Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming

Neural Networks (NNs) can provide major empirical performance improvements for robotic systems, but they also introduce challenges in formally analyzing those systems’ safety properties. In particular, this work focuses on estimating the forward reachable set of closed-loop systems with NN controllers. Recent work provides bounds on these reachable sets, yet the computationally efficient approaches provide overly conservative bounds (thus cannot be used to verify useful properties), whereas tighter methods are too intensive for online computation. This work bridges the gap by formulating a convex optimization problem for reachability analysis for closed-loop systems with NN controllers. While the solutions are less tight than prior semidefinite program-based methods, they are substantially faster to compute, and some of the available computation time can be used to refine the bounds through input set partitioning, which more than overcomes the tightness gap. The proposed framework further considers systems with measurement and process noise, thus being applicable to realistic systems with uncertainty. Finally, numerical comparisons show that our approach based on linear programming and partitioning can give 10× reduction in conservatism in $\frac{1}{2}$ of the computation time compared to the state-of-the-art, and the ability to handle various sources of uncertainty is highlighted on a quadrotor model.

This benchmark suite presents a detailed description of a series of closed-loop control systems with artificial neural network controllers. In many applications, feed-forward neural networks are heavily involved in the implementation of controllers by learning and representing control laws through several methods such as model predictive control (MPC) and reinforcement learning (RL). The type of networks that we consider in this manuscript are feed-forward neural networks consisting of multiple hidden layers with ReLU activation functions and a linear activation function in the output layer. While neural network con- trollers have been able to achieve desirable performance in many contexts, they also present a unique challenge in that it is difficult to provide any guarantees about the correctness of their behavior or reason about the stability a system that employs their use. Thus, from a controls perspective, it is necessary to verify them in conjunction with their corresponding plants in closed-loop. While there have been a handful of works proposed towards the verification of closed-loop systems with feed-forward neural network controllers, this area still lacks attention and a unified set of benchmark examples on which verification techniques can be evaluated and compared. Thus, to this end, we present a range of closed-loop control systems ranging from two to six state variables, and a range of controllers with sizes in the range of eleven neurons to a few hundred neurons in more complex systems.

Applying neural networks as controllers in dynamical systems has shown great promises. However, it is critical yet challenging to verify the safety of such control systems with neural-network controllers in the loop. Previous methods for verifying neural network controlled systems are limited to a few specific activation functions. In this work, we propose a new reachability analysis approach based on Bernstein polynomials that can verify neural-network controlled systems with a more general form of activation functions, i.e., as long as they ensure that the neural networks are Lipschitz continuous. Specifically, we consider abstracting feedforward neural networks with Bernstein polynomials for a small subset of inputs. To quantify the error introduced by abstraction, we provide both theoretical error bound estimation based on the theory of Bernstein polynomials and more practical sampling based error bound estimation, following a tight Lipschitz constant estimation approach based on forward reachability analysis. Compared with previous methods, our approach addresses a much broader set of neural networks, including heterogeneous neural networks that contain multiple types of activation functions. Experiment results on a variety of benchmarks show the effectiveness of our approach.

Automated image classification is an essential task of the computer vision field. The tagging of images into a set of predefined groups is referred to as image classification. The implementation of computer vision to automate image classification would be beneficial because manual image evaluation and identification can be time-consuming, particularly when there are many images of different classes. Deep learning approaches are proven to overperform existing machine learning techniques in many fields in recent years, and computer vision is one of the most notable examples. The very deep neural network (VDCNN) is a powerful deep learning model for image classification, and this paper examines it briefly using MNIST handwritten digit dataset. This dataset is used to prove the efficacy of very deep neural networks over other deep learning models. The proposed study aims to comprehend the very deep neural network architecture used to accomplish a handwritten digit recognition task. The feasibility of the proposed model is evaluated using mean accuracy, validation accuracy, and standard deviation. The study results of the very deep neural network model are compared to a convolutional neural network and convolutional neural network with batch normalization. According to the results of the comparison study, very deep neural networks achieve high accuracy of 99.1% for a handwritten dataset. The outcome of the proposed work is used to interpret how well a very deep neural network performs when compared to the other two models of deep neural networks. This proposed architecture may be used to automate the classification of handwritten digits dataset.

Experimental validation of saliency maps for understanding deep neural networks for weld penetration prediction

Video hyperlinking represents a classical example of multimodal problems. Common approaches to such problems are early fusion of the initial modalities and crossmodal translation from one modality to the other. Recently, deep neural networks, especially deep autoencoders, have proven promising both for crossmodal translation and for early fusion via multimodal embedding. A particular architecture, bidirectional symmetrical deep neural networks, have been proven to yield improved multimodal embeddings over classical autoencoders, while also being able to perform crossmodal translation. In this work, we focus firstly at evaluating good single-modal continuous representations both for textual and for visual information. Word2Vec and paragraph vectors are evaluated for representing collections of words, such as parts of automatic transcripts and multiple visual concepts, while different deep convolutional neural networks are evaluated for directly embedding visual information, avoiding the creation of visual concepts. Secondly, we evaluate methods for multimodal fusion and crossmodal translation, with different single-modal pairs, in the task of video hyperlinking. Bidirectional (symmetrical) deep neural networks were shown to successfully tackle downsides of multimodal autoencoders and yield a superior multimodal representation. In this work, we extensively tests them in different settings, with different single-modal representations, within the context of video-hyperlinking. Our novel bidirectional symmetrical deep neural networks are compared to classical autoencoders and are shown to yield significantly improved multimodal embeddings that significantly (alpha=0.0001) outperform multimodal embeddings obtained by deep autoencoders with an absolute improvement in precision at 10 of 14.1% when embedding visual concepts and automatic transcripts and an absolute improvement of 4.3% when embedding automatic transcripts with features obtained with very deep convolutional neural networks, yielding 80% of precision at 10.

In this study, the deep multi-layered group method of data handling (GMDH)-type neural network algorithm using revised heuristic self-organization method is proposed and applied to medical image diagnosis of liver cancer. The deep GMDH-type neural network can automatically organize the deep neural network architecture which has many hidden layers. The structural parameters such as the number of hidden layers, the number of neurons in hidden layers and useful input variables are automatically selected to minimize prediction error criterion defined as Akaike’s information criterion (AIC) or prediction sum of squares (PSS). The architecture of the deep neural network is automatically organized using the revised heuristic self-organization method which is a type of the evolutionary computation. This new neural network algorithm is applied to the medical image diagnosis of the liver cancer and the recognition results are compared with the conventional 3-layered sigmoid function neural network.

Robust vibration control of smart composite beams using neural networks was studied. Linear quadratic Gaussian with loop transfer recovery (LQG/LTR) methodology was used to design a robust controller on the basis of the state space model of the system. The state space model of the system was obtained using the finite-element method and mode superposition. The finite-element model was based on a higher-order shear deformation theory which included the lateral strains. The mode superposition method was used to transform the coupled finite-element equations of motion in the physical coordinates into a set of reduced uncoupled equations in the modal coordinates. The performance of the LQG/LTR controller was verified for various arbitrary initial conditions. A system of neural networks was then trained to emulate the robust controller. The neural network system was trained using the backpropagation algorithm. After suitable training, the NN (neural network) controller was shown to effectively control the vibrations of the composite beam. A robustness study including the effects of tip mass, structural parameter variation, and loss of a sensor input was performed. The NN controller is shown to provide robustness and control capabilities equivalent to that of the LQG/LTR controller.

Self-mixing interferometry (SMI) is superior to other laser interferometry methods due to its simplicity and compactness. However, SMI signals are often complex and difficult to process due to interference in the form of variations in the effective reflectivity of the target, noisy signals, complex signal shapes, and other dependencies. Deep neural networks have been a very popular area of research in computer artificial intelligence in recent years, allowing more implicit features to be uncovered than traditional shallow machine learning. It has been shown that convolutional and back-propagation neural networks can be used for SMI signal processing. There are also studies that have used machine learning genetic algorithms for absolute distance measurement. Based on the above, this study used convolutional neural networks to form a deep neural network for absolute distance measurement based on SMI technology. We first trained the deep convolutional neural network at different feedback strengths. The results of the Convolutional Neural Network (CNN) model showed a coefficient of determination of 0.9987. which is consistent with the required model. The trained network was then used to estimate absolute distances with and without the addition of noise. The comparison proves that the proposed method is noise-proof and has high adaptability for measurements under different conditions.

A robust neural network (NN) output feedback scheme is proposed for the motion control of rigid robots. A dynamic NN observer is presented to estimate the joint speeds. The stability of a closed-loop system composed of a robot, a NN observer, and a NN controller is proven. The NN weights in both the observer and the controller are tuned online, with no off-line learning phase required. Most importantly, we can guarantee the boundness of the estimated velocities, the position tracking errors, and the NN weights. Also no exact knowledge of the robot dynamics is required so that the NN controller is model-free and so applicable to any type of rigid robot. When compared with adaptive-type controllers, we do not require persistent excitation conditions, linearity in the unknown system parameters, or the tedious computation of a regression matrix. Thus the new NN approach represents an improvement over adaptive techniques.

Multi-label learning based deep transfer neural network for facial attribute classification

In this paper, adaptive neural network (NN) control is investigated for a class of multiinput and multioutput (MIMO) nonlinear systems with unknown bounded disturbances in discrete-time domain. The MIMO system under study consists of several subsystems with each subsystem in strict feedback form. The inputs of the MIMO system are in triangular form. First, through a coordinate transformation, the MIMO system is transformed into a sequential decrease cascade form (SDCF). Then, by using high-order neural networks (HONN) as emulators of the desired controls, an effective neural network control scheme with adaptation laws is developed. Through embedded backstepping, stability of the closed-loop system is proved based on Lyapunov synthesis. The output tracking errors are guaranteed to converge to a residue whose size is adjustable. Simulation results show the effectiveness of the proposed control scheme.

Semiconductor optical amplifier (SOA) is a promising solution for future O-band optical amplification. However, the SOA-induced nonlinearity will affect the quality of the signals and cause bit errors. Nowadays, the neural network is a hot topic in communication networks and has been considered a practical algorithm for signals equalization. In this paper, we investigate the performance of the three types of neural networks: deep neural networks (DNN), convolutional neural network (CNN), and long short term memory (LSTM) neural networks for the compensation of the SOA-induced nonlinearity. We first compare the fitting ability of the three neural networks on the saturated response of SOA. Based on the simulation result, LSTM has the best nonlinear expressiveness for the SOA-induced nonlinearity. Subsequently, we do some experimental validations in an intensity modulation and direct-detection (IM/DD) system, where the 100 Gbit/s pulse-amplitude-modulation-4 (PAM4) signals are transmitted over 80 km at O-band using an 18 GHz electro-absorption modulated laser (EML). The bit-error-ratio (BER) results show that the cell states in LSTM play a crucial role in nonlinearity compensation. Furthermore, LSTM offers the best equalization performance and can achieve a receiver sensitivity of −18.0 dBm.

The article deals with the problem of path planning in a two-dimensional environment based on deep learning neural networks. Deep neural networks require large amounts of data and place high computational requirements on computing tools. The lack of sufficient data leads to a decrease in the accuracy of the neural network, and high computational requirements at the learning stage limit the use of this technology in engineering practice. In this paper, the forms of representation of the environment for the input of a neural network are studied. Vector form allows to reduce the amount of information supplied to the input of a neural network, but it leads to the need to use more complex neural networks. In this article, a combined form of representation is proposed, including a vector global and local map layout. The vector part of the map includes the position of the robot, the position of the target point and a description of obstacles. The local raster map describes the area closest to the robot. Using numerical research, the effectiveness of this form of data representation for a precise neural network is shown, compared with the raster representation of the map. In this article, two structures of neural networks are studied, one of which uses 8 possible directions of movement, and the other uses 3 possible directions of movement. It is shown that when using 3 possible directions, the cycling of trajectories planned by the neural network is eliminated, which leads to an increase in accuracy.

The present paper investigates an advanced control system of the wind energy conversion using an Artificial Neural Network (ANN) controller, which is developed in two modes: the offline mode, required for testing different sets of neural network parameters to find the optimal neural network controller (structure, activation function, and training algorithm), and the online mode where the optimal ANN controller is used in the five-phase PMSG based wind turbine system. The required data to generate the ANN model is obtained from a conventional PI controller. The controller inputs are the error between reference and actual speed (e), and the error change (de = e(k)-e(k-1)). As for the ANN output, it is the decreasing or increasing electromagnetic torque (dTe*). The studied neural network controller has been tested and validated using different wind speed values. Results and analysis are presented, and the contribution has been demonstrated.