## How to decouple PID Controllers

Decoupling PID controllers can be done on the basis of the following model:

$$PV = \frac{K_1}{T_{p,2} s + 1} e^{-T_{DT,1} T_s} MV_1 + \frac{K_2}{T_{p,2} s + 1} e^{-T_{DT,2} T_s} MV_2 + \frac{K_3}{T_{p,3} s + 1} e^{-T_{DT,3} T_s} MV_3$$
Where:
$$PV =$$ Process Variable
$$T_p =$$ Process Time Constant
$$T_{dt} =$$ Process Delay
$$s =$$ La Place variable
$$MV =$$ Manipulated Variable.

This model can easily be identified, using the PID-Tuner.

Assume that $$K_1 = -1, K_2 = 2, K_3 = 3.$$ The other parameters are not relevant for the decoupler design. Furthermore, assume that a PID controller adjusts $$MV_2$$ to control $$PV_2$$. The figure below shows the PID with decoupler that would result. This design consists of two steps:

First, a decoupler gain is calculated for every ‘other’ MV that is not adjusted by the PID controller,
on the basis of the assumption that the dynamics can be ignored. In this example:

$$DEC_1 = -\frac{K_1}{K_2} = 0.5$$ $$DEC_3 = -\frac{K_3}{K_2} = -1.5$$

In the second step, the decoupler with the highest absolute gain is searched for and selected as the one to which decoupling will be applied to. In the example, $$DEC_2$$ has the highest absolute gain, and therefore, $$MV_3$$ would be selected for decoupling, with a gain equal to -1.5.