Regional output feedback stabilization of semilinear time‐fractional diffusion systems in a parallelepipedon with control constraints

This article is concerned with the regional output feedback stabilization problems for semilinear time‐fractional diffusion systems in a 1≤n?dimensional parallelepipedon with control inequality constraints. For this, the spectrum decomposition method is used to derive a finite‐dimensional fractional ordinary differential equation (ODE) system that captures the dominant dynamics of the considered system. With this ODE system, we propose a finite‐dimensional fractional compensator to guarantee that the constrained closed‐loop semilinear systems are Mittag‐Leffler stable in some subregions of their evolution domains. An example is finally included to illustrate our results.     

Adaptive Mittag‐Leffler Stabilization of a Class of Fractional Order Uncertain Nonlinear Systems

This paper deals with the stabilization of a class of commensurate fractional order uncertain nonlinear systems. The fractional order system concerned is of the strict‐feedback form with uncertain nonlinearity. An adaptive control scheme combined with fractional order update laws is proposed by extending classical backstepping control to fractional order backstepping scheme. The asymptotic stability of the closed‐loop system is guaranteed under the construction of fractional Lyapunov functions in the sense of generalized Mittag‐Leffler stability. The fractional order nonlinear system investigated can be stabilized asymptotically globally in presence of arbitrary uncertainty. Finally illustrative examples and numerical simulations are performed to verify the effectiveness of the proposed control scheme.     

Consensus Problem with a Reference State for Fractional‐Order Multi‐Agent Systems

In this paper, the consensus problem of fractional‐order multi‐agent systems with a reference state is studied under fixed directed communication graph. At the beginning, the convergence speeds of fractional‐order multi‐agent systems are investigated based on the Mittag‐Leffler function. Then, a common consensus control law and a consensus control law based on error predictor are proposed, and it is shown that the consensus tracking can be achieved using the above control laws when a communication graph has a directed spanning tree. Finally, the convergence speeds of fractional‐order systems are compared, and it is discovered that the convergence of systems is faster using the control law based on error predictor than using the common one.     

Anti‐windup design of active disturbance rejection control for sampled systems with input delay

A novel anti‐windup design of active disturbance rejection control (ADRC) is proposed for industrial sampled systems with input delay and saturation. By using a generalized predictor to estimate the delay‐free system output, a modified extended state observer is designed to simultaneously estimate the system state and disturbance, which could become an anti‐windup compensator when the input saturation occurs. Accordingly, a feedback controller is analytically designed for disturbance rejection. By proposing the desired closed‐loop transfer function for the set‐point tracking, a prefilter is designed to tune the tracking performance while guaranteeing no steady‐state output tracking error. A sufficient condition for the closed‐loop system stability is established with proof for practical application subject to the input delay variation. Illustrative examples from the literature are used to demonstrate the effectiveness and merit of the proposed control design.     

Output‐based disturbance rejection control for 1‐D anti‐stable Schrödinger equation with boundary input matched unknown disturbance

In this paper, we are concerned with the output feedback control design for a system (plant) described by a boundary controlled anti‐stable one‐dimensional Schrödinger equation. Our output measure signals are the displacements at both side. An untraditional infinite‐dimensional disturbance estimator is developed to estimate the disturbance. Based on the estimator, we propose a state observer that is exponentially convergent to the original system and then design a stabilizing control law consisting of two parts: The first part is to compensate the disturbance by using its approximated value and the second part is to stabilize the observer system by applying the classical backstepping approach. The resulting closed‐loop system is shown to be exponentially stable with guaranteeing that all internal systems are uniformly bounded. An effective output‐based disturbance rejection control algorithm is concluded. An application, namely, a cascade of ODE–wave systems, is investigated by the developed control algorithm. Numerical experiments are carried out to illustrate the effectiveness of the proposed control law. Copyright © 2017 John Wiley & Sons, Ltd.     

