What Is a Control System? A Friendly First Look at Feedback

From adjusting a shower to a room thermostat: what feedback is, the parts of every control loop, open vs closed loop, your first transfer function, and what a good step response looks like.

Put your hand under a running tap and try to get the water just right. Too cold, so you turn the hot knob a little. Now it's too hot, so you ease it back. After a few tries it feels perfect, and you stop fiddling.

You just ran a control system. Your hand was the sensor, your brain compared what you felt with what you wanted, and your fingers on the knob did something about the difference. Then you checked again. That habit of measuring, comparing and correcting, over and over, is the whole subject in one sentence. Everything else we'll study builds on it.

A room that keeps itself warm

Here is the same idea in a machine you already know, the room thermostat.

You set 22 °C. The thermostat reads 21.4 °C, works out that the room is 0.6 °C short, and turns the heater up. Heat leaks out through the window, the room cools a little, and the thermostat notices and responds again. Nobody has to stand there watching it.

The parts of every feedback loop

Engineers draw this loop as a block diagram. Once you learn to read one, you'll see the same shape in cruise control, aircraft autopilots, robot arms and the voltage regulator in your phone charger.

Let's name each block, using the thermostat as our example:

Setpoint $r(t)$: what we want. Here it's 22 °C. Plant: the thing being controlled. Here it's the room itself. Output $y(t)$: what we actually get, the room temperature. Sensor: measures the output. It's the thermometer inside the thermostat. Comparator: subtracts to find the error, $e(t) = r(t) - y(t)$. Controller: decides what to do about the error. Actuator: does the physical work. Here it's the heater. Disturbance: anything that pushes the output around without asking, like an open window on a cold night.

Open loop and closed loop

A bread toaster is open loop. It heats for a fixed time and never checks whether the bread is actually brown. That's simple and cheap, but if the bread is frozen you get pale toast.

A thermostat is closed loop, because the output is fed back and compared with the goal. Closed loops cost a sensor and a bit of thinking, but they fix their own mistakes and shrug off disturbances. That is the main reason feedback is everywhere.

Our first formula: the transfer function

To predict how a system will behave, we describe each block with a transfer function. It is the ratio of output to input after both have been moved into the Laplace domain (the variable $s$), which turns messy differential equations into algebra.

A simple heater and room behave roughly like a first-order system:

Here $K$ is the gain (how far the output finally moves for each unit of input) and $\tau$ is the time constant (how quickly it gets there). When we close the loop with a controller $C(s)$ and a sensor $H(s)$, the whole system becomes:

Don't worry if that looks heavy today. We'll build it step by step in the coming posts. For now, just notice the $1 +$ in the denominator. That is where feedback lives.

What does a good response look like?

A favourite test is the step response: suddenly change the setpoint from 0 to 1 and watch the output. The graph below shows three systems answering the same step.

The first-order curve rises smoothly and never overshoots. The lightly damped system is quick, but it overshoots by about half and wobbles for a long time, which is like a shower that swings from freezing to scalding. The well damped one reaches the target fast with only a tiny overshoot, and that's usually what we aim for.

How much a second-order system overshoots depends only on its damping ratio $\zeta$ (zeta):

Designing a controller is largely the art of choosing where you want to sit on that chart.

Try this before the next post

Look around you today and find three feedback loops. For each one, name the setpoint, sensor, controller and actuator. Cruise control, the float valve in a water tank and your own body temperature are good places to start. Write your answers in the comments and I'll reply.

Next time we'll turn the thermostat into equations and draw its first real step response by hand.