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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 pdfLecture 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:
EE263: Introduction to Linear Dynamical Systems
Stephen Boyd, Stanford University.
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.
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.
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.
The EE263 course reader is one pdf file consisting of a cover page together with the lecture slides, support notes and homework exercises below.
Linear algebra review
Orthonormal sets of vectors and QR factorization
Regularized least-squares and Gauss-Newton method
Least-norm solutions of underdetermined equations
Autonomous linear dynamical systems
Solution via Laplace transform and matrix exponential
Eigenvectors and diagonalization
Jordan canonical form
Linear dynamical systems with inputs and outputs
Example: Aircraft dynamics
Symmetric matrices, quadratic forms, matrix norm, and SVD
Example: Quantum mechanics
Controllability and state transfer
Observability and state estimation
Summary and final comments
All lectures, in 2up format, in one pdf file
Additional background notes.
All EE263 homework problems in one file.