Classical Control Theory

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

$\begin{align*} u(t)={ \begin{cases} 1&{\text{if }}t \ge 0 \newline 0&{\text{if }}t \lt 0 \newline \end{cases}} \qquad u[n]={ \begin{cases} 1&{\text{if }}n \ge 0 \newline 0&{\text{if }}n \lt 0 \newline \end{cases}} \end{align*}$

1.2. Impulse unit function (in descrete domain)

Impulse function

$\begin{align*} & \delta[n]={ \begin{cases} 1&{\text{if }}n = 0 \newline 0&{\text{if }}n \ne 0 \newline \end{cases}} \newline & u[n] = \sum_{k = 0}^{\infty} \delta[n-k] \qquad \text{for step function} \newline & x[n] = \sum_{k = - \infty}^{\infty} x[k] \delta[n-k] \qquad \text{for arbitrary function} \end{align*}$

We will use this relation later on.

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

LTI system

An example of LTI system.

LTI example

1.4. Laplace transformation

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)

impulse responses

$\begin{align*} & \delta [n]\overset L \longrightarrow h[n] \qquad \text{ OR } \qquad y[n] = T\{x[n]\} \newline & x[n] = \sum_{k = - \infty}^{\infty} x[k] \delta[n-k] \newline & \Rightarrow y[n]=T \{ \sum_{k = - \infty}^{\infty} x[k] \delta[n-k] \} \overset {LTI} \longrightarrow \sum_{k = - \infty}^{\infty} x[k] L \{\delta[n-k] \} \newline & \Rightarrow y[n] = \sum_{k = - \infty}^{\infty} x[k] h[n-k] := x[n]*h[n] \quad \text{(This is convolution)} \end{align*}$

Now, if we take a Laplace transformation of $y[n]$, we have.

$\begin{align*} & x [n]\overset L \longrightarrow X[s] \newline & h [n]\overset L \longrightarrow H[s] \qquad \text{(where h[n] is the impulse response of the system)} \newline & y [n] = x[n]*h[n] \overset L \longrightarrow Y[s] = X[s] . H[s] \end{align*}$

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

An example of using impulse response in the system.

impulse responses example

From convolution to transfer function.

convolution-to-transfer-function

impulse responses example

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.

impulse responses example

3. Fourier transformation

Fourier transform

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 $\Phi$.

Fourier use sinusoidal

No phase information included.

Fourier no phase included

Using complex number to include phase information into account.

Fourier complex number

$\begin{align*} & F(v) = \cfrac{\sqrt{2}}{2} + \cfrac{\sqrt{2}}{2} j = e^{j \pi/4} \newline & F(v)e^{2 \pi j v t} = e^{j \pi/4} e^{2 \pi j v t} \end{align*}$

4. References

Brian Douglas Youtube channel, "Classical Control Theory" series
Dang Quang Hieu, Digital Signal Processing lecture note

Back to top

Classical Control Theory

1. Background

1.1. Step function

1.2. Impulse unit function (in descrete domain)

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

1.4. Laplace transformation

2. Transfer function

3. Fourier transformation

4. References

Index