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Accurate representation of the atomic force microscopy (AFM) system is not only necessary to achieve control objectives, but it is also beneficial for detecting the nanomechanical properties of the samples. To this end, this paper addresses the issue of controller design for the AFM system based on an accurate nonaffine nonlinear distributed-parameters model in which flexibility and distributed mass effects of the microcantilever beam are considered properly. First, a T-S fuzzy model is derived for this dynamical model of the AFM system in order to simplify the procedure of controller design. Then, a fuzzy model-based controller is designed to suppress the chaos and attenuate the disturbance in the AFM system through the linear matrix inequality (LMI) formulation. Moreover, by considering some criteria for disturbance rejection and transient performance, and some constraints on control input and states, new stabilization conditions are proposed based on a fuzzy Lyapunov function. Finally, simulation results are represented to demonstrate the effectiveness of the proposed method.

This letter aims to investigate the robust stability of an interval fractional-order plant with an interval time delay by a general form of fractional-order controllers. Based on a simple graphical procedure, necessary and sufficient criteria are proposed to analyze the robust stability of an interval fractional-order plant. Then, the idea of “robust stability testing function” is extended to analyze the robust stability of the considered type of systems. Besides, an upper frequency bound is provided which is useful for the simulation purposes and it helps to reduce the computational cost. Finally, two examples are presented to verify the usefulness of the obtained results.

In this article, a Takagi–Sugeno fuzzy model is applied to deal with the problem of observer-based control design for nonlinear time-delayed systems with fractional-order [Formula: see text]. By applying the Lyapunov–Krasovskii method, a fuzzy observer–based controller is established to stabilize the time-delayed fractional-order Takagi–Sugeno fuzzy model. Also, the problem of disturbance rejection for the addressed systems is studied via the state-feedback method in the form of a parallel distributed compensation approach. Furthermore, sufficient conditions for the existence of state-feedback gains and observer gains are achieved in the terms of linear matrix inequalities. Finally, two numerical examples are simulated for the validation of the presented methods.

This paper deals with the robust stability analysis of a class of incommensurate fractional order quasi-polynomials with a general type of interval uncertainties. The concept of the general type of interval uncertainties means that all the coefficients and orders of the fractional order quasi-polynomials have interval uncertainties. Generally, the computational complexity of specifying the robust stability of such a quasi-polynomial is shown in this paper. To this end, the robust stability is studied by Principle of Argument theorem. In fact, by presenting two theorems and three lemmas it is shown that the robust stability of a fractional order quasi-polynomial involving the general type of uncertainty can be simply investigated without needing to plot its value set by heavy computations. Examples are attested the validity of the paper results.

This paper proposes a novel method for the simultaneous estimation of the coefficients and the delay term of a delayed fractional order system. Because of the practicality aspect of the non-ideal step inputs, such inputs are used in this paper for the first time to identify a fractional order system. To this end, the proposed identification procedure is separately described for two types of fractional order systems, i.e., including both non-delayed and delayed systems. For the non-delayed system, a fractional order integral approach is developed, and for the delayed system, a filtering approach is investigated to make the delay term to be explicitly appeared in the parameters vector. In simulation results, some illustrative examples, covering both non-delayed and delayed systems, are given to demonstrate the validity of the proposed method.

This paper focuses on the issue of stabilization of nonlinear fractional order time-delay systems in the presence of fault. A general class of nonlinear fractional order systems is considered in which fault causes variations in the system dynamic and actuators. Besides, time-varying delay is noticed in the system equations that is of high importance in control of real-life systems. In order to facilitate the problem of controller design for such systems, a precise method for modeling complex nonlinear systems, namely, Takagi–Sugeno fuzzy model, is adopted. Regarding a Caputo derivative–based Lyapunov–Krasovskii functional, new delay-dependent stabilization conditions are established in the form of linear matrix inequalities. Eventually, two examples including a truck–trailer and Lorenz system are simulated in order to evaluate the results of this research work.

In this paper, a new value set based approach is presented to study the robust stability of a fractional order plant with parametric uncertainty of the interval type by using a fractional order proportional integral controller. The concept of interval uncertainty means that the parameters of the system transfer functions are uncertain parameters that each adopts a value in a real interval. Based on the zero exclusion principle, it is necessary to check whether the value set of the characteristic polynomial for the infinite frequency range [Formula: see text] includes the origin or not. To this end, in this paper, an auxiliary function is defined. The sign of this auxiliary function within a finite frequency range verifies whether the origin is included in the value set of the characteristic polynomial or not. The upper bound and the lower bound of this finite frequency range are obtained using the triangle inequality. Finally, the results of the numerical examples are provided, which confirm the effectiveness of the paper results.

