Efficient Reachability Analysis of Closed-Loop Systems with Neural Network Controllers

There has been an increasing interest in using neural networks in closed-loop control systems to improve performance and reduce computational costs for on-line implementation. However, providing safety and stability guarantees for these systems is challenging due to the nonlinear and compositional structure of neural networks. In this paper, we propose a novel forward reachability analysis method for the safety verification of linear time-varying systems with neural networks in feedback interconnection. Our technical approach relies on abstracting the nonlinear activation functions by quadratic constraints, which leads to an outer-approximation of forward reachable sets of the closed-loop system. We show that we can compute these approximate reachable sets using semidefinite programming. We illustrate our method in a quadrotor example, in which we first approximate a nonlinear model predictive controller via a deep neural network and then apply our analysis tool to certify finite-time reachability and constraint satisfaction of the closed-loop system.

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.

A high-level state model of the national airspace system (NAS) has been developed and refined. That model provides guidance on the effect and interactions of sources of uncertainty in the NAS. The model also provides guidance on the possible effect of NextGen concepts and technologies on the NAS. The model is being coded as a high-level, abstract simulation that, when provided with specific uncertainty distributions, can also identify the emergent effect of those changes in the uncertainty distributions. The results of that analysis and simulation provide guidance for prioritizing the development and adoption of NextGen concepts and technologies for the NAS. I. Introduction he National Airspace System (NAS) is a highly complex, socio-technical system. At its lowest level, it consists of many thousands of entities, including aircraft and other sophisticated technology, passengers, pilots, and air traffic controllers. Modeling and simulation of such a system is difficult due to the large number of entities to be modeled, the infinite (or nearly so) number of degrees of freedom within the system, and the high degree of possible interactivity in the system. As a result, existing models/simulations must simplify the system by eliminating from consideration certain aspects, and therefore cannot be completely accurate in determining the effect of uncertainties on the performance of the NAS. To try to fill this gap in capability, a model of the NAS was developed, with the purpose of determining the effect of the various sources of uncertainty on the system performance of the NAS. This model is state-based, and models the system from a high level (Landry and Archer, 2013). The model is believed to be complete. Given this model, one can reason about the effects of uncertainty on the NAS, as well as the interactions of these sources of uncertainty (Landry and Archer, 2013). In addition, if one characterizes the effect of different operational concepts and technologies being considered under NextGen in terms of their effect on those sources of uncertainty, one can then reason about the relative benefits of those different concepts and technologies. In this work, we focus on the effect of concepts and technologies on the performance of the NAS. Following a brief discussion of the model, an enumeration of the technologies and concepts under consideration for NextGen will be presented, including their anticipated degree of effect on various sources of uncertainty. Lastly, the results of simulation exercises to test the model will be presented. II. Model of the NAS: Framework

This paper presents investigations into the development of neural network (NN) control of a two-link flexible robot manipulator with load 0.5 kg. A dynamic model of the system is derived using a combined Euler-Lagrange and assumed mode methods (AMM). The controller reduces nonlinearities problem that can be efficiently solved using NN control. The system responses namely hub angular position and deflection responses and end-point acceleration at both links are obtained and analysed. The performances of the controllers are assessed in terms of the input tracking controller capability of the system. Moreover, the robustness of the NN control schemes is discussed. Finally, a comparative assessment of the control strategies is presented. The results show that NN controller performs give increasing profiles that compared with PID control.

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.

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.

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.

Improved spatiotemporal monitoring of soil salinity using filtered kriging with measurement errors: An application to the West Urmia Lake, Iran

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.

In this paper, we present experimental validation of a novel method for controlling a robotic manipulator using an experience-based Neural Network (NN) controller derived from Deterministic Learning (DL) techniques. Studies are performed on simulated and real-world environments to quantify performance of this controller compared (as a baseline case) to traditional methods such as Proportional-Integral-Derivative (PID) control. Both of these controller types (NN and PID) are integrated into a 7 Degrees of Freedom (DOF) manipulator system, having the controller available to use on all 7 joints simultaneously. The NN controller utilizes adaptive learning strategies focusing on the state-feedback case. The learning algorithms train Radial Basis Function Neural Networks (RBFNN) to enhance tracking performance along a non-linear recurrent reference trajectory. With convergence of NN weights within the RBFNN, the unmodeled manipulator dynamics are guaranteed to be identified and stored for use with the experience-based algorithms without online adaptation. This paper details the process of integrating a NN controller with system state feedback and employing learning/experience based algorithms in a manner that host system computations are not affected beyond a pre-determined sampling rate.

Probabilistic neural data fusion for learning from an arbitrary number of multi-fidelity data sets

Differential Evolution to Solve the Lot Size Problem in Stochastic Production Systems

To achieve fast four-quadrant operation and smooth starting and acceleration, vector control is used in the design of the stepping motor servo system drive. The control performance of the stepping motor servo system drive still influenced by the uncertainties, such as external load disturbance, nonlinearity and motor parameter variations. Neural network is used as a supervisor to deal with nonlinearity and uncertainties of the system. Here, we propose a control strategy which combines the advantages of the integral-proportional (IP) controller, neural network and vector control, for the stepping motor position servo system drive.

ABSTRACTThis paper proposes an Adaptive Neural Network (NN) controller for the real-time pressure control in water distribution systems. Pressure control is one of the main technical options that can be implemented by a water utility to increase the hydraulic and energy efficiency of systems. The network adopted the Levenberg–Marquardt backpropagation algorithm, being responsible for maintaining the pump head at an optimal value, eliminating the excess pressure of the system. The advantage of the approach is that, once the network is trained, it allows instantaneous evaluation of solutions at any desired number of points; thus, spending little computing time. The controller was applied in the experimental setup, and the results showed excellent performance regarding pressure regulation. Finally, it is expected that the NN controller can be easily implemented in similar water distribution systems.

A neural-network-based adaptive control (NNAC) design method is proposed to control an induction servomotor. In this NNAC design, a neural network (NN) controller is investigated to mimic a feedback linearization control law; and a compensation controller is designed to compensate for the approximation error between the feedback linearization control law and the NN controller. The interconnection weights of the NN can be online tuned in the sense of the Lyapunov stability theorem; thus, the stability of the control system can be guaranteed. Additionally, in this NNAC system design, an error estimation mechanism is investigated to estimate the bound of approximation error so that the chattering phenomenon of the control effort can be reduced. Simulation and experimental results show that the proposed NNAC servomotor control systems can achieve favorable tracking and robust performance with regard to parameter variations and external load disturbances.