The objective of tuning rules is to achieve a “tight” control, meaning that the controller responds quickly to changes in setpoints or disturbances without excessive cycling. However, tight control is not always necessary or desirable, especially for level controls. With this in mind, the focus now turns to the expected performance of a tightly tuned loop. Performance is evaluated based on how closely the controlled variable follows the setpoint during setpoint changes and disturbances. Various mathematical criteria have been used to study this problem, such as integral of error (IE), integral of the absolute error (IAE), integral of the error squared (ISE), integral of the absolute error multiplied by time (ITAE), peak error, and others. These studies assume mathematically simple process models and welldefined disturbances, and they present results with three significant figures. However, real processes and disturbances are often more complex and not welldefined mathematically. Despite this, these studies provide helpful generalities, which will be presented.
Consider a scenario where a controller is in manual mode and a disturbance causes the controlled variable to respond, as illustrated in Figure 3.1. The same disturbance is applied when the controller is in automatic mode, as shown in the same figure. The numerical values for the settings provided are indicative only. Now, it’s essential to note that when the loop is tightly tuned, the controlled variable will oscillate with a deviation roughly equal to the deviation that would have occurred without control. This deviation will persist for a little more than units of time, which is approximately equal to two apparent or real dead times. The reason for this is that it takes one dead time for the controller to detect the disturbance and another dead time for it to respond to it.
Therefore, the natural period can be regarded as a metric for rating the loop’s performance in the case mentioned above. All else being equal, a loop with a of one minute will typically perform twice as well as a loop with a of two minutes. This is a simplification, but it encapsulates the core of the concept of performance. In industrial control, we don’t typically worry about deviations from expectations of 10%, 20%, or even 30%. Even getting close within a factor of two is significantly better than many systems can achieve without these ideas. For a single loop, shorter periods are always preferable if minimizing deviation from the setpoint is the objective.
In the case where integral action is incorporated into the system depicted in Figure 3.1, the outcomes shown in Figure 3.2 are obtained. The peak error arises slightly later, and the use of integral action amplifies the propensity to cycle, which occurs at a slightly longer period.
Figure 3.1 [For an upset, increasing gain in a proportionalonly controller reduces the error, and the tendency to cycle is increased. The maximum deviation for a tightly tuned controller occurs at a time slightly longer than .]
A second crucial point to note from the above example is that, if the controller is tightly tuned, the controlled variable will take about time to return to the setpoint. The specific multiplier may vary slightly, but the important idea is that the time to recover is proportional to . Thus, shortening the natural period has the double benefit of reducing both the size and duration of the error. This is a fortunate situation where achieving one desirable outcome leads to the achievement of another.
Figure 3.2 [Integral action restores the controlled variable to the setpoint. Reducing the integral time increases the tendency to cycle, and at a longer period than for proportionalonly control.]
To achieve minimal deviation from the setpoint, shorter natural periods are more desirable for a single loop. The numerical value of the gain is not as significant as the natural period. This is evident in the case of a temperature loop with different transmitter spans, where the ultimate gain may differ by a factor of 100, but the performance remains the same as long as the controller can reach the ultimate gain for stability.
Improvements in performance may result in using a higher gain, but the gain itself does not cause the improvement. Therefore, the natural period, not the gain, is the primary factor that determines the performance of a loop. Performance of a loop is uniquely tied to the natural period and improves as the natural period decreases.
It is important to note that a control loop has limitations in reducing the effects of upsets. The controlled variable will typically deviate for a time approximately equal to two dead times before the controller can react. If the loop is tightly tuned, it may take a tad longer than for the controlled variable to recover to the setpoint. However, if the natural period is shorter, the deviation and recovery times will also be shorter, resulting in improved performance.
Response to Cyclic Upsets
In the example provided, the nature of the upset was not specified in terms of its numerical value or its type. Instead, it was assumed that the upset occurred and persisted for a certain period. The presented ideas are significant, but there is another concept that needs to be presented differently.
Consider a hypothetical upset that oscillates sinusoidally instead of being constant. This situation will be examined under three conditions:

 Disturbance period short relative to natural period.
It is clear that if the disturbance has a short period, the control system will not be able to respond fast enough to have a noticeable impact. In this case, control action would be ineffective. For example, consider taking a shower where the hot water pressure changes every half second. It would be impossible to adjust the temperature quickly enough to counteract the changes.  Disturbance period long relative to natural period.
After some consideration, it becomes clear that control can still provide some benefit, albeit not as much as in the previous case. As the period of the upset increases, the integral action of the controller has more time to work, so the benefit of control is proportional to the period.For instance, let’s take the example of a shower where the hot water pressure changes sinusoidally with a period of one minute. Although you may not have time to wash, you will be able to compensate for the changes in pressure quite well. However, if the period is shorter, say half a second, you won’t be able to compensate fast enough, rendering the control action ineffective.  Disturbance period (or near) the natural period.
This case is presented last as it is more complex and not as intuitive. In this situation, control action actually worsens the situation, leading to more errors than without control. This occurs because the controller output is oscillating in the opposite direction of what it should be. An analogy often used is that of a handheld mass suspended by a spring. If the hand holding the spring is disturbed at the right frequency, the mass will move much more than the hand. While this analogy may be helpful, it is not necessary to fully understand the concept. Another example within the range of normal experience is difficult to come up with. The shower example is not applicable here as the brain would recognize that it is making things worse and would stop trying to adjust.
