Adaptive neural control of nonlinear MIMO systems with time-varying output constraints

Effects of Two-Channel Noise and Packet Loss on Performance of Information Time Delay Systems

The performance limitations of multiple-input multiple-output (MIMO) information time delay system (ITDS) with packet loss, codec and white Gaussian noise (WGN) are investigated in this article. By using the spectrum decomposition technique, inner-outer factorization, and partial factorization techniques, the expression of performance limitations is obtained under the two-degree-of-freedom (2DOF) compensator. The theoretical analysis results demonstrate that the system performance is related to the time delay, non-minimum phase (NMP) zeros, unstable zeros and their directions in a given device. In addition, WGN, packet loss and codec also impact the performance. Finally, the theoretical results are verified by simulation examples. Simulation results show that packet loss rate and encoding and decoding have a greater impact on system performance.

Traffic Signal Control With Adaptive Online-Learning Scheme Using Multiple-Model Neural Networks

This article proposes a new traffic signal control algorithm to deal with unknown-traffic-system uncertainties and reduce delays in vehicle travel time. Unknown-traffic-system dynamics are approximated using a recurrent neural network (NN). To accurately identify the traffic system model, an online-learning scheme is developed to switch among a set of candidate NNs (i.e., multiple-model NNs) based on their estimation errors. Then, a bank of optimal signal-timing controllers is designed based on the online identification of the traffic system. Simulation studies have been carried out for the obtained control strategies using multiple-model NNs, and the desired results have been obtained. Moreover, compared with the widely used actuated traffic signal control schemes, it is shown that the proposed method can reduce vehicle travel delays and improve traffic system robustness.

Adaptive Neural Control of Nonlinear Nonstrict Feedback Systems With Full-State Constraints: A Novel Nonlinear Mapping Method

In this work, a neural-networks (NNs)-based adaptive asymptotic tracking control scheme is presented for a class of uncertain nonstrict feedback nonlinear systems with time-varying full-state constraints. First, we construct a novel exponentially decaying nonlinear mapping to map the constrained system states to new system states without constraints. Instead of the traditional barrier Lyapunov function methods, the feasible conditions which require the virtual control signals satisfying the constraint requirements are removed. By employing the Nussbaum design method to eliminate the effect of unknown control gains, the general assumption about the signs of the unknown control gains is relaxed. Then, the nonstrict feedback form of the system can be pulled back to the strict feedback form through the basic properties of radial basis function NNs. Simultaneously, the intermediate control signals and the desired controller are constructed by the backstepping process and the Nussbaum design method. The designed controller can ensure that all signals in the whole closed-loop system are bounded without the violation of the constraints and hold the asymptotic tracking performance. In the end, a practical example about a brush dc motor driving a one-link robot manipulator is given to illustrate the effectiveness of the proposed design scheme.

Prescribed Finite-Time Adaptive Neural Tracking Control for Nonlinear State-Constrained Systems: Barrier Function Approach

The purpose of this article is to present a novel backstepping-based adaptive neural tracking control design procedure for nonlinear systems with time-varying state constraints. The designed adaptive neural tracking controller is expected to have the following characters: under its action: 1) the designed virtual control signals meet the constraints on the corresponding virtual control states in order to realize the backstepping design ideal and 2) the output tracking error tends to a sufficiently small neighborhood of the origin with the prescribed finite time and accuracy level. By combining the barrier Lyapunov function approach with the adaptive neural backstepping technique, a novel adaptive neural tracking controller is proposed. It is shown that the constructed controller makes sure that the output tracking error converges to a small neighborhood of the origin with the prespecified tracking accuracy and settling time. Finally, the proposed control scheme is further tested by simulation examples.

