This paper provides an overview of system identification using orthonormal basis function models, such as those based on Laguerre, Kautz, and generalised orthonormal basis functions. The paper is separated in two parts. In this first part, the mathematical foundations of these models as well as their advantages and limitations are discussed within the context of linear and robust system identification. The second part approaches the issues related with non-linear models. The discussions comprise a broad bibliographical survey of the subjects involving linear models within the orthonormal basis functions framework. Theoretical and practical issues regarding the identification of these models are presented and illustrated by means of a case study involving a polymerisation process.

This paper provides an overview of system identification using orthonormal basis function models, such as those based on Laguerre, Kautz, and generalised orthonormal basis functions. The paper is separated in two parts. The first part of the paper approached issues related with linear models and models with uncertain parameters. Now, the mathematical foundations as well as their advantages and limitations are discussed within the contexts of non-linear system identification. The discussions comprise a broad bibliographical survey of the subject and a comparative analysis involving some specific model realisations, namely, Volterra, fuzzy, and neural models within the orthonormal basis functions framework. Theoretical and practical issues regarding the identification of these non-linear models are presented and illustrated by means of two case studies.

An approach to obtain Takagi-Sugeno (TS) fuzzy models of nonlinear dynamic systems using the framework of orthonormal basis functions (OBFs) is presented in this paper. This approach is based on an architecture in which local linear models with ladder-structured generalized OBFs (GOBFs) constitute the fuzzy rule consequents and the outputs of the corresponding GOBF filters are input variables for the rule antecedents. The resulting GOBF-TS model is characterized by having only real-valued parameters that do not depend on any user specification about particular types of functions to be used in the orthonormal basis. The fuzzy rules of the model are initially obtained by means of a well-known technique based on fuzzy clustering and least squares. Those rules are then simplified, and the model parameters (GOBF poles, GOBF expansion coefficients, and fuzzy membership functions) are subsequently adjusted by using a nonlinear optimization algorithm. The exact gradients of an error functional with respect to the parameters to be optimized are computed analytically. Those gradients provide exact search directions for the optimization process, which relies solely on input-output data measured from the system to be modeled. An example is presented to illustrate the performance of this approach in the modeling of a complex nonlinear dynamic system.

An improved approach to determine exact search directions for the optimization of Volterra models based on Generalized Orthonormal Bases of Functions (GOBF) is proposed. The proposed approach extends the work in [7], where a novel, exact technique for optimizing the GOBF parameters (poles) for Volterra models of any order was presented. The proposed extensions take place in two different ways: (i) the formulation here is derived in such a way that each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases (rather than a single, common basis), each of which is parameterized by an individual set of poles intended for representing the dominant dynamic of the kernel along a particular dimension; and (ii) a novel, more computationally efficient method to analytically and recursively calculate the search directions (gradients) for the bases poles is derived. A simulated example is presented to illustrate the performance of the proposed approach. A comparison between the proposed method, which uses asymmetric kernels with multiple orthonormal bases, and the original method, which uses symmetric kernels with a single basis, is presented.

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