The proportional controller

The controller is intended to keep the plant variable, 6p, at or near to the setpoint value, G,. The plant variable passes through a measurement system, which produces the value G , which is fed back in a negative sense to give the error, e  

The error signal is passed to the controller, which multiplies it by the gain, k, and transmits the resultant control signal, c, to the plant. Note, however, that the signal sent to the plant will be limited between a minimum value, Cmin. and a maximum value, Cm , so that it may be calculated using a limiting function lim (normally provided as standard within a simulation language)  

Process plant custom sometimes expresses the controller gain in units of proportional band (PB). The proportional band is defined as the percentage of the 

For example, suppose that the measured variable is the level in a tank, that the measurement transducer has an effective range of 0.5 m and that the output from the controller is a current signal in the range 4-20 mA. If the proportional band is set at 250%, then the gain is given by 

Simulating this directly, the equation for the controller output may be calculated as 

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Fig. 2. Flow sheet of lecithin producing unit. Crude soybean oil is heated in the preheater, 1, to 80°C, mixed with 2% water in the proportion control unit, 2, and intensively agitated in 3. The mixture goes to a dweUing container, 4, and is then centrifuged after a residence time of 2—5 min. The degummed oil flows without further drying to the storage tanks. The lecithin sludge is dried in the thin-film evaporator, 6, at 100°C and 6 kPa (60 mbar) for 1—2 min and is discharged after cooling to 50—60°C in the cooler, 8. 9 and 10 are the condenser and vacuum pump, respectively.

The Ziegler and Nichols closed-loop method requires forcing the loop to cycle uniformly under proportional control. The natural period of the cycle—the proportional controller contributes no phase shift to alter it—is used to set the optimum integral and derivative time constants. The optimum proportional band is set relative to the undamped proportional band P , which produced the uniform oscillation. Table 8-4 lists the tuning rules for a lag-dominant process. A uniform cycle can also be forced using on/off control to cycle the manipulated variable between two limits. The period of the cycle will be close to if the cycle is symmetrical the peak-to-peak amphtude of the controlled variable divided by the difference between the output limits A, is a measure of process gain at that period and is therefore related to for the proportional cycle ... 

The error for the proportional control algorithm is determined from the actual position of the carriage PULSEcar-... 

Kcar is the gain used in the proportional control algorithm for the carriage. 

SIGN is used to determine whether the carriage position is incrementing or decremented. TIME is the sampling time of the proportional control algorithm. 

Proportional control can be based on the temperature of the third stage. Here FO is the base flow rate, KC is the proportional controller gain, and TSET is the temperature set point. Note that in order to guard against the unrealistic condition of negative flow, a limiter condition on F should be inserted into the DYNAMIC region. This can be accomplished with ISIM by the following statement... 

We expect a system with only a proportional controller to have a steady state error (or an offset). A formal analysis will be introduced in the next section. This is one simplistic way to see why. Let s say we change the system to a new set point. The proportional controller output, p = ps + Kce, is required to shift away from the previous bias ps and move the system to a new steady state. For p to be different from ps, the error must have a finite non-zero value.3... 

We ll skip the proportional controller, which is just Gc = Kc. Again, do the plots using sample numbers with MATLAB as you read the examples. 

Using the Ziegler-Nichols tuning parameters, we repeat the proportional controller system Bode plot ... 

If we want to increase the margin, we either have to reduce the value ofKc or increase One possibility is to keep = 1.58 min and repeat the Bode plot calculation to find a new Kc which may provide a gain margin of, say, 2 (6 dB), as in the case of using only the proportional controller. To do so, we first need to find the new ultimate gain using the PI controller ... 

In the proportional control mode, the final control element is throttled to various positions that are dependent on the process system conditions. For example, a proportional controller provides a linear stepless output that can position a valve at intermediate positions, as well as "full open" or "full shut." The controller operates within a band that is between the 0% output point and the 100% output point and where the output of the controller is proportional to the input signal. 

With proportional control, the final control element has a definite position for each value of the measured variable. In other words, the output has a linear relationship with the input. Proportional band is the change in input required to produce a full range of change in the output due to the proportional control action. Or simply, it is the percent change of the input signal required to change the output signal from 0% to 100%. 

The proportional controller is reverse-acting so that the control valve throttles down to reduce steam flow as the hot water outlet temperature increases the control valve will open further to increase steam flow as the water temperature decreases. 

In the proportional control mode, the final control element is throttled to various positions that are dependent on the process system conditions. 

The proportional band is the change in input required to produce a full range of change in the output due to the proportional control action. 

The main advantage of the proportional control mode is that an immediate proportional output is produced as soon as an error signal exists at the controller as shown in Figure 22. The proportional controller is considered a fast-acting device. 

This immediate output change enables the proportional controller to reposition the final control element within a relatively short period of time in response to the error. 

The main disadvantage of the proportional control mode is that a residual offset error exists between the measured variable and the setpoint for all but one set of system conditions. 

