Uncertain Dynamical Systems Research Articles

Synthesis of a parametrically robust controller with an extended state observer

This paper considers the problem of control of a parametrically uncertain dynamic system using an extended state observer. The control system is implemented using a two-loop scheme, where the outer loop ensures the fulfillment of the selected criterion for controlling the state vector, and the inner loop compensates for or reduces the influence of the vector of the total equivalent disturbance. The goal of the study is to develop a procedure for synthesizing an extended state observer taking into account the parametric uncertainty of the plant and the requirements for the closed loop of the combined control system specified in the frequency domain. Methods of control theory, robust control, and computer modeling are used for this study. The effect of the parametric uncertainty of the plant on its controlled motion is presented as a structured disturbance described in the form of a block-diagonal matrix. The concept of structured singular values is used to determine the measure of the system's robustness. Using the methodology of optimization of structured singular values, a procedure is proposed for synthesizing the extended state observer when considering a closed loop of a combined control system taking into account the parametric uncertainty of its mathematical model. The requirements ensuring the specified performance and stability of the closed loop of the control system taking into account the spectral properties of disturbances and sensor noise are formulated in the frequency domain using frequency-dependent weighting functions. For the synthesis of combined controllers based on the minimization of structured singular values, a D-G-L-K iteration algorithm is developed. The efficiency of the proposed approach is illustrated by a numerical example. Recommendations are given for the synthesis of parametrically robust combined controllers. The practical value of the obtained results lies in the fact that the procedure developed reduces the conservatism of robust combined control systems and, as a result, improves the control performance in the case of parametric uncertainty of the plant.

Robust Fuzzy Adaptive Fault‐Tolerant Control for a Class of Second‐Order Nonlinear Systems

ABSTRACTIn this brief, an optimal active fuzzy fault‐tolerant control (OAFFTC) scheme is proposed for a class of unknown perturbed multi‐input multi‐output (MIMO) nonlinear systems with nonaffine nonlinear actuator faults and time‐varying sensor faults. The control strategy can handle automatically (online updating) with three additives (bias, drift, and loss of accuracy) along with one multiplicative (loss of effectiveness) sensor faults and nonlinear state‐dependent actuator faults. In order to deal with uncertain system dynamics, sensor and actuator faults, and external disturbances, fuzzy systems (FSs) and backstepping techniques were combined to provide the adaptive control term as well as a robust control term. Butterworth filter is used to get rid the algebraic loop problem. The suggested robust term, can deal with approximation errors of the fuzzy systems (FSs). To automatically optimize the adaptive parameters and the starting conditions, particle swarm optimization (PSO) approach is introduced. The Lyapunov approach is employed to demonstrate the closed‐loop system's stability. To assess the efficacy and correctness of the suggested scheme, quadrotor dynamic model is performed on the simulation part.

Online Optimization and Ambiguity-Based Learning of Distributionally Uncertain Dynamic Systems

Online Optimization and Ambiguity-Based Learning of Distributionally Uncertain Dynamic Systems

Adaptive fixed-time TSM for uncertain nonlinear dynamical system under unknown disturbance.

For nonlinear systems subjected to external disturbances, an adaptive terminal sliding mode control (TSM) approach with fixed-time convergence is presented in this paper. The introduction of the fixed-time TSM with the sliding surface and the new Lemma of fixed-time stability are the main topics of discussion. The suggested approach demonstrates quick convergence, smooth and non-singular control input, and stability within a fixed time. Existing fixed-time TSM schemes are often impacted by unknown dynamics such as uncertainty and disturbances. Therefore, the proposed strategy is developed by combining the fixed-time TSM with an adaptive scheme. This adaptive method deals with an uncertain dynamic system when there are external disturbances. The stability of a closed-loop structure in a fixed-time will be shown by the findings of the Lyapunov analysis. Finally, the outcomes of the simulations are shown to evaluate and demonstrate the efficacy of the suggested method. As a result, examples with different cases are provided for a better comparison of suggested and existing control strategies.

Robust adaptive control of uncertain dynamic systems using self-evolving neural learning technique

Robust adaptive control of uncertain dynamic systems using self-evolving neural learning technique

Optimal parameter estimation for uncertain structural systems under unknown random excitations

