Question

The electrical system shown in the figure converts input source current \( i_s(t) \) to output voltage \( v_o(t) \). \[ \text{Current } i_L(t) \text{ in the inductor and voltage } v_C(t) \text{ across the capacitor are taken as the state variables, both assumed to be initially equal to zero, i.e., } i_L(0) = 0 \text{ and } v_C(0) = 0. \text{ The system is} \]

• A. completely state controllable as well as completely observable
• B. completely state controllable but not observable
• C. completely observable but not state controllable
• D. neither state controllable nor observable

Hint:

For electrical systems with coupled inductors and capacitors, check the controllability and observability matrices to determine if the system is controllable and observable.

Answer 1

The Correct Option is D

In this electrical system, the state variables are \( i_L(t) \) and \( v_C(t) \). The system's behavior involves an inductor and a capacitor, which are coupled, and the input and output are related in a way that does not allow both complete state controllability and observability. Analyzing the system’s dynamics, we find that the system is neither completely state controllable nor observable because there is not enough independent information to control both state variables or observe them fully from the output. 
Step 1: For a system to be state controllable, the controllability matrix must be full rank, and for observability, the observability matrix must also be full rank. In this case, due to the system's structure and coupling between the inductor and capacitor, neither condition is satisfied. 
Step 2: Hence, the correct answer is option (D), "neither state controllable nor observable." 
Final Answer: (D) neither state controllable nor observable