Uncertain Strict-feedback Nonlinear Systems Research Articles

Adaptive finite time command filtered backstepping control scheme of uncertain strict-feedback nonlinear systems

This paper proposes an adaptive finite time command filtered backstepping control scheme for single‐input and single‐output (SISO) uncertain strict-feedback nonlinear systems. The problem of the explosion of complexity existing in the adaptive backstepping control design and the singularity problem are solved by using the proposed adaptive finite time control method. In addition, the compensating signals are introduced to deal with the effect of the known filtering errors caused by the command filters. By using the finite time Lyapunov stability theory, the proposed adaptive finite time control strategy guarantees that all the signals in the closed-loop system are practical finite time stable, and the tracking errors converge to an arbitrarily small neighbourhood of the origin in finite time. Simulation and experimental results of the DC-DC buck converter are given to illustrate the effectiveness of the proposed adaptive finite time control scheme.

Adaptive Fuzzy Tracking Control With Global Prescribed-Time Prescribed Performance for Uncertain Strict-Feedback Nonlinear Systems.

For strict-feedback systems with mismatched uncertainties, adaptive fuzzy control techniques are developed to provide global prescribed performance with prescribed-time convergence. First, a class of prescribed-time prescribed performance functions are designed to quantify the performance constraints of the tracking error. Additionally, a novel error transformation function is provided to eliminate the initial value limitations and resolve the singularity issue in previous research. To ensure the convergence of the tracking error into a prescribed bounded region within a prescribed time and satisfactory transient performance, controllers with or without approximating structures are established. Notably, the settling time and initial condition of the prescribed performance function are completely independent of the initial tracking error and system parameters, thereby improving upon existing results. Furthermore, the disadvantage of the semi-global boundedness of tracking error induced by dynamic surface control can be eliminated through the use of a novel Lyapunov-like energy function. Finally, the effectiveness of the proposed strategies is validated through numerical simulations performed on practical examples.

Dynamic Event-Based Adaptive Fixed-Time Control for Uncertain Strict-Feedback Nonlinear Systems With State Constraints.

In this article, the event-triggered fixed-time tracking control is investigated for uncertain strict-feedback nonlinear systems involving state constraints. By employing the universal transformed function (UTF) and coordinate transformation techniques into backstepping design procedure, the proposed control scheme ensures that all states are constrained within the time-varying asymmetric boundaries, and meanwhile, the undesired feasibility condition existing in other constrained controllers can be removed elegantly. Different from the existing static event-triggered mechanism, a dynamic event-triggered mechanism (DETM) is devised via constructing a novel dynamic function, so that the communication burden from the controller to actuator is further alleviated. Furthermore, with the aid of adaptive neural network (NN) technique and generalized first-order filter, together with Lyapunov theory, it is proved that the states of closed-loop system converge to small regions around zero with fixed-time convergence rate. The simulation results confirm the benefits of developed scheme.

Event-triggered impulsive tracking control for uncertain strict-feedback nonlinear systems via the neural-network-based backstepping technique

This paper studies the problem of event-triggered impulsive tracking control (ETITC) for uncertain strict-feedback nonlinear systems (USFNSs). In contrast to existing impulsive control schemes, this paper incorporates the neural-network (NN)-based backstepping technique into impulsive control design, such that stronger nonlinearities and uncertainties are allowed to be included in the concerned systems. The proposed state-feedback ETITC scheme guarantees that all the signals of the closed-loop system are bounded and the tracking error ultimately converges to an adjustable bounded region, while also avoiding the Zeno behavior. In addition, by constructing a NN-based observer, this paper further develops an output-feedback ETITC scheme to extend its investigation to the scenario where full states are not available for impulsive control design. Finally, an illustrative example involving a chemical reactor system is presented to demonstrate the effectiveness of our control schemes.

Global regulation for uncertain feedforward nonlinear systems based on event-triggered impulsive control: a time-varying gain scaling approach

This paper addresses the problem of global regulation for uncertain feedforward nonlinear systems using event-triggered impulsive control. By constructing a state transformation with a time-varying scaling gain, both the state-feedback and output-feedback event-triggered impulsive control strategies are established. It is proven that the proposed impulsive controllers ensure the global regulation of the concerned systems while avoiding the Zeno behaviour. Furthermore, the derived results of this work are generalised to a broader class of uncertain feedforward nonlinear systems incorporating time-varying nonlinearities, as well as to a class of uncertain strict-feedback nonlinear systems. It is noteworthy that this paper is the first attempt to design an impulsive regulating controller for feedforward nonlinear systems. Finally, a simulation example is presented to verify the effectiveness of our theoretical results.

Performance-Based Distributed Control of Multiagent Systems: A Dual Phase Approach.