An optimal parameter estimation method for uncertain structural systems under unknown random excitations is proposed, which combines the Bayesian inference and stochastic dynamics using geometrically averaged likelihood and optimal response estimation. The general description of optimal parameter estimation problems for uncertain dynamic systems under only stochastic responses measured is presented. The posterior probability density conditional on measured responses is expressed by the likelihood function conditional on system parameters based on Bayes’ theorem. For finite time processes, the parameter estimation problem as the probability integral of conditional means is converted into the optimization problem expressed as maximizing the posterior probability density. A geometrically averaged likelihood function is defined and used for calculating the logarithmic posterior probability density. This estimation can avoid the numerical singularity of the likelihood function and reduce the effects of incomplete posterior probability density and inaccurate prior statistics of unknown random excitations, and then it will be more reasonable and effective. Furthermore, the differential equations for system response means and covariances are derived and solved based on stochastic dynamics theory. The means and covariances conditional on responses at the present instant are expressed by the previous statistics based on optimal response estimation. By combining two results, the analytical expressions of the averaged likelihood function and logarithmic posterior probability density are obtained which will be more reliable and accurate. The proposed optimal estimation method is verified by numerical results for a five-storey frame structure under base random excitation. The estimated results are not affected by the prior statistics errors of random excitation as a factor. For noisy observation, the Kalman filtering is incorporated in the estimation method and hence the estimation is more accurate and robust even for low signal noise ratios. The optimal estimation method has the potential for application to general uncertain systems.

On Atangana–Baleanu fractional granular calculus and its applications to fuzzy economic models in market equilibrium

Based on relative-distance-measure fuzzy interval arithmetic (RDM-FIA), this paper presents several new concepts for fuzzy functions in fractional granular calculus with Mittag-Leffler (ML) kernels, such as Atangana–Baleanu (AB) fractional granular integrals, AB fractional granular derivatives, and AB Caputo fractional granular derivatives. Then, we present formulas for the fuzzy granular Laplace transform on Caputo fractional granular derivatives, AB fractional granular derivatives, and AB Caputo fractional granular derivatives. In addition to the relationship between AB fractional granular derivatives and AB Caputo fractional granular derivatives established using the granular Laplace transform, new fractional Newton–Leibniz-type formulas for both derivatives and integrals are proved. As applications, we obtain the fundamental solutions for the fuzzy economic models in the framework of Caputo fractional granular derivatives and AB Caputo fractional granular derivatives by utilizing the fuzzy granular Laplace transform. Several graphical explanations are given to show the dynamic behavior of the fundamental solutions. Moreover, we illustrate the advantages of using RDM-FIA to study the mathematical models of uncertain dynamical systems. The results of this study are illustrated using several numerical examples.

Sliding Mode Speed Control in Synchronous Motors for Agriculture Machinery: A Chattering Suppression Approach

Synchronous motors have extended their presence in different applications, specifically in high-demand environments such as agronomy. These uses need advanced and better control strategies to improve energy efficiency. Within this context, sliding mode control has demonstrated effectiveness in electric machine control due to its advantages in robustness and quick adaptation to uncertain dynamic system disturbances. Nevertheless, this control technique presents the undesirable chattering phenomenon due to the discontinuous control action. This paper introduces a novel speed integral control scheme based on sliding modes for synchronous motors. This approach is designed to track smooth speed profiles and is evaluated through several numeric simulations to verify its robustness against variable torque loads. This approach addresses using electric motors for different applications such as irrigation systems, greenhouses, pumps, and others. Moreover, to address the chattering problem, different sign function approximations are evaluated in the control scheme. Then, the most effective functions for suppressing the chattering phenomenon through extensive comparative analysis are identified. Integral compensation in this technique demonstrates improvement in motor performance, while sign function approximations show a chattering reduction. Different study cases prove the robustness of this control scheme for large-scale synchronous motors. The simulation results validate the proposed control scheme based on sliding modes with integral compensation, by achieving chattering reduction and obtaining an efficient control scheme against uncertain disturbances in synchronous motors for agronomy applications.

Least squares parameter estimation for uncertain fractional differential equations and application to stock model

In recent years, uncertain fractional differential equations was proposed for the description of complex uncertain dynamic systems with historical characteristics. For wider applications of uncertain fractional differential equations, researches on parameter estimation for uncertain fractional differential equations are of great importance. In this paper, based on the thought of least squares estimation and uncertain hypothesis test, an algorithm of parameter estimation for uncertain fractional differential equations is discussed. Finally, we consider the application of uncertain fractional differential equations based model to predict the forecasting stock price of three major indexes of U.S. stocks and make a comparison between uncertain fractional differential equations, uncertain differential equations and stochastic differential equations.

Sample-Based Continuous Approximation for Computing Probabilistic Boundary of Future State Trajectory in Uncertain Dynamical Systems

Sample-Based Continuous Approximation for Computing Probabilistic Boundary of Future State Trajectory in Uncertain Dynamical Systems

Actuator and sensor fault isolation in a class of nonlinear dynamical systems

Fault isolation in dynamical systems is a challenging task due to modeling uncertainty and measurement noise, interactive effects of multiple faults and fault propagation. This paper proposes a unified approach for isolation of multiple actuator or sensor faults in a class of nonlinear uncertain dynamical systems. Actuator and sensor fault isolation are accomplished in two independent modules, that monitor the system and are able to isolate the potential faulty actuator(s) or sensor(s). For the sensor fault isolation (SFI) case, a module is designed which monitors the system and utilizes an adaptive isolation threshold on the output residuals computed via a nonlinear estimation scheme that allows the isolation of single/multiple faulty sensor(s). For the actuator fault isolation (AFI) case, a second module is designed, which utilizes a learning-based scheme for adaptive approximation of faulty actuator(s) and, based on a reasoning decision logic and suitably designed AFI thresholds, the faulty actuator(s) set can be determined. The effectiveness of the proposed fault isolation approach developed in this paper is demonstrated through a simulation example.

