Error from setpoint

FIGURE 15.3 Block diagram of a generalized feedback system e is the error from setpoint, c is the controller output, and u is the manipulated variahle. 

The symbol in Figure 15.3 represents a summation block. The negative sign on the measurement of the controlled variable results in forming the difference between the setpoint and the measured value of the controlled variable, which is referred to as the error from setpoint. A block diagram of an open-loop process involves only the actuator, process, and sensor without the feedback of the measurement of the controlled variable to the controller. 

Another way to represent the controller gain is the proportional band PB), which is an approach that was in more common use 10 to 15 years ago. Proportional band can be expressed (as a percent) in terms of when is in dimensionless form. For example, the controller output and the error from setpoint can be scaled 0 to 100%, yielding a dimensionless K ... 

Note that Equation (15.7) is written as a reverse-acting controller. A reverse-acting controller subtracts Ac(t) from c(t - At). The velocity form for the derivative on the error from setpoint is given by... 

Another popular version of the velocity form of the PID can be developed by eliminating proportional action for setpoint changes. Noticing that the proportional part of Equation (15.6) is simply the difference between y(t) and y(t - At) when the setpoint remains unchanged leads one to replace the difference between errors from setpoint with the difference between measured values of the controlled variable in the velocity form of the PID controller, i.e.,... 

The position form of the PID algorithm calculates the absolute value of the output of the controller, whereas the velocity form calculates the change in the controller output that should be added to the current level of the controller output. The position and velocity modes are different forms of the same equation therefore, they are generally equivalent. The velocity form is usually used industrially. In general, DCSs offer the velocity form of the PID controller in three versions the velocity form in which P, I, and D are based on the error from setpoint [Equation (15.8)] the form in which only P and I are based on the error from setpoint [Equation (15.6)] and the form in which only integral action is based on the error from setpoint [Equation (15.9)]. 

Remember that the standard deviation is based on the error from the average valne of a set of data, whereas this statistic is based on the error from setpoint. 

Derivative action based on error from the setpoint instead of the measurement. 

An integrating term, which ensures that the long-term error approaches zero. This term guarantees that there will be no offset or long-term deviation from setpoint. 

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. 

Catch decimal errors by software or procedure. For example, have the control system logic trap and prevent setpoint changes, for example, from 6% to 61%, when a change from 6.0 to 6.1% is intended. 

Figure 12 shows the input to output, characteristic waveform for a two position controller that switches from its "OFF" state to its "ON" state when the measured variable increases above the setpoint. Conversely, it switches from its "ON" state to its "OFF" state when the measured variable decreases below the setpoint. This device provides an output determined by whether the error signal is above or below the setpoint. The magnitude of the error signal is above or below the setpoint. The magnitude of the error signal past that point is of no concern to the controller. 

When the error equals zero, the controller provides a 50%, or 9 psi, signal to the control valve. As the error goes above and below this point, the controller produces an output that is proportional to the magnitude of the error, determined by the value of the proportional band. The control valve is then capable of being positioned to compensate for the demand disturbances that can cause the process to deviate from the setpoint in either direction. 

Feedback control. The traditional way to control a process is to measure the variable that is to be controlled, compare its value with the desired value (the setpoint to the controller) and feed the difference (the error) into a feedback controller that will change a manipulated variable to drive the controlled variable back to the desired value. Information is thus fed back from the controlled variable to a manipulated variable, as sketched in Fig. 1.7. 

The part of the control loop that we will spend most of our time with in this book is the controller. The job of the controller is to compare the process signal from the transmitter with the setpoint signal and to send out an appropriate signal to the control valve. We will go into more detail about the performance of the controller in Sec. 7.2. In this section we will describe what kind of action standard commercial controllers take when they sec an error. 

Derivative aetion can be used on either the error signal (SP — PM) or just the process measurement (PM). If it is on the error signal, step changes in set-point will produce large bumps in the control valve. Therefore, in most process control applications, the derivative action is applied only to the PM signal as it enters the controller. The P and I action is then applied to the difference between the setpoint and the output signal from the derivative unit (see Fig. 7.12). 

Most of the control systems we have discussed, simulated, and designed thus far in this book have been feedback control devices. A deviation of an output variable from a setpoint is detected. This error signal is fed into a feedback controller that changes the manipulated variable. The controller makes no use of any information about the source, magnitude, or directiop of the disturbance that has caused the output variable to change. 

Equation (4) is a feedback control algorithm for both setpoint and load changes which computes the new velocity from the current velocity, current and desired outlet temperatures, and estimated and known system parameters. Storage of previous values of the manipulated variable and error are not required for the algorithm of Equation (4). 

The steady-state gain is not equal to unity. Figure 15.24 shows setpoint changes for three different values of K K. The steady-state value differs from the setpoint value, which indicates offset. Offset is the error between the new setpoint and the new steady-state controlled variable value. Also note that as increases, the offset is reduced. 

The well-known deficiency of the proportional controller is that an error is necessary for the controller to sustain a non-zero output. Accordingly the plant variable, Bp, will always be offset by a certain amount from the setpoint, 0,. The proportional plus integral (P-fl) controller avoids this problem by adding in an integral term that will build up as long as an error remains. Figure 22.2 shows the arrangement. 

A proportional-only feedback controller changes its output signal, CO, in direct proportion to the error signal E, which is the difference between the setpoint signal SP and the process variable signal PV coming from the transmitter. 

Figure 24.7. The proportional-plus-derivative controller. Derivative action is accomplished by a shunt capacitor C across Rf. When deviation from the setpoint is rapid, the low reactance of the capacitor causes less negative feedback—hence, greater amplifier gain. The derivative time resistor Ra allows adjustment of the magnitude of derivative control action to a given rate of change of the error signal. Courtesy of the Foxboro Company.

The next window shows that a Comparator block subtracts the process variable (%) from the setpoint (%). The third window shows that a Multiply block generates the product of the controller gain and the error. The fourth window generates the temperature controller output by adding the output of the previous block to the output of the Lag l block. The next window shows that a Multiply block converts the percent-of-scale signal to a reboiler duty (shown in GJ/h). The final window gives the low selector. Note that the reboiler duties of the two inputs from the two controllers are in GJ/h while the output has units of MMkcal/h. Inputl comes from the normal temperature controller, and it is the lower of the two at design conditions. 

In the LTOP mode, each SCS relief valve is designed to protect the reactor vessel given a single failure in addition to the failure that i iitiated the pressure transient. The event initiating the pressure transient is considered to result from either an operator error or equipment malfunction. The SCS relief valve system is independent of a loss of offsite power. Each SCS relief valve is a self actuating spring-loaded liquid relief valve which does not require control circuitry. The valve opens when the RCS pressure exceeds its setpoint. 

F. The analysis reflects consideration of plant instrumentation error and safety valve setpoint uncertainty. For example, all safety valves are assumed to open at their maximum opening pressure. This extends the period of time before energy can be removed from the system. The reactor trip setpoint errors are always assumed to act in such a manner that delays the reactor trip and results in maximum pressurization. 

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