Ultimate controller tuning

We have yet to tackle the PI controller. There are, of course, different ways to find a good integral time constant. With frequency response, we have the handy tool of the Ziegler-Nichols ultimate cycle tuning relations. So with Kcu = 60 and cocg = 3.3 rad/min, we find by referring to... 

The two measurement lags are included so that reasonable controller tuning constants can be determined. The reactor itself is only net second-order (first-order polynomial in the numerator and third-order polynomial in the denominator), so the theoretical ultimate gain would be infinite if lags were not included. The linear model is used in the following section to explore stability. 

Click the Closed loop ATV bullet, start the simulation mnning, and click the Start test button. After several cycles, click the Pause button to stop the simulation and click the Finish test button at the bottom of the Tune window (see Fig. 3.79). The ultimate gain (3.73) and the ultimate period (4.8 min) are displayed, as shown in the left side of Figure 3.80. To calculate the controller tuning constants, click the Tuning parameters page tab on the Tune window and select either Ziegler-Nichols or Tyreus-Luyben. 

The next part involves controller tuning. We must determine the tuning constants for the controllers in the plant. While this task is often performed by using heuristics and experience, it can sometimes be a nontrivial exercise for certain loops. We recommend using a relay-feedback test that determines the ultimate gain and period for the control loop, from which controller settings can be calculated (Luyben and Luyben, 1997). 

The critical product-quality and safety-constraint loops were tuned by using a relay -feedback test to determine ultimate gains and periods. The Tyreus-Luyben PI controller tuning constants were then implemented. Table 11.12 summarizes transmitter and valve spans and gives controller tuning constants for the important loops. Proportional control was used for all liquid levels and pressure loops. 

Find the ultimate gain and period of the closedloop three-CSTR system with a PID controller tuned at t/ = To = 1. Make a root locus plot of the system. 

The four composition control loops each have a 5 min deadtime. They were tuned using a sequential method. Because reboiler heat input affects aU of the controlled variables fairly quickly, the Xb(x/<2r loop was tuned first with the other three controllers on manual. Relay-feedback testing was used to find the ultimate gain and period. Tyreus-Luyben tuning rules were used. Next, since reflux affects all compositions, the Xixt)IR loop was tuned using the same procedure with the Xb(j) Q loop on automatic. Then the Xs(x> loop was tuned with the two loops on automatic. Finally, the ypio(x) loop was tuned with the other three loops on automatic. Table 12.2 gives controller tuning results for aU four loops. 

This dynamic test gives very accurate estimates of the ultimate gain and ultimate period of the loop, which can then be used for controller tuning. The method inserts an on-off relay in the feedback loop that positions the controller output signal at a specified percent higher or lower than the initial steady-state value. For example, if the OP signal to the valve is 50% at steady state and the specified displacement is 5%, the controller output will fluemate back and forth between 45 and 55%. 

The auto-tune variation or ATV technique of Astrom is one of a number of techniques used to determine two important system constants called the ultimate period and the ultimate gain. Tuning values for proportional, integral and derivative controller parameters may be determined from these two constants. All methods for determining the ultimate period and ultimate gain involve disturbing the system and using the disturbance response to extract the values of these constants. 

There is a slight hump in the Ui curve at tray 13 shown in Figure 12.43. We tested a control structure that used Fqa to control the tray 5 temperature and Fqb to control the tray 13 temperature. These two loops were successftilly tuned, but the tray 13 controller had a very large ultimate period (55 min). Table 12.3 gives the controller tuning parameters. 

Ziegler-Nichols Continuous Cycling (empirical tuning with closed loop test) Increase proportional gain of only a proportional controller until system sustains oscillation. Measure ultimate gain and ultimate period. Apply empirical design relations. 

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A key to the successful application of a PID control is the tuning of parameters, Xp, Tp and Tp in Equation 13.5. To tune them properly, the Ziegler-Nichols method is used, which includes an ultimate-gain method and a step-response method. 

Figure 13.6 compares the typical response of P, PI, and PID controllers against the step change of a set-point. As can be seen, a well-tuned PID controller realizes Table 13.2 Determination of parameters of PID control by ultimate gain method. 

Ziegler-Nichols (ZN) and Tyreus-Luyben (TL) PI tunings are evaluated. Ultimate gain and frequency are obtained by performing relay-feedback tests. Temperature control loops have three 20-s lags. The pressure control loop has two 30-s lags. There is a... 

The first controller GCuS) sees a coupled openloop transfer function in which the second controller is nested, because the controller Gq ,> is on automatic. The 67 i(v) controller is tuned by finding the ultimate gain and period and using the Tyreus-Luyben settings. The settings of this PI controller are Kc2 = 2.13 and r/2 = 1.94. 

The purpose of a controller is to keep the controlled variable as close to its set point as possible. Meeting this objective is a function of the nature of the process, the nature of the disturbances, and of tuning. Disturbances arise from three different sources load, set point changes, and noise. The controller has no impact on noise, other than possibly amplifying it and passing it onto the final actuator, which can cause excessive wear and ultimate failure. 

A few comments about the method are warranted. The controlled (dominant) variables, Ycd, should be measured such that they belong to the set Yd for rapid control. Similarly, the manipulators in the feedback control loops should belong to the set, Ud. The feedback controllers should have integral action (PI controllers). These can be tuned with minimal information (e.g., ultimate gain and frequency from a relay test). The model Ms is usually quite simple and can be developed from operating data using statistical regressions. This works because the model includes all the dominant variables of the system, Y d, as independent variables by way of their setpoints, Y. The definition of domi-... 

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