Continuous stochastic approximation with semi-Markov switchings in the diffusion approximation scheme

The sufficient conditions of convergence are obtained for a continuous stochastic approximation procedure in the diffusion approximation scheme under balance conditions imposed on singular perturbations of the regression function in a semi-Markov environment. To this end, a singular perturbation problem is solved for the asymptotic representation of a compensating operator of a Markov renewal process. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 170–178, July–August 2007.     

Continuous procedure of stochastic approximation with singular perturbation under balance conditions

Sufficient conditions for the convergence of a continuous stochastic approximation procedure are established for the case where the regression function depends on a Markovian environment and has a singular perturbation that satisfies balance conditions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 133–139, May–June 2006.     

Fluctuations of a stochastic system under an asymptotic diffusive perturbation

A stochastic singularly perturbed system is considered. Sufficient conditions of the asymptotic normality are obtained for the dynamic system in the vicinity of its equilibrium point. To this end, a solution to the singular perturbation problem is constructed for an asymptotic representation of the generator of a Markov renewal process. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 104–109, September–October 2008.     

On the viscous–viscous and the viscous–inviscid interactions in Computational Fluid Dynamics

We consider here the exterior boundary value problem for compressible viscous flow around airfoils. In a first approximation, the viscosity effects are neglected at some distance to the airfoil. The unbounded domain is decomposed by an artificial boundary into a bounded computational domain (near field) and an associated far field. The complete system of conservation laws, modelling viscous flow in the near field is coupled with simplified models for inviscid flow in the far field. The use of the heterogeneous domain decomposition method including physically and mathematically justified transmission conditions at the artificial interface provides one with a quite accurate approximate solution, modelling the viscous–inviscid interaction between the two model zones. However, such a solution does not take into account the viscosity in the far field and does not satisfy the natural transmission conditions at the artificial interface (i.e. continuity of the solution and of the normal flux). In order to get some information for the a-posteriori improvement of this solution, we introduce one-dimensional transmission-boundary value problems, obtained by an appropriate dimensional reduction of the coupled problems from CFD. The one-dimensional problems are analyzed in the framework of singular perturbation theory. We consider formal asymptotic expansions to construct appropriate boundary layer corrections of the coupled problem modelling the viscous–inviscid interaction. Our one-dimensional analysis seems to allow an extension to higher dimensions and therefore could be used in the computation of the solution to the compressible Navier–Stokes problem by updating the solution of the approximation by a (degenerate) Navier–Stokes/Euler problem with boundary layer viscosity correction terms. Received: 22 February 1999 / Accepted: 17 June 1999     

Singular perturbation approximation of balanced systems

This paper relates the singular perturbation approximation technique for model reduction to the direct truncation technique if the system model to be reduced is stable, minimal and internally balanced. It shows that these two methods constitute two fully compatible model-reduction techniques for a continuous-time system, and both methods yield a stable, minimal and internally balanced reduced-order system with the same L_∞-norm error bound on the reduction. Although the upper bound for both reductions is the same, the direct truncation method tends to have smaller errors at high frequencies and larger errors at low frequencies, while the singular perturbation approximation method will display the opposite character. It also shows that a certain bilinear mapping not only preserves the balanced structure between a continuous-time system and an associated discrete-time system, but also preserves the slow singular perturbation approximation structure. Hence the continuous-time results on the singular perturbation approximation of balanced systems are easily extended to the discrete-time case. Examples are used to show the compatibility and the differences in the two reduction techniques for a balanced system     

Chebyshev approximation of functions by the sum of a polynomial and an expression with a nonlinear parameter and endpoint interpolation

Sufficient existence conditions are established for the uniform Chebyshev (minimax) approximation of a function by the sum of a polynomial and an expression with a nonlinear parameter with the minimum absolute error and interpolation at the interval endpoints. An algorithm for determining the parameters of such an approximation using the Remez algorithm is proposed. The application of the iterative method to calculating the nonlinear parameter is substantiated. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 64–75, January–February 2009.     

