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.

Power series generalized nonlinear models

We introduce in this paper a new class of discrete generalized nonlinear models to extend the binomial, Poisson and negative binomial models to cope with count data. This class of models includes some important models such as log-nonlinear models, logit, probit and negative binomial nonlinear models, generalized Poisson and generalized negative binomial regression models, among other models, which enables the fitting of a wide range of models to count data. We derive an iterative process for fitting these models by maximum likelihood and discuss inference on the parameters. The usefulness of the new class of models is illustrated with an application to a real data set. (C) 2008 Elsevier B.V. All rights reserved.

Generalized solutions of a nonlinear parabolic equation with generalized functions as initial data

In [H. Brezis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pure Appl. (9) (1983) 73-97.] Brezis and Friedman prove that certain nonlinear parabolic equations, with the delta-measure as initial data, have no solution. However in [J.F. Colombeau, M. Langlais, Generalized solutions of nonlinear parabolic equations with distributions as initial conditions, J. Math. Anal. Appl (1990) 186-196.] Colombeau and Langlais prove that these equations have a unique solution even if the delta-measure is substituted by any Colombeau generalized function of compact support. Here we generalize Colombeau and Langlais` result proving that we may take any generalized function as the initial data. Our approach relies on recent algebraic and topological developments of the theory of Colombeau generalized functions and results from [J. Aragona, Colombeau generalized functions on quasi-regular sets, Publ. Math. Debrecen (2006) 371-399.]. (C) 2009 Elsevier Ltd. All rights reserved.

Stabilizing controller design for uncertain nonlinear systems using fuzzy models

A Lyapunov-based stabilizing control design method for uncertain nonlinear dynamical systems using fuzzy models is proposed. The controller is constructed using a design model of the dynamical process to be controlled. The design model is obtained from the truth model using a fuzzy modeling approach. The truth model represents a detailed description of the process dynamics. The truth model is used in a simulation experiment to evaluate the performance of the controller design. A method for generating local models that constitute the design model is proposed. Sufficient conditions for stability and stabilizability of fuzzy models using fuzzy state-feedback controllers are given. The results obtained are illustrated with a numerical example involving a four-dimensional nonlinear model of a stick balancer.

Generalized neighbor-interaction models induced by nonlinear lattices

It is shown that the tight-binding approximation of the nonlinear Schrodinger equation with a periodic linear potential and periodic in space nonlinearity coefficient gives rise to a number of nonlinear lattices with complex, both linear and nonlinear, neighbor interactions. The obtained lattices present nonstandard possibilities, among which we mention a quasilinear regime, where the pulse dynamics obeys essentially the linear Schrodinger equation. We analyze the properties of such models both in connection to their modulational stability, as well as in regard to the existence and stability of their localized solitary wave solutions.

On the resolution of the generalized nonlinear complementarity problem

Minimization of a differentiable function subject to box constraints is proposed as a strategy to solve the generalized nonlinear complementarity problem (GNCP) defined on a polyhedral cone. It is not necessary to calculate projections that complicate and sometimes even disable the implementation of algorithms for solving these kinds of problems. Theoretical results that relate stationary points of the function that is minimized to the solutions of the GNCP are presented. Perturbations of the GNCP are also considered, and results are obtained related to the resolution of GNCPs with very general assumptions on the data. These theoretical results show that local methods for box-constrained optimization applied to the associated problem are efficient tools for solving the GNCP. Numerical experiments are presented that encourage the use of this approach.

Sensitivity, specificity and predictive values of linear and nonlinear indices of heart rate variability in stable angina patients

Background: Decreased heart rate variability (HRV) is related to higher morbidity and mortality. In this study we evaluated the linear and nonlinear indices of the HRV in stable angina patients submitted to coronary angiography. Methods. We studied 77 unselected patients for elective coronary angiography, which were divided into two groups: coronary artery disease (CAD) and non-CAD groups. For analysis of HRV indices, HRV was recorded beat by beat with the volunteers in the supine position for 40 minutes. We analyzed the linear indices in the time (SDNN [standard deviation of normal to normal], NN50 [total number of adjacent RR intervals with a difference of duration greater than 50ms] and RMSSD [root-mean square of differences]) and frequency domains ultra-low frequency (ULF) ≤ 0,003 Hz, very low frequency (VLF) 0,003 - 0,04 Hz, low frequency (LF) (0.04-0.15 Hz), and high frequency (HF) (0.15-0.40 Hz) as well as the ratio between LF and HF components (LF/HF). In relation to the nonlinear indices we evaluated SD1, SD2, SD1/SD2, approximate entropy (-ApEn), α1, α2, Lyapunov Exponent, Hurst Exponent, autocorrelation and dimension correlation. The definition of the cutoff point of the variables for predictive tests was obtained by the Receiver Operating Characteristic curve (ROC). The area under the ROC curve was calculated by the extended trapezoidal rule, assuming as relevant areas under the curve ≥ 0.650. Results: Coronary arterial disease patients presented reduced values of SDNN, RMSSD, NN50, HF, SD1, SD2 and -ApEn. HF ≤ 66 ms§ssup§2§esup§, RMSSD ≤ 23.9 ms, ApEn ≤-0.296 and NN50 ≤ 16 presented the best discriminatory power for the presence of significant coronary obstruction. Conclusion: We suggest the use of Heart Rate Variability Analysis in linear and nonlinear domains, for prognostic purposes in patients with stable angina pectoris, in view of their overall impairment. © 2012 Pivatelli et al.; licensee BioMed Central Ltd.

