A Constructive Approach for Neural Network Approximation Sets in Adaptive Control of Strict-Feedback Systems.

Nonlinear mathematical-programming-based design optimization can be an elegant method. However, the calculations required to generate the merit function, constraints, and their gradients, which are frequently required, can make the process computational intensive. The computational burden can be greatly reduced by using approximating analyzers derived from an original analyzer utilizing neural networks and linear regression methods. The experience gained from using both of these approximation methods in the design optimization of a high speed civil transport aircraft is the subject of this paper. The Langley Research Center's Flight Optimization System was selected for the aircraft analysis. This software was exercised to generate a set of training data with which a neural network and a regression method were trained, thereby producing the two approximating analyzers. The derived analyzers were coupled to the Lewis Research Center's CometBoards test bed to provide the optimization capability. With the combined software, both approximation methods were examined for use in aircraft design optimization, and both performed satisfactorily. The CPU time for solution of the problem, which had been measured in hours, was reduced to minutes with the neural network approximation and to seconds with the regression method. Instability encountered in the aircraft analysis software at certain design points was also eliminated. On the other hand, there were costs and difficulties associated with training the approximating analyzers. The CPU time required to generate the input-output pairs and to train the approximating analyzers was seven times that required for solution of the problem.

In this paper, the design of a neural network (NN) based adaptive model-based event-triggered control of an uncertain single input single output (SISO) nonlinear discrete time system in affine form is presented. The controller uses an adaptive estimator consisting of a single-layer NN not only to approximate the internal dynamics of an affine nonlinear discrete-time system but also to provide an estimate of the state vector during inter event interval. The NN weights of the adaptive NN estimator are tuned in a aperiodic manner at the event trigger instants unlike periodic updates in standard adaptive neural network (NN) control. A dead zone operator is used to reset the event trigger error to zero as long as the system states continue to remain in a bounded region due to NN reconstruction errors. Lyapunov method is used to derive the event trigger condition, prove uniform ultimate boundedness (UUB) of the NN weight estimation error and system states.

An online adaptive optimal control is proposed for continuous-time nonlinear systems with completely unknown dynamics, which is achieved by developing a novel identifier-critic-based approximate dynamic programming algorithm with a dual neural network (NN) approximation structure. First, an adaptive NN identifier is designed to obviate the requirement of complete knowledge of system dynamics, and a critic NN is employed to approximate the optimal value function. Then, the optimal control law is computed based on the information from the identifier NN and the critic NN, so that the actor NN is not needed. In particular, a novel adaptive law design method with the parameter estimation error is proposed to online update the weights of both identifier NN and critic NN simultaneously, which converge to small neighbourhoods around their ideal values. The closed-loop system stability and the convergence to small vicinity around the optimal solution are all proved by means of the Lyapunov theory. The proposed adaptation algorithm is also improved to achieve finite-time convergence of the NN weights. Finally, simulation results are provided to exemplify the efficacy of the proposed methods.

SummaryThis paper studies the problem of stabilizing reference trajectories (also called as the trajectory tracking problem) for underactuated marine vehicles under predefined tracking error constraints. The boundary functions of the predefined constraints are asymmetric and time‐varying. The time‐varying boundary functions allow us to quantify prescribed performance of tracking errors on both transient and steady‐state stages. To overcome difficulties raised by underactuation and nonzero off‐diagonal terms in the system matrices, we develop a novel transverse function control approach to introduce an additional control input in backstepping procedure. This approach provides practical stabilization of any smooth reference trajectory, whether this trajectory is feasible or not. By practical stabilization, we mean that the tracking errors of vehicle position and orientation converge to a small neighborhood of zero. With the introduction of an error transformation function, we construct an inverse‐hyperbolic‐tangent‐like barrier Lyapunov function to show practical stability of the closed‐loop systems with prescribed transient and steady‐state performances. To deal with unmodeled dynamic uncertainties and external disturbances, we employ neural network (NN) approximators to estimate uncertain dynamics and present disturbance observers to estimate unknown disturbances. Subsequently, we develop adaptive control, based on NN approximators and disturbance estimates, that guarantees the prescribed performance of tracking errors during the transient stage of on‐line NN weight adaptations and disturbance estimates. Simulation results show the performance of the proposed tracking control.

