Signals and Systems

9.8.3 Properties of STM, ϕ(t)

1. ϕ(0) = e^A0 = I
2. ϕ(t) = eAt = (e−At)⁻¹ = [ϕ(−t)]⁻¹
3. ϕ⁻¹(t) = ϕ(−t)
4. ϕ(t₁ + t₂) = e^(t₁+t₂) = e^At₁e^At₂ = ϕ(t₁)ϕ(t₂) = e^At₂e^At₁ = ϕ(t₂)ϕ(t₁) t₁ + t₂
5. [ϕ(t)]ⁿ = (e^At)ⁿ = e^Ant = ϕ(nt)
6. ϕ(t₁ - t₂)ϕ(t₂ - t₀) = e^{A(t₁ - t₂)}e^{A(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 ...