Nonlinear Input-output Systems Research Articles

Entropy of Generating Series for Nonlinear Input-Output Systems and Their Interconnections

This paper has two main objectives. The first is to introduce a notion of entropy that is well suited for the analysis of nonlinear input-output systems that have a Chen-Fliess series representation. The latter is defined in terms of its generating series over a noncommutative alphabet. The idea is to assign an entropy to a generating series as an element of a graded vector space. The second objective is to describe the entropy of generating series originating from interconnected systems of Chen-Fliess series that arise in the context of control theory. It is shown that one set of interconnections can never increase entropy as defined here, while a second set has the potential to do so. The paper concludes with a brief introduction to an entropy ultrametric space and some open questions.

State space construction for continuous time transfer function matrices via Nerode equivalence

AbstractIn this paper, we show that a minimal state space realization in Jordan canonical form for linear multivariable continuous‐time systems described by rational transfer function matrices could be obtained in a natural and basic way by using the concept of Nerode equivalence. While the minimal state space realization is known, the contribution of this note is to provide an alternative realization procedure which is directly introduced in a simple and self‐contained manner. Both scalar and multivariable cases in the continuous‐time setting are discussed. The presentation also provides a concrete construction of rational vector function with preassigned poles to interpolate any 2‐norm bounded analytical vector function in the open left‐half of the complex plane. The basic idea of Nerode equivalence is that the state can be identified with a corresponding equivalence class of inputs. For a linear finite dimensional time‐invariant continuous‐time system, the zero state is identified with the kernel of certain Hankel operator. This characterization via Nerode equivalence class sheds light on the construction of state for general nonlinear input–output systems.

USE OF NEURAL NETWORKS TO MAKE COMPLEX DECISIONS TO OPERATE ELECTRIC TRANSPORT UNDER UNCERTAINTY

The use of neural networks to solve the problems of insolubility and the solution of complex computational equations becomes a common practice in academic circles and industry. It has been shown that, despite the complexity, these problems can be formulated as a set of equations, and the key is to find zeros from them. Zero Neural Networks (ZNNs), as a class of neural networks specially designed to find zeros of equations, have played an indispensable role in online decision-changing problems over time in recent years, and many fruitful research results have been documented in literature. The purpose of this article is to provide a comprehensive overview of ZNN studies, including ZNN continuous time and discrete time models for solving various problems, and their application in motion planning and superfluous manipulator management, chaotic system tracking, or even population control in mathematical biological sciences. Considering the fact that real-time performance is in demand for time-varying problems in practice, analysis of the stability and convergence of various ZNN models with continuous time is considered in a unified form in detail. In the case of solving the problems of discrete time, procedures are summarized for how to discriminate a continuous ZNN model and methods for obtaining an accuracy decision. Approaches based on the neural network to address various nodal tasks have attracted considerable attention in many areas. For example, an adaptive fuzzy controller based on a neural network is constructed for a class of nonlinear systems with discrete time with a dead zone with discrete time in. An applied decentralized circuit, based on a neural network, is presented for multiple nonlinear input and multiple output systems (MIMO) using the methods of the reverse step in. Such a scheme guarantees a uniform limiting limit of all signals in a closed system relative to the average square. In order to overcome the structural complexity of the nonlinear feedback structure, uses the method of dividing variables for the decomposition of unknown functions of all state variables into the sum of smooth functions of each dynamic error.

Nonlinear system identification for multivariable control via discrete-time Chen–Fliess series

The nonlinear system identification problem is solved for a multivariable nonlinear input–output system that can be represented in terms of a Chen–Fliess functional expansion. The problem is formulated in terms of a regression which is linear in the parameters and has a nonlinear regressor. An inductive implementation of the nonlinear regressor is developed using the underlying noncommutative algebraic and combinatorial structures. The method is demonstrated in an adaptive control application involving a two-input, two-output Lotka–Volterra dynamical system.

