Nonlinear Single-input Single-output Research Articles

An experimental approach to multi-input multi-output nonlinear active vibration control of a clamped sandwich beam

Large amplitude vibrations are often associated with geometric nonlinearity. These nonlinear systems are usually controlled using linear controllers, such as positive position feedback (PPF). Nonlinear control has also often been limited to single-input single-output (SISO) architectures. The present study develops a nonlinear PPF controller implemented with both a SISO and a multiple-input multiple-output (MIMO) architecture on a real experimental set-up, consisting of a clamped composite sandwich beam. The controller aimed to reduce the vibration amplitude observed when the structure was subject to a stepped-sine excitation in the frequency neighborhood of the first natural frequency for different force excitation levels. The developed nonlinear controller is based on a linear PPF controller to which a cubic term was added since the system under study is of hardening type. The control parameters were selected by optimization of the reduction of the vibration at low excitation amplitude. The nonlinear MIMO and SISO versions of the controller were compared to linear MIMO and SISO versions, which omitted the cubic term but maintained the other parameters. The nonlinear controllers were found to outperform the linear ones, and the MIMO versions of each controller also outperformed the SISO ones. Overall, the experimental evidence shows that a nonlinear MIMO PPF controller should be preferred for the application under study.

Online learning of stable robust adaptive controllers design based on data-dependent feedback linearization with application to rotary inverted pendulum

This study introduces an online (supervised) learning method to design nonlinear auto-regressive moving average (NARMA) controllers for feedback-linearized nonlinear single-input single-output (SISO) systems. The algorithm ensures Schur stability of the overall closed-loop system and provides adaptiveness and robustness for the NARMA controllers. The first stage of the method derives, in a data-dependent way, a feedback-linearized model of the nonlinear plant by using its input and output sample pairs. The method’s second stage, which constitutes the novel part of the presented study, builds up an online learning scheme for the linear auto-regressive moving average (ARMA) controller based on an already learned feedback-linearized model of the nonlinear plant. During online supervised learning, ARMA parameters of the feedback-linearized SISO plant model and the closed-loop ARMA model are computed by minimizing the plant identification and the closed-loop system tracking errors. Both errors are defined as ℓ1,ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\ell}}_{1,{\\varvec{\\varepsilon}}}$$\\end{document}, namely ε-insensitive loss functions that provide NARMA controller the robustness against noise and outliers. The proposed online learning control algorithm is applied to a rotary inverted pendulum model and to a real rotary inverted pendulum setup. The tracking performance of the developed controller is compared with those of the linear quadratic regulator and coupled sliding mode controller in terms of mean square error.

Disturbance Observer-Based Terminal Sliding Mode Tracking Control for a Class of Nonlinear SISO Systems with Input Saturation

This paper focuses on the trajectory tracking control for general nonlinear single-input single-output (SISO) systems in which the output is not directly related to the control input. To address the tracking problem with the consideration of possible model uncertainty, external disturbance, and control input saturation, we employ the input-output feedback linearization technique and design a finite-time disturbance observer-based terminal sliding mode controller to improve the tracking performance and enhance the robustness. The stability analysis is carried out by using the Lyapunov method. To alleviate the chattering while achieving an acceptable control performance, a boundary layer method is adopted for the trade-off between the high-frequency control actions and the bounded unavoidable nonzero steady-state error. The proposed method is evaluated on the two typical nonlinear systems, which are fully linearizable and partially linearizable, respectively, and compared to the state-of-the-art method in terms of tracking and robustness through comprehensive numerical simulations. The results show that the proposed method not only renders the estimated disturbance error tends to be zero in finite time, but also has superiority in the fast reaction to disturbance and small tracking error without high-frequency chattering.

Neural network-based iterative learning control for trajectory tracking of unknown SISO nonlinear systems

This paper proposes a neural network-based (NN-based) data-driven iterative learning control (ILC) algorithm for the tracking problem of nonlinear single-input single-output (SISO) discrete-time systems with unknown models and repetitive tasks. The control objective is to make the output of the system track the reference trajectory in each iteration process. Therefore, at each relative time during every iteration process, a generalized regression neural network (GRNN) is used as the estimator to solve the key parameters of the system, and a radial basis function neural network (RBFNN) is used as the controller to solve the control input. Compared with the traditional ILC algorithm, the two complex solving processes, i.e., dynamic linearization and criterion function minimization, are replaced and simplified into the iterative training of GRNNs and RBFNNs. The proposed algorithm is out-of-the-box and uses a point-to-point method to calculate the control input for each relative time of the system iteration, driving the tracking error of the system to approach zero. In addition, it is proved that the tracking error of the system under the proposed control algorithm is uniformly ultimately bounded. Finally, a numerical example shows the effectiveness and superiority of the control algorithm, and a path tracking experiment of an unmanned vehicle further verifies its practicability.

