Showing papers on "Nonlinear system published in 2021"

Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators

Abstract: It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN with a single hidden layer can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators or complex systems from streams of scattered data. Here, we thus extend this theorem to DNNs. We design a new network with small generalization error, the deep operator network (DeepONet), which consists of a DNN for encoding the discrete input function space (branch net) and another DNN for encoding the domain of the output functions (trunk net). We demonstrate that DeepONet can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations. We study different formulations of the input function space and its effect on the generalization error for 16 different diverse applications. Neural networks are known as universal approximators of continuous functions, but they can also approximate any mathematical operator (mapping a function to another function), which is an important capability for complex systems such as robotics control. A new deep neural network called DeepONet can lean various mathematical operators with small generalization error.

Observer-Based Neuro-Adaptive Optimized Control of Strict-Feedback Nonlinear Systems With State Constraints

Abstract: This article proposes an adaptive neural network (NN) output feedback optimized control design for a class of strict-feedback nonlinear systems that contain unknown internal dynamics and the states that are immeasurable and constrained within some predefined compact sets. NNs are used to approximate the unknown internal dynamics, and an adaptive NN state observer is developed to estimate the immeasurable states. By constructing a barrier type of optimal cost functions for subsystems and employing an observer and the actor-critic architecture, the virtual and actual optimal controllers are developed under the framework of backstepping technique. In addition to ensuring the boundedness of all closed-loop signals, the proposed strategy can also guarantee that system states are confined within some preselected compact sets all the time. This is achieved by means of barrier Lyapunov functions which have been successfully applied to various kinds of nonlinear systems such as strict-feedback and pure-feedback dynamics. Besides, our developed optimal controller requires less conditions on system dynamics than some existing approaches concerning optimal control. The effectiveness of the proposed optimal control approach is eventually validated by numerical as well as practical examples.

Neural-Network-Based Event-Triggered Adaptive Control of Nonaffine Nonlinear Multiagent Systems With Dynamic Uncertainties

Abstract: This article addresses the adaptive event-triggered neural control problem for nonaffine pure-feedback nonlinear multiagent systems with dynamic disturbance, unmodeled dynamics, and dead-zone input. Radial basis function neural networks are applied to approximate the unknown nonlinear function. A dynamic signal is constructed to deal with the design difficulties in the unmodeled dynamics. Moreover, to reduce the communication burden, we propose an event-triggered strategy with a varying threshold. Based on the Lyapunov function method and adaptive neural control approach, a novel event-triggered control protocol is constructed, which realizes that the outputs of all followers converge to a neighborhood of the leader’s output and ensures that all signals are bounded in the closed-loop system. An illustrative simulation example is applied to verify the usefulness of the proposed algorithms.

Fuzzy Adaptive Finite-Time Fault-Tolerant Control for Strict-Feedback Nonlinear Systems

Abstract: This article devotes to investigating the issue of fuzzy adaptive control for a class of strict-feedback nonlinear systems with nonaffine nonlinear faults. The computational complexity is reduced by adopting the dynamic surface control technique. Under the framework of finite-time stability, a novel fault-tolerant control strategy is designed so that the closed-loop system is semiglobally practically finite-time stable, and the tracking error converges to a small residual set in a finite time. Finally, simulation studies for an electromechanical system are shown to verify the feasibility of the presented approach.

Observer-Based Fuzzy Adaptive Finite-Time Containment Control of Nonlinear Multiagent Systems With Input Delay

Abstract: This article is concerned with the finite-time containment control problem for nonlinear multiagent systems, in which the states are not available for control design and the control input contains time delay. Fuzzy-logic systems (FLSs) are used to approximate the unknown nonlinear functions and a novel distributed fuzzy state observer is proposed to obtain the unmeasured states. Under the framework of cooperative control and finite-time Lyapunov function theory, an observer-based adaptive fuzzy finite-time output-feedback containment control scheme is developed via the adaptive backstepping control design algorithm and integral compensator technique. The proposed adaptive fuzzy containment control method can ensure that the closed-loop system is stable and all followers can converge to the convex hull built by the leaders in finite time. A simulation example is provided to confirm the effectiveness of the proposed control method.

