Design of unknown‐input reduced‐order observers for a class of nonlinear fractional‐order time‐delay systems

This paper considers the design of reduced‐order state observers for fractional‐order time‐delay systems with Lipschitz nonlinearities and unknown inputs. By using the Razumikhin stability theorem and a recent result on the Caputo fractional derivative of a quadratic function, a sufficient condition for the asymptotic stability of the observer error dynamic system is presented. The stability condition is obtained in terms of linear matrix inequalities, which can be effectively solved by using existing convex algorithms. Numerical examples and simulation results are given to illustrate the effectiveness of the proposed design approach.     

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.     

Analysis and design of discrete‐time charge domain filters with complex conjugate poles

This paper proposes a discrete‐time charge domain filter that achieves complex conjugate poles in the transfer function of the filter. To achieve complex conjugate poles, local feedbacks are inserted around two successive discrete‐time integrators. The feedback path is implemented through a transconductance cell which applies a continuous time current into the integrators. Analytical models have been proposed to approximate the behavior of the filter. These models confirm that the structure is capable of realizing complex poles and thus can be used to synthesize any type of filter structures such as Butterworth, Chebysheve, etc. To show the effectiveness of the proposed architecture, Butterworth filters of order 2 and 4 operating at 50MS/s are designed and implemented in 180‐nm CMOS technology with 1.8‐V power supply. The effect of circuit nonidealities on the performance of the filter is analyzed and verified through simulations. Simulation results show that a conventional charge domain filter can be simply extended to implement complex conjugate poles while the noise and linearity performance of the filter are also improved. Copyright © 2016 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.     

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.     

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.     

Time‐varying linear systems and input–output stability

One useful strategy for establishing input–output stability is to seek conditions on the system map so that a certain lower‐bound condition is met. We show here that, for a large class of time‐varying linear systems, the lower‐bound condition for stability is in fact necessary. Copyright © 2000 John Wiley & Sons, Ltd.     

Stability analysis and robust synchronization of fractional‐order competitive neural networks with different time scales and impulsive perturbations

This article mainly examine a class of robust synchronization, global stability criterion, and boundedness analysis for delayed fractional‐order competitive type‐neural networks with impulsive effects and different time scales. Firstly, by endowing the robust analysis skills and a new class of Lyapunov‐Krasovskii functional approach, the error dynamical system is furnished to be a robust adaptive synchronization in the voice of linear matrix inequality (LMI) technique. Secondly, by ignoring the uncertain parameter terms, the existence of equilibrium points are established by means of topological degree properties, and the solution representation of the considered network model are provided. Thirdly, a novel global asymptotic stability condition is proposed in the voice of LMIs, which is less conservative. Finally, our analytical results are justified with two numerical examples with simulations.     

Constructive approximation of non‐linear discrete‐time systems

It is known that large classes of approximately‐finite‐memory maps can be uniformly approximated arbitrarily well by the maps of certain non‐linear structures. As an application, it was proved that time‐delay networks can be used to uniformly approximate arbitrarily well the members of a large class of causal nonlinear dynamic discrete‐time input–output maps. However, the proof is non‐constructive and provides no information concerning the determination of a structure that corresponds to a prescribed bound on the approximation error. Here we give some general results concerning the problem of finding the structure. Our setting is as follows. There is a large family 𝒢 of causal time‐invariant approximately‐finite‐memory input‐output maps G from a set S of real d‐vector‐valued discrete‐time inputs (with d⩾1) to the set of ℝ‐valued discrete‐time outputs, with both the inputs and outputs defined on the non‐negative integers 𝒵₊. We show that for each ϵ>0, any Gϵ𝒢 can be uniformly approximated by a structure map H(G, ·) to within tolerance ϵ, and we give analytical results and an example to illustrate how such a H(G, ·) can be determined in principle. Copyright © 2000 John Wiley & Sons, Ltd.     

Mittag‐Leffler state estimator design and synchronization analysis for fractional‐order BAM neural networks with time delays

This paper deals with the extended design of Mittag‐Leffler state estimator and adaptive synchronization for fractional‐order bidirectional associative memory neural networks with time delays. By the aid of Lyapunov direct approach and Razumikhin‐type method, a suitable fractional‐order Lyapunov functional is constructed and a new set of novel sufficient condition are derived to estimate the neuron states via available output measurements such that the ensuring estimator error system is globally Mittag‐Leffler stable. Then, the adaptive feedback control rule is designed, under which the considered FBNNs can achieve Mittag‐Leffler adaptive synchronization by means of some fractional‐order inequality techniques. Moreover, the adaptive feedback control may be utilized even when there is no ideal information from the system parameters. Finally, two numerical simulations are given to reveal the effectiveness of the theoretical consequences.     

Finite‐time dissipativity‐based filter design for networked control systems

This paper investigates the problem of finite‐time boundedness and dissipativity‐based filter design for networked control systems together with parameter uncertainties and random packet dropouts. The packet transmission information is defined by using Bernoulli distributed white sequence which characterizes the measurement conditions. Some new sufficient conditions are established to ensure that the filtering error system is stochastically finite‐time bounded and strictly finite‐time dissipative. These sufficient conditions to design the filter parameters are derived by using linear matrix inequalities and reciprocally convex approach. Finally, an example is given to validate the effectiveness of the proposed filter design.     

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.     

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.     

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.     

