Control Problem For Nonlinear Systems Research Articles

Ultimately bounded control for nonlinear systems under denial-of-service attacks: an accumulation-based event-triggered mechanism

This paper investigates the control problem for nonlinear systems under denial-of-service (DoS) attacks. In contrast to traditional event-triggered mechanisms (ETMs), an accumulation-based ETM, which is robust to abrupt signals, is employed to schedule the signal transmission between sensors and controllers, thereby saving more communication resources. Due to the openness of network-based transmission, DoS attacks are encountered during data transmissions, where the DoS attacks considered in this paper follow the Bernoulli distribution. This paper aims to design a controller for nonlinear systems considering the impact of DoS attacks and accumulation-based ETMs, ensuring that the controlled system state is ultimately bounded in the mean square. Firstly, sufficient conditions are provided to guarantee that the system state can achieve the given control performance, along with an explicit expression for the ultimate bound. Then, the controller gains are obtained by solving certain linear matrix inequalities. Finally, two simulation examples are conducted to validate the effectiveness of the designed controller.

Adaptive Dynamic Programming for Robust Event-Driven Tracking Control of Nonlinear Systems With Asymmetric Input Constraints.

This article considers the robust dynamic event-driven tracking control problem of nonlinear systems having mismatched disturbances and asymmetric input constraints. Initially, to tackle the asymmetric constraints, a novel nonquadratic value function is constructed for the original system. This makes the asymmetrically constrained tracking control problem transformed into an unconstrained optimal regulation problem. Then, a dynamic event-driven mechanism is proposed. Meanwhile, the event-driven Hamilton-Jacobi-Bellman equation (ED-HJBE) is developed for the optimal regulation problem in order to acquire the optimal control with distinctly decreased computational burden. To solve the ED-HJBE, a single critic neural network (CNN) is designed in the adaptive dynamic programming framework. Meanwhile, the gradient descent method is employed to update the CNN's weights. After that, both the weight estimation error and the tracking error are proved to be uniformly ultimately bounded via Lyapunov's direct method. Finally, simulations of the spring-mass-damper system and the pendulum plant are separately utilized to validate the established theoretical claims.

Machine learning model‐based optimal tracking control of nonlinear affine systems with safety constraints

AbstractThis work focuses on the development of a machine learning (ML) model‐based framework for safe optimal tracking control of a class of nonlinear control‐affine systems to ensure simultaneous closed‐loop stability and safety. Specifically, a novel multilayer feedforward neural network (FNN) with a control‐affine architecture is designed to model nonlinear dynamic systems. Subsequently, a model‐based reinforcement learning (RL) framework is presented, utilizing a novel cost function with Control Lyapunov‐Barrier Function (CLBF) properties, to learn both the control policy and the optimal value function for an infinite‐horizon optimal tracking control problem for nonlinear systems with safety constraints. The efficacy of the proposed methodology is demonstrated through simulations of a one‐link robot manipulator and a chemical process example.

Novel flexible fixed-time stability theorem and its application to sliding mode control nonlinear systems.

For the fixed-time nonlinear system control problem, a new fixed-time stability (FxTS) theorem and an integral sliding mode surface are proposed to balance the control speed and energy consumption. We discuss the existing fixed time inequalities and set up less conservative inequalities to study the FxTS theorem. The new inequality differs from other existing inequalities in that the parameter settings are more flexible. Under different parameter settings, the exact upper bound on settling time in four cases is discussed. Based on the stability theorem, a new integral sliding mode surface and sliding mode controller are proposed. The new control algorithm is successfully applied to the fixed-time control of chaotic four-dimensional Lorenz systems and permanent magnet synchronous motor systems. By comparing the numerical simulation results of this paper's method and traditional fixed-time sliding mode control (SMC), the flexibility and superiority of the theory proposed in this paper are demonstrated. Under the same parameter settings, compared to the traditional FxTS SMC, it reduces the convergence time by 18%, and the estimated upper bound of the fixed time reduction in waiting time is 41%. In addition, changing the variable parameters can improve the convergence velocity.

Observer-based fuzzy tracking control of nonlinear systems with intermittent output constraints

This paper pays attention to the output tracking control problem of nonlinear systems where the output is uniquely measurable and subject to an intermittent constraint. A multiplicative transformation based on the shifting-replaying function is introduced together with barrier Lyapunov function to solve the intermittent output constraint. A state observer independent of the matching condition of control gains is designed via the non-uniform state transformation to relax the assumption of full-state measurability. By virtue of the estimated states, we propose an output feedback controller that only involves one fuzzy adaptive parameter to restrain the effect of unknown system nonlinearities. Through Lyapunov stability analysis, it is proved that the proposed controller can make all signals in the closed-loop system remain semi-globally uniformly ultimately bounded. Two simulation examples are provided to verify the effectiveness of the proposed method, and also the superiority over the existing ones.

