FCPN Approach for Uncertain Nonlinear Dynamical System with Unknown Disturbance

A novel neural network approximation scheme that is especially appropriate for adaptive control of nonlinear dynamical systems is proposed. In light of the new function approximation characterization, a neuro adaptive control framework for continuous- and discrete-time nonlinear uncertain dynamical systems is also presented. Specifically, the proposed neural network control framework is Lyapunov-based and, unlike standard neural network controllers guaranteeing ultimate boundedness, the framework guarantees partial asymptotic stability of the closed-loop system, that is, asymptotic stability with respect to part of the closed-loop system states associated with the system plant dynamics. The neuro adaptive controllers are constructed without requiring explicit knowledge of the system dynamics other than the assumption that the plant dynamics are continuous and piecewise continuously differentiable, and that the approximation error of uncertain system nonlinearities lie in a small gain-type norm bounded conic sector. This allows us to show that the standard neural network controllers are in fact capable of achieving partial asymptotic stability around the system equilibrium point for continuous-and discrete-time uncertain systems.

This paper has proposed a novel class of the stable discrete-time multilayer perceptron (MLP) neural observer for a class of discrete-time multiple-input multiple-output uncertain nonlinear systems. It is trained online and is used to estimate the states of the system especially when there is no prior knowledge about the dynamics of the system and can be applied as a basis for designing an intelligent control. The weight updating for the MLP is the developed backpropagation algorithm. Online training, convergence of the observer error to neighborhood of origin, robustness against uncertainties and disturbance, low computational expense and fast convergence rate are the main qualities of the suggested method. Lyapunov’s direct method is used to guarantee the stability of the proposed system. To demonstrate the performance of the discrete-time MLP observer, two discrete-time nonlinear dynamic systems are simulated in MATLAB/Simulink. Simulation results confirm the proficiency of the system even at different operating conditions and the presence of disturbance and measurement noise.

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

This article addresses the robust stability for a class of nonlinear uncertain discrete-time systems with convex polytopic of uncertainties. The system to be considered is subject to both interval time-varying delays and convex polytopic-type uncertainties. Based on the augmented parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent conditions for the robust stability are established in terms of linear matrix inequalities. An application to robust stabilization of nonlinear uncertain discrete-time control systems is given. Numerical examples are included to illustrate the effectiveness of our results. MSC:15A09, 52A10, 74M05, 93D05.

Due to the under-actuated feature, the reference signals using the combination of the system outputs, whose number is larger than that of reference signal, are designed so that the number of control inputs and sliding surfaces is the same, and that the uncontrollable mode is indirectly controlled. Under the uncertain environment, the sliding-mode under-actuated control (SMUC) with the satisfaction of suitable condition is designed to asymptotically track the reference signal. Otherwise, a bounded tracking result is obtained for the mild condition. In this situation, an on-line fuzzy modeling for the uncertainty is employed to design a fuzzy model-based sliding-mode under-actuated control (FSMUC) to improve the system performance; e.g., the bounded tracking result of SMUC becomes an asymptotical tracking. The proposed hybrid fuzzy sliding-mode under-actuated control (HFSMUC) combining SMUC and FSMUC with a transition can be applied to a class of generalized, under-actuated and uncertain nonlinear systems, e.g., the trajectory tracking control of a differential mobile robot (DMR). Finally, the simulations of the HFSMUC system are presented to confirm the efficiency and effectiveness of the proposed control.

A direct hybrid adaptive control framework for nonlinear uncertain hybrid dynamical systems is developed. The proposed hybrid adaptive control framework guarantees attraction of the closed-loop system states associated with the hybrid plant states in the face of parametric system uncertainty. A numerical example is provided to demonstrate the efficacy of the proposed approach.

An adaptive hybrid neural control scheme is presented for uncertain non-linear discrete-time Systems (UNLDTS) with non-symmetric dead-zone input and unknown disturbances. This work aims to design an efficient control scheme for the proposed systems in the case of non-symmetric dead-zone as the input due to the presence of which the control of such systems becomes very complex and difficult. The system is converted into an n-step ahead predictor and an adaptive compensative term is introduced to overcome the non-symmetric dead-zone present in the system. A hybrid neural network controller is constructed for the control of the proposed class of systems. The designed controller is proved to be semi-globally uniformly ultimately bounded with the assistance of Lyapunov theory and the error is proved to approach very close to zero. The effectiveness of the proposed scheme is validated through two simulation examples and one example has been inspired from a real word system named continuous stirred tank reactor taken in discrete-time form which ensures the applicability of this controller in real-world applications.

