Myke King continues his detailed series on process control, seeking to inspire chemical engineers to exploit untapped opportunities for improvement

**The proportional-integral-derivative (PID) control algorithm has been the regulatory controller of choice for around 85 years. Initially provided by pneumatic instrumentation, and later by electronic analog, the advent of digital control systems facilitated a range of modifications. Industry has yet to fully exploit many of the valuable features now available. To understand these missed opportunities we need to develop the algorithm from first principles.**

The proportional action is defined by

*M* is the controller output; using our example of the fired heater in last month’s article, it is the set-point of the fuel flow controller. *E* is the error – the deviation of the process variable* (PV)* from the set-point *(SP)*. While not all control system vendors have adopted it, the now recognised definition is *PV – SP*. This is part of the standard published by the ISA (formerly the Instrument Society of America, now the International Society of Automation). Traditionally, text books will use *SP – PV*; and accounts for differences in sign when formulae from different sources are compared.

The algorithm includes the first of our three tuning parameters – the controller gain (*K _{c}*). The term

As presented, the algorithm is in the *full position* form. To eliminate *C* most control systems use the velocity form. We obtain this by differentiation.

Modern control systems are digital, operating at a fixed *scan interval* *(ts)*. So we make an approximation to give the *incremental* form.

The main purpose of proportional control is to respond to changes in set-point. In the first control interval, after a change, ∆*E* will be equal to ∆*SP*. The controller will generate a step change in output (known as the *proportional kick*) equal to *K _{c} ∆SP* .

However, the *PV* will not reach the set-point. Instead there will be a sustained error, known as *offset*. To understand why this occurs, consider the level controller illustrated as Figure 1. Assume that the process is at steady state and that the controller error is zero. If the inlet flow is then increased by *f* then, to achieve steady state, the controller must increase the outlet flow also by *f*.

No matter how large we make *K _{c}* , we cannot reduce

Integral action gets its name from the full position version of the algorithm. Integrating gives

For many applications the PI algorithm will give adequate control. However, when we later come to tune the controller, we’ll show that the addition of derivative action permits *K _{c}* to be significantly increased and so resolve process disturbances more quickly. Derivative action is based on the rate of change of error. Adding it to the full position PI algorithm

This adds our third tuning parameter – *derivative time (T _{d})*. If the error is zero, no action will be taken by the proportional or integral actions. But, if it is changing quickly, an error will surely exist in the future. Derivative action anticipates this; indeed, it was once called

Differentiating and converting to digital control

This is generally known as the *ideal* version of the algorithm. An alternative version adds derivative action to the PI controller by replacing *E* with the *projected error* *(E’)* – again anticipating the need to take corrective action.

This results in the algorithm

If derivative action is included *(T _{d}> 0)* then changing either

In principle, both the ideal and interactive algorithms are implemented in the control system as derived. However, the digital approximation causes a problem with the derivative action. Imagine that the process has been steady for some time and the operator causes a controller error by changing the set-point. Ignoring, for the moment, the action taken by the proportional and integral parts of the algorithm, the changes made by the derivative action are shown in Table 1. It causes a *derivative spike* that has a duration of one controller scan interval and a magnitude of *K _{c}T_{d}E/ts*. The value of

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