Double-integrator and its control (1): motivation

Antonio Sala, UPV

Difficulty: ** ,       Relevance: ,      Duration: 17:54

Summary:

This video discusses the relevance in engineering practice of studying the control of a ‘double integrator’ system, transfer function $1∕s^2$.

Indeed, the motion of a mass $M\frac{d^{2}p}{d{t}^2}=F$ has such dynamics, if there is no friction... and, well, in technological applications of motion control one should try to make such friction as small as possible, at least in principle. Angular motion also has double integrator dynamics, changing masses for moments of inertia and forces for torques, obviously.

The transfer function $1∕s^2$ is an ‘abstraction’: in many technological solutions, directly manipulating a force or torque is not possible, and an internal electronic system for ‘torque’ or ‘pressure’ control will be required, for example. This is called cascade control and is VERY frequently used to simplify and divide the complexity of a practical control problem into different ‘hierarchical levels’ where each level assumes that the lower slave controllers behave correctly so that they follow their references for force, torque, pressure, etc.

As a final example, in lateral position control of a drone, the non-vertical component of the force is approximately the weight times the sine of the roll angle; lateral attitude control of a drone may assume an $A∕s^2$ model if there is internally a control system ‘much faster’ than the attitude control system that achieves a given pitch/roll angle using the appropriate accelerometers and gyroscopes.

All of this video is about the concept of a double integrator (low-friction motion control) in the abstract sense, although the ‘real technology’ of ‘motion control’ applications requires many more implementation details that are not the goal of a first introductory course in control theory, obviously.

In applications with high friction, gearboxes, etc., it is sometimes more appropriate to think of my manipulated variable as the ‘velocity of motion’, that is, to control position thinking of the transfer function $1∕s$ (simple integrator); this type of abstraction is called ‘kinematic control’, in materials on robotics, for example. This is not considered in the present video.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.