论文题目：维时滞动力系统的稳定性分析
作者简介:
王在华，男，1964年01月出生，1996年09月师从南京航空航天大学胡海岩教授，于2000年03月获博士学位。
摘
要
近年来，时滞动力系统受到数学力学及各工程技术领域的高度重视。一方面是由于时滞普遍存在于受控系统。另一方面是时滞的存在对系统的动力学行为有至关重要的影响，如使得系统失稳等。另外，时滞反馈控制也常用来改善系统的特性，如控制混沌。然而，由于描述时滞系统的数学模型是具有无穷维解空间的时滞微分方程，其各种动态特性的研究都有相当的困难，对高维且含待定参数的系统的研究更是如此。本文系统地研究了高维时滞动力系统的稳定性。研究内容包括如下几个方面：全时滞稳定性问题，稳定性切换问题，区间稳定性的检验问题，系统方程降维化简并应用于稳定性分析问题，以及利用Pade逼近研究小时滞动力系统的稳定性。目的是建立有效的稳定性分析方法和判据，并应用于车辆主动底盘系统的稳定性分析。
基于我国数学家建立的广义Sturm判别法提出了含待定参数的高维时滞动力系统的全时滞稳定性分析的一种简单而系统的方法。首先，由Routh-Hurwitz判据可得零时滞系统稳定的充要条件。其次，我们将全时滞稳定性的问题转化为判定含参数的高次代数方程有无实根。依据广义Sturm判别法，我们计算由某多项式及其导数定义的Bezout矩阵（即判别矩阵）的顺序主子式而导出该系统的判别式序列，由此依据广义Sturm判别法中的符号法则可方便地列出由不等式组描述的系统全时滞稳定的充分必要条件，所得条件是易于检验的代数判据。当系统含有1～3个待定设计参数时，该方法可方便地在参数空间中给出稳定区和不稳定区的图形表述。按现有的方法，对某些简单的低维系统可得全时滞稳定性的充分必要条件，或利用不等式估计技巧可给出一些充分条件。而本文的方法可以按一种统一、简洁的方式获得全时滞稳定性的充分必要条件。
本文研究了时滞对系统稳定性的影响，利用广义Sturm判别法讨论了系统随着时滞从无到有逐渐增大而发生的稳定性切换问题。根据上述判别式序列的符号表和Routh-Hurwitz条件，可以将系统的参数空间分为若干子空间。在不同的子空间中，系统具有不同的稳定性切换次数，从零到任意有限多次，并且系统若发生稳定性切换则必最终切换到不稳定。由广义Sturm序列理论可确定上述多项式的实根的个数。当上述多项式无重实根时，稳定性切换次数主要取决于两方面：一是该多项式的正根的个数，二是该多项式的每个正根所对应的临界时滞的差的大小。如果该多项式至多有一个正的单根，则稳定性的切换次数为0或1，这由无时滞系统是不稳定的还是稳定的来决定。而当该多项式有二个及二个以上的正根时，稳定性切换次数要由各正根对应的临界时滞序列中相邻两项差的大小关系来确定。如果系统确实发生稳定性切换，那么适当增加时滞量的大小可使一个稳定的状态变为不稳定，或使一个不稳定状态变为稳定状态。
本文研究了具公约时滞的多时滞高维时滞动力系统的区间稳定性。这个问题可转化为研究多胞形特征拟多项式族的鲁棒稳定性。在常微分方程(多项式区间族)情形，其稳定性可由四个特殊的多项式的稳定性完全决定。这个结论对特征拟多项式区间族无效。但由棱边检验定理，研究区间稳定性的关键工作是判断各棱边对应的单参数特征拟多项式无纯虚根。和现有方法需检验无穷多个特征拟多项式是否稳定来判断该特征拟多项式族的稳定性不同，本文提出了一种具有限检验集的解析检验方法。其基本思路是先利用多项式方程组的Dixon结式理论导出使各棱边对应的单参数特征拟多项式有纯虚根时，参数值应满足的一个等式条件，即所谓的Dixon结式等于零，然后由广义Sturm判别法验证对所有参数值，Dixon结式不等于零，或使结式等于零的解不产生特征拟多项式的纯虚根。
由于许多受控动力系统中的时滞很短，在动力学分析中属于小量。此时利用Taylor逼近将系统方程化为常微分方程来讨论是一种很自然的想法，但分析表明，采用这种逼近常导致错误的结论。本文以单自由度时滞反馈系统为例用特征根法研究了其稳定性，采用Pade有理分式逼近特征方程中的指数函数，证明在短时滞条件下可以将具时滞反馈控制的单自由度振动控制系统的稳定性转化为研究一6次多项式的稳定性，并给出了使这种转化有效的参数估计。结果表明，该方法在一定时滞范围内可获得非常满意的结果。在此基础上，采用Kharitonov定理讨论了短时滞反馈系统的区间稳定性。数值计算表明，这种方法是成功的。原则上，该方法也适宜于多自由度时滞动力系统。
工程中常常遇到具刚-柔子结构且带有时滞的机械系统或机械-结构系统，本文基于中心流形定理对这类系统提出了一种新的简化方法。首先，通过引入适当的奇异小参数和状态变量变换，将系统方程化为具有时滞的奇异扰动方程。通常，这个小参数可取为柔性子系统的固有频率与刚性子系统的固有频率之比。然后，引入快变时间尺度，将奇异扰动方程化为临界稳定的泛函微分方程。当刚性子系统仅有负实部的特征根时，利用泛函微分方程的中心流形定理可知，原系统的（局部）动力学可由一个用常微分方程（组）描述的维数和柔性子结构状态变量个数相同的系统来刻画。由此，我们研究了具刚-柔子结构的多自由度时滞动力系统的稳定性，得到原系统的稳定性条件。数值计算表明，该方法是一种精度相当高的有效方法。就多自由度时滞系统的稳定性问题来说，按常规方法得到的稳定性条件不是偏于保守就是难以检验，而本方法能在较大参数范围内给出实用的稳定性条件。由简化的低维方程，或结合使用泛函微分方程的Normal
Form计算，还可完成多自由度的具刚-柔子结构的时滞动力系统的局部分岔分析。
关键词：时滞，全时滞稳定性，稳定性切换，区间稳定性，系统化简，多项式理论。
Abstract
Time delay systems have received great attention over the past decades in mathematics, mechanics and all technical fields. It lies in the following three facts. One is that time delays exist in most controlled practical systems. The second is that time delay may deteriorate the performance of a system, say stability. The third is that time delay can also be used to improve the performance of a system such as controlling chaos. On the other hand, the existence of time delay results in infinite solution space of the system. As a result, it is usually very difficult to analyze the dynamics of the system, especially when uncertain parameters are involved or the system is of high-dimensional. This dissertation presents a systematical study on stability analysis for high dimensional systems with time delays, including delay-independent stability, stability switches, interval stability, stability test by using Pade approximation, and system reduction by using center manifold theorem with application to stability analysis. It aims at developing effective analytical methods and criteria for stability analysis as well as application to the active chassis of ground vehicles. The five topics are studied in from chapter 1 to chapter 5 respectively.
