Control loops can be affected by interactions and nonlinearities, making their behavior more complex than the clean patterns that have been described. It is important to understand these clean patterns so that you can have an idea of what to expect in the absence of interactions and nonlinearities. In my experience, it has been helpful to think about the open-loop step response in terms of resistance ( $R$ ) and inductance ( $L$ ), and then use that understanding to interpret how interactions and nonlinearities can affect stability. I encourage you to adopt this approach when faced with situations that don’t quite fit the norm.

Figure 8.1 [Tuning of interacting loops can be difficult.]

The pressure control valve and flow control valve in this system have a mutual effect on flow, and the flow control valve also affects pressure. This interaction between the two control systems can lead to complications, as tuning one system will impact the other. This is particularly problematic when the natural periods of both loops are nearly identical.

To address interaction problems, it can be useful to consider the effect of tight tuning on the open-loop step response of the other loop. The Ziegler and Nichols open-loop tuning rules can then be applied to determine whether the desired gain or integral has changed based on the values of R, L, and RL.

There are various strategies for addressing interaction problems, and the following list is not exhaustive:

Determine whether one of the control loops is unnecessary. In some cases, such as the example mentioned, it may be possible to eliminate one of the loops, such as the pressure control loop.
Reconfigure which valve is controlled by which variable. This can be a complex task that may require specialized knowledge.
Decide which variable is more important and tune the associated loop to be more tightly controlled. For instance, in the example provided, the flow control loop could be made the primary focus.
If a computer-based control system is used, it might be possible to implement decoupling terms in the programming or configuration to minimize the interaction. However, this may be too advanced for a novice to tackle.

When there is a problem with interaction, it is often because the natural periods of two loops are similar. If the natural periods are significantly different, there is less likelihood of problems arising. Therefore, it is important to understand what the natural period is, what determines it, and how to change it when solving control problems.

Tuning interacting systems can be challenging. It is possible to arrive at different sets of tuning parameters for one loop depending on whether the other loop is on manual or automatic and tightly tuned. Switching from automatic to manual, or vice versa, can cause one loop to become unstable. The purpose of this section on interactions is to raise awareness of potential issues rather than provide solutions. Unless a more sophisticated approach is warranted, it is recommended to tune for the worst case scenario, which is the condition most likely to produce cycling, and accept the performance at other conditions.

Nonlinearities

Up until this point, the assumption has been made that the system is linear, which means that it can be described by a set of ordinary linear differential equations. Simply put, doubling a change produces exactly twice the result of the previous change. However, in reality, no physical system is truly linear. Despite this, it is still common practice to analyze and approach systems as if they were linear, as it is simpler and often acceptable.

Nonlinearities, which exist in many different forms, can cause issues with loop performance, but most often do not significantly affect it. In cases where performance is reduced, it is typically difficult to notice. However, there are a few types of nonlinearities that cannot be ignored. By understanding their characteristics, you can decide whether to ignore them or take action to minimize their effects. Nonlinearities can be categorized into two types: process and hardware. Hardware nonlinearities can be further divided into continuous and discontinuous, with the distinction becoming clearer as you continue reading.

Process

Nonlinearities in processes are common and diverse, and can arise in various areas such as heat transfer, mass transfer, and chemistry. These nonlinearities can be continuous, where the relationship between variables changes smoothly over a range of values, or they can be discontinuous, where the relationship abruptly changes at certain values. However, in many cases, the assumption of linearity is made for simplicity, and conclusions are drawn from that. For example, in pH control problems, linear concepts are used despite the severe continuous nonlinearity of the pH titration curve. For important problems, computer simulation techniques can be used to account for process nonlinearities.

Hardware, Continuous Nonlinearities

Hardware (and software) nonlinearities in the continuous category are usually intentional, aimed at compensating for some other nonlinearity. The goal is typically to make the overall system linear, or at least closer to linear. An example of this is a square root extractor on a differential pressure transmitter. With the flexibility of computer-based controllers, there is a wide selection of algorithms that can be applied to achieve this goal. Whether a continuous hardware nonlinearity is deliberate or not, its effect is analyzed in a loop much like a process nonlinearity. The system is assumed to be linear over a small operating range, and extreme conditions are examined for potential problems.

The real challenge comes with discontinuous nonlinearities, which can wreak havoc. They can reverse the effects of standard tuning rules and procedures, requiring you to forget or at least relearn the exceptions to linear system behavior.