A New Approach to Nonlinear State Observation for Affine Control Dynamical Systems

In this work, a Nonlinear Higher Order Extended State Observer (NHOESO) is presented to replace the Linear Extended State Observer (LESO) used in Conventional Active Disturbance Rejection Control (C-ADRC) solutions. In the NHOESO, the standard LESO is completed with a two-term smooth nonlinear function with saturation-like characteristics. The proposed novel NHOESO enables precise observation of the generalized disturbances with higher-order derivatives. The stability of the NHOESO is examined with the aid of the Lyapunov method. A simulation of an uncertain nonlinear Single-Input–Single-Output (SISO) system with time-varying external disturbances confirms that the proposed NHOESO copes well with the generalized disturbance, which is not true for other ESOs.

Parameterization of a Novel Nonlinear Estimator for Uncertain SISO Systems with Noise Scenario

Dynamic observers are commonly used in feedback loops to estimate the system’s states from available control inputs and measured outputs. The presence of measurement noise degrades the performance of the observer and consequently degrades the performance of the controlled system. This paper presents a novel nonlinear higher-order extended state observer (NHOESO) for efficient state and disturbance estimation in presence of measurement noise for nonlinear single-input–single-output systems. The proposed nonlinear function allows a fast reconstruction of the system’s states and is robust against uncertainties and measurement noise. An analytical parameterization technique is proposed to parameterize the coefficients of the proposed nonlinear higher-order extended state observer in the case of measurement noise in the output signal. Several scenarios are simulated to demonstrate the effectiveness of the proposed observer.

Event-triggered adaptive approximately optimal tracking control of a class of non-affine SISO nonlinear systems via output feedback

In this paper, an event-triggered adaptive approximately optimal tracking control approach is proposed for a class of non-affine nonlinear single-input single-output (SISO) systems via output feedback. With the help of fuzzy logic systems (FLSs), a fuzzy state observer is designed for the estimation of internal states by approximating unknown nonlinear functions, where a low-pass filter (LPF) is added for the algebraic loop problem. Then, the output feedback control approach, in the backstepping framework and the event-triggered mechanism, is presented. It contains the adaptive backstepping control and the approximately optimal control via adaptive dynamic programming (ADP) technology, and the computation cost and transmission load are reduced while guaranteeing that the performance index of the system is approximately minimised. Finally, two simulation examples are provided to verify the effectiveness of the proposed approach.

Adaptive Output Feedback Tracking Control for a Class of Nonlinear Time-Varying State Constrained Systems With Fuzzy Dead-Zone Input

This article proposes an adaptive fuzzy controller for a class of uncertain strict-feedback nonmatching nonlinear single-input single-output systems with fuzzy dead zone and full time-varying state constraints. The states considered here are immeasurable and full states of the systems are constrained in a bounded set with time-varying regions. Following the adaptive backstepping design framework, the tangent barrier Lyapunov functions are introduced to the integrated design to address the problems in such systems. Fuzzy logic systems are used to identify the unknown smooth functions and unknown parameters. An input-driven observer is designed to estimate the immeasurable states. To distinguish the conventional deterministic dead zone models, the output of dead zone is uncertainty. The form of indeterminate dead zone as a combination of a liner and a disturbance-like term is extended by the fuzzy algorithms. Even though the output of dead zone is fuzzy and adopting the integrated design, the proposed fuzzy controller can ensure that all the signals in the closed-loop systems are semiglobal uniformly ultimately bounded and guarantee the tracking performance. Finally, simulation results are shown to verify the effectiveness and reliability of the proposed approach.

Constrained online trajectory planning for nonlinear flat SISO systems using a switched state variable filter

The problem of constrained Online Trajectory Planning (OTP) for Flatness-based Feedforward Controllers (FFC) is discussed for online tracking problems with a-priori unknown reference signals for the system output. As plant, nonlinear minimum phase Single Input Single Output (SISO) systems with input constraints as well as polytopic constraints on the flat state are assumed. The goal is to achieve both, fast reference tracking and a prioritization-based constraint satisfaction despite avoiding an online optimization. Therefore, the OTP is set up by means of a switched State Variable Filter (SSVF). The switching results from the limitation of the filter’s highest derivative in accordance to the above requirements and is derived with a piecewise solution of an underlying optimization problem. The paper shows the derivation of the SSVF, discusses necessary stability conditions of the proposed concept, and finally demonstrates the SSVF’s performance by means of a simulation study. A comparison to a respective optimal control problem solution and model predictive controller (MPC) is furthermore provided.

