## Nonlinear Unknown Input Observability: the General Analytic Solution

In this talk I provide the analytic solution of an important open problem in control theory. Specifically, I provide the analytic procedure to obtain the state observability of nonlinear systems in presence of multiple unknown inputs. This problem, called the **Unknown Input Observability** (UIO) problem, was introduced in the seventies.

As for the observability rank condition (that is the analytic procedure to obtain the observability in absence of unknown inputs), the analytic solution of the nonlinear UIO problem is based on the computation of the observable codistribution by a recursive and convergent algorithm. As in the standard case of only known inputs, the observable codistribution is obtained by recursively computing the Lie derivatives along the vector fields that characterize the dynamics. However, in correspondence of the unknown inputs, the corresponding vector fields must be suitably rescaled. Additionally, the Lie derivatives must also be computed along a new set of vector fields that are obtained by recursively performing suitable Lie bracketing of the vector fields that define the dynamics. The analytic derivations and all the proofs necessary to analytically derive the algorithm and its convergence properties and to prove their general validity are very complex and they are based on new concepts borrowed from theoretical physics (specifically, from the standard model of particle physics and from the theory of general relativity). In practice, these proofs requires the introduction of the Group of Invariance of Observability and the twofold role of time in system theory. In the seminar, I only provide the strategy of the proof and an intuitive description of the above concepts.

The analytic solution is illustrated by checking the observability of several nonlinear systems driven by multiple known inputs and multiple unknown inputs, ranging from planar robotics up to advanced navigation systems in 3D that can be important also in the framework of neuroscience.