Dual closed‐loop tracking control for wheeled mobile robots via active disturbance rejection control and model predictive control

A dual closed‐loop tracking control is proposed for a wheeled mobile robot based on active disturbance rejection control (ADRC) and model predictive control (MPC). In the inner loop system, the ADRC scheme with an extended state observer (ESO) is proposed to estimate and compensate external disturbances. In the outer loop system, the MPC strategy is developed to generate a desired velocity for the inner loop dynamic system subject to a diamond‐shaped input constraint. Both effectiveness and stability analysis are given for the ESO and the dual closed‐loop system, respectively. Simulation results demonstrate the performances of the proposed control scheme.     

Sliding mode control and active disturbance rejection control to the stabilization of one‐dimensional Schrödinger equation subject to boundary control matched disturbance

In this paper, we are concerned with the boundary stabilization of a one‐dimensional anti‐stable Schrödinger equation subject to boundary control matched disturbance. We apply both the sliding mode control (SMC) and the active disturbance rejection control (ADRC) to deal with the disturbance. By the SMC approach, the disturbance is supposed to be bounded only. The existence and uniqueness of the solution for the closed‐loop system is proved and the ‘reaching condition’ is obtained. Considering the SMC usually requires the large control gain and may exhibit chattering behavior, we develop the ADRC to attenuate the disturbance for which the derivative is also supposed to be bounded. Compared with the SMC, the advantage of the ADRC is not only using the continuous control but also giving an online estimation of the disturbance. It is shown that the resulting closed‐loop system can reach any arbitrary given vicinity of zero as time goes to infinity and high gain tuning parameter goes to zero. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite‐time stability of fractional order impulsive switched systems

This paper considers the finite‐time stability of fractional order impulsive switched systems. First, by using the fractional order Lyapunov function, Mittag–Leffler function, and Gronwall–Bellman lemma, two sufficient conditions are given to verify the finite‐time stability of fractional order nonlinear systems. Then, the concept of finite‐time stability is extended to fractional order impulsive switched systems. A sufficient condition is given to verify the finite‐time stability of fractional order impulsive switched systems by combining the method of average dwell time with fractional order Lyapunov function. Finally, two numerical examples are provided to illustrate the theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.     

Lebesgue‐p NORM Convergence OF Fractional‐Order PID‐Type Iterative Learning Control for Linear Systems

This paper discusses first‐ and second‐order fractional‐order PID‐type iterative learning control strategies for a class of Caputo‐type fractional‐order linear time‐invariant system. First, the additivity of the fractional‐order derivative operators is exploited by the property of Laplace transform of the convolution integral, whilst the absolute convergence of the Mittag‐Leffler function on the infinite time interval is induced and some properties of the state transmit function of the fractional‐order system are achieved via the Gamma and Bata function characteristics. Second, by using the above properties and the generalized Young inequality of the convolution integral, the monotone convergence of the developed first‐order learning strategy is analyzed and the monotone convergence of the second‐order learning scheme is derived after finite iterations, when the tracking errors are assessed in the form of the Lebesgue‐p norm. The resultant convergences exhibit that not only the fractional‐order system input and output matrices and the fractional‐order derivative learning gain, but also the system state matrix and the proportional learning gain, and fractional‐order integral learning gain dominate the convergence. Numerical simulations illustrate the validity and the effectiveness of the results.     

A backstepping‐based low‐and‐high gain design for marine vehicles

A backstepping control design for marine vehicles was described in (Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs, and Underwater Vehicles. Marine Cybernetics AS: Trondheim, Norway, 2002). Under a backstepping feedback law, global asymptotic stability of the closed‐loop system can be shown under the assumption of unlimited actuation. This paper addresses the issues that arise in the implementation of a backstepping feedback law by saturating actuators. First, for a given backstepping feedback law, an estimate of the domain of attraction is given for the resulting closed‐loop system under actuator saturation. A high gain component is then constructed and augmented to the original backstepping feedback law. This additional high gain component is shown not to shrink the estimate of the domain of attraction but to possess the ability to improve the closed‐loop response and to reject disturbance. Copyright © 2008 John Wiley & Sons, Ltd.     

