Kaleb Ben Naveed Notes
Preface
1
System Modelling
1.1
Introduction
1.2
State, input, and dynamics
1.3
What is a dynamical system?
1.4
State variables
1.5
Inputs
1.6
Linear systems
1.7
Nonlinear systems
1.8
How to tell if a system is linear or nonlinear
1.9
Linearization
1.10
Control-affine systems
1.11
Degrees of freedom
1.12
Fully actuated systems
1.13
Underactuated systems
1.14
Another underactuated example: quadrotor
1.15
Time-invariant and time-varying systems
1.16
Disturbances and uncertainty
1.17
Output models
1.18
Summary
2
Different Dynamic Models
2.1
Example 1: Single Integrator
2.2
Example 2: Double Integrator
2.3
Example 3: Unicycle Model
2.4
Example 4: 2D Dynamic Bicycle Model
2.4.1
Global kinematics
2.4.2
Slip angles
2.4.3
Vertical loads
2.4.4
Lateral tire forces
2.4.5
Longitudinal forces
2.4.6
Body-frame dynamics
2.5
Example 5: Quadrotor Dynamics
3
Controllers
3.1
Example 1: PID Controller for a Unicycle
3.2
Example 2: Geometric Controller for Quadrotor
4
Optimization
4.1
Epigraph Form
4.2
Example 1: Basic Epigraph Form
4.3
Why does this work?
4.4
Geometric meaning of the epigraph
4.5
Example 2: Absolute Value Objective
4.6
Epigraph form for constrained optimization
4.7
Example 3: Constrained Epigraph Form
4.8
Combining constraints using a maximum
4.9
Inner and outer problem view
4.10
Example 4: Inner and Outer View
4.11
Why
\(z\)
is subtracted, not multiplied
4.12
Feasibility interpretation
4.13
Monotonicity property
4.14
Convexity property
4.15
Example 5: Maximum Objective
4.16
Relation to minimax problems
4.17
Relation to Lagrangian methods
4.18
Important properties
4.18.1
Property 1: Same optimizer
4.18.2
Property 2: Converts objective into a constraint
4.18.3
Property 3: Useful for maximum objectives
4.18.4
Property 4: Preserves convexity
4.18.5
Property 5: Gives an upper-bound interpretation
4.19
Common mistake
4.20
Summary
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Notes
Notes
Kaleb Ben Naveed
2026-04-25
Preface
Here is the list of topics.