University:
Kentucky State University
Course:
MAT 172 | Accelerated Contemp. Math
Academic year:

2021
Views:

489

Pages:

7
Author:

Athena Harrison

System Response with Additional Poles It must be emphasized that the formulas describing percent overshoot, settling time, and peak time were derived only for a system with two complex poles and no zeros. If a system has more than two poles or has zeros, we cannot use the formulas to calculate the performance specifications that we derived. However, under certain conditions, a system with more than two poles or with zeros can be approximated as a second-order system that has just two complex dominant poles. 1 System Response with Additional Poles Once we justify this approximation, the formulas for percent overshoot, settling time, and peak time can be applied to these higher-order systems by using the location of the dominant poles. In this section, we investigate the effect of an additional pole on the second-order response. In the next section, we analyse the effect of adding a zero to a two-pole system. 2 System Response with Additional Poles Let us now look at the conditions that would have to exist in order to approximate the behaviour of a three-pole system as that of a two-pole system. Consider a three-pole system with complex poles and a third pole on the real axis. How much farther from the dominant poles does the third pole have to be for its effect on the second-order response to be negligible? 3 System Response with Additional Poles The answer of course depends on the accuracy for which you are looking. This course assumes that if the real pole is five times farther to the left than the dominant poles, we assume that the system is represented by its dominant second-order pair of poles-“ Five times” rule of thumb. 4 System Response with Additional Poles Let us look at an example that compares the responses of two different three pole systems with that of a second-order system. PROBLEM: Find the step response of each of the transfer functions shown below and compare them. 5 System Response with Additional Poles SOLUTION: The step response, Ci(s), for the transfer function, Ti(s), can be found by multiplying the transfer function by 1/s, a step input, and using partial-fraction expansion followed by the inverse Laplace transform to find the response, ci(t). 6 System Response with Additional Poles The three responses are plotted below. Notice that c2(t), with its third pole at −10 and farthest from the dominant poles, is the better approximation of c1(t), the pure second-order system response; c3(t), with a third pole close to the dominant poles, yields the most error. 7

System Response with Additional Poles

System Response with Additional Poles - Page 3

System Response with Additional Poles - Page 4

System Response with Additional Poles - Page 5

System Response with Additional Poles - Page 6

System Response with Additional Poles - Page 7

of 7

Description

Free up your schedule!

Take 5 seconds to unlock