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If this identity is to hold for fdl arbitrary values of x(0), we must have
<P(t) - AtP(t) : 0 (5.181)
This shows that the state transition matrix q>(t) is a solution to the homogeneous
state Eq. (5.176).
Determination of state transition matrix. Take the Laplace transformation
of Eq. (5.176),
sx(s) - x(0) : Ax(s)
(5.182)
x(s) = (sl - A)-lx(0) (5.183)
Here, we assume that (sl - A)-] exists,i.e., (sl - A) is nonsingular. Then,
x(t) = L-l[(sl - A)-llx(0) (5.184)
for t > O. Comparing Eqs. (5.184) and (5.178), we get
q>(t) = L-l[(sl - A)-l] (5.185)
Let
x(t) = eArx(0) (5.186)
The matrix exponentialis given by
[ntn
eA' =I+At+ A2 +...+ A~+-- (5.187)
rr
j
L
s
t
;t
ain
a2n
a3n
N
.
ann
510 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
where I is the identity matrix. We note that Eq. (5.186) satisfies the homogeneous
state Eq. (5.176). Hence,
A2t2 A t
(p(t) = eAr = I+At + 2~ + " '+ r7r +- " (5.188)
Using this, the solution of the complete nonhomogeneous state Eq. (5.171) can be
expressed as1.3
and the output
x(t) = Q(t)x(0) + [,' <P(t - r)Bu(r)dr
(5.189)
y(t) = C [q>(,)x(0) + [r q>(t _ r)B.(,) d,] +Du (5.190)
The integral in Eq. (5.189) is the convolution integral, which was introduced
earlier in Eq. (5.39). The first term on the right-hand side of Eq. (5.189) represents
the solution to the homogeneous part of the state equation and gives the free (tran-
sien0 response. The second term represents Lhe forced response and is independent
of the initial conditions x(0).
Properties of state transition matrix. The state transition matrix cP(t) has
the following properties. The proof of these identities is left as an exercise to the
reader.
q>(0) -- 1
<p-l(t) = q>(-t)
<P(t2 - tI)Q(tl - t0) : <p(t2 - t0)
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