## Steam

BOTTOM ^"PRODUCT

FIGURE 21.1 Sampled-data control in feed rate or feed composition occurs. A set-point change requires a different response of the controlled variable than a load disturbance requires. Consequently a controller tuned for good response to a set-point change may be unsatisfactory for load disturbances. When a conventional discrete PI or PID algorithm is used as shown in Figure 21.2, the parameters can be specified for either good servo or regulator control, but not for both. Some have proposed that two separate sets of tuning parameters be stored in the computer. This approach requires additional computer logic and space.

### Dual Algorithm

A better approach is to implement a "dual" sampled-data algorithm that is structured to handle both set-point and load disturbances simultaneously, as shown in Figure 21.3.29 The DL(z) part of the algorithm is designed to achieve good regulator control and the Ds(z) part to achieve good servo control. These algorithms are derived in the following manner.

The equation for the controlled variable in sampled notation is derived from Figure 21.3. We assume the dynamics of the continuous-flow controller are fast enough to be neglected.

GpH(z) is the z-transform of the product of the process transfer function and zero-order hold. The output of the zero-order hold is the last value of the computer output, which is held constant until the next sample time. GlL{z) is the z-transform of the product of the load transfer function and the load variable. Dl{z) is determined from equation (21.3) by setting C^z) equal to zero.

GlL(z) and C(z) have to be specified before DL(z) can be calculated. Therefore, some knowledge of the type of load is necessary—whether it is a step or ramp function. The time response of C(z) to the load disturbance can be specified to meet any number of criteria as long as DL(z) is physically realizable. C(z) is in the form of a series of negative powers of z. The power of z corresponds to the number of sample points following the disturbance. The coefficient of z is the deviation from the initial value of C(z).

The set-point compensation part of the algorithm Ds(z) is determined from equation (21.3) by setting GlL(z) equal to zero. This says that the load variable does not change.

_ C(z)[l + Dl(z)G„H(z)] A(Z) - Csa(z)DL(z)GpH(z) (2L5)

The appropriate terms must be substituted into equation (21.5) to calculate Ds{z). Dl(z) has been calculated already from equation (21.4). Csct(z) is the z-