Uncertain Nonlinear Discrete-time 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.

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.

Finite-time stability and observer-based control for nonlinear uncertain discrete-time switched system

Finite-time stability and observer-based control for nonlinear uncertain discrete-time switched system

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.

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

Robust preview control of nonlinear uncertain discrete-time time-delayed singular systems via sliding mode approach

This paper addresses the problem of preview sliding mode control for a class of nonlinear uncertain discrete-time singular systems with time-delays. A difference operator approach is used to deal with the controlled system such that the derived augmented systems including preview information of reference signal can be constructed. A suitable surface is proposed in the sense of admissibility by the free-weighed matrix technique. Based on this, a sliding mode controller with preview information is constructed incorporating with the tracking error to ensure that the closed-loop control system reaches the designed sliding mode surface within finite time. To apply the proposed control scheme to the practical engineering systems, the sliding mode tracking controller with preview information is developed for a DC motor based on the proposed control approach. The comparative experiment results demonstrate that the preview information of reference signal can improve the tracking performance.

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

Hybrid neural network controller for uncertain nonlinear discrete-time systems with non-symmetric dead zone and unknown disturbances

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

Preview repetitive control for polytopic nonlinear discrete-time systems.

In this study, a preview repetitive control (PRC) strategy was developed for uncertain nonlinear discrete-time systems subjected to a previewable periodic reference signal. The proposed preview repetitive controller was designed such that the system output tracked a previewable periodic reference signal even with model uncertainties and nonlinear terms. An augmented two-dimensional (2D) model was constructed based on the 2D model approach and state augmented technique. Second, considering the state unmeasured and periodic tracking reference signal, a static output PRC law was designed using the linear matrix inequality (LMI) techniques. Finally, the effectiveness of the proposed controller was verified through two illustrative examples.

A set-theoretic generalization of dissipativity with applications in Tube MPC

This paper introduces a framework for analyzing a general class of uncertain nonlinear discrete-time systems with given state-, control-, and disturbance constraints. In particular, we propose a set-theoretic generalization of the concept of dissipativity for systems that are affected by external disturbances. The corresponding theoretical developments build upon set based analysis methods and lay a general theoretical foundation for a rigorous stability analysis of economic tube model predictive controllers.

Sensor Bias Fault Diagnosis for a Class of Nonlinear Uncertain Hybrid Systems

This paper presents a sensor bias fault diagnosis approach for a class of hybrid systems with nonlinear uncertain discrete-time dynamics, measurement noise, and autonomous and controlled mode transitions. The proposed approach uses an observer based on a modified hybrid automaton framework and a fault detection scheme that employs a filtering method tighter mode-dependent thresholds for the detection of sensor faults (even with small magnitude). An autonomous guard events identification (AGEI) module is also developed and linked with both the fault detection scheme and the hybrid observer to eliminate any false alarms due to autonomous mode transitions and allow effective mode estimation. Finally, an adaptive sensor fault estimation scheme is included, which is activated once a fault is detected to isolate and estimate the sensor bias fault magnitude.

Stabilization of uncertain nonlinear discrete-time switched systems with state delays: A constrained robust model predictive control approach

This paper studies the problem of robust stabilization for a class of nonlinear discrete-time switched systems with polytopic uncertainties and unknown state delay. Moreover, the control signal is assumed to be constrained. The objective of the proposed controller is to stabilize the switched system under arbitrary switching signals based on the switched Lyapunov function approach. Therefore, based on the constrained robust model predictive control method and an appropriate Lyapunov–Krasovskii functional, the sufficient conditions to guarantee the asymptotical stability of the switched system are developed as linear matrix inequalities. Through online solving an optimization problem, the predictive state-feedback controller is designed. Furthermore, in this delay-dependent approach, only the upper bound of time-delay should be known. Appropriate transient response, ability to handle constraints, and nonconstrained switching signal are the other advantages of the proposed method. Finally, the performance of the proposed approach is compared with a similar approach through a numerical example. As well as, to show the applicability of the proposed controller, it is applied to a drinking water supply network, as an application example.

Computational method for estimating the domain of attraction of discrete-time uncertain rational systems

Using linear matrix inequality (LMI) conditions, we propose a computational method to generate Lyapunov functions and to estimate the domain of attraction (DOA) of uncertain nonlinear (rational) discrete-time systems. The presented method is a discrete-time extension of the approach first presented in [39], where the authors used Finsler’s lemma and affine annihilators to give sufficient LMI conditions for stability. The system representation required for DOA computation is generated systematically by using the linear fractional transformation (LFT). Then a model simplification step not affecting the computed Lyapunov function (LF) is executed on the obtained linear fractional representation (LFR). The LF is computed in a general quadratic form of a state and parameter dependent vector of rational functions, which are generated from the obtained LFR model. The proposed method is compared to the numeric n-dimensional order reduction technique proposed in [11]. Finally, additional tuning knobs are proposed to obtain more degrees of freedom in the LMI conditions. The method is illustrated on two benchmark examples.

