Solving polynomial systems via symbolic-numeric reduction to geometric involutive form

We briefly survey several existing methods for solving polynomial systems with inexact coefficients, then introduce our new symbolic-numeric method which is based on the geometric (Jet) theory of partial differential equations. The method is stable and robust. Numerical experiments illustrate the performance of the new method.     

Hybrid singular systems of differential equations

This work develops hybrid models for large-scale singular differential system and analyzes their asymptotic properties. To take into consideration the discrete shifts in regime across which the behavior of the corresponding dynamic systems is markedly different, our goals are to develop hybrid systems in which continuous dynamics are intertwined with discrete events under random-jump disturbances and to reduce complexity of large-scale singular systems via singularly perturbed Markov chains. To reduce the complexity of large-scale hybrid singular systems, two-time scale is used in the formulation. Under general assumptions, limit behavior of the underlying system is examined. Using weak convergence methods, it is shown that the systems can be approximated by limit systems in which the coefficients are averaged out with respect to the quasi-stationary distributions. Since the limit systems have fewer states, the complexity is much reduced.     

Topologically guaranteed univariate solutions of underconstrained polynomial systems via no-loop and single-component tests

We present an algorithm which robustly computes the intersection curve(s) of an underconstrained piecewise polynomial system consisting of n equations with n+1 unknowns. The solution of such a system is typically a curve in Rⁿ⁺¹. This work extends the single solution test of Hanniel and Elber (2007) [6] for a set of algebraic constraints from zero-dimensional solutions to univariate solutions, in Rⁿ⁺¹. Our method exploits two tests: a no-loop test (NLT) and a single-component test (SCT) that together isolate and separate domains D where the solution curve consists of just one single component. For such domains, a numerical curve tracing is applied. If one of those tests fails, D is subdivided. Finally, the single components are merged together and, consequently, the topological configuration of the resulting curve is guaranteed. Several possible applications of the solver, namely solving the surface–surface intersection problem, computing 3D trisector curves, flecnodal curves or kinematic simulations in 3D are also discussed.     

基于逆系统方法的广义非线性系统控制及电力系统应用

对一般广义非线性系统的反馈线性化问题,通过将代数变量视为虚拟输入,给出了构造性递归求逆算法,实现了系统的精确反馈线性化.作为该算法的应用,讨论了带有非线性负荷的三节点电力系统的励磁控制问题,设计了非线性励磁控制器.仿真结果表明了方法的有效性.     

Robust stability of systems with application to singular perturbations

In this paper we will give a simple approach to determining conditions for stability of feedback systems subject to perturbations in the operators describing these systems. The approach is based on techniques used in functional analysis, and provides an alternative development and generalization of some conditions for the linear time-invariant case that have appeared in the literature very recently. As an example of the application of the conditions, we consider the determination of finite regions of stability for singularly perturbed systems.     

Order reduction of complex systems described by TSK fuzzy models based on singular perturbations method

This article deals with the concept of order reduction of linear complex systems described by TSK fuzzy models. The use of singular perturbations technique for process modelling and the choice of Benrejeb arrow form characteristic matrix provide the decoupling of dynamics of linear systems in the continuous and discrete case. An original contribution is to apply nonconventional TSK fuzzy approach on a direct current motor in order, on one hand, to highlight the order reduction of this system presented locally in the form of linear models and, on the other hand, to show the efficiency of the proposed approaches.     

基于不完全转移率的连续Markov 跳变奇异系统的H∞ 控制

研究一类不完全转移率信息的Markov跳变奇异系统的H∞控制问题,提出连续Markov跳变奇异系统的新型有界实引理,并将其推广到不完全转移率条件.进一步设计H∞状态反馈控制器,使得闭环系统在转移率部分未知的条件下随机可容许,且满足H∞性能H.所得结论涵盖了奇异矩阵模态依赖情形,且表示为严格线性矩阵不等式形式,利于工程实现.最后,通过仿真算例表明了所提出方法的有效性和优越性.