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CourseEE26320 lectures24 h 27 min

Introduction to Linear Dynamical Systems

Stephen Boyd · Stanford University

Start with lecture 1 At Stanford Engineering Everywhere

About

Introduction to applied linear algebra and linear dynamical systems, with applications to circuits, signal processing, communications, and control systems.

Topics include: Least-squares aproximations of over-determined equations and least-norm solutions of underdetermined equations. Symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and right eigenvectors, and dynamical interpretation. Matrix exponential, stability, and asymptotic behavior. Multi-input multi-output systems, impulse and step matrices; convolution and transfer matrix descriptions. Control, reachability, state transfer, and least-norm inputs. Observability and least-squares state estimation.

Prerequisites: Exposure to linear algebra and matrices (as in Math. 103). You should have seen the following topics: matrices and vectors, (introductory) linear algebra; differential equations, Laplace transform, transfer functions. Exposure to topics such as control systems, circuits, signals and systems, or dynamics is not required, but can increase your appreciation.

Lectures

  1. Overview Of Linear Dynamical Systems, Why Study Linear Dynamical Systems?, Examples Of Linear Dynamical Systems, Estimation/Filtering Example, Linear Functions And Examples Lecture 1 · 1 h 17 min · Video
  2. Linear Functions (Continued), Interpretations Of Y=Ax, Linear Elastic Structure, Example, Total Force/Torque On Rigid Body Example, Linear Static Circuit Example, Illumination With Multiple Lamps Example, Cost Of Production Example,… Lecture 2 · 1 h 6 min · Video
  3. Linearization (Continued), Navigation By Range Measurement, Broad Categories Of Applications, Matrix Multiplication As Mixture Of Columns, Block Diagram Representation, Linear Algebra Review, Basis And Dimension, Nullspace Of A Matrix Lecture 3 · 1 h 19 min · Video
  4. Nullspace Of A Matrix(Continued), Range Of A Matrix, Inverse, Rank Of A Matrix, Conservation Of Dimension, 'Coding' Interpretation Of Rank, Application: Fast Matrix-Vector Multiplication, Change Of Coordinates, (Euclidian) Norm, Inner… Lecture 4 · 1 h 14 min · Video
  5. Orthonormal Set Of Vectors, Geometric Interpretation, Gram-Schmidt Procedure, General Gram-Schmidt Procedure, Applications Of Gram-Schmidt Procedure, 'Full' QR Factorization, Orthogonal Decomposition Induced By A, Least-Squares Lecture 5 · 1 h 15 min · Video
  6. Least-Squares, Geometric Interpretation, Least-Squares (Approximate) Solution, Projection On R(A), Least-Squares Via QR Factorization, Least-Squares Estimation, Blue Property, Navigation From Range Measurements, Least-Squares Data Fitting Lecture 6 · 1 h 16 min · Video
  7. Least-Squares Polynomial Fitting, Norm Of Optimal Residual Versus P, Least-Squares System Identification, Model Order Selection, Cross-Validation, Recursive Least-Squares, Multi-Objective Least-Squares Lecture 7 · 1 h 16 min · Video
  8. Multi-Objective Least-Squares, Weighted-Sum Objective, Minimizing Weighted-Sum Objective, Regularized Least-Squares, Laplacian Regularization, Nonlinear Least-Squares (NLLS), Gauss-Newton Method, Gauss-Newton Example, Least-Norm Solutions… Lecture 8 · 1 h 16 min · Video
  9. Least-Norm Solution, Least-Norm Solution Via QR Factorization, Derivation Via Langrange Multipliers, Example: Transferring Mass Unit Distance, Relation To Regularized Least-Squares, General Norm Minimization With Equality Constraints,… Lecture 9 · 1 h 9 min · Video
  10. Examples Of Autonomous Linear Dynamical Systems, Finite-State Discrete-Time Markov Chain, Numerical Integration Of Continuous System, High Order Linear Dynamical Systems, Mechanical Systems, Linearization Near Equilibrium Point,… Lecture 10 · 1 h 12 min · Video
  11. Solution Via Laplace Transform And Matrix Exponential, Laplace Transform Solution Of X_^ = Ax, Harmonic Oscillator Example, Double Integrator Example, Characteristic Polynomial, Eigenvalues Of A And Poles Of Resolvent, Matrix Exponential,… Lecture 11 · 1 h 9 min · Video
  12. Time Transfer Property, Piecewise Constant System, Qualitative Behavior Of X(T), Stability, Eigenvectors And Diagonalization, Scaling Interpretation, Dynamic Interpretation, Invariant Sets, Summary, Markov Chain (Example) Lecture 12 · 1 h 14 min · Video
  13. Markov Chain (Example), Diagonalization, Distinct Eigenvalues, Digaonalization And Left Eigenvectors, Modal Form, Diagonalization Examples, Stability Of Discrete-Time Systems, Jordan Canonical Form, Generalized Eigenvectors Lecture 13 · 1 h 13 min · Video
  14. Jordan Canonical Form, Generalized Modes, Cayley-Hamilton Theorem, Proof Of C-H Theorem, Linear Dynamical Systems With Inputs & Outputs, Block Diagram, Transfer Matrix, Impulse Matrix, Step Matrix Lecture 14 · 1 h 18 min · Video
  15. DC Or Static Gain Matrix, Discretization With Piecewise Constant Inputs, Causality, Idea Of State, Change Of Coordinates, Z-Transform, Symmetric Matrices, Quadratic Forms, Matrix Nom, And SVD, Eigenvalues Of Symmetric Matrices,… Lecture 15 · 1 h 9 min · Video
  16. RC Circuit (Example), Quadratic Forms, Examples Of Quadratic Form, Inequalities For Quadratic Forms, Positive Semidefinite And Positive Definite Matrices, Matrix Inequalities, Ellipsoids, Gain Of A Matrix In A Direction, Matrix Norm,… Lecture 16 · 1 h 13 min · Video
  17. Gain Of A Matrix In A Direction, Singular Value Decomposition, Interpretations, Singular Value Decomposition (SVD) Applications, General Pseudo-Inverse, Pseudo-Inverse Via Regularization, Full SVD, Image Of Unit Ball Under Linear… Lecture 17 · 1 h 17 min · Video
  18. Sensitivity Of Linear Equations To Data Error, Low Rank Approximations, Distance To Singularity, Application: Model Simplification, Controllability And State Transfer, State Transfer, Reachability, Reachability For Discrete-Time LDS Lecture 18 · 1 h 15 min · Video
  19. Reachability, Controllable System, Lest-Norm Input For Reachability, Minimum Energy Over Infinite Horizon, Continuous-Time Reachability, Impulsive Inputs, Least-Norm Input For Reachability Lecture 19 · 1 h 10 min · Video
  20. Continuous-Time Reachability, General State Transfer, Observability And State Estimation, State Estimation Set Up, State Estimation Problem, Observability Matrix, Least-Squares Observers, Some Parting Thoughts..., Linear Algebra, Levels… Lecture 20 · 1 h 9 min · Video

Syllabus and materials

At Stanford Engineering Everywhere, as the course publishes them.

Credits and licence

Institution
Stanford University
Department
Electrical Engineering
Teacher
Stephen Boyd
Level
Undergraduate
Language
English
Subjects
Engineering
Source
Stanford Engineering Everywhere
Licence
CC BY-NC-SA 4.0, as Stanford Engineering Everywhere states it. Non-commercial use only. Stanford Engineering Everywhere's terms
Attribution
Stanford Engineering Everywhere, Stanford University. Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, unless otherwise expressly indicated.
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