Alexander Ovseevich

Properties of the optimal ellipsoids approximating the reachable sets of uncertain systems

The ellipsoidal estimation of reachable sets is an efficient technique for the set-membership modelling of uncertain dynamical systems. In the paper, the optimal outer-ellipsoidal approximation of reachable sets is considered, and attention is paid to the criterion associated with the projection of the approximating ellipsoid onto a given direction. The nonlinear differential equations governing the evolution of ellipsoids are analyzed and simplified. The asymptotic behavior of the ellipsoids near the initial point and at infinity is studied. It is shown that the optimal ellipsoids under consideration touch the corresponding reachable sets at all time instants. A control problem for a system subjected to uncertain perturbations is investigated in the framework of the optimal ellipsoidal estimation of reachable sets.

Asymptotic behavior of attainable sets of linear periodic control systems

This paper deals with the attainable sets of linear periodic control systems. The asymptotic behavior of the attainable sets over a long time interval is investigated in terms of shapes of the sets. The shape of a set stands for the totality of all its images under nonsingular linear transformations. It is shown that there exist limits of the shape of attainable sets corresponding to time instants with the same residue modulo the period of the system and that the limit shapes are different if the system includes a stable subsystem.

Bounded feedback controls for linear dynamic systems by using common Lyapunov functions

This paper considers the problem of synthesizing a bounded control of a linear dynamical system satisfy� ing the Kalman controllability condition. An approach is developed which makes it possible to con� struct feedback control laws transferring the system to the origin in finite time. The approach is based on methods of stability theory. The construction is based on the notion of a common Lyapunov function. It is shown that the constructed control remains effective in the presence of uncontrollable perturbations of the system. As an illustration, results of numerically mod�

On equations of ellipsoids approximating attainable sets

This paper is devoted to a search for a guaranteed counterpart of the stochastic Kalman filter. We study the guaranteed filtering of a linear system such that the phase state and external disturbance form a vector subject to an ellipsoidal bound. This seemingly exotic setup can be justified by an analogy with the observation of Gaussian processes. Unfortunately, the resulting guaranteed filtering supplies us an ellipsoid approximating the localization domain for the state vector, but not the localization domain itself, and turns out to be more difficult compared to the Kalman filter. Our main results consist of an explicit evaluation of the Hamiltonians. In principle, this permits us to write explicitly the equations of the guaranteed filter.

A Local Feedback Control Bringing a Linear System to Equilibrium

We design a bounded feedback control that steers a controllable linear system to equilibrium in a finite time. The procedure amounts to solving several linear-algebraic problems, including linear matrix inequalities. We solve these problems in an efficient way. The resulting steering time and the minimum time have the same order of magnitude.

METHODS OF ELLIPSOIDAL ESTIMATION FOR LINEAR CONTROL SYSTEMS

Abstract We present explicit formulas for ellipsoids bounding reachable sets for linear control dynamic systems with geometric bounds on control. We study both locally and globally optimal ellipsoidal estimates with regard to different optimality criteria. In particular, we solve some essentially nonlinear boundary problems related to the search for globally optimal ellipsoids with regard to the volume criterion. It is shown that by using the explicit formulas one can efficiently pass to limits in several asymptotic problems, including passing to the limit when the phase space dimension goes to infinity.

Asymptotic behavior of attainable sets of a linear impulse control system

Abstract We study the long term asymptotic behavior of attainable sets and their shapes of linear time-invariant impulse control systems. We give an exhaustive description of attractors arising and the related dynamics. The results are compared with [4], [3].

BIRTH OF THE SHAPE OF A REACHABLE SET

We address a linear control system under geometric constraints on control and study its reachable sets starting at zero time from the origin. The main result is the existence of a limit shape of the reachable sets as the terminal time tends to zero. Here, a shape of a set stands for the set regarded up to an invertible linear transformation. Both autonomous and nonautonomous cases are considered.

LIMIT SHAPES OF REACHABLE SETS FOR LINEAR IMPULSE CONTROL SYSTEMS

Abstract The main purpose of the paper is to study the asymptotic behavior of reachable sets to linear time-invariant impulsive control systems. The issue is analyzed within the framework of shapes of reachable sets. This approach enables an exhaustive description of attractors arising in the space of shapes and the related dynamics. The results are compared with (Ovseevich, 1991; Figurina and Ovseevich, 1999).

Limit Behavior of Attainable Sets of Linear Systems

Abstract.The main purpose of the paper is to study the asymptotic behavior of reachable sets to linear time-invariant control systems with a bounded control. The issue is analyzed within a framework of shapes of reachable sets. Main results consist in an existence proof, and a rather explicit description of limit shapes of the studied reachable sets as time goes to infinity. We also state a conjecture on the structure of the set of limit shapes in the general non-autonomous case.