Interval Estimation in Nonlinear System with Parametric Uncertainties

The problems of estimating the known linear function of the nonlinear system state vector under the external disturbances, measurement noise, and parametric uncertainties is studied. А solution is provided by interval observers and based on the reduced order model of the original system estimating the prescribed linear function. The interval observer is constructed based on the Jordan canonical form with negative eigenvalues since it has properties which are necessary for correct operation of the interval observer, namely, it is stable and Meltzer. The problem is solved in several steps. At the first step, the nonlinear terms are removed from the original system and the linear reduced order model insensitive to the disturbances is designed. Then it is transformed into the interval observer accounting the parametric uncertainties in the matrix of system dynamic; a proof of correctness of such observer operation is given. The obtained solution is supplemented by addends accounting the parametric uncertainties in the actuators. Then the result is corrected by taking into account the disturbances and measurement noise. To construct the nonlinear interval observer, the nonlinear term is transformed and supplemented into the linear observer, and a proof of correctness of the nonlinear observer operation is given. The obtained interval solution is similar to the confidence interval in mathematical statistics. The theoretical results are illustrated by an example of tree tank systems where the problem of interval estimation of unmeasured component of the state vector is solved. Simulation results based on the package Matlab show the effectiveness of the developed theory.

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