Nonlinear Function Approximation Research Articles

An Optimized Method for Nonlinear Function Approximation Based on Multiplierless Piecewise Linear Approximation

In this paper, we propose an optimized method for nonlinear function approximation based on multiplierless piecewise linear approximation computation (ML-PLAC), which we call OML-PLAC. OML-PLAC finds the minimum number of segments with the predefined fractional bit width of input/output, maximum number of shift-and-add operations, user-defined widths of intermediate data, and maximum absolute error (MAE). In addition, OML-PLAC minimizes the actual MAE as much as possible by iterating. As a result, under the condition of satisfying the maximum number of segments, the MAE can be minimized. Tree-cascaded 2-input and 3-input multiplexers are used to replace multi-input multiplexers in hardware architecture as well, reducing the depth of the critical path. The optimized method is applied to logarithmic, antilogarithmic, hyperbolic tangent, sigmoid and softsign functions. The results of the implementation prove that OML-PLAC has better performance than the current state-of-the-art method.

Quality prediction for polypropylene extrusion based on neural networks

In the polypropylene (PP) industry, melt index (MI) is considered the most important quality variable. Different grades of PP have their specific range of MI. Accurate prediction of MI is essential for efficient monitoring and off-grade reduction. Neural Networks (NN) modelling is proposed as the technique for MI estimation. It has powerful adaptive capabilities in response to nonlinear behaviours. By training the NN, it can discover the relationship between inputs and outputs and makes it capable of function approximation. The goal of this research is to develop NN model to predict the MI based on PP extrusion parameters. Different types of NN such as artificial neural networks (ANN), stacked neural networks (SNN) and deep neural networks were trained and compared to understand their efficiency in solving the problem. The simulation results show that deep neural networks can perform the highest accuracy prediction with the lowest root mean square error (RMSE), followed by SNN and ANN. All three modelling proved that NN can perform non-linear function approximation for polymer extrusion.

Continuous Action Reinforcement Learning From a Mixture of Interpretable Experts.

Reinforcement learning (RL) has demonstrated its ability to solve high dimensional tasks by leveraging non-linear function approximators. However, these successes are mostly achieved by 'black-box' policies in simulated domains. When deploying RL to the real world, several concerns regarding the use of a 'black-box' policy might be raised. In order to make the learned policies more transparent, we propose in this paper a policy iteration scheme that retains a complex function approximator for its internal value predictions but constrains the policy to have a concise, hierarchical, and human-readable structure, based on a mixture of interpretable experts. Each expert selects a primitive action according to a distance to a prototypical state. A key design decision to keep such experts interpretable is to select the prototypical states from trajectory data. The main technical contribution of the paper is to address the challenges introduced by this non-differentiable prototypical state selection procedure. Experimentally, we show that our proposed algorithm can learn compelling policies on continuous action deep RL benchmarks, matching the performance of neural network based policies, but returning policies that are more amenable to human inspection than neural network or linear-in-feature policies.

State estimation and fuzzy sliding mode control of nonlinear Markovian jump systems via adaptive neural network

This paper deals with the problem of Takagi-Sugeno fuzzy model-based state estimation and sliding mode control for nonlinear systems through an adaptive neural network, in which the system parameters follow the Markovian switching rules. In order to deal with the unknown local nonlinearity, a multi-layer neural network is used for the nonlinear function approximation. First, a Lebesgue fuzzy observer with adaptive compensator is designed based on state-dependent fuzzy rules. Second, an integral sliding surface is proposed, based on which the obtained sliding mode dynamics has good property of sliding mode manifold. Third, an H∞ performance with stochastic stability of the sliding mode dynamics and error dynamics are developed in the form of linear matrix inequality. Moreover, reachability of sliding surface in finite-time and maintenance of sliding motion are realized by constructing a fuzzy sliding mode controller. Finally, a simulation study is added to show the validity of the proposed results on the robot manipulator model.

