Interval observers for linear functions of state vectors of linear fractional‐order systems with delayed input and delayed output

This paper addresses the problem of interval observer design for linear functions of state vectors of linear fractional‐order systems, which are subjected to time delays in the measured output as well as the control input. By using the information of both the delayed output and input, we design two linear functional state observers to compute two estimates, an upper one and a lower one, which bound the unmeasured linear functions of state vectors. As a particular case with output delay only, we design a linear functional state observer to estimate (asymptotically) the unmeasured linear functions of state vectors. Existence conditions of such observers are provided, and some of them are translated into a linear programming problem, in which the observers' matrices can be effectively computed. Constructive design algorithms are introduced. Numerical examples are provided to illustrate the design procedure, practicality, and effectiveness of the proposed design method.     

Gramian‐based model order reduction of parameterized time‐delay systems

Time‐delay systems (TDSs) frequently arise in circuit simulation especially in high‐frequency applications. Model order reduction (MOR) techniques can be used to facilitate the simulation of TDSs. On the other hand, many kinds of variations, such as temperature and geometric uncertainties, can have significant impact on the transient responses of TDSs. Therefore, it is important to preserve parametric dependence during the MOR procedure. This paper presents a new parameterized MOR scheme for TDSs with parameter variations. We derive parameterized reduced‐order models (ROMs) for TDSs using balanced truncation by approximating the Gramians in the multi‐dimensional space of parameters. The resulting ROMs can preserve the parametric dependence, making it efficient for repeated simulations under different parameter settings. Numerical examples are presented to verify the accuracy and efficiency of our proposed algorithm. Copyright © 2012 John Wiley & Sons, Ltd.     

Composite learning fuzzy synchronization for incommensurate fractional‐order chaotic systems with time‐varying delays

This paper presents a composite learning fuzzy control to synchronize two different uncertain incommensurate fractional‐order time‐varying delayed chaotic systems with unknown external disturbances and mismatched parametric uncertainties via the Takagi‐Sugeno fuzzy method. An adaptive controller together with fractional‐order composite learning laws is designed based on both a parallel distributed compensation technology and a fractional Lyapunov criterion. The boundedness of all variables in the closed‐loop system and the Mittag‐Leffler stability of tracking error can be guaranteed. T‐S fuzzy systems are provided to tackle unknown nonlinear functions. The distinctive features of the proposed approach consist in the following: (1) a supervisory control law is designed to compensate the lumped disturbances; (2) both the prediction error and the tracking error are used to estimate the unknown fuzzy system parameters; (3) parameter convergence can be ensured by an interval excitation condition. Finally, the feasibility of the proposed control strategy is demonstrated throughout an illustrative example.     

Adaptive output tracking for a class of non‐linear time‐delay systems

In this paper the output tracking control problem for a class of non‐linear time delay systems with some unknown constant parameters is addressed. Such a problem is solved in the case that the non‐linear time‐delay system has full delay relative degree and stable internal dynamics. It is supposed moreover that the output and its time derivatives until n?1, where n is the length of the state vector (euclidean part), do not depend explicitly on the unknown parameters. This work is the first step towards the application of the methodologies of adaptive control for non‐linear delayless systems, based on tools of differential geometry, to non‐linear time‐delay systems too. Copyright © 2004 John Wiley & Sons, Ltd.     

An adaptive regulation method for a class of uncertain time‐delay systems

In this paper, a new adaptive robust stabilization scheme is proposed for uncertain neutral time‐delay systems. No upper bounds on the uncertainties are assumed to be available. An update law is first used to find estimates of these upper bounds. A state‐feedback controller is then designed, which is shown to stabilize the underlying system under some mild conditions. The asymptotic stability of the state trajectories is proved using the Lyapunov–Krasovskii approach. An example is provided, which demonstrates the efficacy of the proposed adaptive control scheme. Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive stabilization of uncertain nonlinear time‐delay systems with quantized input and output

This paper investigates the problem of global output feedback stabilization for a class of uncertain nonlinear time‐delay systems with unknown time delay in the states and the input in which both the input and the output are logarithmically quantized. The nonlinear functions of such systems are not completely known and satisfying certain bounded condition depending on the unmeasured states and the input. We construct a new dynamic high‐gain observer where only an output quantization instead of the output is available for measurement to dominate the unknown nonlinear functions view as external disturbances. A scaled change of coordinates and an appropriate Lyapunov‐Krasovskii functional are derived to achieve the global stabilization in the sense that all the states of such systems are defined, bounded in the maximal interval [0, +∞), and converge to the zero equilibrium. A numerical example is provided to illustrate the result.     