Introducing a Novel Model-Free Multivariable Adaptive Neural Network Controller for Square MIMO Systems

In this study, a novel Multivariable Adaptive Neural Network Controller (MANNC) is developed for coupled model-free n-input n-output systems. The learning algorithm of the proposed controller does not rely on the model of a system and uses only the history of the system inputs and outputs. The system is considered as a 'black box' with no pre-knowledge of its internal structure. By online monitoring and possessing the system inputs and outputs, the parameters of the controller are adjusted. Using the accumulated gradient of the system error along with the Lyapunov stability analysis, the weights' adjustment convergence of the controller can be observed, and an optimal training number of the controller can be selected. The Lyapunov stability of the system is checked during the entire weight training process to enable the controller to handle any possible nonlinearities of the system. The effectiveness of the MANNC in controlling nonlinear square multiple-input multiple-output (MIMO) systems is demonstrated via three simulation studies covering the cases of a time-invariant nonlinear MIMO system, a time-variant nonlinear MIMO system, and a hybrid MIMO system, respectively. In each case, the performance of the MANNC is compared with that of a properly selected existing counterpart. Simulation results demonstrate that the proposed MANNC is capable of controlling various types of square MIMO systems with much improved performance over its existing counterpart. The unique properties of the MANNC will make it a suitable candidate for many industrial applications.

Observer-based adaptive neural tracking control for a class of nonlinear systems with prescribed performance and input dead-zone constraints

This paper investigates the problem of output feedback neural network (NN) learning tracking control for nonlinear strict feedback systems subject to prescribed performance and input dead-zone constraints. First, an NN is utilized to approximate the unknown nonlinear functions, then a state observer is developed to estimate the unmeasurable states. Second, based on the command filter method, an output feedback NN learning backstepping control algorithm is established. Third, a prescribed performance function is employed to ensure the transient performance of the closed-loop systems and forces the tracking error to fall within the prescribed performance boundary. It is rigorously proved mathematically that all the signals in the closed-loop systems are semi-globally uniformly ultimately bounded and the tracking error can converge to an arbitrarily small neighborhood of the origin. Finally, a numerical example and an application example of the electromechanical system are given to show effectiveness of the acquired control algorithm.

Fuzzy Approximation-Based Finite-Time Control for a Robot With Actuator Saturation Under Time-Varying Constraints of Work Space

A finite-time control method is presented for n -link robots with actuator saturation under time-varying constraints of work space. Barrier Lyapunov functions (BLFs) are designed for ensuring that the robot remains under time-varying constraints of the work space. In order to deal with asymmetric saturation nonlinearity, we transform asymmetric saturation into a symmetric one by using a hyperbolic tangent function, which is introduced to avoid the discontinuous problem existing in the auxiliary system-based saturation method. Combining fuzzy-logic systems (FLSs) with the backstepping technique, a finite-time control policy is designed for ensuring the stability of the closed-loop system. With the use of the Lyapunov stability theory, all the error signals are proved to be semiglobal finite-time stable (SGFS). Finally, the experiment is carried out to verify the effectiveness of the finite-time method.

Performance-Guaranteed Fault-Tolerant Control for Uncertain Nonlinear Systems via Learning-Based Switching Scheme

This article is concerned with the challenge of guaranteeing output constraints for fault-tolerant control (FTC) of a class of unknown multi-input single-output (MISO) nonlinear systems in the presence of actuator faults. Most industrial systems are equipped with redundant actuators and a fault detection-isolation mechanism for accommodating unexpected actuator faults. To simplify the system design and reduce the risk of false alarm or missed detection brought by the detection unit, a learning-based switching function scheme is proposed to automatically activate different sets of actuators in a rotational manner without human intervention. By this means, no explicit fault detection mechanism is needed. An additional step has been made to guarantee that the system output remains in user-defined time-varying asymmetric output constraints all the time during the occurrence of failures by utilizing error transformation techniques. The stability of the transformed system can equivalently deliver the result that the original system output stays in the required bounds. Hence, system crash or further catastrophic outcomes can be avoided. A neural network is integrated to embody the adaptive FTC design for dealing with unknown system dynamics. The dynamic surface control (DSC) technique is also invoked to decrease complexity. Furthermore, the stability analysis is carried out by the standard Lyapunov approach to guarantee that all the signals of the closed-loop system are semiglobally uniformly ultimately bounded. Finally, the simulation results are provided to verify the effectiveness of the proposed scheme.