QUALITY PREDICTION WITH NEURAL NETWORK TECHNIQUES FOR POLYPROPYLENE PRODUCTION VIA THE SPHERIPOL PROCESS

In the polypropylene (PP) industry, melt index (MI) is the most important quality variable. Different grades of PP have their specific range of MI. Accurate prediction of MI is essential for efficient monitoring and off-grade reduction. Artificial Neural Network (ANN) models are proposed as the technique for MI estimation. It has powerful adaptive capabilities in response to nonlinear behaviour. In this research, ANN models for PP polymerization to predict the MI based on reactor parameters were developed. Three types of ANN models, the single hidden layer ANN (shallow ANN), stacked neural network (SNN) and deep learning are compared. The simulation results show that deep learning can perform better than shallow ANN and SNN by considering the accuracy of the prediction and detection of process fluctuation. All three model have proven that ANN are able to perform non-linear function approximation. Thus, ANN models are effective for supporting MI prediction such as for soft-sensors and process optimization in the polymer industry.

A Boundary Regression Model for Nested Named Entity Recognition

Recognizing named entities (NEs) is commonly treated as a classification problem, and a class tag for a word or an NE candidate in a sentence is predicted. In recent neural network developments, deep structures that map categorized features into continuous representations have been adopted. Using this approach, a dense space saturated with high-order abstract semantic information is unfolded, and the prediction is based on distributed feature representations. In this paper, the positions of NEs in a sentence are represented as continuous values. Then, a regression operation is introduced to regress the boundaries of NEs in a sentence. Based on boundary regression, we design a boundary regression model to support nested NE recognition. It is a multiobjective learning framework that simultaneously predicts the classification score of an NE candidate and refines its spatial location in a sentence. This model was evaluated on the ACE 2005 Chinese and English corpus and the GENIA corpus. State-of-the-art performance was experimentally demonstrated for nested NE recognition, which outperforms related works about 5% and 2% respectively. Our model has the advantage to resolve nested NEs and support boundary regression for locating NEs in a sentence. By sharing parameters for predicting and locating, this model enables more potent nonlinear function approximators to enhance model discriminability.

Improved Deep Multi-Patch Hierarchical Network For Handling Saturation In Image Deblurring

The most active study topic in the field of computer vision is dealing with saturation in image deblurring. The Deep Multi Patch Framework (DMPHN) is primarily utilized in dynamic scene deblurring, and its fine-to-coarse hierarchical representation results in lower processing costs. The contribution of the framework in gathering and processing local residual information of blur at the courser levels in the hierarchy makes the dynamic scene deblurring successful. The existing framework uses the traditional CNN with the convolutional filter, which is a generalized linear model (GLM) that goes along with a nonlinear activation function to scan the input. When the samples of latent concepts are not linearly separable then the linear CNN cannot abstract the good representations. Because saturation in an image contradicts the linear properties of the blur model, the present Deep Multi Patch architecture with linear CNN is unable to collect the necessary local information of blur in an image. In CNN, the higher-level layer maps to the larger regions of the input image by combining the lower-level concepts from each local patch. Due to the framework’s incapacity to retain information at lower levels, noticeable ringing artifacts and poor deblurring performance resulted. We designed a nonlinear function approximator for this section, which was influenced by the good abstraction capabilities of ”Network-in-Network”. The nonlinear features of saturated pixels and blur are successfully captured by this upgraded DMPHN with a nonlinear function approximator. Compared to the current DMPHN model, the suggested approach greatly reduced the number of trainable parameters while improving the time cost. On the GoPro, Synthetic, and Natural Saturated Blurred Image datasets, the upgraded DMPHN network achieves superior deblurring performance with better PSNR and SSIM than the state-of-the-art.

Analytic Deep Neural Network-Based Robot Control

Neural networks have been extensively used in robot control for various applications because of their powerful capability in approximation of nonlinear functions. However, existing literature on feedback control of robots mainly focuses on shallow networks where the analysis is developed for the output weights only and the linearity in parameters is often a requirement. This is due to the fact that convergence analysis is difficult for deep networks. Since stability and convergence are critical in robot control, our main aim is to develop a theoretical framework for using deep networks in robotics in a safe and predictable manner. In this article, we use a deep network to approximate the Jacobian matrix of a robot with unknown kinematics. An analytic layer-wise deep learning framework is proposed where the deep network is progressively built and trained, and the convergence of the tracking error is guaranteed during the online learning process. The experimental results for tracking control tasks performed on an industrial robot are given to illustrate the effectiveness of the proposed method.