Sensitivity, specificity and predictive values of linear and nonlinear indices of heart rate variability in stable angina patients

Abstract Background Decreased heart rate variability (HRV) is related to higher morbidity and mortality. In this study we evaluated the linear and nonlinear indices of the HRV in stable angina patients submitted to coronary angiography. Methods We studied 77 unselected patients for elective coronary angiography, which were divided into two groups: coronary artery disease (CAD) and non-CAD groups. For analysis of HRV indices, HRV was recorded beat by beat with the volunteers in the supine position for 40 minutes. We analyzed the linear indices in the time (SDNN [standard deviation of normal to normal], NN50 [total number of adjacent RR intervals with a difference of duration greater than 50ms] and RMSSD [root-mean square of differences]) and frequency domains ultra-low frequency (ULF) ≤ 0,003 Hz, very low frequency (VLF) 0,003 – 0,04 Hz, low frequency (LF) (0.04–0.15 Hz), and high frequency (HF) (0.15–0.40 Hz) as well as the ratio between LF and HF components (LF/HF). In relation to the nonlinear indices we evaluated SD1, SD2, SD1/SD2, approximate entropy (−ApEn), α1, α2, Lyapunov Exponent, Hurst Exponent, autocorrelation and dimension correlation. The definition of the cutoff point of the variables for predictive tests was obtained by the Receiver Operating Characteristic curve (ROC). The area under the ROC curve was calculated by the extended trapezoidal rule, assuming as relevant areas under the curve ≥ 0.650. Results Coronary arterial disease patients presented reduced values of SDNN, RMSSD, NN50, HF, SD1, SD2 and -ApEn. HF ≤ 66 ms2, RMSSD ≤ 23.9 ms, ApEn ≤−0.296 and NN50 ≤ 16 presented the best discriminatory power for the presence of significant coronary obstruction. Conclusion We suggest the use of Heart Rate Variability Analysis in linear and nonlinear domains, for prognostic purposes in patients with stable angina pectoris, in view of their overall impairment.

LOCAL DIGITAL CONTROL OF POWER ELECTRONIC CONVERTERS IN A DC MICROGRID BASED ON A-PRIORI DERIVATION OF SWITCHING SURFACES

In power electronic basedmicrogrids, the computational requirements needed to implement an optimized online control strategy can be prohibitive. The work presented in this dissertation proposes a generalized method of derivation of geometric manifolds in a dc microgrid that is based on the a-priori computation of the optimal reactions and trajectories for classes of events in a dc microgrid. The proposed states are the stored energies in all the energy storage elements of the dc microgrid and power flowing into them. It is anticipated that calculating a large enough set of dissimilar transient scenarios will also span many scenarios not specifically used to develop the surface. These geometric manifolds will then be used as reference surfaces in any type of controller, such as a sliding mode hysteretic controller. The presence of switched power converters in microgrids involve different control actions for different system events. The control of the switch states of the converters is essential for steady state and transient operations. A digital memory look-up based controller that uses a hysteretic sliding mode control strategy is an effective technique to generate the proper switch states for the converters. An example dcmicrogrid with three dc-dc boost converters and resistive loads is considered for this work. The geometric manifolds are successfully generated for transient events, such as step changes in the loads and the sources. The surfaces corresponding to a specific case of step change in the loads are then used as reference surfaces in an EEPROM for experimentally validating the control strategy. The required switch states corresponding to this specific transient scenario are programmed in the EEPROM as a memory table. This controls the switching of the dc-dc boost converters and drives the system states to the reference manifold. In this work, it is shown that this strategy effectively controls the system for a transient condition such as step changes in the loads for the example case.