Generalized Multiscale RBF Networks and the DCT for Breast Cancer Detection

The use of the multiscale generalized radial basis function (MSRBF) neural networks for image feature extraction and medical image analysis and classification is proposed for the first time in this work. The MSRBF networks hold a simple and flexible architecture that has been successfully used in forecasting and model structure detection of input-output nonlinear systems. In this work instead, MSRBF networks are part of an integrated computer-aided diagnosis (CAD) framework for breast cancer detection, which holds three stages: an input-output model is obtained from the image, followed by a high-level image feature extraction from the model and a classification module aimed at predicting breast cancer. In the first stage, the image data is rendered into a multiple-input-single-output (MISO) system. In order to improve the characterisation, the nonlinear autoregressive with exogenous inputs (NARX) model is introduced to rearrange the available input-output data in a nonlinear way. The forward regression orthogonal least squares (FROLS) algorithm is then used to take advantage of the previous arrangement by solving the system as a model structure detection problem and finding the output layer weights of the NARX-MSRBF network. In the second stage, once the network model is available, the feature extraction takes place by stimulating the input to produce output signals to be compressed by the discrete cosine transform (DCT). In the third stage, we leverage the extracted features by using a clustering algorithm for classification to integrate a CAD system for breast cancer detection. To test the method performance, three different and well-known public image repositories were used: the mini-MIAS and the MMSD for mammography, and the BreaKHis for histopathology images. A comparison exercise was also made between different database partitions to understand the mammogram breast density effect in the performance since there are few remarks in the literature on this factor. Classification results show that the new CAD method reached an accuracy of 93.5% in mini-Mammo graphic image analysis society (mini-MIAS), 93.99% in digital database for screening mammography (DDSM) and 86.7% in the BreaKHis. We found that the MSRBF networks are able to build tailored and precise image models and, combined with the DCT, to extract high-quality features from both black and white and coloured images.

Non‐linear model‐order reduction based on tensor decomposition and matrix product

In this study, based on tensor decomposition and matrix product, the authors investigate two model-order reduction (MOR) methods for the quadratic-bilinear (QB) systems which are equivalently transformed from the non-linear input–output systems. Since the quadratic term coefficient of the QB system can be considered as the matricisation of a tensor, they propose two computationally efficient ways to obtain the reduced system by using tensor calculus. First, the Tucker decomposition of tensors is used to deal with the quadratic term coefficient of the QB system. The transformational matrix is constructed by applying the analysis of matrix product. Then, they get the reduced QB system which can match the first several expansion coefficients of the original output. Besides, they propose another MOR method based on the CANDECOMP/PARAFAC decomposition. These two methods can avoid large computational complexity in the process of computing the reduced system. Moreover, the error estimation and stability of the authors' MOR methods are discussed. The efficiency of their MOR methods is illustrated by two numerical examples and they show their competitiveness when compared to the proper orthogonal decomposition method.

Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping

<p style='text-indent:20px;'>We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

Supervised-Learning-Aided Communication Framework for MIMO Systems With Low-Resolution ADCs

This paper considers a multiple-input-multiple-output (MIMO) system with low-resolution analog-to-digital converters (ADCs). In this system, we propose a novel communication framework that is inspired by supervised learning. The key idea of the proposed framework is to learn the non-linear input-output system, formed by the concatenation of a wireless channel and a quantization function used at the ADCs, for data detection. In this framework, a conventional channel estimation process is replaced by a system learning process, in which the conditional probability mass functions (PMFs) of the nonlinear system are empirically learned by sending the repetitions of all possible data signals as pilot signals. Then the subsequent data detection process is performed based on the empirical conditional PMFs obtained during the system learning. To reduce both the training overhead and the detection complexity, we also develop a supervised-learning-aided successive-interference-cancellation method. In this method, a data signal vector is divided into two subvectors with reduced dimensions. Then these two subvectors are successively detected based on the conditional PMFs that are learned using artificial noise signals and an estimated channel. For the case of one-bit ADCs, we derive an analytical expression for vector-error-rate of the proposed framework under perfect channel knowledge at the receiver. Simulations demonstrate the detection error reduction of the proposed framework compared to conventional detection techniques that are based on channel estimation.

