GATE EC 2014 Set 2 — Question 58
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Control Systems → State-Space Analysis → Controllability & Observability
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Question
Consider the state space system expressed by the signal flow diagram shown in the figure.
The corresponding system is
The corresponding system is
Correct answer
(A) always controllable
Solution
From the given signal flow diagram, we can write the state equations. Let the states be corresponding to the outputs of the integrators .From the diagram:
In matrix form, the state equation is:So, the state matrix and the input matrix .This system is in the controllable canonical form.
The controllability matrix is:
The determinant of is . Since , the system is always controllable, regardless of the values of .
In matrix form, the state equation is:So, the state matrix and the input matrix .This system is in the controllable canonical form.
1.Controllability: A system in controllable canonical form is always controllable. To verify, we can compute the controllability matrix .
The controllability matrix is:
The determinant of is . Since , the system is always controllable, regardless of the values of .
2.Observability: The output equation is . So, . The observability depends on the values of and . For example, if , the system is not observable. Thus, it is not always observable.
3.Stability: The stability of the system depends on the eigenvalues of the matrix , which are the roots of the characteristic equation . The characteristic equation is . The stability depends on the values of (e.g., using the Routh-Hurwitz criterion). It is not always stable or always unstable.
Based on the analysis, the system is always controllable.The final answer is Continue learning with Success Tracker
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