Classical Integer Research Articles

Fractional Dynamics of Computer Virus Propagation

We propose a fractional model for computer virus propagation. The model includes the interaction between computers and removable devices. We simulate numerically the model for distinct values of the order of the fractional derivative and for two sets of initial conditions adopted in the literature. We conclude that fractional order systems reveal richer dynamics than the classical integer order counterpart. Therefore, fractional dynamics leads to time responses with super-fast transients and super-slow evolutions towards the steady-state, effects not easily captured by the integer order models.

A Domain Decomposition Method for Time Fractional Reaction-Diffusion Equation

The computational complexity of one-dimensional time fractional reaction-diffusion equation is O(N 2 M) compared with O(NM) for classical integer reaction-diffusion equation. Parallel computing is used to overcome this challenge. Domain decomposition method (DDM) embodies large potential for parallelization of the numerical solution for fractional equations and serves as a basis for distributed, parallel computations. A domain decomposition algorithm for time fractional reaction-diffusion equation with implicit finite difference method is proposed. The domain decomposition algorithm keeps the same parallelism but needs much fewer iterations, compared with Jacobi iteration in each time step. Numerical experiments are used to verify the efficiency of the obtained algorithm.

Fractional Modeling of the AC Large-Signal Frequency Response in Magnetoresistive Current Sensors

Fractional calculus is considered when derivatives and integrals of non-integer order are applied over a specific function. In the electrical and electronic domain, the transfer function dependence of a fractional filter not only by the filter order n, but additionally, of the fractional order α is an example of a great number of systems where its input-output behavior could be more exactly modeled by a fractional behavior. Following this aim, the present work shows the experimental ac large-signal frequency response of a family of electrical current sensors based in different spintronic conduction mechanisms. Using an ac characterization set-up the sensor transimpedance function Zt(if) is obtained considering it as the relationship between sensor output voltage and input sensing current, . The study has been extended to various magnetoresistance sensors based in different technologies like anisotropic magnetoresistance (AMR), giant magnetoresistance (GMR), spin-valve (GMR-SV) and tunnel magnetoresistance (TMR). The resulting modeling shows two predominant behaviors, the low-pass and the inverse low-pass with fractional index different from the classical integer response. The TMR technology with internal magnetization offers the best dynamic and sensitivity properties opening the way to develop actual industrial applications.

Solution to automatic generation control problem using firefly algorithm optimized IλDµ controller

Present work focused on automatic generation control (AGC) of a three unequal area thermal systems considering reheat turbines and appropriate generation rate constraints (GRC). A fractional order (FO) controller named as IλDµ controller based on crone approximation is proposed for the first time as an appropriate technique to solve the multi-area AGC problem in power systems. A recently developed metaheuristic algorithm known as firefly algorithm (FA) is used for the simultaneous optimization of the gains and other parameters such as order of integrator (λ) and differentiator (μ) of IλDµ controller and governor speed regulation parameters (R). The dynamic responses corresponding to optimized IλDµ controller gains, λ, μ, and R are compared with that of classical integer order (IO) controllers such as I, PI and PID controllers. Simulation results show that the proposed IλDµ controller provides more improved dynamic responses and outperforms the IO based classical controllers. Further, sensitivity analysis confirms the robustness of the so optimized IλDµ controller to wide changes in system loading conditions and size and position of SLP. Proposed controller is also found to have performed well as compared to IO based controllers when SLP takes place simultaneously in any two areas or all the areas. Robustness of the proposed IλDµ controller is also tested against system parameter variations.

Uncertain multi-product newsboy problem with chance constraint

A multi-product newsboy problem with uncertain demand and uncertain storage space is investigated under a warehousing chance constraint. It is assumed that indeterminacy may appear in demand and storage space due to rough estimation of their distributions by decision-maker. Uncertainty theory provides a new tool to deal with the human uncertainty. We develop an uncertain expected value programming model (UEVPM) based on uncertainty theory. Uncertain variables (functions from uncertainty space to the set of real numbers) are used to describe subjective estimation. The objective function is to maximize the expected profit of newsboy at a predetermined confidence level. Furthermore, we convert the chance constrained uncertain programming model to its equivalent deterministic form so as to be solved by classic integer programming method. Finally, three numerical examples are given to illustrate the potential usefulness of the study.

On Estimation of the Bivariate Poisson INAR Process

In a recent article, Pedeli and Karlis (2010) examined the extension of the classical Integer–valued Autoregressive (INAR) model to the bivariate case. In the present article, we examine estimation methods for the case of bivariate Poisson innovations. This is a simple extension of the classical INAR model allowing for two discrete valued time series to be correlated. Properties of different estimators are given. We also compare their properties via a small simulation experiment. Extensions to incorporate covariate information is discussed. A real data application is also provided.

