In this chapter, the closed-loop control system will be introduced for SISO plants. We start by calculating the transfer functions from the input signals (command r, disturbance d , and noise n) to the plant output y and to the internal quantities (the control error r-y and the control u). The most basic requirement for a feedback loop is that all these transfer functions have to be stable. Then the control loop is said to be internally stable. Next, we consider the closed-loop system for constant inputs and require that in this case the steady-state control error vanishes, if no noise is present. The leads to the PID controller, which is the most important controller in classical control theory. The main tool for synthesizing such a controller uses the Nyquist stability theorem, which plays an important role in modern control theory, too. It is presented in Sect. 3.4.
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