Neuroadaptive Fault-Tolerant Control Under Multiple Objective Constraints With Applications to Tire Production Systems

Many manufacturing systems not only involve nonlinearities and nonvanishing disturbances but also are subject to actuation failures and multiple yet possibly conflicting objectives, making the underlying control problem interesting and challenging. In this article, we present a neuroadaptive fault-tolerant control solution capable of addressing those factors concurrently. To cope with the multiple objective constraints, we propose a method to accommodate these multiple objectives in such a way that they are all confined in certain range, distinguishing itself from the traditional method that seeks for a common optimum (which might not even exist due to the complicated and conflicting objective requirement) for all the objective functions. By introducing a novel barrier function, we convert the system under multiple constraints into one without constraints, allowing for the nonconstrained control algorithms to be derived accordingly. The system uncertainties and the unknown actuation failures are dealt with by using the deep-rooted information-based method. Furthermore, by utilizing a transformed signal as the initial filter input, we integrate dynamic surface control (DSC) into backstepping design to eliminate the feasibility conditions completely and avoid off-line parameter optimization. It is shown that, with the proposed neuroadaptive control scheme, not only stable system operation is maintained but also each objective function is confined within the prespecified region, which could be asymmetric and time-varying. The effectiveness of the algorithm is validated via simulation on speed regulation of extruding machine in tire production lines.

Guaranteed Synchronization Performance Control of Nonlinear Time-Delay MIMO Multiagent Systems With Actuator Faults

This paper addresses the synchronization control problem of leader-follower multiagent systems with each follower described by a class of high-order nonlinear multiple-input-multiple-output (MIMO) dynamics in the presence of time delays and actuator faults. A distributed synchronization scheme with guaranteed synchronization performance based on the radial basis function neural network (RBF NN) is introduced. We propose an augmented quadratic Lyapunov function by incorporating the lower bounds of control gain matrices and the actuator healthy indicator, and the problems caused by the unknown time-varying control gain matrices, actuator faults, and coupling terms among agents are solved. Meanwhile, the output of followers can track that of the leader and the steady state, and the transient performance of synchronization can be guaranteed, while all the other signals in the closed-loop system are guaranteed to be bounded. Finally, numerical analysis has been carried out to verify the effectiveness of the proposed controller.

Adaptive Neural Network Fixed-Time Leader-Follower Consensus for Multiagent Systems With Constraints and Disturbances

This article is concerned with fixed-time leader-follower consensus problem for multiagent systems (MASs) with output constraints, unknown control direction, unknown system dynamics, unknown external disturbance, and dead-zone control input. First, a fixed-time distributed observer is presented for each follower to estimate the leader's states. Next, using a modified nonlinear mapping, an output-constrained system is transformed into an unconstrained system. Then, by adopting adding a power integrator technique, radial basis function neural network (RBFNN) approximation, and adaptive method, the ideal fixed-time stable virtual control protocol is derived and the issues of unknown control direction, unknown system dynamics, and unknown external disturbance are addressed. Finally, the actual control protocol is developed using the bound of dead-zone parameters. It is shown that the proposed control scheme achieves fixed-time leader-follower consensus of the studied MAS. The presented control protocol is applied to the leader-follower consensus of inverted pendulums and simulation results verify its effectiveness.

Adaptive Neural-Network Control of MIMO Nonaffine Nonlinear Systems With Asymmetric Time-Varying State Constraints

In this paper, a novel robust adaptive barrier Lyapunov function (BLF)-based backstepping controller has been proposed for a class of interconnected, multi-input-multi-output (MIMO) unknown nonaffine nonlinear systems with asymmetric time-varying (ATV) state constraints. The design involves a neural-network-based online approximator to cope with uncertain dynamics of the system. To tune its weights, a novel adaptive law is proposed based on the Hadamard product. A theorem has also been proposed to have the bounds on virtual control signals beforehand. This theorem eliminates the need for tedious offline computation for the feasibility condition on the virtual controller in BLF-based controller design. To overcome the problem of unknown control gain in the nonaffine system, Nussbaum gain has been used during the design. A simulation study on the robot manipulator in task space has been performed to illustrate the effectiveness of the proposed methodology.

Event-triggered adaptive finite-time tracking control for full state constraints nonlinear systems with parameter uncertainties and given transient performance

This paper investigates event-triggered finite-time tracking control problem for full state constraints nonlinear systems with uncertain parameters. Considering a class of full state constraints nonlinear systems, a new finite-time barrier Lyapunov function (FTBLF) is constructed, and it is utilized to achieve finite-time tracking control while each state constraints are not violated. Further, to reduce communication resource burden, a time-varying threshold event-triggered mechanism is proposed. Meanwhile, by integrating prescribed exponential function into FTBLF, the transient performance can be guaranteed and free from influences of event-triggered control input. Finally, on the basic of backstepping design, an event-triggered adaptive finite-time tracking control method is developed. The proposed method guarantees that tracking error tends to a small adjustable set and its trajectory is within specified bound, while full state constraints are never violated. Two examples are given to demonstrate the control effect.

