Observability: A New Theory Based on the Group of Invariance

Observability is a critical concept in control theory. Loosely speaking, the state of a system is observable if knowledge of its inputs and outputs during a given time interval permit determination of that state. For linear time-invariant systems, there is a simple criterion for observability: that the observability matrix (conceptually straightforward and easy to compute) should have full rank. For more general systems with all the inputs known there are similar criteria. But for nonlinear systems with unknown inputs, the problem is a good deal harder. This monograph offers a new approach to observability in nonlinear systems, one that was developed by the author.

 

In his preface, the author describes how he, as a theoretical physicist, came to work in this area of control theory. He was studying the problem of using monocular vision and inertial sensors to make a drone fly autonomously in situations like search and rescue after natural disasters. This brought him into some very complex estimation problems with nonlinear observability.

 

The author intends to provide a complete theory of observability using his new and broader approach. This includes extension of the observability rank condition to nonlinear systems that may have unknown inputs. New derivations are also provided for existing results in nonlinear observability.

 

The primary problem that the theory of observability studies is determining whether an input-output system provides the information necessary to estimate the state. Such information comes from sensors that provide measurements of system inputs and outputs. One way to formulate this problem quantitatively is to ask whether it is possible to determine the initial state of a system given system inputs and outputs over a given time interval.

 

The book begins with several examples that illustrate observability for simple systems and identifies regions of state space that are unobservable. These examples describe the motion of a vehicle in space and try to identify intuitively which states are observable.

 

Where the author departs from standard control theory is his contention that there is a group of invariance that is inherent in the concept of observability and that this group is critical for approaching observability for nonlinear systems with unknown inputs. To prepare readers for an exposition of this new advanced approach he devotes the second chapter to a discussion of manifolds, tensors and Lie groups.

 

The development begins with the simpler questions on nonlinear observability where there are no unknown inputs. Here the groups of invariance of observability are abelian, and the tensors that appear in systems with unknown inputs are just real numbers. The author applies his approach to re-establish known results in this case. Then he moves on to nonlinear systems with unknown inputs. Here his results are new. 

 

Although the book has many fairly elementary examples, some very complex examples appear that illustrate the more advanced material. These often involve inertial navigation for aerial vehicles. Unknown inputs arise, for example, from an accelerometer that measures acceleration along only one axis, or an inertial measurement system that measures velocity or rotation rate but not both.

 

This monograph is clearly aimed at specialists. Its natural audience would be control theorists, but they may find the level of abstraction very challenging. 

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Bill Satzer (

bsatzer@gmail.com

), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and material science. He did his PhD work in dynamical systems and celestial mechanics.

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