Good Luck on the Final
Use the forum for final questions!
Linear Dynamical Systems (sometimes also called Linear Operator Theory refers to a mathematical representation of a physical system that can be represented by a set of 1st order differential equations or 1st order difference (or recursion) equations for discrete time systems. Generally, these systems can be written in a very simple (and very overloaded form) of:
The study of these linear systems started historically in the 1960's and required a Ph.D. in math as a necessary prerequisite. Most of the applications at the time were to aerospace control problems (such as rocket guidance). Today, these types of systems are studied extensively, and applications range from controls to economics. Frequently, these problems are cast as dual problems: design (where the input vector is altered to reach a desired output) and estimation (where a set a sensor measurements are processed to estimate the state of the system).
The only prerequisites for this class are exposure to Linear Algebra and Differential Equations (AMS/ENG 27 fulfills these just fine). A class on circuits (EE 70), controls (EE 154), and/or dynamics (PHYS 5/6) would be useful, but are by no means critical. The only other prerequisites are a willingness to do the work, which will be hard at times.
This course is based on the Introduction to Linear Dynamical Systems sequence (EE263 and EE363), offered at Stanford by Professor Stephen Boyd. Lecture notes are taken from his published lecture notes, "EE263: Introduction to Linear Dynamical Systems," Fall 2004.
I would like to acknowledge the tremendous help and generosity of Prof. Stephen Boyd of Stanford University in teaching the subject matter to me, for all of his help with the slides, the homeworks, and the course materials. I would also like to thank Prof. Ed Carryer at Stanford University for pioneering this video capture technology, and helping me to set it up. Without their help and inspiration, this class would not be here.
Index of class resources
General Class Information class and section times, instructor and TA information
Lecture Video Video files of the lectures, and download information for the right codec.
Handouts homework problem sets, homework solutions, other helpful handouts
WebForum - for announcements, general discussion, and help
The technology to record these videos is supported by a grant from the Center for Teaching Excellence (CTE), and it is an experiment. Feedback as to the utility, and the usability of these videos would be highly appreciated. The basic hardware required is a tablet PC with the Office Tablet PC extensions, and a standard headset to capture the lecturers voice. Additionally, a program called Camtasia is used to capture the entire sequence into a standard movie format that can then be viewed at a later time for review and additional study.
You may view these lectures at any time, but do not distribute them beyond the UCSC environment. These lectures have been created using the Camtasia software, and can be played through the Camtasia player software, downloadable for free from techsmith here, or through the standard windows media player with the techsmith codec. A Mac OSX version of the codec can be found here that allows playback of the files.
- Lecture #0, 22-Sept-2005, Introduction to Linear Dynamical Systems.
- Lecture #1, 27-Sept-2005, Linear Functions.
- Lecture #2, 29-Sept-2005, Linear Algebra Review.
- Lecture #3, 05-Oct-2005, Linear Algebra Review (con't).
- Lecture #4, 06-Oct-2005, Orthonormal Vectors and QR Factorization
- Lecture #5, 11-Oct-2005, Least Squares Approximate Solution
- Lecture #6, 14-Oct-2005, Least Squares Applications
- Lecture #7, 18-Oct-2005, Multi-Objective Least Squares
- Lecture #8, 20-Oct-2005, Min-Norm Solutions
- Lecture #9, 25-Oct-2005, Autonomous Linear Dynamical Systems
- Lecture #10, 27-Oct-2005, Laplace Transforms and State Transition Matrices
- Lecture #11, 1-Nov-2005, The Matrix Exponential
- Lecture #12, 3-Nov-2005, Eigenvalues and Eigenvectors
- Lecture #13, 8-Nov-2005, Jordan Canonical Form
- Lecture #14, 10-Nov-2005, Cayley-Hamilton and LDS with I/O
- Lecture #15, 15-Nov-2005, Impulse and Step Response matrices
- Lecture #16, 17-Nov-2005, Symmetric Matrices and Quadratic Form
- Lecture #17, 22-Nov-2005, Singular Value Decomposition
- Lecture #18, 29-Nov-2005, Controllability and Reachability
- Lecture #19, 1-Dec-2005, Observability
Homeworks are handed out in class, and are due back either in class or in my office, 337B Engineering 2, at 6 PM on the Tuesday of the following week. Homeworks will only be accepted at the beginning of class, not at the end of class. Homeworks turned in late will be receive half the total points once the solution set has been posted. Cooperation and collaboration on the homeworks is encouraged, but this is NOT licence to copy. The work you turn in should be your own.
