Revisiting four approximation methods for fractional order transfer function implementations: Stability preservation,time and frequency response matching analyses

Due to high computational load of ideal realization of fractional order elements, fractional order transfer functions are commonly implemented via integer-order, limited-band approximate models. An important side effect of such a non-ideal fractional order controller function realization for control applications is that the approximate fractional order models may deteriorate practical performance of optimal control tuning methods. Two major concerns come out for approximate realization in fractional-order control. These are stability preservation and model response matching properties. This study revisits four fundamental fractional order approximation methods, which are Oustaloup's method, CFE method, Matsuda's method and SBL fitting method, and considers stability preservation, time and frequency response matching performances. The study firstly presents a detailed review of Oustaloup's method, CFE method, Matsuda's method. Then, a modified version of SBL fitting method is presented. The stability preservation properties of approximation methods are investigated according to critical root placements of corresponding approximation method. Stability issue is highly significant for control applications. For this reason, a detailed analysis and comparision of stability preservation properties of these four approximation methods are investigated. Moreover, approximate implementations of an optimally tuned FOPID controller function are performed according to these four methods and compared for closed loop control of a large time delay system. Findings of this study indicate a fact that approximate models can considerably influence practical performance of optimally tuned FOPID control systems and ignorance of limitations of approximation methods in optimal tuning solutions can significantly affect real world performances.     

Characterizing Strong Stability Preserving Additive Runge-Kutta Methods

Space discretization of some time dependent partial differential equations give rise to ordinary differential equations containing additive terms with different stiffness properties. In these situations, additive Runge-Kutta (additive RK) methods are used. For additive RK methods the curve of absolute monotonicity gives stepsize restrictions for monotonicity. Necessary conditions for nontrivial curves of absolute monotonicity are the nonnegativity of the additive RK coefficients and some inequalities on some incidence matrices. In this paper we characterize strong stability preserving additive Runge-Kutta methods giving some order barriers and structural properties. Research supported by the Ministerio de Educación y Ciencia, Project MTM2005-03894.     

Identification from step responses with transient initial conditions

Over the last few years there have been some interesting developments in system identification from step and step-like responses. New methods have been proposed to simultaneously estimate model parameters and the delay from step responses that require the process to be at steady state when the step is applied. To deal with transient initial conditions, methods based on pulse responses and the so called piece-wise step tests and iterative algorithm based on a single step response have also been proposed. In this article we propose a new method to estimate model parameters and the delay from a single step response in the presence of transient initial conditions, which does not require iteration. The performance of the proposed procedure is demonstrated through simulations as well as applications on experimental data from a continuous heating tank process and a mixing process.     

Shape-preserving computation in economic growth models

We point out the shape characteristics — monotonicity and concavity — of the value functions of optimal economic growth problems. We introduce the concept of shape preservation in approximating the value functions. We also present a shape-preserving algorithm to compute the solutions of continuous-state optimal economic growth problems. Numerical results show that shape-preserving interpolation methods are superior to others with less-sophisticated interpolation in the sense of smaller approximation errors.Scope and PurposeUnder standard conditions on a risk-averse utility, the value function of an optimal economic growth problem is known to possess the shape characteristics-monotonicity and concavity. As the closed form solutions are rarely available, the only way to solve for the value function is numerically. However, there are no numerical methods which guarantee to preserve the shape features in the course of approximation. In this article, we introduce the usage of shape preservation and present a shape-preserving interpolation in numerical dynamic programming.     

Semi-discrete approximations for stochastic differential equations and applications

In this paper, we propose a new point of view in numerical approximation of stochastic differential equations. By using Ito–Taylor expansions, we expand only a part of the stochastic differential equation. Thus, in each step, we have again a stochastic differential equation which we solve explicitly or by using another method or a finer mesh. We call our approach as a semi-discrete approximation. We give two applications of this approach. Using the semi-discrete approach, we can produce numerical schemes which preserves monotonicity so in our first application, we prove that the semi-discrete Euler scheme converge in the mean square sense even when the drift coefficient is only continuous, using monotonicity arguments. In our second application, we study the square root process which appears in financial mathematics. We observe that a semi-discrete scheme behaves well producing non-negative values.     

Incompressible Navier–Stokes solutions by a triangular spectral/p element projection method

The application of the fractional step projection method recently proposed by Guermond and Quartapelle to the numerical approximation of unsteady Navier–Stokes solutions by means of a spectral/p element method is considered. In particular we illustrate the second-order pressure correction technique and evaluate its accuracy properties in some test cases. Stability with respect to the compatibility condition between the approximation spaces for velocity and pressure is also addressed. The high (spectral) accuracy in space and the second-order accuracy in time are verified by two simple test cases with analytical solution. A more interesting problem is solved showing the ability of the method to produce very accurate results also for problems in complex geometries.     

Proportional stabilization and closed-loop identification of an unstable fractional order process

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.     