Finite-Time Tracking Control for a Class of Uncertain Strict-Feedback Nonlinear Systems With State Constraints: A Smooth Control Approach

This article is concerned with the finite-time tracking control problem for a class of strict-feedback nonlinear systems involving state constraints, unknown nonlinearities, and nonvanishing disturbances. Unlike the literature that mainly focuses on a C⁰ finite-time controller, in this article, a novel C¹ smooth finite-time adaptive neural network (NN) controller is proposed by employing a smooth switch between the fractional and cubic form state feedback. The proposed controller not only avoids the singularity but also makes it possible to implement the dynamic surface control (DSC) technique. By applying the adaptive NN control technique, together with barrier Lyapunov functions (BLFs) and a generalized first-order filter including both linear and fractional terms, the desired fast finite-time control performance of the closed-loop nonlinear systems can be guaranteed, and meanwhile, the state constraints are never violated. Under the proposed control scheme, the tracking control problems of nonlinear systems with output constraint and full-state constraints are, respectively, discussed. It is explicitly shown that all the internal error signals are driven to converge into small regions in a finite time. Finally, the effectiveness of the control scheme is also confirmed by the applications to the control of a second-order nonlinear system and an uncertain ship autopilot.

Adaptive Neural Quantized Control of MIMO Nonlinear Systems Under Actuation Faults and Time-Varying Output Constraints

In this article, a neural network (NN)-based robust adaptive fault-tolerant control (FTC) algorithm is proposed for a class of multi-input multi-output (MIMO) strict-feedback nonlinear systems with input quantization and actuation faults as well as asymmetric yet time-varying output constraints. By introducing a key nonlinear decomposition for quantized input, the developed control scheme does not require the detailed information of quantization parameters. By imposing a reasonable condition on the gain matrix under actuation faults, together with the inherent approximation capability of NN, the difficulty of FTC design caused by anomaly actuation can be handled gracefully, and the normally used yet rigorous assumption on control gain matrix in most existing results is significantly relaxed. Furthermore, a brand new barrier function is constructed to handle the asymmetric yet time-varying output constraints such that the analysis and design are extremely simplified compared with the traditional barrier Lyapunov function (BLF)-based methods. NNs are used to approximate the unknown nonlinear continuous functions. The stability of the closed-loop system is analyzed by using the Lyapunov method and is verified through a simulation example.

Command Filter-Based Adaptive NN Control for MIMO Nonlinear Systems With Full-State Constraints and Actuator Hysteresis

This article studies the issue of adaptive neural network (NN) control for strict-feedback multi-input and multioutput (MIMO) nonlinear systems with full-state constraints and actuator hysteresis. Radial basis function NNs (RBFNNs) are introduced to approximate unknown nonlinear functions. The command filter is adopted to solve the issue of "explosion of complexity." By applying a one-to-one nonlinear mapping, the strict-feedback system with full-state constraints is converted into a new pure-feedback system without state constraints, and a novel NN control method is proposed. The stability of the closed-loop system is proved via the Lyapunov stability theory, and the tracking errors converge to small residual sets. The simulation results are given to confirm the validity of the proposed method.

Distributed Synchronization Control of Nonaffine Multiagent Systems With Guaranteed Performance

This paper deals with the synchronization control problem in the leader-follower format of a class of high-order nonaffine nonlinear multiagent systems under a directed communication protocol. A novel adaptive neural distributed synchronization scheme with guaranteed performance is proposed. The main contribution lies in the fact that both nonaffine agent dynamics, which basically makes most existing agent dynamics as special cases, and guaranteed synchronization performance are taken into account. The difficulty lies mainly in the nonaffine terms and coupling terms due to the interactions of agents. To overcome this challenge, an augmented quadratic Lyapunov function by incorporating the lower bounds of control gains is proposed. The problems resulting from the nonaffine dynamics and the coupling terms among agents are solved by incorporating the special property of radial basis function neural network into the derivative of the augmented quadratic Lyapunov function. The unknown nonaffine terms are addressed by using an indirected neural network approach. A nonlinear mapping is built to relate the local consensus error to a new one, which is subsequently stabilized via Lyapunov synthesis. As a result, the proposed approach can ensure the outputs of all follower agents to track the outputs of the leader, while the synchronization performance bounds can be quantified on both transient and steady-state stages. All other signals in the closed loop are ensured to be semiglobally, uniformly, and ultimately bounded. Finally, the effectiveness of the proposed controller is verified through a heterogeneous four-agent example.