New methods for the estimation of Takagi-Sugeno model based extended Kalman filter and its applications to optimal control for nonlinear systems

This paper describes new approaches to improve the local and global approximation (matching) and modeling capability of Takagi–Sugeno (T-S) fuzzy model. The main aim is obtaining high function approximation accuracy and fast convergence. The main problem encountered is that T-S identification method cannot be applied when the membership functions are overlapped by pairs. This restricts the application of the T-S method because this type of membership function has been widely used during the last 2 decades in the stability, controller design of fuzzy systems and is popular in industrial control applications. The approach developed here can be considered as a generalized version of T-S identification method with optimized performance in approximating nonlinear functions. We propose a noniterative method through weighting of parameters approach and an iterative algorithm by applying the extended Kalman filter, based on the same idea of parameters’ weighting. We show that the Kalman filter is an effective tool in the identification of T-S fuzzy model. A fuzzy controller based linear quadratic regulator is proposed in order to show the effectiveness of the estimation method developed here in control applications. An illustrative example of an inverted pendulum is chosen to evaluate the robustness and remarkable performance of the proposed method locally and globally in comparison with the original T-S model. Simulation results indicate the potential, simplicity, and generality of the algorithm. An illustrative example is chosen to evaluate the robustness. In this paper, we prove that these algorithms converge very fast, thereby making them very practical to use.

A new approach to fuzzy estimation of Takagi-Sugeno model and its applications to optimal control for nonlinear systems

An efficient approach is presented to improve the local and global approximation and modelling capability of Takagi-Sugeno (T-S) fuzzy model. The main aim is obtaining high function approximation accuracy. The main problem is that T-S identification method cannot be applied when the membership functions are overlapped by pairs. This restricts the use of the T-S method because this type of membership function has been widely used during the last two decades in the stability, controller design and are popular in industrial control applications. The approach developed here can be considered as a generalized version of T-S method with optimized performance in approximating nonlinear functions. A simple approach with few computational effort, based on the well known parameters' weighting method is suggested for tuning T-S parameters to improve the choice of the performance index and minimize it. A global fuzzy controller (FC) based Linear Quadratic Regulator (LQR) is proposed in order to show the effectiveness of the estimation method developed here in control applications. Illustrative examples of an inverted pendulum and Van der Pol system are chosen to evaluate the robustness and remarkable performance of the proposed method and the high accuracy obtained in approximating nonlinear and unstable systems locally and globally in comparison with the original T-S model. Simulation results indicate the potential, simplicity and generality of the algorithm.

Variable structure control with chattering elimination and guaranteed stability for a generalized T-S model

In this paper, a fuzzy based Variable Structure Control (VSC) with guaranteed stability is presented. The main objective is to obtain an improved performance of highly non-linear unstable systems. The main contribution of this work is that, firstly, new functions for chattering reduction and error convergence without sacrificing invariant properties are proposed, which is considered the main drawback of the VSC control. Secondly, the global stability of the controlled system is guaranteed.The well known weighting parameters approach, is used in this paper to optimize local and global approximation and modeling capability of T-S fuzzy model.A one link robot is chosen as a nonlinear unstable system to evaluate the robustness, effectiveness and remarkable performance of optimization approach and the high accuracy obtained in approximating nonlinear systems in comparison with the original T-S model. Simulation results indicate the potential and generality of the algorithm. The application of the proposed FLC-VSC shows that both alleviation of chattering and robust performance are achieved with the proposed FLC-VSC controller. The effectiveness of the proposed controller is proven in front of disturbances and noise effects.

Optimal Control Using Feedback Linearization for a Generalized T-S Model

In this paper, a fuzzy feedback linearization is used to control nonlinear systems described by Takagi-Suengo (T-S) fuzzy systems. In this work, an optimal controller is designed using the linear quadratic regulator (LQR). The well known weighting parameters approach is applied to optimize local and global approximation and modelling capability of T-S fuzzy model to improve the choice of the performance index and minimize it. The approach used here can be considered as a generalized version of T-S method. Simulation results indicate the potential, simplicity and generality of the estimation method and the robustness of the proposed optimal LQR algorithm.

A Generalized Fuzzy Linguistic Model for Predicting Component Concentrations in an Optical Gas Sensing System

Motivated by environmental protection concerns, monitoring the flue gas of thermal power plant is now often mandatory due to the need to ensure that emission levels stay within safe limits. Optical based gas sensing systems are increasingly employed for this purpose, with regression techniques used to relate gas optical absorption spectra to the concentrations of specific gas components of interest (NOx, SO2 etc.). Accurately predicting gas concentrations from absorption spectra remains a challenging problem due to the presence of nonlinearities in the relationships and the high-dimensional and correlated nature of the spectral data. This article proposes a generalized fuzzy linguistic model (GFLM) to address this challenge. The GFLM is made up of a series of “If-Then” fuzzy rules. The absorption spectra are input variables in the rule antecedent. The rule consequent is a general nonlinear polynomial function of the absorption spectra. Model parameters are estimated using least squares and gradient descent optimization algorithms. The performance of GFLM is compared with other traditional prediction models, such as partial least squares, support vector machines, multilayer perceptron neural networks and radial basis function networks, for two real flue gas spectral datasets: one from a coal-fired power plant and one from a gas-fired power plant. The experimental results show that the generalized fuzzy linguistic model has good predictive ability, and is competitive with alternative approaches, while having the added advantage of providing an interpretable model.