Fractional‐order–dependent global stability analysis and observer‐based synthesis for a class of nonlinear fractional‐order systems

This paper focuses on proposing novel conditions for stability analysis and stabilization of the class of nonlinear fractional‐order systems. First, by considering the class of nonlinear fractional‐order systems as a feedback interconnection system and applying small‐gain theorem, a condition is proposed for L₂‐norm boundedness of the solutions of these systems. Then, by using the Mittag‐Leffler function properties, we show that satisfaction of the proposed condition proves the global asymptotic stability of the class of nonlinear fractional‐order systems with fractional order lying in (0.5, 1) or (1.5, 2). Unlike the Lyapunov‐based methods for stability analysis of fractional‐order systems, the new condition depends on the fractional order of the system. Moreover, it is related to the H_∞‐norm of the linear part of the system and it can be transformed to linear matrix inequalities (LMIs) using fractional‐order bounded‐real lemma. Furthermore, the proposed stability analysis method is extended to the state‐feedback and observer‐based controller design for the class of nonlinear fractional‐order systems based on solving some LMIs. In the observer‐based stabilization problem, we prove that the separation principle holds using our method and one can find the observer gain and pseudostate‐feedback gain in two separate steps. Finally, three numerical examples are provided to demonstrate the advantage of the novel proposed conditions with the previous results.     

Chattering‐free adaptive sliding mode control for continuous‐time systems with time‐varying delay and process disturbance

This paper concerns the controller design for continuous‐time linear systems with time‐varying delay and process disturbance. A novel adaptive sliding mode control law is mainly proposed to attract the sliding mode to first‐order sliding surface within a finite time; afterwards, the uniformly ultimately bounded stability of the closed‐loop system on the sliding surface is simultaneously guaranteed. In addition, the chattering phenomena can be conveniently excluded if the disturbance is a low‐intensity process. Once the high‐intensity disturbance is involved, the state variation can be significantly reduced as well. Furthermore, by the technique of a novel exponential free‐matrix technique, the convergence rate of the closed‐loop system can be conveniently preregulated. Numerical example is provided to demonstrate the effectiveness of the proposed method.     

Composite Finite‐time Convergent Guidance Law for Maneuvering Targets with Second‐order Autopilot Lag

This paper aims to develop a new finite‐time convergent guidance law for intercepting maneuvering targets accounting for second‐order autopilot lag. The guidance law is applied to guarantee that the line of sight (LOS) angular rate converges to zero in finite time and results in a direct interception. The effect of autopilot dynamics can be compensated based on the finite‐time backstepping control method. The time derivative of the virtual input is avoided, taking advantage of integral‐type Lyapunov functions. A finite‐time disturbance observer (FTDOB) is used to estimate the lumped uncertainties and high‐order derivatives to improve the robustness and accuracy of the guidance system. Finite‐time stability for the closed‐loop guidance system is analyzed using the Lyapunov function. Simulation results and comparisons are presented to illustrate the effectiveness of the guidance strategy.     

Finite‐time stabilization for a class of nonlinear systems with time‐varying delay

This paper investigates the finite‐time stabilization problem for a class of nonlinear systems with time‐varying delay. Under suitable assumptions, a new state feedback control law is designed by using the adding a power integrator technique. By constructing an appropriate Lyapunov‐Krasovskii functional, it is shown that the corresponding closed‐loop system is finite‐time stable. Two simulation examples are given to verify the effectiveness of the proposed scheme.     

Stability analysis of interconnected nonlinear fractional‐order systems via a single‐state variable control

This paper is devoted to investigating the stability of interconnected nonlinear fractional‐order systems via a single‐state variable control. First of all, based on stability theory, the Grönwall‐Bellman lemma and the Mittag‐Leffler function, the relevant stability results are derived. The obtained results are general and can further extend the application range. Meanwhile, an improved single‐state variable control method is introduced. The control scheme only needs to control some state variable of the system or some subsystem(s) to realize and any additional restrictions are not added. Finally, the effectiveness of the obtained results is demonstrated by several typical examples. Besides, by comparison, simulation results show that the proposed control method can indeed decrease the design and control cost and improve flexibility of control.     

