A New Finite Time Convergence Condition for Super‐Twisting Observer Based on Lyapunov Analysis

A new convergence condition is proposed for the super‐twisting sliding mode observer in this paper, where Lyapunov stability analysis is used as the main method to get the new convergence condition. The super‐twisting sliding mode observer is designed to obtain unknown system states of the second order nonlinear system with bounded uncertainties and disturbances. By involving a quadratic Lyapunov function, the Lyapunov approach is applied to the stability analysis of the super‐twisting observer, from which a new convergence condition is obtained to guarantee the finite time convergence of the observer. Simulation results of a pendulum and a rigid manipulator are included to demonstrate the effectiveness of the new convergence condition.     

Discrete‐Time Super‐Twisting Guidance Law with Actuator Faults Consideration

This paper proposes a robust fault‐tolerant guidance law against unknown maneuvering targets based on discrete‐time sliding mode control theory. To address this problem, a time‐delay observer is designed to estimate the lumped disturbance, which includes target maneuver as well as actuator faults. A robust discrete‐time guidance law is then synthesized based on the discrete‐time super‐twisting algorithm. Due to the principle of the super‐twisting algorithm, the presented guidance law is a naturally chattering‐free formulation. Detailed stability analysis shows that the line‐of‐sight angular rate under the proposed guidance law can be stabilized in a small region around zero. Simulation results are also provided to verify the effectiveness of the proposed approach.     

Global finite‐time stabilization of planar nonlinear systems with disturbance

This paper proposes a continuous global finite‐time controller for a class of planar systems with disturbance. The proposed controller consists of nominal and compensated parts. The nominal part, designed by an homogeneity‐based technique, takes care only of the nominal system. The closed‐loop nominal system is asymptotically stable and satisfies negative homogeneity of degree. However, using a finite‐time convergent second order sliding mode super‐twisting algorithm, the compensated part enables cancelling of the disturbance which is time‐varying or unbounded as long as its derivative is bounded. The combination of the nominal and compensated parts makes the planar system globally finite‐time stable. Simulation results show the effectiveness of the proposed method. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Adaptive disturbance observer‐based finite‐time continuous fault‐tolerant control for reentry RLV

The control effectors of reusable launch vehicle (RLV) can produce significant perturbations and faults in reentry phase. Such a challenge imposes tight requirements to enhance the robustness of vehicle autopilot. Focusing on this problem, a novel finite‐time fault‐tolerant control strategy is proposed for reentry RLV in this paper. The key of this strategy is to design an adaptive‐gain multivariable finite‐time disturbance observer (FDO) to estimate the synthetical perturbation with unknown bounds, which is composed of model uncertainty, external disturbance, and actuator fault considered as the partial loss of actuator effectiveness in this work. Then, combined with the finite‐time high‐order observer and differentiator, a continuous homogeneous second‐order sliding mode controller based on the terminal sliding mode and super‐twisting algorithm is designed to achieve a fast and accurate RLV attitude tracking with chattering attenuation. The main features of the integrated control strategy are that the adaptation algorithm of FDO can achieve non‐overestimating values of the observer gains and the second‐order super‐twisting sliding mode approach can obtain a more elegant solution in finite time. Finally, simulation results of classical RLV (X‐33) are provided to verify the effectiveness and robustness of the proposed fault‐tolerant controller in tracking the guidance commands. Copyright © 2017 John Wiley & Sons, Ltd.     

Heterogeneous consensus of higher‐order multi‐agent systems with mismatched uncertainties using sliding mode control

A robust consensus controller is proposed for heterogeneous higher‐order nonlinear multi‐agent systems, when the agent dynamics are involved with mismatched uncertainties. A distributed consensus protocol based on a time‐varying nonhomogeneous finite‐time disturbance observer and sliding mode control is designed to realize the network consensus of higher‐order multi‐agent systems. The time‐varying finite‐time disturbance observer overcomes the problem of peaking value near the initial time caused by the constant gain one and is designed to estimate the uncertainties and to mitigate the effect of mismatched uncertainties during the sliding mode. To eliminate the chattering phenomenon and ensure finite‐time convergence to the sliding surface, the control law is designed by using the super twisting algorithm. Finally numerical simulations are given to illustrate the validity of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