Prescribed performance adaptive fault‐tolerant control for nonlinear systems with actuator faults and input dead‐zone

AbstractThis paper investigates the prescribed performance adaptive fault‐tolerant control problem for nonlinear systems with actuator faults and input dead‐zone (IDZ). The partial loss of effectiveness fault and bias fault are considered. First, the asymmetric nonlinear IDZ is transformed into an affine function in the backstepping process. Second, the radial basis function neural network technique is applied to deal with the unknown uncertainty of the studied system. The state observer is designed to estimate the unmeasurable states in the nonlinear systems. Third, a fault‐tolerant controller is developed such that all signals are semi‐globally uniformly ultimately bounded for the closed‐loop system, and the tracking error remains within the specified performance requirements even though the actuator faults and IDZ occur. Finally, an electromechanical system example is provided to exhibit the utilizability of the obtained control technique.

Guaranteed cost tracking control of constrained input nonlinear uncertain systems using event-based ADP

This article investigates the guaranteed cost robust tracking control problem of nonlinear systems subjected to input constraint and unmatched uncertainty. The event-based adaptive dynamic programming (ADP) approach is utilized to address this problem. First, the tracking error and reference trajectory are combined to form an augmented uncertain system. Then, by decomposing the uncertainty into the matched and unmatched parts, the original tracking problem is converted into the optimal regulation problem of an auxiliary system. The cost function for the auxiliary system is defined, and the associated Hamilton–Jacobi–Bellman (HJB) equation is solved using a single critic neural network (NN). Moreover, a novel event-triggering rule is formulated, and it is shown that the designed event-based controller guarantees that the tracking error is uniformly ultimately bounded. The derivation of event-based guaranteed cost and its relation with the time-based counterpart is presented. The exclusion of the infamous Zeno behavior is guaranteed. The uniform ultimate boundedness of the critic weight estimation error is shown. Finally, the effectiveness of the proposed event-triggered ADP method is illustrated through simulations of the spring–mass–damper system and Van der Pol’s oscillator with unmatched uncertainty.

Cooperative-Critic Learning-Based Secure Tracking Control for Unknown Nonlinear Systems With Multisensor Faults.

This article develops a cooperative-critic learning-based secure tracking control (CLSTC) method for unknown nonlinear systems in the presence of multisensor faults. By introducing a low-pass filter, the sensor faults are transformed into "pseudo" actuator faults, and an augmented system that integrates the system state and the filter output is constructed. To reduce design costs, a joint neural network Luenberger observer (NNLO) structure is established by using neural network and input/output data of the system to identify unknown system dynamics and sensor faults online. To achieve the optimal secure tracking control, an augmented tracking system is formed by integrating the dynamics of tracking error, reference trajectory, and filter output. Then, a novel cost function is designed for the augmented tracking system, which employs the fault estimation and the discount factor. The Hamilton-Jacobi-Bellman equation is solved to obtain the CLSTC strategy through an adaptive critic structure with cooperative tuning laws. Besides, the Lyapunov stability theorem is utilized to prove that all signals of the closed-loop system converge to a small neighborhood of the equilibrium point. Simulation results demonstrate that the proposed control method has good fault tolerance performance and is suitable for solving secure control problems of nonlinear systems with various sensor faults.

Dynamic Threshold Finite-Time Prescribed Performance Control for Nonlinear Systems With Dead-Zone Output.

This article investigates the tracking control problem for nonlinear systems. An adaptive model is proposed to represent the dead-zone phenomenon and solve its control challenge with a Nussbaum function in conjunction. Drawing inspiration from the existing prescribed performance control schemes, a novel dynamic threshold scheme is developed that fuses a proposed continuous function with a finite-time performance function. A dynamic event-triggered strategy is applied to reduce the redundant transmission. The proposed time-varying threshold control strategy has fewer updates than the traditional fixed threshold and improves the efficiency of resource utilization. A command filter backstepping approach is employed to prevent the complexity explosion faced by the computation. The suggested control strategy ensures that all system signals are bounded. The validity of the simulation results has been verified.

