Dynamical System Models

Contents 
Chapter One Components 
1.1 Introduction  
1.2 Translational Mechanical Variables 
1.3 Translational and Rotational Mechanical Components 18 
1.3.1 Translational Spring 24 
1.3.2 Translational Mass 27 
1.3.3 Translational Converters 29 
1.3.4 Representation of Translational Mechanical 
Systems by Linear Graphs 33 
1.3.5 Relationships between Translational and 
Rotational Mechanical Interactions 39 
1.3.6 Rotational Mechanical Components 41 
1.3.7 Mechanical Transformers 51 
1.4 Fluid Components 52 
1.4.1 Fluid Capacitance 53 
1.4.2 Fluid Inertance 59 
1.4.3 Fluid Dissipatance 61 
1.4.4 Representation of Lumped Fluid Systems by 
Linear Graphs 63 
1.4.5 Wave Propagation in Fluid Systems 64 
1.5 Thermal System Components 66 
1.6 Electrical System Components 74 
1.6.1 Capacitor 77 
1.6.2 Inductor 80 
1.6.3 Converters 83 
1.6.4 Representation of Lumped Electrical Systems by 
Linear Graphs 85 
1.6.5 Wave Propagation in Electrical Systems 86 
Chapter Two Spaces 
2.1 Sets 90 
2.2 Metric Spaces 92 
2.3 Linear Vector Spaces 92 
2.4 Linear Independence and Bases 94 
2.5 Inner Product 96 
2.6 Matrix Representation of Linear Operators in an 
n-dimensional Linear Vector Space 98 
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Contents 
2.7 Dual Basis and the Projection Theorem 100 
2.8 Matrix Representation of Change of Basis in an 
n-dimensional Euclidean Vector Space 101 
2.9 Coordinate Transformations 102 
2.10 Representation of an Operator Matrix in a Different 
Basis 102 
2.11 Eigenvalues and Eigenvectors of a Linear Operator 
Matrix Representation 103 
2.12 Scalar-valued Functions of Vectors 109 
2.13 Range Space, Null Space, Rank and Nullity of a Linear 
Operator 110 
2.14 Vector Differentiation 112 
2.15 Differentiation and Integration of Matrices and 
Determinants 114 
2.16 Direct Sum of Matrices 115 
2.17 Extremum Values of Vector Functions Subject to 
Constraints 116 
2.18 Extremum Characteristics of Eigenvalues 120 
2.19 Matrix Functions 124 
2.20 Miscellaneous Notes 125 
Chapter Three Transform Models 
3.1 Laplace Transform 126 
3.2 Transfer Function Relationships 143 
3.2.1 Convolution 145 
3.2.2 Asymptotic Relations between Weighting 
Function and Transfer Function 149 
3.2.3 Response to Sinusoidal Input 150 
3.2.4 Determination of System Response from Transfer 
Function Pole and Zero Distribution 151 
3.2.5 Relationships between Real and Imaginary Part 
of Transfer Function 154 
3.2.6 Minimum-phase Transfer Functions 158 
3.3 Block Diagrams 159 
3.3.1 Block Diagram Conventions 159 
3.3.2 Block Diagram Manipulations 163 
3.4 Signal-flow Graphs 170 
3.4.1 Signal-flow Graph Conventions 170 
3.4.2 Signal-flow Graph Manipulations 172 
3.4.3 Mason's Circuit Rule 177 
3.5 Nyquist's Stability Criterion 184 
3.5.1 Complex Plane Mappings 184 
3.5.2 Open- and Closed-loop Transfer Function 
Relationships 187 
Contents 
IX 
3.6 
3.5.3 Basic Closed-loop Stability Theorem 
3.5.4 Simple Form of Nyquist's Criterion 
3.5.5 General Form of Nyquist's Criterion 
3.5.6 Relative Stability Criteria 
Root Locus Method 
3.6.1 General Rules for Construction of Root Loci 
Optimal Linearization and the Describing Function 
z-transforms for Discrete Systems 
3.8.1 Basis Sequences 
3.8.2 Discrete Operators 
3.8.3 z-transform 
3.8.4 Convolution of Sequences 
3.8.5 z-transfer Functions 
3.7 
3.8 
188 
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191 
192 
194 
197 
201 
206 
206 
206 
207 
214 
215 
Chapter Four Hamiltonian Models 
4.1 Fundamental Processes of the Calculus of Variations 217 
4.1.1 Conditions for Stationary Values of Definite 
Integral 220 
