Question

Consider a system \(S\) represented in state space as: \[ \frac{dx}{dt} = \begin{bmatrix} 0 & -2
1 & -3 \end{bmatrix} x + \begin{bmatrix} 1
0 \end{bmatrix} r, \quad y = \begin{bmatrix} 2 & -5 \end{bmatrix} x. \] Which of the state space representations given below has/have the same transfer function as that of \(S\)?

• A. \(\frac{dx}{dt} = \begin{bmatrix} 0 & 1
  -2 & -3 \end{bmatrix} x + \begin{bmatrix} 0
  1 \end{bmatrix} r, \quad y = \begin{bmatrix} 1 & 2 \end{bmatrix} x.\)
• B. \(\frac{dx}{dt} = \begin{bmatrix} 0 & 1
  -2 & -3 \end{bmatrix} x + \begin{bmatrix} 1
  0 \end{bmatrix} r, \quad y = \begin{bmatrix} 0 & 2 \end{bmatrix} x.\)
• C. \(\frac{dx}{dt} = \begin{bmatrix} -1 & 0
  0 & -2 \end{bmatrix} x + \begin{bmatrix} -1
  3 \end{bmatrix} r, \quad y = \begin{bmatrix} 1 & 1 \end{bmatrix} x.\)
• D. \(\frac{dx}{dt} = \begin{bmatrix} -1 & 0
  0 & -2 \end{bmatrix} x + \begin{bmatrix} 1
  1 \end{bmatrix} r, \quad y = \begin{bmatrix} 1 & 2 \end{bmatrix} x.\)

Hint:

State-space representations are equivalent if a similarity transformation exists between their matrices, ensuring identical transfer functions. Verify by checking the existence of a transformation matrix \(T\).

Answer 1

The Correct Option is A

Step 1: State-space equivalence condition.
Two systems are considered equivalent in terms of their transfer function if their state-space representations are related through a similarity transformation. This means there exists a transformation matrix \(T\) such that the state-space matrices satisfy: \[ A' = TAT^{-1}, \, B' = TB, \, C' = CT^{-1}, \, D' = D. \] Step 2: Analyze the given options.
- For options (1) and (3), it can be confirmed that a transformation matrix \(T\) exists, satisfying the similarity transformation criteria. This ensures the transfer functions for these systems are identical. - For options (2) and (4), no valid transformation matrix \(T\) exists to relate the state matrices. Hence, the transfer functions for these systems differ. Step 3: Verify the transfer functions.
Using the similarity transformation, the transfer function equivalence can be checked for each option. Only options (1) and (3) yield equivalent transfer functions. Final Answer: \[ \boxed{{(1, 3) }} \]