Robust stability region of fractional order PIλ controller for fractional order interval plant

In this article, by using the fractional order PI^λ controller, we propose a simple and effective method to compute the robust stability region for the fractional order linear time-invariant plant with interval type uncertainties in both fractional orders and relevant coefficients. The presented method is based on decomposing the fractional order interval plant into several vertex plants using the lower and upper bounds of the fractional orders and relevant coefficients and then constructing the characteristic quasi-polynomial of each vertex plant, in which the value set of vertex characteristic quasi-polynomial in the complex plane is a polygon. The D-decomposition method is used to characterise the stability boundaries of each vertex characteristic quasi-polynomial in the space of controller parameters, which can obtain the stability region by varying λ orders in the range (0,?2). These regions of each vertex plant are computed by using three stability boundaries: real root boundary (RRB), complex root boundary (CRB) and infinite root boundary (IRB). The method gives the explicit formulae corresponding to these boundaries in terms of fractional order PI^λ controller parameters. Thus, the robust stability region for fractional order interval plant can be obtained by intersecting stability region of each vertex plant. The robustness of stability region is tested by the value set approach and zero exclusion principle. Our presented technique does not require sweeping over the parameters and also does not need linear programming to solve a set of inequalities. It also offers several advantages over existing results obtained in this direction. The method in this article is useful for analysing and designing the fractional order PI^λ controller for the fractional order interval plant. An example is given to illustrate this method.     

Research progress of the fractional Fourier transform in signal processing

While solving a heat conduction problem in 1807, a French scientist Jean Baptiste Jo-seph Fourier, suggested the usage of the Fourier theorem. Thereafter, the Fourier trans-form (FT) has been applied widely in many scientific disciplines, and has played i…     

On linear stability of predictor–corrector algorithms for fractional differential equations

This paper deals with the numerical approximation of differential equations of fractional order by means of predictor–corrector algorithms. A linear stability analysis is performed and the stability regions of different methods are compared. Furthermore the effects on stability of multiple corrector iterations are verified.     

Legendre multiwavelet functions for numerical solution of multi-term time-space convection–diffusion equations of fractional order

Engineering with Computers - In this study, the Ritz–Galerkin method based on Legendre multiwavelet functions is introduced to solve multi-term time-space convection–diffusion equations...     

Stability of standing waves for the fractional Schrödinger–Choquard equation

In this paper, we consider the stability of standing waves for the fractional Schrödinger–Choquard equation with an $L^2$-critical nonlinearity. By using the profile decomposition of bounded sequences in $H^s$ and variational methods, we prove that the standing waves are orbitally stable. We extend the study of Bhattarai for a single equation (Bhattarai, 2017) to the $L^2$-critical case.     

Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation

Engineering with Computers - This paper introduces a new version for the nonlinear Ginzburg–Landau equation derived from fractal–fractional derivatives and proposes a computational...     

Effect of fractional parameter on plane waves of generalized magneto–thermoelastic diffusion with reference temperature-dependent elastic medium

The present paper is concerned with the investigation of disturbances in a homogeneous, isotropic reference temperature-dependent elastic medium with fractional order generalized thermoelastic diffusion. The formulation is applied to the generalized thermoelasticity based on the fractional time derivatives under the effect of diffusion. The analytical expressions for displacement components, stresses, temperature field, concentration and chemical potential are obtained in the physical domain by using the normal mode analysis technique. These expressions are calculated numerically for a copper-like material and depicted graphically. Effect of fractional parameter and presence of diffusion is analyzed theoretically and numerically. Comparisons are made with the results predicted by the fractional and without fractional order in the presence and absence of diffusion.     

Homotopy analysis method for solving multi-term linear and nonlinear diffusion–wave equations of fractional order

In this paper we have used the homotopy analysis method (HAM) to obtain solutions of multi-term linear and nonlinear diffusion–wave equations of fractional order. The fractional derivative is described in the Caputo sense. Some illustrative examples have been presented.     