Discrete-time approximations of Fliess operators

A convenient way to represent a nonlinear input-output system in control theory is via a Chen-Fliess functional expansion or Fliess operator. The general goal of this paper is to describe how to approximate Fliess operators with iterated sums and to provide accurate error estimates for two different scenarios, one where the series coefficients are growing at a local convergence rate, and the other where they are growing at a global convergence rate. In each case, it is shown that the error estimates are achievable in the sense that worst case inputs can be identified which hit the error bound. The paper then focuses on the special case where the operators are rational, i.e., they have rational generating series. It is shown in this situation that the iterated sum approximation can be realized by a discrete-time state space model which is a rational function of the input and state affine. In addition, this model comes from a specific discretization of the bilinear realization of the rational Fliess operator.

Dendriform–Tree Setting for Fliess Operators

This paper provides a dendriform-tree setting for Fliess operators with matrix-valued inputs. This class of analytic nonlinear input-output systems is convenient, for example, in quantum control. In particular, a description of such Fliess operators is provided using planar binary trees. Sufficient conditions for convergence of the defining series are also given.

Nonlinear model order reduction based on tensor Kronecker product expansion with Arnoldi process

In this paper, we present a model order reduction (MOR) method for large nonlinear input–output systems based on tensor Kronecker product expansion with Arnoldi process. We first approximate the nonlinear system with a quadratic form at single-point expansion, and then use a tensor Kronecker product analysis to it. Constructing the projection matrix through solving a linear equation, a reduced quadratic system is produced, which can match the first several expansion coefficients of the original output. What is more, it can preserve the stability and passivity under some certain conditions as well. The error estimation is also well discussed. Finally, the robust behavior of our MOR method is successfully illustrated via two numerical examples.

A methodology for equivalent modeling of distribution system based on nonlinear model reduction

This paper presents a new method to construct the low-order equivalent model for distribution system, which is very essential in large power system simulations. The distribution system connected to a substation is seen from the transmission power system as a nonlinear input–output system. In order to cope with the computation complexity, nonlinear model reduction technique based on covariance matrix is employed to obtain a new low-order dynamic system, which can closely represent the dynamic characteristics of the detailed system. The absence of drastic simplifications, direct consideration of the distribution network and the proper preservation of the input–output behavior of the detailed distribution system allow the equivalent model to attain good accuracy. Simulations and analysis are carried out on WSCC system to verify the validity of the proposed method. The simulation results show that this new method could attain high accuracy of simulation of distribution system under various operating conditions, thus improving the precision of power system stability analysis.

Perfect adaptation of general nonlinear systems

Perfect adaptation describes the ability of a biological system to restore its biological function precisely to the pre-perturbation level after being affected by the environmental disturbances. Mathematically, a biological system with perfect adaptation can be modelled as an input-output nonlinear system whose output, usually determining the biological function, is asymptotically stable under all input disturbances concerned. In this paper, a quite general input-output mathematical model is employed and the ‘functional’ of biological function (FBF) - output Lyapunov function - is explored to investigate its perfect adaptation ability. Sufficient condition is established for the systems with FBF to achieve perfect adaptation. Then a sufficient and necessary condition is obtained for the linear systems to possess an output Lyapunov function. Furthermore, it is shown that the ‘functional’ of receptors activity exists in the perfect adaptation model of E. coli chemotaxis. Different with the existing mathematical surveys on perfect adaptation, most of which are based on the standpoint of control theory, we first investigate this problem using ways of nonlinear systems analysis.

A robust nonlinear observer-based approach for distributed fault detection of input–output interconnected systems

This paper develops a nonlinear observer-based approach for distributed fault detection of a class of interconnected input–output nonlinear systems, which is robust to modeling uncertainty and measurement noise. First, a nonlinear observer design is used to generate the residual signals required for fault detection. Then, a distributed fault detection scheme and the corresponding adaptive thresholds are designed based on the observer characteristics and, at the same time, filtering is used in order to attenuate the effect of measurement noise, which facilitates less conservative thresholds and enhanced robustness. Finally, a fault detectability condition characterizing quantitatively the class of detectable faults is derived.

System identification of point-process neural systems using probability based Volterra kernels.

