The electrical system shown in the figure converts input source current \( i_s(t) \) to output voltage \( v_o(t) \). 
\[ \text{Current } i_L(t) \text{ in the inductor and voltage } v_C(t) \text{ across the capacitor are taken as the state variables, both assumed to be initially equal to zero, i.e., } i_L(0) = 0 \text{ and } v_C(0) = 0. \text{ The system is} \]
The electrical system shown in the figure converts input source current \( i_s(t) \) to output voltage \( v_o(t) \). \[ \text{Current } i_L(t) \text{ in the inductor and voltage } v_C(t) \text{ across the capacitor are taken as the state variables, both assumed to be initially equal to zero, i.e., } i_L(0) = 0 \text{ and } v_C(0) = 0. \text{ The system is} \]
Show Hint
- completely state controllable as well as completely observable
- completely state controllable but not observable
- completely observable but not state controllable
- neither state controllable nor observable
The Correct Option is D
Solution and Explanation
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
Top Questions on Control Systems
The Block diagram for a control system is shown below:
- Bode diagram is a plot of (AR is amplitude ratio; $\omega$ is frequency factor and $\phi$ is phase angle):
- The transfer function of a PID controller is:
- The characteristic equation for the system is \[ s^3 + 9s^2 + 26s + 12(2 + K_c) = 0 \] Using Routh test, the value of Kc that will keep the system stable is:
- The Bode diagram stability criterion is applicable when:
Questions Asked in GATE EC exam
- Consider the function \( f: \mathbb{R} \to \mathbb{R} \), defined as \[ f(x) = 2x^3 - 3x^2 - 12x + 1. \] Which of the following statements is/are correct? (Here, \( \mathbb{R} \) is the set of real numbers.)
- GATE EC - 2025
- Calculus
Eight students (P, Q, R, S, T, U, V, and W) are playing musical chairs. The figure indicates their order of position at the start of the game. They play the game by moving forward in a circle in the clockwise direction.
After the 1st round, the 4th student behind P leaves the game.
After the 2nd round, the 5th student behind Q leaves the game.
After the 3rd round, the 3rd student behind V leaves the game.
After the 4th round, the 4th student behind U leaves the game.
Who all are left in the game after the 4th round?
Here are two analogous groups, Group-I and Group-II, that list words in their decreasing order of intensity. Identify the missing word in Group-II.
Abuse \( \rightarrow \) Insult \( \rightarrow \) Ridicule
__________ \( \rightarrow \) Praise \( \rightarrow \) Appreciate
Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable \( X \) denote the sum of the outcomes obtained. The expectation of \( X \) is _________ (rounded off to two decimal places).
- GATE EC - 2025
- Simple diode circuits
The 12 musical notes are given as \( C, C^\#, D, D^\#, E, F, F^\#, G, G^\#, A, A^\#, B \). Frequency of each note is \( \sqrt[12]{2} \) times the frequency of the previous note. If the frequency of the note C is 130.8 Hz, then the ratio of frequencies of notes F# and C is: