Page 808 - The Mechatronics Handbook

0066_Frame_C25 Page 7 Wednesday, January 9, 2002 7:05 PM

Under mild technical conditions involving controllability and observability of the state space model, the
transfer function representations in (25.13) and (25.15) are similar in case the state space model in (25.2)
is derived from the differential equation (25.1) and vice versa.

25.2 Dynamic Response

The Laplace and z-transform offer the possibility to compute the dynamic response of a dynamic system
by means of algebraic manipulations. The analysis of the dynamic response gives insight into the dynamic
behavior of the system by addressing the response to typical test signals such as impulse, step, and sinusoid
excitation of the system.
The response can be computed for relatively simple continuous or discrete dynamical systems given
by low order differential or difference equations. Both the state space model and the transfer function
descriptions provide helpful representations in the analysis of a dynamic system. The result are presented
in the following.

Pulse and Step Response
A possible way to evaluate the response of a dynamic system is by means of pulse and step based test signals.
For continuous time systems an input impulse signal is defined as a δ function

 ∞, t = 0
u imp t() := d t() = 
 0, t ≠ 0

with the property
∞
∫ t=−∞ ft()d t() = f 0()

where f(t) is an integrable function over (−∞, ∞). Although an impulse signal is not practical from an
experiment point of view, the computation or simulation of the impulse response gives insight into the
transient behavior of the dynamical system.
With the properties of the impulse function δ(t) mentioned above, the Laplace transform of the impulse
function is given by

∞
{
L d t()} = d s() = ∫ d t()e – st t d = e – s0 = 1
t=0
Hence the output y(s) due to an impulse input is given by y imp (s) = G(s)u imp (s) = G(s)δ(s) = G(s). As a
result, an immediate inverse Laplace transform of the continuous time transfer function G(s),

1
–
y imp t() = L { Gs()}
gives the dynamic response y imp (t) of the system to an impulse input response.
The computation of the step response is done in a similar way. In (25.9), the Laplace transform of the
step signal

 0, t < 0
u step t() := 
 1, t ≥ 0
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