Sergey A. Nazin

Ellipsoidal parameter or state estimation under model uncertainty

Ellipsoidal outer-bounding of the set of all feasible state vectors under model uncertainty is a natural extension of state estimation for deterministic models with unknown-but-bounded state perturbations and measurement noise. The technique described in this paper applies to linear discrete-time dynamic systems; it can also be applied to weakly non-linear systems if non-linearity is replaced by uncertainty. Many difficulties arise because of the non-convexity of feasible sets. Combined quadratic constraints on model uncertainty and additive disturbances are considered in order to simplify the analysis. Analytical optimal or suboptimal solutions of the basic problems involved in parameter or state estimation are presented, which are counterparts in this context of uncertain models to classical approximations of the sum and intersection of ellipsoids. The results obtained for combined quadratic constraints are extended to other types of model uncertainty.

Rejection of bounded exogenous disturbances by the method of invariant ellipsoids

Rejection of the bounded exogenous disturbances was first studied by the l1-optimization theory. A new approach to this problem was proposed in the present paper on the basis of the method of invariant ellipsoids where the technique of linear matrix inequalities was the main tool. Consideration was given to the continuous and discrete variants of the problem. Control of the “double pendulum” was studied by way of example.

Rejection of Bounded Disturbances via Invariant Ellipsoids Technique

In this paper an approach based on invariant ellipsoids is applied to the problem of persistent disturbance rejection by means of static state-feedback control. Dynamic system is supposed to be linear time-invariant and affected by unknown-but-bounded exogenous disturbances. Synthesis of an optimal controller that returns a minimum of the size of the corresponding invariant ellipsoid is reduced to one-dimensional convex minimization with LMI constraints. The problem is considered in continuous and discrete time cases

Interval Parameter Estimation under Model Uncertainty

This paper is devoted to the estimation of parameters of linear multi-output models with uncertain regressors and additive noise. The uncertainty is assumed to be described by intervals. Outer-bounding interval approximations of the non-convex feasible parameter set for uncertain systems are obtained. The method is based on the calculation of the interval solution for an interval system of linear algebraic equations and provides parameter estimators for models with a large number of measurements.

Ellipsoidal estimation under model uncertainty

Abstract Ellipsoidal outer-bounding under model uncertainty is a natural extension of state estimation for models with unknown-but-bounded errors. The technique described in this paper applies to linear discrete-time dynamic systems. Many difficulties arise because of the non-convexity of feasible sets. Analytical optimal or suboptimal solutions are presented, which are counterparts in this context of uncertainty to classical approximations of the sum and intersection of ellipsoids.

Estimation of Parameters in Linear Multidimensional Systems under Interval Uncertainty

we suggested a model of competitiveness of brand product, which takes into account 10 factors. The model is based on 52 fuzzy rules of type. Potential of application of the model for control of competitiveness of brand product is shown. We state a problem of learning of fuzzy model of competitiveness on the basis of experimental data.

On Convergence of External Ellipsoidal Approximations of the Reachability Domains of Discrete Dynamic Linear Systems

The ellipsoid technique is widely used in the guaranteed estimation for approximation of the reachability domains of dynamic systems. The present paper considered the issues of external ellipsoidal estimation of the current and limiting reachability sets of a stable discrete dynamic linear system. Recurrent estimation algorithms using the criterion of minimum trace of the “weighted” ellipsoid matrix were developed for these systems, and their limiting properties were considered.

Ellipsoid-based parametric estimation in the linear multidimensional systems with uncertain model description

Parametric estimation under uncertainty of the plant model description was considered within the framework of the ellipsoidal approach to the problems of guaranteed estimation. The unknown multidimensional plant whose parameter vector should be estimated was assumed to be linear and static, and uncertainty of its “input-output” model, to have both additive and multiplicative components. The external ellipsoidal approximations of the nonconvex information sets guaranteeing that the vector of possible plant parameters is contained in them were constructed from the results of observations. The method of their construction comes to semidefinite programming, that is, to minimization of the linear function constrained by the linear matrix inequalities which are readily realized by the numerical methods.

Interval technique for parameter estimation under model uncertainty

Abstract This paper is devoted to the problem of estimation of parameters for linear multi-output models with uncertain regressors and additive noise. The uncertainty is assumed to be described by intervals. Outer-bounding interval approximations of the non-convex feasible parameter set for uncertain system are obtained. The proposed method is based on the calculation of the interval solution for an interval system of linear algebraic equations and provides the parameter estimators for models with large number of measurements.

Guaranteed ellipsoidal state estimation for uncertain MIMO models

Abstract This paper extends to uncertain models the classical ellipsoidal outer-bounding of the set of all feasible state vectors with unknown but bounded state perturbations and measurement noise. The technique applies to linear discrete-time dynamic systems and could be extended to weakly non-linear systems. Combined quadratic constraints on model uncertainty and additive disturbances are considered in order to simplify analysis and to allow an analytical solution of the basic problems involved in parameter or state estimation. The results obtained for combined quadratic constraints are extended to other types of model uncertainty.