Page 60 - Biosystems Engineering

Biosystems Analysis and Optimization 41

only be written as an explicit function of the applied force f(t) by solv-
ing the differential equation [Eq. (2.10)], which is often a tedious and
time-consuming task. Thanks to the separation of signals and sys-
tems in the Laplace domain, the transfer function G(s), which con-
verts the applied force signal F(s) into a position signal X(s), can be
obtained by simply inverting Eq. (2.23):

Gs() = Xs() = 1 (2.24)
2
Fs() ms + cs + k
The transfer function G(s) in Eq. (2.24) is a rational polynomial in the
complex variable s. This system is said to be of the second order, because
the highest power of the Laplace variable s present in the denominator of
the transfer function G(s) is 2, which corresponds to the differential equa-
tion of the second order in the time domain [Eq. (2.10)].

Subsystems in Series
For a system consisting of two subsystems connected in series, the
input signal of the second subsystem u (t) equals the output signal
2
of the first subsystem y (t). In the time domain combination of the
1
transfer functions of both subsystems would mean incorporating
the first differential equation into the second differential equation.
Thanks to the separation of signals and systems in the Laplace
domain, combination of different subsystems placed in series also
becomes a simple algebraic operation. For each subsystem the trans-
fer function can be defined as a ratio of the output signal Y (s) over
k
the input signal U (s). The transfer function of the system G(s) con-
k
sisting of two subsystems connected in series can then be defined as
the product of the transfer functions of the subsystems G (s) and
1
G (s):
2
()
Ys () Ys Ys ()
Gs () = 2 = 2 1 = Gs G ()s (2.25)
()
Us () Ys Us () 2 1 1
()
1 1 1
For example, let us add a second mass m to the previously described
2
mass–spring–damper system (Fig. 2.4), which is connected to the first
mass, now noted as m for reason of clarity, by a hydraulic actuator
1
oriented in the same direction as the spring and damper and with a
time-dependent length x(t) (Fig. 2.5). In this system, the oil flow to the
actuator q(t) can be seen as the input signal, and the position x (t) of
2
the second mass m (e.g., height of the implement) can be seen as the
2
output signal.
The hydraulic actuator subsystem converts the oil flow to the
actuator q(t) into an actuator length x(t). Because the speed of the
actuator shaft, and not the position, is proportional to the oil flow q(t),