Adaptive Fuzzy Practical Fixed-Time Tracking Control of Nonlinear Systems

Abstract: This article investigates an adaptive practical fixed-time control strategy for the output tracking control of a class of strict feedback nonlinear systems. By utilizing a backstepping algorithm, finite-time Lyapunov stable theory, and fuzzy logic control, a novel adaptive practical fixed-time controller is constructed. Fuzzy logic systems are introduced to approximate the unknown items of the system. Theoretical analysis proves that under the presented control strategy, the closed-loop system is practically fixed-time stable, and the tracking error converges to a small neighborhood of the origin within a fixed-time interval, in which the convergence time has no connection with the initial states of the system. In the meantime, all the signals of the closed-loop system are bounded. Finally, a numerical example is presented to indicate the feasibility and effectiveness of the proposed method.

Quantized Static Output Feedback Fuzzy Tracking Control for Discrete-Time Nonlinear Networked Systems With Asynchronous Event-Triggered Constraints

Abstract: In this article, the $\mathcal {H}_{\infty }$ static output feedback tracking control problem is studied for discrete-time nonlinear networked systems subject to quantization effects and asynchronous event-triggered constraints. The Takagi–Sugeno (T–S) fuzzy model is utilized to represent the investigated nonlinear networked systems. A novel asynchronous event-triggered strategy is given to reduce the network communication burdens in both communication channels from the plant to the controller and from the reference model to the controller. The objective of this article is to design a quantized event-triggered tracking controller such that the resulting system is asymptotically stable and the given $\mathcal {H}_{\infty }$ tracking performance is guaranteed. The sufficient design conditions for the tracking controller are formulated in the form of the linear matrix inequalities (LMIs). Furthermore, a simulation example will be utilized to show the effectiveness of the developed design strategy.

Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: one-lump-multi-stripe and one-lump-multi-soliton types

Abstract: Interaction solutions between lump and soliton are analytical exact solutions to nonlinear partial differential equations. The explicit expressions of the interaction solutions are of value for analysis of the interacting mechanism. We analyze the one-lump-multi-stripe and one-lump-multi-soliton solutions to nonlinear partial differential equations via Hirota bilinear forms. The one-lump-multi-stripe solutions are generated from the combined solution of quadratic functions and N exponential functions, while the one-lump-multi-soliton solutions from the combined solution of quadratic functions and N hyperbolic cosine functions. Within the context of the derivation of the lump solution and soliton solution, necessary and sufficient conditions are presented for the two types of interaction solutions, respectively, based on the combined solutions to the associated bilinear equations. Applications are made for a (2+1)-dimensional generalized KdV equation, the (2+1)-dimensional NNV system and the (2+1)-dimensional Ito equation.

Neural-Based Decentralized Adaptive Finite-Time Control for Nonlinear Large-Scale Systems With Time-Varying Output Constraints

Abstract: This paper addresses the adaptive finite-time decentralized control problem for time-varying output-constrained nonlinear large-scale systems preceded by input saturation. The intermediate control functions designed are approximated by neural networks. Time-varying barrier Lyapunov functions are used to ensure that the system output constraints are never breached. An adaptive finite-time decentralized control scheme is devised by combining the backstepping approach with Lyapunov function theory. Under the action of the proposed approach, the system stability and desired control performance can be obtained in finite time. The feasibility of this control strategy is demonstrated by using simulation results.

A Novel Finite-Time Control for Nonstrict Feedback Saturated Nonlinear Systems With Tracking Error Constraint

Abstract: This article investigates the neural network-based finite-time control issue for a class of nonstrict feedback nonlinear systems, which contain unknown smooth functions, input saturation, and error constraint. Radial basis function neural networks and an auxiliary control signal are adopted to identify unknown smooth functions and deal with input saturation, respectively. The issue of error constraint is solved by combining the performance function and error transformation. Based on the backstepping recursive technique, a neural network-based finite-time control scheme is developed. The developed control scheme can ensure that the closed-loop system is semi-globally practically finite-time stable. Finally, the validity of theoretical results is verified via simulation studies.

Reduced Adaptive Fuzzy Tracking Control for High-Order Stochastic Nonstrict Feedback Nonlinear System With Full-State Constraints

Abstract: This paper focuses on the design of a reduced adaptive fuzzy tracking controller for a class of high-order stochastic nonstrict feedback nonlinear systems with full-state constraints. In the proposed approach, reduced fuzzy systems are used to approximate uncertain functions which involve all state variables and a high-order tan-type barrier Lyapunov function (BLF) is considered to deal with full-state constraints of the controlled system. With this BLF and a combination of the reduced fuzzy control and adding a power integrator, a novel control scheme is constructed to ensure that tracking error is within a very small range of the origin almost surely, meanwhile, the constraints on the system states are not breached almost surely during the operation. Two examples are proposed to show the effectiveness of the design scheme.