An improved continuous sliding mode controller for flexible air‐breathing hypersonic vehicle

An improved continuous sliding mode control algorithm is proposed for a flexible air‐breathing hypersonic vehicle (FAHV), including nonsingular fast fixed‐time sliding surface (NFFS) and dual‐layer adaptive continuous twisting reaching law (DACTL). Firstly, the nonlinear control‐oriented model of FAHV is processed using input/output feedback linearization method with the significant flexible effects modeling as unknown matched disturbances. Secondly, a novel NFFS is improved from conventional fixed‐time sliding surface by adjusting power exponent to accelerate convergence rate. In the meanwhile, in order to avoid singularity aroused by fractional power term, an exponential convergent sliding surface is switched when tracking error approaches zero. Thirdly, a DACTL is proposed to realize finite‐time convergence of sliding mode variable with higher convergence precision and less chattering. Dual‐layer adaptive law is utilized to adjust the gain in DACTL based on equivalent control concept so as to enhance robustness automatically and avoid overestimation of control gain. Meanwhile, disturbances can be compensated without knowledge of Lipschitz constants. Ultimately, simulations on longitudinal control of FAHV demonstrate the control algorithm proposed is superior to conventional quasi‐continuous sliding mode controller in the aspect of convergence accuracy and chattering suppression.     

Fixed‐time sliding mode cooperative control for multiagent networks via event‐triggered strategy

In this article, the problem of event‐triggered‐based fixed‐time sliding mode cooperative control is addressed for a class of leader‐follower multiagent networks with bounded perturbation. First, a terminal integral sliding mode manifold with fast convergent speed is designed. Then, a distributed consensus tracking control strategy based on event‐triggered and sliding mode control is developed that guarantees the multiagent networks achieve consensus within a fixed time which is independent of initial states of agents in comparison with the finite‐time convergence. Furthermore, the update frequency of control law can be considerably reduced and Zeno behavior can be removed by utilizing the proposed event‐triggered control algorithm. Simulation examples are used to show the effectiveness of the new control protocol.     

Fault‐tolerant spacecraft attitude control under actuator saturation and without angular velocity

In this paper, a fault‐tolerant control scheme is proposed to control the attitude of a rigid spacecraft subject to external disturbances and multiple system uncertainties, as well as actuator faults and saturation. More challengingly, it is assumed that the angular velocity is unavailable. A super‐twisting observer with time‐varying gain is firstly designed to accurately estimate the angular velocity in finite time. The choice of the time‐varying gain is dependent on a state‐norm estimate. Then, using the information from the observer (estimate of angular velocity), a fault‐tolerant controller is proposed, where an adaptive law is introduced to address the unknown loss of effectiveness and neural networks are used to approximate the unknown nonlinear functions. It is proved that the attitude orientations converge to the desired values at a fixed time. Finally, a simulation example is utilized to verify the effectiveness of the proposed scheme.     

On the design of nonautonomous fixed‐time controllers with a predefined upper bound of the settling time

Recently, there has been a great deal of attention in a class of finite‐time stable dynamical systems, called fixed‐time stable, that exhibit uniform convergence with respect to its initial condition, that is, there exists an upper bound for the settling‐time (UBST) function, independent of the initial condition of the system. Of particular interest is the development of stabilizing controllers where the desired UBST can be selected a priori by the user since it allows the design of controllers to satisfy real‐time constraints. Unfortunately, existing methodologies for the design of controllers for fixed‐time stability exhibit the following drawbacks: on the one hand, in methods based on autonomous systems, either the UBST is unknown or its estimate is very conservative, leading to over‐engineered solutions; on the other hand, in methods based on time‐varying gains, the gain tends to infinity, which makes these methods unrealizable in practice. To bridge these gaps, we introduce a design methodology to stabilize a perturbed chain of integrators in a fixed‐time, with the desired UBST that can be set arbitrarily tight. Our approach consists of redesigning autonomous stabilizing controllers by adding time‐varying gains. However, unlike existing methods, we provide sufficient conditions such that the time‐varying gain remains bounded, making our approach realizable in practice.     

