2.1 Introduction

The ideal continuous time domain PID controller for a SISO process is expressed in the Laplace domain as follows:

with

and with K _c=proportional gain, T _i=integral time constant and T _d=derivative time constant. If T _i= ? and T _d=0 (i.e. P control), then it is clear that the closed loop measured value, y, will always be less than the desired value, r (for processes without an integrator term, as a positive error is necessary to keep the measured value constant, and less than the desired value). The introduction of integral action facilitates the achievement of equality between the measured value and the desired value, as a constant error produces an increasing controller output. The introduction of derivative action means that changes in the desired value may be anticipated, and thus an appropriate correction may be added prior to the actual change. Thus, in simplified terms, the PID controller allows contributions from present controller inputs, past controller inputs and future controller inputs.

Many variations of the PI and, in particular, the PID controller structure have been proposed. As Tan et al. (1999a) suggest, one important reason for the non-standard structures is due to the transition of the controllers from pneumatic implementation through electronic implementation to the present microprocessor implementation. The variations in the controller structures are detailed below.

2.2 PI Controller Structures

Tuning rules have been detailed for seven PI controller structures:

1. Ideal controller

   
   

2. Ideal controller in...