t)
(t)
-^1 (t)=e-^ At=(-t)
(t)=eAt=l-^1 C(s I-A)-^1 D
x
(t)=
(t) x(0)=A(t) x(0)=Ax(t)
x(0)=(0) x(0)=x(0)
(t)=A(t), (0)=I
(t)
x(t)=(t) x(0)
x
=Ax
x(t)=eAt x(0)
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