Simulation of a viscoelastic flexible multibody system using absolute nodal coordinate and fractional derivative methods

This paper employs a new finite element formulation for dynamics analysis of a viscoelastic flexible multibody system. The viscoelastic constitutive equation used to describe the behavior of the system is a three-parameter fractional derivative model. Based on continuum mechanics, the three-parameter fractional derivative model is modified and the proposed new fractional derivative model can reduce to the widely used elastic constitutive model, which meets the continuum mechanics law strictly for pure elastic materials. The system equations of motion are derived based on the absolute nodal coordinate formulation (ANCF) and the principle of virtual work, which can relax the small deformation assumption in the traditional finite element implementation. In order to implement the viscoelastic model into the absolute nodal coordinate, the Grünwald definition of the fractional derivative is employed. Based on a comparison of the HHT-I3 method and the Newmark method, the HHT-I3 method is used to solve the equations of motion. Another particularity of the proposed method based on the ANCF method lies in the storage of displacement history only during the integration process, reducing the numerical computation considerably. Numerical examples are presented in order to analyze the effects of the truncation number of the Grünwald series (fading memory phenomena) and the value of several fractional model parameters and solution convergence aspects. An erratum to this article can be found at     

Distributed solving Sylvester equations with fractional order dynamics

This paper addresses distributed computation Sylvester equations of the form AX + XB = C with fractional order dynamics. By partitioning parameter matrices A, B and C, we transfer the problem of distributed solving Sylvester equations as two distributed optimization models and design two fractional order continuous-time algorithms, which have more design freedom and have potential to obtain better convergence performance than that of the existing first order algorithms. Then, rewriting distributed algorithms as corresponding frequency distributed models, we design Lyapunov functions and prove that the proposed algorithms asymptotically converge to an exact or least squares solution. Finally, we validate the effectiveness of the proposed algorithms by providing a numerical example     

Online tuning of derivative order term of variable-order fractional proportional–integral–derivative controllers for the first-order time delay systems

In this study, an online tuning strategy for the fractional derivative order term of the variable-order fractional proportional–integral–derivative (PID) controller is proposed for processes with dead time. The classical step response is divided into regions, and meta-rules are developed for each region in order to improve the control performance. To achieve the goals of the meta-rules, a set of equations that are the functions of absolute error and model parameters are proposed to manipulate the fractional order derivative during the process. These equations can handle the changes in model parameters since the coefficients of these equations are functions of model parameters. On both simulation studies and experimental results on the active suspension system, we show that the proposed method improves the time domain performance criteria both in relation to reference tracking and load disturbance rejection. Moreover, the robustness of the proposed method has also been tested and analyzed for the dead time variation within the process.     

ε-Optimality and duality for multiobjective fractional programming

Using the scalar ε-parametric approach, we establish the Karush-Kuhn-Tucker (which we call KKT) necessary and sufficient conditions for an ε-Pareto optimum of nondifferentiable multiobjective fractional objective functions subject to nondifferentiable convex inequality constraints, linear equality constraints, and abstract constraints. These optimality criteria are utilized as a basis for constructing one duality model with appropriate duality theorems. Subsequently, we employ scalar exact penalty function to transform the multiobjective fractional programming problem to an unconstrained problem. Under this case, we derive the KKT necessary and sufficient conditions without a constraint qualification for ε-Pareto optimality of multiobjective fractional programming.     

Existence of ground state solutions of Nehari–Pohozaev type for fractional Schrödinger–Poisson systems with a general potential

In this paper, we consider the following fractional Schrödinger–Poissonproblem

where $0<s\le t<1$ and $2s+2t>3$, the potential $V\left(x\right)$ is weakly differentiable and $f\in C(\mathbb{R},\mathbb{R})$. By introducing some new tricks, we prove that the problem admits a ground state solution of Nehari–Pohozaev type under mild assumptions on $V$ and $f$. The results here extend the existing study.     

Robustness analysis and design of fractional order Iλ Dμ controllers using the small gain theorem

ABSTRACT

In this paper, a simple method is proposed to tune the parameters of Fractional Integral-Fractional Derivative (FIFE) I^λ D^μ controllers based on the Bode diagram. The proposed technique provides a practical approach for tuning FIFE controllers to compensate stable plants. Using the small gain theorem and based on the sensitivity functions analysis, it is proved that by applying the designed FIFE controller the robustness of the compensated system in the presence of plant uncertainties is improved in comparison to the PI controller in a similar structure. Moreover, the closed-loop phase margin and gain crossover frequency are adjustable by tuning the free controller parameters. Simulation results are presented to demonstrate the simplicity of application and effectiveness of the tuned controller.