In this article, we investigate the distributed tracking control problem for networked uncertain nonlinear strict-feedback systems with unknown time-varying gains under a directed interaction topology. A dual phase performance-guaranteed approach is established. In the first phase, a fully distributed robust filter is constructed for each agent to estimate the desired trajectory with prescribed performance such that the control directions of all agents are allowed to be nonidentical. In the second phase, by establishing a novel lemma regarding Nussbaum function, a new adaptive control protocol is developed for each agent based on backstepping technique, which not only steers the output to track the corresponding estimated signal asymptotically with arbitrarily prescribed transient response but also extends the application scope of the proposed control scheme largely since the unknown control gains are allowed to be time-varying and even state-dependent. In such a way, the underlying problem is tackled with the output tracking error converging into an arbitrarily preassigned residual set exhibiting an arbitrarily predefined convergence rate. Besides, all the internal signals are ensured to be semi-globally ultimately uniformly bounded (SGUUB). Finally, two examples are provided to illustrate the effectiveness of the co-designed scheme.

Observer-based adaptive neural network control design for nonlinear systems under cyber-attacks through sensor networks

This paper considers the adaptive neural tracking control for uncertain nonlinear strict-feedback systems under state-dependent cyber-attacks through sensor networks. As a distinctive feature, by constructing a novel adaptive neural observer, the estimates of system states are used for feedback control instead of the compromised system states corrupted by sensor attacks, such that the assumption on the derivative of attack weight can be removed from the adaptive controller design process. Then, an observer-based control scheme is developed such that all the system signals are bounded and the tracking error converges to a small neighborhood of zero. During the procedure of control design, adaptive backstepping technique is combined with radial basis function neural networks (RBFNNs) to construct controllers. Finally, two examples validate the efficacy of the proposed control scheme.

Neural Network Boundary Approximation for Uncertain Nonlinear Spatiotemporal Systems and Its Application of Tracking Control.

This brief addresses the neural network (NN) approximation problem for uncertain nonlinear systems with time-varying parameters (that is, unknown nonlinear spatiotemporal systems). Due to the fact that the unknown spatiotemporal functions cannot be directly approximated by NNs, a so-called time-varying parameter extraction is given to separate time-varying parameters from uncertain nonlinear spatiotemporal functions. By using the supremum of Euler norm of the extracted time-varying parameters, the nonlinear spatiotemporal function is mapped to an unknown state-based boundary function, which can be approximated by NNs. Based on the time-varying parameter extraction, an adaptive neural tracking control law is designed for uncertain strict-feedback nonlinear spatiotemporal systems, which guarantees the convergence of the tracking error with a trajectory performance. The effectiveness of the designed method is verified by simulations.

Output feedback command filtered fuzzy controller design for uncertain strict-feedback nonlinear systems with unmodeled dynamics and event-triggered strategy

An adaptive control problem is considered for a class of nth-order uncertain nonlinear systems with unmodeled dynamics. Compared with the traditional backstepping method, this article improves the systems performance by introducing command filtered and event-triggered. The former primarily enhances the flexibility of controller design, while the latter employs a variable threshold triggering mechanism to achieve savings in communication resources. Simultaneously, the estimation error is compensated by the introduced hyperbolic tangent function. In addition, fuzzy logic system (FLS) is primarily used for observer design and approximation of unknown functions. The conclusion that the closed-loop signals are bounded is obtained through stability analysis. Finally, the numerical simulation results provide a comparative validation of the effectiveness of the proposed control scheme.

Event-triggered prescribed-time tracking for uncertain nonlinear systems with unknown control gain and output constraints

This paper uses a prescribed-time event-triggered approach to explore the control of uncertain strict-feedback nonlinear systems with output constraints and an unknown time-varying control gain. The proposed method utilizes a universal barrier Lyapunov function (BLF) to ensure compliance with the output constraints. Furthermore, a technique known as congelation of variables is employed to handle the unknown time-varying control gain. Approximating the unknown dynamics using radial basis function neural network (RBFNN), a neural prescribed-time controller is designed based on the backstepping approach. Additionally, the event-triggered mechanism is employed to conserve computational and communication resources. With this scheme, all signals are guaranteed to converge to a compact set within the prescribed time frame. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed algorithm.

Finite-time adaptive optimal control of uncertain strict-feedback nonlinear systems based on fuzzy observer and reinforcement learning

This paper proposes an adaptive optimal control strategy of finite-time control for high-order uncertain strict-feedback nonlinear systems. Firstly, a reinforcement learning (RL) based an optimal control scheme is employed to design a optimal controller, to achieve global optimisation. Additionally, considering the unmeasurable states, we construct a fuzzy observer and utilise fuzzy logic systems to approximate the unknown functions. Meanwhile, the inclusion of command filtering and time-based control simplifies the controller design and enhances the system's response rapidity. Finally, the effectiveness and feasibility of the proposed approach are validated through a numerical simulation and a single link-robot system simulation.

