Introduction to Control Systems

A control system is an arrangement of components that makes a physical process behave in a desired way. This introduction sets out the vocabulary and the performance measures that the rest of the series builds on.

The basic loop

A closed-loop system has four recurring roles.

• The plant is the process being controlled, such as a motor, a vehicle or a heating element.
• The sensor measures the plant output.
• The controller compares the measurement with the reference and computes a command from the error.
• The actuator applies that command to the plant.

Disturbances act on the plant from outside, and measurement noise corrupts the sensor signal. A good design keeps the output close to the reference despite both.

What makes a controller good

Engineers usually state requirements in terms of a few properties.

1. Stability. Bounded inputs and disturbances should produce bounded outputs.
2. Tracking. The output should follow the reference, with small steady-state error.
3. Disturbance rejection. Outside forces should have little lasting effect.
4. Robustness. Performance should survive model errors and parameter changes.

These properties trade against each other. Pushing for fast tracking tends to reduce robustness, which is why design is more than choosing a large gain.

Reading a step response

A standard reference model is the second-order system

$$G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2},$$

with natural frequency $\omega_n$ and damping ratio $\zeta$. For $0 < \zeta < 1$ its unit-step response overshoots the final value by

$$M_p = \exp\!\left(\frac{-\pi\zeta}{\sqrt{1 - \zeta^2}}\right),$$

and a common approximation for the 2% settling time is

$$T_s \approx \frac{4}{\zeta\,\omega_n}.$$

For $\zeta = 0.5$ the overshoot is about 16.3%. These formulas apply to this specific model. Real systems often have additional poles and zeros, so the numbers are a starting point for design, not a guarantee.

import math

zeta, wn = 0.5, 2.0
overshoot = math.exp(-math.pi * zeta / math.sqrt(1 - zeta**2))
settling_time = 4 / (zeta * wn)

print(f"overshoot: {overshoot:.1%}, settling time: {settling_time:.1f} s")

Where this series goes next

The next article closes the loop around the simplest possible plant, a first-order system, and shows exactly what proportional feedback changes: the speed of the response and its sensitivity to model errors and disturbances, at the price of a steady-state offset. The article after it introduces state-space models, which describe systems with several internal variables and connect stability to the eigenvalues of a single matrix. Planned entries build on it to treat stability and state-feedback design.