Multi-dimensional Taylor Network Research Articles

A two-stage fault diagnosis strategy for air handling units via a backpropagation multidimensional Taylor network fitter and a novel statistical process control

Faults in Heating, Ventilation, and Air Conditioning Air Handling Units can lead to increased energy consumption and failure to meet human comfort standards. Accurate and prompt fault diagnosis of Air Handling Units is therefore critical. Although widely-used deep learning methods have shown high precision in fault diagnosis, they present two main drawbacks: 1) deep learning approaches are often “black box” models lacking interpretability and rely on labeled data; 2) their complex structures entail substantial computational resources, which may not be conducive to timely fault diagnosis. To overcome these challenges, this paper introduces a method for diagnosing faults in air handling units that combines a novel statistical process control strategy, a backpropagation multidimensional Taylor network fitter, and a D-matrix approach. The backpropagation multidimensional Taylor network, using a polynomial network with only one hidden layer, approximates nonlinear functions. Compared to deep learning models, it maintains high accuracy with a simpler structure, enabling rapid fault diagnosis. Additionally, the novel statistical process control strategy employs the Markov property for fault detection, characterized by low false alarm rates, interpretability, robustness, minimal labeled data requirements, and suitability for real-time monitoring. Experimental results validate the effectiveness of our method in diagnosing faults in air handling units, achieving an average precision of 99.9%, surpassing several commonly used techniques.

Adaptive prescribed performance tracking control for uncertain nonlinear systems with unknown backlash-like hysteresis

In this paper, an adaptive prescribed performance tracking control(PPTC) problem of uncertain nonlinear systems with unknown hysteresis input is investigated. For the purpose of implementing the PPTC, a performance function and an error conversion function are introduced. Moreover, a backlash-like hysteresis model is adopted to describe the hysteresis nonlinearity, which makes the controller design feasible. Additionally, by converting the backlash-like hysteresis model into a linear model with bounded error, the difficulties caused by hysteresis behavior on controller design are settled. Thus, integrating multi-dimensional Taylor network(MTN) approximation technique into adaptive backstepping method, an adaptive control scheme for uncertain nonlinear systems is proposed. Apart from ensuring that all signals of the closed-loop system keep bounded, the proposed control scheme not only makes the tracking error converge to an arbitrarily small neighborhood around the origin, but also guarantees the tracking error trajectory within the limits set by PPTC. Ultimately, two simulations are applied to verify the validity of the proposed control scheme.

Dynamic event-triggered adaptive tracking control for stochastic nonlinear systems with deferred time-varying constraints

This article investigates the problem of adaptive tracking control for stochastic nonlinear systems with deferred state constraints. The novel unified universal barrier function (UUBF) and deferred funnel error transformation are developed to handle various state constraints and adjust the tracking error more precisely, respectively. Multi-dimensional Taylor network (MTN) and dynamic event-triggered mechanism (DETM) are employed to design controller in the backstepping process, which can achieve full-state constraints and the tracking control after the preassigned time while conserving more communication resources. Finally, two simulation examples are presented to verify the effectiveness of the proposed control scheme.

Adaptive multi-dimensional Taylor network tracking control of time-varying delay nonlinear systems subject to input saturation

For the tracking control problem of a class of nonlinear systems with input saturation constraint and time-varying delay, an adaptive tracking control scheme based on multi-dimensional Taylor network (MTN) is designed, and the proposed control scheme has the advantages of simple structure and small computation. A Gaussian error function is introduced to transform the input saturation into a linear model with bounded error. The Lyapunov-Krasovskii function is constructed, and the nonlinearity of the system is approximated using MTN, and an adaptive control strategy is constructed based on the backstepping method. The results show that the proposed control method can both ensure that all signals in the closed-loop system are bounded and achieve convergence of the tracking error to a small neighborhood near the origin. Numerical simulations verify the effectiveness of the method.

Multi-dimensional Taylor network-based adaptive fixed-time tracking control for a class of nonlinear systems with input delay

For the nonlinear systems with input delay, an adaptive fixed-time tracking control strategy based on multi-dimensional Taylor network (MTN) is proposed for the first time. First, to address the main challenge brought by input delay, an auxiliary system is constructed, which transforms the input delay systems into non-delay systems. Second, the MTNs are introduced into the backstepping design process, which reduce the design difficulty by approximating unknown nonlinear functions. Thus, a control scheme subject to low complexity is obtained. Third, aside from ensuring that the tracking error converges within a small neighbourhood of the origin within a fixed-time, the designed control scheme not only makes all signals in the closed-loop system remain bounded, but also avoids the problem of finite-time control relying on initial state. Finally, three simulations are given to prove the feasibility and superiority of the proposed control scheme.