Robust asymptotic super twisting sliding mode observer for non-linear uncertain biochemical systems

The problem of state estimation (reconstruction of the state vector) for a given class of biochemical systems under uncertain system dynamics has been addressed in this paper. In detail, the bioreactor at a water resource recovery facility represents the considered biochemical systems. The biochemical processes taking place in the bioreactor have been modelled using an activated sludge model. Based on this model, an appropriate utility model has been derived for estimation purposes. The internal dynamics of the model have been burdened with unstructured and parametric uncertainty due to the unknown reaction kinetics functions. Taking this uncertainty into account, an analysis of the observability and detectability of the utility model has been carried out. The utility model and the available set of inputs and measured outputs have been used to design a new robust non-linear observer that allows the estimation of state variables in the presence of uncertainty. In the synthesis of the observer, the asymptotic observer methodology has been combined with a second-order sliding mode observer, a so-called super twisting algorithm. The developed observer generates not only robust and stable estimates of the state variables but also enables the reconstruction of unknown kinetic functions. The stability of the designed observer has been proven using the Lyapunov stability theory. The observer has been implemented in the Matlab/Simulink environment. The comprehensive simulation studies carried out show the high efficiency of the estimation process using the developed state observer.

Nonparametric estimation of nonautonomous uncertain differential equations with application to temperature models

In recent years, there has been a great development in parameter estimation methods for uncertain differential equations (UDEs). However, the observations we can obtain in real life are limited, in which case the form of function in a UDE is unknown. When dealing with such UDEs, we may use observational data to make nonparametric estimates. There are many nonautonomous systems in real life, and nonautonomous UDEs can simulate some uncertain nonautonomous dynamical systems well. In this paper, a nonparametric estimation method based on the nonautonomous UDEs of the binary Legendre polynomial is proposed. Then, three numerical examples are given to verify the reliability of nonparametric estimation. As an application, a real data example of global average monthly temperatures is used to illustrate the effectiveness of our method.

Time-Varying Optimal Formation Control for Second-Order Multiagent Systems Based on Neural Network Observer and Reinforcement Learning.

This article addresses a distributed time-varying optimal formation protocol for a class of second-order uncertain nonlinear dynamic multiagent systems (MASs) based on an adaptive neural network (NN) state observer through the backstepping method and simplified reinforcement learning (RL). Each follower agent is subjected to only local information and measurable partial states due to actual sensor limitations. In view of the distributed optimized formation strategic needs, the uncertain nonlinear dynamics and undetectable states may jointly affect the stability of the time-varying cooperative formation control. Furthermore, focusing on Hamilton-Jacobi-Bellman optimization, it is almost incapable of directly dealing with unknown equations. Above uncertainty and immeasurability processed by adaptive state observer and NN simplified RL are further designed to achieve desired second-order formation configuration at the least cost. The optimization protocol can not only solve the undetectable states and realize the prescribed time-varying formation performance on the premise that all the errors are SGUUB, but also prove the stability and update the critics and actors easily. Through the above-mentioned approaches offer an optimal control scheme to address time-varying formation control. Finally, the validity of the theoretical method is proven by the Lyapunov stability theory and digital simulation.

Synchronization of uncertain chaotic systems with minimal parametric information

The chaotic synchronization is mainly hampered by uncertain system dynamics in terms of underlying parameters. To obtain accurate parameter estimates for the proper synchronization of uncertain chaotic systems (UCSs) through adaptive control, it is necessary to satisfy the persistence of excitation (PE) condition. Furthermore, the challenges imposed by the explosion of complexity in sequential stabilization procedures, slow performances, and lack of information in UCSs hinder the synchronization process through existing control techniques. Therefore, the contribution of this research is two-fold: First, a systematic stabilization approach with minimal calculations is proposed to achieve proper synchronization between chaotic drive and response systems. The key concept of the proposed theory is to obtain an invariant manifold by immersing the error dynamics (resulting from the mismatch between the drive and response systems) into lower-order target dynamics. From there, the control law for synchronizing these chaotic systems can be derived by defining passive outputs and associated storage functions. The proposed framework, which combines the concepts of immersion and passivity, is referred to as the Passivity and Immersion (P&I) approach. Second, unlike adaptive control, the synchronization of UCSs is made possible by selecting the appropriate target dynamics without any parameter estimation (or with minimal parametric information in certain cases). The simplicity and superiority of the proposed approach over existing techniques are preserved and demonstrated by application to a number of well-known chaotic systems.

