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Signals and Systems by Smarajit Ghosh

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9.8.3 Properties of STM, ϕ(t)

  1. ϕ(0) = eA0 = I
  2. ϕ(t) = eAt = (eAt)−1 = [ϕ(−t)]−1
  3. ϕ−1(t) = ϕ(−t)
  4. ϕ(t1 + t2) = e(t1+t2) = eAt1eAt2 = ϕ(t1)ϕ(t2) = eAt2eAt1 = ϕ(t2)ϕ(t1) t1 + t2
  5. [ϕ(t)]n = (eAt)n = eAnt = ϕ(nt)
  6. 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

 

image

 

The initial state is image

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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