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Mathematical description of linear dynamical systems pdf

linear and dynamical systems optimization and games answers and the mathematical description of linear dynamical systems kalman pdf
Lecture Notes for EE263 Stephen Boyd Introduction to Linear Dynamical Systems Autumn 2007-08 Copyright Stephen Boyd. Limited copying or use for educational purposes is fine, but please acknowledge source, e.g., “taken from Lecture Notes for EE263, Stephen Boyd, Stanford 2007.”Contents Lecture 1 – Overview Lecture 2 – Linear functions and examples Lecture 3 – Linear algebra review Lecture 4 – Orthonormal sets of vectors and QR factorization Lecture 5 – Least-squares Lecture 6 – Least-squares applications Lecture 7 – Regularized least-squares and Gauss-Newton method Lecture 8 – Least-norm solutions of underdetermined equations Lecture 9 – Autonomous linear dynamical systems Lecture 10 – Solution via Laplace transform and matrix exponential Lecture 11 – Eigenvectors and diagonalization Lecture 12 – Jordan canonical form Lecture 13 – Linear dynamical systems with inputs and outputs Lecture 14 – Example: Aircraft dynamics Lecture 15 – Symmetric matrices, quadratic forms, matrix norm, and SVD Lecture 16 – SVD applications Lecture 17 – Example: Quantum mechanics Lecture 18 – Controllability and state transfer Lecture 19 – Observability and state estimation Lecture 20 – Some final comments Basic notation Matrix primer Crimes against matrices Solving general linear equations using Matlab Least-squares and least-norm solutions using Matlab ExercisesEE263 Autumn 2007-08 Stephen Boyd Lecture 1 Overview • course mechanics • outline & topics • what is a linear dynamical system? • why study linear systems? • some examples 1–1 Course mechanics • all class info, lectures, homeworks, announcements on class web page: www.stanford.edu/class/ee263 course requirements: • weekly homework • takehome midterm exam (date TBD) • takehome final exam (date TBD) Overview 1–2Prerequisites • exposure to linear algebra (e.g., Math 103) • exposure to Laplace transform, differential equations not needed, but might increase appreciation: • control systems • circuits & systems • dynamics Overview 1–3 Major topics & outline • linear algebra & applications • autonomous linear dynamical systems • linear dynamical systems with inputs & outputs • basic quadratic control & estimation Overview 1–4Linear dynamical system continuous-time linear dynamical system (CT LDS) has the form dx =A(t)x(t)+B(t)u(t), y(t) =C(t)x(t)+D(t)u(t) dt where: • t∈R denotes time n • x(t)∈R is the state (vector) m • u(t)∈R is the input or control p • y(t)∈R is the output Overview 1–5 n×n • A(t)∈R is the dynamics matrix n×m • B(t)∈R is the input matrix p×n • C(t)∈R is the output or sensor matrix p×m • D(t)∈R is the feedthrough matrix for lighter appearance, equations are often written x˙ =Ax+Bu, y =Cx+Du • CT LDS is a first order vector differential equation • also called state equations, or ‘m-input, n-state, p-output’ LDS Overview 1–6Some LDS terminology • most linear systems encountered are time-invariant: A, B, C, D are constant, i.e., don’t depend on t • when there is no input u (hence, no B or D) system is called autonomous • very often there is no feedthrough, i.e., D = 0 • when u(t) and y(t) are scalar, system is called single-input, single-output (SISO); when input & output signal dimensions are more than one, MIMO Overview 1–7 Discrete-time linear dynamical system discrete-time linear dynamical system (DT LDS) has the form x(t+1) =A(t)x(t)+B(t)u(t), y(t) =C(t)x(t)+D(t)u(t) where • t∈Z =0,±1,±2,... • (vector) signals x, u, y are sequences DT LDS is a first order vector recursion Overview 1–8Why study linear systems? applications arise in many areas, e.g. • automatic control systems • signal processing • communications • economics, finance • circuit analysis, simulation, design • mechanical and civil engineering • aeronautics • navigation, guidance Overview 1–9 Usefulness of LDS • depends on availability of computing power, which is large & increasing exponentially • used for – analysis & design – implementation, embedded in real-time systems • like DSP, was a specialized topic & technology 30 years ago Overview 1–10Origins and history • parts of LDS theory can be traced to 19th century • builds on classical circuits & systems (1920s on) (transfer functions . . . ) but with more emphasis on linear algebra • first engineering application: aerospace, 1960s • transitioned from specialized topic to ubiquitous in 1980s (just like digital signal processing, information theory, . . . ) Overview 1–11 Nonlinear dynamical systems many dynamical systems are nonlinear (a fascinating topic) so why study linear systems? • most techniques for nonlinear systems are based on linear methods • methods for linear systems often work unreasonably well, in practice, for nonlinear systems • if you don’t understand linear dynamical systems you certainly can’t understand nonlinear dynamical systems Overview 1–12y y Examples (ideas only, no details) • let’s consider a specific system x˙ =Ax, y =Cx 16 with x(t)∈R , y(t)∈R (a ‘16-state single-output system’) • model of a lightly damped mechanical system, but it doesn’t matter Overview 1–13 typical output: 3 2 1 0 −1 −2 −3 0 50 100 150 200 250 300 350 t 3 2 1 0 −1 −2 −3 0 100 200 300 400 500 600 700 800 900 1000 t • output waveform is very complicated; looks almost random and unpredictable • we’ll see that such a solution can be decomposed into much simpler (modal) components Overview 1–14t 0.2 0 −0.2 0 50 100 150 200 250 300 350 1 0 −1 0 50 100 150 200 250 300 350 0.5 0 −0.5 0 50 100 150 200 250 300 350 2 0 −2 0 50 100 150 200 250 300 350 1 0 −1 0 50 100 150 200 250 300 350 2 0 −2 0 50 100 150 200 250 300 350 5 0 −5 0 50 100 150 200 250 300 350 0.2 0 −0.2 0 50 100 150 200 250 300 350 (idea probably familiar from ‘poles’) Overview 1–15 Input design add two inputs, two outputs to system: x˙ =Ax+Bu, y =Cx, x(0) = 0 16×2 2×16 where B∈R , C ∈R (same A as before) 2 problem: find appropriate u :R →R so that y(t)→y = (1,−2) + des simple approach: consider static conditions (u, x, y constant): x˙ = 0 =Ax+Bu , y =y =Cx static des solve for u to get:  
LeonBrown,Hawaii,Researcher
Published Date:14-07-2017

