Local and global bifurcations in simple Takagi-Sugeno fuzzy systems

The relevance of bifurcation analysis in Takagi-Sugeno (T-S) fuzzy systems is emphasized mainly through examples. It is demonstrated that even the most simple cases can show a great variety of behaviors. To understand the richness of asymptotic dynamics one can find in T-S systems, a methodology is proposed by invoking bifurcation theory. Several local and global bifurcations (some of them, degenerate) are detected and summarized in the corresponding bifurcation diagrams. It is claimed,that this kind of careful analysis could help to cope with some criticism raised regarding the blind use of fuzzy systems     

分段线性连续系统中的同宿分岔

对于平面上分段线性的连续系统研究了同宿轨的存在性及同宿分岔问题.该系统同宿轨的存在性可以归结为两种情况：一种是由一个可见鞍点和一个可见焦点（或中心）组成的系统;另一种是由两个稳定性相反的结点重合于原点组成的系统.本文对第一种情况给出了同宿轨存在的充要条件,并研究了相应的同宿分岔问题.     

Local bifurcations analysis of a state-dependent delay differential equation

In this paper, a first-order equation with state-dependent delay and a nonlinear right-hand side is considered. The conditions of the existence and uniqueness of the solution of the initial value problem are supposed to be executed.The task is to study the behavior of solutions of the considered equation in a small neighborhood of its zero equilibrium. The local dynamics depends on real parameters, which are coefficients of the right-hand side decomposition in a Taylor series.The parameter, which is a coefficient at the linear part of this decomposition, has two critical values that determine the stability domain of the zero equilibrium. We introduce a small positive parameter and use the asymptotic method of normal forms in order to investigate local dynamics modifications of the equation near each two critical values. We show that the stability exchange bifurcation occurs in the considered equation near the first of these critical values, and the supercritical Andronov–Hopf bifurcation occurs near the second of these values (provided the sufficient condition is executed). Asymptotic decompositions according to correspondent small parameters are obtained for each stable solution. Next, a logistic equation with state-dependent delay is considered to be an example. The bifurcation parameter of this equation has the only critical value. A simple sufficient condition of the occurrence of the supercritical Andronov–Hopf bifurcation in the considered equation near a critical value has been obtained as a result of applying the method of normal forms.     

A classification of generic families of control-affine systems and their bifurcations

We study control-affine systems with n − 1 inputs evolving on an n-dimensional manifold for which the distribution spanned by the control vector fields is involutive and of constant rank (equivalently, they may be considered as 1-dimensional systems with n − 1 inputs entering nonlinearly). We provide a complete classification of such generic systems and their one-parameter families. We show that a generic family for n > 2 is equivalent (with respect to feedback or orbital feedback transformations) to one of nine canonical forms which differ from those for n = 2 by quadratic terms only. We also describe all generic bifurcations of 1-parameter families of systems of the above form.     

Stability region bifurcations of nonlinear autonomous dynamical systems: Type‐zero saddle‐node bifurcations

The behavior of stability regions of nonlinear autonomous dynamical systems subjected to parameter variation is studied in this paper. In particular, the behavior of stability regions and stability boundaries when the system undergoes a type‐zero sadle‐node bifurcation on the stability boundary is investigated in this paper. It is shown that the stability regions suffer drastic changes with parameter variation if type‐zero saddle‐node bifurcations occur on the stability boundary. A complete characterization of these changes in the neighborhood of a type‐zero saddle‐node bifurcation value is presented in this paper. Copyright © 2010 John Wiley & Sons, Ltd.     

A frequency-domain approach to bifurcations in control systems with saturation

Bifurcation analysis of nonlinear control systems supplies a global perspective of the system behaviour modes. In this paper, it is shown that this analysis can be implemented with very elementary and classical tools, such as frequency-domain graphical methods. The methodology proposed is illustrated analysing the effects of a saturation nonlinearity on a system designed with a linear control law. The full morphology of qualitative behaviours and of polar plots for two-dimensional systems is displayed, pointing out how bifurcations give rise to the boundaries between regions with different qualitative behaviour modes. Also a case of a three-dimensional system is analysed.     

Resonant terms and bifurcations of nonlinear control systems with one uncontrollable mode

In this paper we provide a simple algorithm of feedback design for systems with uncontrollable linearization with only quadratic degeneracy, such as transcritical and saddle-node bifurcations. This approach avoids the computation of nonlinear normal forms. It is based only on quadratic invariants which can be determined directly from the quadratic terms in the uncontrollable dynamics.     

The feasibility of axiomatically-based expert systems

We distinguish axiomatically-based expert systems, whose design and implementation are guided by one or more axiomatically-based theories of decision-making (e.g., decision theory, Bayesian probability theory, maximum entropy theory), from traditional expert systems. An analysis of the knowledge acquisition and computational needs of axiomatically-based expert systems is presented. An explicit quantitative comparison is made between the actual knowledge acquisition effort required to build an existing expert system, and the effort that would be required to build an analogous axiomatically-based advice system. The costs and benefits of the axiomatic approach are discussed. The analysis suggests that the small additional cost of knowledge acquisition for the axiomatic approach are outweighed by the long-term benefits this approach provides.     

A study of local bifurcations of forced oscillations in dynamical systems

We consider the problem of local bifurcations in neighborhoods of stationary states of dynamical systems with a periodic external influence. We study basic scenarios of the bifurcational behavior of a system. We show asymptotic formulas for the resulting oscillations and recommend how to construct the solutions.     

性能维持的增广可行域Tube鲁棒经济模型预测控制

针对未知但有界扰动作用下的约束线性系统,提出一种性能维持的增广可行域Tube经济模型预测控制(tube economic model predictive control,TEMPC)策略.首先考虑经济性能优化目标和鲁棒稳定控制目标,构造TEMPC优化问题的隐式收缩约束,并对系统状态和控制约束进行紧缩Tube设计,给出增广可行域优化问题的数学描述;然后,引入线性分解增广名义终端状态和终端罚函数,扩大优化问题的初始可行域,在此基础上应用终端“三要素”和收缩原理,建立TEMPC策略的递推可行性和闭环系统关于最优经济平衡点有界稳定性的充分性条件,进而证明闭环性能在原初始可行域上的不变性;最后,通过对比仿真结果验证所提出策略的有效性和优越性.     

Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems

A family of polynomial differential systems describing the behavior of a chemical reaction network with generalized mass action kinetics is investigated. The coefficients and monomials are given by graphs. The aim of this investigation is to clarify the algebraic-discrete aspects of a Hopf bifurcation in these special differential equations. We apply concepts from toric geometry and convex geometry. As usual in stoichiometric network analysis we consider the solution set as a convex polyhedral cone and we intersect it with the deformed toric variety of the monomials. Using Gröbner bases the polynomial entries of the Jacobian are expressed in different coordinate systems. Then the Hurwitz criterion is applied in order to determine parameter regions where a Hopf bifurcation occurs. Examples from chemistry illustrate the theoretical results.     

Local shape blending using coherent weighted regions

We present a novel local shape blending method that maps a sparse configuration of facial markers captured from an actor onto target models. The advantage of local shape blending methods is that, given a small set of key shapes for each local region, their combination can generate various facial expressions. However, they have the common problem that they use the pre-determined (fixed) regions and compute the combination of local key shapes for each region independently of each other. So, they have a risk of breaking natural correlations between the regions. We present a stochastic method of computing the regions and the blending weight vectors simultaneously. To do so, we formulate local shape blending as a problem of finding an optimal distribution of blending weight vectors of all control points in MAP?CMRF framework.     

