Systems In Strict-feedback Form Research Articles

Improved Command Filtered Backstepping Control for Uncertain Nonlinear Systems with Time-Delay

This paper presents an alternative method to solve the control problem of an uncertain nonlinear system in strict-feedback form with a time delay. Instead of using Lyapunov–Krasovskii functionals, ordinary Lyapunov functionals are used to design the controllers. In order to address the completely unknown uncertainties of the system, including the unmodeled dynamics, time-delay nonlinearities, and external disturbances, command filters are applied to reconstruct the estimations of such uncertainties, and the negative feedback of these estimations can be used to reduce the influence of such uncertainties on the system. With the help of the backstepping technique and the Lyapunov stability criterion, it is proved that the system output tracks the target signal with a small error, and contrastive simulation results verify our method’s effectiveness.

Continual online learning-based optimal tracking control of nonlinear strict-feedback systems: application to unmanned aerial vehicles

A novel optimal trajectory tracking scheme is introduced for nonlinear continuous-time systems in strict feedback form with uncertain dynamics by using neural networks (NNs). The method employs an actor-critic-based NN backstepping technique for minimizing a discounted value function along with an identifier to approximate unknown system dynamics that are expressed in augmented form. Novel online weight update laws for the actor and critic NNs are derived by using both the NN identifier and Hamilton-Jacobi-Bellman residual error. A new continual lifelong learning technique utilizing the Fisher Information Matrix via Hamilton-Jacobi-Bellman residual error is introduced to obtain the significance of weights in an online mode to overcome the issue of catastrophic forgetting for NNs, and closed-loop stability is analyzed and demonstrated. The effectiveness of the proposed method is shown in simulation by contrasting the proposed with a recent method from the literature on an underactuated unmanned aerial vehicle, covering both its translational and attitude dynamics.

Barrier Lyapunov Function-Based Safe Reinforcement Learning for Autonomous Vehicles With Optimized Backstepping.

Guaranteed safety and performance under various circumstances remain technically critical and practically challenging for the wide deployment of autonomous vehicles. Safety-critical systems in general, require safe performance even during the reinforcement learning (RL) period. To address this issue, a Barrier Lyapunov Function-based safe RL (BLF-SRL) algorithm is proposed here for the formulated nonlinear system in strict-feedback form. This approach appropriately arranges and incorporates the BLF items into the optimized backstepping control method to constrain the state-variables in the designed safety region during learning. Wherein, thus, the optimal virtual/actual control in every backstepping subsystem is decomposed with BLF items and also with an adaptive uncertain item to be learned, which achieves safe exploration during the learning process. Then, the principle of Bellman optimality of continuous-time Hamilton-Jacobi-Bellman equation in every backstepping subsystem is satisfied with independently approximated actor and critic under the framework of actor-critic through the designed iterative updating. Eventually, the overall system control is optimized with the proposed BLF-SRL method. It is furthermore noteworthy that the variance of the attained control performance under uncertainty is also reduced with the proposed method. The effectiveness of the proposed method is verified with two motion control problems for autonomous vehicles through appropriate comparison simulations.

Exponentially and finite-time stability using backstepping method for a nonlinear system in strict-feedback form

Exponentially and finite-time stability using backstepping method for a nonlinear system in strict-feedback form

Distributed Event-Triggered Control of Networked Strict-Feedback Systems via Intermittent State Feedback

It poses technical difficulty to achieve stable tracking even for single mismatched nonlinear strict-feedback systems when intermittent state feedback is utilized. The underlying problem becomes even more complicated if such systems are networked with directed communication and state-triggering setting. In this work, we present a fully distributed adaptive tracking control scheme for multiple agent systems in strict-feedback form using triggered state from the agent itself and the triggered states from the neighbor agents. To circumvent the non-differentiability of virtual controllers stemming from state-triggering, we first develop a distributed continuous control scheme under regular state feedback, upon which we construct the distributed event-triggered control scheme by replacing the states in the preceding scheme with the triggered ones. Several useful lemmas are introduced to allow the stability condition to be established with such replacement, ensuring that all the closed-loop signals are semi-globally uniformly ultimately bounded (SGUUB), with the output tracking error converging to a residual set around zero. Besides, with proper choices of the design parameters, the tracking performance in the mean square sense can be improved. Numerical simulation verifies the benefits and efficiency of the proposed method.

