Wagner Caradori do Amaral

Brief Optimal expansions of discrete-time Volterra models using Laguerre functions

This work is concerned with the optimization of Laguerre bases for the orthonormal series expansion of discrete-time Volterra models. The aim is to minimize the number of Laguerre functions associated with a given series truncation error, thus reducing the complexity of the resulting finite-dimensional representation. Fu and Dumont (IEEE Trans. Automatic Control 38(6) (1993) 934) indirectly approached this problem in the context of linear systems by minimizing an upper bound for the error resulting from the truncated Laguerre expansion of impulse response models, which are equivalent to first-order Volterra models. A generalization of the work mentioned above focusing on Volterra models of any order is presented in this paper. The main result is the derivation of analytic strict global solutions for the optimal expansion of the Volterra kernels either using an independent Laguerre basis for each kernel or using a common basis for all the kernels.

Brief Constrained robust predictive controller for uncertain processes modeled by orthonormal series functions

The present work focuses on robust predictive control (RPC) of uncertain processes and proposes a new approach based on orthonormal series function modeling. In such unstructured modeling, the output signal is described as a weighted sum of orthonormal functions that uses approximative information about the time constant of the process. Due to an efficient uncertainty representation, this kind of modeling is advantageous in the RPC context, even for constrained systems and processes with integral action. The stability of the closed-loop system is guaranteed by the setting of sufficient conditions for the selection of the controller prediction horizon. Simulation results are presented to illustrate the performance of this new RPC algorithm.

Brief paper: Choice of free parameters in expansions of discrete-time Volterra models using Kautz functions

This work tackles the problem of modeling nonlinear systems using Volterra models based on Kautz functions. The drawback of requiring a large number of parameters in the representation of these models can be circumvented by describing every kernel using an orthonormal basis of functions, such as the Kautz basis. The resulting model, so-called Wiener/Volterra model, can be truncated into a few terms if the Kautz functions are properly designed. The underlying problem is how to select the free-design complex poles that fully parameterize these functions. A solution to this problem has been provided in the literature for linear systems, which can be represented by first-order Volterra models. A generalization of such strategy focusing on Volterra models of any order is presented in this paper. This problem is solved by minimizing an upper bound for the error resulting from the truncation of the kernel expansion into a finite number of functions. The aim is to minimize the number of two-parameter Kautz functions associated with a given series truncation error, thus reducing the complexity of the resulting finite-dimensional representation. The main result is the derivation of an analytical solution for a sub-optimal expansion of the Volterra kernels using a set of Kautz functions.

Fuzzy models within orthonormal basis function framework

Presents a framework for fuzzy modeling of dynamic systems using orthonormal basis functions in the representation of the model input signals. The main objective of using orthonormal bases is to overcome the task of estimating the order and time delay of the process. The result is a nonlinear moving average fuzzy model which, consequently, has no feedback of prediction errors. Although any technique of fuzzy modeling can be used in the proposed framework, a relational approach is considered. The performance of fuzzy models with orthonormal basis functions is illustrated by examples and the results are compared with those provided by conventional fuzzy models and Volterra models.

Technical communique: A note on the optimal expansion of Volterra models using Laguerre functions

This work tackles the problem of expanding Volterra models using Laguerre functions. A strict global optimal solution is derived when each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases, each of which parameterized by an individual Laguerre pole intended for representing the dominant dynamic of the kernel along a particular dimension. It is proved that the solution derived minimizes the upper bound of the squared norm of the error resulting from the practical truncation of the Laguerre series expansion into a finite number of functions. This is an extension of the results in Campello, Favier and Amaral [(2004). Optimal expansions of discrete-time Volterra models using Laguerre functions. Automatica, 40, 815-822.], where an optimal solution was obtained for the usual yet particular case in which a single Laguerre pole is used for expanding a given kernel along all its dimensions. It is also proved that the particular and extended solutions are equivalent to each other when the Volterra kernels are symmetric.

