Uncertain Discrete-time Nonlinear Systems Research Articles

A compact meta-learned neuro-fuzzy technique for noise-robust nonlinear control

Neuro-fuzzy systems show promise for adaptive control but can become complex due to the need to learn many parameters. This paper presents a resilient nonlinear controller that combines a simplified neuro-fuzzy system (Simp_NFS) and simplified neural network (Simp_NN) with only two meta-learnable parameters. This architecture enables fast and stable adaptation in uncertain nonlinear discrete-time systems. Simp_NFS utilizes interpretable hyperplane-based rules without antecedent parameters, simplifying the learning process to consequent weights. Simp_NN reduces complexity by replacing hidden-output weights with their mean. The hybrid auto-adaptive controller (HAC) combines the advantages of Simp_NFS and Simp_NN, significantly reducing the number of adaptive parameters compared to standard neuro-fuzzy methods for real-time control with limited resources. Simp_NFS provides structural adaptivity to handle uncertainties, while Simp_NN ensures reliable disturbance attenuation. The stability of HAC is proven using Lyapunov analysis. Extensive testing on challenging single-input single-output (SISO) and multi-input multi-output systems (MIMO) demonstrates that HAC improves performance by up to 82.55% compared to existing techniques. Key innovations include an ultra-compact meta-learned architecture, transparent online evolution of hyperplane clusters, and enhanced modeling capability for nonlinear uncertain systems. This interpretable neuro-fuzzy approach could enhance autonomy and safety by maintaining model transparency. The implementation of HAC is publicly available on GitHub at https://github.com/m-ferdaus/HAC.

Adaptive Fault-Tolerant Tracking Control for Discrete-Time Nonstrict-Feedback Nonlinear Systems With Stochastic Noises

This paper discusses the problem of fault-tolerant tracking control for discrete-time nonstrict-feedback nonlinear systems in the presence of stochastic noises and actuator faults. The system is characterized by a discrete-time nonstrict-feedback structure and multiplicative stochastic noise related to the states, which poses a challenge for the design and analysis of the fault-tolerant controller. Moreover, the considered actuator faults include multiplicative and additive faults without prior information. By integrating the properties of the backstepping framework and neural networks, and by introducing an adaptive fault compensation term, a novel fault-tolerant tracking control strategy is proposed. The effects of faults are compensated and the difficulties caused by the system structure are overcome, avoiding the algebraic loop problem and overcoming the causal contradiction. Given specific parameters, the designed fault-tolerant controller can ensure that the tracking error converges to an adjustable region regarding the origin and that all system signals are uniformly bounded concerning the mean-square sense. The simulation examples illustrate the effectiveness of the designed fault-tolerant control method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —As system complexity has been increasing, practical systems are often affected by faults when performing tracking control. Fault-tolerant control can provide acceptable robustness and improve system safety. On the one hand, many practical systems are nonlinear, difficult to model mathematically and accurately, and affected by stochastic noises. On the other hand, with digital control, the object is a discrete-time system. In practical applications, the upper bounds of the faults and the noises are often unknown. In this paper, a novel adaptive fault-tolerant control method is proposed for the more general uncertain discrete-time nonlinear systems, i.e., nonstrict-feedback nonlinear systems, with stochastic noises and actuator faults. Neural networks and an adaptive fault compensation term are integrated into the fault-tolerant controller. By adjusting the controller parameters properly, it is ensured that the tracking error converges to a small neighborhood of zero in the event of a fault occurring. In future research, sensor faults, input constraints, optimal control design, and applications in multi-agent systems will be considered.

Set-valued state estimators for a class of uncertain discrete-time nonlinear systems

Set-membership state estimation problem for a certain class of discrete-time nonlinear systems is tackled in this work. First, this problem is formulated as a reachability analysis problem. Then, to avoid the wrapping effect, explicit expression of the general solution of this class of systems is used to design new set-valued state estimation schemes. Moreover, for practical considerations, two periodically re-initialized algorithms based on both interval analysis and zonotope computation are introduced. The merit of the proposed approach is illustrated on a numerical example and its performance is compared to that of other methods borrowed from the literature.