Model-free intelligent critic design with error analysis for neural tracking control

The core of the optimal tracking control problem for nonlinear systems is how to ensure that the controlled system tracks the desired trajectory. The utility functions in previous studies have different properties which affect the final tracking effect of the intelligent critic algorithm. In this paper, we introduce a novel utility function and propose a Q-function based policy iteration algorithm to eliminate the final tracking error. In addition, neural networks are used as function approximator to approximate the performance index and control policy. Considering the impact of the approximation error on the tracking performance, an approximation error bound for each iteration of the novel Q-function is established. Under the given conditions, the approximation Q-function converges to the finite neighborhood of the optimal value. Moreover, it is proved that weight estimation errors of neural networks are uniformly ultimately bounded. Finally, the effectiveness of the algorithm is verified by the simulation example.

Adaptive tracking control for nonlinear systems with uncertain control gains and its application to a TCP/AQM network

This paper is concerned with the adaptive tracking control problem for nonlinear systems with virtual control coefficients including known and unknown items. The known items are employed for controller design directly, such that more information is utilized to achieve better performance. To deal with the unknown items, a novel real control law is firstly constructed by introducing an auxiliary system. The proposed controller is designed and applied to an uncertain TCP/AQM network system, which guarantees the practical boundedness of all the signals in the closed-loop system. Finally, the effectiveness and practicability of the developed control strategy are validated by simulation results.

Observer‐based adaptive neural fault‐tolerant control for nonlinear systems with prescribed performance and input dead‐zone

AbstractThis article investigates the adaptive fault‐tolerant tracking control problem for nonlinear systems with prescribed performance and input dead‐zone. First, a new composite variable is constructed by using the characteristics of actuator fault and input dead‐zone for the modeling purpose. Second, an adaptive neural network observer is designed to estimate the system states in the presence of inaccurate feedback information. Third, the proposed control strategy effectively counteracts the effects of sensor failure and unknown nonlinear functions, which makes the tracking error confined within the performance boundary and all the signals of the closed‐loop system semi‐globally uniformly ultimately bounded. Finally, an application oriented example is provided to demonstrate the effectiveness of the proposed control algorithm.

Adaptive Dynamic Programming for Nonlinear-Constrained H∞ Control

This article considers the <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> control problem of nonlinear systems having unavailable dynamics and asymmetric saturating actuators. Initially, such an <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> control problem is converted into the zero-sum game with a nonquadratic cost function being introduced. Then, in order to solve the Hamilton–Jacobi–Isaacs equation arising in the zero-sum game, a simultaneous policy iteration (SPI) algorithm is developed under the adaptive dynamic programming framework. Meanwhile, it is proved that the convergence of the SPI algorithm in essence amounts to the convergence of the sequential PI algorithm. To implement the SPI algorithm, the critic, the actor, and the perturbation neural networks (NNs) are, respectively, constructed to estimate the cost function, the control policy, and the perturbation. The three NNs’ weights are simultaneously determined by using the least-squares method together with the Monte Carlo integration technique. A remarkable characteristic of such an SPI algorithm is that arbitrary control policies and perturbations are applicable in the learning process. This makes system’s information be able to be replaced by the data collected along system’s trajectories in advance. More importantly, the persistence of the excitation condition is not required. Finally, simulations of two nonlinear examples are given to validate the present SPI algorithm.

Reduced-Order Observer-Based Preassigned Finite-Time Control of Nonlinear Systems and Its Applications

In this article, a preassigned finite time control problem of nonlinear systems in strict-feedback form is investigated. From the perspective of arbitrary settling time, an appropriate preassigned finite-time performance function (PFPF) is constructed, and the preassigned finite-time stability (PAFS) is established, where the settling (convergence) time is not only completely unconcerned with initial conditions and design parameters but also more flexible. Furthermore, the backstepping technique and reduced-order observer are used to obtain the preassigned finite-time control scheme. The stability criteria of PAFS are developed to guarantee that the output can quickly converge to an arbitrarily small zone in preassigned time, and all signals of the closed-loop control system are PAFS. In the end, simulation examples verify the effectiveness of the presented method.

Computer Modeling of Robust Control Problems for Nonlinear Systems Using Feedback

The control problem of nonlinear systems with feedback is one of the most important problems in the theory of control. In particular, it requires modeling of complex control laws at the design stage of the controller with feedback that meets the strict requirements of modern technologies. The lack of a universal approach to the control law, due to the problems of stability, tracking and reduction of deviations in feedback control, requires different methods in computer modeling. The article discusses the computer modeling of robust control problems and stabilization of inherently nonlinear systems with state or output feedback. With the help of the Matlab GUI software package, the sample examples under consideration were performed and the results were obtained

Adaptive asymptotic tracking control for nonlinear systems with actuator fault and prescribed performance based on event‐triggered mechanism

