Composite Control of Nonlinear Singularly Perturbed Systems via Approximate Feedback Linearization

This article is devoted to the problem of composite control design for continuous nonlinear singularly perturbed(SP)system using approximate feedback linearization(AFL)method.The essence of AFL method lies in the feedback linearization only of a certain part of the original nonlinear system.According to AFL approach,we suggest to solve feedback linearization problems for continuous nonlinear SP system by reducing it to two feedback linearization problems for slow and fast subsystems separately.The resulting AFL control is constructed in the form of asymptotic composition(composite control).Standard procedure for the composite control design consists of the following steps:1)system decomposition,2)solution of control problem for fast subsystem,3)solution of control problem for slow subsystem,4)construction of the resulting control in the form of the composition of slow and fast controls.The main difficulty during system decomposition is associated with dynamics separation condition for nonlinear SP system.To overcome this,we propose to change the sequence of the design procedure:1)solving the control problem for fast state variables part,2)system decomposition,3)solving the control problem for slow state variables part,4)construction of the resulting composite control.By this way,fast feedback linearizing control is chosen so that the dynamics separation condition would be met and the fast subsystem would be stabilizable.The application of the proposed approach is illustrated through several examples.

UNIQUENESS OF SOLUTIONS OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR THIRD QRDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].

Interval Type-2 Fuzzy Passive Filtering for Nonlinear Singularly Perturbed PDT-Switched Systems and Its Application

The problem of designing a passive filter for nonlinear switched singularly perturbed systems with parameter uncertainties is explored in this paper.Firstly,the multiple-time-scale phenomenon is settled effectively by introducing a singular perturbation parameter in the plant.Secondly,the interval type-2 fuzzy set theory is employed where parameter uncertainties are expressed in membership functions rather than the system matrices.It is worth noting that interval type-2 fuzzy sets of the devised filter are different from the plant,which makes the design of the filter more flexible.Thirdly,the persistent dwell-time switching rule,as a kind of time-dependent switching rules,is used to manage the switchings among nonlinear singularly perturbed subsystems,and this rule is more general than dwell-time and average dwell-time switching rules.Next,sufficient conditions are provided for guaranteeing that the filtering error system is globally uniformly exponentially stable with a passive performance.Furthermore,on the basis of the linear matrix inequalities,the explicit expression of the designed filter can be obtained.Finally,a tunnel diode electronic circuit is rendered as an example to confirm the correctness and the validity of the developed filter.

ON THE SINGULAR PERTURBATION OF A NONLINEAR ORDINARY DIFFERENTIAL EQUATION WITH TWO PARAMETERS

In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and the remainders have been estimated.

Shock Wave Solution for a Class of Nonlocal Nonlinear Singularly Perturbed Boundary Value Problems with Turning Point

The nonlocal nonlinear boundary value problem with turning point is considered. Leading the stretched variables the formal asymptotic solution of the original problem is constructed. And by using the theory of differential inequalities the existence and uniformly validity of solution for the original boundary value problems are proved. The contribution of this paper is that, the asymptotic behavior of solution is studied by using a simple and special method.

OPTIMAL CONTROL OF SINGULARLY PERTURBED NONLINEAR STATE REGULATOR PROBLEM

In this paper, we consider the optimal control of singularly perturbed nonlinear state regulator by diagonalization techniques. Under the suitable conditions we proved that the given problem has a unique solution (ξ(t,ε,h0), η(t,ε.h0)), moreover, the optimal control u0(t) and the optimal cost J (e) are given as well.

Singularly perturbed Neumann problem for fractional Schrdinger equations

This paper is concerned with a Neumann type problem for singularly perturbed fractional nonlinear Schrdinger equations with subcritical exponent. For some smooth bounded domain ? R^n, our boundary condition is given by∫_?u(x)-u(y)/|x-y|^(n+2s)dy = 0 for x ∈ R^n\?.We establish existence of non-negative small energy solutions, and also investigate the integrability of the solutions on Rn.

SOLVING NONLINEAR DELAY-DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH SINGULAR PERTURBATION VIA BLOCK BOUNDARY VALUE METHODS

Block boundary value methods(BBVMs)are extended in this paper to obtain the numerical solutions of nonlinear delay-differential-algebraic equations with singular perturbation(DDAESP).It is proved that the extended BBVMs in some suitable conditions are globally stable and can obtain a unique exact solution of the DDAESP.Besides,whenever the classic Lipschitz conditions are satisfied,the extended BBVMs are preconsistent and pth order consistent.Moreover,through some numerical examples,the correctness of the theoretical results and computational validity of the extended BBVMs is further confirmed.

THE APPLICATION OF INTEGRAL EQUATIONS TO THE NUMERICAL SOLUTION OF NONLINEAR SINGULAR PERTURBATION PROBLEMS

The nonlinear singular perturbation problem is solved numerically on nonequidistant meshes which are dense in the boundary layers. The method presented is based on the numerical solution of integral equations [1]. The fourth order uniform accuracy of the scheme is proved. A numerical experiment demonstrates the effectiveness of the method.