Page 14 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

xi¥ Contents

A-10 A Useful Test Function . . . . . . . . . . . . . . . . . 352
A-ll Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
B Complex Numbers . . . . . . . . . . . . . • . . • . . • . . • . . . . 357
B-1 Complex Conjugates . . . . . . . . . . . . . . . . . . . 359
B-2 Complex Numbers as Vectors or Phasors . 360
8-3 Euler's Equation . . . . . . . . . . . . . . . . . . . . . . 361
8-4 An Application to a Problem in Chapter 4 . 364
8-5 The Full Monty . . . . . . . . . . . . . . . . . . . . . . . . 366
8-6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

C Spectral Analysis . . . . . . . . . • . . • . . • . . • . . • . . • . . • . 369
C-1 An Elementary Discussion of the Fourier
Transform as a Data-Fitting Problem 369
C-2 Partial Summary . . . . . . . . . . . . . . . . . . . . . . 373
C-3 Detecting Periodic Components . . . . . . . . . 374
C-4 The Line Spectrum . . . . . . . . . . . . . . . . . . . . 374
C-5 The Exponential Form of the Least Squares
Fitting Equation . . . . . . . . . . . . . . . . . . . . . . . 376
C-6 Periodicity in the Trme Domain . . . . . . . . . 378
C-7 Sampling and Replication . . . . . . . . . . . . . . 378
C-8 Apparent Increased Frequency Domain
Resolution via Padding . . . . . . . . . . . . . . . . . 379
C-9 The Variance and the Discrete Fourier
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
C-10 Impact of Increased Frequency Resolution
on. V~ability of the Power Spectrum . . . . . 382
C-11 Aliasmg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
C-12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

D Infinite and Taylor's Series • . . • . . • . . • . . • . . • . . • . 385
D-1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
E Application of the Exponential Function to
Differential Equations . . . . . . . . . . • . . • . . • . . • . . . . 389
E-1 First-Order Differential Equations . . . . . . . . 389
E-2 Partial Summary . . . . . . . . . . . . . . . . . . . . . . . 391
E-3 Partial Solution of a Second-Order
Differential Equation . . . . . . . . . . . . . . . . . . . 391
E-4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
F The Laplace Transform . . . . • . . • . . • . . • . . • . . • . . • . 395
F-1 Laplace Transform of a Constant
(or a Step Change) . . . . . . . . . . . . . . . . . . . . . 396
F-2 Laplace Transform of a Step at a Trme
Greater than Zero . . . . . . . . . . . . . . . . . . . . . . 396
F-3 Laplace Transform of a Delayed Quantity 397