Problem #1: a) (3 pts) Using Laplace transform solve for x(t) in problem B-2-23 in your textbook (System Dynamics, Ogata, 4th edition). Please show all the steps in your solution. b) (1 pt) Using the Initial Value Theorem (IVT) and Final Value Theorem (FVT), find the initial value and final value (xss) of x(t). Use ωn = 0.5 rad/s. Hint: 𝐼𝑉𝑇 = lim𝑠→ ∞ 𝑠𝑋(𝑠) and 𝐹𝑉𝑇 = lim𝑠→ 0 𝑠𝑋(𝑠) c) (1 pt) Using Matlab, for the frequency ωn = 0.5 rad/s, plot x(t). What is the IV and FV from the plot?

### Solution Overview

To solve this problem, we will break it down into parts as requested:

- **(a)** Use the Laplace transform to solve for \( x(t) \).
- **(b)** Use the Initial Value Theorem (IVT) and Final Value Theorem (FVT) to find the initial and final values.
- **(c)** Use MATLAB to plot \( x(t) \) for \( \omega_n = 0.5 \, \text{rad/s} \) and find the initial and final values from the plot.

We will need to assume some general form for the equation in problem B-2-23 since the exact problem is not available. Assuming the system is a standard second-order system, we can follow a step-by-step approach.

### (a) Laplace Transform Solution

For a general second-order system described by:

\[
\ddot{x}(t) + 2 \zeta \omega_n \dot{x}(t) + \omega_n^2 x(t) = u(t)
\]

where:
- \( \omega_n \) is the natural frequency,
- \( \zeta \) is the damping ratio,
- \( u(t) \) is the input (could be a unit step or another function).

Taking the Laplace transform of both sides (assuming zero initial conditions):

\[
s^2 X(s) + 2 \zeta \omega_n s X(s) + \omega_n^2 X(s) = \frac{U(s)}{s}
\]

This is the standard form for a second-order system with input \( U(s) \).

Now solve for \( X(s) \):

\[
X(s) = \frac{U(s)}{s \left( s^2 + 2 \zeta \omega_n s + \omega_n^2 \right)}
\]

#### Example Input: Unit Step Function

For a unit step input, \( U(s) = \frac{1}{s} \), so:

\[
X(s) = \frac{1}{s \left( s^2 + 2 \zeta \omega_n s + \omega_n^2 \right)}
\]

Use partial fraction decomposition if necessary to simplify and then take the inverse Laplace transform to find \( x(t) \).

### (b) Initial and Final Value Theorems

Using the provided formulas:

- **Initial Value Theorem (IVT):**

\[
\lim_{s \to \infty} s X(s)
\]

This will give the initial value \( x(0) \).

- **Final Value Theorem (FVT):**

\[
\lim_{s \to 0} s X(s)
\]

This will give the final steady-state value \( x_{\text{ss}} \).

We will substitute the expression for \( X(s) \) into these formulas to find the initial and final values.

### (c) MATLAB Plot

To plot \( x(t) \) using MATLAB, we will:
1. Define the system transfer function using \( \omega_n = 0.5 \, \text{rad/s} \) and a given damping ratio \( \zeta \) (which could be assumed as critically damped, underdamped, etc.).
2. Use the `step` function in MATLAB to plot the step response of the system.
3. From the plot, determine the initial and final values.

---

This is an overview of the approach to solve the problem. If you would like more details on any step, such as solving the Laplace transform, performing MATLAB calculations, or more specific examples, please let me know!

### Expansion Questions:
1. How do the damping ratio \( \zeta \) and natural frequency \( \omega_n \) affect the system's behavior?
2. What are the conditions for the system to be critically damped, underdamped, and overdamped?
3. How is partial fraction decomposition used in solving Laplace transforms of second-order systems?
4. What is the physical significance of the Initial Value Theorem and Final Value Theorem in system analysis?
5. Can you derive the response of an underdamped second-order system using Laplace transforms?

**Tip:** For second-order systems, the damping ratio \( \zeta \) plays a critical role in determining whether the system oscillates and how fast it returns to equilibrium.

Problem #1: a) (3 pts) Using Laplace transform solve for x(t) in problem B-2-23 in your textbook (System Dynamics, Ogata, 4th edition). Please show all the steps in your solution. b) (1 pt) Using the Initial Value Theorem (IVT) and Final Value Theorem (FVT), find the initial value and final value (xss) of x(t). Use ωn = 0.5 rad/s. Hint: IVT = lim𝑠→∞ 𝑠𝑋(𝑠) and FVT = lim𝑠→0 𝑠𝑋(𝑠) c) (1 pt) Using Matlab, for the frequency ωn = 0.5 rad/s, plot x(t). What is the IV and FV from the plot?

This solution covers the use of Laplace transform to solve for x(t) in a second-order system, using the Initial Value Theorem (IVT) and Final Value Theorem (FVT). It also demonstrates how to plot the system's response using MATLAB for ωn = 0.5 rad/s, showing key concepts in system dynamics and control theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation