Multi-dimensional Taylor Network Research Articles

Novel adaptive controller design for a class of switched nonlinear systems subject to input delay using multi‐dimensional Taylor network

AbstractFor the tracking control problem of a class of switched nonlinear systems with input delay, this article proposes an adaptive tracking control scheme based on multi‐dimensional Taylor network (MTN) approach. First, Padé approximation and Laplace transform are employed to overcome the problem of input delay, and a new variable is introduced. Second, the MTNs are used to approximate the uncertain nonlinear structures in the process of controller design via backstepping. The results show that the controller designed in this article can keep all the signals in the closed‐loop system are bounded, meanwhile, the output tracking error can be converged to arbitrarily small. It should be pointed out that MTN‐based approach is applied to switched nonlinear systems subject to input delay for the first time. Finally, three examples are presented to verify the effectiveness of the proposed control strategy.

Multi-dimensional Taylor network adaptive predictive control for single-input single-output nonlinear systems with input time-delay

In this study, an adaptive predictive control approach based on the multi-dimensional Taylor network (MTN) is proposed for the real-time tracking control of single-input single-output nonlinear systems with input time-delay. Two MTNs are used to implement the accurate tracking control. First, to compensate for the influence of time-delay, MTN is taken as a predictor and the damped recursive least squares algorithm is used as its online learning algorithm. Second, a feed-forward MTN controller is developed on the basis of the proportional–integral–derivative controller, and the closed-loop errors between the reference input and the system output are directly chosen to be the MTN controller’s inputs. The back propagation algorithm is introduced for its learning which can update its weights online at stable learning rate by the errors caused by the system’s uncertain factors. Convergence and stability analysis are given to guarantee the performance of our proposed approach. Finally, two examples are given to verify the effectiveness of the proposed approach.

Design of decentralized adaptive control approach for large-scale nonlinear systems subjected to input delays under prescribed performance

For the first time, the issue of input delay and prescribed performance control is investigated in the same framework for large-scale nonlinear systems in this study, and an original adaptive decentralized control method is proposed by taking advantage of multi-dimensional Taylor network (MTN) method. Firstly, the problem of input delays is solved by introducing new variables, and a new form of coordinate transformation is introduced before controller design, which simplified the control system. Secondly, the problem of prescribed performance control is coped with by integrating the idea of prescribed performance into the Lyapunov functions of first step of backstepping of each subsystem. Thirdly, MTNs are employed to evaluate the combination of unknown functions, and then, a decentralized MTN-based adaptive control scheme is developed by way of backstepping technology. The theoretical analysis indicates that the proposed control scheme can implement the expected tracking goals under the condition of meeting the prescribed performance control. Finally, two examples are given to show the validity and rationality of the proposed control method.

Design Method for a Higher Order Extended Kalman Filter Based on Maximum Correlation Entropy and a Taylor Network System

This paper proposes one new design method for a higher order extended Kalman filter based on combining maximum correlation entropy with a Taylor network system to create a nonlinear random dynamic system with modeling errors and unknown statistical properties. Firstly, the transfer function and measurement function are transformed into a nonlinear random dynamic model with a polynomial form via system identification through the multidimensional Taylor network. Secondly, the higher order polynomials in the transformed state model and measurement model are defined as implicit variables of the system. At the same time, the state model and the measurement model are equivalent to the pseudolinear model based on the combination of the original variable and the hidden variable. Thirdly, higher order hidden variables are treated as additive parameters of the system; then, we establish an extended dimensional linear state model and a measurement model combining state and parameters via the previously used random dynamic model. Finally, as we only know the results of the limited sampling of the random modeling error, we use the combination of the maximum correlation estimator and the Kalman filter to establish a new higher order extended Kalman filter. The effectiveness of the new filter is verified by digital simulation.

Adaptive finite-time control for stochastic nonlinear systems using multi-dimensional Taylor network

In this paper, the problem of adaptive finite-time multi-dimensional Taylor network (MTN) control for a class of stochastic nonlinear systems is investigated. By combining the MTN-based approximate method and adaptive backstepping technique, a novel adaptive finite-time MTN control scheme is proposed. In this scheme, the MTNs are used to approximate the unknown nonlinear functions of the systems. The finite-time Lyapunov stability theory is utilized to prove the stability of the close-loop system. The proposed scheme can ensure that all signals in the closed-loop system are bounded in probability and the tracking error converges to a small neighborhood of the origin in a finite time. Three simulation examples are presented to show the effectiveness of the control scheme. It should be pointed that the adaptive MTN controller proposed in this paper has the advantages of fast computational speed and good real-time performance thanks to the simple structure of the MTN.

