Modeling and Control

If a control loop is oscillating, would it be best to increase or decrease the controller gain?

The standard answer of decreasing the controller gain is right for a decaying or growing oscillation in a relatively fast self-regulating loop (loop whose PV quickly goes to a steady value when in manual and disturbances have dissipated). If the oscillation is banging between set point limits of a secondary loop or output limits of any loop, then you could end up with an equal amplitude oscillation for an unstable loop and the best thing to do is to first decrease the controller gain until it settles down.

If the oscillation amplitude does not decay but is relatively constant and the loop is staying well within its set point and output limits, the oscillation is probably a limit cycle caused by stick-slip, or a resolution limit in the control valve. Decreasing the controller gain will not reduce the oscillation amplitude but will make its period longer. Over a narrow time range, this may make the trend appear smoother but the longer oscillation period is less filtered out by downstream volumes and is consequently more likely to appear in the product. Here a well mixed downstream volume divided by the throughput flow acts like a filter time constant.

If you have an integrating loop (a loop whose PV ramps away from the set point when in manual) or a runaway loop (a loop whose PV accelerates away from the set point when in manual), decreasing the controller gain can make the oscillation worse if you were below the low controller gain limit. Note that the oscillations are extremely slow and may not be noticeable over a trend for a single shift. The minimum controller gain for an integrating loop is approximately 4 divided by the product of the reset time and integrating process gain. The minimum controller gain for a runaway loop is approximately the inverse of the process gain.

For integrating loops, if you are near the limit, the controller gain should be increased if the reset time is decreased to prevent an oscillation, which is counter intuitive. With real processes, the dynamics can change so any tuning should be thoroughly tested and the user must be well below the high gain limit that causes instability. Lambda tuning prevents violating the low gain limit for integrating processes. To avoid getting too close to the high gain limit, Lambda must be larger than the largest possible total loop dead time.

There are many important types of loops that have an integrating response besides level, such as batch chemical and fermenter dissolved oxygen, pH, overhead pressure, and temperature. Extremely exothermic batch and continuous reactors (e.g. polymerization reactors) can have a runaway response.

There are 8 different choices for controller structures in a modern DCS. Which one is the best for your application and does it take a download to change it?

One choice is for integral only controller (no proportional or derivative action). The most common I-only control is a valve position controller (VPC) which is trying to slowly optimize the position of a valve to save energy or to provide better resolution and rangeability. For example, the furthest open air feed valve position for trains of reactors becomes the PV of a VPC whose set point is the maximum useable throttle position (e.g. 60% for a rotary valve) and whose output sets the air compressor speed or guide vanes. Then there is the classic big and small valve setup where the VPC keeps the small valve in its mid throttle range for fine adjustments by making coarse adjustments to the big valve. Slow I-only tuning is used to minimize the interaction between the VPC and the existing process loops. The VPC is difficult tune and the problem is better solved with model predictive control (see my Advanced Application Note 2).

Proportional-derivative controllers are sometimes used on highly exothermic reactors with a runaway response or for batch temperature or batch pH control where the response in integrating and in only one direction. For example, when heating up a batch with no vaporization or heat loss, the temperature will only rise and cannot drop. Similarly, when adding a base to a batch with no reagent consumption or escape, the pH will only rise and cannot drop. Here integral action causes overshoot from which there is no recovery. Pulse width modulation of a proportional only pH controller output has been successfully used to mimic titration in a batch vessel which can be thought of as very large beaker.

You can get proportional-integral control by simply setting the rate to zero. This still leaves many choices as to structure most of which involve whether you want proportional or derivative action on error.

Since a change in structure require a download, once you have decided you want both proportional and integral action, by choosing the “two degrees of freedom controller” you can tune or write to the set point weights online. Furthermore the transition of the weights can be smooth because you can change them to any value between 0 and 1 whereas a change in structure makes a discrete switch between 0 and 1.

Dinner is waiting on me so we will have to wait till next week for more about weights.

The “Two Degrees of Freedom” structure functionally can give you the smooth transition online between 4 choices of PID structure. The user can adjust the set point weights for proportional action and derivative action between zero and one.

If the set point weights are zero. there is no step or bump from a set point change for proportional (gain) and derivative (rate) action, respectively. Zero weights slow down the response to a set point change because you are relying on integral action. For processes with small time constants (e.g. flow, liquid pressure, liquid blending, inline temperature and composition, and sheet thickness), the response is smoother and the likelihood of an overshoot is reduced. However, for large process time constants (e.g. continuous vessel temperature and composition), the time to get to set point can be too long. For an integrating processes (e.g. batch vessel temperature and composition), the controller output must drive past the final settling value and is best achieved by proportional action on the set point change. The set point weights can be increased from zero to give an effect similar to a set point filter to work a compromise between a smooth and fast response.

For cascade loops, do we want to tune the secondary the loop for a set point response?

As you have probably surmised by now from previous blogs the answer to my question is unexpected. The typically desired set point response (smooth, gradual, with no overshoot) when applied to the secondary loop is not generally best for the purposes of the primary loop. A set point filter or weight on the secondary loop is counter productive. For cascade control, the secondary loop should respond immediately to the requests of the primary loop. In fact, a zero set point weight on proportional action makes the cascade response worse than if the cascade was eliminated. This assessment does not take into account the beneficial compensation of nonlinearities and feedforward offered by a secondary loop.

While the ability of a primary loop to reject load upsets is affected by a set point weight or filter on a secondary loop, this is not the case for the primary loop or a single loop assuming these loop set points are constant during the load change.

After watching a PI controller take an incredibly long time to cycle through a very small resolution limit, I figured it would be helpful to develop an equation that would provide an estimate of the period of the limit cycle (sustained equal amplitude oscillation) from a valve resolution limit (e.g. stick-slip). The attached equation shows that while dead band in itself does not cause a limit cycle in a single self-regulating loop, dead band can have a profound effect on the period of the oscillation. Since on-off rotary valves disguised as throttling valves can have dead bands more than 10 times the resolution limit, drawn out limit cycles is not an uncommon phenomenon. The following file tells the story.

Valve Limit Cycle Period