Discrete-time Nonlinear Dynamical Systems Research Articles

Robust analysis for data-driven model predictive control

Here the idea of data driven is introduced in model predictive control to establish our proposed data-driven model predictive control. Considering one first-order discrete time nonlinear dynamical system, the main essence of data driven means the actual output value in cost function for model predictive control is identified through input–output observed data in case of unknown but bounded noise and martingale difference sequence. After substituting the identified actual output in cost function, the total cost function in model predictive control is reformulated as its standard form, i.e. one quadratic program problem with input and output constraints. Then semidefinite relaxation scheme is used to derive a lower bound for its optimal value, and the robust counterpart of an uncertain quadratic program is reduced to one conic quadratic problem. The above semidefinite relaxation scheme and conic quadratic problem correspond to the similar robust analysis based on convex optimization theory. Finally, one simulation example is used to prove the efficiency of our proposed theory.

Tractable higher-order under-approximating AE extensions for non-linear systems

We consider the problem of under and over-approximating the image of general vector-valued functions over bounded sets, and apply the proposed solution to the estimation of reachable sets of uncertain non-linear discrete-time dynamical systems. Such a combination of under and over-approximations is very valuable for the verification of properties of controlled systems. Over-approximations prove properties correct, while under-approximations can be used for falsification. Coupled, they provide a measure of the conservatism of the analysis. This work introduces a general framework relying on approximations of robust ranges of vector-valued functions, formulated as AE extensions, that can be interpreted as quantified propositions where universal quantifiers (A) precede existential quantifiers (E). This framework allows us to extend for under-approximation many precision refinements that are classically used for over-approximations, such as affine approximations, Taylor models, quadrature formulae and preconditioning methods. We end by evaluating the efficiency and precision of our approach, focusing on the application to the analysis of discrete-time dynamical systems.

A Switching Control Scheme With Increment Estimate of Unmodeled Dynamics

This article presents a new switching control scheme for controlling a class of nonlinear discrete-time dynamical systems. The key idea behind the proposed control techniques lies in the decomposition of unmodeled dynamics, that is, the unmodelled dynamics are decomposed as a sum of a known function depending on the data from the posterior unmodeled dynamics measurement and an unknown increment. The control algorithm is based on a novel estimation algorithm for the increment of unmodeled dynamics, which contributes two nonlinear controllers. The theoretical results on both convergence and stability of the closed-loop system are given. The system performance is evaluated by some simulation results.

The Problem of Invariance in Nonlinear Discrete-Time Dynamic Systems

The paper considers the problem of invariance with respect to the unknown input for discrete-time nonlinear dynamic systems. To solve the problem, the algebraic approaches, called algebra of functions and logic–dynamic approach, are used. Such approaches assume that description of the system may contain non-differentiable functions. Necessary and sufficient conditions of solvability the problem are obtained. Moreover, procedures which find the appropriate functions and matrices are developed. Some applications of such invariance in fault detection and isolation, disturbance decoupling problem, and fault-tolerant control are considered.

Self-Evolving Neural Control for a Class of Nonlinear Discrete-Time Dynamic Systems With Unknown Dynamics and Unknown Disturbances

In this article, a novel self-evolving general regression neural network (SEGRNN) is designed for tracking control of a class of discrete-time dynamic systems with unknown dynamics and unknown external disturbances. The proposed controller starts from scratch and automatically adjusts its structure and parameters online to solve the tracking control problem. The proposed controller can add, prune, and replace nodes online according to the control task, external disturbance, and the design specifications. A robustifying control term is also added to SEGRNN's output to mitigate the effects of the external disturbance. The concept of a data reservoir is proposed where a record of the deleted nodes is stored for any future recall, if they are seen to be significant again. Unlike most of the previously proposed self-evolving systems, our controller offers user-friendly design parameters to suit a variety of real-world systems. Lyapunov stability analysis is utilized to study the stability of the suggested controller and to determine an appropriate learning rate for the SEGRNN weights. A continuous stirred-tank reactor simulation example is employed to verify the performance of the proposed controller. The performance of the proposed controller is also compared with a variety of controllers, including adaptive radial basis functional networks, adaptive feed-forward neural networks, adaptive fuzzy logic system, proportional integral derivative controller, sliding-mode controller, and iterative learning controller. Finally, a dc motor platform is used to experimentally validate the controller performance.

