Symmetries, Control and Invariant Tracking

Prof. Pierre Rouchon, Centre Automatique et Systemes, Ecole des Mines de Paris

The first point is a natural generalization of the indirect vector-field control method of induction motors. This method shows how to extract from symmetries additional control variables. It has been invented by Blaschke, an Electrical Engineer, in the seventies. It relies on rotational invariance and adds one control variable, the "stator velocity" $\omega_s$. For a general control problem admitting a symmetry group with $r$ dimensional orbits, the analogue of this method adds $r$ control variables.

The second point concerns output tracking y=h(x) of a control system $\dot{x}=f(x,u)$. Tracking a reference trajectory $t\mapsto y_r(t)$ via classical decoupling and input/ouput linearization techniques provides a linear and stable closed-loop dynamics for the tracking error $e=y-y_r$. For many physical systems, the intrinsic character of $y-y_r$ is not guaranteed. It is well known that when $y\in SO(3)$ is the orientation of a satellite, the "good" tracking error is $y (y_r)^{-1}$. When $y$ is the outlet concentration of a chemical reactor, the case is less classical. $y$ belongs to the simplex of molar fractions (each component lies in $[0,1]$ and the sum is $1$). The difference $y-y_r$ has no physical meaning. One has to measure the tracking error in a different way in order to be invariant with respect to units changes: the tracking controller must remain unchanged if instead of using mole fractions we use mass fractions. Such natural invariance properties can be ensured if we can have an invariant way to measure the tracking error. For a general system admitting a symmetry group (up to static feedback), it is possible to derive (under some regularity assumptions relative to the group action) a collection of independent invariants, functions of $y$ and $y_r$, ... $y_r^{\nu}$ for some derivation order $\nu$ measuring the tracking error. The construction of such invariants is based on the Darboux-Cartan moving frame method. Several examples of invariant errors will be given (chemical reactor and units changes, non holonomic systems with $SE(2)$ invariance, mechanical system with Galilean invariance, ... ). Application of input/ouput linearization techniques with these invariant errors as outputs automatically yields to invariant tracking controllers.