On Observer Design for Nonlinear Caputo Fractional‐Order Systems

The observer design problem for integer‐order systems has been the subject of several studies. However, much less interest has been given to the more general fractional‐order systems, where the fractional‐order derivative is between 0 and 1. In this paper, a particular form of observers for integer‐order Lipschitz, one‐sided Lipschitz and quasi‐one‐sided Lipschitz systems, is extended to the fractional‐order calculus. Then, the obtained states estimates are used for an eventual feedback control, and the separation principle is tackled. The effectiveness of the proposed scheme is shown through simulation for two numerical examples.     

Fractional‐Order Dynamic Output Feedback Sliding Mode Control Design for Robust Stabilization of Uncertain Fractional‐Order Nonlinear Systems

A robust fractional‐order dynamic output feedback sliding mode control (DOF‐SMC) technique is introduced in this paper for uncertain fractional‐order nonlinear systems. The control law consists of two parts: a linear part and a nonlinear part. The former is generated by the fractional‐order dynamics of the controller and the latter is related to the switching control component. The proposed DOF‐SMC ensures the asymptotical stability of the fractional‐order closed‐loop system whilst it is guaranteed that the system states hit the switching manifold in finite time. Finally, numerical simulation results are presented to illustrate the effectiveness of the proposed method.     

Finite‐Time Stability of Linear Caputo‐Katugampola Fractional‐Order Time Delay Systems

In this paper, a finite‐time stability procedure is suggested for a class of Caputo‐Katugampola fractional‐order time delay systems. Sufficient conditions are derived to prove this fact. Numerical results are provided to demonstrate the validity of our theoretical results.     

Semi‐Global Output Feedback Stabilization for a Class of Uncertain Nonlinear Systems

This paper addresses the problem of semi‐global stabilization by output feedback for a class of nonlinear systems whose output gains are unknown. For each subsystem, we first design a state compensator and use the compensator states to construct a control law to stabilize the nominal linear system without the perturbing nonlinearities. Then, combining the output feedback domination approach with block‐backstepping scheme, a series of homogeneous output feedback controllers are constructed recursively for each subsystem and the closed‐loop system is rendered semi‐globally asymptotically stable.     

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.     

Full‐Order and Reduced‐order Observer Design for a Class of Fractional‐order Nonlinear Systems

The paper is concerned with problem of the full‐order and reduced‐order observer design for a class of fractional‐order one‐sided Lipschitz nonlinear systems. By introducing a continuous frequency distributed equivalent model and using indirect Lyapunov approach, the sufficient condition for asymptotic stability of the full‐order observer error dynamic system is presented. Furthermore, the proposed design method was extended to reduced‐order observer design for fractional‐order nonlinear systems. All the stability conditions are obtained in terms of LMI, which are less conservative than some existing ones. Finally, a numerical example demonstrates the validity of this approach.     

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.     

非线性系统基于增益观测器的全局输出反馈镇定

讨论几类非线性控制系统可输出反馈指数镇定的充分条件;特别地,在可输出反馈指数镇定的条件下讨论控制器参数的有效范围;数值模拟证实所设计的控制器厦其调节参数的有效性。     

Output Feedback Terminal Sliding Mode Control for a Class of Second Order Nonlinear Systems

In this study, a novel output feedback terminal sliding mode control (TSMC) approach is proposed for a class of second order nonlinear systems in light of the equivalent output injection sliding mode observer (SMO) method and TSMC principle. The SMO method is applied to reconstruct full states in finite time and the non‐singular TSMC algorithm is designed to stabilize system states to equilibrium points in finite time. The corresponding stability analysis is presented. An indispensable illustrative example is bench tested to validate the effectiveness of the proposed approach.     

Output Feedback Stabilization for a Class of Stochastic High‐Order Feedforward Nonlinear Systems with Time‐Varying Delay

This paper considers the output feedback control problem for a class of stochastic high‐order feedforward nonlinear systems with time‐varying delay. Compared with existing works, the features of our system include different bounded time‐varying delays, more general high‐order power and homogeneous feedforward growth conditions. Firstly, we use the adding one power integrator technique to construct an output feedback controller without nonlinearities. Then, by introducing a scaling gain into the controller and choosing an appropriate Lyapunov–Krasovskii functional, the closed‐loop system can be rendered globally asymptotically stable in probability. A simulation example is provided to illustrate the effectiveness of the designed controller.     

Global Robust Stabilization via Sampled‐Data Output Feedback for Nonlinear Systems with Uncertain Measurement and Control Gains

This paper studies the problem of using a sampled‐data output feedback controller to globally stabilize a class of nonlinear systems with uncertain measurement and control gains. A reduced‐order observer and a linear output control law, both in the sampled‐data form, are designed without the precise knowledge of the measurement and control gains except for their bounds. The observer gains are chosen recursively in a delicate manner by utilizing the output feedback domination approach. The allowable sampling period is determined by estimating and restraining the growth of the system states under a zero‐order‐hold input with the help of the Gronwall–Bellman Inequality. A DC–DC buck power converter as a real‐life example will be shown by numerical simulations to demonstrate the effectiveness of the proposed control method.     

