In this final module, you'll learn about continuous-time reachability and general state transfer concepts. The module discusses observability and state estimation processes, providing a comprehensive conclusion to the course's topics and offering insights into future learning paths.
This module introduces the fundamental concepts of linear dynamical systems, explaining their significance and real-world applications. You'll explore various examples, including estimation and filtering, and gain a deeper understanding of linear functions.
This module continues the discussion on linear functions, delving into practical examples such as linear elastic structures and static circuits. You'll learn about linearization and first-order approximation methods, enhancing your skills in interpreting linear relationships.
This module focuses on linearization techniques, particularly in navigation via range measurements. You will review matrix multiplication and learn about the concepts of basis, dimension, and nullspace, reinforcing your understanding of linear algebra.
Continuing from the previous module, this section explores the nullspace of a matrix, discussing its range, rank, and the conservation of dimension. You'll also learn about matrix inverses and their applications in matrix-vector multiplication.
This module introduces orthonormal sets of vectors, including their geometric interpretations and applications. You will learn the Gram-Schmidt procedure and how to perform QR factorization, essential for various applications in least-squares problems.
This module covers the principles of least-squares methods, providing geometric interpretations and the projection onto the column space of a matrix. You'll also explore least-squares estimation techniques and their applications in navigation and data fitting.
This module focuses on least-squares polynomial fitting, discussing optimal residual norms and model order selection. You will also learn about cross-validation and recursive least-squares, which are vital for model accuracy and performance.
This module delves into multi-objective least-squares, discussing weighted-sum objectives and regularized approaches. You will learn about nonlinear least-squares methods and their practical applications, enhancing your ability to solve complex optimization problems.
This module addresses least-norm solutions, illustrating how they can be achieved through QR factorization and Lagrange multipliers. You'll explore practical examples and concepts related to regularized least-squares, enhancing your understanding of norm minimization.
This module presents examples of autonomous linear dynamical systems, including discrete-time Markov chains and numerical integration techniques. Youâll gain insights into high-order systems and their linearization near equilibrium points.
In this module, you will learn about solutions via Laplace transforms and matrix exponentials. It includes examples like the harmonic oscillator and double integrator, emphasizing the relationship between eigenvalues and system stability.
This module discusses the time transfer property in linear systems, exploring qualitative behaviors, stability, and the role of eigenvectors in diagonalization. You will also understand invariant sets and their implications in system dynamics.
This module provides an in-depth look at Markov chains, covering diagonalization, distinct eigenvalues, and their applications in stability analysis for discrete-time systems, including insights into Jordan canonical form and generalized eigenvectors.
This module introduces Jordan canonical form, discussing its role in linear dynamical systems and the Cayley-Hamilton theorem. You will learn about generalized modes and how to represent systems with inputs and outputs effectively.
This module explores the concept of the DC or static gain matrix within linear systems, including discretization techniques and causality principles. You will also learn about symmetric matrices and their eigenvalues, with practical examples to enhance comprehension.
This module examines an RC circuit as a practical example, discussing quadratic forms and their properties. Youâll learn about positive definiteness, matrix inequalities, and their significance in understanding system behavior.
This module discusses the gain of a matrix in a specific direction, introducing singular value decomposition and its applications. You will learn about the pseudo-inverse and the image of unit balls under linear transformations, emphasizing the importance of sensitivity in linear equations.
This module explores the sensitivity of linear equations to data errors, discussing low-rank approximations and applications in model simplification. You will gain insights into controllability, state transfer, and their roles in discrete-time linear dynamical systems.
This module covers reachability in linear dynamical systems, defining controllable systems and the concept of least-norm inputs for achieving reachability. You will explore minimum energy solutions and the implications for continuous-time systems.
In this final module, you'll learn about continuous-time reachability and general state transfer concepts. The module discusses observability and state estimation processes, providing a comprehensive conclusion to the course's topics and offering insights into future learning paths.