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Lecture Notes on Linear System Theory

Lecture Notes on Linear System Theory, linear system theory: Introduction, introduction to Algebra, introduction to Analysis, time varying linear systems Solutions, Time invariant linear systems, Solutions and transfer functions,. | Lecture Notes on Linear System Theory John Lygeros and Federico A. Rampond Automatic Control Laboratory ETH Zurich CH-8092 Zurich Switzerland lygeros@control.ee.ethz.ch Department of Information Engineering University of Brescia Via Branze 38 25123 Brescia Italy federico.ramponi@unibs.it January 3 2015 Contents 1 Introduction 1 1.1 Objectives of the course. 1 1.2 Proof methods . 2 1.3 Functions and maps. 8 2 Introduction to Algebra 11 2.1 Groups. 11 2.2 Rings and fields. 12 2.3 Linear spaces. 15 2.4 Subspaces and bases. 17 2.5 Linear maps . 21 2.6 Linear maps generated by matrices. 24 2.7 Matrix representation of linear maps . 25 2.8 Change of basis . 27 3 Introduction to Analysis 33 3.1 Norms and continuity . 33 3.2 Equivalent norms . 35 3.3 Infinite-dimensional normed spaces. 38 3.4 Completeness. 39 3.5 Induced norms and matrix norms . 43 3.6 Ordinary differential equations . 46 3.7 Existence and uniqueness of solutions . 51 3.7.1 Background lemmas . 51 3.7.2 Proof of existence . 53 3.7.3 Proof of uniqueness . 56 4 Time varying linear systems Solutions 59 4.1 Motivation Linearization about a trajectory . 59 i 4.2 Existence and structure of solutions . 61 4.3 State transition matrix. 63 5 Time invariant linear systems Solutions and transfer functions 70 5.1 Time domain solution. 70 5.2 Semi-simple matrices. 71 5.3 Jordan form . 74 5.4 Laplace transforms . 77 6 Stability 85 6.1 Nonlinear systems Basic definitions. 85 6.2 Linear time varying systems. 93 6.3 Linear time invariant systems. 95 6.4 Systems with inputs and outputs . 98 6.5 Lyapunov equation . 100 7 Inner product spaces 104 7.1 Inner product . 104 7.2 The space of square-integrable functions . 106 7.3 Orthogonal complement . 107 7.4 Adjoint of a linear map . 109 7.5 Finite rank lemma . 112 7.6 Application Matrix pseudo-inverse . 114 8 Controllability and observability 118 8.1 Nonlinear systems . 118 8.2 Linear time varying systems Controllability . 121 8.3 Linear time varying systems Minimum energy .