Event-Triggered Adaptive Control of Uncertain Strict-Feedback Nonlinear Systems Using Fully Actuated System Approach.

In most existing results, event-triggered controllers are designed based on the backstepping design approach for uncertain strict-feedback nonlinear systems (SFNSs). However, the transmitted signals in the event-triggered scheme (ETS) are discontinuous, which makes the repetitive differentiation of virtual control signals undefined. To overcome this deficiency, this article designs an event-triggered adaptive controller for uncertain SFNSs based on the fully actuated system (FAS) approach. Since the system states and the adaptive parameters are only updated at each triggering instant, the original dynamics cannot be completely removed by using the FAS approach, leading to that the asymptotic stability of the control system is difficult to be guaranteed. To handle such a problem, an ETS with the adaptive parameters is constructed based on Lyapunov method to compensate the effect of triggering. As a result, the asymptotic stability of the system can be guaranteed in the presence of nonlinearities without the global Lipschitz condition, and Zeno behavior can be avoided by using the contradiction method. Furthermore, a positive lower bound for interevent intervals can be got by adding a constant into the ETS, which ensures that the system is practically stabilizable under the bounded nonlinearities. Finally, two simulation examples are presented to demonstrate the superiority and effectiveness of the proposed approach.

Neural Network-Based Nonconservative Predefined-Time Backstepping Control for Uncertain Strict-Feedback Nonlinear Systems.

To handle the tracking problem of uncertain strict-feedback nonlinear systems with matched and mismatched composite disturbances, this article studies a predefined-time backstepping controller by resorting to a Lyapunov-based predefined-time dynamic paradigm, a regulation function, and neural networks (NNs). Moreover, an adding-absolute-value (ADV) technique is adopted in the design process to remove the control singularity. Theoretical analyses prove the boundedness of all closed-loop system signals and the predefined-time convergence of the tracking error into an arbitrarily small vicinity of the origin. The proposed controller exhibits four advantages: 1) the actual convergence time is precisely predefined by only one design parameter irrespective of the initial conditions, and the control energy is economized; 2) no unbounded terms are adopted for predefining the actual convergence time, thus avoiding numerical overflow problem under limited memory space and gaining strong noise-tolerant ability; 3) the peaking tracking error and control input magnitude can be effectively reduced by appropriately setting parameters of the regulation function; and 4) the controller is continuous and nonsingular everywhere. Finally, a practical example of a single-link manipulator is presented to validate the efficacy and superiority of our predefined-time controller.

Adaptive neural DSC for nonlinear switched systems with prescribed performance and input saturation

This paper solves the problem of an adaptive neural dynamic surface control for a class of uncertain strict-feedback nonlinear systems with guaranteed transient and steady-state performance under arbitrary switchings. First, by utilizing the prescribed performance control, the prescribed tracking control performance can be ensured, while the requirement for the initial error is removed. Second, radial basis function (RBF) neural networks (NNs) are used to handle unknown nonlinear functions, the Gaussian error function is employed to represent a continuous differentiable asymmetric saturation model and the dynamic surface control (DSC) technique is used to overcome the problem of 'explosion of complexity' inherent in the control design. At last, by using the common Lyapunov function method in combination with the backstepping technology, a common adaptive neural controller is constructed. The designed controller overcomes the problem of the over-parameterization, and further alleviates the computational burden. Under the proposed common adaptive controller, all the signals in the closed-loop system are bounded, and the prescribed transient and steady tracking control performance are guaranteed under arbitrary switchings. Simulation studies demonstrate the effectiveness of the proposed method.

Prescribed performance event-triggered fuzzy optimal tracking control for strict-feedback nonlinear systems

The prescribed performance event-triggered optimal tracking control problem is considered for a class of uncertain strict-feedback nonlinear systems with external disturbances. First, the disturbance observers are constructed to estimate external disturbances. Then, the controller contains an adaptive fuzzy controller and an optimal compensation term, in which the unknown nonlinearities and cost function are approximated by the fuzzy logic systems. An adaptive fuzzy controller is established by the dynamic surface control (DSC) technique, which is addressed to handle “computation complexity” issue occurred in conventional backstepping approach. Based on adaptive dynamic programming (ADP) method, an optimal compensation term is developed by minimizing the cost function. Furthermore, communication load is reduced via embedding event-triggered mechanism. In addition, the tracking error can be restricted in the prescribed region with the aid of the prescribed performance control. Thus, the whole control scheme can not only ensure that the tracking error converges to a boundary but also achieve optimization, reduce the communication burden and avoid “computation complexity” issue. The boundedness of all signals in the closed-loop is proved. Finally, the simulation examples illustrate the validity of the designed control scheme.