Reliable stabilization of discrete-time nonlinear singular systems with time-varying delays

The problem of delay-dependent reliable control for uncertain discrete-time nonlinear singular system with time-varying delays is investigated. A new set of delay-dependent sufficient conditions are established for the existence of a reliable controller which ensures the singular system to be regular, causal (absence of impulses) and stable for all admissible parameter uncertainties. First, the controller for singular system in the presence of known actuator fault matrix is designed. Further, the result is extended to study the robust stabilization of singular systems with unknown actuator fault matrix. The sensor failures of actuator are considered by a variable, which is varying in a given interval. The conditions for the existence of proposed controller design are established in the form of linear matrix inequality (LMI), which can be easily solved via standard numerical software. Finally, numerical examples with simulation results are provided to illustrate the usefulness of control design and results are compared with existing results to show the less conservatism of the proposed theory.

Robust Observer-Based Proportional Derivative Fuzzy Control Approach for Discrete-Time Nonlinear Descriptor Systems with Transient Response Requirements

This paper proposes an observer-based proportional Derivative (O-BPD) fuzzy controller for uncertain discrete-time nonlinear descriptor systems (NDSs). Representing NDSs with the Takagi–Sugeno fuzzy model (T-SFM), the proportional derivative (PD) feedback method can be utilized in the fuzzy controller design via the Parallel Distributed Compensation (PDC) concept, such that the noncausal problem and impulse behavior are avoided. A fuzzy observer is proposed to obtain unmeasured states to fulfill the PD fuzzy controller. Moreover, uncertainties and transient response performances are taken into account for the NDSs. Then, a stability analysis process and corresponding stability conditions are derived from the Lyapunov theory with the robust control method and the pole constraint. Different from existing research, the Singular Value Decomposition (SVD) and the projection lemma are utilized to transfer the stability conditions into the Linear Matrix Inequation (LMI) form. Because of this reason, the conservatism of the analysis process can be reduced by eliminating the restriction on the positive definite matrix in the Lyapunov function. By giving the proper center and radius parameters of the pole constraint in the O-BPD fuzzy controller design process, the expected transient responses can be obtained for different designers and different practical applications. Finally, the effectiveness and applicability of the proposed O-BPD fuzzy controller are demonstrated by two examples of the simulation.

Fractional-Order Model-Free Adaptive Control with High Order Estimation

This paper concerns an improved model-free adaptive fractional-order control with a high-order pseudo-partial derivative for uncertain discrete-time nonlinear systems. Firstly, a new equivalent model is obtained by employing the Grünwald–Letnikov (G-L) fractional-order difference of the input in a compact-form dynamic linearization. Then, the pseudo-partial derivative (PPD) is derived using a high-order estimation algorithm, which provides more PPD information than the previous time. A discrete-time model-free adaptive fractional-order controller is proposed, which utilizes more past input–output data information. The ultimate uniform boundedness of the tracking errors are demonstrated through formal analysis. Finally, the simulation results demonstrate the effectiveness of the proposed method.

Optimal Adaptive Tracking Control of Partially Uncertain Nonlinear Discrete-Time Systems Using Lifelong Hybrid Learning.

This article addresses a multilayer neural network (MNN)-based optimal adaptive tracking of partially uncertain nonlinear discrete-time (DT) systems in affine form. By employing an actor-critic neural network (NN) to approximate the value function and optimal control policy, the critic NN is updated via a novel hybrid learning scheme, where its weights are adjusted once at a sampling instant and also in a finite iterative manner within the instants to enhance the convergence rate. Moreover, to deal with the persistency of excitation (PE) condition, a replay buffer is incorporated into the critic update law through concurrent learning. To address the vanishing gradient issue, the actor and critic MNN weights are tuned using control input and temporal difference errors (TDEs), respectively. In addition, a weight consolidation scheme is incorporated into the critic MNN update law to attain lifelong learning and overcome catastrophic forgetting, thus lowering the cumulative cost. The tracking error, and the actor and critic weight estimation errors are shown to be bounded using the Lyapunov analysis. Simulation results using the proposed approach on a two-link robot manipulator show a significant reduction in tracking error by 44% and cumulative cost by 31% in a multitask environment.

Reinforcement learning event-triggered output feedback control for uncertain nonlinear discrete systems

In this paper, a novel reinforcement learning (RL)-based event-triggered (ET) output feedback control algorithm is proposed for a class of uncertain strict-feedback nonlinear discrete-time systems. In contrast to traditional RL-based control methods, we proposed an ET output feedback controller based on the backstepping technique, where the transmission cost can be efficiently conserved. Then, in light of the radial basis function (RBF) neural network (NN), various critic NNs are constructed to approximate the critic functions in each step. Furthermore, with the backing of the proposed ET mechanism, a sampled output feedback controller is addressed to guarantee that the tracking errors and all signals of the closed-loop system are semi-global uniformly ultimately bounded (SGUUB). Finally, a simulation example is presented to demonstrate the effectiveness of the control strategy.

