Feedback and control

A heater holds 40 °C at 40% power while the room is at 20 °C. A window opens and the room falls to 10 °C. Will the same action still hold the target? An action's effect depends on the environment; feedback adjusts the action using the observed result. This chapter starts with arithmetic. The next derives the time evolution from energy balance.

Separate the target, action and result

Let $r$ be the target, $y$ the actual temperature, $y_m$ the sensor reading, and $u$ the heating duty ratio: 0 is off and 1 is full power. Define error as $e=r-y_m$. The controller maps observations into an action; the plant turns that action into a physical response.

Open-loop operation follows a predetermined action; closed-loop operation uses output measurements to revise it. Timed heating can be appropriate when its errors and operating conditions meet the task. Feedback adds sensing, computation and actuation, along with new failure paths.

Why a fixed action stops working

Assume maximum heating power of 1000 W and heat-loss coefficient of 20 W/K. At equilibrium, heating equals heat loss:

$$1000u=20(y-y_a),y=y_a+50u.$$

With ambient $y_a=20$, $u=0.4$ gives 40 °C. With ambient 10, it gives 30 °C. This is the eventual equilibrium; temperature does not jump there when the window opens.

Keep a baseline $u_b=0.4$ and add proportional feedback:

$$u=u_b+K_p(r-y_m).$$

For accurate measurements, no saturation and $K_p=0.04{\mathrm{K}}^{-1}$, the new equilibrium satisfies $y=10+50[0.4+0.04(40-y)]$. Thus $y=36.67$ °C and $u=0.5333$. Feedback helps, but leaves 3.33 K error: proportional action needs that error to sustain the additional heating. PID control introduces an integral state that can retain the correction.

Predict the final temperatures for all three strategies before changing the ambient temperature. Compare the temperature and action timelines. The model has thermal capacitance 2000 J/K, no measurement noise, and an ambient step at 100 s. Every strategy starts at 40 °C.

Feedforward and feedback

If ambient temperature is measured and the equilibrium model is accurate, compute

$$u_{\mathrm{ff}}=\frac{20(r-y_a)}{1000}.$$

An ambient fall from 20 to 10 °C changes this action immediately from 0.4 to 0.6. It need not wait for an output error. This depends on accurate measurements, parameters and available power. If the actual heat-loss coefficient is 25 W/K, using 20 underestimates the necessary action.

A common combination is $u=u_{\mathrm{ff}}+u_{\mathrm{fb}}$: feedforward anticipates known changes; feedback corrects residual error. Neither automatically identifies sensor bias. Perfect cancellation in this experiment results from the exact match between the feedforward calculation and the plant model.

Negative feedback is not a stability guarantee

For heating, a low measured temperature causes more heating, which raises temperature and opposes the original error. A cooling actuator requires a different controller sign. The entire chain determines feedback direction.

Time matters too. Old sensor readings and delayed actuators can make a controller keep correcting an error that has already changed. Later chapters separate feedback direction, stability and response quality. A delay-free first-order thermal plant with positive proportional negative feedback does not oscillate merely because its positive gain increases; an oscillation explanation must identify additional dynamics or delay.

Also separate tracking from disturbance rejection. Changing the target from 40 to 45 °C tests tracking. Keeping the target at 40 while cooling the room tests rejection. These signals enter different paths, so one reference-step test does not establish complete performance.

Check your understanding

A sensor always reads 2 °C high. If feedback eventually makes its reading 40 °C, what is the actual temperature?

With ambient 10 °C, a 1000 W heater and 20 W/K heat loss, can a stronger controller reach 70 °C?

Further reading

The python-control cruise-control example combines a plant, load changes, feedback and actuator limits. Transfer the modeling approach; the thermal parameters and calculations here are independently constructed.