Unknown Continuous Functions Research Articles

Practical preset time fault‐tolerant control of uncertain Euler–Lagrange systems with input saturation and guaranteed performance

AbstractThe practical preset time fault‐tolerant control (FTC) problem is explored in this article for uncertain Euler–Lagrange systems with input saturation and guaranteed performance. To cope with the uncertainty of the Euler–Lagrange systems, the adaptive neural network (NN) is exploited to approximate the unknown continuous function. Most existing results that consider input saturation and actuator faults simultaneously need to design compensation strategies separately, which increases the complexity of control algorithms. To overcome the above obstacle, the Nussbaum gain technique is used to deal with the effects of input saturation and actuator faults in this article. Besides, with the help of error transformation technology and speed function, the proposed control algorithm can ensure that the tracking error converges within the preset time and its overshoot is constrained within the prescribed performance boundaries. Furthermore, the boundedness of all closed‐loop system signals is confirmed. Finally, comparative simulation results are depicted to highlight the superiority of the designed control algorithm.

Finite-Time Stability Control of Uncertain Nonlinear Systems With Self-Limiting Control Terms.

In this brief, we define a self-limiting control term, which has the function of guaranteeing the boundedness of variables. Then, we apply it to a finite-time stability control problem. For nonstrict feedback nonlinear systems, a finite-time adaptive control scheme, which contains a piecewise differentiable function, is proposed. This scheme can eliminate the singularity of derivative of a fractional exponential function. By adding a self-limiting term to the controller and the virtual control law of each subsystem, the boundedness of the overall system state is guaranteed. Then the unknown continuous functions are estimated by neural networks (NNs). The output of the closed-loop system tracks the desired trajectory, and the tracking error converges to a small neighborhood of the equilibrium point in finite time. The theoretical results are illustrated by a simulation example.

Adaptive interval type-2 fuzzy fixed-time control for nonlinear MIMO fast-slow coupled systems with user-defined tracking performance

This paper focuses on the adaptive interval type-2 (IT2) fuzzy fixed-time self-triggered control issue for a class of multi-input and multi-output (MIMO) nonlinear fast-slow coupled systems (FSCSs) with preassigned tracking performance requirements. At first, the steady-state error and dynamic response of the investigated system are synthetically weighed by introducing an improved fixed-time prescribed performance function into coordinate transformation. Second, an IT2 fuzzy adaptive self-trigged controller is designed, where the unknown continuous functions are approximated by the interval type-2 fuzzy logic systems (IT2FLSs). In addition, the 1-bit encoding-decoding mechanism (EDM) is embedded in the signal transmission channel between the controller and the actuator, which also provides a feasible solution to further reduce network bandwidth consumption under the framework of the self-triggered protocol. Stability analysis indicates that the developed control method ensures that all states of the closed-loop system are practical fixed-time bounded, and the tracking error evolves to a pre-specified performance envelope within a predetermined time. Finally, a numerical example and an application example are provided to demonstrate the effectiveness and superiority of the proposed method.

Adaptive Fuzzy Appointed-Time Control for Uncertain Systems Bounded by Positive Functions

This brief presents an appointed-time prescribed performance control design for uncertain nonlinear systems via adaptive fuzzy technique. Compared with existing results, the controller achieves appointed-time tracking control for higher-order nonlinear systems with uncertainties bounded by positive functions. Firstly, barrier Lyapunov functions integrated with appointed-time prescribed performance functions are designed to cope with the requirements of output constraints and quantitative performance constraints. Then, adaptive fuzzy systems are constructed to approximate the unknown continuous upper-bound functions of the uncertainties. Subsequently, through the backstepping framework, a novel adaptive fuzzy appointed-time control scheme is proposed for the uncertain nonlinear system bounded by positive functions. Thus, the appointed settling time, predetermined steady-state error, and quantitative transient-state performance of the closed-loop system are ensured. Finally, the fixed-time convergence of the closed-loop system is proved via Lyapunov analysis, while comparative simulations are conducted to illustrate the superiority of the proposed scheme.