Adaptive NN Control Without Feasibility Conditions for Nonlinear State Constrained Stochastic Systems With Unknown Time Delays

In the novel, an adaptive neural network (NN) controller is developed for a category of nonlinear stochastic systems with full state constraints and unknown time delays. The control quality and system stability suffer from the problems of state time delays and constraints which frequently arises in most real plants. The considered systems are transformed into new constrained free systems based on nonlinear mappings, such that full state constraints are never violated and the feasibility conditions on virtual controllers (the values of virtual controllers and its derivative are assumed to be known) are removed. To compensate for unknown time delayed uncertainties, the exponential type Lyapunov-Krasovskii functionals (LKFs) are employed. NNs are utilized to approximate unknown nonlinear functions appearing in the design procedure. In addition, by employing dynamic surface control (DSC) technique and less adjustable parameters, the online computation burden is lightened. The control method presented can achieve the semiglobal uniform ultimate boundedness of all the closed-loop system signals and the satisfactions of full state constraints by rigorous proof. Finally, by presenting simulation examples, the efficiency of the presented approach is revealed.

Robust State/Output-Feedback Control of Coaxial-Rotor MAVs Based on Adaptive NN Approach

The coaxial-rotor micro-aerial vehicles (CRMAVs) have been proven to be a powerful tool in forming small and agile manned-unmanned hybrid applications. However, the operation of them is usually subject to unpredictable time-varying aerodynamic disturbances and model uncertainties. In this paper, an adaptive robust controller based on a neural network (NN) approach is proposed to reject such perturbations and track both the desired position and orientation trajectories. A complete dynamic model of a CRMAV is first constructed. When all system states are assumed to be available, an NN-based state-feedback controller is proposed through feedback linearization and Lyapunov analysis. Furthermore, to overcome the practical challenge that certain states are not measurable, a high-gain observer is introduced to estimate the unavailable states, and then, an output-feedback controller is developed. Rigorous theoretical analysis verifies the stability of the entire closed-loop system. In addition, extensive simulation studies are conducted to validate the feasibility of the proposed scheme.

Neural Network Controller Design for a Class of Nonlinear Delayed Systems With Time-Varying Full-State Constraints

This paper proposes an adaptive neural control method for a class of nonlinear time-varying delayed systems with time-varying full-state constraints. To address the problems of the time-varying full-state constraints and time-varying delays in a unified framework, an adaptive neural control method is investigated for the first time. The problems of time delay and constraint are the main factors of limiting the system performance severely and even cause system instability. The effect of unknown time-varying delays is eliminated by using appropriate Lyapunov-Krasovskii functionals. In addition, the constant constraint is the only special case of time-varying constraint which leads to more complex and difficult tasks. To guarantee the full state always within the time-varying constrained interval, the time-varying asymmetric barrier Lyapunov function is employed. Finally, two simulation examples are given to confirm the effectiveness of the presented control scheme.

Neuro-Adaptive Control With Given Performance Specifications for Strict Feedback Systems Under Full-State Constraints

In this paper, we investigate the tracking control problem for a class of strict feedback systems with pregiven performance specifications as well as full-state constraints. Our focus is on developing a feasible neural network (NN)-based control method that is able to, under full-state constraints, force the tracking error to converge into a prescribed region within preset finite time and further reduce the error to a smaller and adjustable residual set, while confining the overshoot within predefined small level. Based on two consecutive error transformations governed by two auxiliary functions, named with behavior-shaping function and asymmetric scaling function, respectively, a novel approach to achieve given performance specifications is developed under certain bound condition on the transformed error, such condition, along with the full-stated constraints, is guaranteed by imbedding barrier Lyapunov function (BLF) into the back-stepping design. Furthermore, asymmetric output constraints are maintained with a single symmetric BLF, simplifying the procedure of stability analysis. All internal signals including the stimulating inputs to the NN unit are ensured to be bounded. Both theoretical analysis and numerical simulation verify the effectiveness and the benefits of the design.

