Carlo Novara

Set membership prediction of nonlinear time series

In this paper, a prediction method for nonlinear time series based on a set membership (SM) approach is proposed. The method does not require the choice of the functional form of the model used for prediction, but assumes a bound on the rate of variation of the regression function defining the model. At the contrary, most of the existing prediction methods need the choice of a functional form of the regression function or of state equations (piecewise linear, quadratic, etc.) and this choice is usually the result of heuristic searches. These searches may be quite time consuming, and lead only to approximate model structures, whose errors may be responsible of bad propagation of prediction errors, especially for the multistep ahead prediction. Moreover, the method proposed in this paper assumes only that the noise is bounded, in contrast with statistical approaches, which rely on noise assumptions such as stationarity, ergodicity, uncorrelation, type of distribution, etc. The validity of these assumptions may be difficult to be reliably tested in many applications and is certainly lost in presence of approximate modeling. In the present SM approach, using a result developed in a previous paper, the values of the bounds on the gradient of the regression function and on the noise can be suitably assessed to verify the validity tests. Two almost optimal prediction algorithms are then derived, the second one having improved optimal properties over the first one, at the expense of an increased computational complexity. The method is tested and compared with other literature methods on the well-known Wolf Sunspot Numbers series, widely used in the time series literature as a benchmark test, and on the prediction of vertical dynamics of vehicles with controlled suspensions. A simulation example is also presented to investigate how much conservative the SM approach may be in the most adverse situation where data are generated by a linear autoregressive (AR) model driven by i.i.d. gaussian white noise and the SM prediction is compared with the optimal statistical predictor, which makes use of the exact assumptions.

Guest Editorial Special Issue on Applied LPV Modeling and Identification

The nine regular and three brief papers in this special issue present the current state-of-the-art in the neighboring fields of Linear Parametrically Varying (LPV) modeling and system identification.

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

Consider a nonlinear system with input u and outputs y, z. Assume that ut and yt are measured for all times t and that zt is measured only for t les Tm, but it is of interest to know zt for t > Tm. 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 t, t > Tm is a system using (ut, y t) as inputs and producing as output an estimate of zt , the idea of directly identifying an estimator model from the available noisy data (utildet, ytildet) and ztildet in the time interval (0, Tm) 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

Input design for structured nonlinear system identification

This paper is concerned with the input design problem for a class of structured nonlinear models. This class contains models described by an interconnection of known linear dynamic systems and unknown static nonlinearities. Many widely used model structures are included in this class. The model class considered naturally accommodates a priori knowledge in terms of signal interconnections. Under certain structural conditions, the identification problem for this model class reduces to standard least squares. We treat the input design problem in this situation. An expression for the expected estimate variance is derived. A method for synthesizing an informative input sequence that minimizes an upper bound on this variance is developed. This reduces to a convex optimization problem. Features of the solution include parameterization of the expected estimate variance by the input distribution, and a graph-based method for input generation.

Unified Set Membership theory for identification, prediction and filtering of nonlinear systems

The problem of making inferences from data measured on nonlinear systems is investigated within a Set Membership (SM) framework and it is shown that identification, prediction and filtering can be treated as specific instances of the general presented theory. The SM framework presents an alternative view to the Parametric Statistical (PS) framework, more widely used for studying the above specific problems. In particular, in the SM framework, a bound only on the gradient of the model regression function is assumed, at difference from PS methods which assume the choice of a parametric functional form of the regression function. Moreover, the SM theory assumes only that the noise is bounded, in contrast with PS approaches, which rely on noise assumptions such as stationarity, uncorrelation, type of distribution, etc. The basic notions and results of the general inference making theory are presented. Moreover, some of the main results that can be obtained for the specific inferences of identification, prediction and filtering are reviewed. Concluding comments on the presented results are also reported, focused on the discussion of two basic questions: what may be gained in identification, prediction and filtering of nonlinear systems by using the presented SM framework instead of the widely diffused PS framework? why SM methods could provide stronger results than the PS methods, requiring weaker assumptions on system and on noise?

Parametric identification of structured nonlinear systems

In this paper, identification of structured nonlinear systems is considered. Using linear fractional transformations (LFT), the a priori information regarding the structural interconnection is systematically exploited. A parametric approach to the identification problem is investigated, where it is assumed that the linear part of the interconnection is given and the input to the nonlinear part is measurable. An algorithm for the identification of the nonlinear part is proposed. The uniqueness properties of the estimate provided by the algorithm are examined. It is shown that the estimate converges asymptotically to its true value under a certain persistence of excitation condition. Two simulated examples and a real-data example are presented to show the effectiveness of the proposed algorithm.

Computation of local radius of information in SM-IBC identification of nonlinear systems

System identification consists in finding a model of an unknown system starting from a finite set of noise-corrupted data. A fundamental problem in this context is to asses the accuracy of the identified model. In this paper, the problem is investigated for the case of nonlinear systems within the Set Membership-Information Based Complexity framework of [M. Milanese, C. Novara, Set membership identification of nonlinear systems, Automatica 40(6) (2004) 957-975]. In that paper, a (locally) optimal algorithm has been derived, giving (locally) optimal models in nonlinear regression form. The corresponding (local) radius of information, providing the worst-case identification error, can be consequently used to measure the quality of the identified model. In the present paper, two algorithms are proposed for the computation of the local radius of information: The first provides the exact value but requires a computational complexity exponential in the dimension of the regressor space. The second is approximate but involves a polynomial (quadratic) complexity.

Direct Filtering: A New Approach to Optimal Filter Design for Nonlinear Systems

Optimal filters for nonlinear systems are in general difficult to derive or implement. The common approach is to use approximate solutions such as extended Kalman filters, ensemble filters or particle filters. However, no optimality properties can be guaranteed by these approximations, and even the stability of the estimation error cannot often be ensured. Another relevant issue is that, in most practical situations, the system whose variables have to be estimated is not known, and a two-step procedure is adopted, based on model identification from data and filter design from the identified model. However, the designed filter may display large performance deteriorations in the case of modeling errors. In this paper, a new approach overcoming these issues is proposed, allowing the design of optimal filters for nonlinear systems in both the cases of known and unknown system. The approach is based on the direct filter design from a set of data generated by the system. Either experimental or simulated data can be used for design. A bound on the number of data necessary to ensure a given filter accuracy is also provided, showing that the proposed approach is not affected by the curse of dimensionality.

Linear Parameter-Varying System Identification: New Developments and Trends

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Brief paper: The filter design from data (FD2) problem: Nonlinear Set Membership approach

In the paper, we consider the problem of designing a filter that, operating on noisy measurements of input u and output y of a dynamical system, gives estimates (possibly optimal in some sense) of some other variable of interest z. A large body of literature exists, which investigates this problem assuming that the system equations relating u, y and z are known. However, in most practical situations, the system equations are not (completely) known, but a data set composed of noisy measurements of u, y and z is available. In such situations, a two-step procedure is typically adopted: a model is identified from the set of measured data, and the filter is designed on the basis of the identified model. In this paper, we propose an alternative solution, which uses the available data set of measured u, y and z not for the identification of system dynamics, but for the direct design of the filter. Such a direct design is investigated within the Nonlinear Set Membership framework. In the case of full observability, an almost optimal filter is derived, where optimality refers to minimizing a worst-case estimation error. In the case of partial observability, conditions are given for which the direct design is guaranteed to give bounded estimation error. Three examples are presented, related to the Lorenz chaotic system the first two, and to an automotive application the third one.