Eigenvalue regions for positive systems

Positive systems are systems in which the state variables take only nonnegative values for all times. In this paper, we derive eigenvalue regions for discrete and continuous-time positive linear systems by resorting to the immediately available information on the values of the main diagonal entries of the system matrix.     

含参量分数阶微分系统的基本分岔分析

本文中,我们讨论了含参量分数阶微分系统的基本分岔,即跨临界分岔、折叠分岔与音叉分岔.首先,根据分数阶Lyapunov方法,讨论了含参量分数阶微分系统的稳定性,并给出了这些基本分岔的相图.其次,根据Taylor展式与隐函数定理,研究了分数阶微分系统的规范形,从而求出这些基本分岔的拓扑规范形.     

Parameter functionalization and its application to the problem of local bifurcations in dynamic systems

The method of parameter functionalization suggested by M.A. Krasnosel’skii for solving problems with parameters and continuums of fixed points is considered. A general scheme of constructing the functionals in the bifurcation problem of small solutions to operator equations is suggested. As an application, we consider problems of local bifurcations in dynamic systems that are topical for the control theory: bifurcations of double equilibrium and forced oscillations and bifurcations of cycles of discrete systems. New sufficient criteria of bifurcations are indicated, an iteration procedure for constructing the solutions and their asymptotic representations are elaborated, and new stability conditions are stated.     

Control bifurcations

A parametrized nonlinear differential equation can have multiple equilibria as the parameter is varied. A local bifurcation of a parametrized differential equation occurs at an equilibrium where there is a change in the topological character of the nearby solution curves. This typically happens because some eigenvalues of the parametrized linear approximating differential equation cross the imaginary axis and there is a change in stability of the equilibrium. The topological nature of the solutions is unchanged by smooth changes of state coordinates so these may be used to bring the differential equation into Poincare/spl acute/ normal form. From this normal form, the type of the bifurcation can be determined. For differential equations depending on a single parameter, the typical ways that the system can bifurcate are fully understood, e.g., the fold (or saddle node), the transcritical and the Hopf bifurcation. A nonlinear control system has multiple equilibria typically parametrized by the set value of the control. A control bifurcation of a nonlinear system typically occurs when its linear approximation loses stabilizability. The ways in which this can happen are understood through the appropriate normal forms. We present the quadratic and cubic normal forms of a scalar input nonlinear control system around an equilibrium point. These are the normal forms under quadratic and cubic change of state coordinates and invertible state feedback. The system need not be linearly controllable. We study some important control bifurcations, the analogues of the classical fold, transcritical and Hopf bifurcations.     

Local ISS of large-scale interconnections and estimates for stability regions

We consider interconnections of locally input-to-state stable (LISS) systems. The class of LISS systems is quite large, in particular it contains input-to-state stable (ISS) and integral input-to-state stable (iISS) systems.Local small-gain conditions both for LISS trajectory and Lyapunov formulations guaranteeing LISS of the composite system are provided in this paper. Notably, estimates for the resulting stability region of the composite system are also given. This in particular provides an advantage over the linearization approach, as will be discussed.     

Dielectrophoretically assembled particles: feasibility for optofluidic systems

This work presents the dielectrophoretic manipulation of sub-micron particles suspended in water and the investigation of their optical responses using a microfluidic system. The particles are made of silica and have different diameters of 600, 450, and 250 nm. Experiments show a very interesting feature of the curved microelectrodes, in which the particles are pushed toward or away from the microchannel centerline depending on their levitation heights, which is further analyzed by numerical simulations. In doing so, applying an AC signal of 12 V_p–p and 5 MHz across the microelectrodes along with a flow rate of 1 μl/min within the microchannel leads to the formation of a tunable band of particles along the centerline. Experiments show that the 250 nm particles guide the longitudinal light along the microchannel due to their small scattering. This arrangement is employed to study the feasibility of developing an optofluidic system, which can be potentially used for the formation of particles-core/liquid-cladding optical waveguides.