The goal is to carry on basic research in nonlinear control theory, which has a firm base in applied mathematics. The group's research activities are directed towards developing theoretical and symbolic computation tools for modelling, analysis and synthesis of nonlinear control systems.

The group has made significant contributions in developing algebraic methods for discrete-time nonlinear control systems. A universal algebraic formalism has been developed that unifies the study of very different problems. In this formalism, sequences of subspaces of the differential forms, associated with the control system, are defined and provide a lot of information about the structural properties of the system. During recent years we have applied the algebraic methods for the study of a number of fundamental properties of a control system; like feedback linearizability, accessibility, identifiability, system (input-output and transfer) equivalence, irreducibility and realizability of the system in the classical state space form. Most procedures have been implemented in the computer algebra system Mathematica.

Our main effort has been devoted to the development of tools and techniques to fill the gap between nonlinear system identification and controller design fields stemming from the use of non-complementary models, i.e. NARX versus state space models, in those areas. Firstly we have obtained the necessary and sufficient realizability conditions for NARX model in the classical state space forms as well as the constructive procedure (up to finding the integrating factors) for constructing observable and accessible state space equations. We have suggested a wide subclass of NARX-models that admit a state space description, and studied the realizability properties of bilinear, quadratic and associative i/o equations. We have also proved that typical neural network(NN)-based NARX type models do not admit a classical state space description and have suggested a new class that can be easily realized in the classical state space form. Also this allows one to simplify the controller design task in the i/o domain.

Secondly, we have developed output feedback control laws directly for the i/o models. It is an alternative to be used when the nonlinear i/o model cannot be transformed into the state space form. The solutions for system linearization and decoupling into subsystems have been obtained. Advantages and limitations of the algebraic approach in comparison with other methods have also been studied and clarified.

Our main effort is at present targeted towards the polynomial approach which is built upon the algebraic formalism based on differential one-forms and extends directly the results from the linear theory to the nonlinear domain. The basic difference is that unlike the linear case the skew polynomials related to the nonlinear system belong to a non-commutative polynomial ring and the polynomial equations relate the differentials of inputs and outputs, not the inputs and outputs themselves. The polynomial approach allows also to extend the design methods that are based on the transfer function to the nonlinear domain.

At present, not much application-oriented research is carried out in the group. As an intermediate solution, a nonlinear control system package for computer-aided modelling, analysis and synthesis of control systems on the basis of a computer algebra system Mathematica is being developed. With such a toolkit the prospective user is able to deal with more realistic problems. Most important functions of the NLControl toolkit are made available for browser-based usage via webMathematica tools. Future important activity is to make new contacts with industrial partners, through our present academic international co-operation, and also with groups in Estonia.

The group has strong international links as evidenced by the joint publications