Superiority of q-Chlodowsky operators versus fuzzy systems and neural networks: Application to adaptive impedance control of electrical manipulators

This paper introduces a novel application of q-Chlodowsky operators in the approximation of unknown nonlinear functions including uncertainties, un-modeled dynamics, and external disturbances. In fact, q-Chlodowsky operators play the role of basis functions with unknown coefficients. Furthermore, an effective model-free observer is designed for estimation of the task-space velocity signals of the end-effector. The results illustrate that the performances of both radial basis functions neural networks (RBFNN) and the q-Chlodowsky-based approach are nearly the same due to the universal approximation property of both estimators, while the adaptive fuzzy controller needs optimal tuning which is time consuming. Therefore, compared with fuzzy systems and neural networks, the proposed scheme is superior in terms of simplicity and is less computational due to the state-free basis functions in the regressor vector. Simulation results on a 2-DOF electrical manipulator effectively verify the efficiency of the proposed strategy.

Fast and Accurate Approximation Methods for Trigonometric and Arctangent Calculations for Low-Performance Computers

In modern computers, complicated signal processing is highly optimized with the use of compilers and high-speed processing using floating-point units (FPUs); therefore, programmers have little opportunity to care about each process. However, a highly accurate approximation can be processed in a small number of computation cycles, which may be useful when embedded in a field-programmable gate array (FPGA) or micro controller unit (MCU), or when performing many large-scale operations on a graphics processing unit (GPU). It is necessary to devise algorithms to obtain the desired calculated values without an accelerator or compiler assistance. The residual correction method (RCM) developed here can produce simple and accurate approximations of certain nonlinear functions with minimal multiply–add operations. In this study, we designed an algorithm for the approximate computation of trigonometric and inverse trigonometric functions, which are nonlinear elementary functions, to achieve their fast and accurate computation. A fast first approximation and a more accurate second approximation of each function were created using RCM with a less than 0.001 error using multiply–add operations only. This achievement is particularly useful for MCUs, which have a low power consumption but limited computational power, and the proposed approximations are candidate algorithms that can be used to stabilize the attitude control of robots and drones, which require real-time processing.

Application of Artificial Neural Network in Chemical Separation Process

The chemical separation process is a necessary unit operation to obtain qualified purity products. The design and operation control of the separation device are very important. Due to the numerous parameters that affect the separation efficiency and product purity, it is very difficult to optimize by traditional methods. Artificial neural network (ANN) has strong fault tolerance because of its self-learning, self-organization, self-adaptive and strong nonlinear function approximation ability. ANN can be used to map the complex nonlinear relationship between dependent variables and independent variables, and can be used for the design and control of chemical separation devices. This paper summarizes the characteristics of artificial neural network and several important artificial neural network models, and discusses the application of artificial neural network in different chemical separation processes.

Using Reinforcement Learning to Control Traffic Signals in a Real-World Scenario: An Approach Based on Linear Function Approximation

Reinforcement learning is an efficient, widely used machine learning technique that performs well in problems with a reasonable number of states and actions. This is rarely the case regarding control-related problems, as for instance controlling traffic signals, where the state space can be very large. One way to deal with the curse of dimensionality is to use generalization techniques such as function approximation. In this paper, a linear function approximation is used by traffic signal agents in a network of signalized intersections. Specifically, a true online SARSA <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\lambda)$ </tex-math></inline-formula> algorithm with Fourier basis functions (TOS( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> )-FB) is employed. This method has the advantage of having convergence guarantees and error bounds, a drawback of non-linear function approximation. In order to evaluate TOS( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> )-FB, we perform experiments in variations of an isolated intersection scenario and a scenario of the city of Cottbus, Germany, with 22 signalized intersections, implemented in MATSim. We compare our results not only to fixed-time controllers, but also to a state-of-the-art rule-based adaptive method, showing that TOS( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> )-FB shows a performance that is highly superior to the fixed-time, while also being at least as efficient as the rule-based approach. For more than half of the intersections, our approach leads to less congestion and delay, without the need for the knowledge that underlies the rule-based approach.