In recent years, neural networks have achieved remarkable progress in various fields and have also drawn much attention in applying them on scientific problems. A line of methods involving neural networks for solving partial differential equations (PDEs), such as Physics-Informed Neural Networks (PINNs) and the Deep Ritz Method (DRM), has emerged. Although these methods outperform classical numerical methods in certain cases, the optimization problems involving neural networks are typically non-convex and non-smooth, which can result in unsatisfactory solutions for PDEs. In contrast to deterministic neural networks, the hidden weights of random neural networks are sampled from some prior distribution and only the output weights participate in training. This makes training much simpler, but it remains unclear how to select the prior distribution. In this paper, we focus on Barron type functions and approximate them under Sobolev norms by random neural networks with clear prior distribution. In addition to the approximation error, we also derive bounds for the optimization and generalization errors of random neural networks for solving PDEs when the solutions are Barron type functions.

This paper proposes a finite-time adaptive backstepping control for an n-link flexible joint manipulator based on neural network approximation. In each recursive step, an adaptive virtual controller or practical controller is designed to guarantee that all the state errors can converge into a small region within a finite time. Besides, two simple neural networks are employed to approximate and compensate for the lumped uncertainties, and the finite time stability analysis is provided based on Lyapunov synthesis. Finally, simulation results show the effectiveness of the proposed method.

PurposeBuilding cost is an important part of construction projects, and its correct estimation has important guiding significance for the follow-up decision-making of construction units.Design/methodology/approachThis study focused on the application of back-propagation (BP) neural network in the estimation of building cost. First, the influencing factors of building cost were analyzed. Six factors were selected as input of the estimation model. Then, a BP neural network estimation model was established and trained by ten samples.FindingsAccording to the experimental results, it was found that the estimation model converged at about 85 times; compared with radial basis function (RBF), the estimation accuracy of the model was higher, and the average error was 5.54 per cent, showing a good reliability in cost estimation.Originality/valueThe results of this study provide a reliable basis for investment decision-making in the construction industry and also contribute to the further application of BP neural network in cost estimation.

This paper addresses the learning control problem for a group of robot manipulator systems in the presence of homogeneous nonlinear uncertain dynamics, where all the robots have an identical system structure but the reference signals to be tracked are different. Our control law has two objectives: one is tracking control, and the other is learning/identification. To this end, deterministic learning theory is incorporated with consensus theory to find a common neural network (NN) approximation of the nonlinear uncertain dynamics for the multi-robot system. Specifically, we first present a control scheme called cooperative deterministic learning using adaptive NNs to enable the robotic agents to track their respective reference trajectories on one hand, and to exchange their estimated NN weights online through networked communication on the other. As a result, a consensus about one common NN approximation for the nonlinear uncertain dynamics is achieved for all the agents. Thus, the trained distributed NNs have better generalization capability than those obtained by existing techniques. Numerical simulations on a team of two-degrees-of-freedom robot manipulators have been conducted to demonstrate the effectiveness of the proposed results.

Neural network-based adaptive fault-tolerant control for nonlinear systems with unknown backlash-like hysteresis and unmodeled dynamics

In theory, recent results in nonparametric regression show that neural network estimates are able to achieve good rates of convergence provided suitable assumptions on the structure of the regression function are imposed. However, these theoretical analyses cannot explain the practical success of neural networks since the theoretically studied estimates are defined by minimizing the empirical L_2 risk over a class of neural networks and in practice, solving this kind of minimization problem is not feasible. Consequently, the neural networks examined in theory cannot be implemented as they are defined. This means that neural network in applications differ from the ones that are analyzed theoretically. In this thesis we narrow the gap between theory and practice. We deal with neural network regression estimates for (p,C)-smooth regression functions m that satisfy a projection pursuit model. We construct three implementable neural network estimates and show that each of them achieve up to a logarithmic factor the optimal univariate rate of convergence. Firstly, for univariate regression functions with p contained in [-1/2,1] we construct a neural network estimate with one hidden layer where the weights are learned via gradient descent. The starting weights are randomly chosen from an interval independently of the data. The interval is large enough to guarantee that the estimate is close to a piecewise constant approximation. Secondly, for multivariate regression functions with p contained in (0,1] we construct a neural network estimate with one hidden layer where the weights are learned via gradient descent. The initial weights are chosen from specific intervals dependently on the data and the projection directions. This choice guarantees that the estimate is close to a piecewise constant approximation. The projection directions are repeatedly chosen randomly. Lastly, for multivariate regression functions with p>0 we construct a multilayer neural network estimate. The value of the inner weights are prescribed dependently on the projection directions by a new approximation result for a projection pursuit model by piecewise polynomials. The outer weights are chosen by solving a linear equation system. The projection directions are repeatedly chosen randomly. Since we are able to show a rate of convergence that is independent of the dimension of the data our second and third estimates are able to circumvent the curse of dimensionality.

