Strict-feedback Nonlinear Systems Research Articles

Event-Triggered Observers and Distributed H∞ Control of Physically Interconnected Nonholonomic Mechanical Agents in Harsh Conditions

In this article, event-triggered (ET) observers and ET-distributed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\boldsymbol {\mathcal {H}}_{\boldsymbol \infty }}$ </tex-math></inline-formula> controllers are investigated for physically interconnected nonholonomic mechanical agents in harsh conditions, such as skidding, slipping, and dead-zone disturbances. The agents’ models are presented by strict-feedback nonlinear large-scale systems, but unlike the existing studies, unknown dynamics with feedback outputs are considered, and assumptions of polynomial-type nonlinearities for physical interconnection functions are relaxed. Initially, the observer of unmeasurable states via the outputs is designed. Then, ET augmented controllers are established to transform the physically interconnected system into isolated subsystems connected by a communication network. By utilizing the local information of each agent and adaptive dynamic programming (ADP), ET-distributed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\boldsymbol {\mathcal {H}}_{\boldsymbol \infty }}$ </tex-math></inline-formula> optimal control laws and disturbance rejection laws are derived. The parameters of the ET observers and controllers are synchronously updated online by event-triggering mechanisms; thus, it reduces computational complexity and communication. It is guaranteed that the states in the closed dynamics are <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> -gain bounded and the Zeno behavior is avoided. Additionally, the convergence of cost functions to the near-optimal values is accelerated by a concurrent learning technique, avoiding dependence on the online examination of the persistence of excitation conditions. Finally, the control performance of a nonholonomic multirobot system in a comparative simulation study shows that the proposed observers and controllers are effective.

Prescribed-Time Asymptotic Tracking Control of Strict Feedback Systems With Time-Varying Parameters and Unknown Control Direction

This paper considers the global prescribed-time asymptotic tracking control problem for uncertain strict feedback nonlinear systems with time-varying parameters and an unknown control direction. A novel prescribed-time scaling function is constructed, and the frequently used basic lemma on the Nussbaum function is modified to adapt to the situation where the initial value of the control coefficient is zero. With the help of the nonlinear mapping technique, the congelation of variables method and the backstepping method, an adaptive tracking control method is presented, which exhibits several attractive features: (1) all the closed-loop signals are bounded; (2) for any initial condition, the tracking error can be confined within a prescribed zone in a prescribed time; and (3) the asymptotic convergence of the tracking error is ensured. The effectiveness of the proposed control method is demonstrated by using two simulation examples.

Adaptive Fuzzy Control of Nonlinear Systems With Predefined Time and Accuracy

In this article, an adaptive fuzzy predefined time and accuracy tracking control scheme is proposed for strict-feedback nonlinear systems. A sufficient condition that can be used to determine whether the tracking error converges to a predefined region within a predefined time is first derived based on the definition of predefined time stability. Under the proposed stability criterion, an adaptive fuzzy controller is developed by applying the backstepping technique, and the stability time and convergence accuracy of the tracking error can be set previously, which facilitates the engineering application with requirements for convergence time and accuracy. Especially, to reduce chattering and decrease energy consumptions on the premise of ensuring the stability of the system, an inequality is introduced to construct a continuous term which can replace the sign function. Finally, a practical example is presented to illustrate the feasibility of main results.

Fuzzy Control of Nonlinear Strict-Feedback Systems With Full-State Constraints: A New Barrier Function Approach

This article is concerned with adaptive fuzzy control design of nonlinear strict-feedback systems, in which system functions are unknown and system states are subjected to some constant constraints. The main control issue is to design an adaptive fuzzy controller such that the system output follows the reference signal, meanwhile, all the state variables abide by their constrained requirements. Differing from the existing way to build barrier function, the constructed virtual control signal is employed to build the barrier function. Furthermore, a backstepping-based adaptive fuzzy control design process is presented in this article. Compared to the control schemes in the existing literature on full-state constrained systems, the current control scheme has the following advantages: 1) the built virtual control signals in this article are ensured to satisfy corresponding state constraints, rather than assuming that they do, as in the existing articles; 2) the choice of initial values is independent of the norm of the virtual control signal. So, the conservatism of initial value selection is considerably reduced. It is shown that the proposed adaptive fuzzy controller not only ensures perfect tracking performance, but also ensures that all the states abide by the preassigned state constraints during the operation. At last, simulation study further verifies the efficacy of the proposed control strategy.

