GATE 2022 IN Q.30: solution and answer

Official answer: 3Source: Official answer key, GATE 2022 IN (IIT Kharagpur)

1. A minimal state-space realization has as many states as the order of the transfer function once it is fully reduced. That order is the degree of the denominator after every common pole-zero factor has been cancelled.

2. Check for cancellation. The zeros are at $s=-1,-3$ and the poles are at $s=-5,-7,-9$. No zero coincides with a pole, so no factor cancels and $G(s)$ is already irreducible.

3. Expand the denominator: $(s+5)(s+7)=s^2+12s+35$, and $(s^2+12s+35)(s+9)=s^3+21s^2+143s+315$. Its degree is $3$.

4. Build one realization to confirm. The numerator is $(s+1)(s+3)=s^2+4s+3$, so $G(s)=\frac{s^2+4s+3}{s^3+21s^2+143s+315}$ and the controllable canonical form is $$A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -315& -143& -21\end{array}\right],B=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],C=\left[\begin{array}{ccc}3& 4& 1\end{array}\right],D=0$$

5. This realization is controllable by construction. For a single-input single-output system, the controllable canonical form is also observable exactly when the numerator and denominator share no common factor. That holds here, so the realization is minimal. No realization can use fewer states, and any realization with more states would have a mode that is uncontrollable or unobservable.

6. Minimum number of state variables $=3$.