Adaptive observer for a class of nonlinear systems with time‐varying delays

The adaptive observer design problems have been extensively studied in literature for both linear and nonlinear systems. Some researches have also been carried out on adaptive observer design for linear time‐delay systems, but there is no significant work on adaptive observer design for nonlinear time‐delay systems. In this work, the adaptive observer design problem for a class of nonlinear time‐delay systems is considered. The observer is designed for the nonlinear systems whose nonlinear functions satisfy Lipschitz condition. Like conventional adaptive observers for the systems without time delays, this observer also estimates both states and unknown parameters simultaneously. For this property, it will be very much useful for many real‐time systems where time delays cannot be avoided. The sufficient conditions for existence of the observer are derived using the linear matrix inequality approach. With the help of a numerical example, effectiveness of the proposed observer is demonstrated. Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive finite‐time control for high‐order nonlinear systems with mismatched disturbances

In this paper, adaptive finite‐time control is addressed for a class of high‐order nonlinear systems with mismatched disturbances. An adaptive finite‐time controller is designed in which variable gains are adjusted to ensure finite‐time stabilization for the closed‐loop system. Chattering is reduced by a designed adaptive sliding mode observer which is also used to deal with the mismatched disturbances in finite time. The proposed adaptive finite‐time control method avoids calculating derivative repeatedly of traditional backstepping methods and reduces computational burden effectively. Three numerical examples are given to illustrate the effectiveness of the proposed method.     

An adaptive switching scheme for uncertain discrete time‐delay systems

In this paper, an adaptive switching control algorithm is proposed for the stabilization of uncertain discrete‐time systems with time‐varying delay. It is assumed that the time delay is unknown and time varying, nonetheless bounded with a known bound. It is supposed that the system is highly uncertain, and that a set of controllers are designed (off‐line) to stabilize the system in the whole uncertain parameter space; subsequently, a switching scheme is developed to stabilize the uncertain time‐delay system. A thorough stability analysis for the uncertain time‐delay system under the mentioned control scheme is provided. Furthermore, an upper bound on the allowable rate of change of the system parameters and delay is obtained. Simulation results are presented to show the efficacy of the proposed switching scheme. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive observer for nonlinear fractional‐order systems

In this paper, an adaptive observer is proposed for the joint estimation of states and parameters of a fractional nonlinear system with external perturbations. The convergence of the proposed observer is derived in terms of linear matrix inequalities (LMIs) by using an indirect Lyapunov method.The proposed adaptive observer is also robust against Lipschitz additive nonlinear uncertainty. The performance of the observer is illustrated through some examples including the chaotic Lorenz and Lü's systems. Copyright © 2016 John Wiley & Sons, Ltd.     

Sliding mode disturbance observer control based on adaptive synchronization in a class of fractional‐order chaotic systems

In this paper, a fractional‐order Dadras‐Momeni chaotic system in a class of three‐dimensional autonomous differential equations has been considered. Later, a design technique of adaptive sliding mode disturbance‐observer for synchronization of a fractional‐order Dadras‐Momeni chaotic system with time‐varying disturbances is presented. Applying the Lyapunov stability theory, the suggested control technique fulfils that the states of the fractional‐order master and slave chaotic systems are synchronized hastily. While the upper bounds of disturbances are unknown, an adaptive regulation scheme is advised to estimate them. The recommended disturbance‐observer realizes the convergence of the disturbance approximation error to the origin. Finally, simulation results are presented in one example to demonstrate the efficiency of the offered scheme on the fractional‐order Dadras‐Momeni chaotic system in the existence of external disturbances.     

Robust observer‐based control design for uncertain singular systems with time‐delay

This paper is concerned with the stability analysis and robust dynamic output feedback controller synthesis for uncertain continuous singular systems with time‐delay. First, on the basis of the Lyapunov functional method and by resorting to the delay‐partition technique, improved delay‐dependent sufficient conditions are presented to ensure the nominal unforced system to be admissible (i.e., to be regular, impulse‐free, and stable). Second, with the help of the obtained admissibility criterion, an observer‐based controller is designed by solving a set of LMIs. Finally, the validity and applicability of the proposed approach is shown by examples. Copyright © 2013 John Wiley & Sons, Ltd.     