Adaptive finite‐time control for a class of stochastic nonlinear systems with input saturation constraints: A new approach based on multidimensional Taylor network

AbstractThis article proposes a novel multidimensional Taylor network (MTN)‐based adaptive finite‐time control approach for stochastic nonlinear systems with input saturation constraints. First, by introducing the hyperbolic tangent function, a piecewise smooth function is proposed to approximate the input saturation constraints, which can solve the problem caused by input saturation. Then, with the aid of MTN approximation theorem, a novel adaptive finite‐time control approach is proposed, which can guarantee that all signals of the closed‐loop system are semi‐globally finite‐time stability in probability (SGFTSP), the system output follows the given reference signal, meanwhile, the tracking error converges to a small neighborhood of the origin in finite‐time. Compared with the existing results, the proposed approach has the characteristics of simple structure and low calculation, which can achieve the satisfactory control performance with a reduced computational burden. In the end, the effectiveness of the proposed approach is validated by Lyapunov stability theory and two simulation experiments.

A novel network-based adaptive fault-tolerant control of switched nonlinear systems subject to multiple faults under prescribed performance

It is the first report about fault-tolerant-based prescribed performance control of switched nonlinear systems under multiple faults. The concerned faults include not only external faults but also actuator faults. In the process of backstepping control design, prescribed performance control is fully considered, and the combination of unknown nonlinear functions is estimated by multi-dimensional Taylor network. Finally, the developed adaptive fault-tolerant control strategy guarantees the boundedness of all controlled signals while prescribed tracking performance is satisfied. In an effort to further manifest the validity of the fault-tolerant controller, a numerical simulation and a practical simulation are introduced.

Contraction-Based Stochastic Model Predictive Control for Nonlinear Systems With Input Delay Using Multidimensional Taylor Network

For nonlinear systems with stochastic uncertainties and input delay, the existing control approaches based on the robust mechanism are generally conservative in most practical scenarios. Within this context, a stochastic model predictive control (MPC) scheme based on uncertainty contraction is proposed here to reduce conservativeness and improve real-time performance. In terms of the Lyapunov-Razumikhin theorem, a robust variational controller (RVC) is deduced as the primary controller for the variational dynamic with input delay. A variational vector field in Wasserstein metric space is constructed to detect the minimum geodesic between the uncertainty distribution and the desired one. Integrating RVC along the minimum geodesic, the modified auxiliary controller is developed for tracking the nominal trajectory with less conservatism. Subsequently, the chance constraints of stochastic states are transformed into the deterministic quantile constraints of nominal states. The sparse multi-dimensional Taylor network based on Bayesian compressive sensing is designed to parameterize the ambiguity set in Wasserstein metric space for higher computational efficiency.A tractable MPC scheme is then formulated to achieve the reference index with an improved trade-off between robustness and real-time performance.The stochastic input-to-state stability of the considered system is verified theoretically by the Lyapunov-Krasovskii theorem. The effectiveness of the proposed scheme is confirmed by a numerical simulation derived from a practical industrial process.

Adaptive Multi-Dimensional Taylor Network Tracking Control for a Class of Nonlinear Strict Feedback Systems

Nonlinear systems are very common in real life, but because they are not superposed and homogeneous, there are many difficulties in controlling nonlinear systems. Therefore, an adaptive control method based on a multi-dimensional Taylor network (MTN) is proposed for a class of nonlinear systems with strict feedback so that the output of the system can track the given signal. In order to achieve the control effect, we define a new state variable and transform the strict feedback system. After transformation, the original feedback system has a standard form, and two parameters to be identified are obtained. Then, the state observer is designed, and the two parameters are identified via the approximation of the MTN. On this basis, the controller design and a system stability analysis are completed. The lemma is introduced, and the stability condition is established by using this low-pass filter to ensure that all closed-loop signals are semi-globally uniform and finally bounded and the output tracking error converges to the residual set near zero. Finally, a numerical simulation of a hydraulic system is carried out to verify the effectiveness of the proposed method. Under the three indexes, the proposed method has obvious advantages.

Adaptive tracking control for stochastic nonlinear state-constrained systems with input delay based on multi-dimensional Taylor network

In this paper, an adaptive tracking control strategy based on multi-dimensional Taylor network (MTN) is proposed for a class of stochastic nonlinear state-constrained systems with input delay. Firstly, to avoid constraint violation, the adaptive backstepping approach and barrier Lyapunov function (BLF) are combined in a unified framework. Then, a suitable auxiliary system with the same order as the considered system is constructed to deal with the effect of input delay. In addition, MTN is used to estimate unknown nonlinear functions during the design of the controller. Furthermore, with the help of the Lyapunov stability theorem, the controller presented in this paper can guarantee that all the closed-loop signals are bounded in probability, the system states remain in the defined compact sets, the output signal can track the reference signal successfully, and the tracking error is bounded by the expected bound. Finally, two examples are given to verify the effectiveness of the designed scheme.

