Neural Network Approximation Errors Research Articles

Nonlinear Filtering and Reinforcement Learning Based Consensus Achievement of Uncertain Multi-Agent Systems

Abstract This paper introduces a novel approach for designing estimators to achieve consensus in uncertain multi-agent systems (MAS), even when various fault conditions are present and communication is assumed to be undirected and connected. The method includes an adaptive fault detection technique to detect faults and a unique adaptation in the unscented Kalman filter (UKF) to adjust noise covariance matrices and reconstruct uncertain states in the multi-agent system is proposed in the framework of Q-learning. Additionally, it involves training neural network internal parameters using previous measurements. A Chebyshev neural network (CNN) is employed to model the uncertain plant, and a hyperbolic tangent-based robust control term is used to mitigate neural network approximation errors. This novel approach is known as reinforced UKF (RUKF). The paper also discusses the asymptotic stability of the proposed method and presents numerical simulations to demonstrate its effectiveness with reduced computational load.

SDO-Based Command Filtered Adaptive Neural Tracking Control for MIMO Nonlinear Systems With Time-Varying Constraints.

In this article, an adaptive neural tracking control based on saturation disturbance observer (SDO) and command filter is studied for multiple-input-multiple-output nonlinear systems with time-varying constraints and system uncertainties. By employing neural networks (NNs), the system uncertainties are approximated. The SDO is proposed to estimate the composited disturbances which consist of NN approximation errors and the external bounded disturbances. Compared with the traditional disturbance observer, the SDO can reduce the estimation error to some extent. The control requirements are achieved based on the multiconstraints which contain three layers: 1) prescribed performance functions (PPFs); 2) actual constraints; and 3) virtual constraints. The errors remain within the prescribed small neighborhood of zero by using the PPFs, the error constraints ensure that the time-varying constraints are never violated even if the PPFs are not available, and the virtual constraints are applied in a new time-varying barrier Lyapunov function (TVBLF) to design virtual controllers and controller to solve the singularity problem of the traditional TVBLF. In addition, the command filter is introduced to solve the problem of "explosion of complexity." Finally, a numerical simulation verifies the effectiveness of the proposed scheme for a flight control of unmanned aerial vehicle.

Adaptive Inverse Nonlinear Optimal Control Based on Finite-Time Concurrent Learning and Semidefinite Programming.

This article investigates the problem of inverse optimal control (IOC) for a class of nonlinear affine systems. An adaptive IOC approach is proposed to recover the cost functional using only the system state data, which integrates the finite-time concurrent learning (FTCL) technique and the semidefinite programming (SDP) technique. First, an identifier neural network (NN) is employed to approximate the unknown nonlinear control policy, and an FTCL-based update law is proposed to estimate the weights of the identifier NN online, which removes the traditional persistent excitation (PE) condition. Moreover, the finite-time convergence as well as the uniformly ultimately boundness (UUB) of estimation error of the identifier NN weights are analysed according to whether or not there exists the identifier NN approximation error. Then, with the help of a value NN for approximating the value function, an SDP problem with a quadratic objective function can be set up for determining the weighting matrices of the cost functional. Finally, simulation results are presented to validate the proposed method.

Approximation error of single hidden layer neural networks with fixed weights

Neural networks with finitely many fixed weights have the universal approximation property under certain conditions on compact subsets of the d-dimensional Euclidean space, where approximation process is considered. Such conditions were delineated in our paper [26]. But for many compact sets it is impossible to approximate multivariate functions with arbitrary precision and the question on estimation or efficient computation of approximation error arises. This paper provides an explicit formula for the approximation error of single hidden layer neural networks with two fixed weights.

