微分代数系统的数值仿真算法

介绍了微分代数系统ＤＡＥ的基本概念及仿真算法,特别指出了用ＢＤＦ方法求解高指标常系数线性ＤＡＥ系统时的数值稳定性缺陷。最后,针对飞行器轨道约束实时控制问题,给出了３阶收敛的代数约束算法。     

Self-adjoint differential-algebraic equations

Motivated by the structure which arises e.g., in the necessary optimality boundary value problem of DAE constrained linear-quadratic optimal control, a special class of structured DAEs, so-called self-adjoint DAEs, is studied in detail. It is analyzed when and how this structure is actually associated with a self-conjugate operator. Local structure preserving condensed forms under constant rank assumptions are developed that allow to study the existence and uniqueness of solutions. A structured global condensed form and structured reduced models based on derivative arrays are developed as well. Furthermore, the relationship between DAEs with self-conjugate operator and Hamiltonian systems are analyzed and it is characterized when there is an underlying symplectic flow.     

Bohl exponent for time-varying linear differential-algebraic equations

We study stability of linear time-varying differential-algebraic equations (DAEs). The Bohl exponent is introduced and finiteness of the Bohl exponent is characterised, the equivalence of exponential stability and a negative Bohl exponent is shown and shift properties are derived. We also show that the Bohl exponent is invariant under the set of Bohl transformations. For the class of DAEs which possess a transition matrix introduced in this article, the Bohl exponent is exploited to characterise boundedness of solutions of a Cauchy problem and robustness of exponential stability.     

Minimax state estimation for linear discrete-time differential-algebraic equations

This paper presents a state estimation approach for an uncertain linear equation with a non-invertible operator in Hilbert space. The approach addresses linear equations with uncertain deterministic input and noise in the measurements, which belong to a given convex closed bounded set. A new notion of a minimax observable subspace is introduced. By means of the presented approach, new equations describing the dynamics of a minimax recursive estimator for discrete-time non-causal differential-algebraic equations (DAEs) are presented. For the case of regular DAEs it is proved that the estimator’s equation coincides with the equation describing the seminal Kalman filter. The properties of the estimator are illustrated by a numerical example.     

Hopf bifurcation via Volterra series

The Hopf bifurcation theorem gives a method of predicting oscillations which appear in a nonlinear system when a parameter is varied. There are many different ways of proving the theorem and of using its results, but the way which is probably the most useful, to control and system theorists, uses Nyquist loci in much the same way as the describing function method does. The main advantages of this method are dimensionality reduction, which eases the calculation, and the ability to cope with higher-order approximations than are used in the original Hopf theorem. This paper shows how such an approach to the Hopf bifurcation follows naturally and easily from Volterra series methods. Such use of Volterra series in nonlinear oscillations appears to be new. In many problems, the calculations involved are simplified when the Volterra series approach is taken, so the approach has practical merits as well as theoretical ones.     

Hopf bifurcation index for large scale power systems

In this paper, we propose a novel index for the prediction of the Hopf bifurcation (HB) in power systems. The proposed index has strictly less processing load and less computation time in comparison to former HB indices. It also uses modern control properties of the power system such as system matrix and critical eigenvalue. Therefore, stochastic subspace system identification (SSSI) is used as a tool for estimation and prediction of the proposed HB index. Combining beneficial properties of SSSI methods and the processing advantages of the proposed index, we project a new algorithm for power system monitoring. The algorithm is easy and straight-forward. We conduct several test conditions for the proposed materials using 2-area 4-machine system, New England 10-Machine and IEEE 50-machine system. Simulation outcome expresses good performance of proposed index in comparison to former HB indices. The proposed index has fairly linear behavior, without discontinuities with respect to increases of system load. It also has less computation load.     

时延神经元的Hopf分岔控制

通过对时延神经元的分析得到相应的控制器,使时间延迟作为分岔参数得到有效的控制;由线性化理论确定出Hopf分岔点的位置,得到相应的控制条件,并从理论上证明了控制器的可行性.实例分析表明该理论与仿真结果相一致.     

一类两自由度含间隙系统的Hopf分岔

建立了一类两自由度含间隙系统的力学模型,并研究了该系统的周期运动及运动的受扰运动．通过选取适当的Poincare截面及系统碰撞的周期性条件,建立了系统的Poincare映射．利用Poincare映射数值证明了Hopf圈的存在性,揭示了并讨论了随着参数的变化系统通过Hopf分岔及周期倍化分岔向混沌演化的过程．最后讨论了系统参数对系统动力学行为的影响,为系统的动力学优化设计找到了理论依据．     

Structure of differential-algebraic equations for control problems in mechanics

A problem of control of a mechanical system in a partly specified motion is studied. It is shown that the structure of differential-algebraic equations (DAEs) describing the problem may essentially differ from the structure of governing equations for a constrained mechanical system. The control forces chosen to realize the program constraints may have arbitrary directions with respect to the constraint manifold, and, in the extreme, they may be tangent. Thus, the index of the original DAEs of the problem may exceed three and a special approach to the problem solution has to be undertaken. Such an approach is proposed; a systematic formulation is given for the transformation of the DAEs into index-one DAEs or ODEs. Criteria for solvability of the problem are included.     