Neural information processing involves a series of nonlinear dynamical input/output transformations between the spike trains of neurons/neuronal ensembles. Understanding and quantifying these transformations is critical both for understanding neural physiology such as short-term potentiation and for developing cognitive neural prosthetics. A novel method for estimating Volterra kernels for systems with point-process inputs and outputs is developed based on elementary probability theory. These Probability Based Volterra (PBV) kernels essentially describe the probability of an output spike given q input spikes at various lags t1, t2, …, tq. The PBV kernels are used to estimate both synthetic systems where ground truth is available and data from the CA3 and CA1 regions rodent hippocampus. The PBV kernels give excellent predictive results in both cases. Furthermore, they are shown to be quite robust to noise and to have good convergence and overfitting properties. Through a slight modification, the PBV kernels are shown to also deal well with correlated point-process inputs. The PBV kernels were compared with kernels estimated through least squares estimation (LSE) and through the Laguerre expansion technique (LET). The LSE kernels were shown to fair very poorly with real data due to the large amount of input noise. Although the LET kernels gave the best predictive results in all cases, they require prior parameter estimation. It was shown how the PBV and LET methods can be combined synergistically to maximize performance. The PBV kernels provide a novel and intuitive method of characterizing point-process input-output nonlinear systems.

Faà di Bruno Hopf algebra of the output feedback group for multivariable Fliess operators

Given two nonlinear input–output systems written in terms of Chen–Fliess functional expansions, i.e., Fliess operators, it is known that the feedback interconnected system is always well defined and in the same class. An explicit formula for the generating series of a single-input, single-output closed-loop system was provided by the first two authors in earlier work via Hopf algebra methods. This paper is a sequel. It has four main innovations. First, the full multivariable extension of the theory is presented. Next, a major simplification of the basic setup is introduced using a new type of grading that has recently appeared in the literature. This grading also facilitates a fully recursive algorithm to compute the antipode of the Hopf algebra of the output feedback group, and thus, the corresponding feedback product can be computed much more efficiently. The final innovation is an improved convergence analysis of the antipode operation, namely, the radius of convergence of the antipode is computed.

Functional Series Expansions for Nonlinear Input-Output Systems with Delay

A new type of Chen-Fliess series is first introduced which depends on the input and delayed versions of the input. It is then shown how a class of analytic differential delay systems with a single delay and constant initial conditions has input-output maps representable in terms of this new functional series. As in the classical case, the coefficients can be computed by iterated Lie derivatives, but here the method is applied to an infinite dimensional embedding of the original state space system. Finally, the more technical issue of series convergence is addressed. Sufficient conditions are produced to guarantee convergence in both a local and global sense.

Modeling of Nonlinear Dynamic Systems with Volterra Polynomials

The paper presents a review of the studies that were conducted at Energy Systems Institute (ESI) SB RAS in the field of mathematical modeling of nonlinear input-output dynamic systems with Volterra polynomials. The first part presents an original approach to identification of the Volterra kernels. The approach is based on setting special multi-parameter families of piecewise constant test input signals. It also includes a description of the respective software; presents illustrative calculations on the example of a reference dynamic system as well as results of computer modeling of real heat exchange processes. The second part of the review is devoted to the Volterra polynomial equations of the first kind. Studies of such equations were pioneered and have been carried out in the past decade by the laboratory of ill-posed problems at ESI SB RAS. A special focus in the paper is made on the importance of the Lambert function for the theory of these equations.

A unified approach to generating series for mixed cascades of analytic nonlinear input–output systems

The primary goal is to describe in a unified fashion the generating series for the cascade connection of any two analytic nonlinear input–output systems. In particular, it will be shown that a single general definition of a composition product can be formulated in terms of formal power series to describe any possible cascade connection of analytic integral operators (Fliess operators) and analytic functions provided that the composite system is well defined and resides in one of these two classes. In each case, the radius of convergence of the interconnection is computed.

On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems

A complete analysis is presented of the radii of convergence of the parallel, product, cascade and feedback interconnections of analytic nonlinear input-output systems represented as Fliess operators. Such operators are described by convergent functional series, which are indexed by words over a noncommutative alphabet. Their generating series are therefore specified in terms of noncommutative formal power series. Given growth conditions for the coefficients of the generating series for the subsystems, the radius of convergence of each interconnected system is computed assuming the subsystems are either all locally convergent or all globally convergent. In the process of deriving the radius of convergence for the feedback connection, it is shown definitively that local convergence is preserved under feedback. This had been an open problem in the literature until recently.