Discrete‐time repetitive controller for time‐delay systems with disturbance observer

A novel discrete‐time repetitive controller design for time‐delay systems subject to a periodic reference and exogenous periodic disturbances is presented. The main idea behind the proposed approach is to take advantage of the plant delay in the controller design, and not to compensate for the effect of this delay. To facilitate this concept, we introduce an appropriate time‐delay and a compensator in a positive feedback connection with the plant, such that a generator for periodic signals is constructed. Then a proportional controller is used to stabilize the closed‐loop system. The tracking control capability is thus guaranteed according to the internal model principle (IMP). In addition, to attenuate external periodic disturbances, a disturbance observer (DO) is developed to simultaneously achieve reference tracking and disturbance rejection. The possible fractional delay due to the digital discretization is handled by using a fractional delay filter approximation. The proposed controller has a simple structure, in which only a proportional parameter and a low‐pass filter are required to be chosen. The closed‐loop stability conditions and a robustness analysis under model uncertainties are studied. Numerical simulations and practical experiments on a servo motor system are conducted to verify the feasibility and simplicity of the proposed controller. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

On Finite‐Time Stabilization of Active Disturbance Rejection Control for Uncertain Nonlinear Systems

This paper designs the active disturbance rejection control (ADRC) to achieve finite‐time stabilization for a class of uncertain nonlinear systems. The proposed control incorporates both an extended state observer (ESO) as well as an adaptive sliding mode controller. The ESO is utilized to estimate the full system states and the total uncertainties, and the adaptive strategy is incorporated to deal with the estimation errors. It is proved that, with the application of the proposed control law, semi‐global finite‐time stabilization can be achieved. Effectiveness of the proposed method is illustrated with a numerical example.     

Stabilization of Fractional‐Order Linear Systems with State and Input Delay

This paper describes a variable structure control for fractional‐order systems with delay in both the input and state variables. The proposed method includes a fractional‐order state predictor to eliminate the input delay. The resulting state‐delay system is controlled through a sliding mode approach where the controller uses a sliding surface defined by fractional order integral. Then, the proposed control law ensures that the state trajectories reach the sliding surface in finite time. Based on recent results of Lyapunov stability theory for fractional‐order systems, the stability of the closed loop is studied. Finally, an illustrative example is given to show the interest of the proposed approach.     

Disturbance‐prediction–based control of input time delay systems for rejection of unknown frequency disturbances

In this paper, a novel disturbance rejection approach is presented for a class of input time‐delay systems subject to sinusoidal disturbances with unknown frequency. In particular, an auxiliary observer is proposed to represent the periodic disturbance in a parametric uncertainty form, where the unknown factor related to disturbance frequency can be estimated. Furthermore, the correlation between the future disturbance and the auxiliary observer output is analyzed, such that the future disturbances can be predicted and rejected through the input channel. Based on the aforementioned observer and predictor structure, the overall control architecture can be established as a framework of disturbance‐prediction–based control for systems with input time delays, where the conditions on the asymptotic stability of the closed‐loop systems are also derived. Finally, numerical examples are provided to illustrate the effectiveness of the proposed control approach.     

Resilient H∞ Control Design for Discrete‐Time Uncertain Linear Systems: An Auxiliary System Transformation Approach

This paper studies the resilient (non‐fragile) H∞ output‐feedback control design for discrete‐time uncertain linear systems with controller uncertainty. The design considers parametric norm‐bounded uncertainty in all state‐space matrices of the system, output and controller equations. The paper shows that the resilient H∞ output‐feedback control problem is equivalent to a scaled H∞ output‐feedback control problem of an auxiliary system without any system or controller uncertainty. Using the existing optimal H∞ design to solve the auxiliary system, the design guarantees that the resultant closed‐loop systems are quadratically stable with disturbance attenuation γ for all admissible system and controller uncertainties. A numerical example is given to illustrate the design method and its benefits.