Nonlinear Autoregressive Moving Average-L2 Model Based Adaptive Control of Nonlinear Arm Nerve Simulator System

This paper considers the trouble of the usage of approximate strategies for realizing the neural controllers for nonlinear SISO systems. In this paper, we introduce the nonlinear autoregressive moving average (NARMA-L2) model which might be approximations to the NARMA model. The nonlinear autoregressive moving average (NARMA-L2) model is an precise illustration of the input–output behavior of finite-dimensional nonlinear discrete time dynamical systems in a neighborhood of the equilibrium state. However, it isn't always handy for purposes of neural networks due to its nonlinear dependence on the manipulate input. In this paper, nerves system based arm position sensor device is used to degree the precise arm function for nerve patients the use of the proposed systems. In this paper, neural network controller is designed with NARMA-L2 model, neural network controller is designed with NARMA-L2 model system identification based predictive controller and neural network controller is designed with NARMA-L2 model based model reference adaptive control system. Hence, quite regularly, approximate techniques are used for figuring out the neural controllers to conquer computational complexity. Comparison were made among the neural network controller with NARMA-L2 model, neural network controller with NARMA-L2 model system identification based predictive controller and neural network controller with NARMA-L2 model reference based adaptive control for the preferred input arm function (step, sine wave and random signals). The comparative simulation result shows the effectiveness of the system with a neural network controller with NARMA-L2 model based model reference adaptive control system. Index Terms--- Nonlinear autoregressive moving average, neural network, Model reference adaptive control, Predictive controller DOI: 10.7176/JIEA/10-3-03 Publication date: April 30 th 2020

A Generalized Decomposed State Estimation Approach for Uncertain Constrained Stochastic Nonlinear Systems

In this paper, a decomposed state estimator is developed for stochastic constrained nonlinear discrete-time dynamical systems with uncertain parameters. The proposed estimator can deal with general nonlinear uncertain stochastic systems without any pre-defined specifications on the system structure and/or the measurement model. Moreover, it can handle estimation problems for nonlinear stochastic systems subject to a set of imposed linear and/or nonlinear equality and/or inequality constraints. The mathematical structure of the proposed estimator is developed using multiple projection approach and its performance is analyzed and compared with the global (un-decomposed) structure. The main features of the proposed estimator are: it reduces the processing time, it can handle estimation problems with bad initial conditions of the estimator, it can be implemented on modern parallel processing facilities to further reduce the processing time, and last but not least, it minimizes the rounding off errors and hence improves the numerical stability of the algorithm. Simulations results on different real applications are presented to show the effectiveness of the developed approach, and its improved performance when compared with other techniques reported in the literature.

Stability of Transactive Energy Market-Based Power Distribution System Under Data Integrity Attack

Future smart power distribution system with multiple active consumers and aggregators exchanging real-time electricity pricing and electricity consumption information over a communication network is prone to cyberattack. The goal of this paper is to understand the impact of attacks on pricing/load signals on the physical grid within a transactive energy market framework. First, this paper models the interaction between real-time electricity price and total energy demand in the form of a discrete time nonlinear autonomous dynamical system. Second, equilibrium electricity price and energy demand associated with this coupled dynamical system is derived and conditions for bounded input bounded output (BIBO) stability are identified. Third, a BIBO stable algorithm to design real-time electricity pricing scheme from a techno-economic perspective is developed. Finally, the impact of various levels of false data injection (FDI) attack on price of electricity, demand, and distribution system voltage is investigated. The proposed model is analyzed using simulations on the IEEE 69-bus test system and the impact of FDI attack on both electricity price/demand and distribution grid voltage is quantified. This paper shows that impact of FDI attack on electricity prices is more severe than an attack on electricity demand.

Bayesian filter for nonlinear systems with randomly delayed and lost measurements

In this note, a modified-likelihood Bayesian filter is proposed for a class of discrete-time nonlinear dynamical systems whose outputs are transmitted to the estimator with random delay and dropout. The likelihood function of the filter is computed by marginalizing out the delay variable to extract accurate information from the delayed measurements. Moreover, the measurement update stage of the filtering algorithm is omitted when no new measurement is received. Simulation results are presented to verify the superior performance of the introduced filter compared to some existing ones in the literature.

