Wilbur Wright’s understanding of complex, dynamic
problems contributed to his and his brother Orville’s
successful first airplane flight. Wilbur understood,
for example, that, to turn a bicycle to the left, you must first
turn the handlebars a little to the right and then, as the
bicycle inclines to the left, you must turn them a little to the
left. He understood countersteering. In mathematical language,
the transfer function between the steer torque applied
to the handlebars and the straight-line-path deviation has a
right-half-plane zero, which imposes a limit on maneuverability.
The path deviation has an inverse-response behavior;
that is, in response to a positive step-torque input you apply
to the handlebars, the path deviation is initially positive
and then becomes negative. This effect has contributed to
numerous motorcycle accidents, but countersteering could
prevent these accidents.
To better understand the physical significance of the
poles and zeros of a transfer function, consider a simpler
system, comprising two rigid links and a torsional spring (see
). Assume small displacements. The equations
motion are in matrix
form, along with two transfer functions,
(s) and G1
A pole of a transfer function is a
value of s that makes the denominator
equal to zero, and a zero of a
transfer function is a value of s that
makes the numerator equal to zero.
Systems that have no poles or zeros
in the right half of the complex plane
are minimum-phase systems because
either of the two components of the
frequency response, gain and phase,
contains all the frequency-response information that exists.
This phenomenon, Bode’s gain-phase relationship, stipulates
that systems that have poles in the right half of the plane are
unstable. A nonminimum-phase stable system is one that has a
zero in the right half of the plane. Physical phenomena that give
rise to nonminimum-phase stable behavior include control of the level of a volume of boiling water
and hydroelectric power generation.
The denominators of both transfer
functions are identical. The
double pole at the origin represents
the rigid-body motion of the system.
The complex-conjugate pole pair
represents the natural frequency
associated with the energy-storage
characteristics, including kinetic and
potential energy, of the physical system.
They are independent of the
locations of the sensor (θ0 or θ1) and
the actuator (T). At a frequency of
the complex pole, energy can freely
transfer back and forth between the
kinetic and the potential energy, and
the system behaves as an energy reservoir.
The numerators of the two systems differ greatly. The complex
zero represents the natural frequency associated with the
energy-storage characteristic of a subportion of the system.
The sensor and the actuator impose
artificial constraints that define this
subportion. These constraints include
the resonant frequency of the second
link when the first link is fixed. It
is lower than the natural frequency
of the system, and it corresponds to
the frequency at which the system
behaves as an energy sink, such that
the energy-storage elements of a subportion
of the original system completely
trap the energy that the input
applies. Thus, no output can ever be
detected at the point of measurement. The zero in the right
half of the plane is a nonminimum-phase zero and gives rise
to the same characteristic initial inverse response that Wilbur
Wright observed in the bicycle. The locations of the poles
and the zeros of a transfer function are the result of design
decisions and can make control easy or difficult.
Kevin C Craig, PhD,
is the Robert C Greenheck
chairman in engineering
design and a professor
of engineering at the
College of Engineering
at Marquette University.
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