# Classical Control Theory

Tuyen D. Le April 30, 2022 [Control] #Brian-Douglas## 1. Background

### 1.1. Step function

### 1.2. Impulse unit function (in descrete domain)

We will use this relation later on.

### 1.3. "Linear and Time Invariance" (LTI) systems

An example of LTI system.

### 1.4. Laplace transformation

## 2. Transfer function

Q. What is impulse response $h[n]$ of a system?

A. When the input is a impulse unit function $\delta [n]$ (see "Impulse unit function" section above)

Now, if we take a `Laplace`

transformation of $y[n]$, we have.

We call $H[s]$ is the **transfer function** of the system.

An example of using impulse response in the system.

From convolution to transfer function.

Let take a step back to understand this picture. In this case, $u(t)$ is input, and $x(t)$ is output. If we take $u(t)=\delta (t)$, we should have $x(t)=X(t)$ at the output. Taking **Laplace** transformation both side of the equation, $X(s)$ becomes **Transfer function** of the system.

## 3. Fourier transformation

Q. Why sinusoidal?

A. Because its shape is the same in the `LTI`

system. We only need to care about amplitude $A$ (i.e. **Gain**) and phase $\Phi$.

No phase information included.

Using complex number to include phase information into account.