GATE EE 2015 Set 1 — Question 62
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Control Systems → State-Space Analysis → Controllability & Observability
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Question
In the signal flow diagram given in the figure, and are possible inputs whereas and are possible outputs. When would the SISO system derived from this diagram be controllable and observable?
Correct answer
(B) When u₂ is the only input and y₁ is the only output.
Solution
From the signal flow graph, we define state variables and as the outputs of the integrators. The state equations are:
The system matrix is . The eigenvalues are .
The right eigenvector for is and the left eigenvector is .
The system matrix is . The eigenvalues are .
The right eigenvector for is and the left eigenvector is .
1.Controllability: The input matrix must not be a right eigenvector.
- For only, (assuming branches to both junctions), which is an eigenvector, making it uncontrollable.
- For only, ? No, looking closely at the diagram, only affects the second junction if is the primary input. If is the only input, , which is not an eigenvector, so it is controllable.
- For , , which is not the left eigenvector , so it is observable.
- For , , which is the left eigenvector, making it unobservable.
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