Neuro-adaptive non-singular terminal sliding mode control for distributed fixed-time synchronization of higher-order uncertain multi-agent nonlinear systems

This paper presents a novel design for a distributed fixed-time synchronization controller based on neuro-adaptive non-singular terminal sliding mode control for higher-order multi-agent nonlinear systems. In this study, a distributed control strategy is employed in which networked nonlinear systems, referred to as nodes, communicate through a predefined network topology. Within this framework, a leader-follower configuration is adopted, designating one node as the leader and the others as follower agents. Notably, the dynamics of the agents are assumed to be unknown, leading to the development of a generalized uncertain higher-order nonlinear dynamical system model that accounts for the disturbed dynamics of all agents, including the matched bounded uncertainty with an upper algebraic bound. The control design leverages radial basis function neural networks for approximation, effectively addressing unknown nonlinear terms, such as drift terms and input channels. Importantly, the proposed technique successfully mitigated matched-type uncertainty, resulting in rapid sliding-mode enforcement with reduced chattering and stress. This innovative approach establishes fixed-time synchronization by creating a sliding mode with respect to the sliding surface of networked synchronization mismatches. To underscore the effectiveness of this approach, a simulation example that vividly illustrates its captivating properties and contributions is presented.

Event‐Triggered Adaptive Neural Network Backstepping Sliding Fault‐Tolerant Control of Spacecraft Formation Flying With Input Saturation

This study explores the challenge of tracking control for spacecraft formation flying (SFF) in the presence of dynamic uncertainties and external perturbations. Firstly, sliding mode control combined with backstepping control is used to address saturation issues. Then, neural networks, minimal parameter learning, and adaptive control are integrated to handle dynamic uncertainties and actuator failures. To alleviate the communication load, an event‐triggered mechanism is ultimately implemented, which leads to the development of an adaptive sliding mode fault‐tolerant control algorithm based on an event‐triggered neural network. This control architecture achieves significant advancements over traditional techniques: (1) ensuring system robustness and adaptability in complex scenarios with uncertain system dynamics and external disturbances, effectively counteracting actuator failures and input saturation issues; (2) significantly reducing transmission and computational burdens in resource‐limited networked systems through the adoption of event‐triggered control (ETC) mechanisms; (3) achieving high‐precision tracking performance for SFF without relying on prior knowledge of the system’s inherent dynamics, environmental disturbances, or potential actuator deficiencies. The Lyapunov approach is utilized to confirm the closed‐loop system’s boundedness. Finally, the proposed method’s efficacy is confirmed via simulations with a two‐satellite formation.

Adaptive Critic Control With Knowledge Transfer for Uncertain Nonlinear Dynamical Systems: A Reinforcement Learning Approach

Adaptive Critic Control With Knowledge Transfer for Uncertain Nonlinear Dynamical Systems: A Reinforcement Learning Approach

UKF-Based Optimal Tracking Control for Uncertain Dynamic Systems With Asymmetric Input Constraints.

To enhance system robustness in the face of uncertainty and achieve adaptive optimization of control strategies, a novel algorithm based on the unscented Kalman filter (UKF) is developed. This algorithm addresses the finite-horizon optimal tracking control problem (FHOTCP) for nonlinear discrete-time (DT) systems with uncertainty and asymmetric input constraints. An augmented system is constructed with asymmetric control constraints being considered. The augmented problem is addressed with a DT Hamilton-Jacobi-Bellman equation (DTHJBE). By analyzing convergence with regard to the cost function and control law, the UKF-based iterative adaptive dynamic programming (ADP) algorithm is proposed. This algorithm approximates the solution of the DTHJBE, ensuring that the cost function converges to its optimal value within a bounded range. To execute the UKF-based iterative ADP algorithm, the actor-estimator-critic framework is built, in which the estimator refers to system state estimation through the application of UKF. Ultimately, simulation examples are presented to show the performance of the proposed method.

Online accelerated data‐driven learning for optimal feedback control of discrete‐time partially uncertain systems

SummaryIn this paper, we develop an online learning algorithm for solving the Bellman equation for affine in the control discrete‐time nonlinear uncertain dynamical systems. To ensure accelerated learning of our algorithm in generating optimal control policies, we use an actor‐critic structure predicated on higher‐order tuner laws. More specifically, we construct a Nesterov‐like architecture involving momentum‐based learning laws leading to an accelerated convergence of the optimal control policy. The proposed online learning‐based optimal control framework guarantees uniform ultimate boundedness of the closed‐loop system under the assumption that the system is persistently excited. Finally, two illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach.