Proportional control

Development of control block diagrams

Proportional control dictates ${\displaystyle {\mathit{g}}_{\mathit{c}}\mathit{=}{\mathit{k}}_{\mathit{c}}}$. 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. ${\displaystyle {\mathit{g}}_{\mathit{p}},}$ 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. ${\displaystyle {\mathit{g}}_{\mathit{c}\mathit{l}}=\frac{{\mathit{g}}_{\mathit{p}}{\mathit{g}}_{\mathit{c}}}{1+g_p{g}_c},}$ If the poles of ${\displaystyle {\mathit{g}}_{\mathit{c}\mathit{l}},}$ are stable, then the closed-loop system is stable.

First-order process

For a first-order process, a general transfer function is ${\displaystyle g_p=\frac{k_p}{{\tau }_{p}s+1}}$. Combining this with the closed-loop transfer function above returns ${\displaystyle g_{CL}=\frac{\frac{k_p{k}_c}{{\tau }_{p}s+1}}{1+\frac{k_p{k}_c}{{\tau }_{p}s+1}}}$. Simplifying this equation results in ${\displaystyle g_{CL}=\frac{k_{CL}}{{\tau }_{CL}s+1}}$ where ${\displaystyle k_{CL}=\frac{k_p{k}_c}{1+k_p{k}_c}}$ and ${\displaystyle {\tau }_{CL}=\frac{{\tau }_p}{1+k_p{k}_c}}$. For stability in this system, ${\displaystyle {\tau }_{CL}>0}$; therefore, ${\displaystyle {\tau }_p}$ must be a positive number, and ${\displaystyle k_p{k}_c>-1}$ (standard practice is to make sure that ${\displaystyle k_p{k}_c>0}$).

Introducing a step change to the system gives the output response of ${\displaystyle y(s)=g_{CL}\times \frac{\mathrm{\Delta }R}{s}}$.

Using the final-value theorem,

${\displaystyle \underset{t\to \mathrm{\infty }}{lim}y(t)=\underset{s\searrow 0}{lim}\left(s\times \frac{k_{CL}}{{\tau }_{CL}s+1}\times \frac{\mathrm{\Delta }R}{s}\right)=k_{CL}\times \mathrm{\Delta }R=y(t){|}_{t=\mathrm{\infty }}}$

which shows that there will always be an offset in the system.

Integrating process

For an integrating process, a general transfer function is ${\displaystyle g_p=\frac{1}{s(s+1)}}$, which, when combined with the closed-loop transfer function, becomes ${\displaystyle g_{CL}=\frac{k_c}{s(s+1)+k_c}}$.

Introducing a step change to the system gives the output response of ${\displaystyle y(s)=g_{CL}\times \frac{\mathrm{\Delta }R}{s}}$.

Using the final-value theorem,

${\displaystyle \underset{t\to \mathrm{\infty }}{lim}y(t)=\underset{s\searrow 0}{lim}\left(s\times \frac{k_c}{s(s+1)+k_c}\times \frac{\mathrm{\Delta }R}{s}\right)=\mathrm{\Delta }R=y(t){|}_{t=\mathrm{\infty }}}$

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

Offset error is the difference between the desired value and the actual value, SP − PV 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. While a proportional controller may be tuned (via p0 adjustment, if possible) to eliminate offset error for expected conditions, when a disturbance (deviation from existing state or setpoint adjustment) occurs in the process, corrective control action, based purely on proportional control, will result in an offset error.

Consider an object suspended by a spring as a simple proportional control. The spring will attempt to maintain the object in a certain location despite disturbances that may temporarily displace it. Hooke's law tells us that the spring applies a corrective force that is proportional to the object's displacement. While this will tend to hold the object in a particular location, the absolute resting location of the object will vary if its mass is changed. This difference in resting location is the offset error.

Proportional band

The proportional band is the band of controller output over which the final control element (a control valve, for instance) will move from one extreme to another. Mathematically, it can be expressed as:

${\displaystyle PB=\frac{100}{K_p}}$

So if ${\displaystyle K_p}$, the proportional gain, is very high, the proportional band is very small, which means that the band of controller output over which the final control element will go from minimum to maximum (or vice versa) is very small. This is the case with on–off controllers, where ${\displaystyle K_p}$ is very high and hence, for even a small error, the controller output is driven from one extreme to another.

Advantages

The clear advantage of proportional over on–off control can be demonstrated by car speed control. An analogy to on–off control is driving a car by applying either full power or no power and varying the duty cycle, to control speed. The power would be on until the target speed is reached, and then the power would be removed, so the car reduces speed. When the speed falls below the target, with a certain hysteresis, full power would again be applied. It can be seen that this would obviously result in poor control and large variations in speed. The more powerful the engine, the greater the instability; the heavier the car, the greater the stability. Stability may be expressed as correlating to the power-to-weight ratio of the vehicle.

In proportional control, the power output is always proportional to the (actual versus target speed) error. If the car is at target speed and the speed increases slightly due to a falling gradient, the power is reduced slightly, or in proportion to the change in error, so that the car reduces speed gradually and reaches the new target point with very little, if any, "overshoot", which is much smoother control than on–off control. In practice, PID controllers are used for this and the large number of other control processes that require more responsive control than using proportional alone.