UDE-Based Robust Output Feedback Control With Applications to a Piezoelectric Stage

In this article, an uncertainty and disturbance estimator (UDE)-based robust output feedback control technique without using a state observer is proposed for a general class of nonlinear single-input single-output systems that are bounded-input bounded-output stable and subject to input and output disturbances. Instead of designing a controller for the original system directly, an equivalent system consisting of a first-order linear system plus a lumped uncertainty term is used to represent the input–output relationship. Then, a UDE-based robust controller is designed only using the system output feedback and the bandwidth of the open-loop system, making the design almost modeling-free. A prominent advantage of the proposed approach is that it has converted a challenging robust control problem into an intuitive filter design problem. The effectiveness of the proposed approach is validated on an experimental piezoelectric stage in the presence of hysteresis nonlinearity.

Synthesis of a Multifunctional Tracking System in Conditions of Uncertainty

Class of affine nonlinear single-input single-output systems, where the relative degree of the equivalent form of the input-output is invariant to the presence of external, unmatched disturbances, is formalized. Methods of synthesis of a multifunctional tracking system in the conditions of parametric uncertainty of the control plant model and incomplete measurements are designed for this class of systems. The original method of synthesis of a low dimension observer for estimating mixed variables (these are combinations of state variables, external influences and their derivatives) by measuring only tracking error is designed for information support of discontinuous control. In this observer, using the linear corrective effects with saturation, the method of separating the movements of observation errors is realized. As an illustration of the developed method, an electromechanical control object is considered-an inverted pendulum controlled by a DC motor. The simulation results for the worst case of varying parameters are given.

Multi-dimensional Taylor network modelling and optimal control of SISO nonlinear systems for tracking by output feedback

Abstract In this paper, a multi-dimensional Taylor network (MTN) output feedback tracking control of nonlinear single-input single-output (SISO) systems in discrete-time form is studied. To date, neural networks are generally used to identify unknown nonlinear systems. However, the neuron of neural networks includes the exponential function, which contributes to the complexity of calculation, making the neural network control unable to meet the real-time requirements. In order to identify the controlled object whose model is unknown, the MTN, which requires only addition and multiplication, is utilized for successful real-time control of the SISO nonlinear system based on only its output feedback. Lyapunov analysis proves that output signals in the closed-loop system remain bounded and the tracking error converges to an arbitrarily small neighbourhood around the origin. In contrast to the back propagation (BP) neural network self-adaption reconstitution controller, the edge of the scheme is that the MTN optimal controller promises desirable response speed, robustness and real-time control.

A comparison of manifold regularization approaches for kernel-based system identification

In this paper, we present a simulation study to investigate the role of manifold regularization in kernel-based approaches for nonparametric nonlinear SISO (Single-Input Single-Output) system identification. This problem is tackled as the estimation of a static nonlinear function that maps regressors (that contain past values of both input and output of the dynamic system) to the system outputs. Manifold regularization, as opposite to the Tikhonov one, enforces a local smoothing constraint on the estimated function. It is based on the assumption that the regressors lie on a manifold in the regressors space. This manifold is usually approximated with a weighted graph that connects the regressors. The present work analyzes the performance of kernel-based methods estimates when different choices are made for the graph connections and their respective weights. The approach is tested on benchmark nonlinear systems models, for different connections and weights strategies. Results give an intuition about the most promising choices in order to adopt manifold regularization for system identification.

Adaptive Fuzzy Sliding Mode Control for Nonlinear Uncertain SISO System Optimized by Differential Evolution Algorithm

In this paper, a new adaptive fuzzy sliding mode controller (AFSMC) is proposed for single-input single-output (SISO) nonlinear systems with uncertainties and external disturbances. An adaptive fuzzy model is used to approximate the unknown and uncertain features of a nonlinear system. Furthermore, the fuzzy model parameters are optimally identified with a differential evolution algorithm. The novel AFSMC algorithm is designed using sliding mode control. The adaptive fuzzy law is adaptively generated with constraints based on Lyapunov stability theory to guarantee the asymptotic stability of the closed-loop nonlinear uncertain SISO system. Experimental results are presented to demonstrate that the proposed AFSMC provides a robust and simple approach to effectively control the highly nonlinear uncertain SISO systems.

