The PID controller is the most common form of feedback. It was an essential element of early governors and it became the standard tool when process control emerged in the 1940s. In process control today, more than 95% of the control loops are of PID type, most loops are actually PI control. PID controllers are today found in all areas where control is used. Thecontrollers come in many different forms. There are stand-alone systems in boxes for one or a few loops, which are manufactured by the hundred thousands yearly. PID control is an important ingredient of a distributed control system. The controllers are also embedded in many special-purpose control systems. PID control is often combined with logic, sequential functions, selectors, and simplefunction blocks to build the complicated automation systems used for energy production, transportation, and manufacturing. Many sophisticated control strategies, such as model predictive control, are also organized hierarchically. PID control is used at the lowest level; the multivariable controller gives the setpoints to the controllers at the lower level. The PID controller can thus be said to be the“bread and buttert’t’ of control engineering. It is an important component in every control engineer’s tool box. PID controllers have survived many changes in technology, from mechanics and pneumatics to microprocessors via electronic tubes, transistors, integrated circuits. The microprocessor has had a dramatic inﬂuence on the PID controller. Practically all PID controllers made today are basedon microprocessors. This has given opportunities to provide additional features like automatic tuning, gain scheduling, and continuous adaptation.
6.2 The Algorithm
We will start by summarizing the key features of the PID controller. The “textbook” version of the PID algorithm is described by:
u(t) = K e(t) +
e(τ )dτ + Td
(6.1)where y is the measured process variable, r the reference variable, u is the control signal and e is the control error ( e = ysp − y). The reference variable is often called the set point. The control signal is thus a sum of three terms: the P-term (which is proportional to the error), the I-term (which is proportional to the integral of the error), and the D-term (which is proportional to thederivative of the error). The controller parameters are proportional gain K , integral time Ti , and derivative time Td . The integral, proportional and derivative part can be interpreted as control actions based on the past, the present and the future as is illustrated in Figure 2.2. The derivative part can also be interpreted as prediction by linear extrapolation as is illustrated in Figure2.2. The action of the different terms can be illustrated by the following ﬁgures which show the response to step changes in the reference value in a typical case.
Effects of Proportional, Integral and Derivative Action
Proportional control is illustrated in Figure 6.1. The controller is given by (6.1) with Ti = ∞ and Td = 0. The ﬁgure shows that there is always a steady state error inproportional control. The error will decrease with increasing gain, but the tendency towards oscillation will also increase. Figure 6.2 illustrates the effects of adding integral. It follows from (6.1) that the strength of integral action increases with decreasing integral time Ti . The ﬁgure shows that the steady state error disappears when integral action is used. Compare with the discussion of the “magicof integral action” in Section Section 2.2. The tendency for oscillation also increases with decreasing Ti . The properties of derivative action are illustrated in Figure 6.3. Figure 6.3 illustrates the effects of adding derivative action. The parameters K and Ti are chosen so that the closed-loop system is oscillatory. Damping increases with increasing derivative time, but decreases again when...