Quadratic $G$-BSDEs with convex generators and unbounded terminal conditions

Abstract: In this paper, we first study one-dimensional quadratic backward stochastic differential equations driven by $G$-Brownian motions ($G$-BSDEs) with unbounded terminal values. With the help of a $\theta$-method of Briand and Hu [4] and nonlinear stochastic analysis techniques, we propose an approximation procedure to prove existence and uniqueness result when the generator is convex (or concave) and terminal value is of exponential moments of arbitrary order. Finally, we also establish the well-posedness of multi-dimensional G-BSDEs with diagonally quadratic generators.

Observer-Based Finite-Time Adaptive Fuzzy Control for Nontriangular Nonlinear Systems With Full-State Constraints

Abstract: This article focuses on finite-time adaptive fuzzy output-feedback control for a class of nontriangular nonlinear systems with full-state constraints and unmeasurable states. Fuzzy-logic systems and the fuzzy state observer are employed to approximate uncertain nonlinear functions and estimate the unmeasured states, respectively. In order to solve the algebraic loop problem generated by the nontriangular structure, a variable separation approach based on the property of the fuzzy basis function is utilized. The barrier Lyapunov function is incorporated into each step of backstepping, and the condition of the state constraint is satisfied. The dynamic surface technique with an auxiliary first-order linear filter is applied to avoid the problem of an “explosion of complexity.” Based on the finite-time stability theory, an adaptive fuzzy controller is constructed to guarantee that all signals in the closed-loop system are bounded, the tracking error converges to a small neighborhood of the origin in a finite time, and all states are ensured to remain in the predefined sets. Finally, the simulation results reveal the effectiveness of the proposed control design.

Command-Filter-Based Adaptive Fuzzy Finite-Time Control for Switched Nonlinear Systems Using State-Dependent Switching Method

Abstract: The adaptive fuzzy finite-time tracking control problem of a class of switched nonlinear systems is investigated in this study. Fuzzy logic systems are introduced to handle the unknown nonlinear terms in the considered system. To overcome the drawback in the recursive design method, a finite-time command filter is employed. By constructing a new state-dependent switching law and adaptive fuzzy control signal, the existing restrictions on subsystems of switched systems are relaxed, all subsystems of the considered system are allowed to be unstabilizable. To avoid the Zeno behavior, a new hysteresis switching law is derived. It is proven that all states of the closed-loop system are bounded in finite time under the proposed fuzzy finite-time control scheme. Additionally, the proposed control method is extended to a class of more general switched large-scale nonlinear systems. Finally, two examples are provided to verify the developed method's effectiveness.

Small-Gain Technique-Based Adaptive Neural Output-Feedback Fault-Tolerant Control of Switched Nonlinear Systems With Unmodeled Dynamics

Abstract: In this article, the issue of adaptive neural fault-tolerant control (FTC) is addressed for a class of uncertain switched nonstrict-feedback nonlinear systems with unmodeled dynamics and unmeasurable states. In such a system, the uncertain nonlinear parts are identified by radial basis function (RBF) neural networks (NNs). Also, with the help of the structural characteristics of RBF NNs, the violation between the nontsrict-feedback form and backstepping method is tackled. Then, based on the small-gain technique, input-to-state practical stability (ISpS) theory, and common Lyapunov function (CLF) approach, an adaptive fault-tolerant tracking controller with only three adaptive laws is developed by designing an observer. It is shown that the designed controller can ensure that all the closed-loop signals are bounded under arbitrary switching, while the tracking error can converge to a small area of the origin. Finally, two simulation examples are provided to demonstrate the feasibility of the suggested control approach.

Adaptive Finite-Time Stabilization of Stochastic Nonlinear Systems Subject to Full-State Constraints and Input Saturation

Abstract: In this article, the adaptive finite-time tracking control is studied for state constrained stochastic nonlinear systems with parametric uncertainties and input saturation. To this end, a definition of semiglobally finite-time stability in probability (SGFSP) is presented and a related stochastic Lyapunov theorem is established and proved. To alleviate the serious uncertainties and state constraints, the adaptive backstepping control and barrier Lyapunov function are combined in a unified framework. Then, by applying a function approximation method and the auxiliary system method to deal with input saturation respectively, two adaptive state-feedback controllers are constructed. Based on the proposed stochastic Lyapunov theorem, each constructed controller can guarantee the closed-loop system achieves SGFSP, the system states remain in the defined compact sets and the output tracks the reference signal very well. Finally, a stochastic single-link robot system is established and used to demonstrate the effectiveness of the proposed schemes.