Reduced-Order Observer-Based Preassigned Finite-Time Control of Nonlinear Systems and Its Applications

In this article, a preassigned finite time control problem of nonlinear systems in strict-feedback form is investigated. From the perspective of arbitrary settling time, an appropriate preassigned finite-time performance function (PFPF) is constructed, and the preassigned finite-time stability (PAFS) is established, where the settling (convergence) time is not only completely unconcerned with initial conditions and design parameters but also more flexible. Furthermore, the backstepping technique and reduced-order observer are used to obtain the preassigned finite-time control scheme. The stability criteria of PAFS are developed to guarantee that the output can quickly converge to an arbitrarily small zone in preassigned time, and all signals of the closed-loop control system are PAFS. In the end, simulation examples verify the effectiveness of the presented method.

Global Asymptotic Stability of Nonlinear Strict Feedback System via Backstepping Control Approach

The design of controller for nonlinear system in strict feedback form in which the equilibrium point is the origin is the focus of the study. Backstepping is the technique applied. Recursive design is used in the backstepping control technique. Using the backstepping approach, we are given a reasonably simple method to design the control law for a system with these properties. The closed-loop system’s global asymptotic stability is attained by incorporating a Lyapunov function; in this case, a quadratic one was selected for the study. To demonstrate the viability of the technique, we also performed simulations of the system and the designed control with some initial values, and positive results were reported.

Immersion and invariance based adaptive command filtered backstepping control design for a class of SISO uncertain nonlinear systems in strict-feedback form

In this article, an immersion and invariance based adaptive command filtered backstepping control method (I&I ACFBC) is proposed for a class of single-input and single-output (SISO) uncertain nonlinear systems in strict-feedback form. The problem of the explosion of complexity inherent to the traditional adaptive backstepping control design is solved by applying the proposed adaptive control technique, and the compensating signals are constructed to reduce the effect of the known filtering errors produced by the command filters. The adaptive laws are achieved by an immersion and invariance method. The Lyapunov stability theory is utilized to ensure the boundedness of all the signals in the closed-loop system. The feasibility and validity of the proposed adaptive control scheme are demonstrated by the simulation and experimental results on a DC-DC buck converter system.

Adaptive finite-time fault-tolerant control for switched nonlinear systems with actuator fault and dead-zone via prescribed performance

In this work, an adaptive finite-time fault-tolerant control (FTC) technique for a class of switched nonlinear systems in strict-feedback form is presented. The switched nonlinear systems are considered under the influence of actuator fault, input-dead zone, and external disturbances. A prescribed performance function is used to ensure transient and steady-state control performance. To approximate unknown functions and minimize the negative effect of faults, radial basis function neural networks (RBFNNs) are used. With the help of neural networks and the backstepping method, an adaptive finite-time tracking controller is constructed. For the stability analysis of the proposed approach, the common Lyapunov function (CLF) is employed. Based on the Lyapunov function method, it has been shown that all the signals in the closed-loop system are semi-globally practically finite-time (SGPF)-stable. Also, the tracking error converges to a small residual set with the prescribed performance bounds. Finally, two simulation examples show the effectiveness of the presented control method, one of which is an application to a continuous stirred tank reactor (CSTR).

Hybrid neural network control design for uncertain switched nonlinear systems with external disturbances: Application to ship maneuvering system

In this work, the problem of hybrid neural network control for a class of switched uncertain nonlinear systems in strict-feedback form is considered. To approximate unknown nonlinear functions, an improved whale optimization algorithm (IWOA)-based hybrid neural network controller is introduced. Based on hybrid neural network approximation ability, a new hybrid neural network controller is constructed via the backstepping technique and the common Lyapunov function (CLF) is used for the stability analysis of the proposed model. The suggested method ensures that all signals in the closed-loop system are semi-global uniform ultimate bounded (SGUUB), and the tracking error converges to a small neighborhood of zero. Finally, the proposed scheme is applied to a ship maneuvering system and a numerical example to verify its effectiveness.

Neural-Adaptive Switching Control of Task-Space Objectives on Strict-Feedback Robot

Switching between tasks in robot applications is necessary for autonomous behaviour, requiring a control framework for switching between objectives. Thus, this paper proposes a switching control framework that allows transitions in position-dependent objectives for robot systems in strict-feedback form with adaptive capabilities. The proposed controller integrates artificial neural networks in two ways; one approximates control transforms while another approximates additive terms. Furthermore, the proposed method includes a dynamic damped inverse to ensure the invertible control transform matrices. The proposed controller is derived using the backstepping method and is stable in a Lyapunov sense. The proposed task-switching controller is first demonstrated on a generic robot manipulator. Then, a simulated two-degree-of-freedom manipulator with a camera at the end-effector tasked with position tracking and visual servoing verifies the capabilities of the proposed controller. A baseline switching controller shows that the proposed controller has superior performance, specifically, root mean square (RMS) error and control effort. Also, executing the simulation for multiple cycles shows that the proposed controller's artificial neural network and adaptive laws are stable.

