Singular Fractional Order Research Articles

A highly accurate and efficient Genocchi‐based spectral technique applied to singular fractional order boundary value problems

This article focuses on an efficient and highly accurate approximate solver for a class of generalized singular boundary value problems (SBVPs) having nonlinearity and with two‐term fractional derivatives. The involved fractional derivative operators are given in the form of Liouville–Caputo. The developed algorithm for solving the generalized SBVPs consists of two main stages. The first stage is devoted to an iterative quasilinearization method (QLM) to conquer the (strong) nonlinearity of the governing SBVPs. Secondly, we employ the generalized Genocchi polynomials (GGPs) to treat the resulting sequence of linearized SBVPs numerically. An upper error estimate for the Genocchi series solution in the norm is obtained via a rigorous error analysis. The main benefit of the presented QLM‐GGPs procedure is that the required number of iteration in the first stage is within a few steps, and an accurate polynomial solution is obtained through computer implementations in the second stage. Three widely applicable test cases are investigated to observe the efficacy as well as the high‐order accuracy of the QLM‐GGPs algorithm. The comparable accuracy and robustness of the presented algorithm are validated by doing comparisons with the results of some well‐established available computational methods. It is apparently shown that the QLM‐GGPs algorithm provides a promising tool to solve strongly nonlinear SBVPs with two‐term fractional derivatives accurately and efficiently.

Adaptive Control for group consensus of fuzzy singular fractional order multi-agent systems under non-fragile controller

Adaptive Control for group consensus of fuzzy singular fractional order multi-agent systems under non-fragile controller

Fuzzy control of singular fractional order multi-agent systems with actuator saturation

In this paper, a T-S fuzzy model is established to describe nonlinear fuzzy singular fractional order multi-agent systems (SFOMASs) exhibiting actuator saturation when the order is between 0 and 2. We focus on the case where the system state is difficult to measure or cannot be directly measured, so it needs to be estimated based on other measurable signals. To this end, a fuzzy observer is designed, which can process fuzzy or imprecise input and output signals through fuzzy processing. Then the observer-based leader-following consensus problem of fuzzy SFOMASs with actuator saturation is addressed. To address input saturation, we use some mathematical tools to represent the saturated linear feedback on the convex hull, which is then transformed into seeking the optimal solution of linear matrix inequalities (LMIs). Finally, the effectiveness of the proposed approach is verified through the adoption of a numerical example.

Admissible H$_\infty$ Control of Fuzzy Singular Fractional Order Multi-Agent Systems With External Disturbances

This brief consider the problem about admissible consensus of fuzzy singular fractional order multi agent systems (FSFOMAS). By designing a new distributed fuzzy dynamic output control strategy, the coupled system achieves admissible while satisfying <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_\infty$</tex-math> </inline-formula> performance. Then, when the fuzzy models reduce to ordinary fractional order multi agent systems (FOMAS), a new sufficient condition for determining system consensus is given. The above results all depend only on the eigenvalues of the Laplace matrix and the state matrix of the system and consist of linear matrix inequalities (LMIs). Finally, a numerical example and a practical circuit example can demonstrate the feasibility and validity of the theorems. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Due to the unique memorability of fractional orders, their application to fuzzy multi-agent systems will bring system performance metrics closer to reality. This paper investigates the stabilization of singular fuzzy fractional-order multi-agent systems with the perturbation case, the pulse case. We know that in practical applications a single variable does not accurately describe the phenomenon. Therefore we have introduced the concepts of multi-agent, fuzzy, singular and fractional orders in order to describe the system model more accurately. The main objective is to bring the system into equilibrium in the closest possible approximation to reality. The performance metrics of the system are optimised to bring the system to an optimal state. Finally, two examples are used to illustrate the validity of our theorems.

On a Singular Non local Fractional System Describing a Generalized Timoshenko System with Two Frictional Damping Terms

This paper concerns a nonhomogeneous singular fractional order system, with two frictional damping terms. This system can be considered as a generalization of the so-called Timoshenko system. Results on the existence, uniqueness, and continuous dependence on the solution were obtained via an energy approach, which mainly relies on a priori bounds and density arguments. The approach relies on functional analysis tools and operator theory. Very few results concerning the well-posedness of fractional order Timoshenko systems can be found in the literature. Our results generalize and improve the previous ones and significantly boost the development of the used method.

Generalized Criteria for Admissibility of Singular Fractional Order Systems

Unified frameworks for fractional order systems with fractional order 0&lt;α&lt;2 are worth investigating. The aim of this paper is to provide a unified framework for stability and admissibility for fractional order systems and singular fractional order systems with 0&lt;α&lt;2, respectively. By virtue of the LMI region and GLMI region, five stability theorems are presented. Two admissibility theorems for singular fractional order systems are extended from Theorem 5, and, in particular, a strict LMI stability criterion involving the least real decision variables without equality constraint by isomorphic mapping and congruent transform. The equivalence between the admissibility Theorems 6 and 7 is derived. The proposed framework contains some other existing results in the case of 1≤α&lt;2 or 0&lt;α&lt;1. Compared with published unified frameworks, the proposed framework is truly unified and does not require additional conditional assignment. Finally, without loss of generality, a unified control law is designed to make the singular feedback system admissible based on the criterion in a strict LMI framework and demonstrated by two numerical examples.