Adaptive reinforcement learning optimal tracking control for strict-feedback nonlinear systems with prescribed performance

The reinforcement learning-based prescribed performance optimal tracking control problem is considered for a class of strict-feedback nonlinear systems in this paper. The unknown nonlinearities and cost function are approximated by radial-basis-function (RBF) neural network (NN). The overall controller consists of an adaptive controller and an optimal compensation term. Firstly, the adaptive controller is designed by backstepping control method. Subsequently, the optimal compensation term is derived via policy iteration by minimizing cost function. In addition, depending on the prescribed performance control, the tracking error can be limited in the prescribed area. Therefore, the whole control scheme can effectively guarantee that the tracking error converges to a bound with prescribed performance while the cost function is minimized. The stability analysis shows that all signals in the closed-loop system are bounded. Finally, the effectiveness and advantages of the designed control strategy are illustrated by the simulation examples.

Adaptive Fuzzy Control of Nonlinear Systems With Function Constraints Based on Time-Varying IBLFs

In this article, an adaptive tracking control approach is developed for a class of strict-feedback nonlinear systems with time-varying full state constraints. As a breakthrough in this system, the special function constraints (whose constraint boundary is relevant to both state variables and time) are considered, which are rarely studied by research work. And there is no doubt that this method increases the complexity of designing this scheme. Furthermore, the time-varying integral barrier Lyapunov functions combining with backstepping technique is introduced to break the limitation of traditional methods as well as achieve the full state constraints. Meanwhile, fuzzy logic systems are selected to approximate unknown nonlinear functions. It is verified that all closed-loop signals are bounded and all states are forced in the time-varying boundness. In addition, the proposed control strategy has a good performance. The effectiveness of the theoretical analysis results is proved via a simulation example.

Dynamic Event-Triggered Design With Fixed-Time Performance and Input Dead-Zone

This brief develops a novel fixed-time dynamic event-triggered control design for strict-feedback nonlinear systems with an input dead-zone. Compared with static event-triggered control, a dynamic event-triggering condition is designed to reduce the computation and communication burden. A novel compensation filter is designed to compensate for the input dead-zone and can be utilized to eliminate the influences of input dead-zone on control design. Furthermore, novel command filters (CFs) are introduced to deal with the explosion of complexity. Both the proposed novel compensation filter and CFs can achieve fixed-time performance. The presented method is validated using a jerk circuit system.

Adaptive Control of Uncertain Nonlinear Systems With Discontinuous Input and Time-Varying Input Delay

In this note, we investigate the adaptive quantized or event-triggered control designs for strict-feedback nonlinear systems with time-varying input delay. Because the control signal is discontinuous in the quantized control and event-triggered control, many existing methods to deal with the input delay are no longer applicable. Through constructing an auxiliary tracking error and an auxiliary system, the input-quantized control is implemented for nonlinear systems with unknown control gain and unknown input delay. With the well-designed Lyapunov–Krasovskii functionals and the linear growth condition of input delay, the stability analysis is achieved. The studied method is also applicable to the event-triggered control for systems possessing unknown input delay. The stability analysis shows that all the signals are semiglobally uniformly ultimately bounded (SGUUB). The use of dynamic surface control (DSC) effectively simplifies the controller structure. The quantized and event-triggered control simulations illustrate that the proposed schemes are effective.

Efficient non-conservative realization of dynamic scaling-based controllers via matrix pencils for uncertain nonlinear strict-feedback systems

We propose a new matrix pencil based approach for design of state-feedback and output-feedback controllers for uncertain nonlinear strict-feedback-like systems. While the dynamic controller structure is based on dynamic high-gain scaling, we cast the design procedure within a general matrix pencil structure unlike prior results that utilized conservative algebraic bounds of terms arising in Lyapunov inequalities. The design freedoms in the controller are extracted in terms of generalized eigenvalues associated with matrix pencils formulated to capture detailed structures (locations and state dependences) of uncertain terms in the system dynamics. The approach is developed under both state-feedback (dynamic high-gain scaling based controller) and output-feedback (dual dynamic high-gain scaling based controller and observer) cases and is shown to enable efficient computation of non-conservative bounds with reduced algebraic complexity. The efficacy of the approach is demonstrated through simulation studies. • Matrix pencil approach for design of dynamic high-gain scaling based controllers. • Design procedure in terms of generalized eigenvalues of matrix pencil structures. • Applicable to state- and output-feedback control of strict-feedback-like systems. • Approach enables reduced algebraic complexity and less conservative control designs. • Efficacy of the approach demonstrated through simulations on a numerical example.

