Lecture

Linearization (Continued)

This module delves deeper into the topic of linearization, discussing techniques and applications in navigation through range measurement. It includes:

  • Broad categories of applications for linearization
  • Matrix multiplication as a mixture of columns
  • Block diagram representation of systems
  • A review of linear algebra concepts including basis, dimension, and nullspace

Course Lectures
  • This module introduces the concept of linear dynamical systems, discussing their importance in various fields. It provides illustrative examples to showcase the application of linear functions in practical scenarios.

    Key topics include:

    • Overview of linear dynamical systems
    • Reasons for studying linear dynamical systems
    • Examples from estimation and filtering
  • This module continues the exploration of linear functions by providing various interpretations and applications. It covers examples from different domains including:

    • Linear elastic structures
    • Rigid body mechanics
    • Static circuits
    • Cost of production analysis
    • Network traffic considerations
    • Illumination with multiple lamps

    Additionally, it introduces concepts of linearization and first-order approximation of functions.

  • This module delves deeper into the topic of linearization, discussing techniques and applications in navigation through range measurement. It includes:

    • Broad categories of applications for linearization
    • Matrix multiplication as a mixture of columns
    • Block diagram representation of systems
    • A review of linear algebra concepts including basis, dimension, and nullspace
  • This module continues the discussion on the nullspace of matrices, covering essential concepts such as:

    • Range of a matrix and its implications
    • Inverse and rank of a matrix
    • Conservation of dimension principles
    • Practical applications like fast matrix-vector multiplication
    • Change of coordinates
    • Understanding Euclidean norm and inner product
    • Orthonormal sets of vectors
  • This module focuses on orthonormal sets of vectors, demonstrating their geometric interpretation and applications. Key topics include:

    • Understanding the Gram-Schmidt procedure
    • General Gram-Schmidt applications
    • 'Full' QR factorization and its significance
    • Orthogonal decomposition induced by a matrix
    • Least-squares applications
  • Least-Squares
    Stephen Boyd

    This module introduces the least-squares method, providing a geometric interpretation and explaining its significance in various contexts. Topics covered include:

    • Approximate solutions using least-squares
    • Projection on R(A) space
    • QR factorization method for least-squares
    • Least-squares estimation and its properties
    • Application in navigation from range measurements
    • Data fitting using least-squares
  • This module addresses least-squares polynomial fitting, focusing on various methods and considerations in model identification. Key topics include:

    • Norm of optimal residuals in polynomial fitting
    • System identification through least-squares
    • Model order selection and cross-validation techniques
    • Recursive least-squares approaches
    • Introduction to multi-objective least-squares
  • This module expands on multi-objective least-squares, discussing how to minimize multiple objectives effectively. Topics include:

    • Weighted-sum objective and its minimization
    • Regularized least-squares techniques
    • Laplacian regularization methods
    • Nonlinear least-squares (NLLS) approaches
    • Gauss-Newton method and examples
    • Least-norm solutions of underdetermined equations
  • Least-Norm Solution
    Stephen Boyd

    This module covers least-norm solutions, detailing methods to find solutions in underdetermined systems. It includes:

    • Least-norm solutions through QR factorization
    • Derivation via Lagrange multipliers
    • Practical example: transferring mass unit distance
    • Relation to regularized least-squares
    • General norm minimization with equality constraints
    • Introduction to autonomous linear dynamical systems
    • Block diagram representation of systems
  • This module presents examples of autonomous linear dynamical systems, illustrating their practical applications. Key content includes:

    • Finite-state discrete-time Markov chains
    • Numerical integration techniques for continuous systems
    • High-order linear dynamical systems
    • Mechanical systems and their dynamics
    • Linearization near equilibrium points and along trajectories
  • This module explores solutions via the Laplace transform and matrix exponential methods. It covers:

    • Laplace transform solutions for linear systems
    • Examples: harmonic oscillator and double integrator
    • Characteristic polynomial and eigenvalues
    • Matrix exponential and its time transfer property
  • Time Transfer Property
    Stephen Boyd

    This module focuses on the time transfer property, discussing system behavior over time. Topics include:

    • Piecewise constant systems and their qualitative behavior
    • Stability analysis and its importance
    • Eigenvectors and diagonalization techniques
    • Dynamic interpretations and invariant sets
    • Summary of concepts with an example of Markov chains
  • Markov Chain (Example)
    Stephen Boyd

    This module explores the application of Markov chains as an example of linear dynamical systems. It examines:

    • Diagonalization techniques for system analysis
    • Understanding distinct eigenvalues and their implications
    • Modal form and its applications
    • Examples of diagonalization and system stability
    • Jordan canonical form and generalized eigenvectors
  • Jordan Canonical Form
    Stephen Boyd

    This module introduces the Jordan canonical form, explaining its significance in analyzing linear systems. Key topics include:

    • Understanding generalized modes of a system
    • Cayley-Hamilton theorem and its proof
    • Linear dynamical systems with inputs and outputs
    • Block diagram representation and transfer matrices
    • Impulse and step matrices in system analysis
  • This module focuses on the DC or static gain matrix, discussing its role in system dynamics. It includes:

    • Discretization techniques with piecewise constant inputs
    • Understanding causality and the concept of state
    • Change of coordinates in system representation
    • Z-transform and its applications
    • Symmetric matrices and their properties
    • Quadratic forms and their interpretations
    • Examples, including the RC circuit analysis
  • RC Circuit (Example)
    Stephen Boyd

    This module examines the RC circuit as an example, exploring quadratic forms and their implications. Key points include:

    • Examples of quadratic forms in system analysis
    • Inequalities for quadratic forms and their significance
    • Positive semidefinite and positive definite matrices
    • Matrix inequalities and their applications
    • Understanding ellipsoids and their geometric interpretations
    • Gain of a matrix in a specific direction
    • Properties of matrix norms and their relevance
  • This module explores the concept of gain in a direction, providing insights into singular value decomposition (SVD) and its applications. Key topics include:

    • Understanding gain of a matrix in a specific direction
    • Applications of singular value decomposition
    • General pseudoinverse and its relation to regularization
    • Full SVD and its implications in data analysis
    • Image of the unit ball under linear transformation
    • Sensitivity of linear equations to data error
  • This module addresses the sensitivity of linear equations to data error, exploring concepts of low-rank approximations and model simplification. Topics include:

    • Understanding distance to singularity in systems
    • Application of low-rank approximations
    • Controllability and state transfer concepts
    • Reachability and its significance for discrete-time linear dynamical systems
  • Reachability
    Stephen Boyd

    This module investigates reachability in linear dynamical systems, discussing controllable systems and minimum energy input requirements. Key themes include:

    • Understanding controllable systems and their properties
    • Least-norm input strategies for reachability
    • Minimum energy considerations over infinite horizons
    • Continuous-time reachability and its applications
    • Impulsive inputs and their significance
  • This final module addresses continuous-time reachability, emphasizing general state transfer and observability in linear systems. Topics include:

    • Setting up state estimation problems
    • Understanding observability matrices
    • Least-squares observers and their applications
    • Concluding thoughts on linear algebra and future learning paths