Non-fragile sliding mode observer based fault estimation for interval type-2 fuzzy singular fractional order systems

In this paper, an interval type-2 fuzzy model is presented to describe the nonlinear singular fractional order systems (SFOSs) with . The problem of actuator and sensor fault estimation is addressed by designing a non-fragile sliding mode observer (SMO). Firstly, a new criterion is provided to analyse the admissibility of SFOSs, which does not need to divide the fractional order interval into interval and interval to study them separately. Then, by expanding the dimension of the SFOS, the non-fragile SMO and the sliding surface are constructed and the accurate estimations of faults and the SFOS state simultaneously are achieved. A new reaching law is designed, which can divide the process of the system moving to the sliding surface into three stages. Further the control law is constructed to guarantee the observation error reaches the sliding surface in finite time. Finally, three examples show the effectiveness of the proposed scheme.

Quadratic Admissibility for a Class of LTI Uncertain Singular Fractional-Order Systems with 0 &lt; α &lt; 2

This paper provides a unified framework for the admissibility of a class of singular fractional-order systems with a given fractional order in the interval (0, 2). These necessary and sufficient conditions are derived in terms of linear matrix inequalities (LMIs). The considered fractional orders range from 0 to 2 without separating the ranges into (0, 1) and [1, 2) to discuss the admissibility. Moreover, the uncertain system with the fractional order in the interval (0, 2) is norm-bounded. The quadratic admissibility and general quadratic stability of the system are analyzed, and the equivalence between the two is proved. All the above can be expressed in terms of strict LMIs to avoid any singularity problem in the solution. Finally, the effectiveness of the method is illustrated by three numerical examples.

Adaptive sliding mode consensus control based on neural network for singular fractional order multi-agent systems

In this paper, a suitable state feedback sliding mode controller is designed for the singular fractional order multi-agent systems (SFOMASs) with uncertainty, in order to realize the consensus problem of multi-agent. First, the sliding mode of the designed SFOMAS is in the form of singular systems. The criterion for the admissible consensus of sliding mode is given by using linear matrix inequality (LMI), and an adaptive law based on radial basis function neural network (RBFNN) is established to ensure the accessibility of SFOMASs. Then, a special method is studied to make the sliding mode of the designed SFOMAS normalization. A sufficient condition for the stability and consensus of sliding mode is given by using LMI, and an adaptive law based on RBFNN is established to ensure the accessibility of SFOMAS. Finally, two numerical examples show the applicability of the proposed method.

On Sliding Mode Control for Singular Fractional-Order Systems with Matched External Disturbances

In this paper, we investigate the problem of sliding mode control for singular fractional-order systems that have matched uncertainties. We design an innovative integral sliding mode function and controller based on the normalizable condition. A strict linear matrix inequality-based sufficient condition is obtained for the system’s stability. The normalizable condition is eliminated by updating and developing the control method, and a sufficient and necessary condition is developed for the admissibility of the system. Lastly, verification of our method’s effectiveness is numerically conducted in two instances.

Observer-based adaptive control and faults estimation for T-S fuzzy singular fractional order systems

Observer-based adaptive control and faults estimation for T-S fuzzy singular fractional order systems

Robust stabilization of uncertain rectangular singular fractional order T-S fuzzy systems with the fractional order 0 < α < 1

This paper presents a novel method to investigate the robust stabilization problem of uncertain rectangular singular fractional order Takagi-Sugeno (T-S) fuzzy systems with the fractional order 0 < α < 1. Firstly, the uncertain rectangular singular fractional order T-S fuzzy system is transformed into an augmented uncertain square singular fractional order T-S fuzzy system by designing a new T-S fuzzy dynamic compensator. Secondly, a sufficient condition in the form of linear matrix inequalities (LMI) is obtained for the robust stabilization of the uncertain rectangular singular fractional order T-S fuzzy system. Finally, a numerical example is given to verify the effectiveness of the results proposed.

An infinite number of nonnegative solutions for iterative system of singular fractional order Boundary value problems

In this paper, we consider the iterative system of singular Rimean-Liouville fractional-order boundary value problems with Riemann-Stieltjes integral boundary conditions involving increasing homeomorphism and positive homomorphism operator(IHPHO). By using Krasnoselskii’s cone fixed point theorem in a Banach space, we derive sufficient conditions for the existence of an infinite number of nonnegative solutions. The sufficient conditions are also derived for the existence of a unique nonnegative solution to the addressed problem by fixed point theorem in complete metric space. As an application, we present an example to illustrate the main results.

