December 2005
Intermediate to advanced
475 pages
12h 6m
English
Content preview from Signals and Systems
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9.8.3 Properties of STM, ϕ(t)
- ϕ(0) = eA0 = I
- ϕ(t) = eAt = (e−At)−1 = [ϕ(−t)]−1
- ϕ−1(t) = ϕ(−t)
- ϕ(t1 + t2) = e(t1+t2) = eAt1eAt2 = ϕ(t1)ϕ(t2) = eAt2eAt1 = ϕ(t2)ϕ(t1) t1 + t2
- [ϕ(t)]n = (eAt)n = eAnt = ϕ(nt)
- ϕ(t1 - t2)ϕ(t2 - t0) = eA(t1 - t2)eA(t2 - t0)
= e(t1 - t2 + t2 - t0) = ϕ(t1 - t0)
Example 9.11 A linear time invariant system is characterized by the homogeneous state equation
The initial state is 
Find the resolvent matrix ϕ(s), state transition matrix ϕ(t) and ϕ−1(t) and the solution of the given equation.
Solution
This is an homogeneous ...
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