A Mechanical Feedback Classification of Linear Mechanical Control Systems

Marcin Nowicki,Witold Respondek

doi:10.3390/app112210669

Abstract

We give a classification of linear nondissipative mechanical control system under mechanical change of coordinates and feedback. First, we consider a controllable case that is somehow a mechanical counterpart of Brunovský classification, then we extend the result to all linear nondissipative mechanical systems (not necessarily controllable) which leads to a mechanical canonical decomposition. The classification of Lagrangian systems is given afterwards. Next, we show an application of the classification results to the stability and stabilization problem and illustrate them with several examples. All presented results in this paper are expressed in terms of objects on the configuration space Rn only, while the state-space of a mechanical control system is Rn×Rn consisting of configurations and velocities.

Highlights

In this paper we consider the problem of classification of linear mechanical control systems under mechanical feedback transformations
A classification of controllable linear systems x = Ax + Bu under general linear transformations and general linear feedback has been solved in the celebrated Brunovský classification [1], see [2]
We show that mechanical feedback transformations are perfectly adapted to the class of mechanical systems, namely the classification of mechanical systems is the same if, instead of mechanical feedback transformations, we use all linear feedback transformations

Summary

Introduction

In this paper we consider the problem of classification of linear mechanical control systems under mechanical feedback transformations. We consider the above-defined classification problem in three important cases, namely for controllable and uncontrollable mechanical systems, and for the subclass of Lagrangian systems. We deal with a smaller class of control systems (than general linear systems) and we use more subtle mechanical feedback transformations (than general feedback transformations) and yet the invariants are perfectly analogous to those of the general case They can be computed on a half of the state space, namely using objects defined on the configurations space only. For a survey of port-Hamiltonian systems see [13]

Linear Mechanical Feedback Transformations

Stability and Stabilization

Conclusions