Applied Control Systems 1: autonomous cars: Math + PID + MPC

Name: Applied Control Systems 1: autonomous cars: Math + PID + MPC
Rating: 4.7 (1995 reviews)

Modeling + state space systems + PID + Model Predictive Control + Python simulation: lateral control for autonomous cars

Bestseller

Created byMark Misin Engineering

Last updated 11/2025

English

What you'll learn

mathematical modelling of systems
reformulating models into state-space equations
applying a PID controller to systems (simple magnetic train catching objects)
applying Model Predictive Control (MPC) to systems (autonomous car: lane changing maneuvers)

Course content

8 sections • 148 lectures • 17h 55m total length

Course guide5:07
This lecture gives you a guide for how to study the course in order to take the maximum value out of it.
Intro to Control - how to control systems with a controller 16:52
Intro to Control - how to control systems with a controller 26:34
Open VS Closed Loop System6:36
Explore the key differences between open loop and closed loop systems, highlighting how feedback from sensors and controllers adjusts input to achieve targets like mass flow rate and tank level.
Controlling the water tank in a Python simulation2:52
Intro to a proportional controller4:44
Modelling the water tank 11:45
Modelling the water tank 212:13
Numerical integration applied to the water tank model9:52
Combining math with the control structure7:07
Explore a closed loop water tank system modeled in continuous and discrete mathematics. A proportional controller updates mass flow rate at 50 Hz to track changing volume references.
Water tank simulation - proportional controller2:28
Intro to a PID simulation2:26
Follow up!0:58
PID: Modelling the train with forces 16:27
PID: Modelling the train with forces 29:36
PID: Going from system input to system output using numerical integration10:00
PID: Magnetic train simulation - proportional controller1:59
PID: Proportional controller overshoot explanation 14:39
PID: Proportional controller overshoot explanation 26:28
Explain how a proportional controller yields zero acceleration at zero error, yet the train overshoots because velocity remains nonzero and the position keeps moving past the target.
PID: Proportional controller overshoot explanation 33:40
PID: Intro to Derivative Control10:24
PID: Tuning the controller6:11
PID: Proportional & Derivative controller & magnetic train simulation in Python9:01
PID: Intro to Integral Control4:35
PID: Python magnetic train simulation at an inclination angle1:49
PID: Mathematical modelling of the train with the inclination angle 13:43
PID: Mathematical modelling of the train with the inclination angle 28:45
PID: Proportional, Derivative, Integral Control combined16:03
PID: Magnetic train simulation (inclination angle & PID)2:26
Test your PID fundamental understanding
Intro to (Linux & macOS Terminal) & (Windows Command Prompt)12:54
Python installation instructions0:51
Installing the Python environment and its libraries (Windows 11)5:14
Installing the Python environment and its libraries (Linux Ubuntu)4:43
Installing the Python environment and its libraries (macOS)8:04
PID train code explanation 117:56
PID train code explanation 211:15
PID train code explanation 311:18
Short intro to Python animation tools12:24
Quick code & animation explanation (water tanks)28:29
Codes for the P & PID controllers (Python 3, Numpy & Matplotlib needed)0:06

PID VS Model Predictive Control (MPC) 12:42
Intro to MPC1:10
Getting started with modelling a car5:06
Fundamentals of forces and moments 112:10
Fundamentals of forces and moments 210:18
Setting stage for the car's lateral control 15:09
Setting stage for the car's lateral control 28:50
PID VS Model Predictive Control (MPC) 21:15
Setting stage for the car's lateral control 38:25
Setting stage for the car's lateral control 41:27
Moment calculation exercise

