Page 375 - Mechanical Engineers' Handbook (Volume 2)

366 Mathematical Models of Dynamic Physical Systems

where (t,t ) is the time-varying state transition matrix. This knowledge is typically of little
0
value, however, since it is not usually possible to determine the state transition matrix by
any straightforward method. By analogy with the first-order case, the relationship
(t,t ) exp A( ) d

t
0
t 0
can be proven valid if and only if

t t
A(t) A( ) d A( ) d A(t)
t 0 t 0
that is, if and only if A(t) and its integral commute. This is a very stringent condition for
all but a first-order system and, as a rule, it is usually easiest to obtain the solution using
simulation.
Most of the properties of the fixed transition matrix extend to the time-varying case:
(t,t ) I
0
1
(t,t ) (t ,t)
0
0
(t ,t ) (t ,t ) (t ,t )
1
2
0
2
1
0
(t,t ) A(t) (t,t )
0
0
8.4 Nonlinear Systems
The theory of fixed, linear, lumped-parameter systems is highly developed and provides a
powerful set of techniques for control system analysis and design. In practice, however, all
physical systems are nonlinear to some greater or lesser degree. The linearity of a physical
system is usually only a convenient approximation, restricted to a certain range of operation.
In addition, nonlinearities such as dead zones, saturation, or on–off action are sometimes
introduced into control systems intentionally, either to obtain some advantageous perform-
ance characteristic or to compensate for the effects of other (undesirable) nonlinearities.
Unfortunately, while nonlinear systems are important, ubiquitous, and potentially useful,
the theory of nonlinear differential equations is comparatively meager. Except for specific
cases, closed-form solutions to nonlinear systems are generally unavailable. The only uni-
versally applicable method for the study of nonlinear systems is simulation. As described in
Section 7, however, simulation is an experimental approach, embodying all of the attending
limitations of experimentation.
A number of special techniques are available for the analysis of nonlinear systems. All
of these techniques are in some sense approximate, assuming, for example, either a restricted
range of operation over which nonlinearities are mild or the relative isolation of lower order
subsystems. When used in conjunction with more complex simulation models, however, these
techniques often provide insights and design concepts that would be difficult to discover
through the use of simulation alone. 8

Linear versus Nonlinear Behaviors
There are several fundamental differences between the behavior of linear and nonlinear
systems that are especially important. These differences not only account for the increased