Adaptive neural control for multiagent systems with asymmetric time‐varying state constraints and input saturation

This article investigates the leader‐follower consensus problem of a class of non‐strict‐feedback nonlinear multiagent systems with asymmetric time‐varying state constraints (ATVSC) and input saturation, and an adaptive neural control scheme is developed. By introducing the distributed sliding‐mode estimator, each follower can obtain the estimation of leader's trajectory and track it directly. Then, with the help of time‐varying asymmetric barrier Lyapunov function and radial basis function neural networks, the controller is designed based on backstepping technique. Furthermore, the mean‐value theorem and Nussbaum function are utilized to address the problems of input saturation and unknown control direction. Moreover, the number of adaptive laws is equal to that of the followers, which reduces the computational complexity. It is proved that the leader‐follower consensus tracking control is achieved without violating the ATVSC, and all closed‐loop signals are semiglobally uniformly ultimately bounded. Finally, the simulation results are provided to verify the effectiveness of the control scheme.     

Flatness-based linear output feedback control for disturbance rejection and tracking tasks on a Chua's circuit

A linear output feedback controller is developed for trajectory tracking problems defined on a modified version of Chua's circuit. The circuit modification considers the introduction of a flat input, i.e. a suitable external control input channel guided by (a) the induction of the flatness property on a measurable output signal of the circuit and (b) the physical viability of the control input. A linear active disturbance rejection control based on a high-gain linear disturbance observer, is implemented on a laboratory prototype. We show that the state-dependent disturbance can be approximately, but arbitrarily closely, estimated through a linear high-gain observer, called a generalised proportional integral (GPI) observer, which contains a linear combination of a sufficient number of extra iterated integrals of the output estimation error. Experimental results are presented in the output reference trajectory tracking of a signal generated by an unrelated chaotic system of the Lorenz type. Laboratory experiments illustrate the proposed linear methodology for effectively controlling chaos.     

Output‐feedback tracking in causal nonminimum‐phase nonlinear systems using higher‐order sliding modes

Asymptotic output‐feedback tracking in a class of causal nonminimum phase uncertain nonlinear systems is addressed via sliding mode techniques. Sliding mode control is proposed for robust stabilization of the output tracking error in the presence of a bounded disturbance. The output reference profile and the unknown input/disturbance are supposed to be described by unknown linear exogenous systems of a given order. Local asymptotic stability of the output tracking error dynamics along with the boundedness of the internal states are proven. The unstable internal states are estimated asymptotically via the proposed multistage observer that is based on the method of extended system center. A higher‐order sliding mode observer/differentiator is used for the exact estimation of the input–output states in a finite time. The bounded disturbance is reconstructed asymptotically. A numerical example illustrates the efficiency of the proposed output‐feedback tracking approach developed for causal nonminimum phase nonlinear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

Practical time-varying formation tracking for high-order nonlinear multi-agent systems based on the distributed extended state observer

Practical time-varying formation tracking analysis and design problems for high-order nonlinear multi-agent systems are investigated, where the time-varying formation tracking error is controlled within an arbitrarily small bound. The states of followers form a predefined time-varying formation while tracking the state of the leader with unknown control input. Besides, the dynamics of each agent has heterogeneous nonlinearity and disturbance. First, a distributed extended state observer is constructed to estimate the follower's nonlinearity and disturbance, and the leader's unknown control input simultaneously. A protocol based on the distributed extended state observer is proposed. Second, sufficient conditions for the multi-agent systems to achieve the practical time-varying formation tracking under the protocol are presented by using the Lyapunov stability theory. Third, an approach is derived to design the proposed protocol by solving a linear matrix inequality. Finally, numerical simulation results are given to illustrate the effectiveness of the theoretical results.     

Semi‐global bipartite output consensus of heterogeneous multi‐agent systems subject to input saturation by finite‐time observer

In this paper, the semi‐global bipartite output consensus problem of heterogeneous linear multi‐agent systems is studied. Compared with related works, both cooperative interactions and antagonistic interactions between agents are considered, and the input saturation on each follower is taken into account. First, two distributed finite‐time observers are designed to recover the leader's state. Particularly, the setting time can be independent of any initial states. Due to the antagonistic interactions, estimation values are the same as the leader's state in modulus but may not in sign. Then, the low‐gain feedback technique is used to develop the distributed control law for each follower. Moreover, we summarize a framework for solving the semi‐global bipartite output consensus problem of heterogeneous multi‐agent systems subject to input saturation. Finally, examples are given to illustrate the results.     