Adaptive Neural Control for an Uncertain 2-DOF Helicopter System with Unknown Control Direction and Actuator Faults

In accordance with the rapid development of smart devices and technology, unmanned aerial vehicles (UAVs) have been developed rapidly. The two-degree-of-freedom helicopter system is a typical UAV that is susceptible to uncertainty, unknown control direction and actuator faults. Hence, a novel adaptive neural network (NN), fault-tolerant control scheme is proposed in this paper. Firstly, to compensate for the uncertainty, a radial-basis NN was developed to approximate the uncertain, unknown continuous function in the controlled system, and a novel weight-adaptive approach is proposed to save on computational cost. Secondly, a class of Nussbaum functions was chosen to solve the unknown-control-direction issue to prevent the effect of an unknown sign for the control coefficient. Subsequently, in response to the actuator faults, an adaptive parameter was designed to compensate for the performance loss of the actuators. Through rigorous Lyapunov analyses, the designed control scheme was proven to enable the states of the closed-loop system to be semi-globally uniformly bounded and the controlled system to be stable. Finally, we conducted a numerical simulation on Matlab to further verify the validity of the proposed scheme.

Adaptive fault‐tolerant robust control based on radial basis function neural network for a class of mechanical systems with input constraints

AbstractIn this paper, a robust adaptive fault‐tolerant control (FTC) strategy based on hyperbolic tangent function is proposed for a class of mechanical systems with unknown disturbance and unknown nonlinear friction model. The robust adaptive FTC algorithm proposed in this paper can maintain the system performance even after partial failure occurs in the actuator. Different from the traditional fault‐tolerant control algorithms, the proposed algorithm uses the hyperbolic tangent function and a projection algorithm to limit the control input within a known upper bound. In addition, radial basis function neural network is used to approximate the unknown continuous nonlinear function that contains the friction model in the mechanical system. The control target of the system is realized; and the asymptotic stability of the closed‐loop system is proved via the Lyapunov direct method. Finally, the numerical simulation results also demonstrate the effectiveness of the designed controller.

Riemann Zeta Based Surge Modelling of Continuous Real Functions in Electrical Circuits

Riemann zeta is defined as a function of a complex variable that analytically continues the sum of the Dirichlet series, when the real part is greater than unity. In this paper, the Riemann zeta associated with the finite energy possessed by a 2mm radius, free falling water droplet, crashing into a plane is considered. A modified zeta function is proposed which is incorporated to the spherical coordinates and real analysis has been performed. Through real analytic continuation, the single point of contact of the drop at the instant of touching the plane is analyzed. The zeta function is extracted at the point of destruction of the drop, where it defines a unique real function. A special property is assumed for some continuous functions, where the function’s first derivative and first integral combine together to a nullity at all points. Approximate reverse synthesis of such a function resulted in a special waveform named the dying-surge. Extending the proposed concept to general continuous real functions resulted in the synthesis of the corresponding function’s Dying-surge model. The Riemann zeta function associated with the water droplet can also be modeled as a dying–surge. The Dying- surge model corresponds to an electrical squeezing or compression of a waveform, which was originally defined over infinite arguments, squeezed to a finite number of values for arguments placed very close together with defined final and penultimate values. Synthesized results using simulation software are also presented, along with the analysis. The presence of surges in electrical circuits will correspond to electrical compression of some unknown continuous, real current or voltage function and the method can be used to estimate the original unknown function.

