On the Solution Sets for Uncertain Systems with Phase Constraints

One of the means of modelling a system with an uncertainty in the parameters or in the inputs is to consider a multistage inclusion or a differential inclusion. These types of models may serve to describe an uncertainty for which the only available data is a set-membership description of the admissible constraints on the unknown parameters. A problem under discussion here deals with the specification of the "tube" of all solutions to a nonlinear multistage inclusion that arise from a given set and also satisfy an additional phase constraint. The description of this "solution tube" is important for solving problems of guaranteed estimation of the dynamics of uncertain systems as well as for the solution of other "viability" problems for systems described by equations involving multivalued maps

Set Valued Calculus in Problems of Adaptive Control

This paper deals with feedback control for a linear nonstationary system whose objective is to reach a preassigned set in the state space while satisfying a certain state constraint. The state constraint to be fulfilled cannot be predicted in advance being available only on the basis of observations. It is specified through an adaptive procedure of "guaranteed estimation" and the objective of the basic process is to adapt to this constraint. The problems considered in the paper are motivated by some typical applied processes in environmental, technological, economical studies and related topics. The techniques used for the solution are based on nonlinear analysis for set-valued maps

Exponentially convergent data assimilation algorithm for Navier-Stokes equations

The paper presents a new state estimation algorithm for a bilinear equation representing the Fourier- Galerkin (FG) approximation of the Navier-Stokes (NS) equations on a torus in R2. This state equation is subject to uncertain but bounded noise in the input (Kolmogorov forcing) and initial conditions, and its output is incomplete and contains bounded noise. The algorithm designs a time-dependent gain such that the estimation error converges to zero exponentially. The sufficient condition for the existence of the gain are formulated in the form of algebraic Riccati equations. To demonstrate the results we apply the proposed algorithm to the reconstruction a chaotic fluid flow from incomplete and noisy data

On Viable Solutions for Uncertain Systems

One of the problems that arises in the theory of evolution and control under uncertainty is to specify the set of all the solutions to a differential inclusion that also satisfy a preassigned restriction on the state space variables (the "viability" constraint). The latter set of "viable" trajectories may be described by either a new differential inclusion whose right-hand side is formed with the aid of a contingent cone to the restriction map or by a variety of parametrized differential inclusions each of which has a relatively simple structure. The second approach is described here for a linear-convex differential inclusion with a convex valued restriction on the state space variables

An Observation Theory for Distributed-Parameter Systems

This paper introduces a series of problems on state estimation for parabolic systems on the basis of measurements generated by sensors in the presence of unknown but bounded disturbances. Observability issues, guaranteed filtering schemes for distributed processes and their relation to similar stochastic problems are discussed. The respective problems arise from applied motivations that come, particularly, from ecological and technological issues

Ellipsoidal Techniques: Control Synthesis for Uncertain Systems

This paper deals with a technique of solving the problem of control synthesis under unknown but bounded disturbances that allows an algrithmization with an appropriate graphic simulation. The original theoretical solution scheme taken here comes from the theory introduced by N.N. Krasovski, from the notion of the "alternated integral" of L.S. Pontriagin and the "funnel equation" in the form given by Kurzhanski and Nikonov. The theory is used as a point of application of constructive schemes generated through ellipsoidal techniques developed by the authors. A concise exposition of the latter is the objective of this paper. A particular feature is that the ellipsoidal techniques introduced here do indicate an exact approximation of the original solutions based on set-valued calculus by solutions formulated in terms of ellipsoidal valued functions only

On Noninvertible Evolutionary Systems: Guaranteed Estimates and the Regularization Problem. (Revised Version)

This paper deals with an "inverse problem": the estimation of an initial distribution in the first boundary value problem for the heat equation through some biased information on its solution. Numerically stable solutions to the inverse problem are normally achieved through various regularization procedures. It is shown that these procedures could be treated within a unified framework of solving guaranteed estimation problems for systems with unknown but bounded errors

On Viable Tubes Generated by Synthesized Decision Strategies for Uncertain Systems

One of the basic goals of dynamic models in systems theory is to reflect both the uncertainty in the model and the ability to describe the models' behavior through appropriate decisions (controls). These are generally figured out through the feedback principle on the basis of the on-line position of the system. The aim of such synthesizing control strategies is usually to ensure viability properties and also to achieve some terminal goals despite of the incomplete information about the process. In this paper a mathematical scheme for solving such problems with the techniques of set-valued calculus is given

Ellipsoidal Techniques: The Problem of Control Synthesis

This paper introduces a technique for solving the problem of control synthesis with constraints on the controls. Although the problem is treated here for linear systems only, the synthesized system is driven by a nonlinear control strategy and is therefore generically nonlinear. Taking a scheme based on the notion of extremal aiming strategies of N.N. Krasovski, the present paper concentrates on constructive solutions generated through ellipsoidal-valued calculus and related approximation techniques for set-valued maps. Namely, the primary problem which originally requires an application of set-valued analysis is substituted by one which is based on ellipsoidal-valued functions. This yields constructive schemes applicable to algorithmic procedures and simulation with computer graphics