Filter design from data: direct vs. two-step approaches

Abstract

Consider a nonlinear system with input u and outputs y, z. Assume that u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> and y <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> are measured for all times t and that z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> is measured only for t les T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> , but it is of interest to know z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> for t > T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> . Such situation may arise when the sensor measuring z fails and it is important to recover this variable, e.g., for feedback control. Another case arises when the sensor measuring z is too complex and costly to be used, except for an initial set of experiments. Assuming that z is observable from the couple (u, y), the standard approach consists of a two-step procedure: identify a model and then design an observer/Kalman filter based on the identified model. Observing that an estimator of z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> , t > T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> is a system using (u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> , y <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> ) as inputs and producing as output an estimate of z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t </sup> , the idea of directly identifying an estimator model from the available noisy data (utilde <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> , ytilde <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> ) and ztilde <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> in the time interval (0, T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> ) is investigated in this paper. The two-step procedure is proved to perform, in the case of exact modeling, no better than the direct approach. In the presence of modeling errors, the directly identified filter is proved to be anyway the minimum variance estimator, among the selected approximating filter class. A similar result is not assured by the two-step design, whose performance deterioration due to modeling errors may be significantly larger. Another relevant point is that minimum variance filters for nonlinear systems are in general difficult to derive and/or to implement, and widely used approximate solutions, such as extended Kalman filters, quite often exhibit poor performance. On the contrary, the recent progresses in nonlinear identification methods may allow the direct filter identification. An example related to the Lorenz attractor is presented to demonstrate the effectiveness of the presented approach