Temperature Control Theory Essay, Research Paper

Process control systems serve two different purposes. Generally the first of these purposes is to effect a change in a certain output variable, commonly encountered in the startup of a process. The second of these is to regulate or hold an output variable constant despite any changes in an input variable, which usually cannot be easily managed [1].

In order to discuss process control, in detail it is necessary to define several terms. A control loop is comprised of several variables. Among these are manipulated variables (MV), process variables (PV), and disturbance variables (DV). Manipulated variables refer to variables that are easily controlled, such as stream flowrates. Process variables are those that are desired to be set at a certain datum level, commonly a desired temperature. Disturbance variables create deviations in the process variable from the desired datum level or set point (SP). The controller receives process variable information and in turn attempts to maintain the process variable at the set point.

In particular there are two types of control systems, feedforward and feedback control. Feedback control operates by feeding back process variable data to the controller. In feedback process control, the controller receives process variable data and makes appropriate changes in the manipulated variable via the final control element (FCE). The uniqueness of feedback control is that it utilizes prevailing process variable information in order to determine what measures need to be taken to return the process variable to the set point [2]. Figure I illustrates a typical feedback control loop.

The modus operandi of a feedfoward controller is that the controller receives a direct signal of the disturbance variable. Here the disturbance variable is measured directly rather than the output variable that is desired to be regulated. The disturbance variable is measured before the process is perturbed and the controller endeavors to neutralize the disturbance s effects, usually by some prescribed process model. The controller does not receive any output variable information and consequently has no information pertaining to the actual effect or accuracy of the control action [2]. Figure II depicts a typical feedfoward configuration. A feedback controller was used in this investigation and henceforth only feedback controllers will be discussed.

There are three general categories of feedback controllers, namely proportional control, proportional-plus-integral (PI), and proportional-plus-integral-plus-derivative (PID). Each type will be treated separately.

A proportional-only controller changes its output in such a way that the output is proportional to the deviation of the process variable from the set point. The deviation of the process variable from the set point is referred to as the error and denoted,

Despite the simplicity of proportional control, it has one major deficiency. Under proportional-only control, steady-state offset is observed for non-zero set points [1]. By the definition of steady state, all time derivatives must be equal to zero. It is possible for de(t)/dt to equal zero so long as the error has reached some steady-state value or the error is zero. However, looking back to equation 3 it is apparent that if e(t) is zero so must be the value for the controller output c (t). The difficulty here is that the c (t) can never be zero for non-zero set points; and therefore, for proportional only controllers, steady-state offset will always be observed. Moreover, from equation 3 as the system reaches steady state, the error stabilizes and consequently the controller output, c(t), would also stabilize. Thus at steady state, the proportional only controller does not attempt to counterbalance the error [1].

Under steady-state conditions the controller output and the error will remain constant. Equation 4 implies that the c(t) will change with time until e(t) is zero. Therefore, under integral control c(t) will, at steady state, assume a value such that e(t) reaches zero [4]. Depending upon the capability of the equipment that is being used, this always occurs. However, it is possible for the controller or the final control element to saturate and reach a limiting value. Should saturation occur, the controller or the final control element will become stuck and will be unable to return the controlled variable back to the set-point [4].

Integral control action is seldom used alone, since little correction is achieved until the error signal has endured for a period of time. Since proportional control action occurs immediately at the detection of error, integral control and proportional control are often used in tandem.

Proportional-plus-integral control exhibits the quick response of proportional control and the effective treatment of offset as observed under integral control conditions. Despite this fact, PI control does exhibit some limitations. PI control affords an undulating response pattern and hence can cause disturbances in the system, increasing system instability [4]. Usually small disturbances can be tolerated by virtue of the faster response time. This phenomenon can be corrected by proper tuning of the instrument or by incorporating derivative control into the control system [4].

Unfortunately there is an additional limitation which must be considered when applying integral control. Reset windup is an indigenous hindrance for integral control. Referring back to equation 4, the controller output is subject to change at any point that error exists. Should the error persist, the integral term will tend to grow and the controller will become saturated. As the controller becomes saturated, the integral term will continue to grow larger, resulting in an event referred to as reset windup or, alternatively, integral windup [4]. The following figure qualitatively depicts the occurrence of a step change in set point when using a PI controller [4].

Positive areas account for positive contributions to the integral term whereas the negative areas result in a negative integral term. Reset windup is significant whenever a PI controller experiences sustained error [4]. The startup of a batch process is a common example. Reset windup can also occur due to a large prolonged load disturbance that is significantly outside of the range of the manipulated variable. An example of this type of situation is a control valve being either completely shut or open. Should the aforementioned occur, the controller would be rendered ineffective in returning the error signal to zero [4]. These types of situations are all clearly undesirable. Nevertheless, there are control devices that furnish an anti-reset windup feature. This configuration reduces reset windup by stopping the integral action at the time when the controller is saturated. As the controller becomes unsaturated, the integral action is continued [4].