Neural network-based event-triggered fault detection of discrete-time nonlinear uncertain systems

One of the most common problems in science is to investigate a function describing a system. When the estimate is made based on a classical mathematical model (white-box), the function is obtained throughout solving a differential equation. Alternatively, the prediction can be made by an artificial neural network (black-box) based on trends found in past data. Both approaches have their advantages and disadvantages. Mathematical models were seen as more trustworthy as their prediction is based on the laws of physics expressed in the form of mathematical equations. However, the majority of existing mathematical models include different empirical parameters, and both approaches inherit inevitable experimental errors. Simultaneously, the approximation of neural networks can reproduce the solution exceptionally well if fed sufficient data. The difference is that an artificial neural network requires big data to build its accurate approximation, whereas a typical mathematical model needs several data points to estimate an empirical constant. Therefore, the common problem that developers meet is the inaccuracy of mathematical models and artificial neural networks. Another common challenge is the mathematical models’ computational complexity or lack of data for a sufficient precision of the artificial neural networks. Here we analyze a grey-box solution in which an artificial neural network predicts just a part of the mathematical model, and its weights are adjusted based on the mathematical model’s output using the evolutionary approach to avoid overfitting. The performance of the grey-box model is statistically compared to a Dense Neural Network on benchmarking functions. With the use of Shaffer procedure, it was shown that the grey-box approach performs exceptionally well when the overall complexity of a problem is properly distributed with the mathematical model and the Artificial Neural Network. The obtained calculation results indicate that such an approach could increase precision and limit the dataset required for learning. To show the applicability of the presented approach, it was employed in modeling of the electrochemical reaction in the Solid Oxide Fuel Cell’s anode. Implementation of a grey-box model improved the prediction in comparison to the typically used methodology.

A reactor integrated with separation process has been widely proposed and studied to enhance the production of high value added chemicals. An example of such a process is a pervaporative membrane reactor which is the combination of a reactor and a pervaporation part. The inclusive of a pervaporation can shift the chemical equilibrium of esterification by passing water out of reaction mixture and therefore increase the yield of a desired product. In addition, the temperature of the reactor is one of key factors to define the rate of reactions involved. In this work, a neural network inverse models based control (NIMC) has been designed to control an optimal temperature of a pervaporation membrane reactor. Furthermore, a neural network estimator (NNE) has been incorporated to the strategy as an estimator to estimate the amount of heat released of an esterification reaction. The performance of the designed controller has been evaluated in both nominal and plant/model mismatch cases, and compared with GMC coupled with the Kalman Filter. Simulation study has shown that the NIMC with the NNE can successfully control the reactor temperature at a desired set point in nominal cases and plant/model mismatches. The performance of the NIMC with the NNE is equivalent to that of the GMC coupled with the Kalman Filter. As a result, the NIMC with the NNE is applicable to control the pervaporative membrane reactor without first principle models of the reactor is available. This shows the advantage of the NIMC with NNE over the GMC with the Kalman filter.

In this article, a globally stable adaptive control strategy for uncertain strict-feedback systems is proposed within predefined neural network (NN) approximation sets, despite the presence of unknown system nonlinearities. In contrast to the conventional adaptive NN control results in the literature, a primary benefit of the developed approach is that the barrier Lyapunov function is employed to predefine the compact set for maintaining the validity of NN approximation at each step, thus accomplishing the global boundedness of all the closed-loop signals. Simulation results are performed to clarify the effectiveness of the proposed methodology.

This paper considers the output-feedback control problem for a class of stochastic nonlinear systems with unknown control directions and perturbations. By using radial basis function neural network (RBF NN) approximation approach, the tuning function method and backstepping technique, an adaptive output-feedback controller is successfully constructed to guarantee the closed-loop system to be mean square semi-globally uniformly ultimately bounded (M-SGUUB). A simulation example demonstrates the effectiveness of the proposed scheme.