# 6

# Dynamic Behavior of Closed-Loop Control Systems

It is important to predict the stability and the robustness to uncertainties in the case that a designed controller is applied to a process. This chapter defines the closed-loop transfer function and explains the relationship between the stability and the roots of the characteristic equation. Analysis tools for the Bode plot and the Nyquist plot are also introduced to predict the closed-loop stability by checking the open-loop transfer function. Also, the gain margin and the phase margin are defined to measure how much the closed-loop system is stable.

## 6.1 Closed-Loop Transfer Function and Characteristic Equation

A typical feedback control system has the structure shown in Figure 6.1. Here, *u*(*s*) and *y*(*s*) are the controller output and the process output respectively. *y*_{s}(*s*) and *d*(*s*) denote the setpoint and the disturbance respectively. *d*_{i}(*s*) and *d*_{o}(*s*) are called the input disturbance and the output disturbance respectively. *G*_{c}(*s*) and *G*(*s*) denote the transfer function of the controller and the process respectively. And *G*_{d}(*s*) is the transfer function between the disturbance of *d*(*s*) and the output disturbance of *d*_{o}(*s*). The step setpoint change means that *y*_{s}(*s*) is a step signal. The step input disturbance and the step output disturbance means *d*_{i}(*s*) and *d*_{o}(*s*) are step signals respectively.

### 6.1.1 Closed-Loop Transfer Function

In Figure 6.1, the transfer function from *y*_{s}(*s*) to *y*(*s*) is called the closed-loop transfer function between *y*_{s}