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Introduction to Linear Dynamical Systems

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Introduction to Linear Dynamical Systems

COURSE DESCRIPTION

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. 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.



Lecture 1 - Overview Of Linear Dynamical Systems

Overview Of Linear Dynamical Systems, Why Study Linear Dynamical Systems?, Examples Of Linear Dynamical Systems,
Estimation/Filtering Example, Linear Functions And Examples



Lecture 2 - Linear Functions (Continued)

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,
Network Traffic And Flow Example, Linearization And First Order Approximation Of Functions



Lecture 3 - Linearization (Continued)

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 4 - Nullspace Of A Matrix (Continued)

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
Product, Orthonormal Set Of Vectors



Lecture 5 - Orthonormal Set Of Vectors

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 6 - Least-Squares

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 7 - Least-Squares Polynomial Fitting

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 8 - Multi-Objective Least-Squares

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 Of Undetermined Equations



Lecture 9 - Least-Norm Solution

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, Autonomous Linear Dynamical Systems, Block Diagram



Lecture 10 - Examples Of Autonomous Linear Dynamical Systems

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,
Linearization Along Trajectory



Lecture 11 - Solution Via Laplace Transform And Matrix Exponential

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, Time Transfer Property



Lecture 12 - Time Transfer Property

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 13 - Markov Chain (Example)

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 14 - Jordan Canonical Form

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 15 - DC Or Static Gain Matrix

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,
Interpretations Of Eigenvalues Of Symmetric Matrices, Example: RC Circuit



Lecture 16 - RC Circuit (Example)

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, Properties Of Matrix Norm



Lecture 17 - Gain Of A Matrix In A Direction

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
Transformation, SVD In Estimation/Inversion, Sensitivity Of Linear Equations To Data Error



Lecture 18 - Sensitivity Of Linear Equations To Data Error

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 19 - Reachability

Reachability, Controllable System, Lest-Norm Input For Reachability, Minimum Energy Over Infinite Horizon,
Continuous-Time Reachability, Impulsive Inputs, Least-Norm Input For Reachability



Lecture 20 - Continuous-Time Reachability

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 Of
Understanding, What's Next

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