Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skillsβperfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to Repeated Real Poles
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Today, we are going to discuss repeated real poles. Can anyone explain what we mean by a repeated pole?
I think a repeated pole is when the denominator of our function has the same root multiple times?
Exactly! When we have a root like (s - p1) raised to the power of n, we say that p1 is a repeated pole. This affects how we perform the Partial Fraction Expansion. Why do you think this is important?
Because we need to know how to expand functions properly to find their inverse transforms?
Correct! Understanding how to structure these expansions allows us to solve for coefficients. Let's move to our next point: how do we set up the PFE for repeated poles?
Setting Up the Partial Fraction Expansion
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
The PFE for repeated poles has the form: $$ X(s) = \frac{A_1}{s - p_1} + \frac{A_2}{(s - p_1)^2} + \cdots + \frac{A_n}{(s - p_1)^n} $$. What does this tell you about each A_k?
It suggests that we have to account for each power of the pole separately?
Exactly! Each term corresponds to a power of the pole. This means we need to find coefficients for all these terms. How would we find A_n?
Isn't it something like using the cover-up method?
Great answer! You can apply the cover-up method for the highest power term. What about for the lower power terms?
Finding Coefficients for Lower Power Terms
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
To find the coefficients A_k for lower powers, we need to differentiate the expression. Can anyone articulate the formula we would use?
Is it \( A_k = \frac{1}{(n-k)!} \left( \frac{d^{(n-k)}}{d s^{(n-k)}} \left[ (s - p_1)^{n} X(s) \right] \right) \text{ evaluated at } s = p_1 \)?
You nailed it! That's how we differentiate to find A_k. Remember, this may seem tedious, but it's necessary to fully express your inverse transforms correctly. Why do we say it's tedious?
Because we might have to do this multiple times for different coefficients?
Exactly! Now letβs circle back to why this matters in control theory. Understanding this concept of repeated poles helps us manage complex system behavior effectively.
Applications and Importance
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Why do we care about repeated poles in engineering?
Because many systems have damped or oscillatory behaviors tightly linked to their poles?
Thatβs right! The natural response of systems can be significantly affected by repeated poles. If you were to analyze a system response, how would an undamped response differ from a damped one?
An undamped response will continue indefinitely while damped responses dissipate over time.
Exactly! And repeated poles can indicate such critical behaviors. Can anyone summarize what we learned about repeated poles and their significance?
We learned how to set up the PFE for repeated poles and how to find coefficients necessary for inverse transformation. Understanding this helps us predict system behavior.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, we explore how to apply the Partial Fraction Expansion method to rational functions that have repeated real poles, detailing the appropriate formulation of the expansion and the techniques for finding coefficients for each term.
Detailed
Detailed Summary
In this section, we delve into the Partial Fraction Expansion (PFE) method, specifically for cases involving repeated real poles. When a denominator polynomial has a repeated real root of multiplicity n, the PFE formulation expands into a series of terms for that root. The structure of the expansion for a repeated real pole p1 raised to the power of n is as follows:
$$ X(s) = \frac{A_1}{s - p_1} + \frac{A_2}{(s - p_1)^2} + \cdots + \frac{A_n}{(s - p_1)^n} + \ldots $$
To find the coefficients \( A_k \), especially for the highest power term, the cover-up method can be employed effectively. For lower power terms, differentiation of the expression is required:
$$ A_k = \frac{1}{(n-k)!} \left( \frac{d^{(n-k)}}{d s^{(n-k)}} \left[ (s - p_1)^{n} X(s) \right] \right) \text{ evaluated at } s = p_1 $$
This section also emphasizes the potential complexity of these computations but offers practical strategies, such as using a combination of cover-up and coefficient comparison methods. Furthermore, it prepares students for real-world problem-solving where repeated poles may appear, helping to solidify their understanding of inverse transformations in control system analysis.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Partial Fraction Expansion for Repeated Real Poles
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
If the denominator D(s) has a repeated real root, for example, (s - p1) raised to the power of n (meaning p1 is a root of multiplicity n), then the PFE includes a series of terms for that pole:
X(s) = A1 / (s - p1) + A2 / (s - p1)^2 + ... + An / (s - p1)^n + ... (plus terms for other distinct poles)
Detailed Explanation
In the case of repeated real poles, the denominator D(s) can contain roots that appear multiple times. For example, if 'p1' is a root that repeats 'n' times, we denote this as (s - p1)^n. To handle such repeated roots in the Partial Fraction Expansion (PFE), we express the rational function X(s) in a series of fractions corresponding to each power of the root. This is essential because each of these terms corresponds to a different aspect of the system's behavior around that pole, and it allows us to properly account for the complexity introduced by having multiple poles at the same location.
Examples & Analogies
Think of a playground swing set. If the swing's pivot (the pole) is worn down and starts to wobble, it could swing freely in one direction but face restrictions in another due to the wear at a specific point. This wobble represents a repeated poleβthe swing set's behavior at this pivot is affected not just once but repeatedly, requiring us to analyze its responses in segments (like each rising and falling of the swing) to understand its overall motion.
Finding Coefficients in PFE
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Finding Coefficients (Ak): For the highest power term (An), the cover-up method still works: An = [(s - p1)^n * X(s)] evaluated at s = p1. For lower power terms, derivatives are required:
Ak = [1 / (n - k)!] * (d raised to the power of (n-k) / d s raised to (n-k)) of [(s - p1)^n * X(s)] evaluated at s = p1.
Detailed Explanation
To compute the coefficients for each term in the Partial Fraction Expansion for repeated real poles, we start with the term associated with the highest power, which can be found by multiplying the entire X(s) by (s - p1)^n and evaluating that at s = p1. For the coefficients of lower powers, we take derivatives of this product with respect to 's', applying the factorial to account for the order of the pole we are working with. This requires careful calculations, especially as we move to lower power terms, ensuring that we are accurately capturing the complexity of that repeated pole in our function.
Examples & Analogies
Imagine a complicated cheese cake recipe. The first time you bake it, you might adjust the sugar content directly based on taste. This adjustment represents finding the coefficient for the leading term. However, as you perfect the recipe (like finding lower powers), you might consider how much air to beat into the cream or how long to bake it, which corresponds to the more subtle adjustments of adding derivatives to your calculations. Each small tweak contributes to the overall quality of your cake, just as each coefficient in the PFE is essential for accurately defining this function.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Partial Fraction Expansion: Method to decompose complex rational functions.
-
Repeated Pole: A root of the denominator that appears more than once, affecting the form of the partial fraction expansion.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
An example of finding A_k coefficients for a repeated pole in a Laplace Transform function.
-
Using PFE to break down a rational function with repeated poles to apply the Inverse Laplace Transform effectively.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
-
With repeated poles, don't be slow, find coefficients, make them flow.
π Fascinating Stories
-
Imagine a group of friends (A_1, A_2, A_3) at a party - the repeated pole p1 is the host, welcoming everyone and needing to be reported on how many guests (coefficients) agree with their order - the cover-up keeps things in check.
π§ Other Memory Gems
-
For repeated poles, remember 'DRG': Decompose, Reveal, Grab coefficients!
π― Super Acronyms
PFE = Partial Fraction Expansion
- Perfect for Repeated Terms.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Partial Fraction Expansion
Definition:
A method of decomposing a rational function into simpler fractions that can be more easily inverted.
-
Term: Repeated Pole
Definition:
A root of the denominator polynomial that appears more than once.
-
Term: CoverUp Method
Definition:
A technique for finding coefficients in partial fraction expansions by evaluating the limit at the pole.