Identification of nonlinear block-oriented systems with backlash and saturation

Abstract A new approach to modeling and identification of discrete-time nonlinear dynamic systems with input backlash and output saturation nonlinearities is presented. The proposed three-block cascade mathematical model results from successive applications of the key-term separation principle. This provides special nonlinear model description that is linear in parameters. An iterative technique with internal variable estimation is proposed for estimation of all the model parameters based on measured input/output data and minimizing the least-squares criterion. Illustrative example of cascade system identification with backlash and saturation is included.

The Weighted Nearest Neighbor Estimate for Hammerstein System Identification

This paper concerns the nonparametric identification problem for a class of nonlinear discrete-time dynamical systems that is characterized by its cascade structure. This is a Hammerstein system being a series connection of a nonlinear memoryless element followed by a linear dynamic system. The input–output training data generated from the system are dependent and they do not reveal the strong mixing property. The nonlinear part of the system is recovered with the weighted $k$ -nearest neighbor regression estimate. The a priori information is nonparametric, both the nonlinear characteristic and the impulse response of the linear part are completely unknown and can be of any form. Local and global properties of the estimate are examined. Whatever the probability density of the input signal, the estimate converges at every continuity point of the characteristic as well as in the global sense. We derive the formulas for asymptotic bias and the variance and evaluate the corresponding rate of convergence. The convergence rate is independent of the shape of the input density and is proved to be optimal. These results allow us to find a set of optimal nonnegative weights that further improve the accuracy of our estimation algorithm. Our findings are supported by simulation experiments.

Robust information unscented particle filter based on M‐estimate

Herein, inspired by the unscented particle filter (UPF), an improved particle filter (PF) using an unscented information filter based on M-estimate (UIF-BM) to approximate the importance density function (IDF) is designed. The filtering method is proposed to obtain an accurate state estimation for a non-linear discrete time dynamical system with non-Gaussian system noises and observation noises contaminated by some Gaussian impulsive noises. The PF framework is used as a common method to handle non-Gaussian system noises. In order to obtain a precise IDF, UIF based on M-estimate is used for heavy-tailed observation noises. Meanwhile, instead of the robust Gaussian filter, the robust IF can avoid the numerical problem that zero weight functions cannot be incorporated into the framework. The simulation results indicate the estimation accuracy and efficiency of the proposed filter. Compared with the UIF-BM, PF, and UPF, the superiority of the proposed filter against the non-ideal system and observation noises is obvious.

An iterative method and its application to stable inversion

In this paper, we study convergence and data dependence of SP and normal-S iterative methods for the class of almost contraction mappings under some mild conditions. The validity of these theoretical results is confirmed with numerical examples. It has been observed that a special case of SP iterative method, namely normal-S iterative method, performs better and so the latter is implemented in the stable inversion of nonlinear discrete time dynamical systems to yield convergence results when Picard iterative method diverges. This is also illustrated with a numerical example. Our work extends and improves upon many results existing in the literature.

Learning robot reaching motions by demonstration using nonlinear autoregressive models

This paper presents NAR-RM, a method for learning robot reaching motions from a set of demonstrations using Nonlinear AutoRegressive (NAR) polynomial models. Reaching motions are modeled as solutions to autonomous discrete-time nonlinear dynamical systems, so that the movements started near the data of the demonstrations follow the trained trajectories and always reach and stop at the target. Since NAR models obtained using standard system identification techniques do not always adequately model the reaching motions, in this paper we present a method that uses a least-squares estimator with constraints to impose the location of fixed points in the model. With the imposition of new fixed points it is possible to change the location of the original fixed points of the model, thus allowing the learning of stable reaching motions. We evaluate our method using a library of human handwriting motions, a mobile robot and an industrial manipulator.

A cobweb model with gradient adjustment mechanism: nonlinear dynamics and multistability

ABSTRACTA cobweb model, characterized by boundedly rational producers with a production adjustment mechanism based on the gradient rule, is described by a nonlinear discrete time dynamical system of the plane. Firms do not have a complete knowledge of the demand function and try to infer how the market will respond to their production changes by an empirical estimates of the marginal profits. Analytical conditions for local stability of the market equilibrium are provided, showing that the stability loss of the market equilibrium may give rise to chaotic dynamic as well. When memory is introduced in the production adjustment mechanism, a locally stabilizing effect is revealed as well as a globally qualitatively destabilizing role for memory. This is related to the occurrence of period doubling and Neimark–Sacker bifurcations, the latter being of supercritical nature as analytically proved. Endogenous fluctuations and multistability, with consequent loss of predictability in the long run dynamics, are observed.

