Final answer:
To sketch the root locus, identify poles and zeros of the transfer function, determine the root locus segments on the real axis, calculate asymptotes, find the breakaway point, and sketch the root locus to observe how the poles move as K varies.
Step-by-step explanation:
The student has asked about sketching the root locus for a given system with an open-loop transfer function K G(s)H(s) = K/(-1)(+2)(+3). The root locus is a graphical representation used in control systems engineering to determine the stability of a system as the gain, K, varies. To sketch the root locus, follow these steps:
- Identify the poles and zeros of the transfer function. Poles are at s = -1, s = -2, and s = -3, and there are no zeros.
- Determine the root locus segments on the real axis. Since there are an odd number of poles to the right of each point on the real axis between the poles, these segments will be part of the root locus.
- Calculate the asymptotes, which in this case will be centered at -2 and diverge at angles of ±60° and 180° from the real axis.
- Determine the breakaway point by finding the maximum point of the real part of the poles.
- Sketch the root locus, showing how the poles of the closed-loop system move in the complex plane as K increases from 0 to ∞.
Remember that the root locus provides valuable insight into system stability and helps in the design and analysis of controller gains.