thumb|The [[Centrifugal governor|fly-ball governor is an early example of proportional control. The balls rise as speed increases, which closes the valve, reducing speed until a balance is achieved.]]
Proportional control, in engineering and process control, is a type of linear feedback control system in which a correction is applied to the controlled variable, and the size of the correction is proportional to the difference between the desired value (setpoint, SP) and the measured value (process variable, PV). Two classic mechanical examples are the toilet bowl float proportioning valve and the fly-ball governor.
The proportional control concept is more complex than an on–off control system such as a bi-metallic domestic thermostat, but simpler than a proportional–integral–derivative (PID) control system used in something like an automobile cruise control. On–off control will work where the overall system has a relatively long response time, but can result in instability if the system being controlled has a rapid response time. Proportional control overcomes this by modulating the output to the controlling device, such as a control valve at a level which avoids instability, but applies correction as fast as practicable by applying the optimum quantity of proportional gain.
A drawback of proportional control is that it cannot eliminate the residual SP − PV error in processes with compensation e.g. temperature control, as it requires an error to generate a proportional output. To overcome this the PI controller was devised, which uses a proportional term (P) to remove the gross error, and an integral term (I) to eliminate the residual offset error by integrating the error over time to produce an "I" component for the controller output.
Theory
In the proportional control algorithm, the controller output is proportional to the error signal, which is the difference between the setpoint and the process variable. In other words, the output of a proportional controller is the multiplication product of the error signal and the proportional gain.
This can be mathematically expressed as
:<math>P_{\mathrm{out = K_p\,{e(t)+p0}</math>
where
- <math>p0</math>: Controller output with zero error.
- <math>P_{\mathrm{out</math>: Output of the proportional controller
- <math>K_p</math>: Proportional gain
- <math>e(t)</math>: Instantaneous process error at time t. <math>e(t)=SP - PV</math>
- <math>SP</math>: Set point
- <math>PV</math>: Process variable
Constraints: In a real plant, actuators have physical limitations that can be expressed as constraints on <math>P_{\mathrm{out</math>. For example, <math>P_{\mathrm{out</math> may be bounded between −1 and +1 if those are the maximum output limits.
Qualifications: It is preferable to express <math>K_p</math> as a unitless number. To do this, we can express <math>e(t)</math> as a ratio with the span of the instrument. This span is in the same units as error (e.g. C degrees) so the ratio has no units.
Development of control block diagrams
thumb|Simple feedback control loop2
Proportional control dictates <math>{\mathit{g_c = k_c</math>. From the block diagram shown, assume that r, the setpoint, is the flowrate into a tank and e is error, which is the difference between setpoint and measured process output. <math>{\mathit{g_p,</math> is process transfer function; the input into the block is flow rate and output is tank level.
The output as a function of the setpoint, r, is known as the closed-loop transfer function.
<math>{\mathit{g_{cl} = \frac{\mathit{g_{p}g_{c}{1 + g_{p}g_{c,</math> If the poles of <math>{\mathit{g_{cl},</math> are stable, then the closed-loop system is stable.
First-order process
For a first-order process, a general transfer function is <math>g_p = \frac{k_p}{\tau_p s+1}</math>. Combining this with the closed-loop transfer function above returns <math>g_{CL} = \frac{\frac{k_p k_c}{\tau_p s+1{1+\frac{k_p k_c}{\tau_p s+1</math>. Simplifying this equation results in <math>g_{CL} = \frac{k_{CL{\tau_{CL} s+1}</math> where <math>k_{CL} = \frac{k_p k_c}{1+k_p k_c}</math> and <math>\tau_{CL} = \frac{\tau_p}{1+k_p k_c}</math>. For stability in this system, <math>\tau_{CL} > 0</math>; therefore, <math>\tau_p</math> must be a positive number, and <math>k_p k_c > -1</math> (standard practice is to make sure that <math>k_p k_c > 0</math>).
Introducing a step change to the system gives the output response of <math>y(s) = g_{CL} \times \frac{\Delta R}{s}</math>.
Using the final-value theorem,
<math>\lim_{t \to \infty} y(t) = \lim_{s \,\searrow\, 0} \left( s \times \frac{k_{CL{\tau_{CL} s + 1} \times \frac{\Delta R}{s} \right) = k_{CL} \times \Delta R = y(t)|_{t=\infty}</math>
which shows that there will always be an offset in the system.
Integrating process
For an integrating process, a general transfer function is <math>g_p = \frac{1}{s(s+1)}</math>, which, when combined with the closed-loop transfer function, becomes <math>g_{CL} = \frac{k_c}{s(s+1) + k_c}</math>.
Introducing a step change to the system gives the output response of <math>y(s) = g_{CL} \times \frac{\Delta R}{s}</math>.
Using the final-value theorem,
<math>\lim_{t \to \infty} y(t) = \lim_{s \,\searrow\, 0} \left(s \times \frac{k_c}{s(s+1) + k_c} \times \frac{\Delta R}{s}\right) = \Delta R = y(t)|_{t=\infty}</math>
meaning there is no offset in this system. This is the only process that will not have any offset when using a proportional controller.
Offset error
thumb|300px|Flow control loop. If only used as a proportional controller, then there's always an offset between SP and PV.
Offset error is the difference between the desired value and the actual value, error. Over a range of operating conditions, proportional control alone is unable to eliminate offset error, as it requires an error to generate an output adjustment.