Stabilization and Tracking Control of X‐Z Inverted Pendulum Using Flatness Based Active Disturbance Rejection Control

A flatness based robust active disturbance rejection control technique scheme with tracking differentiator is proposed for the problem of stabilization and tracking control of the X‐Z inverted pendulum known as a special underactuated, non‐feedback linearizable mechanical system. The differential parameterization on the basis of linearizing the system around an arbitrary equilibrium decouples the underactuated system into two lower order systems, resulting in two lower‐order extended state observers. Using a tracking differentiator to arrange the transient process utilizes the problem of stabilization and tracking control and gives a relatively small initial estimation error, which enlarges the range of the controller parameters. The convincing analysis of the proposed modified linear extended state observer is presented to show its high effectiveness on estimating the states and the extended states known as the total disturbances consisting of the unknown external disturbances and the nonlinearities neglected by the linearization. Simulation results on the stabilization and tracking control of the X‐Z inverted pendulum, including a comparative simulation with an all‐state‐feedback sliding mode controller are presented to show the advantages of the combination of flatness and active disturbance rejection control techniques.     

高阶不确定非线性系统的线性自抗扰控制

针对一类具有内部动态和外部扰动未知的SISO高阶非线性系统,提出一种通用的线性自抗扰控制方案.该方案基于单参数调节的高增益观测器思想,分别设计线性跟踪微分器、线性扩张状态观测器和线性状态误差反馈控制律.利用Lagrange中值定理和Cauchy-Schwarz不等式将系统总扰动的微分值转化为关于系统估计和跟踪误差的函数,可以解决因系统控制增益未知所导致的控制量微分值难以预先确定的问题.在此基础上,基于Lyapunov稳定性定理证明闭环系统误差信号有界,并进一步分析得到系统估计和跟踪误差与控制器参数的定量关系,即都可以随观测器增益的增大而达到无限小.仿真比较结果验证了所提出方案的有效性,与韩式自抗扰控制方案相比,该方案结构简单,调节参数少,易于工程实现.     

Trajectory tracking for the magnetic ball levitation system via exact feedforward linearisation and GPI control

The output feedback control of the popular magnetic ball levitation system is addressed from a suitable combination of several complementary viewpoints. We use: first, recent developments on exact feedforward linearisation controllers for nonlinear flat systems to substantially reduce the linear feedback controller efforts through pre-compensation. Second, an on-line ball velocity estimation strategy is proposed by using a model-based integral reconstructor, which is a linear combination of iterated integrals of the input and the output of the system, thus avoiding the use of traditional observers or noisy derivative estimations. Finally, we use a generalised proportional integral (GPI) controller which compensates the errors in the integral reconstructor and further bestows the enhanced robustness on the closed-loop system via output tracking error iterated integration feedback. This methodology only requires the measurements of the position of the levitated ball and of the control input voltage. The proposed feedback regulation scheme is shown to locally guarantee an asymptotically exponentially stable behaviour of the controlled ball position and, definitely, allows for the possibilities of safely carrying out the rest-to-rest trajectory tracking tasks on the ball position. The proposed output feedback controller is actually implemented on a laboratory prototype with excellent experimental results for, both, stabilisation and trajectory tracking tasks.     

领导者–跟随者Thomson跳环系统的自抗扰控制设计

将用电磁方法悬浮起来的一对环构成领导者–跟随者系统实现对光滑参考轨线的跟踪要求. 这一任务通过设计 具有对扰动进行在线估计与补偿功能的自抗扰控制器(ADRC)实现. 本文设计方法与传统ADRC的主要不同在于领导者 和跟随者的跟踪控制器设计都是基于平坦输出的线性化近似系统. 超出线性化近似有效区域的大的高度偏差所导致的 未知非线性则被视为扰动, 这个扰动借助于线性扩张观测器进行在线估计并通过线性反馈控制器进行消除. 实验结果检 验了本文所提方法的有效性.     

