Question

The State Transition Matrix \( \phi(t) \) is given by:

• A. \( \mathcal{L}^{-1}[sI - A] \)
• B. \( \mathcal{L}^{-1}[(sI - A)^{-1}] \)
• C. \( [sI - A]^{-1} \)
• D. \( e^{-At} \)

Hint:

Remember: \( \phi(t) = e^{At} \).
In the Laplace domain, this is \( \Phi(s) = (sI - A)^{-1} \).
To get back to the time domain, always apply the inverse Laplace transform: \( \phi(t) = \mathcal{L}^{-1}[\Phi(s)] \).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the mathematical formula for the State Transition Matrix \( \phi(t) \) in linear state-space system analysis.
The state transition matrix is a matrix that maps the initial state vector at time \( t = 0 \) to the state vector at any future time \( t \) when the input is zero.

Step 2: Key Formula or Approach:
Consider the homogeneous state equation (where input \( u(t) = 0 \)):
\[ \dot{x}(t) = A x(t) \] Taking the Laplace transform on both sides:
\[ s X(s) - x(0) = A X(s) \] Rearranging the terms:
\[ (sI - A) X(s) = x(0) \] Multiplying both sides by the inverse matrix \( (sI - A)^{-1} \):
\[ X(s) = (sI - A)^{-1} x(0) \] Taking the inverse Laplace transform to return to the time domain:
\[ x(t) = \mathcal{L}^{-1}\big[(sI - A)^{-1}\big] x(0) \]

Step 3: Detailed Explanation:
The definition and derivation of the state transition matrix are detailed below:

• Definition of \( \phi(t) \):
- The state response can be written in terms of the state transition matrix \( \phi(t) \) as:
\[ x(t) = \phi(t) x(0) \] - Comparing this with the Laplace derivation, we find:
\[ \phi(t) = e^{At} = \mathcal{L}^{-1}\big[(sI - A)^{-1}\big] \]
• Properties of \( \phi(t) \):
- At \( t = 0 \), it becomes the identity matrix: \( \phi(0) = e^0 = I \).
- The inverse is given by changing the sign of time: \( \phi^{-1}(t) = \phi(-t) = e^{-At} \).
- It satisfies the transition property: \( \phi(t_1 + t_2) = \phi(t_1) \cdot \phi(t_2) \).

• Comparison of options:
- Option A lacks the matrix inversion.
- Option C is the state transition matrix in the s-domain, not the time domain.
- Option D has a negative sign in the exponent, which is incorrect as \( \phi(t) = e^{At} \).

Step 4: Final Answer:
The State Transition Matrix \( \phi(t) \) is mathematically defined as \( \mathcal{L}^{-1}[(sI - A)^{-1}] \).