This study investigates the robust stability of the retarded type of interval fractional order plants with an interval time delay. To this end, the characteristic quasi-polynomial is divided into two terms. The first term is simply the denominator interval polynomial of the open loop system and the second term is the multiplication of the interval delay term in the numerator of the open loop system which is an interval polynomial. Each of these two terms of the characteristic quasi-polynomial makes their own value sets in the complex plane for a given frequency. In this paper, based on these two value sets and by using the zero exclusion principle, the robust stability of the closed loop system by applying a FOPID controller is analyzed. Finally, two numerical examples and an experimental verification are provided to demonstrate the effectiveness of the proposed method in the robust stabilization of fractional order plants with interval uncertainties and interval time delay.

This manuscript provides a frequency-domain control methodology to tune the parameters of a fractional integral-fractional derivative (FIFD) controller, which is developed based on a Smith predictor (SP) control structure. For compensating the interval first order time-delayed processes, this paper proposes a novel technique and provides a practical approach. By profiting from the proposed systematic way, not only the controller parameters can optimally be determined to satisfy the desired control system performance specifications under the perturbed conditions, but also the stability margins of the system, i.e., the phase margin and the gain crossover frequency, are tunable. A necessary and sufficient condition is also achieved for the robust stability of the closed-loop control system. Using simulation, the results obtained by applying the proposed technique are given to show the application and the effectiveness of the paper’s novelties.

A Model Predictive Control (MPC) method, in this research, is investigated to enhance both Load Frequency Control (LFC) and Automatic Voltage Regulator (AVR) of a power system. MPC, as an advanced predictive control method technique, is widely used for industrial applications and systems with certain model dynamics. The controller is first applied to a model of LFC for two areas interconnected non-reheat thermal power system. Then, the optimization problem of the AVR system has been considered into account. Instead of using an individual controller for each zone of the LFC, a single model predictive controller is utilized in the power system to handle the overall frequency and tie-line deviations better. Moreover, to have a realistic stage of the study, the values of the system setting and applied load disturbances in Area 1 are considered different from those parameters in Area 2. To demonstrate the effectiveness of the proposed control strategy, the results of comparisons are provided. The proposed MPC method is compared with the other controllers including, PI and PID in simultaneous performance enhancement of the AVR and LFC units of an interconnected power plant.

In this paper, a simple method is proposed to tune the parameters of Fractional Integral-Fractional Derivative (FIFE) Iλ Dμ controllers based on the Bode diagram. The proposed technique provides a ...

This study proposes an adaptive interval type-2 Takagi–Sugeno–Kang (IT2 TSK) fuzzy system with a supervisory mode to control and stabilise a certain class of non-linear fractional order systems. In this study, a fractional order adaptation law is derived which adjusts the free parameters and bounds them by utilising a projection algorithm. The global Mittag–Leffler stability of the closed-loop system is proved in the sense that all the involved signals are uniformly bounded. Moreover, if the non-linear system tends to be unstable, a supervisory controller starts cooperating with the adaptive IT2 TSK fuzzy controller to guarantee the stability of the closed-loop system. In addition, a new inference mechanism for the adaptive IT2 TSK fuzzy system is introduced for which the antecedent part is chosen as a type-2 fuzzy set and the consequent parameters are represented as interval sets. According to the practical nature of the proposed inference equation, it would be applicable in online and real-time applications. Numerical simulations show the validity and effectiveness of the introduced control strategy for stabilisation and control of a general class of non-linear fractional order systems perturbed by disturbance and uncertainty.

This paper addresses the problem of tracking control and disturbance rejection for nonlinear fractional order systems. To this end, based on Takagi-Sugeno (T-S) fuzzy modeling approach and a fractional order modified repetitive controller, an output feedback strategy is constructed for a class of nonlinear fractional order systems. T-S fuzzy model represents an accurate model of nonlinear fractional order systems and simplifies the procedure of control system design. By introducing a novel Fuzzy Lyapunov Function (FLF), sufficient conditions are established in the framework of Linear Matrix Inequalities (LMIs) to ensure the stability of the closed-loop system. Based on the achieved conditions, it is shown that the external disturbance is attenuated and the output of the system tracks the periodic reference input with an ignorable error. Lastly, three examples including fractional order Chua’s circuit, Lorenz-like and Chen systems are performed to show the efficiency of the method proposed in this paper.

In this paper, a direct adaptive fuzzy controller with compensation signal is presented to control and stabilize a class of fractional order systems with unknown nonlinearities. Based on a Lyapunov function candidate the global Mittag–Leffler stability is proved and a new fractional order adaptation law is derived. The adaptation law adjusts free parameters of the fuzzy controller and bounds them by utilizing a novel fractional order projection algorithm. Furthermore, due to the use of compensation term, the proposed approach does not demand suitable membership functions in the fuzzy system. In addition, the stability of the closed-loop system is guaranteed by utilizing a supervisory controller. Numerical simulations show the validity and effectiveness of the introduced scheme for various fractional order nonlinear models that perturbed by disturbance and uncertainty.

Nowadays, fractional order differential operators, as a generalization for classical differential operators, have established their key-role in modeling, analysis, and implementation of specific circuits and systems in which one typically faces nonlinear behaviors. It enforces to exploit analysis and implementation methods covering simultaneously “fractionality” and “nonlinearity” aspects. This special section, entitled “Nonlinear Fractional Order Circuits and Systems: Advanced Analysis and Effective Implementation,” aims at introducing some of these methods.