Fractional models for modeling complex linear systems under poor frequency resolution measurements

When modeling a linear system in a parametric way, one needs to deal with (i) model structure selection, (ii) model order selection as well as (iii) an accurate fit of the model. The most popular model structure for linear systems has a rational form which reveals crucial physical information and insight due to the accessibility of poles and zeros. In the model order selection step, one needs to specify the number of poles and zeros in the model. Automated model order selectors like Akaikeʼs Information Criterion (AIC) and the Minimum Description Length (MDL) are popular choices. A large model order in combination with poles and zeros lying closer to each other in frequency than the frequency resolution indicates that the modeled system exhibits some fractional behavior. Classical integer order techniques cannot handle this fractional behavior due to the fact that the poles and zeros are lying to close to each other to be resolvable and not enough data is available for the classical integer order identification procedure. In this paper, we study the use of fractional order poles and zeros and introduce a fully automated algorithm which (i) estimates a large integer order model, (ii) detects the fractional behavior, and (iii) identifies a fractional order system. ► Complexity reduction integer order models. ► Automated fractional order model identification. ► Fractional order model selection. ► Dealing with high order systems. ► Dealing with closely spaced poles and zeros.

New Challenges in Fractional Systems

Without any doubt, the recently emerging tools from fractional calculus became successful in a manifold of applications and currently the playground of modern engineering sciences. Fractional order differentiation consists in the generalisation of classical integer differentiation to real or complex orders. From a mathematical point of view, several interpretations of fractional differentiation were proposed, but there is still a deep debate about it. However, all these interpretations demonstrate that fractional order differentiation cannot simply be connected to the slope at one point of the derived function for instance. This lack of interpretation is in fact due to the definition of the fractional order operator. This is a nonlocal operator based on an integral with a singular kernel. The same conclusion can be made for the fractional integrator operator; fractional differentiation operator definition being based on the fractional integrator operator definition. This situation explains why these operators are still not well defined and that several definitions still coexist, which impedes the process of becoming standard tools. Since the first recorded referencework in 1695 up to the present day, many articles have been published on this subject, but much progress is still to be done particularly on the relationship of these different definitions with the physical reality of a system (through taking into account the initial conditions for instance). A fractional order system is a system described by an integro-differential equation involving fractional order derivatives of its input(s) and/or output(s). From a physical point of view, linear fractional order systems are not quite conventional linear systems, and not quite conventional distributed parameter systems. They are in fact halfway between these two classes of systems. Fractional order systems exhibit long memory or hereditary effects. Hence, they are a modelling tool well suited to a wide class of phenomena with nonstandard dynamic behaviour and the applications of fractional order systems are now well accepted in the following disciplines:

New developments in the primal–dual column generation technique

The optimal solutions of the restricted master problems typically leads to an unstable behavior of the standard column generation technique and, consequently, originates an unnecessarily large number of iterations of the method. To overcome this drawback, variations of the standard approach use interior points of the dual feasible set instead of optimal solutions. In this paper, we focus on a variation known as the primal–dual column generation technique which uses a primal–dual interior point method to obtain well-centered non-optimal solutions of the restricted master problems. We show that the method converges to an optimal solution of the master problem even though non-optimal solutions are used in the course of the procedure. Also, computational experiments are presented using linear-relaxed reformulations of three classical integer programming problems: the cutting stock problem, the vehicle routing problem with time windows, and the capacitated lot sizing problem with setup times. The numerical results indicate that the appropriate use of a primal–dual interior point method within the column generation technique contributes to a reduction of the number of iterations as well as the running times, on average. Furthermore, the results show that the larger the instance, the better the relative performance of the primal–dual column generation technique.

Individualization of a pharmacokinetic model by fractional and nonlinear fit improvement

This study presents application of a new linear and nonlinear fractional derivative two compartmental model to the evaluation of individual pharmacokinetics. In the model, the integer order derivatives are replaced by derivatives of real order. A specific nonlinear function is used for the fit improvement of a fractional derivative two compartmental model with the mass balance conservation. The agreement of the values predicted by the proposed model with the values obtained through experiments with bumetanide tablets in human volunteers is shown to be good. Thus, pharmacokinetics of bumetanide can be described well by a linear or a nonlinear two compartmental model with fractional derivatives of the same order proposed here. Parameters in the model are determined by the least squares method and the particle swarm optimization numerical procedure is used. The results show that the linear fractional order two compartmental model for bumetanide is useful improvement of the classical (integer order) two compartmental model and that the nonlinear fractional order model is useful improvement of the linear model in a great number of volunteers.