Convex Temporal Convolutional Network-Based Distributed Cooperative Learning Control for Multiagent Systems.

Due to its great efficiency, scalability, and inclusivity, distributed cooperative learning control has gotten a lot of attention. For complex uncertain multiagent systems, it is challenging to model the uncertainties and exploit the cooperative learning ability of the systems. To address these issues, we proposed a novel convex temporal convolutional network-based distributed cooperative learning control for uncertain discrete-time nonlinear multiagent systems. A new concept of using a convex temporal convolutional network (CTCNet) is proposed for estimating the uncertain agent dynamics in a cooperative way. Unlike previous methods that require adjustment of network weights for different control tasks, the proposed CTCNet can map the high-dimensional input-output space into a deep space spanned by basis features that represent the inherent properties of the system, so it has good robustness for different tasks. Consequently, to improve the control performance, a CTCNet-based distributed cooperative learning control method that shares learned knowledge through the communication topology among adaptive laws of CTCNet is proposed. Furthermore, the asymptotic convergence of system tracking errors to an arbitrarily small neighborhood of the origin is strictly proved. Finally, the simulation results are given to illustrate that our suggested method has higher control accuracy, stronger robustness, and anti-interference ability than the existing methods.

Singularity-free adaptive control of MIMO discrete-time nonlinear systems with general vector relative degrees

This paper develops a singularity-free adaptive tracking control scheme for a general class of multi-input and multi-output uncertain discrete-time nonlinear systems with non-canonical control gain matrices. The estimation of the control gain matrices, especially in some non-canonical forms, may be singular during parameter adaptation, which leads to the singularity problems of the adaptive control laws. This paper employs the matrix decomposition technique to solve the problem under a linearly parameterized adaptive control framework. The state and output feedback cases are addressed, respectively, to ensure closed-loop stability and asymptotic output tracking. Compared with the existing results, the features of the proposed adaptive control scheme include: (i) the proposed control laws do not involve the high-gain issue commonly encountered in robust control methods; (ii) two different filtered tracking error signals are introduced for the state and output feedback cases, respectively. These filters are crucial to avoid causality contradiction of the adaptive control laws commonly encountered in adaptive control of discrete-time systems; and (iii) a future time signal estimation-based adaptive control law is developed to ensure asymptotic output tracking for the output feedback case without requiring the high-gain observer. Finally, an illustrative example is given to verify the validity of the proposed control scheme.

Observer-based model predictive control for uncertain NCS subject to hybrid attacks via interval type-2 T–S fuzzy model

In this study, an observer-based model predictive control (MPC) algorithm is addressed for an uncertain discrete-time nonlinear networked control system (NCS) subject to hybrid malicious attacks by using interval type-2 Takagi–Sugeno (IT2 T–S) fuzzy theory. Hybrid malicious attacks, including two typical attacks, i.e., denial-of-service (DoS) attacks and false data injection (FDI) attacks, are considered in the communication networks. Under DoS attacks, the control signals will be interfered, which cause the degradation of signal-to-interference-plus-noise ratio, then lead to packets loss. Under FDI attacks, the false signals are injected and output signals are modified so that the system performance is deteriorated. For the NCS subject to hybrid attacks, a secure observer that can resist FDI attacks is devised and a fuzzy MPC algorithm that can solve the controller gains is proposed. Besides, by updating the bound of augmented estimation error, the recursive feasibility can be guaranteed. Finally, illustrative examples are given to show the effectiveness of proposed scheme.

A moving horizon state and parameter estimation scheme with guaranteed robust convergence

We propose a moving horizon estimation scheme for joint state and parameter estimation for nonlinear uncertain discrete-time systems. We establish robust exponential convergence of the combined estimation error subject to process disturbances and measurement noise. We employ a joint incremental input/output-to-state stability (δ-IOSS) Lyapunov function to characterize nonlinear detectability for the states and (constant) parameters of the system. Sufficient conditions for the construction of a joint (δ-IOSS Lyapunov function are provided for a special class of nonlinear systems using a persistence of excitation condition. The theoretical results are illustrated by a numerical example.

