Convergence study in extended Kalman filter-based training of recurrent neural networks

An Inexact Sequential Quadratic Programming Method for Learning and Control of Recurrent Neural Networks

This article considers the two-stage approach to solving a partially observable Markov decision process (POMDP): the identification stage and the (optimal) control stage. We present an inexact sequential quadratic programming framework for recurrent neural network learning (iSQPRL) for solving the identification stage of the POMDP, in which the true system is approximated by a recurrent neural network (RNN) with dynamically consistent overshooting (DCRNN). We formulate the learning problem as a constrained optimization problem and study the quadratic programming (QP) subproblem with a convergence analysis under a restarted Krylov-subspace iterative scheme that implicitly exploits the structure of the associated Karush-Kuhn-Tucker (KKT) subsystem. In the control stage, where a feedforward neural network (FNN) controller is designed on top of the RNN model, we adapt a generalized Gauss-Newton (GGN) algorithm that exploits useful approximations to the curvature terms of the training data and selects its mini-batch step size using a known property of some regularization function. Simulation results are provided to demonstrate the effectiveness of our approach.

Event-Based Switching Iterative Learning Model Predictive Control for Batch Processes With Randomly Varying Trial Lengths

Iterative learning model predictive control (ILMPC) has been recognized as an excellent batch process control strategy for progressively improving tracking performance along trials. However, as a typical learning-based control method, ILMPC generally requires the strict identity of trial lengths to implement 2-D receding horizon optimization. The randomly varying trial lengths extensively existing in practice can result in the insufficiency of learning prior information, and even the suspension of control update. Regarding this issue, this article embeds a novel prediction-based modification mechanism into ILMPC, to adjust the process data of each trial into the same length by compensating the data of absent running periods with the predictive sequences at the end point. Under this modification scheme, it is proved that the convergence of the classical ILMPC is guaranteed by an inequality condition relative with the probability distribution of trial lengths. Considering the practical batch process with complex nonlinearity, a 2-D neural-network predictive model with parameter adaptability along trials is established to generate highly matched compensation data for the prediction-based modification. To best utilize the real process information of multiple past trials while guaranteeing the learning priority of the latest trials, an event-based switching learning structure is proposed in ILMPC to determine different learning orders according to the probability event with respect to the trial length variation direction. The convergence of the nonlinear event-based switching ILMPC system is analyzed theoretically under two situations divided by the switching condition. The simulations on a numerical example and the injection molding process verify the superiority of the proposed control methods.

DynaNet: Neural Kalman Dynamical Model for Motion Estimation and Prediction

Dynamical models estimate and predict the temporal evolution of physical systems. State-space models (SSMs) in particular represent the system dynamics with many desirable properties, such as being able to model uncertainty in both the model and measurements, and optimal (in the Bayesian sense) recursive formulations, e.g., the Kalman filter. However, they require significant domain knowledge to derive the parametric form and considerable hand tuning to correctly set all the parameters. Data-driven techniques, e.g., recurrent neural networks, have emerged as compelling alternatives to SSMs with wide success across a number of challenging tasks, in part due to their impressive capability to extract relevant features from rich inputs. They, however, lack interpretability and robustness to unseen conditions. Thus, data-driven models are hard to be applied in safety-critical applications, such as self-driving vehicles. In this work, we present DynaNet, a hybrid deep learning and time-varying SSM, which can be trained end-to-end. Our neural Kalman dynamical model allows us to exploit the relative merits of both SSM and deep neural networks. We demonstrate its effectiveness in the estimation and prediction on a number of physically challenging tasks, including visual odometry, sensor fusion for visual-inertial navigation, and motion prediction. In addition, we show how DynaNet can indicate failures through investigation of properties, such as the rate of innovation (Kalman gain).

Incremental learning of LSTM framework for sensor fusion in attitude estimation

This paper presents a novel method for attitude estimation of an object in 3D space by incremental learning of the Long-Short Term Memory (LSTM) network. Gyroscope, accelerometer, and magnetometer are few widely used sensors in attitude estimation applications. Traditionally, multi-sensor fusion methods such as the Extended Kalman Filter and Complementary Filter are employed to fuse the measurements from these sensors. However, these methods exhibit limitations in accounting for the uncertainty, unpredictability, and dynamic nature of the motion in real-world situations. In this paper, the inertial sensors data are fed to the LSTM network which are then updated incrementally to incorporate the dynamic changes in motion occurring in the run time. The robustness and efficiency of the proposed framework is demonstrated on the dataset collected from a commercially available inertial measurement unit. The proposed framework offers a significant improvement in the results compared to the traditional method, even in the case of a highly dynamic environment. The LSTM framework-based attitude estimation approach can be deployed on a standard AI-supported processing module for real-time applications.

