Page 15 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Contents XY

F-4 Laplace Transform of the Impulse or Dirac
Delta Function . . . . . . . . . . . . . . . . . . . . . . . . 398
F-5 Laplace Transform of the Exponential
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
F-6 Laplace Transform of a Sinusoid . . . . . . . . 399
F-7 Final Value Theorem . . . . . . . . . . . . . . . . . . . 400
F-8 Laplace Transform Tables . . . . . . . . . . . . . . 400
F-9 Laplace Transform of the Time Domain
Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
F-10 Laplace Transform of Higher Derivatives 401
F-11 Laplace Transform of an Integral . . . . . . . . 402
F-12 The Laplace Transform Recipe . . . . . . . . . . 403
F-13 Applying the Laplace Transform to the First-
Order Model: The Transfer Function . . . . . 404
F-14 Applying the Laplace Transform to the First-
Order Model: The Impulse Response 404
F-15 Applying the Laplace Transform to the
First-Order Model: The Step Response . . . 406
F-16 Partial Fraction Expansions Applied to Laplace
Transforms: The First-Grder Problem . . . . 406
F-17 Partial Fraction Expansions Applied to Laplace
Transforms: The Second-Order Problem . . 408
F-18 A Precursor to the Convolution Theorem 409
F-19 Using the Integrating Factor to Obtain the
Convolution Integral . . . . . . . . . . . . . . . . . . 410
F-20 Application of the Laplace Transform to a
First-Order Partial Differential Equation 413
F-21 Solving the Transformed Partial Differential
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
F-22 The Magnitude and Phase of the Transformed
Partial Differential Equation . . . . . . . . . . . . 417
F-23 A Brief History of the Laplace Transform 418
F-24 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
G Vectors and Matrices • . . • . . . . . . . . . . . . . . • . . • . • • . 421
G-1 Addition and Multiplication of Matrices 423
G-2 Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . 424
G-3 State-Space Equations and Laplace
Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
G-4 Transposes and Diagonal Matrices 427
G-5 Determinants, Cofactors, and Adjoints
of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
G-6 The Inverse Matrix . . . . . . . . . . . . . . . . . . . . 429
G-7 Some Matrix Calculus . . . . . . . . . . . . . . . . . 432
G-8 The Matrix Exponential Function
and Infinite Series . . . . . . . . . . . . . . . . . . . . . 432