Single-input Single-output Nonlinear Systems Research Articles

Perturbation-based Extremum Seeking Control Design for a Class of Nonlinear Systems

In this paper, we try to address the problem of output (performance) function by applying the Perturbation-based Extremum Seeking Control (PESC) approach to the observer of the Single-Input Single-Output (SISO) nonlinear systems. In this work, PESC is applied so that the performance function can reach its maximum value. We apply tow controllers design that will take care of maximizing of the cost function. First controller is designed in the availability of full and unknown variables which are fed to the objective function, and the second controller is designed when the full state availability is removed, and these variables become known by applying the High Gain Observer (HGO) model to estimate the system variables. The construction of a seeking algorithm is used to drive the system variables and the observer output to the desired set-points that maximize the value of an objective (performance) function. In addition, Lyapunov's stability theorem and the perturbation theory including the averaging method are used in the design of the extremum seeking controller structure to check the stability of the system. Finally, the simulation results show the performance of the proposed procedures.

A Data-Driven Approach to Set-Theoretic Model Predictive Control for Nonlinear Systems

In this paper, we present a data-driven model predictive control (DDMPC) framework specifically designed for constrained single-input single-output (SISO) nonlinear systems. Our approach involves customizing a set-theoretic receding horizon controller within a data-driven context. To achieve this, we translate model-based conditions into data series of available input and output signals. This translation process leverages recent advances in data-driven control theory, enabling the controller to operate effectively without relying on explicit system models. The proposed framework incorporates a robust methodology for managing system constraints, ensuring that the control actions remain within predefined bounds. By means of time sequences, the controller learns the underlying system dynamics and adapts to changes in real time, providing enhanced performance and reliability. The integration of set-theoretic methods allows for the systematic handling of uncertainties and disturbances, which are common when the trajectory of a nonlinear system is embedded inside a linear trajectory state tube. To validate the effectiveness of our DDMPC framework, we conduct extensive simulations on a nonlinear DC motor system. The results demonstrate significant improvements in control performance, highlighting the robustness and adaptability of our approach compared to traditional model-based MPC techniques.

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.

Machine Learning Based System Identification with Binary Output Data Using Kernel Methods

Within the realm of machine learning, kernel methods stand out as a prominent class of algorithms with widespread applications, including but not limited to classification, regression, and identification tasks. Our paper addresses the challenging problem of identifying the finite impulse response (FIR) of single-input single-output nonlinear systems under the influence of perturbations and binary-valued measurements. To overcome this challenge, we exploit two algorithms that leverage the framework of reproducing kernel Hilbert spaces (RKHS) to accurately identify the impulse response of the Proakis C channel. Additionally, we introduce the application of these kernel methods for estimating binary output data of nonlinear systems. We showcase the effectiveness of kernel adaptive filters in identifying nonlinear systems with binary output measurements, as demonstrated through the experimental results presented in this study.

Metaheuristic Procedures for the Determination of a Bank of Switching Observers toward Soft Sensor Design with Application to an Alcoholic Fermentation Process

The present work focused on the development of soft sensors for single-input single-output (SISO) nonlinear dynamic systems with unknown physical parameters using a switching observer design. Toward the development of more accurate soft sensors, as compared with hard sensors, an extended design methodology for the determination of a bank of operating points satisfying the dense web principle was proposed, where for the determination of the bank of operating points and the observer parameters, a metaheuristic procedure was developed. To validate the results of the metaheuristic algorithm, the case of an alcoholic fermentation process was studied as a special case of the present approach. For the nonlinear model of the process, an observer-based soft sensor was developed using the metaheuristic procedure. First, the accuracy of the linear approximant of the process with respect to the original nonlinear model was investigated. Second, the I/O reconstructability of the linear approximant was verified. Third, based on the linear approximant, an observer was designed for the estimation of the non-measurable variable. Fourth, considering that the observer is designed upon the linear approximant, the linear approximant model parameters are derived through identification, for different operating points, upon the nonlinear model. Fifth, the observers corresponding to the different operating points, constitute a bank of observers. The design was completed using a data-driven rule-based system, performing stepwise switching between the observers of the bank. The efficiency of the proposed metaheuristic algorithm and the performance of the switching scheme were demonstrated through a series of computational experiments, where it was observed that the herein-proposed approach was more than two orders of magnitude more accurate than traditional single-step approaches of transition from one operating point to another.

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 two stage Active Disturbance Rejection Control design for under-actuated nonlinear systems

A two-stage Active Disturbance Rejection Control (ADRC) design procedure is presented for the output trajectory tracking control problem in Single Input Single Output (SISO) nonlinear systems. Aside from the given scalar system output, the plant is assumed to exhibit a second, physically measurable, internal scalar output variable. The system output is controlled in an ADRC manner from an implicit static, or dynamic, nonlinear feedback controller using, in general, a nonlinear differential expression of the internal output variable. A desired internal output trajectory is thus generated. The internal output variable, in turn, is controlled from the original input and forced to track the desired trajectory synthesized in the previous step by the auxiliary controller. Each ADRC controller design is based on a simplified model of the corresponding subsystem, usually relegating interaction variables and additive nonlinearities to be part of a local (total) disturbance input term. The control scheme is reduced to a set of linear ADRC controllers with, possibly, cancellation of nonlinear input gains which are assumed to be known. Two illustrative aerospace examples are presented where the proposed procedure is illustrated via digital computer simulations.