Velocity Limiting

The occurrence of velocity limiting is common in valves, whether operated by an electrical motor, a pneumatic valve positioner, or a pneumatic controller without a valve positioner. To determine whether velocity limiting will cause stability issues, it is important to understand the role of lags in the loop.

As previously discussed in Chapter 2 and shown in Figure 2.3, the gain is set proportional to $1/RL$ . If the product of $R$ and $L$ increases, then a lower gain should be used. Failure to adjust the gain may cause the loop to cycle. As explained in Chapter 4, Figure 4.6, $L$ is a combination of all the smaller lags in the loop, and $R$ is the response of the longest lag in the loop.

Now consider the effects of velocity limiting. A valve positioner may be capable of a 2% change with a first-order lag of 0.3 seconds. However, for larger changes, such as 50%, the valve may experience velocity limiting, resulting in a time constant closer to 5 seconds. Figure 8.2 illustrates this concept.

Figure 8.2 [Velocity limiting will make a short lag look like a long lag, and this can cause instability.]

The occurrence of velocity limiting, which can happen in a valve controlled by an electrical motor, a pneumatic valve positioner or a pneumatic controller without a valve positioner, can result in instability of a loop. The effect of lags in the loop plays a key role in determining whether velocity limiting will cause stability problems. The gain of the loop is proportional to $1/RL$ , and if the product of $R$ times $L$ increases due to velocity limiting, the loop may become unstable if the gain is not adjusted accordingly.

When velocity limiting occurs, the effective time constant may change from a small value to a larger value, making the apparent dead time, $L$ , larger. If the loop had been tuned tightly before, the increase in $RL$ could result in instability and cause the loop to cycle at a constant amplitude until the gain is reduced or the controller is switched to manual mode. Putting the controller in manual mode briefly can restore stability, but the loop may become unstable again if an upset large enough to trigger the velocity limiting occurs.

Whether velocity limiting causes instability or not depends on whether its lag for small upsets was contributing to the apparent dead time. If the high performance of the loop is not required, the gain may be reduced to ensure stability for the worst-case scenario. For a pneumatic valve positioner, a booster between the positioner and the valve motor may be used as a possible fix if high performance is required.

Dead Band

Dead band is often the primary cause of tuning difficulties in control loops, and is commonly associated with valve stem packing friction. Although it can occur in other situations as well, many believe that a valve positioner is the solution to all problems stemming from packing friction. While it does improve the dynamic response for small changes, dead band can still be an issue for changes of less than 1%. In such cases, the controller output may request a 1/4% or 1% change, but the valve fails to move, or moves more slowly than expected, creating the appearance of dead time that can last several seconds. The resulting non-ideal positioning is typically not a source of tuning or stability problems, although under certain circumstances it can be.

A perfect positioner would move the valve by the exact percentage requested by the controller output. In contrast, an imperfect positioner might only move the valve by 1/2% or 1/4%. While this reduces the gain and introduces some small lags, as depicted in Figure 8.3, it does not typically cause any significant issues.

Figure 8.3 [A positioned valve does not have a linear response for small changes.]

When using digital control systems and magnified displays, it is possible to tune the loop at much smaller amplitudes compared to analogue systems and non-magnified displays. However, if the tuning is performed manually at very small controller output changes, there is a chance that the assumed gain between the controller output and valve motion may not be accurate. This may result in inaccurate determination of the parameters $R$ and $L$ from Figure 2.3, which affects the tuning settings. If the tuning is done in automatic mode, the gain established for small changes may be too high for large changes, causing instability. To resolve this problem, the gain can be reduced or a positioner with higher gain can be used for small changes.

Dead band can also occur when the valve is not positioned, and this may result in the same phenomenon at larger amplitudes. In this case, installing a positioner is the first remedial step. The more important effect of dead band is when using control with integral action or when the process integrates. With even a small amount of dead band, there will always be a small error, and the integral action keeps moving the controller output until the valve moves. This may cause the valve to move too much, eventually reversing the error. The integrating action then moves the valve back, but again too far. This creates a small cycle that can be observed in the process. On a slower loop, the presence of dead band is not easily recognised as the lag between the valve movement and process response is more obscured. The cycle may not be visible in normal records, but its effects will be observed elsewhere in the process.