Proportional plus rate control is a control mode in which a derivative section is added to the proportional controller. 

Assuming that Gi and H are constant and written as Ki and K2 respectively, that is the time constants of the final control element and measuring element are negligible in comparison with those of the process, and that the proportional controller has a gain Kc,... 

The steady-state gain of 0.9 can be included in two ways either start the overall AR plot at AR = 0.9 instead of AR = 1, or plot AR/0.9 on the vertical scale and start at AR = 1. In the latter case, the gain of the proportional controller calculated for the given... 

Most controllers are calibrated in minutes (or minutes/repeat, a term that comes from the test of putting into the controller a fixed error and seeing how long it takes the integral action to ramp up the controller output to produce the same change that a proportional controller would make when its gain is 1 the integral repeats the action of the proportional controller). 

Notice that the denominators of all of these closedloop transfer functions are identical. Notice also that the steadystatc gain of the closedloop servo transfer function PM/SP is not unity i.e., there is a steadystate offset. This is because of the proportional controller. We can calculate the PM/SP ratio at steadystate by letting s go to zero in Eq. (10.8). 

Ashkin and Dziedzic (1977) used the radiation pressure force of a laser beam to levitate microdroplets with the apparatus presented in Fig. 15. A polarized and electro-optically modulated laser beam illuminated the particle from below. The vertical position of the particle was detected using the lens and split photodiode system shown. When the particle moved up or down a difference signal was generated then a voltage proportional to the difference and its derivative were added, and the summed signal used to control an electro-optic modulator to alter the laser beam intensity. Derivative control serves to damp particle oscillations, while the proportional control maintains the particle at the null point. 

Suppose the temperature control of a bioreactor using heat supply with a proportional controller. When a proportional controller is tuned at a set point of 30 °C, as long as the set point remains constant, the temperature will remain at 30 °C successfully. Then, if the set point is changed to 40 °C, the proportional controller increases the output (heat supply) proportional to the error (temperature difference). Consequently, a heat supply will continue until the temperature gets to 40 °C and would be off at 40 °C. However, the temperature of a bioreactor will not reach 40 °C because a heat loss from the bioreactor increases due to the temperature increase. Finally, the heat supply matches the heat loss, at this point, the temperature error will remain constant therefore, proportional controller will keep its output constant. Now the system keeps in a steady state, but the temperature of a bioreactor is below its set point. This residual error is called Offset. 

The natural parameterization of a forced periodic trajectory through the forcing term phase provides a very convenient way of closing the control loop and stabilizing unstable periodic trajectories. Consider such a trajectory 0 < < 2n and the proportional control system... 

It can be seen from equation 7.3 and Fig. 7.7 that the controller output will continue to increase as long as e > 0. With proportional control an error (offset) had to be maintained so that the controlled variable (i.e. the temperature at Y—Fig. 7.1) could be kept at a new control point after a step change in load, i.e. in the inlet temperature of the cold stream. This error was required in order to produce an additional output from the proportional controller to the control valve. However, with PI control, the contribution from the integral action does not return to zero with the error, but remains at the value it has reached at that time. This contribution provides the additional output necessary to open the valve wide enough to keep the level at the desired value. No continuous error (i.e. no offset) is now necessary to maintain the new steady state. A quantitative treatment of this is given later (Section 7.9.3). 

To keep the plant at its middle unstable steady state can be achieved by stabilizing the unstable steady state with a simple feedback control loop. For the sake of simplicity, we use a SISO (single input single output) proportional feedback control, in which the dense-phase temperature of the reactor is the controlled measured variable, while the manipulated variable can be any of the input variables of the system Yfa, FCD, etc. We use Yfa as the manipulated variable here. The set-point of the proportional controller is the dense-phase reactor temperature at the desired middle steady state in this case. Our simple SISO control law is... 

The proportional controller is unable to return the controlled variable to the set point following the step load change, as a deviation is required to sustain its output at a value different from its fixed bias b. The amount of proportional offset produced as a fraction of the uncontrolled offset is 1/(1 + KK ), where K is the steady-state process... 

The proportional controller cannot consider the past history or the possible future consequences of an error trend. The gain in DCS control packages is usually adjustable from 0 to 8, whereas in analog controllers it can usually be adjusted from 0.02 to about 25. 

After a permanent load change, the proportional controller is incapable of returning the process back to the set point and an offset results. The smaller the controller s gain, the larger the offset will be. 

The distributed control objectives for this process involve the stabilization of the individual unit holdups (Mr, Me, and Mb), which, according to our prior analysis, should be addressed in the fast time scale. The design of the distributed controllers for the stabilization of the three holdups can easily be achieved, using the large flow rates F, D, and V as manipulated inputs and employing simple proportional controllers - note that only these three flow rates (i.e., the components of u1) affect the fast dynamics. More specifically, the proportional control laws... 

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