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.     

Active disturbance rejection control for nonlinear fractional‐order systems

This paper investigates active disturbance rejection control involving the fractional‐order tracking differentiator, the fractional‐order PID controller with compensation and the fractional‐order extended state observer for nonlinear fractional‐order systems. Firstly, the fractional‐order optimal‐time control scheme is studied to propose the fractional‐order tracking differentiator by the Hamilton function and fractional‐order optimal conditions. Secondly, the linear fractional‐order extend state observer is offered to acquire the estimated value of the sum of nonlinear functions and disturbances existing in the investigated nonlinear fractional‐order plant. For the disturbance existing in the feedback output, the effect of the disturbance is discussed to choose a reasonable parameter in fractional‐order extended state observer. Thirdly, by this observed value, the nonlinear fractional‐order plant is converted into a linear fractional‐order plant by adding the compensation in the controller. With the aid of real root boundary, complex root boundary, and imaginary boot boundary, the approximate stabilizing boundary with respect to the integral and differential coefficients is determined for the given proportional coefficient, integral order and differential order. By choosing the suitable parameters, the fractional‐order active disturbance rejection control scheme can deal with the unknown nonlinear functions and disturbances. Finally, the illustrative examples are given to verify the effectiveness of fractional‐order active disturbance rejection control scheme. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust output feedback stabilization of nonlinear systems with low‐order and high‐order nonlinearities

In this paper, we consider the problem of global output feedback stabilization for a class of nonlinear systems whose nonlinearities are assumed to be bounded by both low‐order and high‐order nonlinearities multiplied by a polynomial‐type output‐dependent growth rate. Instead of the previously proposed dual observer, based on the homogeneous domination approach, a new reduced‐order observer is constructed, which greatly simplifies the closed‐loop controller and is able to cover a more general class of nonlinear systems. Copyright © 2015 John Wiley & Sons, Ltd.     

Unscented Kalman filter for continuous‐time nonlinear fractional‐order systems with process and measurement noises

This study proposes the design of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system involving the process noise and the measurement noise. The nonlinear fractional‐order system is discretized to get the difference equation. According to the unscented transformation, the design method of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system is provided. Compared with the extended Kalman filter, the proposed method can obtain a more accurate estimation effect. For fractional‐order systems containing non‐differentiable nonlinear functions, the method proposed in this paper is still effective. The unknown parameters are also discussed by the augmented vector method to achieve the state estimation and parameter identification. Finally, two examples are offered to verify the effectiveness of the proposed unscented Kalman filter for nonlinear fractional‐order systems.     

On the sliding‐mode control of fractional‐order nonlinear uncertain dynamics

This paper deals with applications of sliding‐mode‐based fractional control techniques to address tracking and stabilization control tasks for some classes of nonlinear uncertain fractional‐order systems. Both single‐input and multi‐input systems are considered. A second‐order sliding‐mode approach is taken, in suitable combination with PI‐based design, in the single‐input case, while the unit‐vector approach is the main tool of reference in the multi‐input case. Sliding manifolds containing fractional derivatives of the state variables are used in the present work. Constructive tuning conditions for the control parameters are derived by Lyapunov analysis, and the convergence properties of the proposed schemes are supported by simulation results. Copyright © 2015 John Wiley & Sons, Ltd.     

An H∞non‐fragile observer‐based adaptive sliding mode controller design for uncertain fractional‐order nonlinear systems with time delay and input nonlinearity

In this paper the problem of non‐fragile adaptive sliding mode observer design is addressed for a class of nonlinear fractional‐order time‐delay systems with uncertainties, external disturbance, exogenous noise, and input nonlinearity. An H_∞ observer‐based adaptive sliding mode control considering the non‐fragility of the observer is proposed for this system. The sufficient asymptotic stability conditions are derived in the form of linear matrix inequalities. It is proven that the sliding surface is reachable in finite time. An illustrative example is provided which corroborates the effectiveness of the theoretical results.     