Backstepping-Based Controller Design for Uncertain Switched High-Order Nonlinear Systems via PI Compensation

This article presents an effective method to address the tracking control problem arising in uncertain switched high-order nonlinear system in strict-feedback form. The system under consideration contains unknown functions, which causally make the asymptotic tracking performance difficult to be achieved. By adopting the adding a power integrator approach in the framework of backstepping, a novel tracking controller is developed to guarantee an asymptotic tracking performance in the presence of the approximation error cased by neural networks (NNs) under arbitrary switching. The main contributions lie in: 1) the article for the first time embeds the backstepping technique in designing a kind of discontinuous controller with proportional integral (PI) compensation and 2) with the help of Filippov’s theory, a new defined system described by differential inclusions can be first obtained by taking some transformations, and then a novel nonsmooth Lyapunov function approach along with its upper right Dini derivative technique is applied to complete the construction of the discontinuous controller. Finally, two simulation examples are exhibited to verify the validity of the proposed design techniques.

Distributed Fuzzy Adaptive Output-Feedback Control of Unknown Nonlinear Multiagent Systems in Strict-Feedback Form.

This article is concerned with the cooperative tracking control problem for heterogeneous multiagent systems in a leader-following form under a directed graph. The dynamics of each following agent is unknown, obeying a strict-feedback form. With the help of fuzzy-logic systems, input filters, and constraint-handling schemes, a fully distributed output-feedback control algorithm is proposed to achieve output synchronization with prescribed performance and guarantee boundedness of signals in the closed-loop systems. In addition, the algorithm exhibits a simplicity control attribute in the sense that: 1) the control design utilizes only relative output measurements, and no extra information needs to be transmitted via the network and 2) the issue of explosion of complexity is addressed, without employing command filters or dynamic surface control techniques. Finally, the simulation results clarify and verify the established theoretical findings.

Inverse Optimal Control of a Class of Nonlinear Multi-Agent Systems With Applications to Ship Slew Rate Control

Abstract This paper deals with the inverse optimal leader-following consensus problem of a class of high-order nonlinear multi-agent systems in strict-feedback form under directed communication topology. By employing the backstepping methodology and introducing an integral type Lyapunov function, a nonlinear distributed control law is constructed using the relative information between neighboring agents. The proposed control law solves the inverse optimal leader-following consensus problem under directed communication graph that contains a spanning tree with the root node being the leader agent. A practical example of ship slew rate control system is given to verify the effectiveness of the theoretical results.

Adaptive fuzzy fixed-time control for a class of strict-feedback stochastic nonlinear systems

This paper studies fixed-time tracking problems for stochastic nonlinear systems in strict-feedback form. Different from previous results, the practical fixed-time bounded theorem for stochastic nonlinear systems is given. The unknown functions of the stochastic nonlinear systems are approximated by the Fuzzy logic system (FLS) which has a universal approximation. Then by using a back-stepping method, a novel adaptive fuzzy fixed-time controller is designed for stochastic nonlinear systems based on the fixed-time bounded theorem. The states of the stochastic nonlinear systems are guaranteed to converge into an equilibrium point contained compact set semi-globally in fixed-time by the designed controller. Finally, a numerical example and a vehicle tracking model example are provided to illustrate the proposed strategy.

A small gain theorem for finite-time input-to-state stability of infinite networks and its applications

