Page 63 - Biosystems Engineering

44 Chapter Two

where X(s) and X (s) are, respectively, the Laplace transforms of the
2
actuator length signal x(t) and the position x (t).
2
Combination of both subsystem transfer functions using Eq. (2.25)
and elimination of X(s) leads to the transfer function G(s) for the total
system:

+
Xs () ms + cs k
2
Gs () = 2 = 1 (2.34)
2
Qs () ( m + m s ) 3 + cs + ks
1 2
2.2.4 Nonlinearities
Unfortunately, the dynamic behavior of many real-life systems cannot
be accurately described by a combination of the nice linear building
blocks mentioned earlier because they also exhibit some nonlinear
behavior. Although modeling linear dynamic systems is fairly straight-
forward using the principles discussed in the previous sections, mod-
eling nonlinear systems is more complex. It has, however, been shown
in the literature that a wide range of nonlinear systems can be mod-
eled as a cascade of linear dynamic systems and nonlinear static
elements (Nelles 2001). When a system has a nonlinear element before
the linear dynamic system, it is called a Hammerstein model, and a sys-
tem with output nonlinearity is known as a Wiener model.
Two architectures are commonly used to describe nonlinear sys-
tems. The first consists of a linear dynamic system followed by a non-
linear static element and another linear dynamic system, known as
the Wiener–Hammerstein model. The second architecture is composed
of a linear dynamic system preceded and followed by nonlinear static
elements, known as the Hammerstein–Wiener model.
In this section, we present an overview of some nonlinear elements
that are common for biologically related systems, together with some
examples of systems containing these nonlinear elements.

Dead Zone
A linear system model describes the output signal as a (frequency-
dependent) multiple of the input. However, many real-life systems
will only react when the input signal is sufficiently large. We can thus
define a band or zone of input signals for which the system remains
unaffected as the dead zone.
For example, in a tooth-wheel connection there typically is some
slop, called backlash, which makes that the first wheel (input) has to
rotate a little before touching the second wheel (output) and put it
into rotation as well. Another example can be found in the dead zone
of a hydraulic valve. Only when the current generates a sufficiently
large magnetic force through its solenoid will the valve open and
generate an oil flow. The static relation between the input and the
output of a system with dead zone is illustrated in Fig. 2.6.