Stabilization of periodic orbits of discrete-time dynamical systems using the Prediction-Based Control: New control law and practical aspects

This paper tackles in the stabilization of periodic orbits of nonlinear discrete-time dynamical systems with chaotic sets. The problem is approximated locally to the stabilization of linear time-periodic systems and the theory of modern control is applied to the Prediction-Based Control, resulting in a new control law. This control law sets all the Floquet multipliers of the stabilized orbit to zero, resulting in fast convergence of trajectories in its vicinity. Another important characteristic of the control law is that no previous knowledge about the periodic orbit is required for stabilization. Using numerical simulations, this control law was analysed and compared to an optimal Delayed Feedback Control evidencing its advantages in theoretical and practical aspects.

Local and global analysis of a speculative housing market with production lag.

We extend the model of Dieci and Westerhoff [J. Evol. Econ. 22(2), 303-329 (2012)], where the authors analyse a speculative housing market populated by heterogeneous interacting agents described by a two dimensional nonlinear discrete time dynamical system. They show the emergence of complicated dynamics through the occurrence of bifurcations for particular parameter combinations. We enlarge their model in several ways. On one hand, we introduce time lag in the supply side and we consider two new scenarios characterised by agents' expectations formation. First, naive expectations instead of perfect foresight are considered, while in the second scenario, we study a mix between the model of Dieci and Westerhoff [J. Evol. Econ. 22(2), 303-329 (2012)] and the one we propose. As a consequence, we, analytically and numerically, explain the appearance of instability in the housing market providing conditions on the parameters that lead to a bifurcation. On the other hand, thanks to further numerical simulations, we conduct a global analysis providing the structure of the basin of attractions of the map showing coexistence of attractors.

Adaptive neural network tracking control-based reinforcement learning for wheeled mobile robots with skidding and slipping

Abstract To track the desired trajectories of the wheeled mobile robot (WMR) with time-varying forward direction, a reinforcement learning-based adaptive neural tracking algorithm is proposed for the nonlinear discrete-time (DT) dynamic system of the WMR with skidding and slipping. And, the typical model is transformed into an affine nonlinear DT system, the constraint of the coupling robot input torque is extended to pseudo dead zone (PDZ) control input. Three neural networks (NNs) are introduced as action NNs to approximate the unknown modeling item, the skidding and the slipping item and the PDZ item, whereas another NN is employed as critic NN to approximate the strategy utility function. Then, the critic and action NN adaptive laws are designed through the standard gradient-based adaptation method. The uniform ultimate boundedness (UUB) of all signals in the affine nonlinear DT WMR system can be ensured, while the tracking error converging to a small compact set by zero. Numerical simulations are conduced to validate the proposed method.

Dynamical systems on weighted lattices: general theory

In this work a theory is developed for unifying large classes of nonlinear discrete-time dynamical systems obeying a superposition of a weighted maximum or minimum type. The state vectors and input-output signals evolve on nonlinear spaces which we call complete weighted lattices and include as special cases the nonlinear vector spaces of minimax algebra. Their algebraic structure has a polygonal geometry. Some of the special cases unified include max-plus, max-product, and probabilistic dynamical systems. We study problems of representation in state and input-output spaces using lattice monotone operators, state and output responses using nonlinear convolutions, solving nonlinear matrix equations using lattice adjunctions, stability and controllability. We outline applications in state-space modeling of nonlinear filtering; dynamic programming (Viterbi algorithm) and shortest paths (distance maps); fuzzy Markov chains; and tracking audio-visual salient events in multimodal information streams using generalized hidden Markov models with control inputs.

An alternative approach for stability analysis of discrete time nonlinear dynamical systems

The paper endeavours to solve the problem of the necessary and sufficient conditions for testing asymptotic stability of the equilibrium state without using a positive definite or semi-definite Lyapunov function for time-invariant nonlinear discrete-time dynamical systems. The solution is based on the concept of the -functions introduced in this paper. As a result, new necessary and sufficient conditions for asymptotic stability of such systems and an estimation or the exact determination of the asymptotic stability domain of the state are established. Examples are worked out to illustrate the results.