Meyer wavelet neural networks to solve a novel design of fractional order pantograph Lane-Emden differential model

The aim of this study is to design a singular fractional order pantograph differential model by using the typical form of the Lane-Emden model. The necessary details of the singular-point, fractional order and shape factor of the designed model are also provided. The numerical solutions of the designed model have been presented using the combination of the fractional Meyer wavelet (FMW) neural networks (NNs) modeling and optimization of global search with genetic algorithm (GA) supported with local search of sequential quadratic programming (SQP), i.e., FMWNN-GASQP. The strength of FMWNN is employed to design an objective function using the differential model along with its initial conditions of the singular fractional order pantograph model. The optimization of this objective function is performed using the integrated competence of GA-SQP. The verification, perfection and authentication of the singular fractional order pantograph model using fractional Meyer computing solver is observed for different cases through comparative studies from the available exact solutions which endorsed its robustness, convergence and stability. Moreover, the statistics observation with necessary explanations further authenticate the performance of the FMWNN-GASQP in terms of accuracy and reliability.

Disturbance estimator based dynamic compensator design for fractional order fuzzy control systems

The robust stabilization problem for singular fractional order time delay T-S fuzzy systems with nonlinearities and unknown external disturbances is addressed in this paper. An equivalent-input-disturbance (EID) estimator is used to estimate the impact of external disturbances and nonlinearities on the system output. Based on this EID approach, a dynamic compensator is designed to solve the stabilization problem of the considered system. Moreover, by considering a relevant Lyapunov-Krasovskii functional candidate and by using Lyapunov technique, the stability conditions in terms of LMIs are acquired for the considered closed-loop system. At last, to validate the effectiveness of the proposed result, two numerical examples are provided.

A Universal Framework of the Generalized Kalman–Yakubovich–Popov Lemma for Singular Fractional-Order Systems

The well-known generalized Kalman–Yakubovich–Popov lemma is widely used in system analysis and synthesis. However, the corresponding theory for singular systems, especially singular fractional-order systems (SFOSs), is lacking. Therefore, many control problems for this type of systems cannot be optimized in limited frequency ranges. In this article, a universal framework of the finite frequency band generalized Kalman–Yakubovich–Popov lemma for SFOSs is established, the bounded real lemma in the sense of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L_{\infty }}$ </tex-math></inline-formula> -norm is derived for different frequency ranges, and the corresponding controller is designed to improve the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L_{\infty } }$ </tex-math></inline-formula> performance index of SFOSs. Three illustrative examples are given to demonstrate the correctness and effectiveness of the theoretical results.

Proportional Plus Derivative State Feedback Control of Takagi-Sugeno Fuzzy Singular Fractional Order Systems

Proportional Plus Derivative State Feedback Control of Takagi-Sugeno Fuzzy Singular Fractional Order Systems

State and Output Feedback Controller Design of Takagi-Sugeno Fuzzy Singular Fractional Order Systems

This paper investigates the admissibility issue of a class of singular fractional order nonlinear systems based on Takagi-Sugeno (T-S) fuzzy model. First, the fuzzy model and the state and output feedback controllers share the different membership functions. Then, we propose the methods about designing the fuzzy state and output feedback controllers, which guarantee the asymptotical stability of closed-loop systems. Next, the equality constraint existing in most of admissibility theorems is eliminated and strict linear matrix inequalities (LMIs) are obtained. Finally, two numerical simulation examples are given to illustrate the effectiveness of the proposed methods.

Time Domain Solution Analysis and Novel Admissibility Conditions of Singular Fractional-Order Systems

This paper investigates the regularity, nonimpulsiveness, stability and admissibility of the singular fractional-order systems with the fractional-order α ∈ (0, 1). Firstly, the structure, existence and uniqueness of the time domain solutions of singular fractional-order systems are analyzed based on the Kronecker equivalent standard form. The necessary and sufficient condition for the regularity of singular fractional-order systems is proposed on the basis of the above analysis. Secondly, the necessary and sufficient conditions of non-impulsiveness as well as stability are obtained based on the proposed time domain solutions of singular fractional-order systems, respectively. Thirdly, two novel sufficient and necessary conditions for the admissibility of singular fractional-order systems are derived including the non-strict linear matrix inequality form and the linear matrix inequality form with equality constraints. Finally, two numerical examples are given to show the effectiveness of the proposed results.

Adaptive Neural Network Sliding Mode Control for Nonlinear Singular Fractional Order Systems with Mismatched Uncertainties

This paper focuses on the sliding mode control (SMC) problem for a class of uncertain singular fractional order systems (SFOSs). The uncertainties occur in both state and derivative matrices. A radial basis function (RBF) neural network strategy was utilized to estimate the nonlinear terms of SFOSs. Firstly, by expanding the dimension of the SFOS, a novel sliding surface was constructed. A necessary and sufficient condition was given to ensure the admissibility of the SFOS while the system state moves on the sliding surface. The obtained results are linear matrix inequalities (LMIs), which are more general than the existing research. Then, the adaptive control law based on the RBF neural network was organized to guarantee that the SFOS reaches the sliding surface in a finite time. Finally, a simulation example is proposed to verify the validity of the designed procedures.