An expression for the voltage response of a current‐excited fractance device based on fractional‐order trigonometric identities

We report a closed‐form expression of the voltage response of a current‐excited fractance device. The derived simple expression is made possible following the introduction of the generalized sine and cosine functions (rmsin_α(t) and cos_α(t)), which are valid on any fractional‐order surface and tend to the normal sin(t) and cos(t) at α = 1 or asymptotically as t→∞. Copyright © 2011 John Wiley & Sons, Ltd.     

Design equations for fractional‐order sinusoidal oscillators: Four practical circuit examples

Four practical sinusoidal oscillators are studied in the general form where fractional‐order energy storage elements are considered. A fractional‐order element is one whose complex impedance is given by Z = a(jω)^±α, where a is a constant and α is not necessarily an integer. As a result, these oscillators are described by sets of fractional‐order differential equations. The integer‐order oscillation condition and oscillation frequency formulae are verified as special cases. Numerical and PSpice simulation results are given. Experimental results are also reported for a selected Wien‐bridge oscillator. Copyright © 2007 John Wiley & Sons, Ltd.     

Robust adaptive control for uncertain time‐delay systems

The problem of robust stabilization for uncertain dynamic time‐delay systems is considered. Firstly a class of time‐delay systems with uncertainties bounded by high‐order polynomials and unknown coefficients are considered. The corresponding controller is designed by employing adaptive method. It is shown that the controller designed can render the closed‐loop system uniformly ultimately bounded stable based on Lyapunov–Krasovskii method and Lyapunov stability theory. Then the proposed adaptive idea is applied to stabilizing a class of large‐scale time‐delay systems with strong interconnections. A decentralized feedback adaptive controller is designed which guarantees the closed‐loop large‐scale systems uniformly ultimately bounded stable. Finally, numerical examples are given to show the potential of the proposed techniques. Copyright © 2005 John Wiley & Sons, Ltd.     

Design and performance study of phase‐locked loop using fractional‐order loop filter

The present work reports the realization of an analog fractional‐order phase‐locked loop (FPLL) using a fractional capacitor. The expressions for bandwidth, capture range, and lock range of the FPLL have been derived analytically and then compared with the experimental observations using LM565 IC. It has been observed that bandwidth and capture range can be extended by using FPLL. It has also been found that FPLL can provide faster response and lower phase error at the time of switching compared to its integer‐order counterpart. Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive neural control for nonstrict‐feedback stochastic nonlinear time‐delay systems with input and output constraints

This paper investigates an adaptive neural tracking control for a class of nonstrict‐feedback stochastic nonlinear time‐delay systems with input saturation and output constraint. First, the Gaussian error function is used to represent a continuous differentiable asymmetric saturation model. Second, the appropriate Lyapunov‐Krasovskii functional and the property of hyperbolic tangent functions are used to compensate the time‐delay effects, the neural network is used to approximate the unknown nonlinearities, and a barrier Lyapunov function is designed to ensure that the output parameters are restricted. At last, based on Lyapunov stability theory, a robust adaptive neural control method is proposed, and the designed controller decreases the number of learning parameters and thus reduces the computational burden. It is shown that the designed neural controller can ensure that all the signals in the closed‐loop system are 4‐Moment (or 2 Moment) semi‐globally uniformly ultimately bounded and the tracking error converges to a small neighborhood of the origin. Two examples are given to further verify the effectiveness of the proposed approach.     

Adaptive observers for a class of multi-output non-linear systems

Sufficient differential geometric conditions are given for the existence of global adaptive observers for a class of multi-output non-linear systems which are linear with respect to a vector of unknown constant parameters. They extend to multi-output systems earlier results on adaptive observers for single-output systems; they also extend to systems with unknown parameters results obtained previously on the existence of non-adaptive observers with linear error dynamics.     

Event-triggered adaptive backstepping tracking control for a class of nonlinear fractional order systems

This article investigates the problem of event-trigger based adaptive backstepping control for a class of nonlinear fractional order systems. By introducing an appropriate transformation of frequency distributed model, the fractional-order indirect Lyapunov method with is obtained. In addition, the event-triggered adaptive controller is developed by employing the event-triggered control approach. Meanwhile, by the proposed adaptive control scheme, all the closed-loop signals are globally uniformly bounded, and the tracking error converges to a small neighborhood of the origin. Finally, simulation results are provided to testify the availability of the presented controller.