662 Chapter 9 / Control Systems Analysis in State SpaceBecause of the convergence of the infinite series the series can be
differentiated term by term to give
The matrix exponential has the property that
This can be proved as follows:
In particular, if s=–t,then
Thus, the inverse of is Since the inverse of always exists, is nonsingular.
It is very important to remember that
To prove this, note that
+
A^2 Bt^3
2!
+
AB^2 t^3
2!
+
B^3 t^3
3!
+p
=I+(A+B)t+
A^2 t^2
2!
+ABt^2 +
B^2 t^2
2!
+
A^3 t^3
3!
eAteBt= aI+At+
A^2 t^2
2!
+
A^3 t^3
3!
+pbaI+Bt+
B^2 t^2
2!
+
B^3 t^3
3!
+pb
e(A+B)t=I+(A+B)t+
(A+B)^2
2!
t^2 +
(A+B)^3
3!
t^3 +p
e(A+B)tZeAteBt, ifABZBA
e(A+B)t=eAteBt, ifAB=BA
eAt e-^ At. eAt eAt
eAte-^ At=e-^ AteAt=eA(t-t)=I
=eA(t+s)
= a
qk= 0Ak
(t+s)k
k!
= a
qk= 0Akaa
qi= 0tisk-i
i! (k-i)!
beAteAs= aa
qk= 0Aktk
k!
baa
qk= 0Aksk
k!
beA(t+s)=eAteAs
= cI+At+
A^2 t^2
2!
+p+
Ak-^1 tk-^1
(k-1)!
+pd A=eAt A
=AcI+At+
A^2 t^2
2!
+p+
Ak-^1 tk-^1
(k-1)!
+pd =AeAt
d
dt
eAt=A+A^2 t+
A^3 t^2
2!
+p+
Aktk-^1
(k-1)!
+p
g
qk= 0 A
ktkk!,
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