Special issue: Fractional signals and systems

Fractional order differentiation is a generalisation of classical integer differentiation to real or complex orders. From a mathematical point of view, several interpretations of fractional differentiation were proposed: all these interpretations demonstrate that fractional order differentiation cannot simply be connected to the local behaviour of the derivative function. This lack of simple interpretation is in fact due to the definition of the fractional order operator. This is a non-local operator based on an integral with a singular kernel. The same conclusion can be made for the fractional integrator operator, fractional differentiation operator definition being based on the fractional integrator operator definition. This situation explains why several definitions still coexist. Since the first recorded reference work in 1695 up to the present day, many articles have been published on this subject, but much progress still to be done particularly on the relationship of these different definitions with the physical reality of a system (through taking into account the initial conditions for instance). A fractional order system is a system described by an integro-differential equation involving fractional order

Robustness evaluation of fractional order control for varying time delay processes

In this paper, we investigate the robustness of a methodology to design fractional order PI controllers combined with Smith Predictors, for varying time delay processes. To overcome the drawback of possible instability associated with Smith Predictor control structures, mainly due to the changes in the time delay, the design focuses on ensuring robustness of the closed loop system against time delay uncertainties. The proposed method is based on time-domain performance specifications—more accessible to the process engineer, rather than the more abstract notions related to the frequency domain. A second advantage of the proposed method relies on additional robustness to plant uncertainties, achieved by maximizing open-loop gain margin. The convergence problems associated with optimization techniques, previously used in fractional order controller designs, are eliminated by an iterative procedure in computing the gain margin. The simulation example provided demonstrates the efficiency of the proposed method, in comparison to classical integer order PI controller.

Mixed Integer Evolution Strategies for Parameter Optimization

Evolution strategies (ESs) are powerful probabilistic search and optimization algorithms gleaned from biological evolution theory. They have been successfully applied to a wide range of real world applications. The modern ESs are mainly designed for solving continuous parameter optimization problems. Their ability to adapt the parameters of the multivariate normal distribution used for mutation during the optimization run makes them well suited for this domain. In this article we describe and study mixed integer evolution strategies (MIES), which are natural extensions of ES for mixed integer optimization problems. MIES can deal with parameter vectors consisting not only of continuous variables but also with nominal discrete and integer variables. Following the design principles of the canonical evolution strategies, they use specialized mutation operators tailored for the aforementioned mixed parameter classes. For each type of variable, the choice of mutation operators is governed by a natural metric for this variable type, maximal entropy, and symmetry considerations. All distributions used for mutation can be controlled in their shape by means of scaling parameters, allowing self-adaptation to be implemented. After introducing and motivating the conceptual design of the MIES, we study the optimality of the self-adaptation of step sizes and mutation rates on a generalized (weighted) sphere model. Moreover, we prove global convergence of the MIES on a very general class of problems. The remainder of the article is devoted to performance studies on artificial landscapes (barrier functions and mixed integer NK landscapes), and a case study in the optimization of medical image analysis systems. In addition, we show that with proper constraint handling techniques, MIES can also be applied to classical mixed integer nonlinear programming problems.

A conformal mapping based fractional order approach for sub-optimal tuning of PID controllers with guaranteed dominant pole placement

A novel conformal mapping based fractional order (FO) methodology is developed in this paper for tuning existing classical (Integer Order) Proportional Integral Derivative (PID) controllers especially for sluggish and oscillatory second order systems. The conventional pole placement tuning via Linear Quadratic Regulator (LQR) method is extended for open loop oscillatory systems as well. The locations of the open loop zeros of a fractional order PID (FOPID or PIλDμ) controller have been approximated in this paper vis-à-vis a LQR tuned conventional integer order PID controller, to achieve equivalent integer order PID control system. This approach eases the implementation of analog/digital realization of a FOPID controller with its integer order counterpart along with the advantages of fractional order controller preserved. It is shown here in the paper that decrease in the integro-differential operators of the FOPID/PIλDμ controller pushes the open loop zeros of the equivalent PID controller towards greater damping regions which gives a trajectory of the controller zeros and dominant closed loop poles. This trajectory is termed as “M-curve”. This phenomena is used to design a two-stage tuning algorithm which reduces the existing PID controller’s effort in a significant manner compared to that with a single stage LQR based pole placement method at a desired closed loop damping and frequency.

Comment to the discussion

In this paper, fractional order system observability is discussed. A representation of these systems that involves a classical linear integer system and a system described by a parabolic equation is used to define the system real state and to conclude that the system state is approximately observable. However, it is also shown that the pseudo state space representation, usually encountered in the literature for fractional systems, can be used to design Luenberger like observers that permits an estimation of important variables in the system. Such an observer is finally used in order to estimate the temperature in a thermal system from a temperature measure at a different abscissa.