Finite‐time output feedback stabilization of nonlinear high‐order feedforward systems

This paper considers the global finite‐time output feedback stabilization of a class of nonlinear high‐order feedforward systems. By using the homogeneous domination method together with adding a power integrator method and overcoming several troublesome obstacles in the design and analysis, a global finite‐time output feedback controller with reduced‐order observer is recursively designed to globally finite‐time stabilize nonlinear high‐order feedforward systems. Copyright © 2015 John Wiley & Sons, Ltd.     

Improved exponential observer design for one‐sided Lipschitz nonlinear systems

This paper investigates the exponential observer design problem for one‐sided Lipschitz nonlinear systems. A unified framework for designing both full‐order and reduced‐order exponential state observers is proposed. The developed design approach requires neither scaling of the one‐sided Lipschitz constant nor the additional quadratically inner‐bounded condition. It is shown that the synthesis conditions established include some known existing results as special cases and can reduce the intrinsic conservatism. For design purposes, we also formulate the observer synthesis conditions in a tractable LMI form or a Riccati‐type inequality with equality constraints. Simulation results on a numerical example are given to illustrate the advantages and effectiveness of the proposed design scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

Back‐stepping stabilization of fractional‐order triangular system with applications to chaotic systems

This paper addresses the stabilization of fractional‐order nonlinear lower triangular systems and applies the obtained results to the synchronization of fractional‐order chaotic systems (FOCSs). Three cases are considered as follows: (i) FOCSs have a lower triangular structure themselves. (ii) FOCSs can be converted into the form in full by appropriate nonsingular coordinate transformations. (iii) FOCSs can be transformed into the structure in part by the conversions. It is shown that the above three cases combining with back‐stepping approach are useful for the control of FOCSs. Different from the previous literature, the system structure studied is more concise with more applications. A numerical example is presented to show the validity of the proposed results.     

Fault estimation observer design for discrete‐time systems in finite‐frequency domain

This paper proposes a framework of fault estimation observer design in finite‐frequency domain for discrete‐time systems. First, under the multiconstrained idea, a full‐order fault estimation observer in finite‐frequency domain is designed to achieve fault estimation by using the generalized Kalman–Yakubovich–Popov lemma to reduce conservatism generated by the entire frequency domain. Then, a reduced‐order fault estimation observer is constructed, which results in a new fault estimator to realize fault estimation using current output information. Furthermore, by introducing slack variables, improved results on full‐order fault estimation observer and reduced‐order fault estimation observer design with finite‐frequency specifications are obtained such that different Lyapunov matrices can be separately designed for each constraint. Simulation results are presented to illustrate the advantages of the theoretic results obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust observer design for uncertain one‐sided Lipschitz systems with disturbances

In this paper, we study the robust observer design problem for a class of uncertain one‐sided Lipschitz systems with disturbances. Not only the system matrices but also the nonlinear functions are assumed to be uncertain. The nominal models of nonlinearities are assumed to satisfy both the one‐sided Lipschitz condition and the quadratically inner‐bounded condition. By utilizing a novel approach, our observer designs are robust against unknown nonlinear uncertainties and system and measurement noises. The new approach also relaxes some conservativeness in related existing results, ie, less conservative observer design conditions are obtained. Furthermore, the problem of designing reduced‐order observers is considered in case the output measurement is not subject to uncertainty and disturbance. Two examples are provided to show the efficiency and advantages of our results over existing works.     

