Page 309 - Mechanical Engineers' Handbook (Volume 2)

Mechanical Engineers’ Handbook: Instrumentation, Systems, Controls, and MEMS, Volume 2, Third Edition.

Edited by Myer Kutz

Copyright  2006 by John Wiley & Sons, Inc.

CHAPTER 10
MATHEMATICAL MODELS OF
DYNAMIC PHYSICAL SYSTEMS
K. Preston White, Jr.
Department of Systems and Information Engineering
University of Virginia
Charlottesville, Virginia
1 RATIONALE 300 5.1 Transform Methods 322
5.2 Transient Analysis Using
2 IDEAL ELEMENTS 302 Transform Methods 328
2.1 Physical Variables 303 5.3 Response to Periodic Inputs
2.2 Power and Energy 303 Using Transform Methods 332
2.3 One-Port Element Laws 304
2.4 Multiport Elements 307 6 STATE-VARIABLE METHODS 340
6.1 Solution of the State
3 SYSTEM STRUCTURE AND Equation 342
INTERCONNECTION LAWS 311 6.2 Eigenstructure 350
3.1 A Simple Example 311
3.2 Structure and Graphs 311 7 SIMULATION 352
3.3 System Relations 313 7.1 Simulation—Experimental
3.4 Analogs and Duals 314 Analysis of Model Behavior 352
7.2 Digital Simulation 353
4 STANDARD FORMS FOR
LINEAR MODELS 314 8 MODEL CLASSIFICATIONS 359
4.1 I/O Form 315 8.1 Stochastic Systems 359
4.2 Deriving the I/O Form— 8.2 Distributed-Parameter
An Example 316 Models 364
4.3 State-Variable Form 318 8.3 Time-Varying Systems 365
4.4 Deriving the ‘‘Natural’’ State 8.4 Nonlinear Systems 366
Variables—A Procedure 320 8.5 Discrete and Hybrid
4.5 Deriving the ‘‘Natural’’ State Systems 376
Variables—An Example 320
4.6 Converting from I/O to REFERENCES 382
‘‘Phase-Variable’’ Form 321
BIBLIOGRAPHY 382
5 APPROACHES TO LINEAR
SYSTEMS ANALYSIS 321

1 RATIONALE
The design of modern control systems relies on the formulation and analysis of mathematical
models of dynamic physical systems. This is simply because a model is more accessible to
study than the physical system the model represents. Models typically are less costly and
less time consuming to construct and test. Changes in the structure of a model are easier to
implement, and changes in the behavior of a model are easier to isolate and understand. A
model often can be used to achieve insight when the corresponding physical system cannot,
because experimentation with the actual system is too dangerous or too demanding. Indeed,
a model can be used to answer ‘‘what if’’ questions about a system that has not yet been
realized or actually cannot be realized with current technologies.
300