Sysquake Remote Live

Proportional Controller

The graphics below represent a 3rd order system with a proportional controller. You can change the controller gain (and hence the graphics) by entering a new value in the text field, or directly by clicking somewhere on the the root locus (the black line of the top left figure). Please see below for more explanations, and visit https://calerga.com for more informations about Sysquake Remote.

With a proportional controller, the input of the system is proportional to the difference between the desired output (named reference or set-point) and the measured output. If the system is a d.c. electrical motor, for instance, and the measured output is the rotor velocity, the voltage applied as input is larger when the measured velocity is much lower than the set-point.

A feedback loop is always a compromise between several contradictory goals: among them, let's mention robustness, i.e. the property to keep good performance even when the system isn't quite the same as the model used to design the controller; set-up time, which measures how fast the controlled system reacts to changes of the set-point; damping, which measures how the controlled system forgets previous actions; and perturbation rejection.

There are several ways to look at these performance criteria. In the graphics below, the Root locus shows where the roots of the controlled system (the little triangles) may be; the system is stable when all of them are inside the blue circle, and unstable when at least one of them is outside. The Tracking Step Response is a simulation of the system when the set-point (in blue) goes from 0 to 1; the Sensitivity shows how a perturbation added to the output of the system is filtered out by the controller as a function of the frequency; and the Nyquist diagram is another way to look at the frequency response: it should leave the point at -1 to its left for a stable controlled system.

All these graphics seem completely unrelated. But they aren't! If you change the gain of the controller, you can see all of the graphics changing. You cannot simultaneously have a very good step response (with the system output following quickly and accurately the set-point) without making the system unstable.