Adaptive Optimal Control of UAV Brake Based on Parameter Estimation

A new parameter estimation method for a UAV braking system with unknown friction parameters is suggested. The unknown part containing friction is separated from the coupling system. The law of parameter updating is driven by the estimation error extracted by the auxiliary filter. A controller is developed to guarantee the convergence of the parameter estimation and tracking errors simultaneously. Furthermore, a designed performance index function enables the system to track the desired slip rate optimally. The optimal value of the performance function is updated by a single critic neural network (NN) to obtain comprehensive optimization of the control energy consumption, dynamic tracking error, and filtering error during braking. In addition, the unknown bounded interference and neural network approximation errors are compensated by robust integral terms. Simulation results are presented to confirm the effectiveness of the proposed control scheme.

Disturbance observer-based full-state constrained control for robotic systems with dead zone

This paper studies the trajectory tracking control problem of uncertain manipulators with full-state constraints and dead-zone inputs. By utilizing the backstepping method and barrier Lyapunov functions, a novel disturbance observer-based full-state constrained control approach is proposed. It is noted that the dead-zone inputs can be expressed in terms of a nonlinear disturbance term and a nominal part. The unknown dynamic uncertainty of robotic system is approximated by the adaptive neural network, and the disturbance observer is integrated into the control design to compensate neural network approximation errors, nonlinear disturbances caused by dead-zone inputs, and external disturbance. A barrier Lyapunov function (BLF) is introduced to illustrate the boundedness of signals within the robotic control system. Finally, the effectiveness of the proposed control scheme is illustrated through simulation results of a two-link manipulator.

Reinforced unscented Kalman filter for consensus achievement of uncertain multi‐agent systems subject to actuator faults

AbstractIn this paper, actuator fault detection and reconstruction in consensus tracking of uncertain multi‐agent systems (MAS) is addressed. The communication is assumed to be connected undirected. An adaptive fault detection method is developed to detect actuator faults. A novel‐reinforced unscented Kalman filter (RUKF) is employed to reconstruct the faults by adjusting the noise covariance matrices of unscented Kalman filter (UKF) as well as to train neural network internal parameters by providing a set of previous measurements. A Chebyshev neural network (CNN) is incorporated to learn the uncertain plant. To prevent the neural network approximation errors a hyperbolic tangent function‐based robust control term is applied. The Lyapunov stability approach guarantees the stability of the proposed RUKF, which runs in conjunction with robust control method. Lastly, numerical simulations are presented to show the effectiveness of the proposed RUKF under actuator abrupt, intermittent, and transient fault conditions.

Observer-Based Event-Triggered Tracking Control for Discrete-Time Nonlinear Systems Using Adaptive Critic Design

In this article, a new output-feedback event-triggered (ET) tracking control strategy is investigated for discrete-time strict-feedback nonlinear systems based on the adaptive critic design (ACD). An effective state observer is designed to obtain current states information, which can remove the restriction that the system can be converted to the input–output model. Subsequently, dissimilar to the n-step forward predictor with multiple neural networks (NNs) and n-step delays, variable substitution technique is used to avoid time delays and calculation burden of multiple NNs. ACD structure is constructed to acquire the optimal control strategy by critic and action NNs. An effective iterative strategy is adopted to deal with the approximation error of action NN. In this way, the update law of the controller is constructed, which is more reasonable and easier to be implemented. Meanwhile, ET mechanism is placed between the senor and the controller. A novel ET condition is constructed to moderate the burden of network communications and realize the ideal control performance. All the state signals are guaranteed to be bounded. Numerical simulations are performed to display the feasibility of the presented approach.

Safe Reinforcement Learning-Based Robust Approximate Optimal Control for Hypersonic Flight Vehicles

In this paper, a safe reinforcement learning-based robust approximate optimal controller (SRL-RAOC) is proposed with the framework of actor-critic for hypersonic flight vehicles. First, we develop a system transformation based on barrier function, which transforms the issue of full-state safety constraints into an unconstrained optimization problem. Next, the safe reinforcement learning algorithm is employed to design the approximate optimal controller under an online actor-critic framework. Further, a robustifying term is presented to compensate for the neural network approximation errors introduced by the actor-critic framework. Hereafter, the stability analysis of the closed-loop system is accomplished by utilizing the Lyapunov technique, which shows that SRL-RAOC can guarantee that the equilibrium point is asymptotically stable. Moreover, the control input obtained by SRL-RAOC is demonstrated to be close to the optimal control input within a small bound. Finally, a numerical example is provided to demonstrate the effectiveness of SRL-RAOC.

