Control theory repository

Requirements

Python 3.6+.
Python Control Systems Library pip install control.
- Documentation: Python Control Systems Library.
- Functions: functions reference
NumPy pip install numpy.
SciPy pip install scipy.
Matplotlib pip install matplotlib.

Fundamentals

Modeling

In physics and engineering, mathematical models are used to abstract the real world. This abstraction is done through first-principles equations such as:

Kirchhoff's circuit laws
1. Current law: $\sum_{k=1}^n I_k = 0$
2. Voltage law: $\sum_{k=1}^n V_k = 0$
Newton's laws of motion
- $\vec{F} = m \vec{a}$
Maxwell's equations
1. $\nabla\cdot E=\frac{\rho}{\epsilon_0}$
2. $\nabla\cdot B=0$
3. $\nabla\times E=-\frac{\partial B}{\partial t}$
4. $\nabla\times B=\mu_0\left(J+\epsilon_0\frac{\partial E}{\partial t}\right)$
etc.

All these formulas describe systems, a collection of interconnected parts that form a larger more complex whole and also receive and yield signals, which are also mathematically modeled. The usefulness of the models depends on the accuracy, i.e., how close it is to the real system and signals.

Signals

A signal is a measurable (physical) function whose values describe a physical quantity. The mathematical model of a signal is a function (mapping from a domain $\mathcal{D}$ to a range $\mathcal{R}$, i.e. $f(\cdot):\mathcal{D}\to\mathcal{R}$ ) or formula that describes the physical quantity of interest.

Generally the domain is the one-dimensional space of the positive reals (time $t\in\mathbb{R}^+$), although there are multidimensional signals (more than one independent variable).

Classification of signals

The signals can be classified into several types, but the main ones are:

graph LR;
    P[Classification of signals]:::green;

    P --> N[Nature of independent variable]:::purple;
    P --> S[Symmetry]:::purple;
    P --> Pe[Periodicity]:::purple;
    P --> R[Range of the function]:::purple;
    P --> I[Number of independent variable]:::purple;
    P --> F[Functional definition]:::purple;
    P --> E[Energy and power]:::purple;

    N --> C[Continuous-time signals]:::blue;
    N --> D[Discrete-time signals]:::blue;
    C --> A[Analog signals]:::yellow;
    D --> Di[Digital signals]:::yellow;

    S --> Ev[Even signals]:::blue;
    S --> O[Odd signals]:::blue;
    S --> As[Asymmetric signals]:::blue;

    Pe --> Per[Periodic signals]:::blue;
    Pe --> No[Non-periodic signals]:::blue;

    R --> Re[Real valued signals]:::blue;
    R --> Im[Complex valued signals]:::blue;

    I --> M[Multidimensional signals]:::blue;
    I --> On[One-dimensional signals]:::blue;

    F --> St[Stochastic signals]:::blue;
    F --> De[Deterministic signals]:::blue;

    E --> En[Energy signals]:::blue;
    E --> Po[Power signals]:::blue;

	%% Colors
	classDef green  fill:#58fc71, stroke:#000;
	classDef yellow  fill:#fcd158, stroke:#000;
	classDef purple  fill:#c668fc, stroke:#000;
    classDef blue  fill:#68d2fc, stroke:#000;

Nature of independent variable:
- Continuous-time (CT) signals: are defined at every time instant in a time interval of interest, and its amplitude can assume any value in a continuous range.
- Discrete-Time (DT) Signals: are defined only at discrete time instants, and its amplitude can assume any value in a continuous range.
  - Digital Signals: its amplitude can assume a value only from a finite given set.
Symmetry:
- Even signals: are symmetric around the origin, i.e. $f(x) = f(-x)$
Periodicity:
- Periodic: the signal is periodic, i.e. it repeats itself periodically.
  - $x(t)=x(nT)$ where $n$ is an integer and $T$ is the period of the signal.
  - $x[n]=x[n+N]$ where $N$ is the number of samples.
- Non-periodic: if doesn't satisfy the periodicity condition, the signal is non-periodic.
Note: All CT sinusoidal signals are periodic, but not all DT sinusoidal signals are periodic. See signals notebook.

Transformations

There are 3 basic transformations of signals:

Shift
Scale
Reversal

Resources and references

Awesome control theory repo: Very useful to find resources (e.g., books, videos, lectures, etc.) about control theory.
Control theory slides: Slides about control theory made by Sergei Savin.
Engineering media: A website with a lot of resources about control theory made by Brian Douglas.

cychitivav / control-theory