Continuous feedback control

There are three basic types of controllers that are commonly used for continuous feedback control. The details of construction of the analog devices and the programming of the digital devices vary from one manufacturer to the next, but their basic functions are essentially the same. 

Both u(x) and Jopt(x) are discontinuous at the origin. Therefore, stability theorems that rely on continuity cannot be used. Yet, it is simple to check by inspection that the feedback law of Eq. (71) (with either sign chosen) is asymptotically stabilizing. However, the continuous feedback control law u(x) = 0, resulting in the closed-loop system... 

To realize a continuous feedback control of rotating excitation waves, the parameter modulation / (t) is determined to be proportional to the average of a system variable over a certain spatial domain S. For instance, the feedback signal I t) is computed according to [47, 48, 50-52] ... 

To determine the drift velocity field for an arbitrarily shaped spatial domain, we divide it into a set of small subdomains. Each subdomain is treated as a one-point detector generating a feedback signal with a phase shift given by Eq. (9.46). Finally, the resulting drift velocity field is derived according to Eqs. (9.45) and (9.42) as the sum over all drift vectors induced by single subdomains. This superposition principle unifies and simplifies the study of different algorithms for continuous feedback control considerably. 

Between the measuring device and the final control element comes the controller (Figure 13.1b). Its function is to receive the measured output signal ym(t) and after comparing it with the set point ysp to produce the actuating signal c(t) in such a way as to return the output to the desired value ysp. Therefore, the input to the controller is the error e(t) = ysp - ym(t), while its output is c(t). The various types of continuous feedback controllers differ in the way they relate e(t) to c(t). 

Besides these uses, the computer has application in fermentation processes for continuous nonprejudicial monitoring and (most important) continuous feedback control and dynamic optimization of the process. 

Some of the inherent advantages of the feedback control strategy are as follows regardless of the source or nature of the disturbance, the manipulated variable(s) adjusts to correct for the deviation from the setpoint when the deviation is detected the proper values of the manipulated variables are continually sought to balance the system by a trial-and-error approach no mathematical model of the process is required and the most often used feedback control algorithm (some form of proportional—integral—derivative control) is both robust and versatile. 

Classical Feedback Control. The majority of controllers ia a continuous process plant is of the linear feedback controller type. These controllers utilize one or more of three basic modes of control proportional (P), iategral (I), and derivative (D) action (1,2,6,7). In the days of pneumatic or electrical analogue controllers, these modes were implemented ia the controller by hardware devices. These controllers implemented all or parts of the foUowiag control algorithm ... 

In continuous processes where automatic feedback control has been implemented, the feedback mechanism theoretically ensures that product quality is at or near the set point regardless of process disturbances. This, of course, requires that an appropriate manipulated variable has been identified for adjusting tne product quality. However, even under feedback control, there may be daily variations of product quahty because of disturbances or equipment or instrument malfunctions. These occurrences can be analyzed using the concepts of statistical quahty control. 

Roliani, S. and Paine, K., 1991. Feedback control of CSD in a continuous cooling crystallizer. Canadian Journal of Chemical Engineering, 69, 165. 

Instrumentation is also fitted to provide a continuous display of important variables such as temperature and pH, the power used hy the electric motor, airflow, dissolved oxygen and exhaust gas analysis. Manual or computer feedback control can be based either directly on the signals provided hy the prohes and sensors or on derived data calculated from those signals, such as the respiratory coefficient or the rate of change of pH. Mass spectronomical analysis of exhaust gases can provide valuable physiological information. 

The simple feedback control system below consists of a continuous-flow stirred tank, a temperature measurement device, a controller and a heater. 

Figure 5.156. Feedback control of a simple continuous water heater.

FEEDBACK CONTROL OF A CONTINUOUS WATER HEATER UNITS DEG C,KCAL,KG,L,HR... 

An exothermic reaction involving two reactants is run in a semi-continuous reactor. The heat evolution can be controlled by varying the feed rate of one component. This is done with feedback control with reactor temperature... 

A total continuity equation for the liquid phase is also needed, plus the two controller equations relating pressure to heat input and liquid level to feed flow rate Fq. These feedback controller relationships will be expressed here simply as functions. In later parts of this book we will iscuss these fhnctions in detail. 

Setpoint changes can also be made, particularly in batch processes or in changing from one operating condition to another in a continuous process. These setpoint changes also act as disturbances to the dosedloop system. The function of the feedback controller is to drive the controlled variable to match the new setpoint. The dosedloop response to a setpoint disturbance is called the servo response (from the early applications of feedback control in mechanical servomechanism tracking systems). 

The feedback controller is continuous, and it sees a constant error between samples. Therefore the manipulated variable m, is ramped up or down during the sampling period by the integral action, as sketched in Fig. 18.12u. 

There is a special type of controller, called a Smith predictor or deadtime compensator, that can be applied in either continuous or discrete form. It is basically a special type of model-based controller, in the same family as IMC. Figure 20.6a gives a sketch of a conventional feedback control system. Let s break up the total openloop process into the portion without any deadtime G j,(s) nd deadtime e... 

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