8.2 The root locus and proportional control

Suppose that we simplify the block diagram of Figure 8.1 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 8.1 into a single transfer function \(H\), as shown in [@fig:block-diagram-feedback-02]. 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, \[ G_\text{CL}(s) = \frac{K G(s)}{1 + K G(s) H(s)}\ , \] shown in the block diagram of [@fig:block-diagram-feedback-03], we also have the open-loop transfer functionOpen-loop transfer function \[ K G(s) H(s), \] the concatenation of forward and feedback transfer functions.

Let us break down \(G\) and \(H\) into …