Active Disturbance Rejection Control of a Magnetic Suspension System

This paper proposes a novel output feedback control scheme for robust stabilization and tracking tasks in a magnetic suspension system. Active disturbance rejection control, differential flatness and on‐line asymptotic disturbance estimation are properly used for the proposed control synthesis. The controlled system is subjected to a wide spectrum of unknown significant matched and unmatched disturbances due to external forces and voltages, parametric uncertainties, control and state‐dependent perturbations and possibly input unmodeled dynamics. The effectiveness and robustness of the proposed active disturbance control scheme is verified by computer simulations for the robust tracking of a rest‐to‐rest reference position trajectory specified to firstly stabilize the suspended mass at a desired vertical position and next transfer it to another equilibrium position for both continuous and switched control voltage signals.     

Transverse function control with prescribed performance guarantees for underactuated marine surface vehicles

This paper studies the problem of stabilizing reference trajectories (also called as the trajectory tracking problem) for underactuated marine vehicles under predefined tracking error constraints. The boundary functions of the predefined constraints are asymmetric and time‐varying. The time‐varying boundary functions allow us to quantify prescribed performance of tracking errors on both transient and steady‐state stages. To overcome difficulties raised by underactuation and nonzero off‐diagonal terms in the system matrices, we develop a novel transverse function control approach to introduce an additional control input in backstepping procedure. This approach provides practical stabilization of any smooth reference trajectory, whether this trajectory is feasible or not. By practical stabilization, we mean that the tracking errors of vehicle position and orientation converge to a small neighborhood of zero. With the introduction of an error transformation function, we construct an inverse‐hyperbolic‐tangent‐like barrier Lyapunov function to show practical stability of the closed‐loop systems with prescribed transient and steady‐state performances. To deal with unmodeled dynamic uncertainties and external disturbances, we employ neural network (NN) approximators to estimate uncertain dynamics and present disturbance observers to estimate unknown disturbances. Subsequently, we develop adaptive control, based on NN approximators and disturbance estimates, that guarantees the prescribed performance of tracking errors during the transient stage of on‐line NN weight adaptations and disturbance estimates. Simulation results show the performance of the proposed tracking control.     

Robust sigma–delta generalised proportional integral observer based control of a ‘buck’ converter with uncertain loads

This article describes the design of an observer based robust linear output feedback controller for the regulation and output reference trajectory tracking tasks in switched ‘buck’ converter circuits feeding a completely unknown time-varying load. The state-dependent perturbation effects of the unknown load resistance are on-line estimated by means of a generalised proportional integral (GPI) observer, which represents the dual counterpart of GPI controllers introduced in Fliess, Márquez, Delaleau and Sira-Ramírez (Fliess, M., Márquez, R., Delaleau, E., and Sira-Ramírez, H. (2002), ‘Correcteurs Proportionnels-intégraux Géneralisés’, ESAIM: Control, Optimisation and Calculus of Variations, 7, 23–41). The reconstructed perturbation complements the controller in a cancellation effort which allows the core of the feedback controller to become a traditional proportional derivative (PD) controller. The designed average feedback controller is then implemented via a sigma–delta-modulator, which effectively translates the designed continuous average feedback control input signal into a discrete valued switched input signal driving the converter's input switch and preserving all relevant features of the average design. The Appendix collects some generalities about GPI observers.     

ANALYSIS AND DESIGN OF FEEDBACK CONTROL SYSTEMS WITH TRACKING ERRORS SQUARE

In this study, a command tracking error square control scheme is first proposed for analysis and design of feedback control systems. One of the tracking errors is low‐pass filtered and used in the feedback loop for gain adaptation; the other is used in the forward loop for command tracking control. The overall systems are nonlinear feedback systems, and can be reconfigured to an automatic gain control (AGC) loop with command tracking error input. The stability and robustness of the controlled systems are verified by time response, frequency response, and large parameter variation testing with a simple illustrating example and are finally applied to a complicated electro‐hydraulic velocity servo system with large load disturbance.     

