Vsevolod M. Kuntsevich

Optimization of Linear Systems Subject to Bounded Exogenous Disturbances: The Invariant Ellipsoid Technique

This survey covers a variety of results associated with control of systems subjected to arbitrary bounded exogenous disturbances. The method of invariant ellipsoids reduces the design of optimal controllers to finding the smallest invariant ellipsoid of the closed-loop dynamical system. The main tool of this approach is the linear matrix inequality technique. This simple yet versatile approach has high potential in extensions and generalizations; it is equally applicable to both the continuous and discrete time versions of the problems.

Analysis of Some Classes of Nonlinear Discrete Systems Under Bounded Disturbances

Abstract The constructive solution of the calculating minimal invariant sets problem for some classes of autonomous and nonautonomous discrete systems has been obtained. For the nonautonomous discrete systems the problem is solved on the assumption that only set-valued estimates for additive bounded disturbances are available. The class of systems with bounded nonlinear parts is considered. This class contains systems with saturation, relay systems having a dead zone and others. An example illustrates the proposed method.

Set-Valued Estimation of State and Parameter Vectors within Adaptive Control Systems

The problem under consideration is that of obtaining simultaneously set-valued estimates for state and parameter vectors of linear (in parameters and in phase coordinates) discrete-time systems under uncontrollable bounded disturbances and given bounded noise in measurements.

Design of robust stable controls for nonlinear objects

We consider a problem of discrete control for a class of nonlinear time-varying objects. Only set estimations for object parameters are available. The aim is to design controls that ensure robust stability of closed-loop systems in a given domain of state space. Since the considered class of objects is large enough not to have a stabilizing control, the proposed design method has to verify at the last step if the obtained conditions of robust stability are satisfied for a nonlinear system “in a given domain.”

ANALYSIS OF THE PURSUIT-EVASION PROCESS FOR MOVING PLANTS UNDER UNCERTAIN OBSERVATION ERRORS DEPENDENT ON STATES

The paper presents a solution to the pursuit problem under the presence of bounded (non-stochastic) errors in state measurements for the evader and under uncertain evaders controls bounded within the given compact set. It also provides the worst-case solution conditions, meaning the the worst-case evaders controls and observation errors, and the methods aiming for fulfillment of these conditions. Finally, we calculate the worst-case estimate for the number of discrete-time steps (observations) required for bringing the pursuer in the given neighborhood of evader.

Invariant sets for families of linear and nonlinear discrete systems with bounded disturbances

We use difference inclusions to describe the dynamics of a family of nonlinear discrete systems subject to bounded disturbances. For a family of linear discrete systems, we get an analytic solution of the problem of finding the invariant set, and for families of nonlinear systems, we propose an iterative process that finds their invariant set and converges with the speed of a geometric progression. We also provide illustrative examples.

On Superstable discrete systems

For the class of nonstationary discrete linear systems whose parameters have only multiple estimates, the sufficient condition for their robust stability was obtained on the basis of a discrete analog of the Lyapunov method. If satisfied at each time instant, this condition can be interpreted as the sufficient stability condition for systems with “frozen” coefficients. Robust stability of the stationary systems of the specified class follows from it as a special case.

Optimal Control over the Approach of Conflicting Moving Objects under Uncertainty

A solution of a pursuit-evasion problem is obtained for two controlled moving points for the case where the coordinates of the evader are measured with bounded errors and the controls chosen by the evader are assumed to be known up to their membership in a given compact set. Moreover, the conditions of a guaranteed solution of the pursuit-evasion problem are obtained for the worst values of measurement errors and controls of the evader.

Linear adaptive control for nonstationary uncertain systems under bounded noise

A discrete-time nonstationary linear control system is considered to be given by the algebraic difference equation in the state space. The control system is subject to a bounded additive noise. Uncertain parameters of the system take their values on the given polytopes which evolve in time. The objective is to generate a linear feedback, which provides the minimization of a given performance criterion in adaptive way. In general, the control problem is reduced to the convex programming one of an insignificant computational complexity. Therewith, the control problem can be solved analytically in the case of interval set-valued parameter estimates.

Robust Stability of Linear Dynamic Systems

The problem of robust stability of dynamic systems with its most recent history originating from the pioneering work by V.L. Kharitonov [1] has become one of the most timely problems in control theory in the last few years which was demonstrated in particular by the 11th IFAC World Congress in Tallinn, Estonia, USSR (August 13–17, 1990).