Fault Diagnosis of Uncertain Hybrid Systems

This paper presents an online monitoring and fault diagnosis architecture for hybrid systems with nonlinear uncertain discrete-time dynamics. The architecture features a modular observer based on a modified hybrid automaton framework, that models each subsystem individually and the whole system as a composition of these models. The fault detection scheme can detect both discrete and parametric faults using a mode dependent time-varying threshold that guarantees no false-positives alarms due to modeling uncertainty. The fault isolation scheme can anticipate which fault events may have occurred and dynamically employs only the necessary number of isolation estimators that try to isolate the fault by calculating residual thresholds and approximating the unknown parameter values. Simulation results of a two-tank hybrid system example illustrate the effectiveness of the proposed approach.

RHONN identifier-control scheme for nonlinear discrete-time systems with unknown time-delays

This work presents a neural identifier-control scheme for uncertain nonlinear discrete-time systems with unknown time-delays. This scheme is based on a neural identifier to get a model of the system and a discrete-time block control technique based on sliding modes to generate the control law. The neural identifier is based on a Recurrent High Order Neural Network (RHONN) trained with an Extended Kalman Filter (EKF) based algorithm. Applicability is shown using real-time test results for linear induction motors. Also, a Lyapunov analysis is added in order to prove the semi-globally uniformly ultimately boundedness (SGUUB) of the proposed neural identifier-control scheme.

Robust Global Adaptive Exponential Stabilization of Discrete-Time Systems With Application to Freeway Traffic Control

This paper is devoted to the development of adaptive control schemes for uncertain discrete-time systems, which guarantee robust global exponential convergence to the desired equilibrium point of the system. The proposed control scheme consists of a nominal feedback law, which achieves robust global exponential stability properties when the vector of the parameters is known, in conjunction with a nonlinear dead-beat observer. The proposed adaptive control scheme depends on certain parameter observability assumptions. The obtained results are applicable to highly nonlinear uncertain discrete-time systems with unknown constant parameters. The successful applicability of the obtained results to real control problems is demonstrated by the rigorous application of the proposed adaptive control scheme to uncertain freeway models. A provided example demonstrates the efficiency of the approach.

Dissipativity-Based Reliable Interval Type-2 Fuzzy Filter Design for Uncertain Nonlinear Systems

The problem of dissipativity-based reliable fuzzy filter design for a category of uncertain nonlinear discrete-time systems is investigated based on the interval type-2 (IT2) T–S fuzzy approach. Uncertainties, which exist in considered systems, are described via adopting both lower membership functions and upper membership functions with corresponding weighting coefficients. In the design process, a vital problem, sensor failure, is taken into consideration. To improve the flexibility of filter design, an IT2 fuzzy filter model is employed, in which membership functions are different from those of the system model. Firstly, sufficient criteria are proposed to ensure the asymptotic stability with strict dissipativity of filtering error system. Following the proposed sufficient conditions, sufficient criteria for dissipative IT2 fuzzy filter design are proposed to estimate unknown system dynamics, where cone complementary linear algorithm is introduced to solve the bilinear problem. Secondly, both $$\mathcal{H}_{\infty }$$ and passive IT2 fuzzy filter design algorithms are presented as extensions. Finally, simulation results are given to illustrate the effectiveness of the IT2 fuzzy reliable filtering approach proposed in this paper.

Adaptive Control via Neural Output Feedback for a Class of Nonlinear Discrete-Time Systems in a Nested Interconnected Form.

In this paper, an adaptive output feedback control is framed for uncertain nonlinear discrete-time systems. The considered systems are a class of multi-input multioutput nonaffine nonlinear systems, and they are in the nested lower triangular form. Furthermore, the unknown dead-zone inputs are nonlinearly embedded into the systems. These properties of the systems will make it very difficult and challenging to construct a stable controller. By introducing a new diffeomorphism coordinate transformation, the controlled system is first transformed into a state-output model. By introducing a group of new variables, an input-output model is finally obtained. Based on the transformed model, the implicit function theorem is used to determine the existence of the ideal controllers and the approximators are employed to approximate the ideal controllers. By using the mean value theorem, the nonaffine functions of systems can become an affine structure but nonaffine terms still exist. The adaptation auxiliary terms are skillfully designed to cancel the effect of the dead-zone input. Based on the Lyapunov difference theorem, the boundedness of all the signals in the closed-loop system can be ensured and the tracking errors are kept in a bounded compact set. The effectiveness of the proposed technique is checked by a simulation study.