Point Monotonicity and Radial Convexity of First Order Approximation Mappings

In non-smooth optimization a central role is performed by the concept which replaces the gradient. Several notions of generalized differentiability have been introduced and, within these frameworks, some attempts have been made to characterize non-differentiable functions, by different concepts of generalized monotonicity related to the generalized convexity of the functions. Our purpose is to approach this topic by some kinds of generalized monotonicity and convexity of the first order approximation mappings for the functions, suitable from theoretical and computational points of view.     

Fractional power approximations of elliptic integrals and bessel functions

In the previous papers [1][3], fractional powers were used to approximate elementary functions and their usefulness was proved with experimental results. In the present paper, some further investigations are reported. That is, elliptic integrals in Legendre's canonical form and Bessel functions are approximated by fractional powers. As the fractional power approximation,

f(x) c₀ + c₁x + c₂x^p

is discussed. When all coefficients c₀, c₁, c₂, p are properly assigned, the accuracy of this approximation becomes comparable to that of the Chebyshev approximation using polynomials up to the third degree.     

Linear approximation of transfer function with a pole of fractional power

Two methods for the linear approximation of a transfer function with a pole of fractional power are presented. Analog circuit models are developed, and their frequency response curves and step response curves are compared. It was found that the Padé method gives a better approximation than Wang and Hsia's method within the frequency limit as specified.     

论非对称选择网的活性与安全性

活性与安全性是网系统的重要行为特性.该文的贡献在于为非对称选择网导出其活性与安全性的一般性质.文章讨论了活性具有单调性的非对称选择网活性与安全性的条件,并给出一个多项式时间算法来判定一个给定的非对称选择网是否是活的、安全的与活性满足单调性.文章还讨论了非对称选择网的两个子类(强化I型与强化II型),并给出活性满足单调性时其(结构)活与(结构)安全的充分条件.     

Weakly Monotonic Averaging Functions

Monotonicity with respect to all arguments is fundamental to the definition of aggregation functions. It is also a limiting property that results in many important nonmonotonic averaging functions being excluded from the theoretical framework. This work proposes a definition for weakly monotonic averaging functions, studies some properties of this class of functions, and proves that several families of important nonmonotonic means are actually weakly monotonic averaging functions. Specifically, we provide sufficient conditions for weak monotonicity of the Lehmer mean and generalized mixture operators. We establish weak monotonicity of several robust estimators of location and conditions for weak monotonicity of a large class of penalty‐based aggregation functions. These results permit a proof of the weak monotonicity of the class of spatial‐tonal filters that include important members such as the bilateral filter and anisotropic diffusion. Our concept of weak monotonicity provides a sound theoretical and practical basis by which (monotonic) aggregation functions and nonmonotonic averaging functions can be related within the same framework, allowing us to bridge the gap between these previously disparate areas of research.     

Viable states for monotone harvest models

This paper deals with the control of discrete-time dynamical, monotone both in the state and in the control, in the presence of state and control monotone constraints. A state x is said to belong to the viability kernel if there exists a trajectory, of states and controls, starting from x and satisfying the constraints. Under monotonicity assumptions, we present upper and lower estimates of the viability kernel. Our motivation comes from harvest models, where some monospecies age class models, as well as specific multi-species models (with so-called technical interactions), exhibit monotonicity properties both in the state and in the control. In this context, constraints represent production and preservation requirements to be satisfied for all time, which also possess monotonicity properties. Our results help delineating domains where a viable management is possible. Numerical applications are given for two Chilean fisheries. We obtain upper bounds for production which are interesting for managers in that they only depend on the model’s parameters, and not on the current stocks.     

Deterministic and Stochastic Fractional Order Model for Lesser Date Moth

In this paper, a deterministic and stochastic fractional order model for lesser date moth (LDM) using mating disruption and natural enemies is proposed and analysed. The interaction between LDM larvae, fertilized LDM female, unfertilized LDM female, LDM male and the natural enemy is investigated. In order to clarify the characteristics of the proposed deterministic fractional order model, the analysis of existence, uniqueness, non-negativity and boundedness of the solutions of the proposed fractional-order model are examined. In addition, some sufficient conditions are obtained to ensure the local and global stability of equilibrium points. The occurrence of local bifurcation near the equilibrium points is investigated with the help of Sotomayor’s theorem. Numerical simulations are conducted to illustrate the properties of the proposed fractional order model with respect to the intrinsic growth rate of the LDM larvae, natural enemy’s mortality rate, predation rate, sex pheromone trap parameter, fractional order and environmental noise. The impact of mating disruption on lesser date moth is demonstrated. Also, a numerical approximation method is developed for the proposed stochastic fractional-order model.     

