Tarek Raissi

Set membership state and parameter estimation for systems described by nonlinear differential equations

This paper investigates the use of guaranteed methods to perform state and parameter estimation for nonlinear continuous-time systems, in a bounded-error context. A state estimator based on a prediction-correction approach is given, where the prediction step consists in a validated integration of an initial value problem for an ordinary differential equation (IVP for ODE) using interval analysis and high-order Taylor models, while the correction step uses a set inversion technique. The state estimator is extended to solve the parameter estimation problem. An illustrative example is presented for each part.

Complex Interval Arithmetic Using Polar Form

In this paper, the polar representation of complex numbers is extended to complex polar intervals or sectors; detailed algorithms are derived for performing basic arithmetic operations on sectors. While multiplication and division are exactly defined, addition and subtraction are not, and we seek to minimize the pessimism introduced by these operations. Addition is studied as an optimization problem which is analytically solved. The complex interval arithmetic thus defined is illustrated with some numerical examples which show that in many applications, the polar representation is more advisable.

Set-Membership Identification of Continuous-Time Systems

The identification of continuous-time systems from discrete-time data is investigated in a bounded-error context. Errors are taken as a discrete time signal, additive on output, unknown but bounded with known bounds. The set inversion via interval analysis algorithm is used with validated numerical integration methods. The theory of monotone cooperative systems is also introduced in order to derive tighter enclosures for the solution of the differential equations. Via an illustrative example with simulated data, it is shown that for systems modelled by parabolic differential equations, the cooperativity property only makes it possible to obtain results in a reasonable computation time.

Parameter estimation for non-linear continuous-time systems in a bounded error context

This paper deals with guaranteed parameter estimation in a bounded error context for nonlinear continuous-time systems. Perturbations are assumed bounded but otherwise unknown. The solution is a set of parameter vectors consistent with modelling hypotheses, measured data and prior error bounds. The algorithm proposed in this paper does not suffer from initialization problems encountered in the local optimization methods. The main tools to solve such a problem are the guaranteed techniques for the integration of ordinary differential equations, and set inversion. Both of these techniques use interval analysis. An illustrative example is given.

State Estimation for Nonlinear Continuous Systems in a Bounded-Error Context

Abstract The aim of this paper is to study state estimation for nonlinear continuous-time systems in a bounded-error context. A causal estimator based on prediction-correction approach is proposed. The prediction part consists on a validated integration of an Initial Value Problem for an Ordinary Differential Equation. The correction part uses set inversion. The main tools used are Taylor models and interval analysis. The derived estimator is illustrated on an example.

SET MEMBERSHIP PARAMETER IDENTIFICATION WITH COMPLEX INTERVALS USING POLAR FORMS

Abstract This paper is dedicated to bounded error identification with complex valued non-linear models. Complex intervals are characterized by using polar forms and a new inclusion function is given for the addition of sectors. The latter is expressed as an optimization problem solved analytically. The new complex interval arithmetic is used with actual data and a complex valued non-linear model for the bounded error identification of the thermal properties of materials.