Adaptive modified prescribed performance constraint control for uncertain nonlinear discrete-time systems

In this work, an adaptive output constraint control scheme-based on modified prescribed performance function is investigated for a class of discrete time nonlinear systems subject to unknown parameters and uncertain dynamics. Unlike the traditional adaptive output constraint approach that employs barrier Lyapunov functions, a modified prescribed performance function is first employed to guarantee that the tracking error can convergence to a predefined and asymmetric neighborhood of zero and that the safety constraint bound of the system output is not violated. In order to reduce the effect of unknown parameters, the multiple model set is constructed and it can improve the transient response of systems. The specific features of the established multiple model method are that it not only can enhance the cooperativity of each identification model, but also reduce the computational burden since it does not require to a large number of models. The neural networks are utilized to approximated unknown dynamics. At the same time, the prescribed performance function, slide variable and second adaptive level technology in a unified framework are employed to design the controller. In order to verify the effectiveness of the established control scheme, the presented output constraint method is implemented to two numerical examples.

Model-Free Adaptive Control Based on Fractional Input-Output Data Model

Memory properties of fractional-order operators are considered for an input-output data model for highly uncertain nonlinear systems. The model arises by relating the fractional-order variation of the output to the fractional-order variation of the input; the instantaneous gain is computed through a fuzzy inference network, whose output consequences are adapted online on a gradient descent rule. The fractional-order nature of the proposed model relaxes the stringent conditions on data-driven schemes, allowing instantaneous changes in the output signal with a null variation in the controller. The main contribution consists of taking advantage of the memory properties of fractional-order operators and the flexibility of fuzzy logic rules to construct a data-driven model for highly uncertain discrete-time nonlinear systems. The relevance of the proposed method is assessed through experiments in a real-world scenario.

Efficient Backward Reachability Using the Minkowski Difference of Constrained Zonotopes

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.

Special issue on PID control in the information age: Theoretical advances and applications

Special issue on PID control in the information age: Theoretical advances and applications

Fractional data-driven model for stabilization of uncertain discrete-time nonlinear systems

This paper proposes a novel data-driven control for stabilization of a class of uncertain discrete-time nonlinear systems. The proposed method is based on the compact form dynamic linearization technique, which relates the first variation of the output signal with the fractional-order variation of the input one. Then, a discrete-time controller is proposed, based on the obtained fractional-order data-driven equivalent model. In order to compute the proposed controller and estimator, only input-output data information is considered. The uniform ultimately boundedness of the tracking errors are demonstrated by a formal analysis. Finally, comparison results based on simulations are presented to highlight the effectiveness of the proposed methodology.

Robust Finite-time H∞ Control for Discrete-time Nonlinear Uncertain Singular Systems with Time-delay

This paper deals with the problems of finite-time H∞ control for a class of discrete-time nonlinear singular systems subject to uncertainties and external disturbance. Firstly, the finite-time stability problem is discussed for discrete-time nonlinear uncertain singular systems with time-delay. The sufficient conditions of finite-time stability of discrete-time nonlinear singular systems are established. Then, by using the Lyapunov functional method, a criterion is established to ensure that the closed-loop system with external disturbance is finite-time bounded. We design the controller gain matrix. Finally, we provide a numerical example to illustrate the effectiveness of the proposed results.

Hybrid Neural Network Control for Uncertain Nonlinear Discrete-Time Systems with Bounded Disturbance

Hybrid Neural Network Control for Uncertain Nonlinear Discrete-Time Systems with Bounded Disturbance

Nonfragile H ∞ Control for Uncertain Takagi–Sugeno Fuzzy Systems Under Digital Communication Channels and Its Application

This work addresses the nonfragile <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> controller design problem for a class of discrete-time uncertain nonlinear systems. The norm-bounded uncertainty is contained in the nonlinear plant, which is described by the well-known Takagi–Sugeno (T–S) fuzzy model. The controller gain perturbation is also considered. When the input signal is transmitted from the controller to the system through the digital communication channels, it will be quantized by a static quantizer. The main attention is to design the nonfragile <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> state-feedback controller for the closed-loop quantized uncertain T–S fuzzy system. By introducing some auxiliary scalars and the fuzzy basis-dependent Lyapunov function approach, sufficient conditions are established in the form of linear matrix inequalities (LMIs). The construction for the nonfragile <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> controller can be completed by solving these LMIs. In the end, the applicability and effectiveness of the proposed method have been illustrated by two simulation examples.