system are given below: (input) i(t) = 2u(t) (output) v(oo) (in steady-state) can take any value between 2 to 4 Your cousin suspects the components used in the circuit to be faulty, i.e., their values might be changing over time (e.g., due to temperature at the time of experiment). When your cousin repeated the same experiment with a constant current input of i(t) = 2u(t), they found that the output voltage v(t) could settle down to any value between 2 and 4 during different test runs, i.e., reference R(S) a) Can you use Final Value Theorem to confirm that the resistor is faulty (e.g., temperature-sensitive), with its value varying between 1 and 2? b) Could you help your cousin achieve this by designing a simple proportional controller? feedback controller i(t) = i₁(t) + ₂(t) i2(t) = dv(t) dt G(s) diy(t) v(t) = in(t)R + L dt error input X(s) E(s)=R(S)-Y(s) K(s) output Y(s) Note: In the closed-loop system, X(s) = I(s) is the programmable current input, Y(s) = V(s) is the output voltage, R(s) = 3/s is the reference voltage. You have already identified the plant transfer function K(s). All you need to do is design a proportional gain G, i.e., G(s) = G, such that the steady state relative tracking error is within +10%. For the battery charger to work properly, the output voltage u(t) needs to be maintained within +10% of a target value of 3 unit, i.e., (control objective) v (oo) should stay within +10% of the reference value of 3 unit.
system are given below: (input) i(t) = 2u(t) (output) v(oo) (in steady-state) can take any value between 2 to 4 Your cousin suspects the components used in the circuit to be faulty, i.e., their values might be changing over time (e.g., due to temperature at the time of experiment). When your cousin repeated the same experiment with a constant current input of i(t) = 2u(t), they found that the output voltage v(t) could settle down to any value between 2 and 4 during different test runs, i.e., reference R(S) a) Can you use Final Value Theorem to confirm that the resistor is faulty (e.g., temperature-sensitive), with its value varying between 1 and 2? b) Could you help your cousin achieve this by designing a simple proportional controller? feedback controller i(t) = i₁(t) + ₂(t) i2(t) = dv(t) dt G(s) diy(t) v(t) = in(t)R + L dt error input X(s) E(s)=R(S)-Y(s) K(s) output Y(s) Note: In the closed-loop system, X(s) = I(s) is the programmable current input, Y(s) = V(s) is the output voltage, R(s) = 3/s is the reference voltage. You have already identified the plant transfer function K(s). All you need to do is design a proportional gain G, i.e., G(s) = G, such that the steady state relative tracking error is within +10%. For the battery charger to work properly, the output voltage u(t) needs to be maintained within +10% of a target value of 3 unit, i.e., (control objective) v (oo) should stay within +10% of the reference value of 3 unit.
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ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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![### Educational Content: System Analysis and Control Design
#### System Description
The system's transfer function is given by:
\[ K(s) = \frac{s + 1}{s^2 + s + 1} \]
#### Circuit Diagram
The circuit consists of:
- A current source \( i(t) \).
- A resistor \( R \) and inductor \( L \) in series, forming one branch.
- A capacitor \( C \) in parallel with the load, forming the output voltage \( v(t) \).
#### Equations Governing the System
The system equations are:
- Total current: \( i(t) = i_1(t) + i_2(t) \)
- Capacitor current: \( i_2(t) = C \frac{dv(t)}{dt} \)
- Voltage across the RL branch: \( v(t) = i_1(t)R + L \frac{di_1(t)}{dt} \)
#### Input and Output Conditions
- **Input**: \( i(t) = 2u(t) \)
- **Output**: \( v(\infty) \) (in steady state) can vary between 2 and 4.
#### Problem Statement
Your cousin suspects that the circuit components might be faulty due to potential changes over time, possibly due to temperature variations. The output voltage \( v(t) \) shows variability during different test runs, indicating possible component faults.
1. **Verification using the Final Value Theorem**:
- Can you employ the Final Value Theorem to verify if the resistor is faulty (e.g., temperature-sensitive) with value variations between 1 and 2?
2. **Design of Proportional Controller**:
- Assist in designing a simple proportional controller for stability.
#### Feedback Control System Diagram
- **Block Diagram Components**:
- Input: \( X(s) \)
- Output: \( Y(s) \)
- Error calculation and proportional gain \( G(s) \)
- The closed-loop system requires:
- \( X(s) = I(s) \) — current input.
- \( Y(s) = V(s) \) — output voltage.
- Reference Voltage: \( R(s) = \frac{3}{s} \)
- **Control Objective**:
- Maintain \( v(t) \) within ±10% of the target voltage of 3 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93e18a88-13ba-47d2-8c84-ec9d29397d2a%2F28dfc329-ccb6-494f-b0d8-5a040a00a24a%2F1mkedhx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Content: System Analysis and Control Design
#### System Description
The system's transfer function is given by:
\[ K(s) = \frac{s + 1}{s^2 + s + 1} \]
#### Circuit Diagram
The circuit consists of:
- A current source \( i(t) \).
- A resistor \( R \) and inductor \( L \) in series, forming one branch.
- A capacitor \( C \) in parallel with the load, forming the output voltage \( v(t) \).
#### Equations Governing the System
The system equations are:
- Total current: \( i(t) = i_1(t) + i_2(t) \)
- Capacitor current: \( i_2(t) = C \frac{dv(t)}{dt} \)
- Voltage across the RL branch: \( v(t) = i_1(t)R + L \frac{di_1(t)}{dt} \)
#### Input and Output Conditions
- **Input**: \( i(t) = 2u(t) \)
- **Output**: \( v(\infty) \) (in steady state) can vary between 2 and 4.
#### Problem Statement
Your cousin suspects that the circuit components might be faulty due to potential changes over time, possibly due to temperature variations. The output voltage \( v(t) \) shows variability during different test runs, indicating possible component faults.
1. **Verification using the Final Value Theorem**:
- Can you employ the Final Value Theorem to verify if the resistor is faulty (e.g., temperature-sensitive) with value variations between 1 and 2?
2. **Design of Proportional Controller**:
- Assist in designing a simple proportional controller for stability.
#### Feedback Control System Diagram
- **Block Diagram Components**:
- Input: \( X(s) \)
- Output: \( Y(s) \)
- Error calculation and proportional gain \( G(s) \)
- The closed-loop system requires:
- \( X(s) = I(s) \) — current input.
- \( Y(s) = V(s) \) — output voltage.
- Reference Voltage: \( R(s) = \frac{3}{s} \)
- **Control Objective**:
- Maintain \( v(t) \) within ±10% of the target voltage of 3 units.
Expert Solution
Step 1: Summarize the given information.
For the given RLC circuit, we need to determine:
- variation in resistor values at steady-state, and
- proportional gain G.
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