Adaptive Asymptotic Regulation for Uncertain Nonlinear Stochastic Systems with Time-Varying Delays

In this paper, for a class of uncertain stochastic nonlinear systems with input time-varying delays, an adaptive neural dynamic surface control (DSC) method is proposed. To approximate the unknown continuous functions online, the neural network approximation technique was applied, and based on the DSC scheme, the desired controller was constructed. A compensation system is presented to compensate for the effect of the input delay. The Lyapunov–Krasovskii functionals (LKFs) were employed to compensate for the effect of the state delay. Compared with the existing works, based on using the DSC scheme with the nonlinear filter and stochastic Barbalat’s lemma, the asymptotic regulation performance of this closed-loop system can be guaranteed under the developed controller. To certify the availability for the designed control method, some simulation results are presented.

A topological approach to inferring the intrinsic dimension of convex sensing data

We consider a common measurement paradigm, where an unknown subset of an affine space is measured by unknown continuous quasi-convex functions. Given the measurement data, can one determine the dimension of this space? In this paper, we develop a method for inferring the intrinsic dimension of the data from measurements by quasi-convex functions, under natural assumptions. The dimension inference problem depends only on discrete data of the ordering of the measured points of space, induced by the sensor functions. We construct a filtration of Dowker complexes, associated to measurements by quasi-convex functions. Topological features of these complexes are then used to infer the intrinsic dimension. We prove convergence theorems that guarantee obtaining the correct intrinsic dimension in the limit of large data, under natural assumptions. We also illustrate the usability of this method in simulations.

Adaptive neural asymptotic tracking control for a class of stochastic non-strict-feedback switched systems

The adaptive asymptotic tracking control problem for a class of stochastic non-strict-feedback switched nonlinear systems is addressed in this paper. For the unknown continuous functions, some neural networks are used to approximate them online, and the dynamic surface control (DSC) technique is employed to develop the novel adaptive neural control scheme with the nonlinear filter. The proposed controller ensures that all the closed-loop signals remain semiglobally bounded in probability, at the same time, the output signal asymptotically tracks the desired signal in probability. Finally, a simulation is made to examine the effectiveness of the proposed control scheme.

MT‐filters‐based adaptive quantized dynamic surface control of stochastic nonstrict‐feedback nonlinear systems including input unmodeled dynamics

AbstractIn this article, the issue of stochastic adaptive dynamic surface control (DSC) is discussed for stochastic nonstrict‐feedback constrained nonlinear systems with input quantization and input unmodeled dynamics. Linearly parametrized neural networks are used to estimate unknown continuous functions. With the help of MT‐filters, the unmeasurable system states are observed. An auxiliary signal constructed by the property of unmodeled dynamics is utilized to deal with the dynamic uncertain terms. Input‐quantized actuator is considered to possess quantization and unmodeled dynamics. Using the hyperbolic tangent function as invertible change, the output constrained stochastic nonstrict‐feedback system is transformed into a novel stochastic nonstrict‐feedback system without restrictions. Based on quantized DSC technology and the property of Gaussian function, adaptive neural control is developed for the transformed stochastic nonstrict‐feedback system. The output abides by stochastic constraints in probability. By the Lyapunov synthesis approach, all signals in the whole system are proved to be semi‐global uniform ultimate bounded in probability. The restriction condition is not triggered. The obtained theoretical findings are verified by two numerical examples.

Adaptive tracking control of uncertain large-scale nonlinear time-delay systems with input saturation

This paper investigates the adaptive tracking control problem of uncertain large-scale nonlinear time-delay systems with input saturation. Unknown constants and continuous functions with respect to output or delayed output are allowed in the nonlinear interconnected functions. The dynamic gain control technique is used to study the tracking control problem of considered system, which simplifies design process and form of controller. By introducing a full-order state observer and three dynamic equations, output feedback tracking control scheme is constructed to ensure that all signals of the closed-loop system are globally uniformly ultimately bounded (GUUB). In addition, the tracking error can converge to a small neighbourhood around the origin by selecting appropriate parameters. Finally, a practical example is presented to illustrate the feasibility of the proposed control scheme.

