TI= 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.)
When a controller is too aggressive, the customary practice is to reduce KC. 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:
- 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.
- 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 KC 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 KC.
For a PI controller and an integrating process, the following two observations apply:
1. At low KCvalues, 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 KC.
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.
CECIL L. SMITH is president of Cecil L. Smith, Inc., Baton Rouge, La. E-mail him at firstname.lastname@example.org. This article is based on concepts from his book "Practical Process Control," published by John Wiley & Sons.