Adaptive multi-dimensional Taylor network control for nonlinear stochastic systems with time-delay

In this article, the problem of adaptive multi-dimensional Taylor network control for strict-feedback nonlinear stochastic systems with time-delay is investigated. To overcome the control degradation resulting from the delay terms, the appropriate integral-type Lyapunov–Krasovskii functions are introduced. A novel adaptive multi-dimensional Taylor network control scheme is provided via backstepping technique. The proposed adaptive multi-dimensional Taylor network controller can ensure that all signals in the closed-loop system are bounded in probability and the tracking error eventually converges to a small neighborhood of the origin. Three simulation examples are given to demonstrate the effectiveness of the proposed control scheme.

Adaptive control of a class of stochastic nonlinear systems with full state constraints and input saturation using multi‐dimensional Taylor network

AbstractIn the paper, a new control algorithm to realize the tracking control for a class of stochastic nonlinear systems with both full state constraints and input saturation is put forward using multi‐dimensional Taylor network (MTN) approach. By introducing a continuous function, the input saturation is converted into a linear model with bounded error. With the help of approximation properties of MTN, by proposing barrier Lyapunov functions (BLFs) and involving the idea of state‐constrained control into backstepping technique, a simple and efficient control scheme is developed. The proposed control scheme can guarantee the system states fall in the given constrained bounds as well as keeping the resulting control system stable. The effectiveness of the proposed control method is demonstrated through two examples.

Adaptive decentralized control for large‐scale nonlinear systems with finite‐time output constraints by multi‐ dimensional Taylor network

AbstractThis paper investigates the adaptive decentralized control problem for a class of large‐scale nonlinear systems with finite‐time output constraints. In order to ensure that the tracking errors are constrained within a predefined boundary in finite time, a novel adaptive barrier Lyapunov function (BLF) control method is proposed by combining the modified finite‐time performance function (FTPF) in the first step of backstepping process. Besides, the mean value theorem and regulating functions are employed to handle the difficulties caused by interconnection functions in large‐scale systems. Subsequently, with the approximation performance of multi‐dimensional Taylor network (MTN), a MTN‐based adaptive decentralized tracking control scheme is developed to guarantee that the tracking errors satisfy the prescribed performance and all signals of the closed‐loop systems are bounded. Finally, the stability theory analysis and simulation results demonstrate the effectiveness of the proposed method.

Adaptive multi‐dimensional Taylor network funnel control of a class of nonlinear systems with asymmetric input saturation

SummaryIn view of the result and performance of control are affected by the existence of input constraints and requirements, adaptive multi‐dimensional Taylor network (MTN) funnel control problem is studied for a class of nonlinear systems with asymmetric input saturation in this paper. Firstly, the effect of asymmetric input saturation can overcome by introducing the Gaussian error function, namely, the asymmetric saturation model is represented as a simple linear model with a bounded disturbance. Secondly, MTNs are employed to approximate the unknown functions in the controller design. Then, an adaptive MTN tracking controller is developed by blends the idea of funnel control into backstepping, which can guarantee that the tracking error always meets the given prescribed performance regarding the transient and steady state responses as well as the output of system tracks the give continuous reference signal. Finally, the effectiveness of the proposed control is demonstrated using two examples.

MTN output feedback tracking control for MIMO discrete-time uncertain nonlinear systems

An adaptive controller is developed that is based on the multidimensional Taylor network (MTN). This controller is used for multi-input and multi-output (MIMO) uncertain discrete-time nonlinear systems. This newly developed MTN is dissimilar with the neural network, in which only multiplication and addition are needed for this controller. Thus, real-time control is more easily to be achieved. The theoretical analysis shows that the output errors of the system are convergent and the output signals are semi-globally, uniformly and ultimately bounded. To illustrate the validity of MTN-based adaptive controller (MTNAC), a numerical example is given. The simulation data demonstrate that this MNTAC has better real-time performance and higher robustness compared with neural networks.

Multi-dimensional Taylor network-based adaptive control for nonlinear systems with unknown parameters

This paper presents an adaptive multi-dimensional Taylor network (MTN) control approach for a class of nonlinear systems with unknown parameters. MTN is employed to identify unknown nonlinear characteristics existing in the system, and then a novel adaptive MTN tracking control method is proposed, via backstepping technique. In the controller design, double adaptive laws are designed and appropriate Lyapunov functions are chosen to overcome the difficulties caused by the unknown parameters. The designed controller can guarantee that all the variables in the closed-loop systems are bounded and the tracking error can be arbitrarily small. Finally, simulation results are presented to verify the effectiveness of the proposed approach.

