# The significance of poles and zeros

Kevin Craig
-April 19, 2012

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

**Figure 1**). Assume small displacements. The

**equations**of motion are in matrix form, along with two transfer functions, G

_{0}(s) and G

_{1}(s).

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. For more mechatronics news, visit mechatronicszone.com.*

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