GATE EC 2021 Set 1 — Question 43
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Control Systems → State-Space Analysis → Controllability & Observability
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Question
The electrical system shown in the figure converts input source current to output voltage .
Current in the inductor and voltage across the capacitor are taken as the state variables, both assumed to be initially equal to zero, i.e., and . The system is
Correct answer
(D) neither state controllable nor observable
Solution
Let the state variables be and . The input is and the output is .From the circuit, the current source drives the series combination of the two parallel blocks. Thus, the total current entering each block is .For the first block (Inductor H and Resistor in parallel):
For the second block (Capacitor F and Resistor in parallel):
The state-space representation is:Controllability:
The controllability matrix is .
Since , the rank of is 1 (less than 2). Thus, the system is not controllable.Observability:
The observability matrix is .
Since , the rank of is 1 (less than 2). Thus, the system is not observable.Therefore, the system is neither state controllable nor observable.
For the second block (Capacitor F and Resistor in parallel):
The state-space representation is:Controllability:
The controllability matrix is .
Since , the rank of is 1 (less than 2). Thus, the system is not controllable.Observability:
The observability matrix is .
Since , the rank of is 1 (less than 2). Thus, the system is not observable.Therefore, the system is neither state controllable nor observable.
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