## Interaction Compensation

Interaction occurs in multivariable systems when a change in a manipulative variable causes a deviation in more than one controlled variable.

A classic example is a distillation column in which a rectifying tray temperature is controlled by manipulating reflux, and stripping tray temperature by manipulating steam. A change in reflux causes a deviation in the stripping tray temperature. The steam flow is adjusted to compensate, and it causes a further deviation in the rectifying temperature. If the loops are tighdy tuned, this interaction can result in unstable control. Chapter 20 discusses methods to eliminate interaction for analog controls. The purpose of this section is to illustrate how interaction compensators can be achieved by sampled-data algorithms for the control system discussed above. Basically the interaction compensators will function like feedforward algorithms with reflux and vapor boilup treated as load disturbances on the opposite controlled variables.12 The compensators are used in conjunction with the dual feedback algorithms.

The control system is illustrated in Figure 21.6. The load and set-point algorithms Dss, DLS, DSR, D1J{ are designed according to Section 21.3. The interaction algorithms DIS and Dm are designed as follows. Consider first the stripping section control loop. The controlled variable Xs is given by equation (21.18):

We desire zero change in Xs(z) when a change is made in LR(z). Therefore, the necessary change to make in V(z) is:

Equation (21.19) defines DIS(z):

A similar calculation in the rectifying loop leads to Dm(z):

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