Offset proportional control

The response of a controller to an error depends on its mode. In the proportional mode (P), the output signal is proportional to the detected error, e. Systems with proportional control often exhibit pronounced oscillations, and for sustained changes in load, the controlled variable attains a new equilibrium or steady-state position. The difference between this point and the set point is the offset. Proportional control always results in either an oscillatory behaviour or retains a constant offset error. 

This is a control algorithm that attempts to eliminate the offset (caused by proportional control) between the measurement and the setpoint of the controlled process variable. This control mode remembers how long the measurement has been off the setpoint. 

Offset A sustained deviation between the control points and the set point of a proportional control system. 

Measure the controlled response to a step change in F with proportional control only (set Xi very high). Notice the offset error and its sensitivity to Kp. Change to a low value. Does the offset disappear ... 

Add control (KC positive) and experiment first with proportional control (TI very large) to measure the offset. Add integral control and see the effect. 

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... 

To tackle a problem, consider a simple proportional controller first. This may be all we need (lucky break ) if the offset is small enough (for us to bear with) and the response is adequately fast. Even if this is not the case, the analysis should help us plan the next step. 

On the plus side, the integration of the error allows us to detect and eliminate very small errors. To make a simple explanation of why integral control can eliminate offsets, refer back to our intuitive explanation of offset with only a proportional controller. If we desire e = 0 at steady state, and to shift controller output p away from the previous bias ps, we must have a nonzero term. Here, it is provided by the integral in Eq. (5-5). That is, as time progresses, the integral term takes on a final nonzero value, thus permitting the steady state error to stay at zero. 

If simple proportional control works fine (in the sense of acceptable offset), we may try PD control. Similarly, we may try PID on top of PI control. The additional stabilizing action allows us to use a larger proportional gain and obtain a faster system response. 

The system steady state gain is the same as that with proportional control in Example 5.1. We, of course, expect the same offset with PD control too. The system time constant depends on various parameters. Again, we defer this analysis to when we discuss root locus. 

Example 5.5 Derive the closed-loop transfer function of a system with proportional control and an integrating process. What is the offset in this system ... 

If you come across a proportional controller here, it is only possible if the derivation has ignored the steady state error, or shifted the reference such that the so-called offset is zero. 

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. 

The combination of the two control modes is called the proportional plus reset (PI) control mode. It combines the immediate output characteristics of a proportional control mode with the zero residual offset characteristics of the integral mode. 

Proportional plus reset controllers act to eliminate the offset error found in proportional control by continuing to change the output after the proportional action is completed and by returning the controlled variable to the setpoint. 

Proportional plus reset control eliminates any offset error that would occur with proportional control only. 

As illustrated in Figure 29, the proportional only control mode responds to the decrease in demand, but because of the inherent characteristics of proportional control, a residual offset error remains. Adding the derivative action affects the response by allowing only one small overshoot and a rapid stabilization to the new control point. Thus, derivative action provides increased stability to the system, but does not eliminate offset error. 

For processes that can operate with continuous cycling, the relatively inexpensive two position controller is adequate. For processes that cannot tolerate continuous cycling, a proportional controller is often employed. For processes that can tolerate neither continuous cycling nor offset error, a proportional plus reset controller can be used. For processes that need improved stability and can tolerate an offset error, a proportional plus rate controller is employed. 

Study the response of the system with only proportional control (ti very large) and determine the resultant steady state offset. Add an increasing degree of integral control and study its effect... 

The basic purpose of integral action is to drive the process back to its setpoint when it has been disturbed. A proportional controller will not usually return the controlled variable to the setpoint when a load or setpoint disturbance occurs. This permanent error (SP — PM) is called steadystate error or offset. Integral action reduces the offset to zero. 

B. PROPORTIONAL CONTROLLER. The output of a proportional controller changes only if the error signal changes. Since a load change requires a new control-valve position, the controller must end up with a new error signal. This means that a proportional controller usually gives a steadystate error or offset. This is an inherent limitation of P controllers and why integral action is usually added. 

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). 

Notice in Fig. 13.20 that the curve for the P controller does not approach 0 dB at low frequencies. This shows that there is a steadystate offset with a proportional controller. The curve for the PI controller does go to 0 dB at low frequencies because the integrator drives the closedloop servo transfer function to unity (i.e., no offset). 

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 reason for a permanent offset with a proportional controller can be explained with an example. Suppose the temperature of a reactor is being controlled with a pneumatic system. At the set point, say the valve is 50% open and the flow rate... 

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). 

Choose the best value of Kp and compare the behavior with the behavior for the open loop system without controls, i.e., for Kp = 0.0. Show that it is possible to remove the offset by using a proportional plus integral (PI) controller and find the best value of Kj that can be used with the best value for Kp as obtained above. Plot the change of height with time and compare it with the results of the open-loop system in the case without control and also when the system has only proportional control. 

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 main limitation of plain proportional control is that it cannot keep the controlled variable on set point, because it cannot bring the error to zero. This is because it can only change its output, when there is an error. The resulting deviation is called the offset (Figure 2.30). 

The term T is the integral or reset time setting of the controller. If the bias (b) is zero, this mode acts as a pure integrator, the output of which reaches the value of the step input during the integral time. The integral mode eliminates the offset of plain proportional control because it continuously looks at... 

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. 

Integral Action. Control action is proportional to the sum, or integral, of all previous errors. This controller eliminates offset. Some control textbooks refer to the reset rate, which is defined as 1/t/. 

Response of a proportionally controlled system to a step increase in the set point from T to Tg. In this example the gain is roughly one-half the critical gain and the transient oscillations are well damped. Note the offset from the set-point temperature. 

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