Active fault tolerant control based on compound iterative learning observer for trajectory tracking of autonomous underwater vehicles

This article proposes a novel active non-singular integral terminal sliding mode fault tolerant control scheme based on compound iterative learning observer (CILO-NITSM-FTC) for the trajectory tracking control problems of autonomous underwater vehicles (AUVs), which accompanied by nonlinear dynamics and possible thruster failures. The designed compound iterative learning observer (CILO) can effectively estimate the values of nonlinear dynamics and reconstruct the faults of the thrusters. Compared to the existing adaptive non-singular integral terminal sliding mode-based fault tolerant control (ANITSM-FTC), which is a passive fault tolerant control method essentially, the mentioned CILO-NITSM-FTC in this paper presents faster rate of convergence and more accurate control precision. In the design process of CILO, radial basis function neural network (RBFNN) is adopted to estimate the nonlinear dynamics, then approximation errors of neural network, thruster faults and external disturbances are considered as global disturbances. The estimated values obtained from CILO can be used to perform feedforward compensation on the designed controller to complete the reconstruction of thruster faults. Through rigorous mathematical proof and comparative numerical simulation experiments, the superiority and practicability of CILO-NITSM-FTC proposed in this paper are verified.

Adaptive-critic-based model reference control for unknown nonlinear systems with input constraints

In this paper, the optimal model reference adaptive control (MRAC) problem is studied for the unknown discrete-time nonlinear systems with input constraint under the premise of considering robustness to uncertainty. Through an input constraint auxiliary system, a new adaptive-critic-based MRAC algorithm is proposed to transform the above problem into the optimal regulation problem of the auxiliary error system with lumped uncertainty. In order to realize the chattering-free sliding model control for the auxiliary error system, an action-critic variable is introduced into the adaptive identification learning. In this case, the closed-loop control system is robust to the disturbance and the neural network approximation error. The uniformly ultimate bounded property is proved by the Lyapunov method, and the effectiveness of the algorithm is verified by a simulation example.

Prescribed performance LOS guidance-based dynamic surface path following control of unmanned sailboats

The unmanned sailboat path following control problem with model uncertainties and unknown external disturbances is addressed. Firstly, different sailing strategies are set for the sailboat under three different wind fields in nature for its vulnerability to the true wind field so that the sailboat can complete the path-following task under different wind directions. Secondly, a barrier Lyapunov function with prescribed performance is used to limit position errors and heading errors of the unmanned sailboat within prescribed constraints. A radial-based function neural network technique is then introduced to approximate model uncertainties. Then an adaptive law is introduced to estimate an upper bound on the sum of the external environmental disturbances and neural network approximation errors. A path-following control scheme for the unmanned sailboat is proposed by the backstepping method and combined with the dynamic surface control technique. Finally, stability analysis is performed by Liapunov functions, demonstrating that the proposed control method can guarantee that all error signals in the closed-loop system are semi-global uniform ultimate bounded and that the prescribed performance constraints are not violated. Finally, the effectiveness of the scheme is demonstrated by simulation.

Neuroadaptive Asymptotic Tracking Control With Guaranteed Performance Under Mismatched Uncertainties and Saturated Inputs

It is still an open problem to achieve asymptotic tracking meanwhile maintaining specific performance for nonlinear systems with structurally mismatched uncertainties and strictly constrained inputs. In this work, we present a solution to this problem by using neural network (NN)-based adaptive control embedded with the robust integral of the sign of the error (RISE) technique. Most existing prescribed performance control (PPC) can only ensure uniformly ultimately bounded stability, and the RISE-based control, although capable of achieving asymptotic stability, does not guarantee transient behavior (especially, when the system is in strict-feedback form with saturated input). Here, in this study, we make use of NNs to accommodate the unknown nonlinearities, where the NN approximation error, together with other uncertainties, is fully compensated by using a RISE unit. The constraints imposed on the inputs are addressed by the hyperbolic tangent function, resulting in a solution capable of guaranteeing asymptotic tracking with prescribed transient performance, in the presence of mismatched modeling uncertainties and actuation saturation. A numerical simulation is carried out to verify the effectiveness of the proposed method.