EE263: Introduction to Linear Dynamical Systems

Stephen Boyd, Stanford University.

Archive

This page is an archive of the course as it was taught by Professor Stephen Boyd in 2008. The materials here should be very close to those used for the video lectures.

Video lectures

Prerequisites

Exposure to linear algebra and matrices (as in Math. 103). 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.

Catalog 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. EE263 covers some of the same topics, but is complementary to, CME200.

Course reader

The EE263 course reader is one pdf file consisting of a cover page together with the lecture slides, support notes and homework exercises below.

Lecture slides

  1. Overview

  2. Linear functions

  3. Linear algebra review

  4. Orthonormal sets of vectors and QR factorization

  5. Least-squares

  6. Least-squares applications

  7. Regularized least-squares and Gauss-Newton method

  8. Least-norm solutions of underdetermined equations

  9. Autonomous linear dynamical systems

  10. Solution via Laplace transform and matrix exponential

  11. Eigenvectors and diagonalization

  12. Jordan canonical form

  13. Linear dynamical systems with inputs and outputs

  14. Example: Aircraft dynamics

  15. Symmetric matrices, quadratic forms, matrix norm, and SVD

  16. SVD applications

  17. Example: Quantum mechanics

  18. Controllability and state transfer

  19. Observability and state estimation

  20. Summary and final comments

All lectures, in 2up format, in one pdf file

Support notes

Additional background notes.

Homework problems

All EE263 homework problems in one file.

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