Partial State Constraints-Based Control for Nonlinear Systems With Backlash-Like Hysteresis

This paper focuses on the adaptive control problem for a class of nonlinear single-input single-output lower triangular systems with partial state constraints and unknown backlash-like hysteresis. To prevent the partial states from transgressing the predefined constrained region, a barrier Lyapunov function is presented, whose values will increase to infinity when any of its parameters grows to a given boundary value. To counteract the effect caused by the backlash-like hysteresis, an auxiliary variable with its adaptive mechanism is introduced in the backstepping technique. At the same time, the output of the considered systems can track the reference signal, and all the variables in the design procedure are uniformly ultimate boundedness. In the end, a numerical example is employed to validate the efficiency of the developed theoretical method.

Adaptive Model-Free Control Based on an Ultra-Local Model With Model-Free Parameter Estimations for a Generic SISO System

In this paper, a new structure to design model-free control (MFC) based on the ultra-local model is presented for an unknown nonlinear single-input single-output dynamic system. The proposed structure includes two adaptive laws corresponding to the unknown linear and nonlinear terms. Utilizing the adaptive law for linear term, the controller gain is going to be updated online using a differential Riccati equation. Subsequently, the control policy which includes an optimal term as well as a term for compensating the system unknown dynamics is generated. Here, the two proposed adaptive laws are model-free estimation algorithms, in which the need for any regressor parameter and also the persistent excitation condition is eliminated. Finally, two simulation studies are presented to show that the proposed adaptive MFC (AMFC) policy outperforms the two well-known controllers. Moreover, the AMFC is applied on a Duffing-Holmes chaotic oscillator plant and the convincing performance of the algorithm is observed through the simulation results.

Convergence for SISO ILC Systems with Locally Lipschitz Nonlinear Dynamics

This paper is devoted to iterative learning control (ILC) for single-input single-output (SISO), affine nonlinear systems with locally Lipschitz dynamics and subject to iteration-varying uncertainties arising from external disturbances and initial state shifts. By adopting a P-type update law, a necessary and sufficient condition is proposed to ensure the convergence of nonlinear SISO ILC. It is shown that the ILC process converges robustly with the final error bound depending continuously upon the bounds of iteration-varying uncertainties. Simulations illustrate the validity of the convergence results.

Analysis and synthesis of minimum phase nonlinear SISO systems under external unmatched perturbations

For the nonlinear minimum phase single-input single-output (SISO) systems, this paper formalizes the structure of the equivalent "input---output" representation taking into account the smooth perturbations that do not belong to the control space. Using this representation, we design a basic combined control law ensuring the asymptotic stability of the tracking error. In contrast to the traditional approach with generation of a reference signal and a perturbation, the complex estimation of the external signals and their derivatives employs a sliding mode observer. And finally, we formalize the physical implementability conditions of the invariant tracking system in the case when only the output (controlled variable) and the reference signal are directly measured.

Adaptive Moving Sliding Mode Control for SISO Systems: Application to an Electropneumatic System

This paper aims to propose and develop an adaptive moving sliding mode controller (AMSMC) that can be applied for nonlinear single-input single-output (SISO) systems with external disturbances. The main contribution of this framework consists to overcome the chattering phenomenon problem. The discontinuous term of the classic sliding mode control is replaced by an adaptive term. Moreover, a moving sliding surface is proposed to have better tracking and to guarantee robustness to the external disturbances. The parameters of the sliding surface and the adaptive law are deduced based on Lyapunov stability analysis. An experimental application of electropneumatic system is treated to validate the theoretical results.

Finite memory quadratic Volterra model for the response prediction of a slender marine structure under a Morison load

A quadratic Volterra model with a finite nonlinear memory effect was introduced and applied to the time series prediction of a slender marine structure exposed to the Morison load. First, the unknown nonlinear single-input–single-output dynamic system was identified using the nonlinear autoregressive with exogenous input (NARX) technique based on the prepared datasets of the wave elevation and system response, which was obtained by running nonlinear time domain analysis for a certain short term sea state. The structure of NARX was designed in such a way that the linear part had infinite memory, whereas the nonlinear part had finite memory of a certain length. Second, the frequency domain Volterra kernels, both linear and quadratic, were derived analytically by applying the harmonic probing method to the identified system. To derive the frequency response functions, the sigmoidal function used in NARX to realize the nonlinear relationship between the input and output was expanded to polynomials based on the Taylor series expansion, so that the harmonics of same frequencies were easily matched between the input and output. Finally, the time series of the system response under arbitrarily given short term sea states were predicted using the quadratic Volterra series. The proposed methodology was used to predict the nonlinear dynamic response of a 2-dimentional free standing catenary riser exposed to a random ocean wave load, and the comparison between the prediction and simulation results was made on the probability distribution of the maximum excursion of riser top. The results show that the proposed methodology can successfully capture the nonlinear effects of the dynamic response of a slender marine structure induced by the quadratic term of the Morison formula.