7.2 The root locus and proportional control

Suppose that we simplify the block diagram of figure 7.1 such that controller \(G_C\)⁠, amplifier \(G_A\)⁠, and plant \(G_P\) became just two blocks, a simple controller gain \(K\) and a transfer function \(G\)⁠. Suppose further that we combined the feedback path transfer functions of figure 7.1 into a single transfer function \(H\)⁠, as shown in figure 7.4a. We call \(K G\) the forward transfer functionForward transfer function and \(H\) the feedback transfer functionFeedback transfer function. In addition to the closed-loop transfer function, \[ G_\text{CL}(s) = \frac{K G(s)}{1 + K G(s) H(s)}\ , \tag{7.7}\] shown in the block diagram of figure 7.4b, we also have the open-loop transfer functionOpen-loop transfer function \[ K G(s) H(s), \] the concatenation of forward and feedback transfer functions.