A neural network enhanced generalized minimum variance self-tuning proportional, integral and derivative control algorithm for complex dynamic systems

A neural network enhanced proportional, integral and derivative (PID) controller is presented that combines the attributes of neural network learning with a generalized minimum-variance self-tuning control (STC) strategy. The neuro PID controller is structured with plant model identification and PID parameter tuning. The plants to be controlled are approximated by an equivalent model composed of a simple linear submodel to approximate plant dynamics around operating points, plus an error agent to accommodate the errors induced by linear submodel inaccuracy due to non-linearities and other complexities. A generalized recursive least-squares algorithm is used to identify the linear submodel, and a layered neural network is used to detect the error agent in which the weights are updated on the basis of the error between the plant output and the output from the linear submodel. The procedure for controller design is based on the equivalent model, and therefore the error agent is naturally functioned within the control law. In this way the controller can deal not only with a wide range of linear dynamic plants but also with those complex plants characterized by severe non-linearity, uncertainties and non-minimum phase behaviours. Two simulation studies are provided to demonstrate the effectiveness of the controller design procedure.

Generalized neurofuzzy network modeling algorithms using Bezier-Bernstein polynomial functions and additive decomposition

This paper introduces a new neurofuzzy model construction algorithm for nonlinear dynamic systems based upon basis functions that are Bezier-Bernstein polynomial functions. This paper is generalized in that it copes with n-dimensional inputs by utilising an additive decomposition construction to overcome the curse of dimensionality associated with high n. This new construction algorithm also introduces univariate Bezier-Bernstein polynomial functions for the completeness of the generalized procedure. Like the B-spline expansion based neurofuzzy systems, Bezier-Bernstein polynomial function based neurofuzzy networks hold desirable properties such as nonnegativity of the basis functions, unity of support, and interpretability of basis function as fuzzy membership functions, moreover with the additional advantages of structural parsimony and Delaunay input space partition, essentially overcoming the curse of dimensionality associated with conventional fuzzy and RBF networks. This new modeling network is based on additive decomposition approach together with two separate basis function formation approaches for both univariate and bivariate Bezier-Bernstein polynomial functions used in model construction. The overall network weights are then learnt using conventional least squares methods. Numerical examples are included to demonstrate the effectiveness of this new data based modeling approach.

Generalized beta generated distributions

This article introduces generalized beta-generated (GBG) distributions. Sub-models include all classical beta-generated, Kumaraswamy-generated and exponentiated distributions. They are maximum entropy distributions under three intuitive conditions, which show that the classical beta generator skewness parameters only control tail entropy and an additional shape parameter is needed to add entropy to the centre of the parent distribution. This parameter controls skewness without necessarily differentiating tail weights. The GBG class also has tractable properties: we present various expansions for moments, generating function and quantiles. The model parameters are estimated by maximum likelihood and the usefulness of the new class is illustrated by means of some real data sets.

Response theory for equilibrium and non-equilibrium statistical mechanics: causality and generalized Kramers-Kronig relations

We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.

Numerical convergence of the Block-Maxima approach to the generalized extreme value distribution

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. (Phys. Rev. Lett. 97(21): 210602, 2006) have found analytical results.

On the stability radius of a generalized state-space system

The concept of “distance to instability” of a system matrix is generalized to system pencils which arise in descriptor (semistate) systems. Difficulties arise in the case of singular systems, because the pencil can be made unstable by an infinitesimal perturbation. It is necessary to measure the distance subject to restricted, or structured, perturbations. In this paper a suitable measure for the stability radius of a generalized state-space system is defined, and a computable expression for the distance to instability is derived for regular pencils of index less than or equal to one. For systems which are strongly controllable it is shown that this measure is related to the sensitivity of the poles of the system over all feedback matrices assigning the poles.

Robust control system design for generalized state-space systems

Robustness in multi-variable control system design requires that the solution to the design problem be insensitive to perturbations in the system data. In this paper we discuss measures of robustness for generalized state-space, or descriptor, systems and describe algorithmic techniques for optimizing robustness for various applications.

Symmetric stability of compressible zonal flows on a generalized equatorial β plane

Sufficient conditions are derived for the linear stability with respect to zonally symmetric perturbations of a steady zonal solution to the nonhydrostatic compressible Euler equations on an equatorial � plane, including a leading order representation of the Coriolis force terms due to the poleward component of the planetary rotation vector. A version of the energy–Casimir method of stability proof is applied: an invariant functional of the Euler equations linearized about the equilibrium zonal flow is found, and positive definiteness of the functional is shown to imply linear stability of the equilibrium. It is shown that an equilibrium is stable if the potential vorticity has the same sign as latitude and the Rayleigh centrifugal stability condition that absolute angular momentum increase toward the equator on surfaces of constant pressure is satisfied. The result generalizes earlier results for hydrostatic and incompressible systems and for systems that do not account for the nontraditional Coriolis force terms. The stability of particular equilibrium zonal velocity, entropy, and density fields is assessed. A notable case in which the effect of the nontraditional Coriolis force is decisive is the instability of an angular momentum profile that decreases away from the equator but is flatter than quadratic in latitude, despite its satisfying both the centrifugal and convective stability conditions.

A generalized collectively compact operator theory with an application to integral equations on unbounded domains

In this paper a generalization of collectively compact operator theory in Banach spaces is developed. A feature of the new theory is that the operators involved are no longer required to be compact in the norm topology. Instead it is required that the image of a bounded set under the operator family is sequentially compact in a weaker topology. As an application, the theory developed is used to establish solvability results for a class of systems of second kind integral equations on unbounded domains, this class including in particular systems of Wiener-Hopf integral equations with L1 convolutions kernels