Adaptive tracking control for a class of stochastic non‐linear systems with input delay: a novel approach based on multi‐dimensional Taylor network

In this study, by means of the techniques of adaptive multi-dimensional Taylor network (MTN) control, the tracking problem of a class of stochastic non-linear systems with input delay is studied. Firstly, based on the Pade approximation method, a new variable is employed to overcome the problem of input delay, and MTNs are applied to estimate the non-linear functions in the design of the controller. Secondly, an adaptive MTN controller design scheme is developed in the framework of backstepping. Thirdly, the stability of the closed-loop system is analysed using Lyapunov's stability theory. Finally, the effectiveness of the proposed design approach is demonstrated by two examples.

Adaptive multi-dimensional Taylor network dynamic surface control for a class of strict-feedback uncertain nonlinear systems with unmodeled dynamics and output constraint

Unmodeled dynamics and output constraint are very often experienced in many physical applications, while few studies on them simultaneously are constructed. This study concentrates on the problem of adaptive multi-dimensional Taylor network dynamic surface control (MTN-DSC) for a class of strict-feedback uncertain nonlinear systems subjected to unmodeled dynamics and output constraint. The purpose is to make this study more practical, and design a controller with simple structure to improve the system performance. A barrier Lyapunov function is used to restrict the system output to within the prescribed constraint. At each step during the backstepping design process, an MTN is applied to approach the generated unknown composite function stemming from the unknown functions and uncertain disturbances, rather than individually; this simplifies the structure and reduces the complexity of the controllers. Meanwhile, the computational burden is further reduced, as only one parameter is adjusted online, which has the minimum number of parameters to be adjusted. Additionally, DSC technique is introduced to eliminate the inherent “explosion of complexity” problem in traditional backstepping design procedure. Furthermore, it is proved that all signals of the closed-loop system are semi-globally uniformly ultimately bounded, and that the output constraint is never violated. The validity of the proposed control scheme is illustrated through the numerical and applied simulation examples.

Adaptive sliding mode robust control based on multi‐dimensional Taylor network for trajectory tracking of quadrotor UAV

In this study, multi-dimensional Taylor network (MTN) inspired sliding mode robust control theory based adaptation laws are proposed to realise trajectory tracking for a quadrotor unmanned aerial vehicle. Compared with existing methods, the composite disturbance considered in this study includes external multiple interference and fuselage parameter perturbation, which is more in accord with reality. Through a linear state transformation, the uncertain coefficients and unknown external disturbances are grouped together and the original kinetic model is decoupled into two subsystems of position and attitude that make the control design become feasible. MTNs are used to compensate the lumped non-linearities, and the adaptive sliding mode technique is employed to construct the controller. Different from the controllers based on neural networks, MTN contains only addition and multiplication, which greatly lessens the computational complexity and improves the real-time performance. By the Lyapunov theory, the position tracking error, attitude tracking error and fuselage load identification error are bounded in probability have been proved and the stability of the system is guaranteed. Simulation studies demonstrate the effectiveness and superiority of the proposed trajectory tracking control scheme.

Robust model predictive control based on recurrent multi‐dimensional Taylor network for discrete‐time non‐linear time‐delay systems

This study is concerned with the robust model predictive control (MPC) based on recurrent multi-dimensional Taylor network (RMTN) for the discrete-time non-linear time-delay systems. Regarding the MPC algorithm for the discrete-time time-delay systems, the existing literature only considers the linear case. In this study, by designing the suitable terminal cost and terminal region, the MPC scheme is firstly investigated for the non-linear case. Meanwhile, to reduce the computational burden of MPC using the existing model types (e.g. recurrent neural network) as the identified model, an RMTN possessing the concise structure and high computational efficiency is constructed to approximate the state-space model of the system. After trained by the backpropagation through time algorithm, the RMTN obtaining the high accuracy of prediction over a long-range horizon is capable of serving as the prediction model in the MPC scheme. Furthermore, aiming at alleviating the adverse effect of the inevitable identification error, the tube-based MPC is proposed via leveraging dual-mode MPC to guarantee that actual trajectory is contained within a robust tube. The stability of the considered system is proved theoretically, and numerical simulation is employed to validate the effectiveness of the proposed method.