An Optimized Parallel Implementation of Non-Iteratively Trained Recurrent Neural Networks

Recurrent neural networks (RNN) have been successfully applied to various sequential decision-making tasks, natural language processing applications, and time-series predictions. Such networks are usually trained through back-propagation through time (BPTT) which is prohibitively expensive, especially when the length of the time dependencies and the number of hidden neurons increase. To reduce the training time, extreme learning machines (ELMs) have been recently applied to RNN training, reaching a 99% speedup on some applications. Due to its non-iterative nature, ELM training, when parallelized, has the potential to reach higher speedups than BPTT.

In this work, we present Opt-PR-ELM, an optimized parallel RNN training algorithm based on ELM that takes advantage of the GPU shared memory and of parallel QR factorization algorithms to efficiently reach optimal solutions. The theoretical analysis of the proposed algorithm is presented on six RNN architectures, including LSTM and GRU, and its performance is empirically tested on ten time-series prediction applications. Opt-PR-ELM is shown to reach up to 461 times speedup over its sequential counterpart and to require up to 20x less time to train than parallel BPTT. Such high speedups over new generation CPUs are extremely crucial in real-time applications and IoT environments.

Motion Planning of Autonomous Mobile Robot Using Recurrent Fuzzy Neural Network Trained by Extended Kalman Filter

This paper proposes a novel motion planning method for an autonomous ground mobile robot to address dynamic surroundings, nonlinear program, and robust optimization problems. A planner based on the recurrent fuzzy neural network (RFNN) is designed to program trajectory and motion of mobile robots to reach target. And, obstacle avoidance is achieved. In RFNN, inference capability of fuzzy logic and learning capability of neural network are combined to improve nonlinear programming performance. A recurrent frame with self-feedback loops in RFNN enhances stability and robustness of the structure. The extended Kalman filter (EKF) is designed to train weights of RFNN considering the kinematic constraint of autonomous mobile robots as well as target and obstacle constraints. EKF's characteristics of fast convergence and little limit in training data make it suitable to train the weights in real time. Convergence of the training process is also analyzed in this paper. Optimization technique and update strategy are designed to improve the robust optimization of a system in dynamic surroundings. Simulation experiment and hardware experiment are implemented to prove the effectiveness of the proposed method. Hardware experiment is carried out on a tracked mobile robot. An omnidirectional vision is used to locate the robot in the surroundings. Forecast improvement of the proposed method is then discussed at the end.

Identification and Control for Singularly Perturbed Systems Using Multitime-Scale Neural Networks

Many well-established singular perturbation theories for singularly perturbed systems require the full knowledge of system model parameters. In order to obtain an accurate and faithful model, a new identification scheme for singularly perturbed nonlinear system using multitime-scale recurrent high-order neural networks (NNs) is proposed in this paper. Inspired by the optimal bounded ellipsoid algorithm, which is originally designed for discrete-time systems, a novel weight updating law is developed for continuous-time NNs identification process. Compared with other widely used gradient-descent updating algorithms, this new method can achieve faster convergence, due to its adaptively adjusted learning rate. Based on the identification results, a control scheme using singular perturbation theories is developed. By using singular perturbation methods, the system order is reduced, and the controller structure is simplified. The closed-loop stability is analyzed and the convergence of system states is guaranteed. The effectiveness of the identification and the control scheme is demonstrated by simulation results.

A class of quaternion Kalman filters

The existing Kalman filters for quaternion-valued signals do not operate fully in the quaternion domain, and are combined with the real Kalman filter to enable the tracking in 3-D spaces. Using the recently introduced HR-calculus, we develop the fully quaternion-valued Kalman filter (QKF) and quaternion-extended Kalman filter (QEKF), allowing for the tracking of 3-D and 4-D signals directly in the quaternion domain. To consider the second-order noncircularity of signals, we employ the recently developed augmented quaternion statistics to derive the widely linear QKF (WL-QKF) and widely linear QEKF (WL-QEKF). To reduce computational requirements of the widely linear algorithms, their efficient implementation are proposed and it is shown that the quaternion widely linear model can be simplified when processing 3-D data, further reducing the computational requirements. Simulations using both synthetic and real-world circular and noncircular signals illustrate the advantages offered by widely linear over strictly linear quaternion Kalman filters.

