Enhancing Iterative Learning Control With Fractional Power Update Law

The P-type update law has been the mainstream technique used in iterative learning control(ILC)systems,which resembles linear feedback control with asymptotical convergence.In recent years,finite-time control strategies such as terminal sliding mode control have been shown to be effective in ramping up convergence speed by introducing fractional power with feedback.In this paper,we show that such mechanism can equally ramp up the learning speed in ILC systems.We first propose a fractional power update rule for ILC of single-input-single-output linear systems.A nonlinear error dynamics is constructed along the iteration axis to illustrate the evolutionary converging process.Using the nonlinear mapping approach,fast convergence towards the limit cycles of tracking errors inherently existing in ILC systems is proven.The limit cycles are shown to be tunable to determine the steady states.Numerical simulations are provided to verify the theoretical results.

The Fractional Power Law of Wind Wave Growth in Deep Water for Short Fetch

Combining the 3/2 power law proposed by Toba with the significant wave energy balance equation for wind waves, wave growth in deep water for short fetch is investigated. It is found that the variations of wave height and period with fetch have the form of power function with fractional exponents 3/8 and 1/4 respectively. Using these exponents in the power functions and through data fitting, the concise wind wave growth relations for short fetch are obtained.

Homotopy Analysis Method for a Conservative Nonlinear Oscillator with Fractional Power

In this paper, the homotopy analysis method is applied to deduce the periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to u1/3. By introducing the auxiliary linear operator and the initial guess of solution, the homotopy analysis solving is set up. By choosing the suitable convergence-control parameter, the accurate high-order approximations of solution and frequency for the whole range of initial amplitudes can easily be obtained. Comparison of the results obtained using this method with those obtained by different methods reveals that the former is more accurate, effective and convenient for these types of nonlinear oscillators.

Exact Solitary Wave Solutions of Nonlinear Evolution Equations with a Positive Fractional Power Term

The bounded and smooth solitary wave solutions of 10 nonlinear evolution equations with a positive fractional power term of dependent variable are successfully obtained by homogeneous balance principle and with the aid of sub-ODEs that admits a solution of sech-power or tanh-power type.In the special cases that the fractional power equals to 1 and 2,the solitary wave solutions of more than 10 important model equations arisen from mathematical physics are easily rediscovered.

A Direct Method of Moving Planes to Fractional Power Sub Laplace Equations on the Heisenberg Group

We give the direct method of moving planes for solutions to the conformally invariant fractional power sub Laplace equation on the Heisenberg group.The method is based on four maximum principles derived here.Then symmetry and nonexistence of positive cylindrical solutions are proved.

Existence and uniqueness of S-asymptotically periodic α-mild solutions for neutral fractional delayed evolution equation

In this paper, we investigate a class of abstract neutral fractional delayed evolution equation in the fractional power space. With the aid of the analytic semigroup theories and some fixed point theorems, we establish the existence and uniqueness of the S-asymptotically periodic α-mild solutions. The linear part generates a compact and exponentially stable analytic semigroup and the nonlinear parts satisfy some conditions with respect to the fractional power norm of the linear part, which greatly improve and generalize the relevant results of existing literatures.

The Fractional Investigation of Some Nonlinear Partial Differential Equations by Using an Efficient Procedure

The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo operator with Laplace residual power seriesmethod.It is found that the present technique has a direct and simple implementation to solve the targeted problems.The comparison of the obtained solutions has been done with actual solutions to the problems.The fractional-order solutions are presented and considered to be the focal point of this research article.The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem.Because of the simple implementation,the present technique can be extended to solve other important fractional order problems.

WEAK REGULARIZATION FOR A CLASS OF ILL-POSED CAUCHY PROBLEMS

This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contained in a sector of right-half complex plane and its resolvent is polynomially bounded, the weak regularization for such ill-posed Cauchy problem can be shown by using the quasi-reversibilky method and regularized semigroups. Finally, an example is given.

