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GATE 2015 EE – Question 62

Control Systems · State space model, Solution of state equations of LTI systems · 2 marks · Multiple choice

In the signal flow diagram given in the figure, $u_1$ and $u_2$ are possible inputs whereas $y_1$ and $y_2$ are possible outputs. When would the SISO system derived from this diagram be controllable and observable?

A signal flow diagram with two integrators $\frac{1}{s}$ giving states $x_1$ and $x_2$, interconnected by gains 5, $-2$, 2 and 1, with inputs $u_1$, $u_2$ and outputs $y_1$, $y_2$.
  1. When $u_1$ is the only input and $y_1$ is the only output.
  2. When $u_2$ is the only input and $y_1$ is the only output.
  3. When $u_1$ is the only input and $y_2$ is the only output.
  4. When $u_2$ is the only input and $y_2$ is the only output.

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Show answer and explanation

Correct answer: (B) When $u_2$ is the only input and $y_1$ is the only output.

Explanation

Writing the state equations from the diagram, a single input and a single output give a system that is both controllable and observable only for one pairing. For the pairing of input $u_2$ with output $y_1$, the controllability and observability matrices both have full rank, which matches the official key. The other pairings lose controllability or observability.