Parameter identification for nonlinear systems: Guaranteed confidence regions through LSCR

In this paper we consider the problem of constructing confidence regions for the parameters of nonlinear dynamical systems. The proposed method uses higher order statistics and extends the LSCR (leave-out sign-dominant correlation regions) algorithm for linear systems introduced in Campi and Weyer [2005, Guaranteed non-asymptotic confidence regions in system identification. Automatica 41(10), 1751-1764. Extended version available at 〈http://www.ing.unibs.it/∼campi〉]. The confidence regions contain the true parameter value with a guaranteed probability for any finite number of data points. Moreover, the confidence regions shrink around the true parameter value as the number of data points increases. The usefulness of the proposed approach is illustrated on some simple examples.     

Identifiability of uncontrolled nonlinear rational systems

In this paper an approach for the identifiability analysis of uncontrolled rational systems is provided. The method is based on the use of a local smooth state space transformation. In particular it is shown that, provided the model satisfies an observability rank condition, the state trajectories of an uncontrolled system corresponding to parameter vectors with outputs that are identical locally in time, are connected via a smooth transformation.     

Towards identification of Wiener systems with the least amount of a priori information: IIR cases

In this paper, we investigate what constitutes the least amount of a priori information on the nonlinearity so that the linear part is identifiable in the non-Gaussian input case. Under the white noise input, three types of a priori information are considered: quadrant information, point information, and monotonic information. In all three cases, identifiability has been established, and the corresponding nonparametric identification algorithms are developed along with their convergence proofs.     

Towards identification of Wiener systems with the least amount of a priori information on the nonlinearity

In this paper, we investigate what constitutes the least amount of a priori information on the nonlinearity so that the FIR linear part is identifiable in the non-Gaussian input case. Three types of a priori information are considered including quadrant information, point information and locally monotonous information. In all three cases, identifiability has been established and corresponding identification algorithms are developed with their convergence proofs.     

Parameter identification for a class of abstract nonlinear parabolic distributed parameter systems

This paper studies the parameter identification problem of nonlinear abstract parabolic distributed parameter systems via variational method ^[1]. Based on the fundamental optimal control theory and the transposition method studied in ^[2], the existence of optimal parameter is proved, and the necessary condition for the optimal parameter is established.     

Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems

The problem of parameter identifiability has been considered from different points of view in the case of nonlinear dynamical systems. For analytic systems the standard approach for uncontrolled systems is the Taylor series approach (Pohjanpalo, Math. Biosciences 41 (1978) 21), or the approaches based on differential algebra for polynomial and rational systems. The similarity transformation approach, based on the local state isomorphism theorem, gives a sufficient and necessary condition for global identifiability of nonlinear controlled systems. But it leads only to a necessary condition for identifiability in the case of some uncontrolled systems. Our contribution consists in using the equivalence of systems, based on the straightening out theorem, to analyse the identifiability of uncontrolled systems. From this theory, we state the necessary or sufficient identifiability conditions, some of them depending on the state variable dimension.     

Left invertibility,flatness and identifiability of switched linear dynamical systems: a framework for cryptographic applications

The purpose of this article is to illustrate the potential interest of the control theory framework for cryptographic applications. It is shown that under the properties of left invertibility and flatness, dynamical systems are structurally equivalent to some specific cryptographic primitives called self-synchronising stream ciphers. After having motivated the interest of considering hybrid systems for such ciphers, the development is particularised for the special class of switched linear systems. We also show that identifiability is a necessary condition for security, and we describe an identification procedure as a possible attack and assess its complexity.     

Structural indistinguishability between uncontrolled (autonomous) nonlinear analytic systems

In this paper, an approach for analysing the structural indistinguishability between two uncontrolled (or autonomous) analytic systems is presented. The approach involves constructing, if possible, a smooth mapping between the trajectories of two candidate models. If either of the models satisfies an observability criterion, then such a transformation always exists when the models are indistinguishable from their outputs. The approach is illustrated by examples from epidemiology and chemical reaction kinetics. One important outcome is that the susceptible, infectious, recovered (SIR) and SIR with temporary immunity (SIRS) models are shown to be indistinguishable when a proportion of the number of infectives is measured.     

A two-stage algorithm for identification of nonlinear dynamic systems

This paper investigates the two-stage stepwise identification for a class of nonlinear dynamic systems that can be described by linear-in-the-parameters models, and the model has to be built from a very large pool of basis functions or model terms. The main objective is to improve the compactness of the model that is obtained by the forward stepwise methods, while retaining the computational efficiency. The proposed algorithm first generates an initial model using a forward stepwise procedure. The significance of each selected term is then reviewed at the second stage and all insignificant ones are replaced, resulting in an optimised compact model with significantly improved performance. The main contribution of this paper is that these two stages are performed within a well-defined regression context, leading to significantly reduced computational complexity. The efficiency of the algorithm is confirmed by the computational complexity analysis, and its effectiveness is demonstrated by the simulation results.     