Event-Triggered Adaptive Fuzzy Control for Stochastic Nonlinear Systems With Unmeasured States and Unknown Backlash-Like Hysteresis

Abstract: This article investigates the event-triggered control problem for stochastic nonlinear systems with unmeasured states and unknown backlash-like hysteresis. Based on the fuzzy logic systems, the unknown nonlinear functions can be identified. Then, by utilizing a fuzzy state observer, the unmeasured states of the considered system can be estimated. Moreover, by introducing an event-triggered mechanism, the communication load can be largely reduced. By employing the backstepping control strategy and the adaptive control method, a novel adaptive fuzzy event-triggered control method is constructed. It is shown that whole signals in the closed-loop systems are, ultimately, semiglobally and uniformly bounded in probability. Moreover, the tracking errors and the observer errors are located in a small neighborhood around the origin. Finally, a numerical example is given to confirm the effectiveness of the design scheme.

Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders

Abstract: A common strategy for the dimensionality reduction of nonlinear partial differential equations (PDEs) relies on the use of the proper orthogonal decomposition (POD) to identify a reduced subspace and the Galerkin projection for evolving dynamics in this reduced space. However, advection-dominated PDEs are represented poorly by this methodology since the process of truncation discards important interactions between higher-order modes during time evolution. In this study, we demonstrate that encoding using convolutional autoencoders (CAEs) followed by a reduced-space time evolution by recurrent neural networks overcomes this limitation effectively. We demonstrate that a truncated system of only two latent space dimensions can reproduce a sharp advecting shock profile for the viscous Burgers equation with very low viscosities, and a six-dimensional latent space can recreate the evolution of the inviscid shallow water equations. Additionally, the proposed framework is extended to a parametric reduced-order model by directly embedding parametric information into the latent space to detect trends in system evolution. Our results show that these advection-dominated systems are more amenable to low-dimensional encoding and time evolution by a CAE and recurrent neural network combination than the POD-Galerkin technique.

Distributed Observer-Based Cooperative Control Approach for Uncertain Nonlinear MASs Under Event-Triggered Communication

Abstract: The distributed tracking problem for uncertain nonlinear multi-agent systems (MASs) under event-triggered communication is an important issue. However, existing results only provide solutions that can only ensure stability with bounded tracking errors, as asymptotic tracking is difficult to be achieved mainly due to the errors caused by eventtriggering mechanisms and system uncertainties. In this work, with the aim of overcoming such difficulty, we propose a new methodology. The subsystems in MASs are divided into two groups, in which the first group consists of the subsystems that can access partial output of the reference system and the second one contains all the remaining subsystems. To estimate the state of the reference system, a new distributed eventtriggered observer is firstly designed for the first group based on a combined output observable condition. Then, a distributed eventtriggered observer is proposed for the second group by employing the observer state of the first group. Based on the designed observers, adaptive controllers are derived for all subsystems. It is established that global stability of the closed loop system is ensured and asymptotic convergence of all the tracking errors is achieved. Moreover, a simulation example is provided to show the effectiveness of the proposed method.

Observer-Based Adaptive Sliding Mode Control for Nonlinear Stochastic Markov Jump Systems via T–S Fuzzy Modeling: Applications to Robot Arm Model

Abstract: In this article, the issue of sliding mode control for nonlinear stochastic Markovian jump systems with uncertain time-varying delay is investigated. Considering the system state measurements and the state-dependent disturbances are not available for feedback purposes, an observer-based adaptive control strategy is proposed. Based on the decomposition of the input matrices, the state-space representation of the system is turned into a regular form with the aid of T–S fuzzy models first. Then, a fuzzy observer system is constructed, which could be transformed into two lower order subsystems. By choosing a common linear switching surface, on which it also obtains linear sliding mode dynamics in a simple form. Further, an adaptive controller is synthesized relying on the bounded system delay information to ensure the estimated states driven on the predefined sliding surface and remain the sliding motion. Also, the stochastic stability analysis of the sliding mode dynamics is undertaken with two types of transition rates, and an interesting result reveals that the stability for the dynamics with type of uncertain transition rates may cover the completely known type. Finally, a single-link robot arm model is provided to verify the validity of the proposed method.

