Page 368 - Mechanical Engineers' Handbook (Volume 2)

8 Model Classifications 359

Such problems can be addressed either by neglecting the higher frequency components where
appropriate or by adopting special numerical integration methods for stiff systems.

Selecting an Integration Method
The best numerical integration method for a specific simulation is the method that yields an
acceptable global approximation error with the minimum amount of round-off error and
computing effort. No single method is best for all applications. The selection of an integration
method depends on the model simulated, the purpose of the simulation study, and the avail-
ability of computing hardware and software.
In general, for well-behaved problems with continuous derivatives and no stiffness, a
lower order Adams predictor is often a good choice. Multistep methods also facilitate esti-
mating local truncation error. Multistep methods should be avoided for systems with dis-
continuities, however, because of the need for frequent restarts. Runge–Kutta methods have
the advantage that these are self-starting and provide fair stability. For stiff systems where
high-frequency modes have little influence on the global response, special stiff-system meth-
ods enable the use of economically large step sizes. Variable-step rules are useful when little
is known a priori about solutions. Variable-step rules often make a good choice as general-
purpose integration methods.
Round-off error usually is not a major concern in the selection of an integration method,
since the goal of minimizing computing effort typically obviates such problems. Double-
precision simulation can be used where round-off is a potential concern. An upper bound
on step size often exists because of discontinuities in derivative functions or because of the
need for response output at closely spaced time intervals.

Continuous-System Simulation Languages
Digital simulation can be implemented for a specific model in any high-level language such
as FORTRAN or C. The general process for implementing a simulation is shown in Fig. 26.
In addition, many special-purpose continuous-system simulation languages are commonly
available across a wide range of platforms. Such languages greatly simplify programming
tasks and typically provide for good graphical output.

8 MODEL CLASSIFICATIONS

Mathematical models of dynamic systems are distinguished by several criteria which describe
fundamental properties of model variables and equations. These criteria in turn prescribe the
theory and mathematical techniques that can be used to study different models. Table 11
summarizes these distinguishing criteria. In the following sections, the approaches adopted
for the analysis of important classes of systems are briefly outlined.

8.1 Stochastic Systems
Systems in which some of the dependent variables (input, state, output) contain random
components are called stochastic systems. Randomness may result from environmental fac-
tors, such as wind gusts or electrical noise, or simply from a lack of precise knowledge of
the system model, such as when a human operator is included within a control system. If
the randomness in the system can be described by some rule, then it is often possible to
derive a model in terms of probability distributions involving, for example, the means and
variances of model variables or parameters.