Adaptive output consensus of nonlinear fractional-order multi-agent systems: a fractional-order backstepping approach

This paper presents the distributed control design for a class of fractional-order strict-feedback nonlinear multi-agent systems in the presence of unknown dynamics by employing backstepping strategy. Considering that the information of followers’ states are not fully measurable for feedback design, the fractional-order infinite-dimension neural-network state observer is introduced to estimate the unavailable states. The infinite-dimension neuroadaptive laws are also proposed to eliminate the undesirable effects of the unknown nonlinear functions. Besides, based on the Lyapunov fractional-order stability approach and graph theory, unlike the existing results, a distributed neural adaptive observer-based control architecture is designed to ensure that all the closed-loop network signals are ultimately bounded. Finally, a simulation example is given to demonstrate the validity of the proposed control method.

Adaptive neural network observer control for a class of nonlinear strict-feedback systems with unmeasurable states

This article addresses the backstepping control problem of the nonlinear strict-feedback system with an immeasurable state. To eliminate the effect coming from the immeasurable state variable, a new adaptive state observer method is developed for backstepping design by employing a neural network (NN) approximation strategy. As one of the highlighting contributions, the observer method can be performed with the relaxed condition because it does not require the design parameters to satisfy the Hurwitz equation, therefore it can be more easily applied and extended to serve for nonlinear system control than the existing observer methods. Finally, by integrating the observer dynamic into the backstepping design, the adaptive tracking control is achieved. Lastly, the feasibility of the method is proved by both stability theory and computer simulation.

Asymptotic adaptive output feedback event-triggered control of uncertain strict-feedback nonlinear systems with sensors triggering

In this paper, event-triggered output feedback control of a class of high-order nonlinear strict-feedback systems with parametric uncertainties is investigated, in which both of the controller and the parameter estimator are triggered based on a set of event-triggered conditions. Firstly a new one-step control design framework is proposed for the strict-feedback nonlinear systems, therefore both expressions of the controller and the parameter estimate laws are much more simple than those of the recursive design approaches such as backstepping control. Secondly observers are designed to estimate the unknown states, and a set of event-triggering mechanism is proposed for the sensors such that the states are transmitted through the communication network only at the triggering points. The estimated parameter is obtained without real-time integration due to the event-triggered estimator. It is proved that our proposed control law guarantees the closed-loop system is globally bounded and the system output converges to zero asymptotically. It is also proved that the Zeno behavior is excluded. Simulation results demonstrate the effectiveness of the proposed control scheme.

Finite-time command-filtered backstepping control for nonlinear systems with input delay and time-varying full-state constraints

This paper examines a finite-time command-filtered backstepping control proposed for a class of strict-feedback nonlinear systems with input delay and time-varying asymmetric full-state constraints. The existing control method either ignores the effect of input delay or converges asymptotically in infinite time. A command-filtered backstepping design is used to decrease computational burden. A first-order Levant differentiator is employed to replace the command filter for estimating the virtual control signals in finite time. The time-varying asymmetric barrier Lyapunov function (TVABLF) and command-filtered backstepping design are applied to alleviate the significant challenges caused by the backstepping approach and full-state constraints. A novel finite-time delay compensation mechanism is proposed to remove the impact of input delay. The closed-loop signals are proved to be practical bounded. Simulation results are provided to demonstrate the effectiveness of the proposed control scheme.

Funnel-based adaptive fuzzy finite-time control for non-affine nonlinear systems preceded by unknown actuators

In this paper, an adaptive finite-time funnel control for non-affine strict-feedback nonlinear systems preceded by unknown non-smooth input nonlinearities is proposed. The input nonlinearities include backlash-like hysteresis and dead-zone. Unknown nonlinear functions are handled using fuzzy logic systems (FLS), based on the universal approximation theorem. An improved funnel error surface is utilized to guarantee the steady-state and transient predetermined performances while the differentiability problem in the controller design is averted. Using the Lyapunov approach, all the adaptive laws are extracted. In addition, an adaptive continuous robust term is added to the control input to relax the assumption of knowing the bounds of uncertainties. All the signals in the closed-loop system are shown to be semi-globally practically finite-time bounded with predetermined performance for output tracking error. Finally, comparative numerical and practical examples are provided to authenticate the efficacy and applicability of the proposed scheme.

