A. V. Kuntsevich

An Implementation of Shor's r-Algorithm

Here we introduce a new implementation of well-known Shors r-algorithm with space dilations along the difference of two successive (sub)gradients for minimization of a nonlinear (non-smooth) function (N.Z. Shor, Minimization methods for Non-Differentiable Functions, Springer-Verlag: Berlin, 1985. Springer Series in Computational Mathematics, vol. 3). The modifications made to Shors algorithm are heuristic. They mostly concern the termination criteria and the line search strategy. A large number of test runs indicate that this implementation is very robust, efficient and accurate. We hope that this implementation of Shors r-algorithm will prove to be useful for solving a wide class of non-smooth optimization problems.

Analysis and synthesis of optimal and robustly optimal control systems under bounded disturbances

Abstract Analysis of dynamic (closed-loop control) systems under bounded disturbances by calculating minimal invariant acta is discussed. It is suggested to use the radius of A minimal invariant set as a particular numerical measure of the stability degree of a system and apply such a criterion for optimal linear control synthesis by means of minimizing this radius. It has been proven that the radius of a minimal invariant set reaches its minimum at a nilpotent parameter matrix of a system. The solution to the optimal control synthesis problem can be found, therefore, by calculating the linear feedback parameters providing the nilpotent matrix of the closed-loop system. In the case of scalar controls, the problem has got an analytical solution

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.

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 and Synthesis of Nonlinear Discrete Control Systems under Uncertainty

Abstract With the use of Lyapunov functions chosen as the norm of state vector, we obtain the robust stability sufficient conditions for a wide class of nonlinear, and generally nonstationary, discrete-time control systems with the given set-valued parameter estimates. For a strictly monotone nonlinear function, validation of these conditions is equivalent to solution of a series of combinatorial problem in the state space. Synthesis of robustly stable control systems in a domain is performed on the basis of the obtained sufficient conditions of robust stability.

Analysis of Control Systems with Bounded Nonlinearities and Uncertainties

Abstract The report concerns further development of the invariant sets theory with its application to analysis and synthesis of control systems. Particularly, the results obtained for linear control systems are extended to the class of models including bounded nonlinear components, i.e. those items allowing description with bounded nonlinear functions.

Active identification and control given bounded noise (disturbances)

Problems of active identification and control with bounded disturbances are considered for two classes of plants: dynamic plants and those without memory. Performance criteria of set-membership identification are proposed for each of the classes. For controlled plants without memory, an algorithm of optimal design of experiments is developed which provides a minimum of performance criterion of set-membership identification. Only a suboptimal solution of the problem is obtained for the class of dynamic systems.