feedback and temperature control

7/30/2019 Feedback and Temperature Control

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newton.ex.ac.uk/teaching/CDHW/Feedback/ControlTypes.html 1/3

Physics and Astronomy

Types of Feedback ControlPrevious: System Model.

All the graphs shown in this section use parameter values for the thermal model that are typical of a small

domestic cooker and the set-point temperature Ts is indicated by the red lines.

On-Off Control

This is the simplest form of control, used by almost all domestic thermostats. When the oven is cooler than the

set-point temperature the heater is turned on at maximum power,M, and once the oven is hotter than the set-point temperature the heater is switched off completely. The turn-on and turn-off temperatures are deliberately

made to differ by a small amount, known as the hysteresisH, to prevent noise from switching the heater rapidly

and unnecessarily when the temperature is near the set-point. The fluctuations in temperature shown on the

graph are significantly larger than the hysteresis, as can be confirmed with the interactive simulation, due to the

significant heat capacity of the heating element.

Proportional Control

A proportional controller attempts to perform better than the On-Off type by applying power, W, to the heater

in proportion to the difference in temperature between the oven and the set-point,

wherePis known as theproportional gain of the controller. As its gain is increased the system responds fasterto changes in set-point but becomes progressively underdamped and eventually unstable. The final oven

temperature lies below the set-point for this system because some difference is required to keep the heater

supplying power. The heater power must always lie between zero and the maximumMbecause it can only

source, not sink, heat.

W = P ( ) T

s

T

o
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Proportional+Derivative ControlThe stability and overshoot problems that arise when aproportional controlleris used at high gain can be

mitigated by adding a term proportional to the time-derivative of the error signal,

This technique is known asPD control. The value of the damping constant, D, can be adjusted to achieve a

critically damped response to changes in the set-point temperature, as shown in the next figure.

Too little damping results in overshoot and ringing, too much causes an unnecessarily slow response.

Proportional+Integral+Derivative Control

Although PD control deals neatly with the overshoot and ringing problems associated withproportional control itdoes not cure the problem with the steady-state error. Fortunately it is possible to eliminate this while using

relatively low gain by adding an integral term to the control function which becomes

W = P ( ( ) + D ( ) ) T

s

T

o

d

d t

T

s

T

o

W = P ( ( ) + D ( ) + I ( ) d t )

s o

d

s o s o
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whereI, the integral gain parameter is sometimes known as the controllerreset level. This form of function is

known as proportional-integral-differential, or PID, control. The effect of the integral term is to change the

heater power until the time-averaged value of the temperature error is zero. The method works quite well but

complicates the mathematical analysis slightly because the system is now third-order.

The figure shows that, as expected, adding the integral term has eliminated the steady-state error. The slight

undershoot in the power suggests that there may be scope for further tweaking.

Proportional+Integral ControlSometimes, particularly when the sensor measuring the oven temperature is susceptible to noise or other

electrical interference, derivative action can cause the heater power to fluctuate wildly. In these circumstances it

is often sensible use a PI controller or set the derivative action of a PID controller to zero.

Third-Order Systems

Systems controlled using an integral action controller are almost always at least third-order. Unlike second-

order systems, third-order systems are fairly uncommon in physics but the methods ofcontrol theory make the

analysis quite straightforward. For instance, applying the so-calledRouth-Hurwitz stability criterion, which is

a systematic way of classifying the complex roots of the auxiliary equation for the model, it can be shown that

provided the integral gain is kept sufficiently small then parameter values can be found to give an acceptably

damped response with the error temperature eventually tending to zero if the set-point is changed by a step or

linear ramp in time. Whereas derivative control improved the system damping, integral control eliminates steady-

state error at the expense of stability margin.

Next:Practical Matters.

W=

P ( ( ) + D ( ) + I ( ) d t ) T

s

T

o

d

d t

T

s

T

o

T

s

T

o
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