Consistent initialization and perturbation analysis for abstract differential-algebraic equations

In this paper, we consider linear and time-invariant differential-algebraic equations (DAEs) Eẋ(t) = Ax(t) + f(t), x(0) = x ₀, where x(·) and f(·) are functions with values in Hilbert spaces X and Z. is assumed to be a bounded operator, whereas A is closed and defined on some dense subspace D(A). A transformation to a decoupling form leads to a DAE including an abstract boundary control system. Methods of infinite-dimensional linear systems theory can then be used to formulate sufficient criteria for an initial value being consistent with the given inhomogeneity. We will further derive estimates for the trajectory x(·) in dependence of the initial state x ₀ and the inhomogeneity f(·). In the theory of differential-algebraic equations, this is commonly known as perturbation analysis.     

一类经济模型的分岔周期解

讨论了一类经济模型的稳定性及Hopf分岔．根据特征根给出系统失稳的条件后,利用伪振子法和迭代法得到了Hopf分岔的方向以及周期解的振幅估计,计算式较简洁,最后的数值算例很好地验证了方法的准确性．特别地,当周期解振幅较大时,迭代法的估算更准确．尽管系统失稳后,产生分岔周期解,但适当调整参数大小,仍然可以保证周期解稳定,也就意味着经济的良性发展．     

Finding nondegenerate Hopf bifurcation points for four-dimensional two-parametric systems

We establish a numerically feasible algorithm to find a simplicial approximation A to a certain part of the boundary of the set of stable (or Hurwitz) polynomials of degree 4. Moreover, we have that . Using this, we build an algorithm to find a piecewise-linear approximation to the intersection curve of a given surface contained in ⁴ with . We have also devised an efficient computer program to perform all these operations. The main motivation is to find the curve of nondegenerate bifurcation points in parameter space for a given 2-parametric Hopf bifurcation problem of dimension 4.     

Local feedback stabilization and bifurcation control, I. Hopf bifurcation

Local bifurcation control problems are defined and employed in the study of the local feedback stabilization problem for nonlinear systems in critical cases. Sufficient conditions are obtained for the local stabilizability of general nonlinear systems whose linearizations have a pair of simple, nonzero imaginary eigenvalues. The conditions show, in particular, that generically these nonlinear critical systems can be stabilized locally, even if the critical modes are uncontrollable. The analysis also yields a direct method for computing stabilizing feedback controls. Use is made of bifurcation formulae which require only a series expansion of the vector field.     

Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index

We study optimal control problems for general unstructured nonlinear differential-algebraic equations of arbitrary index. In particular, we derive necessary conditions in the case of linear-quadratic control problems and extend them to the general nonlinear case. We also present a Pontryagin maximum principle for general unstructured nonlinear DAEs in the case of restricted controls. Moreover, we discuss the numerical solution of the resulting two-point boundary value problems and present a numerical example. This research was supported through the Research-in-Pairs Program at Mathematisches Forschungsinstitut Oberwolfach. V. Mehrmann’s research was supported by Deutsche Forschungsgemeinschaft, through Matheon, the DFG Research Center “Mathematics for Key Technologies” in Berlin.     

Hopf bifurcation control: A new approach

In this paper we find conditions to control a Hopf bifurcation in a class of nonlinear systems whose linear approximation has two eigenvalues on the imaginary axis, without assuming that the system is controllable. We use the center manifold theorem to project the dynamics on a two dimensional manifold, and design a controller that permits us to decide the stability and direction of the emerging periodic solution.     

分数阶随机Duffing系统的Hopf分岔

运用正交多项式逼近原理，研究了分数阶随机Duffing系统在零平衡点的Hopf分岔．首先，运用Laguerre正交多项式逼近法将含有随机参数的分数阶Duffing系统转化为等价的确定性系统，然后通过数值计算求得其响应．最后，利用两个引理求得等价系统发生Hopf分岔行为的临界值，并通过数值模拟验证了理论分析结果．     

On the singularity-induced bifurcation theorem

The singularity-induced bifurcation theorem (SIBT) is extended in this note to quasi-linear singular ordinary differential equations. The hypotheses supporting this result are simplified and rewritten in terms of matrix pencils. This approach shows that the SIB follows from a minimal index change at the singularity. The use of a quasi-linear reduction leads to a simple statement of the SIBT for semiexplicit index-1 differential algebraic equations.     

Delayed Hopf bifurcation in time-delayed slow-fast systems

This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed's have different meanings.The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point,but at some other point which is above the bifurcation point by an obvious distance.In a time-delayed system,the evolution of the system depends not only on the present ...     

Asymptotic stability of block boundary value methods for delay differential-algebraic equations

Block boundary value methods are applied to solve a class of delay differential-algebraic equations. We focus on the asymptotic stability of the numerical methods for linear delay differential-algebraic equations with multiple delays. It is shown that A-stable block boundary value methods satisfying a restrictive condition can preserve the asymptotic stability of the analytical solution. Numerical experiments further confirm the effectiveness and stability of the methods.