Hardware-in-the-loop control of glucose in diabetic patients based on nonlinear time-varying blood glucose model

Diabetes Mellitus (DM) is identified as one of the most serious public health challenges which need engineering science to deal with and prevent complications. In this paper, a model of the glucose-insulin system for the virtual type 1 diabetic patient is controlled, which considers factors such as the physical and mental characteristics of the patient. The purpose of this paper is to consider a nonlinear time-varying model of diabetic patients which is affected by Exercise, stress, fatigue, meals, snacks, and sensitivity of insulin (SI) and design an appropriate controller based on the backstepping (BS) technique. In order to make the backstepping controller robust to uncertainties in reality, the level of food, snack, and exercise are considered as unknown continuous functions and the controller is designed such that it can cope with these uncertainties while the blood glucose level is regulated. To verify the performance of the designed controllers, a sample insulin pump is manufactured and the Hardware-In-The-Loop (HIL) experiment is carried out. The result of the HIL test is compared to the in-silico simulation which shows the success of the controller in regulating blood glucose level and preventing the patient from complications of hyperglycemia and hypoglycemia.

Multimodal Dynamic Pricing

We consider a single product dynamic pricing with demand learning. The candidate prices belong to a wide range of a price interval; the modeling of the demand functions is nonparametric in nature, imposing only smoothness regularity conditions. One important aspect of our model is the possibility of the expected reward function to be nonconcave and indeed multimodal, which leads to many conceptual and technical challenges. Our proposed algorithm is inspired by both the Upper-Confidence-Bound algorithm for multiarmed bandit and the Optimism-in-the-Face-of-Uncertainty principle arising from linear contextual bandits. The multiarmed bandit formulation arises from local-bin approximation of an unknown continuous demand function, and the linear contextual bandit formulation is then applied to obtain more accurate local polynomial approximators within each bin. Through rigorous regret analysis, we demonstrate that our proposed algorithm achieves optimal worst-case regret over a wide range of smooth function classes. More specifically, for k-times smooth functions and T selling periods, the regret of our proposed algorithm is [Formula: see text], which is shown to be optimal via the development of information theoretical lower bounds. We also show that in special cases, such as strongly concave or infinitely smooth reward functions, our algorithm achieves an [Formula: see text] regret, matching optimal regret established in previous works. Finally, we present computational results that verify the effectiveness of our method in numerical simulations.This paper was accepted by J. George Shanthikumar, big data analytics.

Observer-based adaptive event-triggered tracking control for nonlinear MIMO systems based on neural networks technique

In this paper, the issue of adaptive neural networks event-triggered control for nonstrict-feedback nonlinear multi-input-multi-output(MIMO) systems containing unmeasured states is investigated. All unmeasured states are approximated by using neural networks observer. Meanwhile, the neural networks are used to estimate the unknown continuous function at each step of recursion. Then, an observer-based adaptive neural networks event-triggered tracking control strategy is proposed based on backstepping technique. The designed controller enables the outputs of the system to track the target trajectory within a small bounded error range, and all signals in the closed-loop system are bounded.

Global output feedback stabilization for a class of nonlinear systems with multiple uncertainties

This paper investigates the problem of global output feedback stabilization for a class of nonlinear systems with multiple uncertainties. A remarkable feature lies in that the system to be considered is not only involved dynamic and parametric uncertainties but also the measurement output affected by an uncertain continuous function, which leads to the obstacles in the constructions of a state observer and a controller. By revamping the double-domination approach with the skillful implantation of a dynamic gain scheme and nonnegative integral functions, a new design strategy is established by which a global output feedback stabilizer together with a novel state observer can be constructed successfully. The novelty of the presented design is attributed to a perspective in dealing with the output feedback stabilization undergone the unknown continuous (time-varying) output function and dynamic/parametric uncertainties. Finally, an illustrative example is provided to illustrate the effectiveness of the theoretical results.