Optimized Design of Parity Relation-Based Residual Generator for Fault Detection: Data-Driven Approaches

Abstract: In the conventional approaches to the design of fault diagnosis systems, little effort is usually paid to the selection of the parity vectors. As a result, the systems’ performance can be significantly affected. In this article, novel approaches are proposed to derive the parity vectors that construct optimized residual generators for linear and nonlinear systems. Based on the analysis on the parity space dimension, a novel parameterization of all parity relation-based residual generators is proposed. An iterative procedure that guarantees minimal regression error is then employed in the search for the optimal parameters. Considering that the traditional parity relation-based approaches are only suitable for linear systems, in this work, the proposed approach is also generalized to deal with strong nonlinearities, with the aid of data-driven Hammerstein function estimation. Furthermore, optimized residual generation algorithms are summarized for offline design and online implementation, the performance of which is evaluated thoroughly with a three-tank system, a numerical nonlinear example, as well as a case study on an industrial hot rolling mill process. Results show that residuals generated by the proposed approaches can significantly improve the sensitivity to small faults, and thus, the fault detection rate is improved compared with the traditional nonoptimized approach.

Command Filter-Based Adaptive Fuzzy Control for Nonlinear Systems With Unknown Control Directions

Abstract: To deal with a class of nonlinear systems with unknown control directions, a command filter-based adaptive tracking controller is designed in this paper. In the design process, fuzzy logic system is required to handle nonlinear functions, command filter is employed to settle the explosion of complexity problem and Nussbaum function is introduced to compensate the influence of unknown directions problem. Finally, the proposed control approach guarantees that error signals converge into bounded compact sets around the origin and all closed-loop signals are bounded. The effectiveness of the presented scheme is illustrated by a simulation example.

Efficient quantum algorithm for dissipative nonlinear differential equations.

Abstract: Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming [Formula: see text], where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity [Formula: see text], where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for [Formula: see text] Finally, we discuss potential applications, showing that the [Formula: see text] condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R.

Robust Adaptive Control Barrier Functions: An Adaptive and Data-Driven Approach to Safety

Abstract: A new framework is developed for control of constrained nonlinear systems with structured parametric uncertainty. Forward invariance of a safe set is achieved through online parameter adaptation and data-driven model estimation. The new adaptive data-driven safety paradigm is merged with a recent adaptive controller for systems nominally contracting in closed-loop. This unification is more general than other safety controllers as contraction does not require the system be invertible or in a particular form. The method is tested on the pitch dynamics of an aircraft with uncertain nonlinear aerodynamics.

Event-Trigger-Based Finite-Time Fuzzy Adaptive Control for Stochastic Nonlinear System With Unmodeled Dynamics

Abstract: This article investigates the problem of finite-time fuzzy adaptive event-triggered control design for stochastic nonlinear nonstrict feedback systems with unmodeled dynamics. The fuzzy logic systems are adopted to identify the unknown nonlinearities and a state observer is designed to estimate the unmeasured states. Using backstepping recursive design and combining it with a varying threshold event-triggered condition, a novel event-triggered-based fuzzy adaptive finite-time control algorithm is developed, where the dynamical signal function is employed to deal with the unmodeled dynamics. A power form of the errors is used to ensure a continuous stabilizer. The semi-global finite-time stability in probability of the closed-loop system is proved based on an It $\hat{\text{o}}$ differential equation and finite-time stability theory. Simulations are provided to verify the effectiveness of the developed control algorithm.

Adaptive Neural Network Control for a Class of Nonlinear Systems With Function Constraints on States.

Abstract: In this article, the problem of tracking control for a class of nonlinear time-varying full state constrained systems is investigated. By constructing the time-varying asymmetric barrier Lyapunov function (BLF) and combining it with the backstepping algorithm, the intelligent controller and adaptive law are developed. Neural networks (NNs) are utilized to approximate the uncertain function. It is well known that in the past research of nonlinear systems with state constraints, the state constraint boundary is either a constant or a time-varying function. In this article, the constraint boundaries both related to state and time are investigated, which makes the design of control algorithm more complex and difficult. Furthermore, by employing the Lyapunov stability analysis, it is proven that all signals in the closed-loop system are bounded and the time-varying full state constraints are not violated. In the end, the effectiveness of the control algorithm is verified by numerical simulation.