## Linear Control System Analysis### OverviewThe block diagram depicted illustrates a linear control system. In this system, \( r(t) \) represents the reference input, while \( n(t) \) signifies the disturbance.### Tasks and Questions1. **Steady-State Error for Unit Ramp Input** - **Task a**: Determine the steady-state value of the error \( e(t) \) as a function of \( K \) when \( n(t) = 0 \) and \( r(t) \) is a unitary ramp. 2. **Conditions for Validity** - **Task b**: Identify conditions on the values of \( K \) to ensure the solution from (a) is valid.3. **Minimum Possible Error** - **Task c**: Evaluate the minimum possible error under the conditions from (a) and (b).4. **Steady-State Output for Unitary Step Disturbance** - **Task d**: Compute the steady-state value of \( y(t) \) when \( r(t) = 0 \) and \( n(t) \) is a unitary step.5. **Steady-State Error for a Specific Input Signal** - **Task e**: Given \( n(t) = 0 \), determine the steady-state value of the error as a function of \( K \) for an input signal of \( 4t \mu_s(t) \).### Block Diagram Components- **R(S)**: Reference input in the Laplace domain.- **2/S**: Integrator block with a gain of 2.- **N(S)**: Input disturbance in the Laplace domain.- **K/((S+2)(S+3))**: Controller/plant transfer function.### Explanation of the Diagram- The diagram exhibits a feedback control system with a disturbance.- The input signal \( R(S) \) is subjected to an integrator with a gain of 2 before being fed into a summation junction, which also takes in the disturbance \( N(S) \).- The resulting signal is then processed by a system described by the transfer function \( \frac{K}{(S+2)(S+3)} \).- The output \( Y(S) \) is generated, representing the controlled system output.This block diagram is crucial for analyzing the control system's response to different inputs and disturbances and for

1.- The block diagram of a linear control system is shown in the Fig., where r(t) is the reference input and n(t), the disturbance. a) Find the steady-state value of e(t), as a function of K, when n(t) = 0 and r(t) is a unitary ramp b) Find the conditions on the values of K so that the solution in a) is valid c) what is the minimum possible error obtained under a) and b) conditions? d) Find the steady-state value of y(t) when r(t) = 0 and n(t) is a unitary step e) Assuming n(t) = 0, what is the steady-state value of the error, as a function of K, for an input signal of 4tµs(t)? N(S) 2/S K/(S+2)(S+3) R(S) +Y(S)

1.- The block diagram of a linear control system is shown in the Fig., where r(t) is the reference input and n(t), the disturbance. a) Find the steady-state value of e(t), as a function of K, when n(t) = 0 and r(t) is a unitary ramp b) Find the conditions on the values of K so that the solution in a) is valid c) what is the minimum possible error obtained under a) and b) conditions? d) Find the steady-state value of y(t) when r(t) = 0 and n(t) is a unitary step e) Assuming n(t) = 0, what is the steady-state value of the error, as a function of K, for an input signal of 4tµs(t)? N(S) 2/S K/(S+2)(S+3) R(S) +Y(S)

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Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Wave Motion

A wave motion is defined as the propagation of disturbances or transfer of energy and momentum from the state of rest or equilibrium to another point in a medium without any actual transfer of matter between them. In a simple explanation, it is the propagation of energy through a medium that is produced due to the repeated vibrations of the particles. Common things we see day to day like waves on water oscillation of a pendulum or bell, light wave, and all the subatomic particles exhibiting wavelike properties. The propagation of waves depends on the medium and their properties. The wave function is the disturbance that produces waves as a function of time. This disturbance oscillates periodically with a fixed frequency and wavelength. There are linear wave motion and non-linear wave motion based on their propagation, motion, and time period where linear wave motion is like sinusoidal waves and non-linear wave motion like a wave motion that gets interrupted in the middle of their propagation.

Question

## Linear Control System Analysis

### Overview
The block diagram depicted illustrates a linear control system. In this system, \( r(t) \) represents the reference input, while \( n(t) \) signifies the disturbance.

### Tasks and Questions
1. **Steady-State Error for Unit Ramp Input**
- **Task a**: Determine the steady-state value of the error \( e(t) \) as a function of \( K \) when \( n(t) = 0 \) and \( r(t) \) is a unitary ramp.

2. **Conditions for Validity**
- **Task b**: Identify conditions on the values of \( K \) to ensure the solution from (a) is valid.

3. **Minimum Possible Error**
- **Task c**: Evaluate the minimum possible error under the conditions from (a) and (b).

4. **Steady-State Output for Unitary Step Disturbance**
- **Task d**: Compute the steady-state value of \( y(t) \) when \( r(t) = 0 \) and \( n(t) \) is a unitary step.

5. **Steady-State Error for a Specific Input Signal**
- **Task e**: Given \( n(t) = 0 \), determine the steady-state value of the error as a function of \( K \) for an input signal of \( 4t \mu_s(t) \).

### Block Diagram Components
- **R(S)**: Reference input in the Laplace domain.
- **2/S**: Integrator block with a gain of 2.
- **N(S)**: Input disturbance in the Laplace domain.
- **K/((S+2)(S+3))**: Controller/plant transfer function.

### Explanation of the Diagram
- The diagram exhibits a feedback control system with a disturbance.
- The input signal \( R(S) \) is subjected to an integrator with a gain of 2 before being fed into a summation junction, which also takes in the disturbance \( N(S) \).
- The resulting signal is then processed by a system described by the transfer function \( \frac{K}{(S+2)(S+3)} \).
- The output \( Y(S) \) is generated, representing the controlled system output.

This block diagram is crucial for analyzing the control system's response to different inputs and disturbances and for

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