Neglect Level Control at Your Peril

Process gain, K_F. This is the effect of a 10% change in the controller output on the slope of the ramp. From Figure 2, a 10% reduction in the controller output causes the slope of the ramp to change from zero to 0.49%/min. So:

K_F = (0.49 %/min)/10% = 0.049 (%/min)/%

A decrease leads to an increase in level, so the process is reverse acting.

Process lag, θ. The material balance suggests the ramp should commence immediately, as indicated by the dashed line in Figure 2. Instead, the slope changes gradually from zero to 0.49 %/min. By the time the slope reaches 0.49 %/min, the actual response lags by 0.4 min.

The process lag shown by the ramp in Figure 2 includes the following:

Control valve lag. A digital system can change its output by 10% very quickly but all final control elements exhibit some lag in responding to a change in their input signal. Rarely are the response characteristics of the final control element well known.

Measurement device lag. This depends on the measurement technology employed and, sometimes, on how the device is installed. Rarely is this lag quantified.

Smoothing of the process variable. When smoothing is applied either within the measurement device or the control system, quantitative values are available. However, with some level measurement technologies, smoothing can be applied externally.

The lag observed in Figure 2 is roughly the sum of these lags. The combined effect often is approximated by a transportation lag or dead time. In the simple approximations of the dynamics of a level process, the process lag θ is considered to be entirely transportation lag.

Unfortunately, for many level loops, conducting a test such as in Figure 2 is impractical due to the presence of noise on the measured level value and variability in the feed flow to the vessel.

Noise and Variable Feed Rate

Figure 4. While level is maintained close to target, discharge valve opening varies significantly.

Testing procedures are available to determine K_F and θ in face of both measurement noise and flow upsets. One approach is to use a pseudo-random binary signal (PRBS) for the output to the control element. Model predictive control technology relies on such tests to find process characteristics. However, such tests are long-duration (days) and difficult to justify for level control applications.

TUNING EQUATIONS
When a proportional-integral (PI) controller is used, the relationships between the tuning coefficients and the process characteristics are:

Controller gain, K_C: inversely proportional to the process gain K_F; inversely proportional to the lag θ.

Reset time, T_I: not affected by K_F; directly proportional to θ.

Especially for large tanks with long residence times, tuning equations often suggest unreasonably large values for K_C. Most tuning relationships link the product K_FK_C (the loop gain) to the process dynamics. The large value for K_C results from two factors:
1. For responsive processes, the tuning equations suggest a large value for K_FK_C. For a vessel with a residence time of 1 hr, a lag of 0.4 min is trivial.
2. For large vessels, the process sensitivity K_F is small.

Using the Ziegler-Nichols tuning equations, the suggested values for the tuning coefficients are:

K_C = 1/(K_F θ) = 0.9/{[0.049 (%/min)/%] × (0.4 min)} = 46 %/%

T_I = 3.33 θ = 3.33 × (0.4 min) = 1.33 min

The performance objective for the Ziegler-Nichols tuning equations is a response with a quarter decay ratio, which usually provides a rapid response to a disturbance. Figure 3 presents the response to a 10-min 50-L/min increase in one feed for the level process in Figure 1. These tuning coefficients maintain the vessel level very close to its set point — the maximum level deviation is approximately 0.2%. The response period, P, is 2.7 min. Also note the feed flow change is translated quickly into a discharge flow change.

A controller gain of 46 %/% is unreasonable in a level controller. A high controller gain amplifies any loop imperfections, such as the consequences of a finite resolution in the measured variable. In the example here the measured level value has resolution of 1 part in 4,000 — this means that 0.025% is the smallest possible change in measured level value. Using a controller gain of 46 %/%, a change of 0.025% in vessel level alters the controller output by (0.025%) × (46 %/%) = 1.15%. This, not surprisingly, leads to the abrupt changes seen in Figure 3, especially as the vessel level approaches its set point. Between the abrupt changes, the controller output exhibits ramp changes. (The finite resolution gives a constant control error that is integrated by the reset mode.)

Figure 4 presents the performance of the level controller with 0.5% noise on the level measurement and a varying feed rate. With the high gain, level is maintained close to its target. In addition, feed flow changes are translated quickly to discharge flow changes (which isn't necessarily beneficial). This comes at the expense of noticeable variability in the discharge valve opening and discharge flow. (The aggressive controller is amplifying the noise in the vessel level.)

Figure 5. As gain decreases, vessel level fluctuates more widely. Click on image to enlarge.

When a controller is too aggressive, the customary practice is to reduce K_C. Figure 5 illustrates the effect on loop performance of decreasing the controller gain from 46 % to 10 % and finally to 2 %. The conventional wisdom is that reducing the controller gain has two effects:

1. The loop responds more slowly. This is clearly the case in Figure 5, with the period increasing from 2.7 min to 9.5 min and finally to 22 min.
2. Any oscillations in the response decay more rapidly. However, this isn't the case in Figure 5. For a gain of 46 %/%, the second peak in the oscillation is barely visible. For a gain of 10 %/%, the second peak is still very small compared to the first one. But for a gain of 2 %/%, the decay ratio is approximately 0.5.

   With regard to the effect of the controller gain on the degree of oscillations, loops for integrating or non-self-regulated processes behave differently, especially at low K_C values. The loop contains two integrators, one in the process and one in the controller. Consider the following possibility:

   • Process dynamics consist of only an integrator (no lag, θ = 0).
   • Controller is integral-only.

   When disturbed, the loop responds with a cycle of constant amplitude. Decreasing the controller gain increases the period of the cycle and its amplitude but the cycle neither grows nor decays. Any additional dynamics in the process (such as the lag exhibited in Figure 2) result in an unstable loop for all values of K_C.

   For a PI controller and an integrating process, the following two observations apply:

   1. At low K_C values, loop behavior approaches that of a loop with an integral-only controller. The cycles in the response have a long period and decay slowly.

   2. If the reset time is less than θ, the loop is unstable for all values of K_C.

   For PI control of an integrating process, continuing to reduce the controller gain results in a slowly decaying cycle with a very long period. Figure 5 clearly illustrates this behavior.

   However, in practice, responses such as those in Figure 5 often are impractical to obtain for a level loop. With noise in the measured variable and frequent changes in flows in or out, the situation in Figure 4 is more typical of most level processes. The next article in this series will examine assessing the performance of the level controller when such factors are present.

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   CECIL L. SMITH is president of Cecil L. Smith, Inc., Baton Rouge, La. E-mail him at cecilsmith@cox.net. This article is based on concepts from his book "Practical Process Control," published by John Wiley & Sons.