Prescribed performance LOS guidance-based dynamic surface path following control of surface vessel with position and heading errors constraint

AbstractConcentrating on a surface vessel with input saturation, model uncertainties and unknown disturbances, a path following the adaptive backstepping control method based on prescribed performance line-of-sight (PPLOS) guidance is proposed. First, a prescribed performance asymmetric modified barrier Lyapunov function (PPAMBLF) is used to design the PPLOS and the heading controller, which make the path following position and heading errors meet the prescribed performance requirements. Furthermore, the backstepping and dynamic surface technique (DSC) are used to design the path following controller and the adaptive assistant systems are constructed to compensate the influence of input saturation. In addition, neural networks are introduced to approximate model uncertainties, and the adaptive laws are designed to estimate the bounds of the neural network approximation errors and unknown disturbances. According to the Lyapunov stability theory, all signals are semi-globally uniformly ultimately bounded. Finally, a 76$\,{\cdot }\,$2 m supply surface vessel is used for simulation experiments. The experimental results show that although the control inputs are limited, the control system can still converge quickly, and both position and heading errors can be limited to the prescribed performance requirements.

A Deep Neural Network Algorithm for Linear-Quadratic Portfolio Optimization With MGARCH and Small Transaction Costs

We analyze a fixed-point algorithm for reinforcement learning (RL) of optimal portfolio mean-variance preferences in the setting of multivariate generalized autoregressive conditional-heteroskedasticity (MGARCH) with a small penalty on trading. A numerical solution is obtained using a neural network (NN) architecture within a recursive RL loop. A fixed-point theorem proves that NN approximation error has a big-oh bound that we can reduce by increasing the number of NN parameters. The functional form of the trading penalty has a parameter ϵ > 0 that controls the magnitude of transaction costs. When ϵ is small, we can implement an NN algorithm based on the expansion of the solution in powers of ϵ. This expansion has a base term equal to a myopic solution with an explicit form, and a first-order correction term that we compute in the RL loop. Our expansion-based algorithm is stable, allows for fast computation, and outputs a solution that shows positive testing performance.

A Theoretical Framework for End-to-End Learning of Deep Neural Networks With Applications to Robotics

Deep Learning (DL) systems are difficult to analyze and proving convergence of DL algorithms like backpropagation is an extremely challenging task as it is a highly non-convex and high-dimensional problem. When using DL algorithms in robotic systems, theoretical analysis of stability, convergence, and robustness is a vital procedure as robots need to operate in a predictable manner to ensure safety. This paper presents the first unified End-to-End (E2E) learning framework that can be applied to both classification problems and real-time kinematic robot control tasks. In the proposed forward simultaneous learning method, the weights of all layers of the fully connected neural networks are updated concurrently. The proposed E2E learning framework uses an adjustable ReLU activation function so that convergence of the output error to a bound, which is dependent on the neural network approximation error, can be ensured. In particular, it is shown that the error between ideal output and estimated output of the network can converge to and stay in a certain bound, and this bound reduces to zero when the approximation error is zero. Therefore, the robustness of the learning system can be ensured even in the presence of the approximation error. Two case studies are done on classification tasks using MNIST and CIFAR10 datasets by using the proposed learning method, and the results show that the E2E learning method achieves comparable test accuracy as the gradient descent method and the main advantage is that convergence can be ensured during training. The framework is also implemented on a UR5e robot with unknown kinematic model, which is the first result on real-time E2E learning of deep neural networks for kinematic control of robots with guaranteed convergence.

