Page 376 - Mechanical Engineers' Handbook (Volume 2)

8 Model Classifications 367

difficulty encountered in the analysis and design of nonlinear systems, but also imply entirely
new types of behavior for nonlinear systems that are not possible for linear systems.
The fundamental property of linear systems is superposition. This property states that
if y (t) is the response of the system to u (t) and y (t) is the response of the system to u (t),
1
2
1
2
then the response of the system to the linear combination a u (t) a u (t) is the linear
1 1
2 2
combination a y (t) a y (t). An immediate consequence of superposition is that the re-
2 2
1 1
sponses of a linear system to inputs differing only in amplitude is qualitatively the same.
Since superposition does not apply to nonlinear systems, the responses of a nonlinear system
to large and small changes may be fundamentally different.
This fundamental difference in linear and nonlinear behaviors has a second consequence.
For a linear system, interchanging two elements connected in series does not affect the overall
system behavior. Clearly, this cannot be true in general for nonlinear systems.
A third property peculiar to nonlinear systems is the potential existence of limit cycles.
A linear oscillator oscillates at an amplitude that depends on its initial state. A limit cycle
is an oscillation of fixed amplitude and period, independent of the initial state, that is unique
to the nonlinear system.
A fourth property concerns the response of nonlinear systems to sinusoidal inputs. For
a linear system, the response to sinusoidal input is a sinusoid of the same frequency, poten-
tially differing only in magnitude and phase. For a nonlinear system, the output will in
general contain other frequency components, including possibly harmonics, subharmonics,
and aperiodic terms. Indeed, the response need not contain the input frequency at all.
Linearizing Approximations
Perhaps the most useful technique for analyzing nonlinear systems is to approximate these
with linear systems. While many linearizing approximations are possible, linearization can
frequently be achieved by considering small excursions of the system state about a reference
trajectory. Consider the nonlinear state equation

˙ x(t) ƒ[x(t),u(t)]
0
together with a reference trajectory x (t) and reference input u (t) that together satisfy the
0
state equation
0
˙ x (t) ƒ[x (t),u (t)]
0
0
Note that the simplest case is to choose a static equilibrium or operating point x as the
reference ‘‘trajectory’’ such that 0 t(x,0). The actual trajectory is then related to the
reference trajectory by the relationships
x(t) x (t) x(t)
0
u(t) u (t) u(t)
0
where x(t) is some small perturbation about the reference state and u(t) is some small
perturbation about the reference input. If these perturbations are indeed small, then applying
Taylor’s series expansion about the reference trajectory yields the linearized approximation
˙x(t) A(t) x(t) B(t) u(t)

where the state and distribution matrices are the Jacobian matrices