Optimized Backstepping Control Using Reinforcement Learning of Observer-Critic-Actor Architecture Based on Fuzzy System for a Class of Nonlinear Strict-Feedback Systems

In this article, a fuzzy logic system (FLS)-based adaptive optimized backstepping control is developed by employing reinforcement learning (RL) strategy for a class of nonlinear strict feedback systems with unmeasured states. For making the virtual and actual controls are optimized solution of the corresponding subsystem, RL of observer-critic-actor architecture based on FLS approximations is constructed in every backstepping step, where the observer aims to estimate the unmeasurable states, and the critic and actor aim to evaluate control performance and perform control behavior, respectively. In the proposed optimized control, on the one hand, the state observer method can avoid to require the design constants making its characteristic polynomial Hurwitz, which is universally demanded in the existing observer methods; on the other hand, the RL is significantly simple in algorithm, because the critic and actor training laws are derived from the negative gradient of a simple positive function, which is produced from the partial derivative of Hamilton–Jacobi–Bellman (HJB) equation, instead of the square of approximated HJB equation. Therefore this optimized scheme can be more easily applied and widely extended. Finally, from two aspects of theory and simulation, it is demonstrated that the desired objective can be fulfilled.

Event-Based Adaptive Neural Control of Nonlinear Systems With Deferred Constraint

In this article, the problem of constant yet deferred output constraint for uncertain strict-feedback nonlinear systems is studied. By “deferred output constraint,” we mean that the system output is free/released from any constraint in the initial interval and then preserves within a bounded region right after a finite time. Due to such a form of output constraint, the normally employed Barrier Lyapunov Function (BLF)-based results become invalid because the corresponding BLF is undefined in the initial period. The problem will be rather complicated yet challenging if computation and communication constraints are taken into account. By developing an error-based nonlinear function and constructing a prescribed-time scaling function, together with the approximate ability of neural networks, a varying threshold-based event-triggering adaptive neural control algorithm is presented such that not only the deferred output constraint can be ensured and the network resources can be saved but also the tracking error is able to converge to a pregiven region in a prescribed time. Simulations are provided to demonstrate the effectiveness of the proposed control.

Prescribed performance dynamic surface fuzzy control for strict‐feedback nonlinear system with actuator fault

In this paper, an adaptive fuzzy control scheme for strict-feedback nonlinear system with finite-time prescribed performance and actuator fault is studied. First, we consider a prescribed performance function that enables the tracking error to converge within a preset interval in a finite time. Subsequently, the fuzzy logic system and dynamic surface control technology are embedded in the backstepping design process, and the purpose of the dynamic surface control technique is to settle the problem of computational explosion when processing the backstepping design. The fuzzy logic system is used to approximate the unknown functions that appear in the system. Finally, a new controller is designed in combination with an updated law which ensures that all the signals in the closed-loop is bounded. Two simulation examples are taken to illustrate the feasibility of the presented control scheme.

Prescribed-time tracking control for nonlinear systems with guaranteed performance

This paper studies the problem of the prescribed-time tracking control for a class of strict-feedback nonlinear systems with guaranteed performance. We will design a controller to achieve the following: one is to achieve the prescribed-time tracking, in which the settling time is not only independent of the design parameters, nor does it depend on the initial conditions, and can be set according to per our will; the other is to achieve prescribed performance composed of both the time-varying output constraint (TVOC) and the non-overshooting response. First, sufficient conditions on the prescribed-time tracking control with guaranteed performance are proposed. Then, based on the method of time-varying Barrier Lyapunov function (BLF) and LGV-backstepping, a recursive algorithm is developed to construct a controller to achieve the prescribed-time tracking control with guaranteed performance. Finally, two examples are provided to demonstrate the effectiveness of the proposed results.

Neuro-adaptive fixed-time control with novel command filter design for nonlinear systems with input dead-zone

This paper designs novel command filters (CFs) and a singularity-free controller with fixed-time performance for strict feedback nonlinear systems with input dead-zone. Compared with traditional filter designs, such as dynamic surface control and CFs, the novel CFs designed in this paper can guarantee fixed-time convergence of filter errors and compensating signals. Moreover, the controller design excludes the singularities of control signals, which might result in severe oscillation and usually appear in the conventional fixed-time controller design. To compensate the input dead-zone, a compensation filter is constructed, where controller design and stability analysis can be simplified. In the end, a simulation example is given to verify the effectiveness of the theoretical results.

Prescribed-time adaptive neural feedback control for a class of nonlinear systems

This paper studies prescribed-time stabilization for a class of unknown strict-feedback nonlinear systems. Adaptive time-varying feedback control method and non-scaling transformation strategies are employed to reduce the computation burden caused by the scaling functions. Besides, the proposed method uses a constrained function to avoid the possibility of an oversized signal when time approaches to the prescribed settling time. Based on the backstepping method, a neural prescribed-time controller is constructed to achieve finite-time regulation. Under the action of the proposed strategy, all the closed-loop signals are bounded and the system is stable in prescribed finite time. Finally, two simulation examples illustrate the effectiveness of the results.