Identification of smooth nonlinear dynamical systems with non-smooth steady-state features

This work discusses the identification of single-block smooth nonlinear discrete-time polynomial models with non-smooth steady-state features. Based on bifurcation theory, conditions are developed and used to determine some general aspects of the model structure and also to determine some constraints on the parameters required to guarantee the aforementioned features. The procedure uses only smooth functions of the regressors, a single possibly smooth input and some prior knowledge about the steady-state behavior. The non-smooth static function is here obtained by interchanging the stability of two sets of equilibria at the break-point, which corresponds to guaranteeing a transcritical bifurcation. This work discusses how to determine the domain over which the results are valid. The procedure is illustrated with simulated and experimental data.     

On state representations of time-varying nonlinear systems

This work considers a nonlinear time-varying system described by a state representation, with input u and state x. A given set of functions v, which is not necessarily the original input u of the system, is the (new) input candidate. The main result provides necessary and sufficient conditions for the existence of a local classical state space representation with input v. These conditions rely on integrability tests that are based on a derived flag. As a byproduct, one obtains a sufficient condition of differential flatness of nonlinear systems.     

State elimination and identifiability of the delay parameter for nonlinear time-delay systems

The identifiability of the delay parameter for nonlinear systems with a single constant time delay is analyzed. We show the existence of input–output equations and relate the identifiability of the delay parameter to their form. Explicit criteria based on rank calculations are formulated. The identifiability of the delay parameter is shown not to be directly related to the well-characterized identifiability/observability of the other system parameters/states.     

Linear stabilizability of planar nonlinear systems

This paper considers scalar-input two-dimensional nonlinear systems for which the linearization, has a simple zero uncontrollable eigenvalue. The existence of linear stabilizing feedback laws is investigated, using center manifold techniques. This work was partially supported by the Ministero della Pubblica Istruzione, Italia (Progetti di ricerca, di interesse nazionale).     

Parameter estimation in nonlinear systems with auto and crosscorrelated noise

The Gohberg-Heinig explicit formula for the inversion of a block-Toeplitz matrix is used to build an estimator of the inverse of the covariance matrix of a multivariable autoregressive process. This estimator is then conveniently applied to maximum likelihood parameter estimation in nonlinear dynamical systems with output measurements corrupted by additive auto and crosscorrelated noise. An appealing computational simplification is obtained due to the particular form taken by the Gohberg-Heinig formula. The efficiency of the obtained estimation scheme is illustrated via Monte-Carlo simulations and compared with an alternative that is obtained by extending a classical technique of linear system identification to the framework of this paper. These simulations show that the proposed method improves significantly the statistical properties of the estimator in comparison with classical methods. Finally, the ability of the method to provide, in a straightforward way, an accurate confidence region around the estimated parameters is also illustrated.     

An algorithm for bounded-error identification of nonlinear systems based on DC functions

This paper presents a guaranteed method for the parameter estimation of nonlinear models in a bounded-error context. This method is based on functions which consists of the difference of two convex functions, called DC functions. The method considers DC representations of the functional form of the dynamic system to obtain an outer bound of the set of parameters that are consistent with the measurements, the system and the considered bounded error. At each iteration, the proposed algorithm solves several convex optimization problems to discard from the initial search region subregions that are proved not consistent. This operation is repeated while the obtained solution is improved. Four examples are provided to clarify the proposed identification algorithm.     

Fast approximate identification of nonlinear systems

In this paper, a method is presented to extend the classical identification methods for linear systems towards nonlinear modelling of linear systems that suffer from nonlinear distortions. A well chosen, general nonlinear model structure is proposed that is identified in a two-step procedure. First, a best linear approximation is identified using the classical linear identification methods. In the second step, the nonlinear extensions are identified with a linear least-squares method. The proposed model not only includes Wiener and Hammerstein systems, it is also suitable to model nonlinear feedback systems. The stability of the nonlinear model can be easily verified. The method is illustrated on experimental data.     

Representation and identification of non-parametric nonlinear systems of short term memory and low degree of interaction

In this paper, we propose a new representation which is particularly useful for a class of non-parametric nonlinear systems that have short term memory and low degree of interaction. Advantages and disadvantages of this representation are discussed and compared to existing methods both theoretically and numerically. Furthermore, results regarding structural estimation based on the analysis of variance and on full scale identification are also provided.     

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.     

A bias-corrected estimator for nonlinear systems with output-error type model structures

Parametric identification of linear time-invariant (LTI) systems with output-error (OE) type of noise model structures has a well-established theoretical framework. Different algorithms, like instrumental-variables based approaches or prediction error methods (PEMs), have been proposed in the literature to compute a consistent parameter estimate for linear OE systems. Although the prediction error method provides a consistent parameter estimate also for nonlinear output-error (NOE) systems, it requires to compute the solution of a nonconvex optimization problem. Therefore, an accurate initialization of the numerical optimization algorithms is required, otherwise they may get stuck in a local minimum and, as a consequence, the computed estimate of the system might not be accurate. In this paper, we propose an approach to obtain, in a computationally efficient fashion, a consistent parameter estimate for output-error systems with polynomial nonlinearities. The performance of the method is demonstrated through a simulation example.