- Background
- Linear State Space Models
- Introduction to Laplace Transforms
- Transfer Functions
- First-Order Behavior
- Integrating System
- Second-Order Behavior
- Lead-Lag Behavior
- Poles and Zeros
- Processes with Dead Time
- Padé Approximation for Dead Time
- Converting State Space Models to Transfer Functions
- Matlab and Simulink
- Summary
- References
- Student Exercises

## 3.9 Poles and Zeros

There are a number of different ways to represent process transfer functions. The "polynomial" form is

The values of *s* that cause the numerator of Equation (3.49) to equal zero are known as the "zeros" of the transfer function. The values of *s* that cause the denominator of Equation (3.49) to equal zero are known as the "poles" of the transfer function.

The "pole-zero" form is

and complex poles (or zeros) must occur in complex conjugate pairs.

The "gain-time constant" form is the one that we use most often for control system design.

where t* _{ni}* is a numerator time constant and t

*is a denominator time constant. This form is normally used when the roots (poles) of the denominator polynomial are real.*

_{pi}#### Example 3.7: Comparison of Various Transfer Function Forms

Consider a transfer function with the following gain-time constant form

The polynomial form is

The gain-polynomial form is

and the pole-zero form is

The zero is 1/10, and the poles are –1/3 and –1/15.

Notice that the zero for Example 3.7 is positive. A positive zero is called a right-half-plane (*RHP*) zero, because it appears in the right half of the complex plane (with real and imaginary axes). *RHP* zeros have a characteristic *inverse response*, as shown in Figure 3-11 for t* _{n}* = -10 (which corresponds to a zero of +0.1).

Also notice that the poles are negative (left-half-plane), indicating a stable process. *RHP* (positive) poles are unstable. Recall that complex poles will yield an oscillatory response. A pole-zero plot of the transfer function in Example 3.7 is shown in Figure 3-13 [the pole locations are (-1/3,0) and (-1/15,0) and the zero location is (1/10,0), with the coordinates (real,imaginary)]. For this system, there is no imaginary component and the poles and zeros lie on the real axis.

**Figure 3-13. Pole-zero location plot for (x, poles; o, zero).**

As poles move further to the left they yield a faster response, and increasing the magnitude of the imaginary portion makes the response more oscillatory. This behavior is summarized in Figure 3-14. Recall also that a process with a pole at the origin (and none in the *RHP*) is known as an *integrating* system; that is, the system never settles to a steady state when a step input change is made.

**Figure 3-14. Effect of pole-zero location on dynamic behavior (x, poles; o, zero). As poles become more negative, the response is faster. As the imaginary/real ratio increases, the response becomes more oscillatory.**

Multiple *RHP* zeros cause multiple "changes in direction"; for example, with two *RHP* zeros, the step response, initially going in one direction, switches direction, then switches back to the initial direction.