Follow up!0:58
The general control structure for the vehicle's lateral control2:31
Car model VS simplified bicycle model 14:59
Car model VS simplified bicycle model 21:41
Car model VS simplified bicycle model 32:52
Ackerman Steering1:52
Longitudinal & lateral velocities of the bicycle model 14:37
Longitudinal & lateral velocities of the bicycle model 23:32
Equations of motion in the lateral direction3:15
Lateral & centripetal acceleration5:28
Centripetal acceleration intuition & mathematical derivation 15:39
Centripetal acceleration intuition & mathematical derivation 28:58
Extra explanation on rotating frames1:36
Centripetal acceleration intuition & mathematical derivation 320:51
Modelling the front wheel of the vehicle 13:41
Rewriting lateral forces in terms of front wheel angles3:39
Modelling the front wheel of the vehicle 22:09
Modelling the front wheel of the vehicle 37:25
Modelling the front wheel of the vehicle 48:10

From equations of motion to state-space equations 11:26
From equations of motion to state-space equations 27:25
From equations of motion to state-space equations 37:17
The meaning of states8:36
Adding extra states to the system8:28
Computing new states in the open loop system 110:29
Computing new states in the open loop system 28:57
Computing new states in the open loop system 34:53
Simplifying systems with small angle assumptions7:29
Nonlinear VS Linear Time Invariant (LTI) models10:47
Connecting LTI matrices with the vehicle's inputs4:41
Getting LTI model using small angle approximation 13:59
Getting LTI model using small angle approximation 26:00
Getting LTI model using small angle approximation 3 + Recap6:51

The objective of this section0:24
Model Predictive Control - Intro7:05
Model Predictive Control - Thrust levels6:15
Model Predictive Control - Cost function12:28
Model Predictive Control - Cost function having several variables 113:28
Model Predictive Control - Cost function having several variables 24:26
Model Predictive Control - Cost function weights6:51
Model Predictive Control - Horizon period10:30
Model predictive control computes a sequence of thrust inputs from a cost function, discusses horizon length tradeoffs, and uses a receding-horizon loop to re-optimize after applying the first input.
Model Predictive Control - measured VS predicted outputs (Kalman Filter)9:20
Recommended reading: Great article about Kalman Filters0:10
Model Predictive Control - Quadratic VS other cost functions 15:26
Explain why a quadratic cost function enables a well defined minimum in model predictive control, unlike linear or cubic forms, with positive weights and finite j_min.
Model Predictive Control - Quadratic VS other cost functions 25:08
Model Predictive Control - Quadratic VS other cost functions 36:57
Compute the gradient of multi-dimensional cost functions and set partial derivatives to zero to locate minima. Quadratic costs with positive weights guarantee the minimum; beware saddle points in higher dimensions.
Model Predictive Control - Quadratic VS other cost functions 46:35
MPC in a nutshell2:02

MPC 1 - continuous VS discrete state space equations11:16
Analyze a bicycle model that forms a continuous LTI system with four states and two outputs. Discretize x_{k+1}=A_d x_k + B_d u_k and y_k=C_d x_k + D_d u_k for MPC.
MPC 2 - deriving discrete state space equations5:38
MPC 3 - discrete form (extra information)0:10
MPC 4 - predicting future states 112:34
MPC 5 - predicting future states 218:04
MPC 6 - reformulating the cost function 111:34
MPC 7 - reformulating the cost function 27:42
MPC 8 - adding extra state to the system (augmentation) 17:51
MPC 9 - adding extra state to the system (augmentation) 29:24
MPC 10 - reason for why only the first input is chosen10:04
MPC 11 -rewriting the error term in form of reference minus output2:19
MPC 12 - expanding the cost function terms 18:21
MPC 13 - expanding the cost function terms 215:00
MPC 14 - expanding the cost function terms 34:54
MPC 15 - expanding the cost function terms 413:40
MPC 16 - reformulating the cost function from polynomial to matrix vector form 13:51
MPC 17 - reformulating the cost function from polynomial to matrix vector form 27:47
MPC 18 - predicting future states (assuming system augmentation)11:04
MPC 19 - cost function reformulation 15:35
MPC 20 - cost function reformulation 27:43
MPC 21 - obtaining the gradient of the cost function to get the inputs 16:24
MPC 22 - obtaining the gradient of the cost function to get the inputs 29:02
Derivation of the gradient of a quadratic vector-matrix form 18:55
Derivation of the gradient of a quadratic vector-matrix form 24:34
Derivation of the gradient of a quadratic vector-matrix form 35:46
Derivation of the gradient of a quadratic vector-matrix form 48:52
Derivation of the gradient of a quadratic vector-matrix form 510:45
Why is H double bar (Hdb) matrix symmetric?