Some exact tests for manifest properties of latent trait models

Item response theory is one of the modern test theories with applications in educational and psychological testing. Recent developments made it possible to characterize some desired properties in terms of a collection of manifest ones, so that hypothesis tests on these traits can, in principle, be performed. But the existing test methodology is based on asymptotic approximation, which is impractical in most applications since the required sample sizes are often unrealistically huge. To overcome this problem, a class of tests is proposed for making exact statistical inference about four manifest properties: covariances given the sum are non-positive (CSN), manifest monotonicity (MM), conditional association (CA), and vanishing conditional dependence (VCD). One major advantage is that these exact tests do not require large sample sizes. As a result, tests for CSN and MM can be routinely performed in empirical studies. For testing CA and VCD, the exact methods are still impractical in most applications, due to the unusually large number of parameters to be tested. However, exact methods are still derived for them as an exploration toward practicality. Some numerical examples with applications of the exact tests for CSN and MM are provided.     

基于下近似单调的变精度粗糙集属性约简方法

单调性在经典粗糙集属性约简过程中发挥着重要的作用。然而,在一些扩展模型中该单调性质并不存在,如变精度粗糙集模型。针对该问题,提出了变精度粗糙集模型中下近似单调约简的定义,下近似单调约简算法打破了传统意义上属性约简保持下近似不发生变化的局限性,认为属性约简可以追求下近似集尽可能增大。同时给出了求得该约简的属性约简方法。实验结果表明,相较于下近似保持约简算法,下近似单调约简算法求得的约简不仅增加了正域规则数目也减少了边界域规则数目,而且提高了数据的分类精度。由此可见,下近似单调约简算法增加了由正域表示的确定性,同时降低了由边界域带来的不确定性。     

Petri网的PP型子网精细化操作性质分析及应用

针对工厂用车间中的若干台机器加工某些部件等这一类业务处理问题,提出了用Petri网精细化操作解决问题的方案。定义了一种PP型子网,用这种子网分别对Petri网系统中的某些库所进行细化,得到更细致、更精确的Petri网系统。研究了Petri网精细化操作的动态性质保持问题,给出这种精细化操作保持活性、有界性、可回复性和公平性的充要条件;本文的结果可为Petri网系统静态和动态性质的考察提供有效途径,为Petri网复杂大系统的分析提供重要手段,并特别适合于一类业务系统的描述和处理,具有一定的实用价值。     

A computational method for solving two‐dimensional nonlinear variable‐order fractional optimal control problems

In this paper, a new class of two‐dimensional nonlinear variable‐order fractional optimal control problems (V‐OFOCPs) is introduced where the variable‐order fractional derivative is defined in the Caputo type. The general procedure for solving theses systems is expanding the state variable and the control variable based on the Legendre cardinal functions in the matrix form. Hence, we derive their operational matrix of derivative (OMD) and operational matrix of variable‐order fractional derivative (OMV‐OFD). More significantly, some properties of these basis functions are proved to be exploited in our approach. Using these achieved results, we simply expand the matrix form of the nonlinear performance index in terms of the Legendre cardinal functions and subsequently convert it to an algebraic equation. We emphasize that it is a valuable advantage of applying cardinal functions in approximation theory. Then, we implement the OMD and the OMV‐OFD of the Legendre cardinal functions to transform the variable‐order fractional dynamical system to a system of algebraic equations. Next, the method of constrained extremum is applied to adjoin the constraint equations including the given dynamical system and the initial‐boundary conditions to the performance index by a set of undetermined Lagrange multipliers. Finally, the necessary conditions of the optimality are derived as a system of nonlinear algebraic equations including the unknown coefficients of the state variable, the control variable and the Lagrange multipliers. The applicability and efficiency of the proposed approach are investigated through the various types of test problems.     

Nonlinear model order reduction with low rank tensor approximation

In this article, two methods of model order reduction based on the low rank approximation of tensor are introduced for the large scale nonlinear problem. We first introduce some definitions and results on tensor extended from matrix theory. Then we show how the general nonlinear system can be converted into the low rank form we treated in this research. We put the model order reduction of it in two frameworks, that is, polynomial framework and moment‐matching framework. In these two frameworks we construct the algorithms correspondingly, and analyze properties of these algorithms, including the preservation of stability, and moment‐matching properties. Next the priorities of these algorithms are presented. Finally we setup several numerical experiments to validate the effectiveness of the algorithms.     

Semiparametric regression with shape-constrained penalized splines

In semiparametric regression models, penalized splines can be used to describe complex, non-linear relationships between the mean response and covariates. In some applications it is desirable to restrict the shape of the splines so as to enforce properties such as monotonicity or convexity on regression functions. We describe a method for imposing such shape constraints on penalized splines within a linear mixed model framework. We employ Markov chain Monte Carlo (MCMC) methods for model fitting, using a truncated prior distribution to impose the requisite shape restrictions. We develop a computationally efficient MCMC sampler by using a correspondingly truncated multivariate normal proposal distribution, which is a restricted version of the approximate sampling distribution of the model parameters in an unconstrained version of the model. We also describe a cheap approximation to this methodology that can be applied for shape-constrained scatterplot smoothing. Our methods are illustrated through two applications, the first involving the length of dugongs and the second concerned with growth curves for sitka spruce trees.