# CONVERGENCE OF NONLINEAR OBSERVERS ON R^N WITH A RIEMANNIAN METRIC

Topic: Observers

Séance du jeudi 13 février 2020, Salle L 226, 14h00-15h30.

Laurent PRALY, CAS, MINES Paristech, PSL University

We propose necessary conditions and sufficient conditions for the convergence of an observer the state of which lives in a copy of the given system’s state space can be established using a Riemannian metric.

The existence of an observer guaranteeing the property that a Riemannian distance between system and observer solutions is non-increasing implies the system is differentially detectable (i.e., the Lie derivative of the Riemannian metric along the system vector field is negative in the space tangent to the output function level sets). This differential detectability property is related to the observability of the system’s linearization along its solutions. This relation is the starting point of techniques for designing a Riemannian metric exhibiting the differential detectability, assuming the system is strongly infinitesimally observable (i.e., each time-varying linear system resulting from the linearization along a solution to the system satisfies a uniform observability property) or is strongly differentially observable (i.e., the mapping state to output derivatives is an injective immersion).

On the other hand, if the observer has an infinite gain margin then the level sets of the output function are geodesically convex. If, furthermore, the correction term in the observer, is of a gradient type then the output function is geodesic monotone, i.e. as a current point moves along a minimizing geodesic, in the state manifold, towards a given point, the “gap” between their images, , in the output manifold, by the output function decreases. When the number of output is equal to 1, this geodesic monotonicity is equivalent to the fact that the output function is affine, i.e. a Riemannian submersion with totally geodesic level sets and integrable orthogonal distribution (maybe after modifying the metric in a way which does not alter the strong differential observability property). This link allows us to give a full description of the family of Riemannian metrics giving this affine property.

Conversely, we establish that, if we have a complete Riemannian metric exhibiting the differential detectability property and making the output function affine, then there exists a semi-globally convergent observer with an infinite gain margin. Actually, already without this last property, we can get a locally convergent observer

Slides (pdf)

Laurent PRALY, CAS, MINES Paristech, PSL University

We propose necessary conditions and sufficient conditions for the convergence of an observer the state of which lives in a copy of the given system’s state space can be established using a Riemannian metric.

The existence of an observer guaranteeing the property that a Riemannian distance between system and observer solutions is non-increasing implies the system is differentially detectable (i.e., the Lie derivative of the Riemannian metric along the system vector field is negative in the space tangent to the output function level sets). This differential detectability property is related to the observability of the system’s linearization along its solutions. This relation is the starting point of techniques for designing a Riemannian metric exhibiting the differential detectability, assuming the system is strongly infinitesimally observable (i.e., each time-varying linear system resulting from the linearization along a solution to the system satisfies a uniform observability property) or is strongly differentially observable (i.e., the mapping state to output derivatives is an injective immersion).

On the other hand, if the observer has an infinite gain margin then the level sets of the output function are geodesically convex. If, furthermore, the correction term in the observer, is of a gradient type then the output function is geodesic monotone, i.e. as a current point moves along a minimizing geodesic, in the state manifold, towards a given point, the “gap” between their images, , in the output manifold, by the output function decreases. When the number of output is equal to 1, this geodesic monotonicity is equivalent to the fact that the output function is affine, i.e. a Riemannian submersion with totally geodesic level sets and integrable orthogonal distribution (maybe after modifying the metric in a way which does not alter the strong differential observability property). This link allows us to give a full description of the family of Riemannian metrics giving this affine property.

Conversely, we establish that, if we have a complete Riemannian metric exhibiting the differential detectability property and making the output function affine, then there exists a semi-globally convergent observer with an infinite gain margin. Actually, already without this last property, we can get a locally convergent observer

Slides (pdf)