AbstractIn this article, the event‐triggered adaptive neural networks tracking control problem for nonlinear systems with prescribed performance and actuator fault is investigated. Meanwhile, bias faults and loss of effectiveness in actuators are taken into consideration, and the effectiveness factor is unknown. In the process of controller design under the framework of backstepping technology, neural networks are implemented to model the unknown terms of systems, and the tracking error is limited to a predefined boundary by using an error transformation. To economize communication resources, an adaptive neural networks event‐triggered control (ETC) strategy is developed, which can ensure that all the closed‐loop signals are bounded, and the tracking error not only confines to a prescribed function but also asymptotically converges to zero. Finally, two simulation examples are presented to further confirm the effectiveness of the proposed control strategy.

Simultaneous moving horizon estimation and control for nonlinear systems subject to bounded disturbances

SummaryIn this work, we address the output–feedback control problem for nonlinear systems under bounded disturbances using a moving horizon approach. The controller is posed as an optimisation‐based problem that simultaneously estimates the state trajectory and computes future control inputs. It minimizes a criterion that involves finite backward and forward horizons with respect to the unknown initial state, measurement noises and control input variables. The main novelty of this work relies on linking the lengths of the forward and backward windows with the closed‐loop stability, assuming detectability and decoding sufficient conditions to assure system stabilizability. It leads to a formulation that does not require to be a Control Lyapunov Function for the terminal cost of the controller. Simulation examples are carried out to compare the performance of solving simultaneously and independently the estimation and control problems. Furthermore, the examples show how the controller influences the length of the estimation window through its gain.

A New Design of Fuzzy Affine Model-Based Output Feedback Control for Discrete-Time Nonlinear Systems

The observer-based output feedback control problem for nonlinear systems is studied through Takagi-Sugeno fuzzy affine models. A novel piecewise fuzzy affine observer is designed to estimate the unmeasurable state variables. To remove the assumptions in the existing works that the system input matrices and/or output matrices should be common, full rank and uncertainties-free, a new approach is developed for the observer-based piecewise output feedback controller synthesis by making sufficient use of Young&#x0027;s inequality with its variance. Through fuzzy piecewise quadratic Lyapunov functions, improved and less conservative results on the output feedback controller design are obtained in terms of linear matrix inequalities, and the restrictions imposed in the existing results can be relaxed. Simulation studies are provided to confirm the effectiveness and conservatism reduction of the developed method.

Optimal Control of Nonlinear Systems With Unsymmetrical Input Constraints and its Application to the UAV Circumnavigation Problem

In this article, a novel design scheme is introduced to solve the optimal control problem for nonlinear systems with unsymmetrical and state-dependent input constraints. By introducing an initial stabilizing control policy as the baseline of the constructed optimal control policy, we remove the assumption in the current study for the adaptive optimal control, that is, the internal dynamics should hold zero when the state of the system is in the origin. Particularly, nonlinear control systems with partially unknown dynamics are investigated and the procedure to acquire the corresponding optimal control policy is presented. The stability for the closed-loop dynamics and the optimality of the obtained control policy are both proved. Besides, we apply the proposed control design framework to solve the optimal circumnavigation problem based on the accumulative Fisher information for a fixed-wing unmanned aerial vehicle (UAV). The control performance of our algorithm is compared with that of the existing circumnavigation control policy in a numerical simulation.

Model-Based Policy Iterations for Nonlinear Systems via Controlled Hamiltonian Dynamics

The infinite-horizon optimal control problem for nonlinear systems is studied. In the context of model-based, iterative learning strategies we propose an alternative definition and construction of the <i>temporal difference error</i> arising in Policy Iteration strategies. In such architectures the error is computed via the evolution of the Hamiltonian function (or, possibly, of its integral) along the trajectories of the closed-loop system. Herein the temporal difference error is instead obtained via two subsequent steps: first the dynamics of the underlying costate variable in the Hamiltonian system is steered by means of a (virtual) control input in such a way that the stable invariant manifold becomes externally attractive. Then, the <i>distance-from-invariance</i> of the manifold, induced by approximate solutions, yields a natural candidate measure for the <i>policy evaluation</i> step. The <i>policy improvement</i> phase is then performed by means of standard gradient descent methods that allows to correctly update the weights of the underlying functional approximator. The above architecture then yields an <i>iterative (episodic) learning</i> scheme based on a scalar, constant <i>reward</i> at each iteration, the value of which is insensitive to the length of the episode, as in the original spirit of Reinforcement Learning strategies for discrete-time systems. Finally, the theory is validated by means of a numerical simulation involving an automatic flight control problem.