Intro to (Linux & macOS Terminal) & (Windows Command Prompt)12:54
Python Simulation Intro1:02
Python installation instructions - Windows 115:14
Python installation instructions - Ubuntu4:43
Python installation instructions - macOS8:04
Intro to the simulator9:17
Recap of the course6:15
Code explanation 1 - general overview9:49
Code explanation 2 - a function for storing the initial variables14:01
Code explanation 3 - a function for generating trajectories18:31
Code explanation 4 - a function for discrete state space matrices6:01
Code explanation 5 - a function for generating the MPC cost function matrices16:23
Code explanation 6 - a function for calculating new states16:44
Code explanation 7 - the MAIN file 115:47
Code explanation 8 - the MAIN file 210:55
Code explanation 9 - the MAIN file 311:37
Code explanation 10 - the MAIN file 43:44
Basic intro to Python animations tools19:00
Discussing the simulation results10:57
Aligning yourself with a fixed reference line SMOOTHLY
PID VS Model Predictive Control (MPC) 39:22
The summary material0:38
The Simulation codes - Model Predictive Control1:08

Requirements

Basic Calculus: Functions, Derivatives, Integrals
Vector-Matrix multiplication

Description

The world is changing! The technology is changing! The advent of automation in our societies is spreading faster than anyone could have anticipated. At the forefront of our technological progress is autonomy in systems. Self driving cars and other autonomous vehicles are likely to be part of our every day lives. How would you like to understand and be able design these autonomous vehicles? How would you like to understand Mathematics behind it?

Welcome! In this course, you will be exposed to one of the most POWERFUL techniques there are, that are able to guide and control systems precisely and reliably.

You are going to DESIGN, MASTER and APPLY:

mathematical models in the form of state-space systems and equations of motion
a PID controller to a simple magnetic train that needs to catch objects that randomly fall from the sky
a Model Predictive Controller (MPC) to an autonomous car in a simple lane changing maneuver on a straight road at a constant forward speed.

You will LEARN the fundamentals and the logic of Modelling, PID and MPC that will allow you to apply it to other systems you might encounter in the future.

You need 3 things when solving an Engineering problem: INTUITION, MATHEMATICS, CODING! You can't choose - you really need them all. After this course, you will master Modelling, PID and MPC in all these 3 ways. That's a promise!

I'm very excited to have you in my course and I can't wait to teach you what I know.

Let's get started!

Who this course is for:

Science and Engineering students
Working Scientists and Engineers
Control Engineering enthusiasts

Applied Control Systems 1: autonomous cars: Math + PID + MPC

What you'll learn

Explore related topics

Course content

Intro to Control - PID controller40 lectures • 4hr 49min

Fundamentals of forces, moments, mass moment of inertia and reference frames10 lectures • 57min

Vehicle modelling for lateral control using equations of motion19 lectures • 1hr 34min

Vehicle's state-space & Linear Time Invariant (LTI) model for lateral control14 lectures • 1hr 37min

Model Predictive Control - Intuition - Rocket example15 lectures • 1hr 37min

Model Predictive Control - Mathematical Derivation - autonomous vehicle example27 lectures • 3hr 49min

Model Predictive Control - Python Simulation - autonomous vehicle22 lectures • 3hr 32min

Last Words1 lecture • 1min

Requirements

Description

Who this course is for: