Proportional controller bias

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... 

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. 

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 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... 

As P approaches zero, the gain of the proportional controller approaches infinity. At 100 percent band, the gain is 1.0. The output of the controller equals the bias when there is no error. 

In the equation describing the proportional controller, the bias b equals the output when the error is zero. This bias may be fixed at the normal value of output, usually 50 percent, or it may be adjusted by hand to match the current load. This adjustment is called manual reset. But because of the proportional relationship between input and output, a change in output by any amount cannot be gained without a corresponding change in error. Should the output of the proportional con-... 

The control system is shown in Fig. 10.16. Note that the forward-loop summation is made conveniently in a proportional controller with remote bias. 

The tuning of proportional level controllers is a trivial job. For example, we could set the bias value at 50 percent of full scale, the setpoint at 50 percent of full scale, and the proportional band at 50. This means that the control valve will be half open when the tank is half full, wide open when the tank is 75 percent full, and completely shut when the tank is 25 percent full. Changing the proportional band to 100 would mean that the tank would be completely full to have the valve wide open and completely empty to have the valve shut. 

In a case-control study in the north of Sweden, Hallquist et al. (1993) compared 188 men and women aged 20-70 years who had thyroid cancer with age- and sex-matched controls (two per case) selected from a register of the local population. The cases were identified retrospectively from a cancer registry and excluded a proportion of patients (19%) who had died by the time of the study. Exposure to potential risk factors, including chlorophenols, was ascertained by postal questionnaire with a supplementary telephone interview if answers were incomplete. The response rates for the cases and controls were 95% and 90%, respectively. Of the 171 cases analysed, 107 had papillary tumours. Four cases and three controls reported exposure to chlorophenols (odds ratio, 2.8 95% CI, 0.5-18). [The Working Group noted that the method of statistical analysis was not the most appropriate for individually matched data, but this is unlikely to have produced serious bias.]... 

The most commonly used controller in the process industries is the three term or PID controller. This controller is a feedback controller and adjusts the manipulated variable in proportion to the change in its output signal, c, from its steady state value (bias), cs, on the basis of a measurement of the error in the controlled variable, s, which is given by... 

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