A first-order system is a dynamic model whose denominator has degree one. Its behavior is dominated by a single pole, so a stable system of this type responds exponentially and does not oscillate by itself.
While in the previous article we saw that the Transfer Function was the “black box” representing our physical plant, today we are going to open the simplest box of all. First-order systems are, without a doubt, the “Hello World” of control theory.
If you missed the previous article where we explained where the Laplace Transform and the Transfer Function come from, now is the time to read it. read more ⯈
Real-world examples
To get an idea of how common they are, we makers deal with first-order systems almost daily without realizing it:
- Heating a heater block (Hotend): You apply a constant voltage to the resistor. The temperature doesn’t jump instantly to 200°C; instead, it rises gradually following a smooth curve until it stabilizes, due to the thermal mass of the block.
- Charging a capacitor (RC circuit): If you connect a resistor and a capacitor to a power supply, the voltage across the capacitor rises gradually as it accumulates charge.
- An idealized Direct Current (DC) motor: If we ignore the electrical dynamics of its windings, a motor subjected to a constant voltage can be approximated by the same smooth curve due to the inertia of its rotor.
The canonical transfer function
Many thermal, hydraulic, mechanical, or electrical systems can be approximated by the first-order canonical form in the Laplace domain:
This model has only two parameters that describe its response. If you know these two values for your plant, you already have a quite useful approximation. Let’s break them down.
Static Gain (K)
The static gain K answers the question: How much does the system move in total?
It is the direct relationship between the input and the output when enough time has passed and the system has stabilized (what we call the steady state).
For example, if we use the power of a heater as input and applying 1 W results in a final temperature 20 °C above ambient, the model gain is K = 20 °C/W. If we apply 2 W, the linear model predicts an increase of 40 °C. It is a scaling factor with its corresponding units.
Time Constant (τ)
The time constant τ (tau) answers the question: How fast does the system move?
This part depends directly on the physics. If we put a huge aluminum block in the 3D printer, the time constant τ will be large (it will take a long time to heat up). If we use a tiny surface-mount resistor, τ will be very small (it will heat up almost instantly).
Mathematically, the time constant τ is the time it takes for a stable first-order system to reach 63.2 % of the total change after applying a step input.
Analysis of the step response
The litmus test in automatic control is to apply a unit step to our plant. This means our input u(t) is at 0, and suddenly, at time t=0, we raise it to 1 (like flipping a switch) and leave it there forever.
The equation in the time domain that describes how our output y(t) rises is:
You don’t need to memorize it, but notice how the time constant works by evaluating this equation at very specific times:
- At t=τ (1 time constant has passed): The system has reached 63.2% of the final value.
- At t=2τ: The system is at 86.5%.
- At t=3τ: The system is at 95.0%.
- At t=4τ: The system is at 98.2%.
- At t=5τ: The system is at 99.3%.
We usually assume that after 4τ or 5τ, the system has finished reacting and is considered to have reached the steady state. So, if heating your hotend to 63% takes 10 seconds (τ=10s), you’ll know it will take almost 50 seconds (5τ) to reach the final stabilized temperature.
Interactive first-order simulator
The best way to understand this is not by reading, but by touching. We have prepared this interactive simulator. Play with the static gain K and the time constant τ, and observe how the exponential curve changes shape.
Pay close attention to the simulator: if you keep τ constant, changing the gain K does not alter the relative speed of the response. It only changes its final value. In this model, the time scale is determined by τ.
Why do we care so much about identifying whether our system is first-order?
Because if we know the plant has no cross-inertias or elements that bounce off each other (like a spring coupled to a mass), we know mathematically that the system will never oscillate by itself. The curve will rise, faster or slower, but it will rise smoothly until it stabilizes.
Knowing this, a PI (proportional-integral) controller is usually a good starting point. Derivative action may not be necessary, although it depends on the delay, noise, disturbances, and response requirements.
But the world is not perfect. Motors drag weights, drones have immense inertias, and airplanes have wings that bend. When the system has the ability to store energy in two different forms that interact with each other (for example, speed and inertia, or capacitor and inductor), things get interesting. We start to bounce.
