Introduction:

The absolute stability and relative stability of the linear time-invariant continuous time closed-loop control system are determined by the location of the closed-loop poles in the s plane. For example, complex closed-loop poles in the left half of the s plane near the jω axis will exhibits oscillatory behavior, and closed-loop poles on the negative real axis will exhibit exponential decay.

Since the complex variables z and s are related by z = e ^Ts , the pole and zero locations in the z plane are related to the pole and zero locations in the s plane. Therefore, the stability of the linear time-invariant discrete-time closed-loop system can be determined in terms of the locations of the poles of the closed-loop pulse transfer function. It is noted that the dynamic behavior of the discrete-time control system depends on the sampling period T. In other words, a change in the sampling period T modifies the pole and zero locations in the z plane and causes the response behavior to change.

In the design of a continuous-time control system, the locations of the poles and zeros in the s plane are very important in predicting the dynamic behavior of the system. Similarly, in designing discrete-time control systems, the locations of the poles and zeros in the z plane are very important. Figure 1 shows the stable region in the s plane and its transformation into the z plane.

In this experiment, we shall investigate how the locations of the poles and zeros in the s plane compare with the locations of the poles and zeros in the z plane.