Question

The transfer function of a system is: \[ \frac{(s + 1)(s + 3)}{(s + 5)(s + 7)(s + 9)}. \] In the state-space representation of the system, the minimum number of state variables (in integer) necessary is _________.

Hint:

The number of state variables in a state-space representation is equal to the number of poles in the transfer function.

Answer 1

Correct Answer: 3

We are given the transfer function: \[ \frac{(s + 1)(s + 3)}{(s + 5)(s + 7)(s + 9)}. \] To find the minimum number of state variables required in the state-space representation, we need to look at the number of poles in the transfer function. Step 1: Identify the poles of the transfer function
The poles are the values of \( s \) that make the denominator of the transfer function equal to zero. From the denominator \( (s + 5)(s + 7)(s + 9) \), we can see that the poles are: \[ s = -5, \, s = -7, \, s = -9. \] Step 2: Minimum number of state variables
In state-space representation, the minimum number of state variables required is equal to the number of poles in the system, as each pole corresponds to a state variable. Since there are 3 poles (\( s = -5, -7, -9 \)), the minimum number of state variables required is \( 3 \).