Assessment and certification of neonatal incubator sensors through an inferential neural network

Measurement and diagnostic systems based on electronic sensors have been increasingly essential in the standardization of hospital equipment. The technical standard IEC (International Electrotechnical Commission) 60601-2-19 establishes requirements for neonatal incubators and specifies the calibration procedure and validation tests for such devices using sensors systems. This paper proposes a new procedure based on an inferential neural network to evaluate and calibrate a neonatal incubator. The proposal presents significant advantages over the standard calibration process, i.e., the number of sensors is drastically reduced, and it runs with the incubator under operation. Since the sensors used in the new calibration process are already installed in the commercial incubator, no additional hardware is necessary; and the calibration necessity can be diagnosed in real time without the presence of technical professionals in the neonatal intensive care unit (NICU). Experimental tests involving the aforementioned calibration system are carried out in a commercial incubator in order to validate the proposal.

Parameter estimation and control for a neural mass model based on the unscented Kalman filter

Recent progress in Kalman filters to estimate states and parameters in nonlinear systems has provided the possibility of applying such approaches to neural systems. We here apply the nonlinear method of unscented Kalman filters (UKFs) to observe states and estimate parameters in a neural mass model that can simulate distinct rhythms in electroencephalography (EEG) including dynamical evolution during epilepsy seizures. We demonstrate the efficiency of the UKF in estimating states and parameters. We also develop an UKF-based control strategy to modulate the dynamics of the neural mass model. In this strategy the UKF plays the role of observing states, and the control law is constructed via the estimated states. We demonstrate the feasibility of using such a strategy to suppress epileptiform spikes in the neural mass model.

Discrete-time neural inverse optimal control for nonlinear systems via passivation

This paper presents a discrete-time inverse optimal neural controller, which is constituted by combination of two techniques: 1) inverse optimal control to avoid solving the Hamilton-Jacobi-Bellman equation associated with nonlinear system optimal control and 2) on-line neural identification, using a recurrent neural network trained with an extended Kalman filter, in order to build a model of the assumed unknown nonlinear system. The inverse optimal controller is based on passivity theory. The applicability of the proposed approach is illustrated via simulations for an unstable nonlinear system and a planar robot.

Class of widely linear complex Kalman filters

Recently, a class of widely linear (augmented) complex-valued Kalman filters (KFs), that make use of augmented complex statistics, have been proposed for sequential state space estimation of the generality of complex signals. This was achieved in the context of neural network training, and has allowed for a unified treatment of both second-order circular and noncircular signals, that is, both those with rotation invariant and rotation-dependent distributions. In this paper, we revisit the augmented complex KF, augmented complex extended KF, and augmented complex unscented KF in a more general context, and analyze their performances for different degrees of noncircularity of input and the state and measurement noises. For rigor, a theoretical bound for the performance advantage of widely linear KFs over their strictly linear counterparts is provided. The analysis also addresses the duality with bivariate real-valued KFs, together with several issues of implementation. Simulations using both synthetic and real world proper and improper signals support the analysis.

Global convergence of online BP training with dynamic learning rate

The online backpropagation (BP) training procedure has been extensively explored in scientific research and engineering applications. One of the main factors affecting the performance of the online BP training is the learning rate. This paper proposes a new dynamic learning rate which is based on the estimate of the minimum error. The global convergence theory of the online BP training procedure with the proposed learning rate is further studied. It is proved that: 1) the error sequence converges to the global minimum error; and 2) the weight sequence converges to a fixed point at which the error function attains its global minimum. The obtained global convergence theory underlies the successful applications of the online BP training procedure. Illustrative examples are provided to support the theoretical analysis.

Robust initialization of a Jordan network with recurrent constrained learning

In this paper, we propose a robust initialization of a Jordan network with a recurrent constrained learning (RIJNRCL) algorithm for multilayered recurrent neural networks (RNNs). This novel algorithm is based on the constrained learning concept of the Jordan network with a recurrent sensitivity and weight convergence analysis, which is used to obtain a tradeoff between the training and testing errors. In addition to using classical techniques for the adaptive learning rate and the adaptive dead zone, RIJNRCL employs a recurrent constrained parameter matrix to switch off excessive contributions from the hidden layer neurons based on weight convergence and stability conditions of the multilayered RNNs. It is well known that a good response from the hidden layer neurons and proper initialization play a dominant role in avoiding local minima in multilayered RNNs. The new RIJNRCL algorithm solves the twin problems of weight initialization and selection of the hidden layer neurons via a novel recurrent sensitivity ratio analysis. We provide the detailed steps for using RIJNRCL in a few benchmark time-series prediction problems and show that the proposed algorithm achieves superior generalization performance.