4.1.2 Stationary Value of Integral with Fixed End- 
points and Several Dependent Variables 223 
4.2 Generalized Coordinates 226 
4.2.1 Generalized Velocities 226 
4.2.2 Generalized Forces 227 
4.3 Primal Form of Hamilton's Postulate and the Set of 
Lagrangian Equations 227 
4.3.1 Lagrangian Equation Set 232 
4.4 Generalized Momenta 238 
4.5 Dual Form of Hamilton's Postulate and the Set of 
Co-Lagrangian Equations 239 
4.6 Conservation of Energy and Momentum 241 
4.7 Hamilton's Equations 243 
4.7.1 Hamilton-Jacobi Equation 244 
4.7.2 Liouville's Theorem 245 
4.8 Hamiltonian Principles for Electrical Networks 246 
4.9 Pontryagin's Equations 248 
4.9.1 Optimal Control Problem 251 
4.9.2 Event Vector and Event Space 251 
4.9.3 The Set of Possible Events 252 
4.9.4 Expanding Wave-fronts in the State Space 253 
4.9.5 Analogy with Huygens' Principle in Geometrical 
Optics 254 
4.9.6 Pontryagin's Maximum Principle 255 
4.9.7 Generation of Optimal Trajectory 255 
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Contents 
4.9.8 Derivation of Pontryagin's Equations 257 
4.9.9 Derivation for the Case When the Control 
Inputs are Unrestricted 262 
4.9.10 Minimization of an Integral Functional of 
System Motion 265 
4.10 Maximal-effort or "Bang-bang" Systems 270 
Chapter Five Network Models 
5.1 Basic Definitions for Linear Graphs 285 
5.2 Interconnective Constraints on Power Variables 287 
5.3 Topological Relationships between Network Variables 290 
5.4 Tellegen's Theorem 296 
5.5 The Dynamical Transformation Matrix 298 
5.6 Analogues, Duals and Dualogues 301 
5.7 Circuit, Vertex and Mixed Transform Analysis Methods 
for Linear Electrical Networks 304 
5.7.1 Circuit Method 308 
5.7.2 Vertex Method 313 
5.7.3 Mixed Method 318 
5.8 Systems Matrix Analysis of Networks 319 
5.9 Lagrangian Equations for Networks 323 
5.10 Co-Lagrangian Equations for Networks 325 
5.11 Special Variational Principles for Networks 329 
5.12 Formulation of Canonical Equation Sets for Linear 
Networks 338 
5.13 Formulation of State Space Equations for Nonlinear 
Networks 363 
5.13.1 Integral Invariants 369 
5.13.2 Derivation of Canonical Equation Set 371 
5.13.3 Construction of Scalar Functions 377 
Chapter Six State Models 
6.1 Analytical Aspects of State Space Equation Sets 385 
6.1.1 Existence and Uniqueness of Solutions 385 
6.1.2 Singular Points and the Liapunov First- 
Approximation Matrix 389 
6.1.3 Simple Trajectory Properties in the State Plane 393 
6.1.4 Analytical Solutions of Linear Equation Sets 396 
6.2 Stability 420 
6.2.1 Liapunov Stability Theory 424 
6.3 Modality 434 
Contents XI 
6.4 Discrete Model Approximation of Linear Constant 
Coefficient Systems 437 
6.5 Functional Matrices 439 
6.5.1 Equivalent Free-motion Systems for Network 
Impulse, Step and Ramp Response 444 
6.6 Generalized Mohr Circles and their Use in Feedback 
Design 446 
6.6.1 Generalized Mohr Circles 447 
6.6.2 Properties of Generalized Mohr Circles 449 
6.7 Controllability and Observability 455 
6.7.1 Controllablity 456 
6.7.2 o bserva bili ty 459 
6.7.3 Decomposition of State Space Systems 461 
6.7.4 Duality and Adjoint Systems 462 
6.7.5 Determination of the Controllable Part of a 
Given Representation 466 
6.7.6 Determination of the Observable Part of a 
Given Representation 469 
6.7.7 Determination of Transfer Function Matrix 
Representations 471 
6.8 Reduction 476 
6.8.1 Optimal Orthogonal Projection on to a Subspace 480 
6.8.2 Optimal Projection Along an Invariant Subspace 483 
Appendix A 
References 
Index 
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490 
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