Sparsity Methods for Systems and Control

Author

Textbook

Table of Contents with Slides and Lecture Videos

• Chapter 1: Introduction
• 1.1 Occam’s Razor
• 1.2 Group Testing
• 1.3 Optimization with l1 Norm
• 1.4 Sparsity Methods for Systems and Control
• Chapter 2: What is Sparsity? (slides, lecture video)
• 2.1 Redundant Dictionary
• 2.2 Underdetermined Systems
• 2.3 The l0 Norm
• 2.4 Exhaustive Search
• 2.5 Sparse Representation for Functions
• 2.6 Further Readings
• Chapter 3: Curve Fitting and Sparse Optimization (slides, lecture video)
• 3.1 Least Squares and Regularization
• 3.2 Sparse Polynomial and l1-norm Optimization
• 3.3 Numerical Optimization by CVX
• 3.4 Further Readings
• Chapter 4: Algorithms for Convex Optimization (slides, lecture video 1, lecture video 2)
• 4.1 Basics of Convex Optimization
• 4.2 Proximal Operator
• 4.3 Proximal Splitting Methods for l1 Optimization
• 4.4 Proximal Gradient Method for l1 Regularization
• 4.5 Generalized LASSO and ADMM
• 4.6 Further Reading
• Chapter 5: Greedy Algorithms (slides)
• 5.1 l0 Optimization
• 5.2 Orthogonal Matching Pursuit
• 5.3 Thresholding Algorithm
• 5.4 Numerical Example
• 5.5 Further Reading
• Chapter 6: Applications of Sparse Representation (slides, lecture video)
• 6.1 Sparse Representations for Splines
• 6.2 Discrete-time Hands-off Control
• 6.3 Further Reading
• Chapter 7: Dynamical Systems and Optimal Control (slides)
• 7.1 Dynamical System
• 7.2 Minimum-time Control
• 7.3 Minimum-time Control of Rocket
• 7.4 Further Reading
• Chapter 8: Maximum Hands-off Control (slides)
• 8.1 L0 Norm and Sparsity
• 8.2 Practical Benefits of Sparsity in Control
• 8.3 Problem Formulation of Maximum Hands-off Control
• 8.4 L1-optimal Control
• 8.5 Equivalence Between L0 and L1 Optimal Controls
• 8.6 Existence of L0-Optimal Control
• 8.7 Maximum Hands-off Control of Rocket
• 8.8 Further Reading
• Chapter 9: Numerical Optimization by Time Discretization (slides)
• 9.1 Time Discretization
• 9.2 Controllability of Discretized Systems
• 9.3 Reduction to Finite-dimensional Optimization
• 9.4 Fast Algorithm by ADMM
• 9.5 MATLAB Programs
• 9.6 Further Reading
• Chapter 10: Advanced Topics
• 10.1 Smooth Hands-off Control by Mixed L1/L2 Optimization
• 10.2 Discrete-valued Control
• 10.3 Time-Optimal Hands-off Control
• 10.4 Further Reading

MATLAB Codes

• For some programs for convex optimization, you need CVX toolbok. CVX can be downloaded from
  http://cvxr.com/cvx/
• I have checked the following programs on MATLAB 2019a.
• To run the programs in Chapter 4 (image denoising with total variation), you need image processing toolbox.

• chap3_curve_fitting_cvx.m (Numerical example in Section 3.3)

• chap4_tv_denoising.m (Total variation denoising example in Section 4.5.2)
• cat.jpg an image to be proccessed
• soft_thresholding.m (soft-thresholding operator)

• MP.m (matching pursuit, MP)
• OMP.m (orthogonal matching pursuit, OMP)
• hard_thresholding.m (hard-thresholding operator)
• supp.m (support of a vector)
• IHT.m (iterative hard-thresholding algorithm, IHT)
• s_sparse_operator.m (s-sparse operator)
• iterative_s_sparse.m (iterative s-sparse algorithm)
• CoSaMP.m (CoSaMP)
• chap5_curve_fitting_greedy.m (numerical example in Section 5.4)

• chap9_sparse_control_cvx.m (L1-optimal control by CVX)
• chap9_sparse_control_ADMM.m (L1-optimal control by ADMM
• chap10_SOAV_control.m (Discrete-valued control by SOAV)
• get_mat2.m (get_mat2.m)