An improved model-free adaptive integral sliding mode control for a class of nonlinear discrete-time systems

For a class of uncertain single-input single-output (SISO) nonlinear discrete-time systems, an improved model-free adaptive integral sliding mode control (MFA-ISMC) algorithm is proposed based on a disturbance observer. The pseudo-partial derivative (PPD) estimation algorithm used to describe the system dynamics is modified and the stability of the closed-loop system is guaranteed. It is shown that the corresponding closed-loop system performance is improved and the effectiveness of the proposed method is demonstrated by a simulation example.

Adaptive regulation for minimum phase systems with unknown relative degree and uncertain exosystems

This paper investigates the adaptive error feedback regulation problem for unknown minimum phase, single-input, single-output linear systems with uncertain exosystems. The design problem of a feedback control which is capable of driving the output tracking error exponentially to zero, on the basis of the output tracking error only, is locally solved by an adaptive controller which continuously updates the frequency estimates and requires only two informations on the uncertain linear system: (i) an upper bound for the relative degree; (ii) the sign of the high-frequency gain. The proposed adaptive controller contains two gains to be tuned: an adaptation gain which should be chosen sufficiently small and a control gain which should be chosen sufficiently large. Extensions to single-input single-output nonlinear systems affected by uncertain biased multisinusoidal disturbances are outlined.

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 Neural Network Control for Uncertain Nonlinear Systems Without State Observer

A robust approximation-based event-triggered control method is presented for single input single output (SISO) nonlinear continuous-time systems with unmeasurable states and external disturbance. In the whole system, just one neural network (NN) is designed to approximate the unknown part in the controller, and output errors are directly used to construct the system controller and event-triggered mechanism to relieve the burden of system communication. The controller with a simple structure is easier to be realized in practical engineering. The system control signals and adaptive parameters are updated only if the event trigger condition is met, and this way further reduces the waste of network resources caused by frequent system sampling. The application of stability theory of Lyapunov proves that the weight estimation of the NN and tracking errors are ultimate and uniform boundedness, and the efficacy of the proposed scheme is verified with numerical results on a robot.

Intelligent Command Filter Design for Strict Feedback Unmodeled Dynamic MIMO Systems With Applications to Energy Systems

This study presents a command filtered control scheme for multi-input multi-output (MIMO) strict feedback nonlinear unmodeled dynamical systems with its applications to power systems. To deal with dynamic uncertainties, a dynamic signal is introduced, together with radial basis function neural networks (RBFNNs) to overcome the influences of the dynamic uncertainties. Command filters (CFs) are used to prevent the explosion of complexity, where the compensating signals can eliminate the effect of filter errors. Compared with single-input single-output strict feedback nonlinear systems, the method proposed in this study has more suitability. In the end, the simulation experiments are carried out by applying the developed algorithm to power systems, where the simulation results verify the efficacy of the approach proposed.

Adaptive Neural Network Sliding Mode Control for a Class of SISO Nonlinear Systems

In this article, a sliding mode control (SMC) is proposed on the basis of an adaptive neural network (NN) for a class of Single-Input–Single-Output (SISO) nonlinear systems containing unknown dynamic functions. Since the control objective is to steer the system states to track the given reference signals, the SMC method is considered by employing the adaptive neural network (NN) strategy for dealing with the unknown dynamic problem. In order to compress the impaction coming from chattering phenomenon (which inherently exists in most SMC methods because of the discontinuous switching term), the boundary layer technique is considered. The basic design idea is to introduce a continuous proportional function to replace the discontinuous switching control term inside the boundary layer so that the chattering can be effectively alleviated. Finally, both Lyapunov theoretical analysis and computer numerical simulation are used to verify the effectiveness of the proposed SMC method.

Nonlinear integral coupling for synchronization in networks of nonlinear systems

This paper presents a novel approach to (controlled) synchronization of networked nonlinear systems. For classes of identical single-input–single-output nonlinear systems and networks, including oscillator networks, we propose a systematic design procedure (with generic as well as constructive conditions) for specifying nonlinear coupling functions that guarantee global asymptotic synchronization of the systems’ (oscillatory) states. The proposed coupling laws are in the form of a definite integral of a nonlinear “coupling gain” function. It can be fit to the system’s nonlinearities and, thus, can avoid cancelling nonlinearities by feedback or high-gain arguments commonly needed for linear (diffusive) coupling laws. As demonstrated by two examples, including a network of FitzHugh–Nagumo oscillators, this design can result in much lower synchronizing coupling gains than for the common case of linear couplings, therewith increasing energy efficiency of the coupling laws and reducing output-noise sensitivity. The resulting coupling structure can be of a varying type, when couplings are activated/deactivated depending on the systems’ outputs without undermining overall synchronization. The approach is based on a novel notion of incremental feedback passivity with a nonlinear gain. In addition to the design contribution, these results provide a new insight into potential synchronization mechanisms in natural and artificial nonlinearly coupled systems.