The problem of model-referenced control for interconnected systems with series non-linearities is investigated. By applying variable structure control theory, we propose a controller which is able to drive the error subsystem between the model state and the plant state to zero. The presented variable structure controller ensures the global reaching condition of the sliding mode of the error system. It shows that the uncertain nonlinear dynamic system also possesses the property of insensitivity to uncertainties and disturbances as a linear system does. Furthermore, the sliding mode can be designed to converge within a specified exponential speed.

A novel interval-data based Takagi-Sugeno fuzzy system is proposed to identify uncertain nonlinear dynamic systems by endowing the classical TS fuzzy system with probability theory and symbolic data analysis. Such systems have variability in their outputs, that is they produce varying responses each time when the same stimuli is applied to them under the same condition. Interval data is generated by repeating the identification experiment multiple times and applying the probabilistic techniques to get soft bounds of output. The interval data is then directly used in the TS fuzzy modelling, giving rise to interval antecedent and consequent parameters. This method does not require any specific assumption on the probability distribution of the random variable that models the uncertainty. The developed procedure is demonstrated for a pneumatic drive system.

Robustness of stochastic expansions for the stability of uncertain nonlinear dynamical systems – Application to brake squeal

Recently, a new robust control algorithm was developed at the Advanced Systems Laboratory. This algorithm extended the variable structure control design to include fuzzy systems and is called fuzzy variable structure control (FVSC). To apply the newly developed algorithm, uncertain nonlinear dynamical systems have been interpreted as fuzzy systems and modeled by fuzzy differential equations. This algorithm was previously applied to the control of a single arm robot using a suboptimal algorithm. This paper extends the previous work to a two joint robot arm and implements it using a computationally optimized algorithm. The simulation and preliminary experimental results are presented at the end.

This paper aims to study the robust nonfragile H ∞ controller design problem for a class of uncertain discrete-time nonlinear singular systems. A Takagi-Sugeno (T-S) fuzzy singular model with norm-bounded uncertainties is employed to represent a nonlinear plant. Basing on observers for the fuzzy model, we can construct a fuzzy controller with uncertainties existing in the implementation process. A augmented fuzzy closed-loop system can be obtained by the descriptor systems approach, which can avoid the appearance of coupling terms between matrices of the system model and the controller. Attention of the paper lies in deriving the design conditions of the observer-based controller by choosing a fixed common and fuzzy Lyapunov function, respectively, such that the admissibility and the prescript H ∞ performance of the augmented closed-loop system can be guaranteed. Finally, the simulation results show the validity of the proposed methods.

A robust adaptive NN output feedback control is proposed to control a class of uncertain discrete-time nonlinear multi-input–multi-output (MIMO) systems. The high-order neural networks are utilized to approximate the unknown nonlinear functions in the systems. Compared with the previous research for discrete-time MIMO systems, robustness of the proposed adaptive algorithm is obvious improved. Using Lyapunov stability theorem, the results show all the signals in the closed-loop system are semi-globally uniformly ultimately bounded, and the tracking errors converge to a small neighborhood of zero by choosing the design parameters appropriately.

ABSTRACTIn this paper, a state observer-based adaptive fuzzy dynamic surface control is developed for uncertain discrete-time non-linear pure-feedback multiple-input-multiple-output (MIMO) systems with network-induced time-delay. The uncertainties are approximated by a set of adaptive fuzzy logic systems, with the adjusted parameters updated by a simplified recursive least squares estimation algorithm, combined with a state observer. For a constant known network-induced time-delay, the proposed modified dynamic surface control utilising the predicted system states, expands the acceptable network-induced time-delay and stable operating range for a discrete-time non-linear pure-feedback MIMO system in the network. The simulation results indicate that the presented method is effective.

Backward reachability analysis is essential to synthesizing controllers that ensure the correctness of closed-loop systems. This article is concerned with developing scalable algorithms that underapproximate the backward reachable sets, for discrete-time uncertain linear and nonlinear systems. Our algorithm sequentially linearizes the dynamics and uses constrained zonotopes for set representation and computation. The main technical ingredient of our algorithm is an efficient way to underapproximate the Minkowski difference between a constrained zonotopic minuend and a zonotopic subtrahend, which consists of all possible values of the uncertainties and the linearization error. This Minkowski difference needs to be represented as a constrained zonotope to enable subsequent computation, but, as we show, it is impossible to find a polynomial-size representation for it in polynomial time. Our algorithm finds a polynomial-size underapproximation in polynomial time. We further analyze the conservatism of this underapproximation technique and show that it is exact under some conditions. Based on the developed Minkowski difference technique, we detail two backward reachable set computation algorithms to control the linearization error and incorporate nonconvex state constraints. Several examples illustrate the effectiveness of our algorithms.