**Dynamic Observers**

Dr. Sergey V. Drakunov

In control theory "observers" are known as dynamic algorithms which allow on-line estimation of the current state of a dynamic system when only an output of this system can be measured. Classical linear observers (also known from 1960ies as Kalman-Bucy or Kalman filters) can provide optimal estimates of a system state in case of extreme uncertainty modeled by white noises. The main limitation of this theory is the assumption of linearity of the model.

On the other hand, the theory of nonlinear observers is very young. In fact, it started actively developing only recently in the last 10-15 years. Its advancement became possible because of theoretical achievements in nonlinear techniques such as Lie algebras and Sliding Modes. Applications of this new theory are not yet widely known mainly due to the use of nonclassical mathematics. The rigorous mathematical description of such systems requires nonstandard understanding of the solution of differential equations and corresponding modifications of Lyapunov stability techniques.

Although the mainstream of the observer theory is being developed for models in the form of differential equations, there is a wide agreement that the concept of observer belongs not only to the theory of nonlinear dynamic systems. It may be a significant part of natural and artificial systems adaptivity and "intelligent" behavior. Deep mathematical results on observers based on nonlinear differential equations may provide some hints how to design observers for other types of models. The seminar will discuss this and other observer related issues, as well as outline the basics and applications of the theory of nonlinear observers.