# Dynamic Response and Transfer Function

## Introduction

##

## Laplace Transform

**Definition**: One-sided Laplace Transform

$$
\mathcal{L}[f(t)](https://www.learnros2.com/control-of-dynamic-systems/s) = \int\_{0}^{\infty}f(t)e^{-st}dt
$$

The inverse of the Laplace is given as follows:

$$
f(t) = \frac{1}{2\pi{}j}\int\_{x\_0-j\infty}^{x\_0+j\infty}\mathcal{L}[f](https://www.learnros2.com/control-of-dynamic-systems/s)e^{st}ds
$$

where $$x\_0$$is a value that lies on the right side of all the singularities of $$\mathcal{L}[f](https://www.learnros2.com/control-of-dynamic-systems/s)$$ in the s-plane.

**Prop:**

$$
\begin{align\*}
(1);;; \mathcal{L}&\[\alpha f\_1 + \beta f\_2] = \alpha \mathcal{L}\[f\_1] + \beta \mathcal{L}\[f\_2] \\
(2);;; \mathcal{L}&[f(t-\lambda)](https://www.learnros2.com/control-of-dynamic-systems/s) = e^{-s\lambda}\mathcal{L}[f](https://www.learnros2.com/control-of-dynamic-systems/s) \\
(3);;; \mathcal{L}&[f(at)](https://www.learnros2.com/control-of-dynamic-systems/s) = \frac{1}{|a|}\mathcal{L}[f](https://www.learnros2.com/control-of-dynamic-systems/\frac{s}{a}) \\
(4);;; \mathcal{L}&[f(t)e^{-at}](https://www.learnros2.com/control-of-dynamic-systems/s) = \mathcal{L}[f](https://www.learnros2.com/control-of-dynamic-systems/s+a) \\
(5);;; \mathcal{L}&[\frac{df}{dt}](https://www.learnros2.com/control-of-dynamic-systems/s) = -f(0^{-}) + s \cdot \mathcal{L}[f](https://www.learnros2.com/control-of-dynamic-systems/s) \\
(6);;; \mathcal{L}&[\int\_{0}^{t}f(\tau)d\tau](https://www.learnros2.com/control-of-dynamic-systems/s) = \frac{1}{s}\mathcal{L}[f](https://www.learnros2.com/control-of-dynamic-systems/s) \\
(7);;; \mathcal{L}&[(f \ast g)(t)](https://www.learnros2.com/control-of-dynamic-systems/s) = \mathcal{L}[f](https://www.learnros2.com/control-of-dynamic-systems/s) \mathcal{L}[g](https://www.learnros2.com/control-of-dynamic-systems/s) \\
(8);;; \mathcal{L}&[f(t)g(t)](https://www.learnros2.com/control-of-dynamic-systems/s) = \frac{1}{2\pi{}j}(\mathcal{L}\[f] \ast \mathcal{L}\[g])(s) \\
(9);;; \mathcal{L}&\[tf(t)] = - \frac{d}{ds}(\mathcal{L}[f](https://www.learnros2.com/control-of-dynamic-systems/s))
\end{align\*}
$$

Note: the equations (5) and (6) are important. They can convert terms in a differential equations into simple algebraic operations. That's one of the reasons when we analyze system behaviors, we work in the state space instead of in the time domain.

## Linear Time-Invariant System

## Dynamic Response

## Analyze System Behavior

TODO partial-Fraction Expansion

<figure><img src="/files/ssErd30iTTAQxEMZhsyX" alt=""><figcaption></figcaption></figure>

<figure><img src="/files/LJy2rE0mya8yop0dk9uq" alt=""><figcaption></figcaption></figure>


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