Continuous Stochastic Optimization with Semi-Markov Switchings in the Diffusion Approximation Scheme

We consider the continuous stochastic optimization procedure with semi-Markov switching in the diffusion approximation scheme with the balance conditions imposed on singular perturbation of the regression function. The sufficient convergence conditions are established for the regression function, which depends on the uniform ergodic semi-Markov process, by using the properties of extended compensating operator of the Markov renewal of the procedure and its asymptotic representation of perturbed Lyapunov function.     

Singular perturbation in piecewise-linear systems

A singular perturbation technique is developed that allows for a decoupling of a continuous piecewise-linear system into slow and fast subsystems. Under the assumption of asymptotic stability, the fast variable is found to decay in the boundary layer to its quasi-steady-state solution, which is given by a continuous implicit function of the slow variable. The solution is found using a finite step algorithm. Sufficient conditions for the approximation to be accurate to an order of O(μ), where μ is a parameter of the system, are given. The technique is illustrated by a numerical example     

Optimal control of ε-coupled and singularly perturbed distributed-parameter systems

Many distributed-parameter systems consist of interconnected subsystems involving fast and slow physical phenomena or reducing to a number of independent subsystems when a scalar parameter ε is zero. The purpose of this paper is to treat the control of such systems by invoking the ε-coupling and singular perturbation approaches developed by Kokotovic and his co-workers for lumped-parameter large-scale systems. In the case of ε- coupled distributed-parameter systems it is shown that the optimal state feedback matrix can be approximated by a Volterra-MacLaurin series with coefficients determined by solving two lower-order decoupled Riccati and linear equations. By using an mth-order approximation of the optimal feedback matrix, one obtains a (2m+1)th order approximation of the optimal performance function. In the singular perturbation approach the result is that for an O(ε²) suboptimal control one must solve two decoupled Riccati equations, one for the fast and one for the slow subsystem, and then construct appropriately the composite control law. By using only the Riccati equation for the slow subsystem, one obtains an O(ε) suboptimal control. The singular perturbation technique is then used to treat interconnected distributed-parameter systems involving may strongly coupled slow subsystems and weakly coupled fast subsystems.     

Higher-Order Properties of the ‘Exchange Rate Dynamics Redux’ Model

In the ‘exchange rate dynamics redux’ model of Obstfeld and Rogoff (1995, Journal of Political Economy, 103(3), 624–660), the short-run and the long-run changes in the net foreign asset are the same. This equivalence is consistent with the first-order linear approximation of the model; but is inconsistent with the long-run consumption smoothing behavior. This paper extends the “redux” model by approximating the changes in the net foreign asset with the second-order perturbation method. This higher-order approximation illustrates that the equivalence does not hold and the difference between the short-run and the long-run changes is of the second order.      

Queuing system evolution in a diffusion approximation scheme

An asymptotic diffusion approximation scheme is investigated as applied to the requirement evolution in semi-Markov queuing systems. In proving the diffusion approximation theorem, the compensating operator of the corresponding extended Markov process is used. This problem is solved with the help of a phase merging procedure. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 136–145, May–June 2009. Original article submitted August 19, 2008.     

On applications of singular perturbation techniques in nonlinear optimal control

The technique of singular perturbations (SPT) has been applied with considerable success in several nonlinear optimal control problems. In many cases the zero-order approximation of the optimal control function has been expressed in a feedback form. This paper deals with topics involved in such closed-loop application, which seem to merit further discussion. It is formally demonstrated that a ‘forced’ singular perturbation model (obtained by artificial insertion of the perturbation parameter) results in the same zero-order composite feedback control solution as a classical singularly perturbed model (where a small parameter of physical significance appears as a consequence of a scaling transformation). The accuracy of the zero-order feedback approximation depends in both cases on the actual time scale separation of the variables. Two inherent limitations of the feedback solution are also pointed out: (1) first and higher-order correction terms of the zero-order approximation have to be computed by a predictive or off-line integration; (2) on-line implementation of SPT control strategy in a terminal boundary layer requires iterative computations. A simple pursuit problem serves as an illustrative example.     

