# The significance of poles and zeros

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 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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