Second order sliding mode block control of single‐phase induction motors

The authors propose a robust nonlinear controller based on a block control linearization technique combined with a second order sliding mode super‐twisting algorithm for controlling the rotor speed of single‐phase induction motors. The block control approach is used to design a sliding manifold in terms of the stator current and its desired value. The super‐twisting sliding mode algorithm is applied then to render the designed manifold be attractive. A nonlinear observer is designed to estimate the unmeasured variables (rotor flux linkages and torque load). Copyright © 2012 John Wiley & Sons, Ltd.     

Iterative Learning Control for Linear Discrete‐Time Systems with Unknown High‐Order Internal Models: A Time‐Frequency Analysis Approach

This work focuses on the iterative learning control (ILC) for linear discrete‐time systems with unknown initial state and disturbances. First, multiple high‐order internal models (HOIMs) are introduced for the reference, initial state, and disturbances. Both the initial state and disturbance consist of two components, one strictly satisfies HOIM and the other is random bounded. Then, an ILC scheme is constructed according to an augmented HOIM that is the aggregation of all HOIMs. For all known HOIMs, an ILC design criterion is introduced to achieve satisfactory tracking performance based on the 2‐D theory. Next, the case with unknown HOIMs is discussed, where a time‐frequency‐analysis (TFA)‐based ILC algorithm is proposed. In this situation, it is shown that the tracking error inherits the unknown augmented HOIM that is an aggregation of all unknown HOIMs. Then, a TFA‐based method, e.g., the short‐time Fourier transformation (STFT), is employed to identify the unknown augmented HOIM, where the STFT could ignore the effect of the random bounded initial state and disturbances. A new ILC law is designed for the identified unknown augmented HOIM, which has the ability to reject the unknown the initial state and disturbances that strictly satisfy HOIMs. Finally, a gantry robot system with iteration‐invariant or slowly‐varying frequencies is given to illustrate the efficiency of the proposed TFA‐based ILC algorithm.     

Adaptive multivariable finite‐time continuous fault‐tolerant control of rigid spacecraft

The problem of fault‐tolerant attitude tracking control for rigid spacecraft in the presence of inertia uncertainties, actuator faults, and external disturbances is investigated in this paper. A novel adaptive finite‐time continuous fault‐tolerant control strategy is developed by combining the fast nonsingular terminal sliding mode surface and the adaptive multivariable super‐twisting algorithm, which improves the robustness while preserving high accuracy and finite‐time convergence. The main features of the control strategy are the double‐layer adaptive algorithm based on equivalent control, which ensures nonoverestimation of the control gain and the continuous controller, which maintains better property of chattering reduction. Finally, the efficiency of the proposed controller is illustrated by numerical simulations.     

Enhancing the settling time estimation of a class of fixed‐time stable systems

In this paper, we provide a new nonconservative upper bound for the settling time of a class of fixed‐time stable systems. To expose the value and the applicability of this result, we present four main contributions. First, we revisit the well‐known class of fixed‐time stable systems, to show the conservatism of the classical upper estimate of its settling time. Second, we provide the smallest constant that the uniformly upper bounds the settling time of any trajectory of the system under consideration. Third, introducing a slight modification of the previous class of fixed‐time systems, we propose a new predefined‐time convergent algorithm where the least upper bound of the settling time is set a priori as a parameter of the system. At last, we design a class of predefined‐time controllers for first‐ and second‐order systems based on the exposed stability analysis. Simulation results highlight the performance of the proposed scheme regarding settling time estimation compared to existing methods.     

Finite‐Time Fault‐Tolerant Control for a Class of Non‐Affine Nonlinear System Using Sliding Mode Disturbance Observer

This study investigates a finite‐time fault‐tolerant control scheme for a class of non‐affine nonlinear system with actuator faults and unknown disturbances. A global approximation method is applied to non‐affine nonlinear system to convert it into an affine‐like expression with accuracy. An adaptive terminal sliding mode disturbance observer is proposed to estimate unknown compound disturbances in finite time, including external disturbances and system uncertainties, which enhances system robustness. Controllers based on finite‐time Lyapunov theory are designed to force tracking errors to zero in finite time. Simulation results demonstrate the effectiveness of proposed method.     