Fluctuations of a stochastic approximation procedure with diffusion perturbation

A stochastic approximation procedure with a singularly perturbed regression function is considered. The form of the limit process is obtained that corresponds to the fluctuation of the stochastic approximation procedure in the neighborhood of an equilibrium point. A generator for the limit process is also constructed. The solution of the singular perturbation problem is given for the asymptotic representation of the generator of a Markov renewal process. The results obtained allow one to extend the possibilities of investigation of the asymptotic behavior of the procedure itself.     

非线性控制系统的近似化方法

介绍了目前几种较常用的非线性控制系统近似化方法包括伪线性方法、扩展线性方法、近似输入／输出线性化、近似反馈线性化、中心流形及平均法等,对有关的理论与应用研究进展作了较全面的综述。     

The remainder in the approximation by a generalized Bernstein operator: a representation by a convex combination of second-order divided differences

We investigate the remainder of the approximation formula of a function f∈C[0,1] by means of a generalized Bernstein operator (1.3), depending on two nonnegative integer parameters, introduced by the author in 1984 in the paper [11]. The remainder is expressed in (2.1)–(2.2) by a formula generalizing the author's earlier representation (1.2) of the remainder in Bernstein's classical approximation formula. In (2.13)–(2.14) an expression is given for the remainder involving a linear functional (2.14) which is a convex combination of second-order divided differences. Received: March 1996 / Accepted: April 1996     

Measurements of the ion-depletion zone evolution in a micro/nano-channel

Ion-depletion and ion-enrichment are the most important phenomena found in a hybrid micro/nano-channel. This depletion process significantly decreases the ion concentration at the anodic end of the nano-channels and lower concentration propagation in the microchannel. The present paper focuses on measurement and analyzes the extension of the depletion zone with time and propagation of concentration perturbation. The systemic experiments were conducted using the nano-porous membranes with different pore diameters (ϕ15 nm, ϕ50 nm), the applied electric voltages (20–100 V) and the buffer solution pH values (pH 5.4, 9.2). According to dimensionless analysis, the extension of the depletion boundary was initially found to be independent of time and to have a non-linear variation for larger times. Based on Nernst–Planck equation, a one-dimensional wave equation is obtained under certain hypothesis, which also shows the propagation of concentration perturbation in depletion zone with constant speed at the initial time.     

基于LMI的小时滞饱和系统稳定域估计方法

基于Pade近似变换,将小时滞饱和系统的稳定域估计转化为估计奇异摄动饱和系统的稳定域问题.证明了此奇异摄动饱和系统的稳定域具有可解耦性,并在此基础上建立LMI优化模型并提出小时滞饱和系统稳定域估计的降阶方法.算例仿真验证了方法的正确性和有效性.     

Linear variational principle for convex vector maximization

For infinitely dimensional convex vector maximization problems, it is proved that there exists, under certain conditions, an arbitrarily small additive perturbation of the performance criterion by a linear continuous operator such that the perturbed problem has efficient solutions. The study was sponsored by the Ukrainian State Fund for Basic Research __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 105–114, March–April 2007.     

A Spectral Element Approximation to Price European Options with One Asset and Stochastic Volatility

We develop a Legendre quadrilateral spectral element approximation for the Black-Scholes equation to price European options with one underlying asset and stochastic volatility. A weak formulation of the equations imposes the boundary conditions naturally along the boundaries where the equation becomes singular, and in particular, we use an energy method to derive boundary conditions at outer boundaries for which the problem is well-posed on a finite domain. Using Heston’s analytical solution as a benchmark, we show that the spectral element approximation along with the proposed boundary conditions gives exponential convergence in the solution and the Greeks to the level of time and boundary errors in a domain of financial interest.