A Generalized Bootstrap Target for Value-Learning, Efficiently Combining Value and Feature Predictions

Estimating value functions is a core component of reinforcement learning algorithms. Temporal difference (TD) learning algorithms use bootstrapping, i.e. they update the value function toward a learning target using value estimates at subsequent time-steps. Alternatively, the value function can be updated toward a learning target constructed by separately predicting successor features (SF)—a policy-dependent model—and linearly combining them with instantaneous rewards. We focus on bootstrapping targets used when estimating value functions, and propose a new backup target, the ?-return mixture, which implicitly combines value-predictive knowledge (used by TD methods) with (successor) feature-predictive knowledge—with a parameter ? capturing how much to rely on each. We illustrate that incorporating predictive knowledge through an ??-discounted SF model makes more efficient use of sampled experience, compared to either extreme, i.e. bootstrapping entirely on the value function estimate, or bootstrapping on the product of separately estimated successor features and instantaneous reward models. We empirically show this approach leads to faster policy evaluation and better control performance, for tabular and nonlinear function approximations, indicating scalability and generality.

Reward-Reinforced Generative Adversarial Networks for Multi-Agent Systems

Multi-agent systems deliver highly resilient and adaptable solutions for common problems in telecommunications, aerospace, and industrial robotics. However, achieving an optimal global goal remains a persistent obstacle for collaborative multi-agent systems, where learning affects the behaviour of more than one agent. A number of nonlinear function approximation methods have been proposed for solving the Bellman equation, which describe a recursive format of an optimal policy. However, how to leverage the value distribution based on reinforcement learning, and how to improve the efficiency and efficacy of such systems remain a challenge. In this work, we developed a reward-reinforced generative adversarial network to represent the distribution of the value function, replacing the approximation of Bellman updates. We demonstrated our method is resilient and outperforms other conventional reinforcement learning methods. This method is also applied to a practical case study: maximising the number of user connections to autonomous airborne base stations in a mobile communication network. Our method maximises the data likelihood using a cost function under which agents have optimal learned behaviours. This reward-reinforced generative adversarial network can be used as a generic framework for multi-agent learning at the system level.

Forward Kinematics for Suspended Under-Actuated Cable-Driven Parallel Robots With Elastic Cables: A Neural Network Approach

Abstract Kinematic analysis of under-constrained cable-driven parallel robots (CDPR) has been a topic of interest because of the inherent coupling between the loop-closure and static equilibrium equations. The non-linearity of the problem is magnified with the addition of the coupling between the cable lengths and their tensions based on the elastic cable model. The paper proposes an unsupervised neural network algorithm to perform real-time forward geometrico-static analysis of such robots in a suspended configuration under the action of gravity. The formulation determines a non-linear function approximation to model the problem and proves to be efficient in solving consecutive and close waypoints in a path. The methodology is applied on a six-degree-of-freedom spatial under-constrained suspended CDPR. Specific comparison results in simulation and hardware to show the effectiveness of the proposed method in tracking a given path are illustrated. Finally, the degree of constraint satisfaction is presented against the results obtained from non-linear least-square optimization.

Reinforcement learning-based close formation control for underactuated surface vehicle with prescribed performance and time-varying state constraints

This paper studies close formation control problem with prescribed performance and time-varying state constraints for a group of 4-degrees-of-freedom (DOF) underactuated surface vehicles (USVs) subject to actuator faults, input saturation and input delay. A finite-time sliding mode control (SMC) scheme based on reinforcement learning (RL) algorithm is introduced to guarantee prescribed formation performance without violating velocity error constraints. By using actor-critic neural network (NN)-based RL algorithm, the actuator faults and system uncertainties are accurately estimated. Afterwards, an exponential decreasing boundary function is developed to suppress overshoot more reasonably, and a novel mechanism of switching gain is given to alleviate chattering inherent in SMC while the RL-based compensation term is constructed to handle the formation accuracy problem caused by the reduced switching gain. Besides, auxiliary nonlinear continuous function and Pade approximation have been successfully applied to process actuator saturation and input delay, respectively. Numerical simulations and experimental results are exhibited to verify the effectiveness and superior formation performance of the proposed control method.