Adaptive Reinforcement Learning Control Based on Neural Approximation for Nonlinear Discrete-Time Systems With Unknown Nonaffine Dead-Zone Input

In this paper, an optimal control algorithm is designed for uncertain nonlinear systems in discrete-time, which are in nonaffine form and with unknown dead-zone. The main contributions of this paper are that an optimal control algorithm is for the first time framed in this paper for nonlinear systems with nonaffine dead-zone, and the adaptive parameter law for dead-zone is calculated by using the gradient rules. The mean value theory is employed to deal with the nonaffine dead-zone input and the implicit function theory based on reinforcement learning is appropriately introduced to find an unknown ideal controller which is approximated by using the action network. Other neural networks are taken as the critic networks to approximate the strategic utility functions. Based on the Lyapunov stability analysis theory, we can prove the stability of systems, i.e., the optimal control laws can guarantee that all the signals in the closed-loop system are bounded and the tracking errors are converged to a small compact set. Finally, two simulation examples demonstrate the effectiveness of the design algorithm.

Neuroadaptive Control of Strict Feedback Systems With Full-State Constraints and Unknown Actuation Characteristics: An Inexpensive Solution

In this paper, we present a neuroadaptive control for a class of uncertain nonlinear strict-feedback systems with full-state constraints and unknown actuation characteristics where the break points of the dead-zone model are considered as time-variant. In order to deal with the modeling uncertainties and the impact of the nonsmooth actuation characteristics, neural networks are utilized at each step of the backstepping design. By using barrier Lyapunov function, together with the concept of virtual parameter, we develop a neuroadaptive control scheme ensuring tracking stability and at the same time maintaining full-state constraints. The proposed control strategy bears the structure of proportional-integral (PI) control, with the PI gains being automatically and adaptively determined, making its design less demanding and its implementation less costly. Both theoretical analysis and numerical simulation validate the benefits and the effectiveness of the proposed method.

Event-Based Adaptive NN Tracking Control of Nonlinear Discrete-Time Systems

This paper is concerned with the simultaneous design of a neural network (NN)-based adaptive control law and an event-triggering condition for a class of strict feedback nonlinear discrete-time systems. The stability and tracking performance of the closed-loop network control system under the event-triggering strategy is formally proven based on the Lyapunov theory in a hybrid framework. The proposed Lyapunov formulation yields an event-triggered algorithm to update the control input and NN weights based on conditions involving the closed-loop state. Different from the existing traditional NN control schemes where the feedback signals are transmitted and executed periodically, the feedback signals are transmitted and executed only when the event-trigger error exceeds the specified threshold, which can largely reduce the communication load. The effectiveness of the approach is evaluated through a simulation example.

Asymptotic Tracking Controller Design for Nonlinear Systems With Guaranteed Performance

In this paper, a novel adaptive control strategy is presented for the tracking control of a class of multi-input-multioutput uncertain nonlinear systems with external disturbances to place user-defined time-varying constraints on the system state. Our contribution includes a step forward beyond the usual stabilization result to show that the states of the plant converge asymptotically, as well as remain within user-defined time-varying bounds. To achieve the new results, an error transformation technique is first established to generate an equivalent nonlinear system from the original one, whose asymptotic stability guarantees both the satisfaction of the time-varying restrictions and the asymptotic tracking performance of the original system. The uncertainties of the transformed system are overcome by an online neural network (NN) approximator, while the external disturbances and NN reconstruction error are compensated by the robust integral of the sign of the error signal. Via standard Lyapunov method, asymptotic tracking performance is theoretically guaranteed, and all the closed-loop signals are bounded. The requirement for a prior knowledge of bounds of uncertain terms is relaxed. Finally, simulation results demonstrate the merits of the proposed controller.

