Question

Consider a system where \( x_1(t), x_2(t), \) and \( x_3(t) \) are three internal state signals and \( u(t) \) is the input signal. The differential equations governing the system are given by: \[ \frac{d}{dt} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} u(t). \] Which of the following statements is/are TRUE?

• A. The signals \( x_1(t), x_2(t), \) and \( x_3(t) \) are bounded for all bounded inputs.
• B. There exists a bounded input such that at least one of the signals \( x_1(t), x_2(t), \) and \( x_3(t) \) is unbounded.
• C. There exists a bounded input such that the signals \( x_1(t), x_2(t), \) and \( x_3(t) \) are unbounded.
• D. The signals \( x_1(t), x_2(t), \) and \( x_3(t) \) are unbounded for all bounded inputs.

Hint:

For systems with decoupled states or eigenvalues equal to zero, the response can become unbounded

Answer 1

The Correct Option is B

The given system is a linear system with a state-space representation. The matrix governing the system is: \[ A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix}. \] From this, we can analyze the system's behavior:

Step 1: Analysis of system stability.
The system is unstable due to the eigenvalue \( \lambda = 0 \) and \( \lambda = \pm 2 \), indicating that the system can become unbounded for certain inputs.

Step 2: Unbounded response for certain inputs.
The state \( x_3(t) \) can become unbounded for a bounded input \( u(t) \), since the third equation is decoupled and has a zero eigenvalue, which can lead to an unbounded solution.

Step 3: Conclusion.
Therefore, there exists a bounded input such that at least one of the signals \( x_1(t), x_2(t), x_3(t) \) becomes unbounded.

Thus, the correct answer is (B).