Adaptive Level Control

Figure 3. Performance of linear PID level controller for a conical tank.

In other applications, level control can be challenging due to shrink and swell (e.g. boiler drums and column sumps) or because of the need for the level to float to avoid upsetting the feed to downstream units (e.g. surge tanks). If the level controller gain is decreased to reduce the reaction to inverse response from shrink and swell or to allow the level to float within alarm limits, the reset time must be increased to prevent slow oscillations.

Adaptive level controllers can not only account for the effect of vessel geometry, but also deal with the changes in process gain from changes in fluid density and nonlinear valves. Even if these nonlinearities are not significant, the adaptive level control with proper tuning rules removes the confusion of the allowable gain window, and prevents the situation of level loops being tuned with not enough gain and too much reset action.

Specific Dynamics for Conical Tank Level

Conical tanks with gravity discharge flow are used as an inexpensive way to feed slurries and solids such as lime, bark and coal to unit operations. The conical shape prevents the accumulation of solids on the bottom of the tank. The Madras Institute of Technology (MIT) at Anna University in Chennai, India, has a liquid conical tank controlled by a distributed control system (DCS) per the latest international standards for the process industry as shown in Figure 1. The use of a DCS in a university lab offers the opportunity for students to become proficient in industrial terminology, standards, interfaces and tools. The DCS allows graduate students and professors to explore the use of industry’s state-of-the-art advanced control tools. Less recognized is the opportunity to use the DCS for rapid prototyping and deployment of leading edge advances developed from university research.

The conical tank with gravity flow introduces a severe nonlinearity from the extreme changes in area. The dependence of discharge flow on the square root of the static head creates another nonlinearity and negative feedback. The process no longer has a true integrating response. In Appendix A online (www.controlglobal.com/1002_LevelAppA.html), the equations for the process time constant (τ_p) and process gain (K_p) are developed from a material balance applicable to liquids or solids. The equations are approximations because the head term (h) was not isolated. Since the radius (r) of the cross-sectional area at the surface is proportional to the height of the level as depicted in Figure 2, it is expected that the decrease in process time constant is much larger than the decrease in process gain with a decrease in level.

Controller Tuning Rules

The lambda controller tuning rules allow the user to provide a closed-loop time constant or arrest time from a lambda factor (λ_f) for self-regulating and integrating processes, respectively. The upper and lower controller gain limits are a simple fall out of the equations and can be readily enforced as part of the tuning rules in an adaptive controller.

For a self-regulating process the controller gain (K_c) and reset time (T_i) are computed as follows from the process gain (K_p), process time constant and process dead time (θ_p):

The upper gain limit to prevent fast oscillations occurs when the closed loop time constant equals to the dead time.

For an integrating process the controller gain (K_c) and reset time (T_i) are computed as follows from the integrating process gain (K_i) and process deadtime (θ_p):

The upper gain limit to prevent fast oscillations occurs when the closed loop arrest time equals the dead time:

The lower gain limit to prevent slow oscillations occurs when the product of the controller gain and reset time is too small.

Opportunities for Adaptive Control of Conical Tank Level

A linear PID controller with the ISA standard structure was tuned for tight level control at 50% level for a detailed dynamic simulation of the conical tank. Figure 3 shows that for setpoints ranging from 10% to 90%, a decrease in process time constant greater than the decrease in process gain at low levels causes excessive oscillations.

An adaptive controller integrated into the DCS was used to automatically identify the process dynamics (process model) for the setpoint changes seen in Figure 3. The adaptive controller employs an optimal search method with re-centering that finds the process dead time, process time constant, and process gain that best fits the observed response. The trigger for process identification can be a setpoint change or periodic perturbation automatically introduced into the controller output or any manual change in the controller output made by the operator.

Figure 4. Process models automatically identified for operating regions.

The process models are categorized into five regions as indicated in Figure 4. The controller gain and reset settings computed from the lambda tuning rules are then automatically used as the level moves from one region to another. This scheduling of the identified dynamics and calculated tuning settings eliminates the need for the adaptive controller to re-identify the process nonlinearity and tuning for different level setpoints. It was found that the use of lambda time, rather than lambda factors, with protection against going outside the controller gain limits helps provide a more consistent tuning criterion. As seen in Figure 5, the adaptive level controller eliminates the oscillations at low levels, and provides a more consistent level response across the whole level range.

Figure 5. Performance of adaptive PID level controller for conical tank.

Adaptive level controllers can eliminate tuning problems from the extreme changes in level control dynamics associated with different equipment designs and operating conditions. The integrated tuning rules prevent the user from getting into the confusing situations of upper and lower gain limits and the associated fast and slow oscillations. The smoother and more consistent response allows the user to optimize the speed of the level loop from fast manipulation of column reflux and reactor or crystallizer feed to slow manipulation of surge tank discharge flow control. 

Greg McMillan is a consultant and ControlTalk columnist.
Sridhar Dasani is a graduate of Madras Institute of Technology (MIT) Anna University in Chennai India.
Dr. Prakash Jagadeesan is an assistant professor at Madras Institute of Technology (MIT) Anna University in Chennai India.