Adaptive Sliding Mode Control of Dynamic Systems Using Double Loop Recurrent Neural Network Structure

In this paper, an adaptive sliding mode control system using a double loop recurrent neural network (DLRNN) structure is proposed for a class of nonlinear dynamic systems. A new three-layer RNN is proposed to approximate unknown dynamics with two different kinds of feedback loops where the firing weights and output signal calculated in the last step are stored and used as the feedback signals in each feedback loop. Since the new structure has combined the advantages of internal feedback NN and external feedback NN, it can acquire the internal state information while the output signal is also captured, thus the new designed DLRNN can achieve better approximation performance compared with the regular NNs without feedback loops or the regular RNNs with a single feedback loop. The new proposed DLRNN structure is employed in an equivalent controller to approximate the unknown nonlinear system dynamics, and the parameters of the DLRNN are updated online by adaptive laws to get favorable approximation performance. To investigate the effectiveness of the proposed controller, the designed adaptive sliding mode controller with the DLRNN is applied to a -axis microelectromechanical system gyroscope to control the vibrating dynamics of the proof mass. Simulation results demonstrate that the proposed methodology can achieve good tracking property, and the comparisons of the approximation performance between radial basis function NN, RNN, and DLRNN show that the DLRNN can accurately estimate the unknown dynamics with a fast speed while the internal states of DLRNN are more stable.

Smooth Neuroadaptive PI Tracking Control of Nonlinear Systems With Unknown and Nonsmooth Actuation Characteristics

This paper considers the tracking control problem for a class of multi-input multi-output nonlinear systems subject to unknown actuation characteristics and external disturbances. Neuroadaptive proportional-integral (PI) control with self-tuning gains is proposed, which is structurally simple and computationally inexpensive. Different from traditional PI control, the proposed one is able to online adjust its PI gains using stability-guaranteed analytic algorithms without involving manual tuning or trial and error process. It is shown that the proposed neuroadaptive PI control is continuous and smooth everywhere and ensures the uniformly ultimately boundedness of all the signals of the closed-loop system. Furthermore, the crucial compact set precondition for a neural network (NN) to function properly is guaranteed with the barrier Lyapunov function, allowing the NN unit to play its learning/approximating role during the entire system operation. The salient feature also lies in its low complexity in computation and effectiveness in dealing with modeling uncertainties and nonlinearities. Both square and nonsquare nonlinear systems are addressed. The benefits and the feasibility of the developed control are also confirmed by simulations.

Adaptive nearly optimal control for a class of continuous-time nonaffine nonlinear systems with inequality constraints

The state inequality constraints have been hardly considered in the literature on solving the nonlinear optimal control problem based the adaptive dynamic programming (ADP) method. In this paper, an actor-critic (AC) algorithm is developed to solve the optimal control problem with a discounted cost function for a class of state-constrained nonaffine nonlinear systems. To overcome the difficulties resulting from the inequality constraints and the nonaffine nonlinearities of the controlled systems, a novel transformation technique with redesigned slack functions and a pre-compensator method are introduced to convert the constrained optimal control problem into an unconstrained one for affine nonlinear systems. Then, based on the policy iteration (PI) algorithm, an online AC scheme is proposed to learn the nearly optimal control policy for the obtained affine nonlinear dynamics. Using the information of the nonlinear model, novel adaptive update laws are designed to guarantee the convergence of the neural network (NN) weights and the stability of the affine nonlinear dynamics without the requirement for the probing signal. Finally, the effectiveness of the proposed method is validated by simulation studies.

Adaptive Neural Control of a Class of Output-Constrained Nonaffine Systems

In this paper, we present a novel tracking controller for a class of uncertain nonaffine systems with time-varying asymmetric output constraints. Firstly, the original nonaffine constrained (in the sense of the output signal) control system is transformed into a output-feedback control problem of an unconstrained affine system in normal form. As a result, stabilization of the transformed system is sufficient to ensure constraint satisfaction. It is subsequently shown that the output tracking is achieved without violation of the predefined asymmetric time-varying output constraints. Therefore, we are capable of quantifying the system performance bounds as functions of time on both transient and steady-state stages. Furthermore, the transformed system is linear with respect to a new input signal and the traditional backstepping scheme is avoided, which makes the synthesis extremely simplified. All the signals in the closed-loop system are proved to be semi-globally, uniformly, and ultimately bounded via Lyapunov synthesis. Finally, the simulation results are presented to illustrate the performance of the proposed controller.