Optimal Expansions of Discrete-Time Volterra Models Using Laguerre Functions

Abstract This paper is concerned with the optimization of Laguerre bases for the orthonormal series expansion of discrete-time Volterra models. Fu and Dumont (1993) approached this problem in the context of linear systems by minimizing an upper bound for the error resulting from the truncated Laguerre expansion of impulse response models, which are equivalent to first-order Volterra models. The present work generalizes the work mentioned above to Volterra models of any order. The main result is the derivation of analytic strict global solutions for the optimal expansion of the Volterra kernels either using an independent Laguerre basis for each kernel or using a common basis for all the kernels.

Modeling and linguistic knowledge extraction from systems using fuzzy relational models

Fuzzy relational models have been widely investigated and found to be an efficient tool for the identification of complex systems. However, little attention has been given to the linguistic interpretation of these models. The use of relational models is recommended since their development follows a natural sequence based on the original ideas about fuzzy sets and fuzzy logic, involving the estimation of the relations existing between linguistic terms which have previously been defined by the user. In the present paper the problem of extracting linguistic knowledge from systems by using relational models is addressed. A new algorithm for the identification of these models which can provide analytical or numerical solutions depending on user requirements is also proposed. Examples are presented showing that both quantitative and qualitative modeling can be effectively achieved by combining the proposed methodologies for identification and extraction of linguistic knowledge from systems.

Exact Search Directions for Optimization of Linear and Nonlinear Models Based on Generalized Orthonormal Functions

A novel technique for selecting the poles of orthonormal basis functions (OBF) in Volterra models of any order is presented. It is well-known that the usual large number of parameters required to describe the Volterra kernels can be significantly reduced by representing each kernel using an appropriate basis of orthonormal functions. Such a representation results in the so-called OBF Volterra model, which has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a nonlinear static mapping given by the Volterra polynomial series. Aiming at optimizing the poles that fully parameterize the orthonormal bases, the exact gradients of the outputs of the orthonormal filters with respect to their poles are computed analytically by using a back-propagation-through-time technique. The expressions relative to the Kautz basis and to generalized orthonormal bases of functions (GOBF) are addressed; the ones related to the Laguerre basis follow straightforwardly as a particular case. The main innovation here is that the dynamic nature of the OBF filters is fully considered in the gradient computations. These gradients provide exact search directions for optimizing the poles of a given orthonormal basis. Such search directions can, in turn, be used as part of an optimization procedure to locate the minimum of a cost-function that takes into account the error of estimation of the system output. The Levenberg-Marquardt algorithm is adopted here as the optimization procedure. Unlike previous related work, the proposed approach relies solely on input-output data measured from the system to be modeled, i.e., no information about the Volterra kernels is required. Examples are presented to illustrate the application of this approach to the modeling of dynamic systems, including a real magnetic levitation system with nonlinear oscillatory behavior.

Hierarchical fuzzy models within the framework of orthonormal basis functions and their application to bioprocess control

Fuzzy models within the framework of orthonormal basis functions (OBF fuzzy models) have been introduced in previous works and shown to be a very promising approach to the areas of nonlinear system identification and control, since they exhibit several advantages over those dynamic model topologies usually adopted in the literature. As fuzzy models, however, they exhibit the dimensionality problem which is the main drawback to the application of neural networks and fuzzy systems to the modeling and control of large-scale systems. This problem has successfully been dealt with in the literature by means of hierarchical structures composed of submodels connected in cascade. In the present paper a hierarchical fuzzy model within the OBF framework is presented. A data-driven hybrid identification method based on genetic and gradient-based algorithms is described in details. A model-based predictive control scheme is also presented and applied to control of a complex industrial process for ethyl alcohol (ethanol) production.

Robust estimation of parametric membership regions

This paper is concerned with the robust identification of linear models when modelling error is bounded. A modified Hopfields neural network is used to calculate a membership set for the model parameters, with the internal parameters of the network obtained using the valid-subspace technique. These parameters can be explicitly computed to guarantee the network convergence. A solution for the robust estimation problem with an unknown-but-bounded error corresponds to an equilibrium point of the network. A comparative analysis with alternative robust estimation methods is provided to illustrate the proposed approach.