Data‐driven consensus control for a class of unknown nonlinear multiagent systems with time delays

In this paper, the distributed output consensus tracking problem for a class of unknown heterogeneous discrete-time single-input single-output nonlinear multiagent systems (MASs) with time delays is considered. For this problem, a distributed data-driven consensus control strategy is proposed by using improved model-free adaptive control with a time delay. First, the unknown nonlinear multiagent time delay system is transformed into a virtual data model with time-varying parameters by using a partial form dynamic linearisation method. Then, a controller with model-free adaptive control and a tracking differentiator are designed. The proposed method uses only the input/output data of neighbouring agents during the control process without mathematically modelling multiple agents. For the time delay system, the boundedness of the closed-loop tracking error is theoretically proven for both the tracking problem and the regulation problem. Furthermore, this result can be generalised to switching topology structures by a rigorous mathematical proof. Numerical simulations verify the effectiveness of the proposed algorithm.

Pattern-Moving-Based Parameter Identification of Output Error Models with Multi-Threshold Quantized Observations

This paper addresses a modified auxiliary model stochastic gradient recursive parameter identification algorithm (M-AM-SGRPIA) for a class of single input single output (SISO) linear output error models with multi-threshold quantized observations. It proves the convergence of the designed algorithm. A pattern-moving-based system dynamics description method with hybrid metrics is proposed for a kind of practical single input multiple output (SIMO) or SISO nonlinear systems, and a SISO linear output error model with multi-threshold quantized observations is adopted to approximate the unknown system. The system input design is accomplished using the measurement technology of random repeatability test, and the probabilistic characteristic of the explicit metric value is employed to estimate the implicit metric value of the pattern class variable. A modified auxiliary model stochastic gradient recursive algorithm (M-AM-SGRA) is designed to identify the model parameters, and the contraction mapping principle proves its convergence. Two numerical examples are given to demonstrate the feasibility and effectiveness of the achieved identification algorithm.

A Robust Sliding-Mode Based Data-Driven Model-Free Adaptive Controller

In this letter, a novel data-driven control algorithm is presented coupling Model-Free Adaptive Control and Sliding Mode Control, which addresses general discrete-time Single-Input Single-Output nonlinear nonaffine systems and is aimed at strengthening standard techniques in the presence of a class of output-dependent perturbations. Use is made of an equivalent dynamic linearization model obtained adopting a dynamic linearization technique based on pseudo-partial derivatives. A stability proof of convergence of the closed loop system is provided, showing that the closed-loop tracking error is an asymptotically vanishing sequence and ensuring boundedness of the I/O sequences. Validation of the technique has been performed using a discrete-time test plant taken from the literature in the presence of perturbations. Simulation results show a remarkable improvement in terms of control authority and of tracking accuracy with respect to recently published analogous approaches.

U-model-based double sliding mode control (UDSM-control) of nonlinear dynamic systems

This study proposes a double sliding mode control-augmented U-control (-control) method for a class of single-input single-output nonlinear systems with internal uncertain parameters, model mismatching and external system noise/disturbance to improve robustness in nonlinear dynamic inversion in the U-control system design. For the configuration, the -control system takes up (1) a double sliding mode dynamic inverter to cancel nonlinearities and dynamics of the plant, (2) a linear invariant controller, the other dynamic inverter of the specified whole desired system performance, so that the whole system dynamic inversion is split into two designs in double feedback loops. For using the framework, this study analyses the associated properties on (1) global stability, (2) double sliding from a switching control driving the states to a sliding band and an equivalent control driving the states to a sliding line and (3) robustness against uncertainties/disturbances and a potential data-driven prototype. To validate the developed control system, it selects bench test examples for simulation studies using Matlab/Simulink, which demonstrates the -control in terms of accuracy, tracking, and robustness. The tests also present a step-by-step design procedure for potential applications.

Fault-tolerant control for model-free networked control systems under DoS attacks

This paper studies the fault-tolerant model-free adaptive control (FT-MFAC) problem for a class of single-input single-output (SISO) nonlinear networked control systems (NCSs) under denial-of-service (DoS) attacks. A novel FT-MFAC framework is established with the consideration of DoS attacks and the sensor fault, in which DoS attacks obeying the Bernoulli distribution randomly happen in the sensor-to-controller channel and the sensor fault is approximated by the radial basis function neural network (RBFNN). Based on the proposed framework, an FT-MFAC algorithm that uses only input/output data is proposed to guarantee that the output tracking error is bounded in the sense of mean square. Finally, the effectiveness of the proposed algorithm is illustrated by a simulation.