 Disturbance period short relative to natural period.
The graph in Figure 3.3 provides an overview of the relationship between tuning variations and the control system’s ability to attenuate periodic disturbances. The center and righthand sections of the curve are affected by tuning changes, with higher gain worsening performance near the natural period but improving attenuation at longer periods, as indicated by the dotted line. Shorter integral times can also improve attenuation, but this comes at the cost of greater resonance near the natural period.
The crucial point to bear in mind is that the control system’s effectiveness in the face of periodic disturbances depends on how the disturbance period relates to the natural period. While disturbances may not always be cyclic, it is still essential to assess the variations in the controlled variable and determine whether they are fast or slow relative to the natural period. This will provide insight into whether the controller has any chance of mitigating the problem.
It is essential to recognize that even a welltuned controller may fail to achieve the desired outcome in some cases. When disturbances are too fast relative to the natural period, it may be challenging to conclude that the controller is not doing its job. In such situations, it is advisable to inform the relevant parties that further tuning efforts will not help and suggest alternative approaches. Loops with long dead times can be particularly challenging, as they can take a long time to recover from disturbances. In these cases, it may be more effective to address the disturbance’s root cause, if possible.
Figure 3.3 [Control action can either help, hurt or do nothing to reduce the effect of a disturbance, depending on whether the disturbance is slower, at or faster than the natural period.]
Factors Affecting the Natural Period
Understanding the factors that determine the natural period is crucial for assessing performance and setting reset and derivative times. This knowledge is especially important when considering the two equations for tuning, as a great revelation is about to be revealed.
for the closedloop testing
Where:
= Integral time
= Natural period
for the openloop testing
Where:
= Apparent dead time
Combine these two equations and you have the most important simple concept in control:
THE NATURAL PERIOD IS APPROXIMATELY 4 TIMES THE APPARENT DEAD TIME
At some point in the future, it may become clear to you how significant the following insight is. Note that the symbol “” signifies an approximation, since the tuning rules themselves are not precise. The tuning rules are based on scientific principles, and it is through those principles that the relationship between and L is validated.
However, it is possible to provide a more precise relationship between and L. Figure 3.4 demonstrates three different scenarios. The first example (Figure 3.4a) can be confirmed by logically analyzing the loop’s behavior over time when cycling. In this case, the equation is exact.
Figure 3.4 [The natural period varies between 2L and 4L, depending on the other lags in the system.]
Before proceeding with the next example, it should be taken as a fact that the step response shown in Figure 3.5 can be divided into a dead time followed by an integrator. In this scenario, the integrator has a specific property that the output lags the input by when the loop is cycling. This requires some explanation. Please refer to Figure 3.5. During the cycling of a loop, the output of an integrator moves at its fastest rate when the input is at its maximum. As a result, the output of the integrator lags the input by a quarter of the period of the sine wave, or . Although it is admittedly more difficult to follow the logic of the signal around the loop than it was for the pure dead time case, the phase lag will always be . The relationship is exact for this example.
Figure 3.5 [When a loop is cycling the output of an integrator lags the input by .]
The apparent dead time is comprised of both pure dead time and small lags in the control loop. However, some processes have a true dead time while others do not. It is important to understand the meaning of certain terms and examples in order to grasp this concept. Interestingly, the largest lag in a control loop, which can sometimes be an integrating element, does not necessarily affect the natural period of the loop, which may seem counterintuitive to some.
Although this concept can be difficult to understand, it is important to note that the speed of a control loop is dependent on the lags within the system. It is also important to understand what is meant by the term “lag.” In order to fully understand this topic, we will first explore the meaning of lags and then come back to connect everything together.
Math / Algebra
At this point, it is important to introduce a concept that is crucial for understanding the mathematics and algebra of control loops. To begin with, it is important to consider what happens when the loop cycles.
During the cycling of a loop, every point within the loop cycles as well. If the cycle returns to a point in the loop and matches the cycle at that point in terms of amplitude and phase, then the cycle will sustain itself. However, if it does not match in amplitude and phase, the cycle will either grow or decay.
While there are methods for determining whether the cycle will improve or deteriorate, they are too complex to be addressed at this point. It is worth noting that the phase lag around the loop is either or minus when it cycles. Half of this phase lag comes from the controller itself. As can be seen in the signalflow diagram for the controller, the controlled variable is subtracted from the setpoint, which is necessary to ensure that the controller acts in the correct direction. This subtraction results in an inverted sine wave and a phase lag of when the loop is cycling. This leaves to be contributed by the lags.
When the loop cycles, each element contributes a specific value to the lag and an amplitude ratio (the ratio of output amplitude to input amplitude). If all the amplitude ratios are multiplied together when the loop cycles, the result is unity and dimensionless.
A controller example, idealized, has no phase lag. A pure dead time has a phase lag of:
(1)
Where:
= dead time
= period
This should be easy to understand. When the dead time and the period are the same, the phase lag is . So what is P when the phase lag is ?
(2)
All you have to do is adjust the controller gain to make the amplitude right, and a cycle at will result.
An integrator example, has a phase lag of all the time, at any period. So the dead time has to contribute only . What is P when the phase lag is ?
(3)