This textbook provides a mathematical introduction to linear systems, with a focus on the continuous-time models that arise in engineering applications such as electrical circuits and signal processing. The book introduces linear systems via block diagrams and the theory of the Laplace transform, using basic complex analysis.
The book mainly covers linear systems with finite-dimensional state spaces. Graphical methods such as Nyquist plots and Bode plots are presented alongside computational tools such as MATLAB. Multiple-input multiple-output (MIMO) systems, which arise in modern telecommunication devices, are discussed in detail. The book also introduces orthogonal polynomials with important examples in signal processing and wireless communication, such as Telatar’s model for multiple antenna transmission. One of the later chapters introduces infinite-dimensional Hilbert space as a state space, with the canonical model of a linear system. The final chapter covers modern applications to signal processing, Whittaker’s sampling theorem for band-limited functions, and Shannon’s wavelet.
Based on courses given for many years to upper undergraduate mathematics students, the book provides a systematic, mathematical account of linear systems theory, and as such will also be useful for students and researchers in engineering. The prerequisites are basic linear algebra and complex analysis.
Author(s): Gordon Blower
Series: Mathematical Engineering
Publisher: Springer
Year: 2023
Language: English
Pages: 416
City: Cham
Preface
Contents
1 Linear Systems and Their Description
1.1 Linear Systems and Their Description
1.2 Feedback
1.3 Linear Differential Equations
1.4 Damped Harmonic Oscillator
1.5 Reduction of Order of Linear ODE
1.6 Exercises
2 Solving Linear Systems by Matrix Theory
2.1 Matrix Terminology
2.2 Characteristic Polynomial
2.3 Norm of a Vector
2.4 Cauchy–Schwarz Inequality
2.5 Matrix Exponential exp(A) or expm (A)
2.6 Exponential of a Diagonable Matrix
2.7 Solving MIMO (A,B,C,D)
2.8 Rational Functions
2.9 Block Matrices
2.10 The Transfer Function of (A,B,C,D)
2.11 Realization with a SISO
2.12 Exercises
3 Eigenvalues and Block Decompositions of Matrices
3.1 The Transfer Function of Similar SISOs (A,B,C,D)
3.2 Jordan Blocks
3.3 Exponentials and Eigenvalues of Complex Matrices
3.4 Exponentials and the Resolvent
3.5 Schur Complements
3.6 Self-adjoint Matrices
3.7 Positive Definite Matrices
3.8 Linear Fractional Transformations
3.9 Stable Matrices
3.10 Dissipative Matrices
3.11 A Determinant Formula
3.12 Observability and Controllability
3.13 Kalman's Decomposition
3.14 Kronecker Product of Matrices
3.15 Exercises
4 Laplace Transforms
4.1 Laplace Transforms
4.2 Laplace Convolution
4.3 Laplace Uniqueness Theorem
4.4 Laplace Transform of a Differential Equation
4.5 Solving MIMO by Laplace Transforms
4.6 Partial Fractions
4.7 Dirichlet's Integral and Heaviside's Expansions
4.8 Final Value Theorem
4.9 Laplace Transforms of Periodic Functions
4.10 Fourier Cosine Transform
4.11 Impulse Response
4.12 Transmitting Signals
4.13 Exercises
5 Transfer Functions, Frequency Response, Realization and Stability
5.1 Winding Numbers
5.2 Realization
5.3 Frequency Response
5.4 Nyquist's Locus
5.5 Gain and Phase
5.6 BIBO Stability
5.7 Undamped Harmonic Oscillator: Marginal Stability and Resonance
5.8 BIBO Stability in Terms of Eigenvalues of A
5.9 Maxwell's Stability Problem
5.10 Stable Rational Transfer Functions
5.11 Nyquist's Criterion for Stability of T
5.12 Nyquist's Criterion Proof
5.13 M and N Circles
5.14 Exercises
6 Algebraic Characterizations of Stability
6.1 Feedback Control
6.2 PID Controllers
6.3 Stable Cubics
6.4 Hurwitz's Stability Criterion
6.5 Units and Factors
6.6 Euclidean Algorithm and Principal Ideal Domains
6.7 Ideals in the Complex Polynomials
6.8 Highest Common Factor and Common Zeros
6.9 Rings of Fractions
6.10 Coprime Factorization in the Stable Rational Functions
6.11 Controlling Rational Systems
6.12 Invariant Factors
6.13 Matrix Factorizations to Stabilize MIMO
6.14 Inverse Laplace Transforms of Strictly Proper Rational Functions
6.15 Differential Rings
6.16 Bessel Functions of Integral Order
6.17 Exercises
7 Stability and Transfer Functions via Linear Algebra
7.1 Lyapunov's Criterion
7.2 Sylvester's Equation AY+YB+C=0
7.3 A Solution of Lyapunov's Equation AL+LA' +P=0
7.4 Stable and Dissipative Linear Systems
7.5 Almost Stable Linear Systems
7.6 Simultaneous Diagonalization
7.7 A Linear Matrix Inequality
7.8 Differential Equations Relating to Sylvester's Equation
7.9 Transfer Functions tf
7.10 Small Groups of Matrices
7.11 How to Convert Complex Matrices into Real Matrices
7.12 Periods
7.13 Discrete Fourier Transform
7.14 Exercises
8 Discrete Time Systems
8.1 Discrete-Time Linear Systems
8.2 Transfer Function for a Discrete Time Linear System
8.3 Correspondence Between Continuous- and Discrete-Time Systems
8.4 Chebyshev Polynomials and Filters
8.5 Hankel Matrices and Moments
8.6 Orthogonal Polynomials
8.7 Hankel Determinants
8.8 Laguerre Polynomials
8.9 Three-Term Recurrence Relation
8.10 Moments via Discrete Time Linear Systems
8.11 Floquet Multipliers
8.12 Exercises
9 Random Linear Systems and Green's Functions
9.1 ARMA Process
9.2 Distributions on a Bounded Interval
9.3 Cauchy Transforms
9.4 Herglotz Functions
9.5 Green's Functions
9.6 Random Diagonal Transformations
9.7 Wigner Matrices
9.8 Pastur's Theorem
9.9 May–Wigner Model
9.10 Semicircle Addition Law
9.11 Matrix Version of Pastur's Fixed Point Equation
9.12 Rank One Perturbations on Green's Functions
9.13 Exercises
10 Hilbert Spaces
10.1 Hilbert Sequence Space
10.2 Hardy Space on the Disc
10.3 Subspaces and Blocks
10.4 Shifts and Multiplication Operators
10.5 Canonical Model
10.6 Hardy Space on the Right Half-Plane
10.7 Paley–Wiener Theorem
10.8 Rational Filters
10.9 Shifts on L2
10.10 The Telegraph Equation as a Linear System
10.11 Exercises
11 Wireless Transmission and Wavelets
11.1 Frequency Band Limited Functions and Sampling
11.2 The Shannon Wavelet
11.3 Telatar's Model of Wireless Communication
11.4 Exercises
12 Solutions to Selected Exercises
Glossary of Linear Systems Terminology
A MATLAB Commands for Matrices
B SciLab Matrix Operations
References
Index