Adaptive Neural Command Filtered Control for Pneumatic Active Suspension With Prescribed Performance and Input Saturation

In this paper, an adaptive neural command filtered backstepping scheme is proposed for the pneumatic active suspension with the vertical displacement constraint of sprung mass and actuator saturation. A quarter car model with a pneumatic spring is first fabricated on the basis of thermodynamic theory to describe the dynamic characteristics. To overcome the lumped unknown nonlinearities and enhance the requirement of modeling precision, the radial basis function neural networks (RBFNNs) are proposed to approximate unknown continuous functions caused by the uncertain body mass and other factors of pneumatic spring. To solve the explosion of complexity problem in the traditional backstepping designs, a proposed command filter control is applied by using the Levant differentiators which approach the derivative of the virtual control signals. Nussbaum gain technique is then incorporated into the controller to avoid the problem of the completely unknown control gain and control directions of a pneumatic actuator. In addition, the prescribed performance function (PPF) is suggested to guarantee that the tracking error of the sprung mass displacement does not violate the constraint boundaries. Based on the command filtered backstepping control with PPF, the Lyapunov theorem is then applied to indicate the system stability analysis. Finally, the comparative simulation examples for the pneumatic suspension are given to verify the effectiveness and reliability of the proposed control.

Adaptive Fuzzy Finite-Time Tracking Control of Stochastic High-Order Nonlinear Systems With a Class of Prescribed Performance

This article investigates the adaptive fuzzy finite-time control problem for a class of high-order stochastic nonlinear systems with a class of exponential type prescribed performance function. It is assumed that the nonlinear functions in the controlled plant are unknown, in which fuzzy logic systems (FLSs) are utilized due to the approximation ability of any unknown continuous functions with arbitrary approximation errors. Based on the FLSs and backstepping design technique, a novel adaptive fuzzy tracking control strategy is proposed to guarantee that the closed-loop nonlinear system is semiglobally finite-time stable in probability via Lyapunov stability theory and It <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\hat{o}$</tex-math></inline-formula> formula. Compared with existing results, the transformed error signal was regarded as a stochastic variable. In addition, the expressions of the first and second-order partial derivatives of the transformed error signals are given in this article. Finally, a simulation example with different covariance values is given to show the effectiveness of the proposed control strategy.

Sliding Mode Dynamic Surface Control for Multi-Machine Power Systems with Time Delays and Dead-Zones

This paper presents a sliding mode dynamic surface control strategy for multi-machine power systems with static var compensator (SVC) to design the controller of the generator excitation system. Firstly, compared with the existing control methods, the combination of sliding mode control and dynamic surface control improves the robustness of the system and avoids the “explosion of complexity” problems. Subsequently, the fuzzy logic system is introduced to estimate the unknown continuous function in power system, and the norm of the weight vector is used to replace the estimation of the entire weight vector. In this way, computational burden will be reduced. In addition, the influences of unknown time delays and dead-zones are considered. Finite cover lemma is introduced to deal with unknown time delays, which enables arbitrarily small tracking performance. The unknown dead-zone is converted into the sum of linear dead zone and bounded nonlinear dead zone. Finally, the simulation results are provided to validate the feasibility and effectiveness of the proposed control strategy.

Adaptive Neural Network Finite-Time Dynamic Surface Control for Nonlinear Systems.

This article addresses the problem of finite-time neural network (NN) adaptive dynamic surface control (DSC) design for a class of single-input single-output (SISO) nonlinear systems. Such designs adopt NNs to approximate unknown continuous system functions. To avoid the "explosion of complexity" problem, a novel nonlinear filter is developed in control design. Under the framework of adaptive backstepping control, an NN adaptive finite-time DSC design algorithm is proposed by adopting a smooth projection operator and finite-time Lyapunov stable theory. The developed control algorithm means that the tracking error converges to a small neighborhood of origin within finite time, which further verifies that all the signals of the controlled system possess globally finite-time stability (GFTS). Finally, both numerical and practical simulation examples and comparing results are provided to elucidate the superiority and effectiveness of the proposed control algorithm.