Adaptive control based on fast online algebraic identification and GPI control for magnetic levitation systems with time-varying input gain

This paper considers the position tracking problem of a voltage-controlled magnetic levitation system (MLS) in the presence of modelling errors caused by uncertainties in the system’s physical parameters. An adaptive control based on fast online algebraic parameter estimation and generalised proportional integral (GPI) output feedback control is considered as a control scheme candidate. The GPI controller guarantees an asymptotically exponentially stable behaviour of the controlled ball position and the possibilities of carrying out rest-to-rest trajectory tracking tasks. The nature of the control input gain in an MLS is that of a state-dependent time-varying gain, reflecting the nonlinear character of the magnetic force with regard to the distance and the properties of the metallic ball. The system gain has therefore been locally approximated using a periodically updated time polynomial function (of second degree), where the coefficients of the polynomial are estimated during a very short period of time. This estimation is achieved using the recently introduced algebraic online parameter estimation approach. The stability of the closed-loop system is demonstrated under the assumption that no external factors cause changes in the parameter during the time interval in which the stability is analysed. Finally, experimental results are presented for the controlled MLS demonstrating the excellent stabilisation and position tracking performance of the control system designed in the presence of significant nonlinearities and uncertainties of the underlying system.     

Cooperative Adaptive Fuzzy Output Feedback Control for Synchronization of Nonlinear Multi‐Agent Systems in the Presence of Input Saturation

This paper considers the leader‐following synchronization problem of nonlinear multi‐agent systems with unmeasurable states in the presence of input saturation. Each follower is governed by a class of strict‐feedback systems with unknown nonlinearities and the information of the leader can be accessed by only a small fraction of followers. An auxiliary system is introduced and its states are used to design the cooperative controllers for counteracting the effect of input saturation. By using fuzzy logic systems to approximate the unknown nonlinearities, local adaptive fuzzy observers are designed to estimate the unmeasurable states. Dynamic surface control (DSC) is employed to design distributed adaptive fuzzy output feedback controllers. The developed controllers guarantee that the outputs of all followers synchronize to that of the leader under directed communication graphs. Based on Lyapunov stability theory, it is proved that all signals in the closed‐loop systems are semiglobally uniformly ultimately bounded (SGUUB), and the tracking error converges to a small neighborhood of the origin. An example is provided to show the effectiveness of the proposed control approach.     

一类严格反馈系统变比例增益精确微分补偿控制

针对包含不确定函数和未知外部扰动的一类严格反馈型非线性系统,提出基于精确扰动观测器的变比例增益自适应模糊控制器.系统中的未知不确定函数由模糊逻辑系统在线逼近,同时将模糊逻辑系统的逼近误差和未知外部扰动定义为总扰动,利用精确扰动观测器进行精确微分补偿控制. 将非线性函数应用于设计可调节的输出反馈增益,有效消除系统的稳态误差,使得系统跟踪误差可以控制在零的任意小邻域内.最后,通过Lyapunov定理证明闭环系统中所有信号均是有界的.数值仿真表明了所提出方案的有效性.     

Adaptive unit vector control of multivariable systems using monitoring functions

An adaptive sliding‐mode unit vector control approach based on monitoring functions to deal with disturbances of unknown bounds is proposed. An uncertain multivariable linear system is considered with a quite general class of nonsmooth disturbances. Global stabilization/tracking is demonstrated using either state or output feedback. The proposed adaptation method makes the control gain less conservative, becoming large enough when the disturbance grows and becoming smaller when it decreases, leading to reduced chattering effects. In contrast to previous methods, the new switching scheme is able to guarantee a prespecified transient time, maximum overshoot, and steady‐state error for multivariable uncertain plants. The proposed technique is applied to the trajectory tracking control of a surface vessel subjected to ocean currents, wind, and waves. Simulations are presented to show the performance of the new adaptation scheme in this adverse scenario of possibly growing, temporarily large, or vanishing exogenous disturbances.     

Robust Fault‐Tolerant Control of Launch Vehicle Via GPI Observer and Integral Sliding Mode Control

A robust fault‐tolerant attitude control scheme is proposed for a launch vehicle (LV) in the presence of unknown external disturbances, mismodeling dynamics, actuator faults, and actuator's constraints. The input‐output representation is employed to describe the rotational dynamics of LV rendering three independently decoupled second order single‐input‐single‐output (SISO) systems. In the differential algebraic framework, general proportional integral (GPI) observers are used for the estimations of the states and of the generalized disturbances, which include internal perturbations, external disturbances, and unknown actuator failures. In order to avoid the defects of the conventional sliding surface, a new nonlinear integral sliding manifold is introduced for the robust fault‐tolerant sliding mode controller design. The stability of the GPI observer and that of the closed‐loop system are guaranteed by Lyapunov's indirect and direct methods, respectively. The convincing numerical simulation results demonstrate the proposed control scheme is with high attitude tracking performance in the presence of various disturbances, actuator faults, and actuator constraints.