On Observability and Pseudo State Estimation of Fractional Order Systems

In this paper, fractional order system observability is discussed. A representation of these systems that involves a classical linear integer system and a system described by a parabolic equation is used to define the system real state and to conclude that the system state is approximately observable. However, it is also shown that the pseudo state space representation, usually encountered in the literature for fractional systems, can be used to design Luenberger like observers that permits an estimation of important variables in the system. Such an observer is finally used in order to estimate the temperature in a thermal system from a temperature measure at a different abscissa.

Application of Assignment Model in PE Human Resources Allocation

The definition of Assignment Model and the Hungarian Method are introduced in this paper and through cases, the application of Assignment Model is elaborated. The Assignment Model is a classic integer linear programming model of 0-1 and it is widely applied in dealing with assignment allocation, personnel selection, the programming of transport system and other practical issues. In the field of PE, the point of assignment gives full play in selecting the proper athletes, assigning tasks and even in recruiting new staff by the human resources, so it is significant to introduce the assignment model to the sports system. However, it still remains a vacuum in applying the assignment model with cases in sports management. This paper applies the assignment model with cases to the sports management, which will provide theoretical foundation for the policy-makers to allocate human resources.

Fractional Order Modeling of a PHWR Under Step-Back Condition and Control of Its Global Power With a Robust ${\rm PI}^{\lambda} {\rm D} ^{\mu}$ Controller

Bulk reduction of reactor power within a small finite time interval under abnormal conditions is referred to as step-back. In this paper, a 500MWe Canadian Deuterium Uranium (CANDU) type Pressurized Heavy Water Reactor (PHWR) is modeled using few variants of Least Square Estimator (LSE) from practical test data under a control rod drop scenario in order to design a control system to achieve a dead-beat response during a stepped reduction of its global power. A new fractional order (FO) model reduction technique is attempted which increases the parametric robustness of the control loop due to lesser modeling error and ensures iso-damped closed loop response with a PI{\lambda}D{\mu} or FOPID controller. Such a controller can, therefore, be used to achieve active step-back under varying load conditions for which the system dynamics change significantly. For closed loop active control of the reduced FO reactor models, the PI{\lambda}D{\mu} controller is shown to perform better than the classical integer order PID controllers and present operating Reactor Regulating System (RRS) due to its robustness against shift in system parameters.

An efficient integer programming formulation for the assignment of base stations to controllers in cellular networks

In mobile networks, the assignment of base stations to controllers when planning the network has a strong impact on network performance. In a previous paper, the authors formulated the assignment of base stations to packet controllers in GSM-EDGE Radio Access Network (GERAN) as a graph partitioning problem, which was solved by a heuristic method. In this paper, an exact method is presented to find optimal solutions that can be used as a benchmark. The proposed method is based on an effective re-formulation of the classical integer linear programming model of the graph partitioning problem, which is solved by the branch-and-cut algorithm in a commercial optimization package. Performance assessment is based on an extensive set of problem instances built from data of a live network. Preliminary analysis shows some properties of the graphs in this problem justifying the limitations of heuristic approaches and the need for more sophisticated methods. Results show that the proposed method outperforms classical heuristic algorithms used for benchmarking, even under runtime constraints. Likewise, it improves the efficiency of exact methods previously applied to similar problems in the cellular field.

Anycast Routing for Survivable Optical Grids: Scalable Solution Methods and the Impact of Relocation

In this paper, we address the issue of resiliency against single link network failures in optical grids and show how the anycast routing principle, which is typical of grids, can be exploited in providing efficient shared path protection. We investigate two different integer linear program models for the full anycast routing problem, deciding on the primary and backup server locations as well as on the lightpaths toward them. The first model is a classical integer linear programming (ILP) model, which lacks scalability. The second model is a large-scale optimization model which can be efficiently solved using column generation techniques. We also design two new heuristics: the first one is an improvement of a previously proposed one which, although providing near optimal solutions, lacks scalability, while the second one is highly scalable, at the expense of reduced accuracy. Numerical results are presented for three mesh networks with varying node degrees. They allow an illustration of the scalability of the newly proposed approaches. Apart from highlighting the difference in performance (i.e., scalability and optimality) among the algorithms, our case studies demonstrate the bandwidth savings that can be achieved by exploiting relocation rather than using a backup path to the original (failure-free) destination site. Numerical results for varying network topologies, as well as different numbers of server sites show that relocation allows bandwidth savings in the range of 7–21%.