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LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 509
Fig. 5.40 Schematic diagram of state-space representation.
(t)
In other words, given the initial conditions x(0), the state transition matrix enables
us to predict the state vector at t > O.
Substituting for x(t) from Eq. (5.178) into state Eq. (5.176), we obtain
<P(t)x(0) - AQ(t)x(0) (5.179)
[<i>(t) - Aq>(t)]x(0) = 0 (5.180)
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
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