We prove a small-gain sufficient condition for (global) finite-time input-to-state stability (FTISS) of infinite networks. The network under consideration is composed of a countable set of finite-dimensional subsystems of ordinary differential equations, each of which is interconnected with a finite number of its “neighbors” only and is affected by some external disturbances. We assume that each node (subsystem) of our network is finite-time input-to-state stable (FTISS) with respect to its finite-dimensional inputs produced by this finite set of the neighbors and with respect to the corresponding external disturbance. As an application we obtain a new theorem on decentralized finite-time input-to-state stabilization with respect to external disturbances for infinite networks composed of a countable set of strict-feedback form systems of ordinary differential equations. For this we combine our small-gain theorem proposed in the current work with the controllers design developed by S. Pavlichkov and C. K. Pang (NOLCOS-2016) for the gain assignment of the strict-feedback form systems in the case of finite networks. The current results address the finite-time input-to-state stability and decentralized finite-time input-to-state stabilization and redesign the technique proposed in recent work S. Dashkovskiy and S. Pavlichkov, Stability conditions for infinite networks of nonlinear systems and their application for stabilization, Automatica. – 2020. – 112. – 108643, in which the case of $\ell_{\infty}$-ISS of infinite networks was investigated. The current paper extends and generalizes its conference predecessor to the case of finite-time ISS stability and decentralized stabilization in presence of external disturbance inputs and with respect to these disturbance inputs. In the special case when all these external disturbances are zeroes (i.e. are abscent), we just obtain finite-time stability and finite-time decentralized stabilization of infinite networks accordingly.

Adaptive Robust Control for Networked Strict-Feedback Nonlinear Systems with State and Input Quantization

Backstepping method is a successful approach to deal with the systems in strict-feedback form. However, for networked control systems, the discontinuous virtual law caused by state quantization introduces huge challenges for its applicability. In this article, a quantized adaptive robust control approach in backsetpping framework is developed in this article for networked strict-feedback nonlinear systems with both state and input quantization. In order to prove the efficiency of the designed control scheme, a novel form of Lyapunov candidate function was constructed in the process of analyzing the stability, which is applicable for the systems with nondifferentiable virtual control law. In particular, the state and input quantizers can be in any form as long as they meet the sector-bound condition. The theoretic result shows that the tracking error is determined by the pregiven constants and quantization errors, which are also verified by the simulation results.

Adaptive prescribed performance control of nonlinear asymmetric input saturated systems with application to AUVs

In this paper, the adaptive prescribed performance tracking control of nonlinear asymmetric input saturated systems in strict-feedback form is addressed under the consideration of model uncertainties and external disturbances. A radial basis function neural network (RBF-NN) is utilized to handle the model uncertainties. By prescribed performance functions, the transient performance of the system can be guaranteed. The continuous Gaussian error function is represented as an approximation of asymmetric saturation nonlinearity such that the backstepping technique can be leveraged in the control design. Based on the Lyapunov synthesis, residual function approximation inaccuracies and external disturbances are compensated by constructed adaptive control laws. As a consequence, all the signals in the closed-loop system are uniformly ultimately bounded and the tracking errors bounded by prescribed functions converge to a small neighbourhood of zero. The proposed method is applied to the autonomous underwater vehicles (AUVs) with extensive simulation results demonstrating the effectiveness of the proposed method.

Fuzzy adaptive output-feedback tracking control for nonlinear strict-feedback systems in prescribed finite time

The current paper addresses the fuzzy adaptive tracking control via output feedback for single-input single-output (SISO) nonlinear systems in strict-feedback form. Under the situation of system states being unavailable, the system output is used to set up the state observer to estimate the real system states. Furthermore, the estimation states are employed to design controller. During the control design process, fuzzy logic systems (FLSs) are used to model the unknown nonlinearities. A novel observer-based finite-time tracking control scheme is proposed via fuzzy adaptive backstepping and barrier Lyapunov function approach. The suggested fuzzy adaptive output feedback controller can force the output tracking error to meet the pre-specified accuracy in a fixed time. Meanwhile, all the closed-loop variables are bounded. Compared to some existing finite-time output feedback control schemes, the developed control strategy guarantees that the settling time and the error accuracy are independent of the uncertainties and can be specified by the designer. At last, the effectiveness and feasibility of the proposed control scheme are demonstrated by two simulation examples.

Robust adaptive fuzzy control for a class of uncertain nonaffine nonlinear systems with unknown control directions

In this paper, the adaptive fuzzy backstepping control problem is considered for a class of single-input single-output (SISO) unknown uncertain nonaffine nonlinear systems in strict-feedback form. Within this approach, Nussbaum gain functions are introduced to solve the problem of unknown control directions. The unknown nonlinear functions are approximated by employing adaptive fuzzy systems. The stability analysis of the closed-loop system in the sense of Lyapunov guarantees the global boundedness property for all the signals and states, and at the same time, steers the tracking error to a small neighborhood of the origin. The feasibility of the developed control approach is illustrated by numerical simulation.