Question

The property of state transition matrix, $\Phi(t_1 + t_2) =$

• A. $\Phi(t_1) + \Phi(t_2)$
• B. $\Phi(t_1) \cdot \Phi(t_2)$
• C. $\Phi(t_1) - \Phi(t_2)$
• D. $\Phi(t_1) / \Phi(t_2)$

Hint:

The properties of the state transition matrix \(\Phi(t) = e^{At}\) are very similar to those of the scalar exponential function \(e^{at}\):
1. \(\Phi(0) = I\)
2. \(\Phi(t_1 + t_2) = \Phi(t_1)\Phi(t_2)\)
3. \(\Phi^{-1}(t) = \Phi(-t)\)

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the algebraic composition property (also called the group property or transition property) of the state transition matrix \(\Phi(t)\) in control systems.

Step 2: Key Formula or Approach:
For a linear time-invariant (LTI) system \(\dot{x}(t) = Ax(t)\), the state transition matrix is defined as:
\[ \Phi(t) = e^{At} \] We need to evaluate the expression for \(\Phi(t_1 + t_2)\).

Step 3: Detailed Explanation:

• Substitute \(t = t_1 + t_2\) into the definition of the state transition matrix:
\[ \Phi(t_1 + t_2) = e^{A(t_1 + t_2)} \]
• Using the properties of the matrix exponential (since the matrices \(At_1\) and \(At_2\) commute, as they are scalar multiples of the same matrix \(A\)):
\[ e^{A(t_1 + t_2)} = e^{At_1 + At_2} = e^{At_1} \cdot e^{At_2} \]
• Since \(\Phi(t_1) = e^{At_1}\) and \(\Phi(t_2) = e^{At_2}\), we have:
\[ \Phi(t_1 + t_2) = \Phi(t_1) \cdot \Phi(t_2) \]
• This property is known as the semi-group or transition property. It physically represents transitioning from the initial state to an intermediate state at \(t_1\), and then from \(t_1\) to \(t_1 + t_2\).

Step 4: Final Answer:
The property is \(\Phi(t_1 + t_2) = \Phi(t_1) \cdot \Phi(t_2)\).