一类二阶非线性系统的有限时间状态反馈镇定方法

针对一类二阶非线性系统的有限时间状态反馈镇定问题进行了讨论. 给出了三种基于连续状态反馈的全局有限时间状态反馈镇定方法. 首先,利用非线性齐次系统性质,设计出一种状态反馈控制器,使得闭环系统渐近稳定并且具有负的齐次度;其次,基于有限时间Lyapunov函数的反步构造法,给出了一种有限时间控制器;最后,利用非奇异终端滑模控制技术,得到了一种使闭环系统有限时间收敛到平衡点的反馈镇定控制器. 仿真结果表明了这些方法的有效性.     

Global Output‐Feedback Stabilization for Stochastic Nonlinear Systems with Function Control Coefficients

This paper investigates the global output‐feedback stabilization for a class of stochastic nonlinear systems with function control coefficients. Notably, the systems in question possess control coefficients that are functions of output, rather than constants; hence, they are different from the existing literature on stochastic stabilization. To solve the control problem, an appropriate reduced‐order observer is introduced to reconstruct the unmeasured system states before a smooth output‐feedback controller is designed using the backstepping method, which guarantees that the closed‐loop system is globally asymptotically stable in probability. This paper combines the related results in the deterministic and stochastic setting and gives the first treatment on the global output‐feedback stabilization for the stochastic nonlinear systems with function control coefficients. A simulation example is given also to illustrate the effectiveness of the proposed approach.     

一类非线性MIMO不确定系统的动态输出反馈镇定

考察其标称系统的相对阶大于{1,1,…,1}同时含匹配和非匹配不确定性的MIMO非线性系统的动态输出反馈镇定问题.文中直接用Lyaunov方法构造一类输出反馈动态补偿器,该补偿器可以实现对所论非线性不确定系统的动态输出反馈渐近镇定.     

New Results on Stabilization of Fractional‐Order Nonlinear Systems via an LMI Approach

This paper considers the systematic design of robust stabilizing state feedback controllers for fractional‐order nonlinear systems. By using the Lyapunov direct method and a recent result on the Caputo fractional derivative of a quadratic function, stabilizability conditions expressed in terms of linear matrix inequalities are derived. The controllers can then be derived by using existing computationally effective convex algorithms. Two numerical examples with simulation results are provided to demonstrate the effectiveness of our results.     

非线性系统的模糊自适应输出反馈控制

针对一类未知非线性系统,考虑系统状态不完全可测的情况,利用Lyapunov综合方法设计了一种基于高增益观测器的模糊鲁棒自适应输出反馈控制器,并证明在一定条件下,所设计的输出反馈控制器能获得状态反馈控制器的性能。     

一类非仿射系统全局镇定控制器的设计

讨论一类多输入多输出非仿射系统的全局可镇定性,其中该系统的自治系统Lyapunov稳定.利用LaSalle不变原理,得到系统全局可镇定的充分条件;基于分离原理和降阶观测器,给出了一类降阶动态输出反馈镇定控制器的设计.     

Practical Output Regulation of Uncertain Strict‐Feedback Form Systems

In this paper, an output feedback control is proposed to solve the practical output regulation problem of a class of nonlinear systems. In the first step of the design procedure, two sets of coordinate transformations are used to convert the output regulation problem of the system in the strict‐feedback form into the regulation problem of the transformed system in the uncertain normal form. Then, for the resulting system, a state feedback in the cast of nested sliding mode control is designed. Finally, by using the nonlinear separation principle, the output feedback controller is achieved by substituting the estimated states, resulting from the high‐gain observer, instead of real states. It can be shown that the states of the closed‐loop system are ultimately bounded. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Stability and Stabilization of Fractional‐Order Systems with Different Derivative Orders: An LMI Approach

Stability and stabilization analysis of fractional‐order linear time‐invariant (FO‐LTI) systems with different derivative orders is studied in this paper. First, by using an appropriate linear matrix function, a single‐order equivalent system for the given different‐order system is introduced by which a new stability condition is obtained that is easier to check in practice than the conditions known up to now. Then the stabilization problem of fractional‐order linear systems with different fractional orders via a dynamic output feedback controller with a predetermined order is investigated, utilizing the proposed stability criterion. The proposed stability and stabilization theorems are applicable to FO‐LTI systems with different fractional orders in one or both of 0 <  α  < 1 and 1 ≤  α  < 2 intervals. Finally, some numerical examples are presented to confirm the obtained analytical results.     

一类线性不可观非线性系统的动态输出反馈镇定

本文研究一类不可观非线性系统的动态输出反馈镇定,基于逼近渐近稳定性的概念,给出了动态输出反馈可镇定的充分条件,本文主要结果的直接推论是零动太逼近渐近稳定的最小相位系统能用动态输出反馈镇定,本文的方法也能处理非最小相位系统。