Page 81 - Modern Control Systems

Section 2.3 Linear Approximations of Physical Systems 55
One immediately notes the equivalence of Equations (2.5) and (2.2) where veloc-
ity v(t) and voltage v(t) are equivalent variables, usually called analogous variables,
and the systems are analogous systems. Therefore the solution for velocity is similar to
Equation (2.4), and the response for an underdamped system is shown in Figure 2.4.
The concept of analogous systems is a very useful and powerful technique for system
modeling. The voltage-velocity analogy, often called the force-current analogy, is a
natural one because it relates the analogous through- and across-variables of the elec-
trical and mechanical systems. Another analogy that relates the velocity and current
variables is often used and is called the force-voltage analogy [21,23].
Analogous systems with similar solutions exist for electrical, mechanical, ther-
mal, and fluid systems. The existence of analogous systems and solutions provides
the analyst with the ability to extend the solution of one system to all analogous sys-
tems with the same describing differential equations. Therefore what one learns
about the analysis and design of electrical systems is immediately extended to an
understanding of fluid, thermal, and mechanical systems.

2.3 LINEAR APPROXIMATIONS OF PHYSICAL SYSTEMS

A great majority of physical systems are linear within some range of the variables.
In general, systems ultimately become nonlinear as the variables are increased with-
out limit. For example, the spring-mass-damper system of Figure 2.2 is linear and
described by Equation (2.1) as long as the mass is subjected to small deflections y(t).
However, if y(t) were continually increased, eventually the spring would be overex-
tended and break. Therefore the question of linearity and the range of applicability
must be considered for each system.
A system is defined as linear in terms of the system excitation and response.
In the case of the electrical network, the excitation is the input current r(t) and the
response is the voltage v(t). In general, a necessary condition for a linear system
can be determined in terms of an excitation x(t) and a response y(t). When the
system at rest is subjected to an excitation Xi(t), it provides a response yi(t). Fur-
thermore, when the system is subjected to an excitation x 2(t), it provides a corre-
sponding response y 2(t). For a linear system, it is necessary that the excitation
Xi(t) + x 2(t) result in a response y x(t) + yz(i). This is usually called the principle
of superposition.
Furthermore, the magnitude scale factor must be preserved in a linear system.
Again, consider a system with an input x{t) that results in an output y(t). Then the
response of a linear system to a constant multiple /3 of an input x must be equal to
the response to the input multiplied by the same constant so that the output is equal
to f3y. This is called the property of homogeneity.

A linear system satisfies the properties of superposition and homogeneity.

A system characterized by the relation y = x 2 is not linear, because the super-
position property is not satisfied. A system represented by the relation y = mx + b
is not linear, because it does not satisfy the homogeneity property. However, this