Global output feedback regulation of nonlinear time delay systems subject to sensor measurement faults

This article studied the global output feedback regulation problem for a class of uncertain nonlinear time delay systems subject to unknown measurement faults on sensors. Different from the existing works, we consider the unknown time‐varying delays on the system states and relax their conservative condition on nonlinear functions. By introducing two novel time‐varying gains, a new global output feedback regulation algorithm is proposed, which ensures control parameters can be chosen flexibly. The proposed linear‐like controller is independent of the unknown time‐varying delays. Moreover, it has a simple structure, which is convenient for the implementation in practice. Based on the Lyapunov stability theory, it is strictly proved that all signals of the resulting closed‐loop system are globally bounded with the designed controller. Finally, a simulation example is presented to illustrate the effectiveness of the proposed output feedback regulation algorithm.     

Reachable set estimation for discrete‐time switched system with application to time‐delay system

This paper addresses the problem of reachable set estimation and synthesis for a class of discrete‐time switched linear systems with time delay and bounded peak disturbance. Combined with the feature of mode‐dependent average dwell time switching, a new algorithm is developed to estimate the reachable set of switched system, which is both quasi‐time‐dependent and mode‐dependent. Then, the proposed method is applied to time‐delay system and a sufficient condition is presented to guarantee the asymptotic stability and estimate the bounding ellipsoid. Furthermore, the quasi‐time‐dependent controller is designed to stabilize the system and restrict the closed‐loop system states to an ellipsoidal bound. Examples are presented to illustrate the effectiveness and advantages of the obtained theorems.     

Fixed‐time synchronization for complex‐valued BAM neural networks with time delays

In this paper, the fixed‐time synchronization for complex‐valued bidirectional associative memory (BAM) neural networks with time delays is studied. Based on the fixed‐time stability, the Lyapunov functional method and some inequality techniques, a new criterion is presented to guarantee that the addressed systems achieve synchronization in fixed time and a more accurate estimation independent of the initial conditions is given for the settling time. Meanwhile, a new nonlinear delayed controller different from the existing ones is designed. In the end, two numerical examples are provided to illustrate the effectiveness of the obtained result.     

Continuous Fixed‐Time Controller Design for Dynamic Systems with Unmeasurable States Subject to Unbounded Disturbances

This paper presents a nonlinear continuous observer‐based controller that provides fixed‐time convergence to the origin (a vicinity of the origin) of a higher‐order dynamic system subject to unbounded disturbances, whose only measurable state is the highest relative degree one. The settling time estimation problem is solved computing an uniform upper bound of the fixed convergence time of the designed controller. Finally, the designed algorithm is verified in a case study of an industrial armature‐controlled DC motor.     

Fixed‐Time Synchronization of a Class of Second‐Order Nonlinear Leader‐Following Multi‐Agent Systems

The fixed‐time synchronization problem for a class of second‐order nonlinear multi‐agent systems with a leader‐follower architecture is investigated in this paper. To achieve the fixed‐time tracking task, the design procedure is divided into two steps. At the first step, a distributed fixed‐time observer is designed for each agent to estimate the leader's state in a fixed time. Then, at the second step, based on the technique of adding a power integrator, a fixed‐time tracking controller for each agent is proposed such that the estimate leader's state can be tracked in a fixed time. Finally, an observer‐based fixed‐time controller is developed such that the leader can be tracked by all the followers in a fixed time, which can be predetermined. Simulations are presented to verify the effectiveness of the proposed approach.     

Distributed fixed‐time consensus for multiagent systems with unknown control directions

This paper investigates the consensus problem for high‐order multiagent systems with unknown control directions and directed communication constraints. To handle the problem of unknown control directions, a logic switching rule is established in the framework of fixed‐time stability. Then, the consensus is achieved in two steps. A group of distributed fixed‐time observers is designed to estimate the reference signals first. Based on these estimates and the designed logic switching rule, a novel control protocol is proposed for each follower system. Different from the existing results, the consensus is achieved with a fixed‐time convergence rate, and the unknown control directions are allowed to be nonidentical for each agent. Finally, simulation results are given to exhibit the validity of the proposed method.