- Homework #1: Introducation to Linear Dynamical Systems, Due 04-Oct-2005
- Homework #2: Some Simple Design and Estimation, Due 11-Oct-2005
- Homework #3: QR Factorization and Gram-Schmidt, Due 18-Oct-2005
- Homework #4: Least Squares and Least Norm Solutions, Due 25-Oct-2005
- Homework #5: Autonomous Linear Dynamical Systems, Due 03-Nov-2005
- Homework #6: Linear Systems and the Matrix Exponential, Due 15-Nov-2005
- Homework #7: Linear Systems with Inputs and Outputs, Due 23-Nov-2005
- Homework #8: Quadratic forms, SVD, Due 29-Nov-2005
- color_perception.m: MATLAB file for Homework #2
- inductor_data.m: MATLAB file for Homework #3
- deconv_data.m: MATLAB file for Homework #3
- emissions_data.m: MATLAB file for Homework #4
- interconn.m: MATLAB file for Homework #7
- time_comp_data.m: MATLAB file for Homework #7
- Homework #1 Solution Set
- Homework #2 Solution Set
- Homework #3 Solution Set
- Homework #4 Solution Set
- Homework #5 Solution Set
- Homework #6 Solution Set
- Homework #7 Solution Set
- Homework #8 Solution Set
Class Presentation Slides
The class lectures use the digital ink capabilities of the TabletPC. The ink is saved back into the presentation, and the presentation is saved to the website for convenience. This year we are using Classroom Presenter rather than PowerPoint. It apprears to be far more stable, and has several nice utilities for the TabletPC. The presentation files are in the .CSD format, and you will need to download Presenter to view them. Presenter can be downloaded here.
- Lecture #0: Introduction to Linear Dynamical Systems
- Lecture #1: Linear Functions
- Lecture #1(b): Example Problems
- Lecture #2: Linear Algebra Review
- Lecture #3: Orthonormal Basis and QR decomposition
- Lecture #4: Least Squares
- Lecture #5: Least Squares Applications
- Lecture #6: Regularized Least Squares
- Lecture #7: Least Norm and Minimum Norm Solutions
- Lecture #8: Autonomous Linear Dynamical Systems
- Lecture #9: The Matrix Exponential
- Lecture #10: Eigenvalues and Eigenvectors
- Lecture #11: The Jordan Form
- Lecture #12: Linear Systems with Inputs and Outputs
- Lecture #13: Symmetric Matrices, Quadratic Forms, SVD
- Lecture #14: Singular Value Decomposition Applications
- Lecture #15: Controllability
- Lecture #16: Observability
General Class Information
- Lecture times:
- Tuesday-Thursday, 10:00 - 11:45 PM, E2-506
- Class Webforum:
- WebForum - for announcements, general discussion, and help
- Textbooks: note that these are NOT required, but are excellent references
- Linear System Theory and Design by Chi-Tsong Chen, Oxford University Press, 1999. ISBN: 0030602890.
- Linear Systems by Thomas Kailath, Prentice-Hall, 1980. ISBN: 0135369614.
- Linear Algebra and its Applications, 3rd Ed. by Gilbert Strang, Brooks Cole, 1988. ISBN: 0155510053.
- Name: Gabriel Hugh Elkaim (email@example.com)
- Phone: 831-459-3054
- Office: Engineering 2, 337B
- Instructor Office Hours:
- Tuesday-Thursday, 2:00 - 4:00 PM, and by appointment
- Teaching Assistants:
- TBD (unlikely to be any)
Plotymeasand ˆyon the same graph. Plot the residual (the diference between thesetwo signals) on a diferent graph, and give its RMS value.Solution.We’ll Form the estimated signal as a linear combination oFf0,... ,f10,g0,... ,g10,ˆy=x1f0+x2f1+···+x11f10+x12g0++x22g10.(Here we are representing the signals as vectors inR500.) We can write this in matrixForm as ˆy=Ax, whereA= [f0f1f10g0g10]∈R500×22.The coe±cientsxare chosen to minimize the RMS deviation between ˆyandymeas,which is the same as minimizing the norm oF the diference. The matrixAis Full rank(i.e., 22), so the best coe±cients are given byxls= (ATA)-1ATymeas.Our estimate oF the original signal isˆy=Axls=A(ATA)-1ATymeas.The Following Matlab code implements this estimation method.% load measured y datasig_est_data;nfcts = 22; ydim = 500;t = 1:ydim;% building A matrixA = zeros(ydim,nfcts);for k = 1:nfcts/2fk = exp(-(t-50*(k-1)).^2/25^2);gk = (t-50*(k-1))/10.*exp(-(t-50*(k-1)).^2/25^2);A(:,k) = fk’; A(:,k+11) = gk’;end% least squares estimationyhat = A*(A\ymeas);residual = ymeas-yhat;RMS = 1/sqrt(500)*norm(residual);figure(1); plot(t,ymeas,’g--’,t,yhat,’k’);xlabel(’time’); ylabel(’fit’);figure(2); plot(t,ymeas-yhat);2