Modern Control Engineering

Example Problems and Solutions 707

Noting that x(t)can also be given by the equation

we obtain or

(9–101)

Next, we shall consider the case where matrix Amay be transformed into a Jordan canonical
form. Consider again the state equation

First obtain a transformation matrix Sthat will transform matrix Ainto a Jordan canonical form
so that

whereJis a matrix in a Jordan canonical form. Now define

Then

The solution of this last equation is

Hence,

Since the solution x(t)can also be given by the equation

we obtain

Note that eJtis a triangular matrix [which means that the elements below (or above, as the case
may be) the principal diagonal line are zeros] whose elements are elt,telt, , and so forth. For
example, if matrix Jhas the following Jordan canonical form:

then

eJt= C

el^1 t

0

0

tel^1 t

el^1 t

0

1

2 t

(^2) el 1 t
tel^1 t
el^1 t
S
J=C
l 1
0
0

1

l 1

0

0

1

l 1

S

1

2 t

(^2) elt
eAt=SeJt S-^1
x(t)=eAt x(0)

x(t)=Sxˆ(t)=SeJt S-^1 x(0)

xˆ(t)=eJt xˆ(0)

xˆ

=S-^1 AS xˆ=Jxˆ

x=Sxˆ

S-^1 AS=J

x# =Ax

eAt=PeDt P-^1 =PF

el^1 t

0

el^2 t







0

eln^ t

VP-^1

eAt=PeDt P-^1 ,

x(t)=eAt x(0)