Tube-Based Model Predictive Control Using Multidimensional Taylor Network for Nonlinear Time-Delay Systems

For nonlinear time-delay systems with uncertainties, the existing approaches to robust model predictive control (MPC) are based on the min-max optimization formulation. Unfortunately, these approaches are generally conservative for most practical problems. For this sake, this article proposes a tube-based MPC consisting of MPC and control contraction metric (CCM) controller. The MPC is utilized as a nominal controller to generate a reference trajectory, while the CCM controller is used as a local ancillary controller to guarantee the actual trajectory to be contained within a robust invariant tube centered along the reference trajectory. Besides, we construct a variational formulation multidimensional Taylor network (MTN) as the basis function to search for the minimal geodesic. With the effective training algorithm, MTN could efficiently approximate the solution with high accuracy. The incremental exponential stability of the considered systems is proved theoretically, and the effectiveness of the proposed method is illustrated by numerical simulation.

Inverse control of single-input/single-output nonlinear time-varying systems with noise disturbances by multi-dimensional Taylor network

In this paper, an inverse control scheme based on the novel dynamic network (multi-dimensional Taylor network (MTN)) is proposed for the real-time tracking control of nonlinear time-varying systems with noise disturbances. Utilized in this scheme are the three MTNs: the adaptive model identifier for system modeling, the adaptive inverse controller for inverse modeling, and the adaptive nonlinear filter for eliminating the noise disturbances, whose weights are modified by the variable forgetting factor recursive least squares (VFF-RLS), back propagation through model (BPTM), normalized least mean square (NLMS) algorithms, respectively. To avoid “compromise”, this scheme is designed into a structure wherein controlling the object dynamic response and eliminating the noise disturbances are implemented in two relatively independent processes. Furthermore, the weight-elimination algorithm is introduced for choice of effective regression items to avoid the dimension explosion, thus overcoming the shortcoming that the number of middle nodes needs to be determined before using the traditional neural network. After a certain number of training, the more streamlined MTNs are observed to contribute to satisfying the real-time requirements of software implementation and engineering application. To ensure that MTN inverse control is strict in theory, the general conditions for the existence of single-input/single-output (SISO) nonlinear inverse systems are identified. Simulation of the MTN inverse control is conducted to confirm the effectiveness of the proposed method.

Adaptive multi-dimensional Taylor network tracking control for a class of switched nonlinear systems with input nonlinearity

In this paper, a multi-dimensional Taylor network (MTN)-based adaptive tracking control approach is proposed for a class of switched nonlinear systems with input nonlinearity. Firstly, the input nonlinearity is assumed to be bounded by a sector interval. Secondly, with the help of MTNs approximating the unknown nonlinear functions, a novel adaptive MTN control scheme has the advantages of low cost, simple structure and real time feature is developed via backstepping technique. It is shown that the tracking error finally converges to a small domain around the origin and all signals in the closed-loop system are bounded. Finally, two examples are given to demonstrate the effectiveness of the proposed control scheme.

Adaptive tracking control of a class of nonlinear systems with unknown dead-zone output: a multi-dimensional Taylor network (MTN)-based approach

ABSTRACT In this paper, an approximation-based adaptive multi-dimensional Taylor network (MTN) control approach is proposed for a class of nonlinear systems with unknown dead-zone output. Firstly, MTNs are utilised to approximate the unknown nonlinearity of the system. Secondly, the relation of the unknown function and the output is establish. Thirdly, integrating adaptive control method and MTNs into the backstepping design process, a novel MTN-based adaptive controller is constructed by introducing a Nussbaum function. The proposed control scheme has the advantages of simple structure and small calculation. Finally, two simulation examples are given to illustrate the effectiveness of the method proposed in this paper.

Adaptive tracking control for a class of stochastic non‐linear systems with input saturation constraint using multi‐dimensional Taylor network

The randomness and input saturation increase computational complexity and impede the tracking performance of stochastic non-linear systems, therefore, it is necessary to build a simple but effective controller. Based on this, a multi-dimensional Taylor network (MTN) is first applied to a class of stochastic non-linear systems with input saturation constraint in this study. Aiming to solve the tracking control problem, a novel MTN-based controller is proposed via backstepping, and the proposed control scheme has some advantages such as simple structure, good real-time performance and easy realisation. With the help of the hyperbolic tangent function approximating the symmetric saturation non-linearity, an adaptive MTN tracking control scheme is constructively designed via backstepping technique. The simulation results are given to illustrate the validity and accuracy of the model of the proposed approach.