Event‐triggered robust control for multi‐player nonzero‐sum games with input constraints and mismatched uncertainties

AbstractIn this article, an event‐triggered robust control (ETRC) method is investigated for multi‐player nonzero‐sum games of continuous‐time input constrained nonlinear systems with mismatched uncertainties. By constructing an auxiliary system and designing an appropriate value function, the robust control problem of input constrained nonlinear systems is transformed into an optimal regulation problem. Then, a critic neural network (NN) is adopted to approximate the value function of each player for solving the event‐triggered coupled Hamilton–Jacobi equation and obtaining control laws. Based on a designed event‐triggering condition, control laws are updated when events occur only. Thus, both computational burden and communication bandwidth are reduced. We prove that the weight approximation errors of critic NNs and the closed‐loop uncertain multi‐player system states are all uniformly ultimately bounded thanks to the Lyapunov's direct method. Finally, two examples are provided to demonstrate the effectiveness of the developed ETRC method.

Advanced neural control technique for autonomous underwater vehicles using modified integral barrier Lyapunov function

This paper presents a novel approach for depth precision control of under-actuated autonomous underwater vehicles (AUV) subject to model uncertainties, ocean currents, and input constraints. Specifically, a transformation is made to convert the input constraint problem into a state constraint problem. Subsequently, an observer-based guidance law is developed to deal with the drift affected by unknown ocean currents by using an extended disturbance observer (EDO). An adaptive neural controller is then designed using the DSC technique and an advanced modified integral barrier Lyapunov function (mIBLF) to guarantee that all states are confined within the given constraint. Besides, a novel nonlinear disturbance observer is introduced to cope with external disturbances and neural network approximation errors. It is proved that all closed-loop signals are uniformly ultimately bounded by Lyapunov stability theory. Finally, comparative simulations are carried out to verify the effectiveness and outstanding characteristics of the proposed method.

Online recorded data-based finite-time composite neural trajectory tracking control for underactuated MSVs.

This paper presents an online recorded data-based composite neural finite-time control scheme for underactuated marine surface vessels (MSVs) subject to uncertain dynamics and time-varying external disturbances. The underactuation problem of the MSVs was solved by introducing the line-of-sight (LOS) method. The uncertain dynamics of MSVs are approximated by the composite neural networks (NNs). A modified prediction error signal is designed by virtue of online recorded data. The weight updating law of NN is driven by both tracking error and prediction error, introducing additional correction information to the weights of NN, thus improving the learning ability of the NN. Furthermore, disturbance observers can be devised to estimate the compound disturbances consisting of the approximation errors of NNs and external disturbances. Moreover, the smooth function is inserted into the design of the control scheme, and the finite-time composite neural trajectory tracking control of MSVs is achieved. The stability of the MSVs trajectory tracking closed-loop control system is guaranteed rigorously by the Lyapunov approach, and the tracking error will converge to the set of residuals around zero within a finite time. The simulation tests on an MSV verify the effectiveness of the proposed control scheme.

Neural network based adaptive sliding mode tracking control of autonomous surface vehicles with input quantization and saturation

The autonomous surface vehicle (ASV) is supposed to be able to adapt unknown environments without human interference. This paper investigates the problem of trajectory tracking control for ASVs in the present of parameter uncertainties, disturbances, input quantization and actuator saturation. The neural network (NN) approximation and an adaptive sliding mode control (SMC) strategy are combined for the ASVs with a priori unknown upper bound of parameter uncertainties, external disturbances and saturation errors. The NN is adopted to approximate the unknown continuous uncertainties and disturbances in the system. The adaptive law for the SMC parameter can guarantee no overestimation of the robust control parameter with respect to unknown upper bounds of the NN approximation errors and the saturation error. By adding the NN approximation into the adaptive SMC laws, the robust control parameter and the chattering can be reduced. Uniformly ultimately bounded (UUB) for sliding variables and tracking errors can be guaranteed by Lyapunov proof. Simulation examples are given to illustrate the improved performance with the proposed approach. The comparisons on the proposed control method with other algorithms are given by using the integrated absolute error (IAE) and integrated time absolute error (ITAE).