The third type of control action is derivative control. The uniqueness of derivative action is that it anticipates the future response of the error signal. The derivative controller accomplishes this anticipation by taking into account the error s rate of change [4]. Consider the situation where a process variable, say pressure, increases ten-fold over a time period of only a few minutes. This is clearly a sharp rise in pressure and could indicate the onset of a predominately uncontrollable system. Were this a manual process the operator would recognize the potential danger and take action to prevent any further pressure increases. Should this process be operated under P or PI control a disaster would surely result. Proportional control and PI control provide no means of predicting or forecasting the future behavior of the process.

The anticipation of the skilled operator can be mimicked mathematically via derivative control. This approach entails setting the controller output so that it is proportional to the rate of change of the controlled variable. Notice that under steady state conditions c(t) would by necessity be equal to cs. Consequently derivative action cannot be effectively used alone. Derivative control is always used with proportional or proportional-plus-integral control. The ability of derivative action to anticipate future error signals enables it to provide a stabilizing effect to proportional and proportional-plus-integral control systems [4].

Derivative control can also produce a settling effect. Namely, derivative control can diminish the time required for the process to reach steady state [4]. This effect is not always desirable. Process systems that are inherent to experience high frequency random fluctuations, such as flow processes, will be adversely affected by derivative action. This arises from the general nature of the derivative action. Any high frequency, random fluctuations will result in a derivative of the controlled variable which acts in an unrestrained fashion. Consequently, derivative action will amplify the noise of a system [4]. A low-pass filtering device, such as a mixing tank can lessen this effect.

The combination of proportional, integral, and derivative control results in the three-mode proportional-plus-integral-plus-derivative controller (PID) [4]. Here as stated before the derivative action allows the controller to predict the future behavior of the error signal. Consider the following situation. The inlet temperature to a process, TI(t) begins to decrease and by consequence the outlet temperature, T(t) does likewise, as depicted in Figure IV. At ta there exist a positive error, which is small. Thus the control action provided by the integral controller is small. Examining the error plot versus time, at time ta, the derivative of the error is large and positive. The PID controller notices the large slope of the error curve, and attempts to adjust the output in such a way as to correct an out of control process. At tb the error is positive and greater than the error at ta. Moreover in addition to the excess action taken by the derivative mode, the controller output is also being further saturated from the increased response of the integral and proportional modes. The slope of the error curve at tb however is negative and the derivative mode begins to subtract from the other two control elements. As a result of this affair the system takes longer to reach the set point.

Previously, only the design of a controller has been considered. Although this is an important element, in order for a controller to be effective, it must be fitted appropriately to the process at hand. This is commonly referred to as tuning. Only the qualitative aspects of tuning will be considered.

Recall that Kc=100/PB. Generally, an increase in the gain, Kc, will tend to cause the controller to react faster, but if the gain is too large the response could display an unwanted undulation and lead to instability [4]. Usually some median value for Kc is preferred to optimize the proportional control. This approach is effective for both PI and PID controllers.

Decreases in the reset rate (recall the definition of reset rate), tI, will cause the controller to respond slowly and sparingly. For very small values of tI, the controlled variable will return very slowly to the set point upon any system load perturbations or set point changes [4].

A qualitative generalization regarding the affect of the derivative rate, tD, is more complicated. Ordinarily small values for tD rectify the response by offsetting large error deviations, reducing the response time, and reducing the degree of oscillatory response. Large values of tD, usually amplify noise and should be avoided. Commonly just as with the proportional gain, a median value for tD is chosen.

References

1) Shilling, David G. Process Dynamics and Control, Holt, Rinehart, and Winston, Inc. New York, NY, 1963.

2) Babatunde, A. Ogunnaike; Ray, W. Harmon. Process Dynamics, Modeling, and Control, Oxford University Press, New York, NY, 1994.

3) Stephanopoulos, George. Chemical Process Control: An Introduction to Theory and Practice, Prentice Hall, Englewood Cliffs, NJ, 1984.

4) Seborg, Dale E.; Edgar, Thomas F.; Mellichamp, Duncan A. Process Dynamics and Control, John Wiley and Sons, Inc. New York, NY, 1989.

5) Smith, Carlos A.; Corripio, Armando B. Principles and Practice of Automatic Process Control, John Wiley and Sons, New York, NY, 1985.

6) Perry, Robert H.; Green, Don W.; Maloney, James O.; Perry s Chemical Engineers Handbook, 7th ed, McGraw-Hill, New York, NY, 1997.

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