Page 1 of 4 The Design of PID Controllers using Ziegler Nichols Tuning Brian R Copeland March 2008 1. Introduction PID controllers are probably the most commonly used controller structures in industry. They do, however, present some challenges to control and instrumentation engineers in the aspect of tuning of the gains required for stability and good transient performance. There are several prescriptive rules used in PID tuning. An example is that proposed by Ziegler and Nichols in the 1940s and described in Section 3 of this note. These rules are by and large based on certain assumed models. 2. PID Controller Structure The PID controller encapsulates three of the most important controller structures in a single package. The parallel form of a PID controller (see Figure 1) has transfer function: )11 ()(sTsTKsKsKKsCd ipdi p+=+=(1) where: Kp := Proportional Gain Ki := Integral Gain Ti := Reset Time =Kp/Ki Kd :=Derivative gain Td := Rate time or derivative time =Kd/Kp Kp Kds sKi+E(s) U(s) Figure 1: Parallel Form of the PID Compensator The proportional term in the controller generally helps in establishing system stability and improving the transient response while the derivative term is often used when it is necessary to improve the closed loop response speed even further. Conceptually the effect of the derivative term is to feed information on the rate of change of the measured variable into the controller action. The most important term in the controller is the integrator term that introduces a pole at s = 0 in the forward loop of the process. This makes the compensated1 open loop system (i.e. original system plus PID controller) a type 1 system at least; our knowledge of steady state errors tells us that such systems are required for perfect steady state setpoint tracking. This is more formally stated in the following theorem: Figure 2 Theorem 1 For the unity feedback system of Figure 2, perfect setpoint control can only be achieved for the controller, C(s), if and only if 1. The open loop forward gain has a steady state gain of infinity i.e. 1 The PID controller is often considered as one member of a family of compensators i.e., devices that can be added to an open loop system to change (compensate) for characteristics which make the achievement of the control objective difficult or impossible. As is the case for the PID controller, compensators are usually cascaded to the input of an existing plant before feedback is applied. This introduces a new set of poles and/or zeros to the picture. Control design proceeds by treating the cascaded pair as the new system to be controlled. C(s) G(s) R(s) Y(s) E(s) Note these definitions! Page 2 of 4 = )()(lim 0ssGsC2. The system is closed loop stable 3. Neither C(s) nor G(s) have zeros at the origin ? ? The final condition of the theorem eliminates the possibility of the loop transfer function cancelling the effect of the integrator pole. Both forms of (1) are used with the second being the most common. The reset and rate times are of special significance in this regard: The reset time is the time taken for the integrator term output to equal the proportional term output in response to a step change in input applied to a PI controller. PI TermEKpEKpETi The rate time is the time taken for the proportional term output to equal the derivative term output in response to a ramp change input applied to a PD controller. PD TermKpTdETdKpTdEu(t)=EtIn addition, the Proportional gain, Kp, is often expressed as a proportional band (%): pKPB1= (2) PB is actually the fractional error change, relative to the error range, required to produce a 100% (full range) change in the proportional term output. Mathematically, we have: = =PBeo PBeo eeooeoKp11.usually is last term Thisx1x/ChangeInput ChangeOutput maxmaxmaxmaxmaxmax(3) In practice PB is expressed as a percentage so pKPB100% = (4) Thus a PB of 5% Kp=20. We also note that: 1. The industry jargon is clearly more practical and useful in this case. The concept of gain, usually quite useful in analysis, is generally harder to grasp than the degree of signal variation (absolute or relative) required to obtain full output swing. 2. In addition, most controllers operate on relative (percentage or per unit) units. In this regard, quantities are scaled relative to their maximum range. This makes it easy to translate from one unit basis to another. For example, if our PV is a temperature in the range 0o - 100o mapped to a current range of 4-20mA, then a PV 20% translates to an absolute temperature of 20o and an equivalent current signal reading of 4+ 16*.2=7.2mA. NB: If analysis is performed on any system the units used will determine the gains in the various transfer boxes. 3. Popular variations of (1) are used to generate the following controllers: Page 3 of 4 Controller Type Kp Ki Kd C(s) P (Proportional) 0 zero zero Kp I (Integral) zero 0 zer