A robust output error identifier for continuous‐time systems

This paper shows that the adaptive output error identifier for linear time‐invariant continuous‐time systems proposed by Bestser and Zeheb is robust vis‐à‐vis finite energy measurement noise. More precisely, it is proven that the map from the noise to the estimation error is –stable—provided a tuning parameter is chosen sufficiently large. A procedure to determine the required minimal value of this parameter is also given. If the noise is exponentially vanishing, asymptotic convergence to zero of the prediction error is achieved. Instrumental for the establishment of the results is a suitable decomposition of the error system equations that allows us to strengthen—to strict—the well‐known passivity property of the identifier. The estimator neither requires fast adaptation, a dead‐zone, nor the knowledge of an upperbound on the noise magnitude, which is an essential requirement to prove stability of standard output error identifiers. To robustify the estimator with respect to non‐square integrable (but bounded) noises, a prediction error‐dependent leakage term is added in the integral adaptation. –stability of the modified scheme is established under a technical assumption. A simulated example, which is unstable for the equation error identifier and the output error identifier of Bestser and Zeheb, is used to illustrate the noise insensitivity property of the new scheme. Copyright © 2014 John Wiley & Sons, Ltd.     

Uniform approximation of time‐invariant non‐linear systems

An Erratum has been published for this article in International Journal of Circuit Theory and Applications 2004; 32(6):633. It is shown that the elements of a large class of time‐invariant non‐linear input–output maps can be uniformly approximated arbitrarily well, over infinite time intervals, using a certain structure that can be implemented in many ways using, for example, radial basis functions, polynomial functions, piecewise linear functions, sigmoids, or combinations of these functions. For the special case in which these functions are taken to be certain polynomial functions, the input–output map of our structure is a generalized finite Volterra series. Results are given for the case in which inputs and outputs are defined on ?. The case in which inputs and outputs are defined on the half‐line ?₊ is also addressed, and in both cases inputs need not be functions that are continuous. Copyright © 2004 John Wiley & Sons, Ltd.     

Adaptive control of discrete‐time nonlinear systems by recurrent neural networks in quasi‐sliding mode like regime

The aim of this study was to design an adaptive control strategy based on recurrent neural networks (RNNs). This neural network was designed to obtain a non‐parametric approximation (identification) of discrete‐time uncertain nonlinear systems. A discrete‐time Lyapunov candidate function was proposed to prove the convergence of the identification error. The adaptation laws to adjust the free parameters in the RNN were obtained in the same stability analysis. The control scheme used the states of the identifier, and it was developed fulfilling the necessary conditions to establish a behavior comparable with a quasi‐sliding mode regime. This controller does not use the regular form of the switching function that commonly appears in the sliding mode control designs. The Lyapunov candidate function to design the controller and the identifier simultaneously requires the existence of positive definite solutions of two different matrix inequalities. As consequence, a class of separation principle was proven when the RNN‐based identifier and the controller were designed by the same analysis. Simulations results were designed to show the behavior of the proposed controller solving the tracking problem for the trajectories of a direct current (DC) motor. The performance of the proposed controller was compared with the solution obtained when a classical proportional derivative controller and an adaptive first‐order sliding mode controller assuming poor knowledge of the plant. In both cases, the proposed controller showed superior performance when the relation between the tracking error convergence and the energy used to reach it was evaluated. Copyright © 2016 John Wiley & Sons, Ltd.     

A note on stability of linear time‐variant electrical circuits having constant eigenvalues

The main aspect of this paper is to show that the stability of linear time‐variant systems cannot be estimated from the location of the eigenvalues. For this purpose, two simple time‐variant electrical circuits are presented, which have constant eigenvalues. As will be shown, the time‐variant circuits can be asymptotically stable although there is a positive eigenvalue and this circuit can be unstable despite negative eigenvalues only. The idea behind is a suited time‐variant state transformation of a linear time‐invariant system. An electrical interpretation of both systems and of state transformations allows for an energetic evaluation from an electrical point of view even though the analytical solution is not necessarily known. Copyright © 2012 John Wiley & Sons, Ltd.     

filtering for switched discrete‐time systems under asynchronous switching: A dwell‐time dependent Lyapunov functional method

In this article, the filtering problem for switched discrete‐time linear systems under asynchronous switching is addressed in the framework of dwell time, where ‘asynchronous switching’ covers more general and practical cases, for example, the switching lags caused by mode identification process are taken into consideration. Firstly, a novel dwell‐time dependent Lyapunov function (DTDLF) is introduced to solve stability and ?₂ gain analysis problems. The main advantage of DTDLF approach is that the derived conditions are all convex in system matrices, so it is convenient to be applied into filter design with performance instead of weighted performance as many other previous results. Thus, on the basis of DTLDF, a dwell‐time dependent filter with time‐varying structure is proposed to achieve the desirable non‐weighted filtering performance. It is notable that the proposed approach can also easily characterize the relationships among filtering performance, dwell time, and asynchronous time. Two examples are provided to validate the theoretical findings in this paper. Copyright © 2014 John Wiley & Sons, Ltd.     

A design method for robust model matching control of nonminimum‐phase discrete‐time systems

This paper presents a design method for robust model matching control of nonminimum‐phase discrete‐time systems. This scheme can robustly control the nominal model in the presence of unmodeled dynamics and can achieve the desired model matching simultaneously. Furthermore, the sufficient condition for stabilizing the nominal model in the presence of the unmodeled dynamics is derived and the existence of bounds for all signals is proved. Finally, computer simulation results are presented to illustrate the effectiveness of the proposed method. © 1999 Scripta Technica, Electr Eng Jpn, 128(2): 36–44, 1999