Observer-based dynamic event-triggered adaptive control for uncertain nonlinear strict-feedback systems

This paper proposes an observer-based dynamic event-triggered adaptive control approach for uncertain nonlinear strict-feedback systems. Initially, an observer for unmeasurable states is constructed. Subsequently, employing backstepping technique, the output-feedback adaptive control law and parameter adaptive law are developed. The tuning function method is used to avoid the over-parameterization problem in parameter adaptive law design. To lower the data exchange rate within the network, a dynamic event-triggered scheme is formulated to allow real-time update control signal. Through Lyapunov theory analysis, the resulting closed-loop system is demonstrated to be stabilized, and the occurrence of Zeno behavior is effectively prevented. Finally, the simulation example validates the obtained result of the presented design approach.

Asymptotic Tracking Control for Uncertain Nonlinear Strict-Feedback Systems With Unknown Time-Varying Delays.

It is nontrivial to achieve asymptotic tracking control for uncertain nonlinear strict-feedback systems with unknown time-varying delays. This problem becomes even more challenging if the control direction is unknown. To address such problem, the Lyapunov-Krasovskii functional (LKF) is used to deal with the time delays, and the neural network (NN) is applied to compensate for the time-delay-free yet unknown terms arising from the derivative of LKF, and then an NN-based adaptive control scheme is constructed on the basis of backstepping technique, which enables the output tracking error to converge to zero asymptotically. Besides, with a milder condition on time delay functions, the notorious singularity issue commonly encountered in coping with time delay problems is subtly settled, which makes the proposed scheme simple in structure and inexpensive in computation. Moreover, all the signals in the closed-loop system are ensured to be semiglobally uniformly ultimately bounded, and the transient performance can be improved with proper choice of design parameters. Both the theoretical analysis and numerical simulation are carried out to validate the relevance of the proposed method.

Adaptive practical prescribed-time formation tracking of networked nonlinear multiagent systems with quantized inter-agent communication

This paper presents an adaptive practical prescribed-time (PPT) control design for distributed time-varying formation tracking of networked uncertain strict-feedback nonlinear systems with quantized inter-agent communication under a directed network. The primary contribution of this study is to develop a time-varying formation tracking strategy to resolve the quantized inter-agent communication problem in the prescribed-time control field. A practical finite-time function is introduced to design an adaptive PPT formation tracker using quantized output information of neighbors, which can be used continuously after a prescribed time. A neural-network-based design methodology that utilizes the quantization-based distributed errors is presented to deal with the unknown virtual and actual control coefficient functions in the command-filtered backstepping design. Distributed adaptive compensating signals are constructed using the practical finite-time function to compensate for command-filter errors, unknown nonlinearities, and quantization errors in the PPT tracking framework. We prove that despite the presence of quantized inter-agent communication, the time-varying formation tracking errors converge to a compact set including the origin within the pre-assigned convergence time, where the compact set can be adjusted by choosing a design parameter of the practical finite-time function, and the prescribed settling time is independent of the initial system conditions and design parameters. The effectiveness of the proposed theoretical approach is confirmed through two simulation examples.

Fuzzy Adaptive Finite-Time Tracking Control for a Class of Nonlinear Systems: An Event-Triggered Quantized Control Scheme

This brief proposes a fuzzy adaptive event-triggered quantized finite-time control (ETQFTC) scheme for uncertain strict-feedback nonlinear systems with full-state constraints. By means of state transformation, filtered backstepping technique and fuzzy adaptive control with the idea of minimum learning parameter (MLP), a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C^{1}$</tex-math></inline-formula> ETQFTC scheme with low differential calculation and few parameter updates is developed. In the proposed scheme, the feasibility condition generally required in the existing state-constrained control, and the control chattering and singularity problem which are common in the existing FTC schemes are removed and avoided respectively. Finally, complete stability analysis and a set of comparative simulations are presented.

Adaptive Robust Output Tracking of Uncertain Strict-Feedback Nonlinear Systems with Full State Constraints via Control Schemes with Simple Structure

In this paper, the problem of output tracking control is considered for a class of uncertain strict-feedback nonlinear systems with virtual control coefficients and full state constraints. It is assumed only that the nonlinear uncertainties of the systems are some smooth functions enough to guarantee the existence of solutions to the differential equations which describe uncertain systems, and that the virtual control coefficients are unknown. Based on backstepping algorithm, by combining the barrier Lyapunov function with an integral inequality reported in the current control literature, a design method is presented whereby a simple adaptive robust output tracking control scheme can be synthesized. The presented design method need not know all the nonlinear upper bounds of uncertainties, which are repeatedly employed at each step of backstepping procedure. It is also shown that the output tracking error of the uncertain nonlinear systems with both unknown virtual control coefficients and full state constraints can converge uniformly exponentially towards a ball which can be as small as desired. Finally, a numerical example is given, and the corresponding simulations are also implemented to demonstrate the simplicity of the proposed method and the validity of the theoretical results.