Dynamic Response and Transfer Function

Introduction

Laplace Transform

Definition: One-sided Laplace Transform

$$\mathcal{L}[f(t)](s) = \int_{0}^{\infty}f(t)e^{-st}dt$$

The inverse of the Laplace is given as follows:

$$f(t) = \frac{1}{2\pi{}j}\int_{x_0-j\infty}^{x_0+j\infty}\mathcal{L}[f](s)e^{st}ds$$

where $x_0$is a value that lies on the right side of all the singularities of $\mathcal{L}[f](s)$ in the s-plane.

Prop:

$$\begin{align*} (1)\;\;\; \mathcal{L}&[\alpha f_1 + \beta f_2] = \alpha \mathcal{L}[f_1] + \beta \mathcal{L}[f_2] \\ (2)\;\;\; \mathcal{L}&[f(t-\lambda)](s) = e^{-s\lambda}\mathcal{L}[f](s) \\ (3)\;\;\; \mathcal{L}&[f(at)](s) = \frac{1}{|a|}\mathcal{L}[f](\frac{s}{a}) \\ (4)\;\;\; \mathcal{L}&[f(t)e^{-at}](s) = \mathcal{L}[f](s+a) \\ (5)\;\;\; \mathcal{L}&[\frac{df}{dt}](s) = -f(0^{-}) + s \cdot \mathcal{L}[f](s) \\ (6)\;\;\; \mathcal{L}&[\int_{0}^{t}f(\tau)d\tau](s) = \frac{1}{s}\mathcal{L}[f](s) \\ (7)\;\;\; \mathcal{L}&[(f \ast g)(t)](s) = \mathcal{L}[f](s) \mathcal{L}[g](s) \\ (8)\;\;\; \mathcal{L}&[f(t)g(t)](s) = \frac{1}{2\pi{}j}(\mathcal{L}[f] \ast \mathcal{L}[g])(s) \\ (9)\;\;\; \mathcal{L}&[tf(t)] = - \frac{d}{ds}(\mathcal{L}[f](s)) \end{align*}$$

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