672 Chapter 9 / Control Systems Analysis in State SpaceHence,Computation of eAt: Method 3. The third method is based on Sylvester’s interpo-
lation method. (For Sylvester’s interpolation formula, see Problem A–9–12.) We shall first
consider the case where the roots of the minimal polynomial of Aare distinct.
Then we shall deal with the case of multiple roots.
Case 1:Minimal Polynomial of AInvolves Only Distinct Roots. We shall assume
that the degree of the minimal polynomial of Aism.By using Sylvester’s interpolation
formula, it can be shown that can be obtained by solving the following determinant
equation:
(9–47)
By solving Equation (9–47) for can be obtained in terms of the Ak(k=0, 1,
2,p, m-1)and the (i=1, 2, 3,p,m). [Equation (9–47) may be expanded, for ex-
ample, about the last column.]
Notice that solving Equation (9–47) for is the same as writing
(9–48)
and determining the (k=0,1,2,p,m-1)by solving the following set of m
equations for the
IfAis an n*nmatrix and has distinct eigenvalues, then the number of to be
determined is m=n.IfAinvolves multiple eigenvalues, but its minimal polynomial has
only simple roots, however, then the number mof to be determined is less than n.
Case 2:Minimal Polynomial ofAInvolves Multiple Roots. As an example, consider
the case where the minimal polynomial of Ainvolves three equal roots
and has other roots that are all distinct. By applying Sylvester’s
interpolation formula, it can be shown that can be obtained from the following
determinant equation:
eAt
Al 4 , l 5 ,p, lmB
Al 1 =l 2 =l 3 B
ak(t)’s
ak(t)’s
a 0 (t)+a 1 (t)lm+a 2 (t)lm^2 +p+am- 1 (t)lmm-^1 =elm^ t
a 0 (t)+a 1 (t)l 2 +a 2 (t)l 22 +p+am- 1 (t)l 2 m-^1 =el^2 t
a 0 (t)+a 1 (t)l 1 +a 2 (t)l 12 +p+am- 1 (t)l 1 m-^1 =el^1 t
ak(t):
ak(t)
eAt=a 0 (t) I+a 1 (t) A+a 2 (t) A^2 +p+am- 1 (t) Am-^1
eAt
eli^ t
eAt,eAt
7
1 1 1 I
l 1
l 2
lm
A
l 12
l 22
lm^2
A^2
p
p
p
p
l 1 m-^1
l 2 m-^1
lmm-^1
Am-^1
el^1 t
el^2 t
elm^ t
eAt
7 = 0
eAt
f(l)
eAt=l-^1 C(s I-A)-^1 D= B
1
0
1
2 A^1 - e-2tB
e-2tR
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