Poles and Zeros of a System

The roots of the transfer function characteristic equation, i.e., the roots of the denom- inator polynomial, are called poles and correspond to the values of s in the Laplace domain and z in the z-domain for which the transfer function becomes infinite. Poles can be used to extract the natural frequencies and damping of the system. Hence,

the location of system poles is defined by the system properties and is independent of the input. The roots of the transfer function numerator are called zeros, since they represent the values of s and z for which the transfer function is equal to zero. Thus, zeros correspond to the values for which the system attenuates the input. Unlike poles, the location of zeros is defined by properties which relate the system input to the system output. Poles and zeros are either real numbers or appear in complex conjugate pairs. They are essential for understanding the system behavior and can be used to design the system response.

Pole-zero plots, which represent the locations of poles and zeros on the complex plane, are used to graphically represent system dynamics and provide qualitative in- sights into the response characteristics of a system. For continuous signals, poles and zeros are plotted upon the complex s-plane, which is shown in Figure 2.1(a). If all poles are located in the left half plane, the system is stable. Stable systems are defined as systems which have bounded output when subjected to bounded input. Complex poles in the left half plane correspond to oscillation with decreasing ampli- tude, whereas poles on the real axis have damping ratio equal to 1 and correspond to a critically damped system. On the other hand, poles located in the right half plane indicate that the system has unstable modes. Unstable modes are modes for which bounded input results in unbounded output.

Poles and zeros of discrete signals can be plotted upon the z-plane, which is shown in Figure 2.1(b). Systems with poles located inside the unit circle are stable, whereas systems with poles outside the unit circle have unstable modes. Figure 2.1(b) contains countour lines of constant frequency and damping, which provide visual ref- erence to pole frequency and damping values. These contour lines are essentially a grid of natural frequencies from 0 to π in steps of π/10 and a grid of damping ratios from 0 to 1 in steps of 0.1.

Causal or nonanticipatory systems depend on past and current inputs but not on future inputs. Noncausal or anticipatory systems, on the other hand, are systems whose output not only depends on past and current inputs, but also on future inputs. If the system input can be uniquely determined by the output, the system is invert- ible. For invertible systems, multiplication of the system matrix by its inverse yields the unit matrix.

A linear time-invariant system is called a minimum-phase system if the system and its inverse are stable and causal. In the s-domain, minimum-phase systems have all poles and zeros in the left half plane. Analogously, a discrete time system is de- fined as minimum-phase if all poles and zeros lie inside the unit circle. The term "minimum-phase" is used because these systems have the least phase lag among all systems with the same magnitude response. Systems which are stable and causal but have unstable inverses are called nonminimum-phase systems. These systems are characterized by the existence of zeros in the right half plane in the s-domain and outside the unit circle in the z-domain.

The term "nonminimum-phase system" stems from the fact that these systems introduce additional phase lag, while the magnitude of the response remains the

Re(s) Im(s) s-plane decreasing decreasing x x n =1 =0 =0 -j n 1- 2 +j n 1- 2 - n sin -1 unstable unstable -1 -0.5 0 0.5 1 Re(z) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Im(z) 0.9 0.8 0.7 0.60.5 0.4 0.30.2 0.1 1π/T 0.9π/T 0.8π/T 0.7π/T 0.6π/T 0.5π/T0.4 π/T 0.3π/T 0.2π/T 0.1π/T 1π/T 0.9π/T 0.8π_/T 0.7π/T 0.6π/T0.5 π/T 0.4π/T 0.3π/T 0.2π_/T 0.1π/T z-plane (b)

same. This phase lag is obvious when considering the step response function of a nonminimum-phase system, which exhibits initial undershoot, zero crossing or over- shoot [96]. Initial undershoot occurs when the step response initially takes values which do not lie between the initial value and the asymptotic value, and then reaches this target value asymptotically. This behavior is equivalent to a response error. In terms of structural response, the structure moves first in the opposite direction of the applied load and reaches the maximum value of the step response with some de- lay. This structural behavior occurs, for instance, in feedback control systems and in structures subjected to forced excitation. Zero crossing refers to the situation in which the step response passes through the value of zero. Finally, overshoot refers to the situation in which the step response takes values greater than and/or smaller than the asymptotic value of the step response.

In particular, it can be shown that the number of nonminimum-phase zeros has different effects on the system response [96]. Table 2.1 summarizes the effect of non- minimum phase zeros for proper and strictly proper systems. The number of zeros and their nature (real or complex numbers) highly affects the step response of the system. More specifically, the number of zeros determines whether the step response has an initial undershoot, a zero passing or an overshoot. Here, an odd number of zeros implies that there is one real nonminimum zero, whereas an even number of zeros implies that there are only complex nonminimum zeros.

Table 2.1: Effect of nonminimum-phase zeros on the response of strictly proper and exactly

proper systems. Effect distinguished into undershoot, zero crossing and overshoot.

Initial undershoot Zero crossing Overshoot Strictly proper If and only ifH(z) has an odd number of positive zeros

IfH(z) has at least one positive zero

IfH(z)− H(∞) has at least one positive zero Exactly

proper

If and only ifH(z)− H(∞) has an odd number

of positive zeros

IfH(z) has at least one positive zero

IfH(z)− H(∞) has at least one positive zero