Adaptive Robust Control of an Industrial Motor-Driven Stage with Disturbance Rejection Ability Based on Multidimensional Taylor Network

Linear motors are widely used in practical engineering fields, but it is difficult to achieve accurate position control because of model uncertainty, external interference, input nonlinearity and other factors. In this paper, a multidimensional Taylor network (MTN) controller with flexible learning robustness is proposed to realize tracking control and make the motor have excellent antijamming ability. The controller is composed of a robust feedback part, a parameter adaptive part and a multidimensional Taylor network control part. This strategy has good track tracking performance and anti-interference ability. In this control strategy, the input of the multidimensional Taylor network is determined only by the referable trajectories, and no model information is required. In addition, the designed multidimensional Taylor network can accurately describe the relationship between state variables and any complex disturbance, which is the basis of disturbance suppression. Finally, the stability of the controller is proved by the Lyapunov theorem. The unknown interference comparison experiment and numerical simulation experiment are carried out on the industrial linear motor platform, respectively. Experimental results show that compared with NNPID, the proposed algorithm has smaller overshoot and better performance under various error statistical dimensions.

Recursive d-step-ahead predictive control of MIMO nonlinear systems with input time-delay via multi-dimensional Taylor network extended from PID

In this paper, a recursive d-step-ahead predictive control scheme based on multi-dimensional Taylor network (MTN) is proposed for the real-time tracking control of multiple-input multiple-output (MIMO) nonlinear systems with input time-delay. The MTN predictive model is designed using a recursive approach to compensate the influence of time-delay, and an extended Kalman filter (EKF) is applied as its learning algorithm. An MTN controller is developed based on a proportional–integral–derivative (PID) controller where the closed-loop errors between the reference input and the system output are set as the MTN controller’s inputs. Then, a back propagation (BP) algorithm, designed to update its weights according to errors caused by system uncertainty, is used as a learning algorithm for the MTN controller. Meanwhile, the convergence of the MTN predictive model and the stability of the closed-loop system are evaluated. Two numerical examples and a practical example – continuous stirred tank reactor (CSTR) process are presented to verify the superiority of the proposed scheme. The experimental results and the computational complexity analysis show that the proposed scheme is effective, promising its desirable robustness, anti-disturbance, tracking and real-time performance.

Multidimensional-Taylor-network-based robust optimal tracking control for MIMO nonlinear discrete-time systems

In practical engineering, system control is indispensable. However, due to the influence of model uncertainty, speed unavailability, input nonlinearity (such as actuator dead zone/fault), and multi-input coupling, the control results are not satisfactory. In this paper, a robust optimal tracking control strategy is proposed for a class of nonlinear multi-input-multi-output discrete-time systems with unknown uncertainties. This control strategy is to minimize the cost function in the process of uncertainty processing and stabilize the closed-loop system by establishing an adaptive controlling approach based on a combination of actor MTN and critic MTN based on the Multi-dimensional Taylor Network (MTN). By using the approximation property of MTN, the optimal control signal is generated by action MTN, which is used to approach the controller, and the cost function is approximated by critic MTN, which is tuned online because the cost function cannot be obtained in hands-on experience. By designing a new cost function, the amount of calculation in the control process is reduced, and the adaptive critic design control idea is integrated into the controller design to deal with the uncertainty of the system. The simulation results verify the effectiveness of the control strategy proposed in the essay.

Adaptive tracking control for a class of stochastic nonlinear systems with input delay using multidimensional Taylor network

AbstractIn this paper, an adaptive tracking control strategy based on multidimensional Taylor network (MTN) is designed for a class of stochastic nonlinear systems with input delay, in which MTN is mainly introduced to approximate uncertain nonlinear functions. Moreover, the effect of input delay is compensated by exploiting an appropriate auxiliary system with the same order as the considered system. By utilizing the backstepping technique and classical adaptive approach, an MTN‐based adaptive tracking control method is proposed, which guarantees that all the signals in the closed‐loop system remain bounded in probability. Furthermore, unlike most existing approaches, the tracking error can be bounded by an explicit expression of design parameters in the mean quartic sense. Finally, two examples are given to verify the validity of the presented scheme.

Adaptive decentralized prescribed performance control for a class of large‐scale stochastic nonlinear systems subject to input saturation and full state constraints

SummaryThis paper focuses on an adaptive decentralized prescribed performance control problem for a class of large‐scale stochastic nonlinear systems with asymmetric input saturation and full state constraints. Firstly, the obstacle of input saturation is overcome by introducing the Gaussian error functions. Secondly, the transient performance of the system output is realized by introducing the asymmetric error transfer functions. Thirdly, the full state constraints are considered in the backstepping control process, and the boundary of state constraints is ensured by constructing barrier Lyapunov functions. Then, the multidimensional Taylor network is employed to approximate the unknown nonlinearity, and an adaptive decentralized controller is designed. Finally, it is shown that the proposed control strategy can ensure that the closed‐loop system is semi‐global ultimately uniformly bounded in probability, and the tracking error of the system can be kept within an adjustable neighborhood of the origin. Two simulation examples are provided to illustrate the feasibility of the proposed control strategy.

