Question

A fairly common basic approach to feedback is to have G(s)=1. This is referred to as a unityfeedback system. For each of the following functions, determine the overall closed-loop transfer function, and then determine if the system is stable. (Where a gain, K, is present, determine any range of K over which the system will be stable.)

H(s)=10/s(s-1)(2s+ 3)

Answer 1

The overall closed-loop transfer function is H(s)=10/s(s-1)(2s+3). The system is stable because all the poles of the transfer function have negative real parts.

Step-by-step explanation:

A unity feedback system has a transfer function of G(s)=1. To determine the overall closed-loop transfer function for H(s)=10/s(s-1)(2s+3), we need to multiply G(s) and H(s) together. Multiplying 1 and H(s) gives us the same transfer function, H(s). So the overall closed-loop transfer function is H(s)=10/s(s-1)(2s+3).

To determine if the system is stable, we need to analyze the poles of the transfer function. The poles are the values of s that make the denominator of the transfer function equal to zero. We set the denominator equal to zero and solve for s:

s(s-1)(2s+3) = 0

This equation has three roots: s=0, s=1, and s=-3/2. All of these roots have negative real parts, which means they are located in the left half of the complex plane. In a unity feedback system, stability is determined by the locations of the poles, and since all the poles of the transfer function have negative real parts, the system is stable.