Section 9–4 / Solving the Time-Invariant State Equation 663
Hence,
The difference between and vanishes if AandBcommute.
Laplace Transform Approach to the Solution of Homogeneous State
Equations. Let us first consider the scalar case:
(9–32)
Taking the Laplace transform of Equation (9–32), we obtain
sX(s)-x(0)=aX(s) (9–33)
where Solving Equation (9–33) for X(s)gives
The inverse Laplace transform of this last equation gives the solution
x(t)=eatx(0)
The foregoing approach to the solution of the homogeneous scalar differential
equation can be extended to the homogeneous state equation:
(9–34)
Taking the Laplace transform of both sides of Equation (9–34), we obtain
sX(s)-x(0)=AX(s)
where Hence,
(sI-A)X(s)=x(0)
Premultiplying both sides of this last equation by (sI-A)–1,we obtain
X(s)=(sI-A)–1x(0)
The inverse Laplace transform of gives the solution Thus,
x(t)=l–1C(sI-A)–1Dx(0) (9–35)
Note that
Hence, the inverse Laplace transform of (sI-A)–1gives
l-^1 C(s I-A)-^1 D =I+At+ (9–36)
A^2 t^2
2!
+
A^3 t^3
3!
+p=eAt
(s I-A)-^1 =
I
s
+
A
s^2
+
A^2
s^3
+p
X(s) x(t).
X(s)=l[x].
x#(t)=Ax(t)
X(s)=
x(0)
s-a
=(s-a)-^1 x(0)
X(s)=l[x].
x
=ax
e(A+B)t eAteBt
+
BA^2 +ABA+B^2 A+BAB- 2 A^2 B- 2 AB^2
3!
t^3 +p
e(A+B)t-eAteBt=
BA-AB
2!
t^2