Adaptive control for stochastic nonlinear systems with time‐varying delays via multidimensional Taylor network

AbstractControl of stochastic nonlinear systems turns out to be notoriously difficult when stochastic uncertainties and time‐varying delays occur simultaneously. This article presents a tractable adaptive control scheme for the stochastic nonlinear system with time‐varying delays. To mitigate the effects of stochastic uncertainties, an adaptive embedded cubature Kalman filter is developed to realize the robust estimation of the state. Unlike the conventional cubature Kalman filter with fixed construction, a semi‐definite programming is designed to adjust the weights of cubature points dynamically. Such programming guarantees the positive definiteness of the error covariance matrix, which enhances the reliability of the filtering procedure. Based on more accurate state estimations, the multidimensional Taylor network (MTN) is utilized to evaluate the dynamic performance under time‐varying delays and approximate the optimal policy in the deterministic policy gradient framework. Adaptive tracking control with high computational efficiency is achieved due to the concise topological structure of MTN. The exponential convergence of the state estimation error and the semi‐globally uniform ultimate boundness of the tracking error are verified theoretically. The effectiveness of the proposed method is confirmed by a numerical simulation based on a practical electric industrial system.

Event-triggered adaptive multi-dimensional Taylor network tracking control for stochastic nonlinear systems

For a class of stochastic nonlinear systems, this paper proposes a novel event-triggered adaptive control scheme by means of multi-dimensional Taylor network (MTN) approach for the first time, which has the advantages of alleviating computational burden and reducing communication frequency. In addition, the event-triggered control (ETC) strategy can effectively save network resource by alleviating the computational burden and reducing the communication frequency. Therefore, the proposed control approach can not only reduce communication frequency but also further alleviate computational burden, thereby saving network resource to a greater extent. The proposed control scheme ensures that all signals of the system are semi-global uniformly ultimately bounded (SGUUB) in probability and the tracking error can be made arbitrarily small by choosing appropriate design parameters. Meanwhile, Zeno behavior can be avoided. Finally, two simulation results are given to illustrate the effectiveness of the proposed scheme.

A novel network-based controller design for a class of stochastic nonlinear systems with multiple faults and full state constraints

In this paper, the control issue of adaptive fault-tolerant is studied for a class of stochastic nonlinear systems with multiple faults and full state constraints, with multiple faults including the actuator faults and the external system fault. The problem with full state constraints are solved by constructing a logarithmic barrier Lyapunov functions (BLFs). By integrating multi-dimensional Taylor network (MTN) technology into the backstepping process, a new adaptive MTN-based fault-tolerant controller is designed. On the basis of considering multiple faults, the proposed control strategy can ensure that all signals in the closed-loop system are semi-global ultimately uniformly bounded (SGUUB) in probability, and all states of the system are constrained within the given boundary. Finally, three simulation examples are given to illustrate the effectiveness and practicability of the proposed control strategy.

Adaptive Multi-Switching-Based Global Tracking Control for Switched Nonlinear Systems With Prescribed Performance

This paper is concerned with the tracking control for a class of switched nonlinear systems subject to prescribed performance. Firstly, a barrier function and a normalized function are introduced to achieve prescribed performance control, which ensures that the tracking error evolves within prescribed boundary. Then, multi-dimensional Taylor network provides a new approach for how to estimate the nonlinearity arising from the backstepping control process. Significantly, the proposed multi-switching-based adaptive controller realizes that all signals in the closed-loop system are globally uniformly ultimately bounded while ensuring asymptotic tracking. Different from most existing network-approximation-based control strategies, the developed method in this paper is not only independent of the initial state, but also can achieve global stability. Finally, it is easy to verify the effectiveness of the proposed control method through two simulations. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This research is motivated by the fact that the control ideas of many practical engineering systems can be provided by making a profound study on switched nonlinear systems. However, most of the existing results focused on semi-global stability of systems. Therefore, this study devotes to develop a novel adaptive prescribed performance control strategy, which can not only ensure global stability of closed-loop system but also realize the asymptotic tracking of the switched nonlinear systems. It is worth noting that the proposed control approach has great significance for many practical systems, such as circuit systems and single-link inverted pendulum systems.

A Backpropagation Multidimensional Taylor Network Model for Measured 3-D Data Fitting

A Backpropagation Multidimensional Taylor Network Model for Measured 3-D Data Fitting