Observers for a class of nonlinear systems with time‐delay

In this paper, the issue of observer designs for a class of nonlinear continuous‐time systems with time‐delay is addressed, where the nonlinear function is not necessarily Lipschitz. It is shown that both full‐order and reduced‐order observers can be obtained by means of the same linear matrix inequality. A numerical example is presented to show the effectiveness of the proposed approach. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Global output‐feedback stabilization for high‐order nonlinear systems with unknown growth rate

This paper considers the global stabilization via time‐varying output‐feedback for a class of high‐order uncertain nonlinear systems with rather weak assumptions. Essentially different from the existing literature, the systems under investigation simultaneously have more serious nonlinearities, unknowns, immeasurableness, and time‐variations, which are indicated from the unknown time‐varying control coefficients and the higher‐order and lower‐order unmeasured states dependent growth with the rate of unknown function of time and output. Recognizing that adaptive technique is quite hard to apply, a time‐varying design scheme is proposed by combining time‐varying approach, certainty equivalence principle and homogeneous domination approach. One key point in the design scheme is the selection of the design functions of time, in order to compensate/capture the serious unknowns and serious time‐variations, and another one is the design of a time‐varying observer to rebuild the unmeasured system states. With the appropriate choice of the involved design functions, the designed controller makes all the signals of the closed‐loop system globally bounded and ultimately converge to zero. Copyright © 2016 John Wiley & Sons, Ltd.     

Design of functional interval observers for non‐linear fractional‐order systems

This paper considers the problem of designing functional interval observers for a class of non‐linear fractional‐order systems with bounded uncertainties. First, interval observers for linear functions of the state vector of the considered system are designed. Then, conditions for the existence of such interval observers are established and an effective algorithm for computing unknown observer matrices is provided in this paper. Finally, numerical examples and simulation results are given to illustrate the effectiveness of the proposed design method.     

A dual‐observer design for global output feedback stabilization of nonlinear systems with low‐order and high‐order nonlinearities

This paper employs a dual‐observer design to solve the problem of global output feedback stabilization for a class of nonlinear systems whose nonlinearities are bounded by both low‐order and high‐order terms. We show that the dual‐observer comprised of two individual homogeneous observers, can be implemented together to estimate low‐order and high‐order states in parallel. The proposed dual observer, together with a state feedback controller, which has both low‐order and high‐order terms, will lead to a new result combining and generalizing two recent results (Li J, Qian C. Proceedings of the 2005 IEEE Conference on Decision and Control, 2005; 2652–2657) and (Qian C. Proceedings of the 2005 American Control Conference, June 2005; 4708–4715). Copyright © 2008 John Wiley & Sons, Ltd.     

Differential flatness‐based ADRC scheme for underactuated fractional‐order systems

This article mainly studies the fractional‐order active disturbance rejection control (FOADRC) schemes for the underactuated commensurate fractional‐order systems (FOSs). The FOADRC framework for linear FOSs‐based fractional proportion integration differentiation is constructed by using the fractional‐order tracking differentiator and the fractional‐order extended state observer, and the necessary conditions for the system to have stable controllers are provided. The FOADRC scheme for underactuated FOSs based on differential flatness is proposed. For underactuated FOSs, a set of flat output expressions with a fixed format is given under the controllable condition of the system. Moreover, making the flat output as the equivalent of the system output is simple and easy to analyze and calculate. Subsequently, the FOADRC scheme is designed by using the flat output. Finally, the scheme proposed in this article is verified by a simulation example.     

Output feedback control for a class of stochastic high‐order nonlinear systems with time‐varying delays

This paper discusses the problem of output feedback stabilization for a more general class of stochastic high‐order nonlinear systems with time‐varying delays. On the basis of a subtle homogeneous observer and controller construction, and the homogeneous domination approach, the closed‐loop system is globally asymptotically stable in probability, by choosing an appropriate Lyapunov–Krasovskii functional. An example is given to illustrate the effectiveness of the proposed design procedure. Copyright © 2013 John Wiley & Sons, Ltd.     

Adaptive fault tolerant control for a class of uncertain fractional‐order systems based on disturbance observer

In this paper, a class of fractional‐order nonlinear systems are considered in the presence of actuator faults. A novel fault tolerant control scheme based on disturbance observer has been presented, where the actuator faults are considered as the system disturbance and can be approximated by the proposed disturbance observer. The developed fault tolerant control guarantees the convergence of the closed‐loop system and the output tracking performance. Finally, a simulation example is presented to verify the effectiveness of the new method.