Simple performance measures
This section defines the simple performance measures that are commonly usedin process control.
Overshoot.
Overshoot is formally defined for the case where the process makes a transition from one operating level to another. Although overshoot can be expressed in engineering units, control engineers most commonly express overshoot in percent.For the response in Figure 3.25, the controlled variable changes by a unitsand overshoots its final value by b units.
Decay Ratio.
The decay ratio reflects the rate of decay of the sinusoidalcomponent of the response. The decay ratio is the ratio of the second peakovershoot c to the first peak overshoot b.
Line - Out Time or Settling Time.
A loop is said to be “lined out” when all variables cease to change. The loop and all of its components are in equilibrium. The lineout time or settling time is the time required for the system to attain equilibrium after a change in one or more of its inputs. In practice, the lineout time is considered to be the time that the response has attained its final value and no oscillations or other change is visible in the response. To quantify the lineout time mathematically, one has to introduce a tolerance for a variable to be considered as having attained its equilibrium value. When the input is a step change in the set point, the tolerance is typically 2% or 5%. The tolerance introduces a band about the final value. The lineout time is the time required for the response to come within that band and to remain within that band thereafter.
Limitations of Simple Performance Measures.
Simple performance measures such as decay ratio and percent overshoot are appealing because theyare easy to apply. We rarely measure peak heights and compute precise values;we use visual estimates instead.However, their limitations must be understood. The decay ratio depends ononly two points on the response; the percent overshoot depends on only onepoint. This gives the possibility that two very different responses could havethe same value of a simple performance measure.This has implications for controller tuning. Both responses in Figure 3.26have a quarter decay ratio, but they are very different. The process is exactlythe same; only the controller tuning is different. We would usually prefer thefaster of the two responses. The controller tuned with a gain of 2.0%/% resultsin a response with a quarter decay ratio; however, the controller is capable of delivering better performance.
The integral criteria
The integral criteria address the deficiencies of the simple performance mea-sures, but at a price. Let us first define the integral criteria and then discusstheir advantages and disadvantages.
General Definition.
An integral criterion is a performance measure that isbased on the integral of some function of the control error and on possiblyother variables (such as time). The general expression is
Integral of Absolute Error ( IAE ).
This integral is the integral of the controlerror, with all error treated as positive.
Integral of Square Error ( ISE ).
The integral of the square error is the coun-terpart to the sum of squares from linear regression
The integral of the square error (ISE) penalizes for large errors more than forsmall errors. But in attempting to minimize this criterion function, responseswith small errors will be accepted provided doing so will reduce the largeerrors. Often the response has a smaller initial overshoot, but the cycle doesnot decay rapidly. Basically several small peaks are tolerated to reduce themagnitude of the first peak. This type of behavior is usually not desired inprocess loops. Integral of Time and Absolute Error ( ITAE ).
The objective of the integralof time and absolute error (ITAE) is to penalize for even small errors thatoccur late in time. This is achieved by including time in the integrand
Even small errors occurring late in time contribute significantly to the integral.Consequently, minimizing this integral criterion generally leads to responseswith a short lineout or settling time. Process engineers generally prefer thistype of behavior. Obtaining a Value for an Integral Criterion.
The integral criteria are not“user-unfriendly.” Simple performance measures such as decay ratio andpercent overshoot are routinely estimated. This is not possible with the integralcriteria. The only way to obtain a value is to evaluate the integral numerically,which in practice requires a computer.We tend to form mental pictures of responses with a certain decay ratio ora certain percent overshoot. However, we cannot do this for the integral cri-teria. Suppose the response of the loop to a certain change in the set point hasan IAE of 27.5. This one number is not especially informative. To some extent,the numbers are only relative. If the tuning parameters are changed such asthe IAE for the same set-point change is 24.1, the new tuning coefficients aresuperior to the old.
Minimizing the Selected Integral Criterion.
Determining the values of the tuning coefficients
that minimize the selected integral criterionis an iterative endeavor. Given starting values for the tuning coefficients, eachiteration consists of the following:1. Obtain a process response to the input change of choice (such as a stepchange in set point).2. Evaluate the integral criterion.3. Select new values for the tuning coefficients.This procedure is repeated until the tuning coefficients that minimize theintegral criterion of choice are found. Computer routines known as multivariable search techniques are available for efficiently performing such minimizations. But even with a good search technique, 50 to 75 iterations are required to tune a PI controller. For a PID controller, the number is in excess of 200. Such undertakings are only feasible on simulations, never on the real processes.
Use of the Integral Criteria.
The availability of simulations basically determines the extent to which integral criteria can be used. Process simulationsare not routinely done, and even when undertaken, they are often not to the detail required for tasks such as controller tuning. To date, the use of the integral criteria has been limited to a few tuning methods. For the simple process models used as the basis for the conventional controller tuning techniques, the results of minimizing the integral can be reduced to a set of equations. But for any model, a computer can determine the tuning coefficients that minimize the selected integral criterion.
Advantages of the Integral Criteria.
An integral criterion does a far better job of quantifying the nature of the response. Every point on the response contributes to the value of the integral criterion. Simple performance measures such as decay ratio or percent overshoot depend on only one or two points on the response. Minimizing an integral criterion gives a unique set of values for the tuning coefficients. For PI or PID control of a given process, many sets of tuning coefficients will give a quarter decay ratio. This is never the case for minimizingan integral criterion.
Most process control presentations begin with process dynamics, the presump-tion being that the design and performance of the controls depend primarily,if not exclusively, on dynamic behavior. We wish to challenge this presumption. In practice, two aspects of process dynamics lead to problems:
Dead time.
The presence of dead time is one of the distinguishing characteristics of process control. The effectiveness of a PID controller goes down rapidly as the dead time increases. Either the dead time has to be reduced or alternative control approaches have to be pursued.
Integrating processes.
The difficulty originates when a short reset time isused in the controller. For integrating processes, the proportional modehas to be the primary mode of control.
The key to a control system that delivers good performance is a P&I diagram that reflects the characteristics of the process. The control structures on the P&I diagram mostly reflect the steady-state characteristics of the process as quantified by the gains or sensitivities. In this regard, problems can surface with regard to process nonlinearities and loop interaction. Control valves aswell as the process can contribute to the nonlinearities. In practice, these problems develop more frequently than problems associated with process dynamics.
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