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Controllability, stabilability, stability

Research in this area is done within the informational approach to control and navigation of moving objects. With this approach determination of parameters of mechanical systems with the help of additional information is reduced to evaluation of the state vector of a dynamic system with measurements. The control is also formed using these measurements. The control and estimation problems can be solved if their solutions exist. This existence conditions are formalized as observability and controllability conditions.

Observability and controllability conditions can be quantified with the measures of observability and controllability. The strongest results are obtained with the stochastic measure of observability. For example, it is proved that order reduction by erasing variables with small measures of observability yields errors of the same order as the measures of the erased variables. Similarly to the measures of observability the measures of controllability for LQG control systems (N.A.Parusnikov, A.A. Golovan) and the measures of disturbance isolation for the game problems with quadratic criteria (Yu.V.Bolotin) were introduced.

Many problems of mechanics and engineering, in particular, control and navigation of moving objects, can be reduced to non-stationary linear control systems. Non-stationarity makes both investigation of structural properties (stability, controllability and observability), and developing algorithms of control more difficult. The theory of resolvable non-stationary linear control systems developed by V.M.Morozov presents a new direction in control theory. This theory was applied to several problems of gyroscopy, correction of INS, orientation of artificial satellites and space vehicles (V.M.Morozov, V.I.Kalenova, A.I.Mozhejko, A.J.Potepalova).

The information approach was applied to stabilize stationary motions of holonomic and non-holonomic mechanical systems. This approach allowed to formulate a new criteria of controllability and observability for linear mechanical systems and to solve a number of applied stabilization problems (V.M.Morozov, V.I.Kalenova, M.A.Salmina, E.N.Shevelev).

The bearing only trajectory estimation problem was investigated using the concept of projective observability. Projective observability was shown to be closely connected with mechanical properties of a system, in particular, with its hamiltonian (conservative) structure. An unbiased estimation algorithm based on the expanded least squares, and applicable both in the observable and non-observable cases was suggested (Yu.V.Bolotin, S.N.Morgunova).

An approach to estimation and control in linear dynamic systems based on the duality theory in convex optimization was developed. This approach provides upper bounds for non-optimality of simplified estimation and control algorithms. These upper bounds can be obtained without solving the initial complex variational problems, which is especially important for systems with after-action. A new formalism for the guaranteed estimation problem for the systems with faults was suggested (A.I.Matasov).

The control resources in real life are usually limited. This poses the problem to determine the set of initial conditions (area of controllability) from which the system can be turned into the desirable operating mode. This problem is closely connected with optimum control. The areas of controllability and the closed-loop areas of attraction were constructed for linear stationary systems under a number of restrictions on the control forces (A.M.Formalsky). Similar questions were considered for the bilinear systems (V.V.Alexandrov).

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