Finite Element Approach for the Solution of First-Order Differential Equations

The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differential equation into a system of algebraic equations by application of the method of weighted residuals in conjunction with a finite element ansatz. However, this procedure is restricted to even-ordered differential equations and leads to symmetric system matrices as a key property of the finite element method. This paper aims in a generalization of the finite element method towards the solution of first-order differential equations. This is achieved by an approach which replaces the first-order derivative by fractional powers of operators making use of the square root of a Sturm-Liouville operator. The resulting procedure incorporates a finite element formulation and leads to a symmetric but dense system matrix. Finally, the scheme is applied to the barometric equation where the results are compared with the analytical solution and other numerical approaches. It turns out that the resulting numerical scheme shows excellent convergence properties.

On the Strongly Damped Wave Equations with Critical Nonlinearities

We study the strongly damped wave equations with critical nonlinearities. By choosing suitable state spaces, we prove sectorial property of the operator matrix together with its adjoint operator, investigate the associated interpolation and extrapolation spaces, analysis the criticality of the nonlinearity with critical growth, and study the higher spatial regularity of the Y-regular solution by bootstrapping.

Adaptive fractional integral terminal sliding mode power control of UPFC in DFIG wind farm penetrated multimachine power system

With an aim to improve the transient stability of a DFIG wind farm penetrated multimachine power system(MPN),an adaptive fractional integral terminal sliding mode power control(AFITSMPC)strategy has been proposed for the unified power flow controller(UPFC),which is compensating the MPN.The proposed AFITSMPC controls the dq-axis series injected voltage,which controls the admittance model(AM)of the UPFC.As a result the power output of the DFIG stabilizes which helps in maintaining the equilibrium between the electrical and mechanical power of the nearby generators.Subsequently the rotor angular deviation of the respective generators gets recovered,which significantly stabilizes the MPN.The proposed AFITSMPC for the admittance model of the UPFC has been validated in a DFIG wind farm penetrated 2 area 4 machine power system in the MATLAB environment.The robustness and efficacy of the proposed control strategy of the UPFC,in contrast to the conventional PI control is vindicated under a number of intrinsic operating conditions,and the results analyzed are satisfactory.

Next generation of power electronic-converter application for energy-conversion and storage units and systems

By a rearrangement of the traditional supply-converter-load system connection,partial-power-processing-based converters can be used to achieve a reduction in size and cost,increase in system efficiency and lower device power rating.The concept is promising for different applications such as photovoltaic arrays,electric vehicles and electrolysis.For photovoltaic applications,it can drive each cell in the array to its maximum power point with a relatively smaller converter;for electric-vehicle applications,both an onboard charger with reduced weight and improved efficiency as well as a fast charger station handling higher power can be considered.By showing different examples of partial-power-processing application for energy-conversion and storage units and systems,this paper discusses key limitations of partial-power-processing and related improvements from different perspectives to show the potential in future power electronic systems.

Non-Existence of Global Solutions for a Fractional Wave-Diffusion Equation

We considered the Cauchy problem for the fractional wave-diffusion equation D^αu-△｜u｜^m-1u＋（-△）^β/2D^γ｜u｜^l-1u=h（x,t）｜u｜^p＋f（x,t）with given initial data and where p〉1,1〈α〈2,0〈β〈2,0〈γ〈1.Nonexistence results and necessary conditions for global existence are established by means of th, test function method. This results extend previous works.

Maximal Attractors for the m-Dimensional Cahn-Hilliard System

In this paper we discuss maximal attractors of the m-dimensional Cahn-Hilliard System in the product spaces (L 2(?)) m and (H 2(?)) m in terms of D. Henry’s general theory and from the viewpoint of compactness and absorptivity of semigroups as R. Temam did. After giving the existence and uniqueness of global solutions, we technically restrict our discussion to some subspaces, give estimates with a new graph norm, and obtain the existence of maximal attractors and some properties of them.

Approximate Controllability of Neutral Functional Differential Systems with State-Dependent Delay

This paper deals with the approximate controllability of semilinear neutral functional differential systems with state-dependent delay. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can apply to the systems involving derivatives of spatial variables. By methods of functional analysis and semigroup theory, sufficient conditions of approximate controllability are formulated and proved. Finally, an example is provided to illustrate the applications of the obtained results.