This paper is devoted to proposing a simple method to tune the parameters of a fractional order PI (FOPI) controller. The studied control scheme is a fractional filtered Smith predictor (FFSP) structure, which can compensate for the inherently long dead time of the industrial processes. The proposed frequency domain scheme would be a practical approach to compensate the stable processes, which can be modeled by a fractional order counterpart of First Order Plus Dead Time (FOPDT) transfer function. Two fractional order filters are also used to improve the reference tracking and to enhance the robustness of the compensated system. The obtained results of practical implementation on a pressure plant and the comparison results are given to demonstrate the effectiveness of the proposed technique.

In this paper, based on the concept of restricted difference basis RDB for a natural number, the general structures of the pseudo state-space realisations of some classes of commensurate fractional-order transfer functions are obtained. It is shown that the number of needed fractional-order capacitors for implementing a transfer function belonging to each of these classes depends on the commensurate order and also the RDB based on which the pseudo state-space realisation of the considered transfer function is obtained. In this regard, three general classes of commensurate fractional-order transfer functions are introduced which all can be implemented by the same number of fractional-order capacitors. Also, four classes of fractional-order transfer functions of commensurate order are introduced which all can be implemented by only one fractional-order capacitor of order . Finally, as a more general case, four classes of fractional-order transfer functions of commensurate order are studied in aspect of the number and the order of the fractional-order capacitors needed for their implementations. It is proved that these classes of commensurate fractional-order transfer functions can be implemented by r − 1 number of fractional-order capacitors of order .

This paper presents a simple analytical method for tuning the parameters of fractional order PI (FOPI) controllers based on Bode’s ideal transfer function. The proposed technique is applicable to stable plants describable by a fractional order counterpart of first order transfer function without time delay. Tuning rules are given in order to improve the robustness of the compensated system in the presence of gain uncertainty in the plant model. Finally, the designed FOPI controller is implemented on a laboratory scale twin rotor helicopter and comparison results are provided to show the effectiveness of the proposed tuning rules.

This article presents a simple frequency domain control technique to tune the parameters of a filtered Smith predictor based fractional integral-fractional derivative (FSP-FIFD) controller. The proposed method provides a practical approach to compensate stable First Order Plus Dead Time (FOPDT) transfer functions based on a filtered Smith predictor (FSP) control structure. In this control structure, an integer order predictor filter is used to improve the noise reduction of the control system. By benefiting from the proposed FSPFIFD controller, not only the phase margin and the gain crossover frequency of the control system are adjustable by tuning the free controller parameters, but also the robustness of the compensated system is enhanced. Finally, the designed FSPFIFD controller is implemented on a laboratory scale pressure plant and the obtained results are compared with those of applying a filtered Smith predictor based PI (FSPPI) controller in a similar structure. The practical results demonstrate the simplicity of application and the effectiveness of the proposed tuning rules.

In this paper, an analytical method for tuning the parameters of the set-point weighted fractional order PID (SWFOPID) controller is proposed. The studied control scheme is the filtered fractional set-point weighted (FFSW) structure. Also to achieve a desired closed-loop performance, a fractional order pre-filter is employed. The proposed method is applicable to stable plants describable by a simple three-parameter fractional order model. Such a model can be considered as the fractional order counterpart of a first order transfer function without time delay. Finally, the proposed method is implemented on a laboratory scale CE 150 helicopter platform and the results are compared with those of applying a filtered fractional order PI (FFOPI) controller in a similar structure. The practical results show the effectiveness of the proposed method.

This paper deals with proportional stabilization and closed-loop step response identification of the fractional order counterparts of the unstable first order plus dead time (FOPDT) processes. At first, the necessary and sufficient condition for stabilizability of such processes by proportional controllers is found. Then, by assuming that a process of this kind has been stabilized by a proportional controller and the step response data of the closed-loop system is available, an algorithm is proposed for estimating the order and the parameters of an unstable fractional order model by using the mentioned data.

This study deals with a generalised version of lead/lag compensators known as fractional-order lead/lag compensators. Exact and simple formulas are found for designing this introduced type of fractional-order compensators in order to provide the required magnitude and phase at a given frequency. Also, the region in the phase-magnitude plane, which is accessible by these compensators, is analytically found. Moreover, numerical examples and experimental results are presented to show the applicability of the achievements of this study in control system design.

In this paper, two integral based methods are proposed to estimate the order and the parameters of simple fractional order models from the noisy step response data. Numerical simulation results show the efficiency of the proposed methods in the presence of the measurement noise.

In this paper, it is shown that by benefiting from a static feedback control signal it is possible to reduce the effect of fractionality of the electrical capacitors on the amplitude of the oscillations produced by a Van der Pol circuit. The averaging method is used in this paper for the behavior analysis of the approximated responses of the under study circuits. Numerical simulation results are presented to confirm the effectiveness of the proposed control technique.