Steady-state offset

Study the response of the system with only proportional control (ti very large) and determine the resultant steady state offset. Add an increasing degree of integral control and study its effect... 

It is evident that, of the 30 modes of the full linear model (with N = 6), 18 are very fast in comparison to the remaining 12 (by 2 orders of magnitude or more). Thus direct modal reduction to a 12th-order model using Davison s method should provide good dynamic accuracy. However, by simply neglecting the non-dominant modes of the system, the contribution of these modes is also absent at steady state, thus leading to possible (usually minor) steady-state offset. Several identical modifications (Wilson et al, 1974) to Davison s... 

Here, gj are model estimates of the coefficients hj. The meaning of Eq. (63) is that the value of the process input u at the end of its horizon should correspond to a steady-state value that would produce zero steady-state offset. 

Adaptation of the design variables to increased 6p should improve performance over most or all of the uncertain parameter set, V, e.g., through increased steady-state offset from the active constraints or increased process capacity. 

Find the closed-loop response of a first-order process using the velocity form of the PI control algorithm. Show that the steady-state offset of the closed-loop response to a unit step change in the set point is zero. 

Figure 1.10 gives the temperature in the first tank and the heat input for three values of controller gain Kc. As gain increases, the dynamics of the system get faster and there is less steady-state offset the final steady-state value of 7i is closer to 150°F. The dynamic responses all show gradual asymptotic trajectories to their final values. There is no overshoot and no oscillation. 

Use Laplace transforms to prove mathematically that a P controller produces steady-state offset and that a PI controller does not. The disturbance is a step change in the load variable. The process openloop transfer functions, Gm and Gi, are both first-order lags with different gains but identical time constants. 

Notice that the denominators of all these closedloop transfer functions are identical. Notice also that the steady-state gain of the closedloop servo transfer function PV/SP is not unity i.e., there is a steady-state offset. This is because of the proportional controller. We can calculate the PV/SP ratio at steady state by letting v go to zero in Eq. (8.8 ). 

A proportional-only controller is used to control the liquid level in a tank by manipulating the outflow. It has been proposed that the steady-state offset of the proportional-only controller could be eliminated by using a combined feedforward-feedback system. 

One easy approach is to use the Ziegler-Nichols value or the Tyreus-Luyben value for reset time. Integral action is utilized only to eliminate steady-state offset, so it is not too critical what value is used as long as it is reasonable, i.e., about the same magnitude as the process time constant. 

Notice in Fig. 11.19 that the curve for the P controller does not approach 0 dB at low frequencies. This shows that there is a steady-state offset with a proportional controller. The curve for the PI controller does go to 0 dB at low frequencies because the integrator drives the closedloop servo transfer function to unity (i.e., no offset). 

Figure 18.3 shows examples of applying this procedure to benzene-toluene columns with different feed points and different feed compositions. Accordingly, trays 7,10, and 5 or 10 are the best control trays in Fig. 18.3a, b, and c, respectively. Figure 18.4, based on the column in Fig. 18.3a, shows how a variation in control tray temperature affects product composition with a correctly located and an incorrectly located control tray. When the temperature variation is caused by a change of pressure or in the concentration of a nonkey component, it will produce a steady-state offset in product composition. A disturbance in the material or energy balance will cause a similar temperature variation until corrected by the control action in this case, the offset will only be temporary. Figure 18.4 shows that the offset in either case is minimized when the control tray is selected in accordance with Tolliver and McCune s procedure (403). A dynamic analysis by these authors (403) indicated that the control tray thus selected tends to have the fastest, most linear dynamics. 

Figure 26.9 shows the step responses for the models in case of balanced realization and truncation and Fig. 26.10 in the case of Hankel norm approximation. It can be seen that in both cases there is a reasonable agreement between the step response of the original model and the reduced model, however, there is some steady-state offset. 

Results for the CSl control structure arc presented in Figures 12.53-12.55. In Figure 12.53 the disturbances are positive and negative 20% step changes in the feed flow-rate at a time of 100 min. The temperature dynamics are very fast and steady state can be reached in a few minutes. The composition dynamics are a little slower and settle in 50 min. However, 0.1 mol% steady-state offsets in product purities are observed. Figure 12.54 shows what happens when composition zo of fresh feed Fq is changed from pure reactant A (zqa = 1) to a mixture of A and B (zqa = 0.95, Zqb = 0.05). When light... 

To eliminate steady-state offsets in product compositions, dual composition control is implemented. Figure 12.56 shows that the distillate composition is controlled by changing the reflux ratio, and the bottoms composition is maintained by adjusting vapor boilup. Because the composition analyzer has a slower response, a measurement dead time of 4 min is assumed in the simulation. Relay-feedback tests are performed to find settings for the PI controllers. Table 12.5 shows that because of the dead time in the composition measurement, the reset times for the top and bottoms loops now become 120 and 102 min, compared to 4.3 and 2.9 min in the case of temperature control. 

For the MeAc system, two temperature control trays (X2 and Tn) show asymmetric responses and somewhat oscillatory behavior is observed for T (Fig. 13.6a). The product composition, Xo acetate in particular, does not settle down 15 h after the 20% feed flowrate change is introduced. Asymmetrical responses are observed for most of the process variables and steady-state offsets ( 0.002mf Fig. 13.6a). For the type II... 

Finally, it is interesting to note that the two systems (MeAc and AmAc) with lower TAC give relatively poor closed-loop responses as evidenced by either slow dynamics or large steady-state offsets. The TACs for these five designs are given in Table 7.5. 

For the MeAc system, the F-FR and F-F control stmctures give slightly better performance for a 20% production rate increase compared to that of the Q-FR control stmcture, as shown in Figure 13.9. The temperatores of the Q-FR control stmcture take 6 h to settle while the F-FR and F-F stmctures need only 3 h. In terms of peak composition error, the F-FR shows the least errors. However, the steady-state offsets are the same for all three control stmctures because the temperature control trays are the same for all three cases. 

Because of the steady-state offsets when using temperature control, it may be necessary to use offset-free composition control in acetic acid esterification. Because of the nature of the... 

Good disturbance rejection can be achieved for a feedrate increase. For the EtAc and IPAc systems (type II flowsheet), little difference in the speed of response, peak error, and steady-state offsets are observed (Figs, 3.6b, 13.6c, 3. 2b, and 13.12c). Thus, there is little incentive to seek composition control for the type II flowsheet. For the BuAc system, good control performance is possible using composition control. The... 

Because of steady-state offsets in the product composition, a one-temperature, one-composition control structure was proposed to maintain on-aim produet quality. This offers an alternative when the two-temperature control structure shows unacceptably large offsets in product composition. 

To eliminate the steady-state offset in acetate purity, we also consider composition control. A parallel cascade control structure is used in which the setpoint of the stripper temperature controller is reset by the output of the composition controller. Thus, we have one composition control loop xb iac-TItr and two ternperamre control loops 2r,str-Tstr and 2/j,rd-Trd, with constant R/F on (CS3, as shown in Fig. 16.38). Relay-feedback tests are performed to find the ultimate gains and ultimate period of each control loop and initial controller parameters are calculated according to = Ku/2> and T/= 2F . Controller settings for aU loops are summarized in Table 16.7. The control performance is tested for feedflow disturbances. Figure 16.44 shows the dynamic responses of CS3 for +20% feedflow changes. The control performance of CS3 is quite similar to that of the temperature control (CS2), except that steady-state error in product purity is eliminated. 

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