Figure 8.4 [Dead band in a valve often results in a small limit cycle, characterised by a triangular wave in the controller output and a square wave in the actual valve motion. Reducing the gain or increasing the integral time increases the period but does not otherwise after the cycle significantly.]

When a cycle is observed, decreasing the gain alone does not address the root cause of the problem. The cycle will simply have a longer period. If the period is recognized, the standard tuning procedure would suggest increasing the integral time. However, doing so will only result in an even longer period. If this phenomenon is not correctly identified as a dead band issue, the controller settings will be significantly off from optimal values and the tuning process will be exceedingly frustrating.

If a problem resulting from dead band is not recognized, it can greatly reduce controller settings from optimal. So, how can dead band be recognized? In fast processes, the behavior shown in Figure 8.4 can be a good indicator, especially when using a monitor with the ability to magnify amplitude and time scales. The controller output will typically be triangular, while the valve motion will be a square wave. However, in slower processes, the observed behavior may not look like Figure 8.4, but you will see a small, relatively fixed amplitude cycle, whose period gets longer as gain is decreased or integral time is increased. The period will be significantly longer than the natural period of approximately $4L$ as shown in Figure 2.3.

In digital systems, the presence of dead band may be observed when making small step changes in manual. With small changes, you may see no response, or the slope may not be proportional to the size of the step change. Also, there may be a longer apparent dead time than expected. To check for dead band, you can make two small changes in manual in the same direction and then reverse direction with the same two small changes. If the process does not repeat itself for the same outputs, it indicates significant dead band.

To fix the problem, you can reduce stem friction, choose a better positioner, or use a booster between the positioner and the valve operator. Boosters do not affect the dead band but can improve small-amplitude dynamics. Look for a positioner with a high gain, as a gain of 50 is too low. The manufacturer can provide this information. The dead band with a positioner is essentially the dead band without a positioner divided by the positioner gain.

To determine if the valve is moving or not, you can place your fingers on the valve stem or mount a dial micrometer to monitor valve movement. However, be aware of the possibility of dead band in linkages, especially when there is a linkage between the positioner and the final flow restriction, such as with a butterfly valve.

In conclusion, dead band is a common cause of controller tuning problems, especially for small excursions. Be aware of its different contributions, especially the small amplitude cycle whose period is significantly longer than the natural period of the loop. While smart valves may minimize these problems, non-smart valves will still be in use for several years.

Valves at Limits

This section discusses the issues that can arise when a valve operates near its limit, typically its closed limit, and with a positioner. This is distinct from integral windup. When a valve hits a stop with a positioner, an additional delay is introduced. The positioner applies full supply or full vent to the valve motor, and when the controller signal returns to scale, the positioner output must change significantly before the valve comes off the stop. This delay can lead to a limit cycle on a fast loop, which can be mitigated by going to manual momentarily or reducing supply pressure to the positioner. If using a program-your-own computer/controller, a program can be written to manage this issue by using feedback from the positioner output. Even without a positioner-saturation problem, it is generally not advisable to have a valve hit a limit, as this can disrupt the tuning assumptions. Therefore, if performance issues arise, it is important to determine if the valve is near its limit, particularly its closed limit.

Integral (Reset) Windup

Integral windup has been a recognized phenomenon since the invention of its function, which was likely in the 1930s. In the early days, there was little that could be done to combat the issue, and the solutions that were developed were awkward. However, as technology progressed, better and more elegant methods were created.

Integral windup can occur with a controller that has reset or integral action, which is present in most controllers. If the valve reaches its limit, whether it be fully open or fully closed, the valve is operating at its maximum capacity. The controller may not realize this, and it will continue to adjust the output based on the integral of the error. The problem arises when the valve needs to return to the desired position on the scale. At this point, the controller output has likely surpassed the valve limit, and the controller needs to “unwind,” which can take a long time. This delay often leads to poor performance and potentially instability.

Instrument manufacturers have developed anti-windup protection to combat this issue. Most electronic and computer controllers available today include this feature, either as a standard or an option. However, pneumatic controllers require external components to address this problem. It is important to note that not all anti-windup features perform the same way. Some require the error signal to reverse sign before the valve can move, while others may cause a valve “kick” from proportional action. It is important to study the details of how a particular vendor implements the protection if you experience repetitive problems with integral windup. It is likely that one vendor would be preferred over another if the windup circumstances tend to be the same.