Neuroadaptive deferred full-state constraints control without feasibility conditions for uncertain nonlinear EASSs

This article studies the neuroadaptive full-state constraints control problem for a class of electromagnetic active suspension systems (EASSs). First, the original constraint system with arbitrary initial values is transformed into a new constraint system with zero initial values by using the shift function method. Then, a new kind of cotangent-type nonlinear state-dependent transition function is constructed to solve the asymmetric time-varying full-state constraints control problem, which eliminates the limitation that the virtual controller needs to satisfy the feasibility conditions in the previous full-state constraints control based on Barrier Lyapunov Function (BLF) and Integral BLF. Furthermore, the neural networks (NNs) are used as nonlinear function approximators to deal with the unknown nonlinear dynamics of EASSs, a neuroadaptive full-state constraints control design method is proposed under the Backstepping recursive design framework. Finally, the effectiveness of the proposed method is verified by a simulation of EASSs with road disturbances.

Analysis and Research on Nonlinear Complex Function Approximation Problem Based on Deep Learning

Shallow models have limited ability to express high-dimensional nonlinear complex functions. Based on deep learning, a Gaussian radial basis function neural network (RBFNN) is proposed, which is an analysis method of nonlinear complex function approximation based on the Gaussian-RBFNN model. The proposed method can approximate single-variable and binary complex nonlinear functions, and the approximation error is less than 0.1, which can achieve the ideal approximation effect. Finally, the proposed method is applied to analyze the nonlinear complex stock prediction approximation problem, and the effectiveness and practicality of the method are further verified. It can be seen that compared with the BP neural network model, this model performs better in average training time, training standard error, Ratt, RATR, and trmse For RBFNN; the training only needs 50 iterations, while the traditional BP needs 7100 training to gradually converge. In the actual function test, after 275 times of training, the convergence speed gradually tends to be stable, and the predicted value is closer to the actual value. Therefore, the model can be used to analyze practical nonlinear complex function approximation problems.

Dynamics modeling of a 2-DoFs cable-driven continuum robot

PurposeContinuum robots modeling, be it from a hard or soft class, is giving rise to several challenges compared with rigid robots. These challenges are mainly due to kinematic redundancy, dynamic nonlinearity and high flexibility. This paper aims initially at designing a hard class of continuum robots, namely, cable-driven continuum robot (CDCR) and equally at developing their kinematic and dynamic models.Design/methodology/approachFirst, the CDCR prototype is constructed, and its description is made. Second, kinematic models are established based on the constant curvature assumption and inextensible bending section. Third, by using the Lagrange method, the dynamic model is derived under some simplifications and based on the kinematic equations, in which the flexible backbone’s elasticity modulus was identified experimentally. Finally, the static model of the CDCR is also derived based on the dynamic model.FindingsNumerical examples are carried out using Matlab software to verify the static and dynamic models. Moreover, the static model is validated by comparing the simulation’s results to the real measurements that have been provided with satisfactory results.Originality/valueTo reduce the complexity of the dynamic model’s expressions and avoid the numerical singularity when the bending angle is close to zero, some simplifications have been taken, especially for the kinetic energy terms, by using the nonlinear functions approximation. Hence, the main advantage of this analytical-approximate solution is that it can be applied in the bending angle that ranges up to 2p with reasonable errors, unlike the previously proposed techniques. Furthermore, the resulting dynamic model has, to some extent, the proprieties of simplicity, accuracy and fast computation time. Ultimately, the obtained results from the simulations and real measurements demonstrate that the considered CDCR’s static and dynamic models are feasible.

A New Adaptive Controller for Nonlinear Systems with Uncertain Virtual Control Gains

This paper addresses the adaptive asymptotic tracking control problem for nonlinear systems whose virtual control gains are unknown nonlinear functions of system states. Only in the first step, the Nussbaum gain technique is utilized to handle the uncertain virtual control gain. In the remaining steps, virtual control gains are dealt with by constructing novel control laws without the approximation of the uncertain nonlinear functions and external disturbances by neural networks or fuzzy logic. New adaptive laws are defined to compensate for unknown virtual control gains, uncertain parameters, and external disturbances. Finally, an adaptive tracking controller is designed and applied to the control of a 3‐order robot system, which guarantees the boundedness of all the signals in the closed‐loop system and asymptotic stability of the tracking error.