In
chapter 1, the delay-independent stability for time delay systems is
investigated. Based on the generalized Sturm criterion, developed by Chinese
mathematicians for determining the number of real roots of polynomials probably
with uncertain parameters, a systematic approach is proposed for the
delay-independent stability analysis. In the generalized Sturm criterion, the
discrimination sequence, defined as the sequence of sub-determinant taken in
order of the Bezout matrix of a certain polynomial and its derivative for the
time delay system, takes the same role as the Sturm sequence does in the
classical Sturm criterion. The discrimination sequence, together with the
Routh-Hurwitz conditions for the corresponding system without time delay, gives
the sufficient and necessary conditions for the delay-independent stability
following the sign rules of the generalized Sturm criterion. The conditions are
in terms of the system parameters algebraically and therefore can be testified
easily. When the system has two to three uncertain parameters, the
delay-independent stable region in the parameter space can be easily plotted.
This method can be applied to high dimensional time delay systems in a concise
and united way.
Chapter
2 deals with the stability switches of time delay systems, a phenomenon that the
stability changes as the time delay increases from zero to infinity. By using
the generalized Sturm criterion and Routh-Hurwitz criterion, the parameter space
of concern can be divided into several regions. It is found that the system in
different parameter regions may have different numbers of stability switches
from null to any finite number. If the system does exhibit change of stability,
the system must eventually become unstable. If the polynomial has none or one
real root, then the system undergoes none stability switches or one stability
switch. When the polynomial has at least two simple real roots, sequences of
critical values of time delays corresponding to each positive root can be
derived. The number of stability switches depends mainly on two factors: one is
the number of real simple roots of a polynomial, the other is the differences
generated by the two neighboring terms in the sequences of critical time delays
corresponding to the simple roots.
The
interval stability, a special kind of robust stability, for dynamic systems with
constant commensurate time delays is studied in chapter 3. According to the edge
theorem for the interval stability of the polytopic family of quasipolynomials,
the tedious task in interval stability test by applying the available methods is
to check whether the certain families of quasipolynomials with one parameter are
asymptotically stable. Based on the Dixon's resultant for polynomials and
generalized Sturm criterion, a new method is proposed in this chapter. One needs
to check the asymptotic stability of finite many quasipolynomials for the
interval stability test by using the proposed method. The basic steps of this
method are as follows. The first step is to derive a necessary condition such
that the families of quasipolynomials with one parameter, corresponding to the
edges of the polytope, have pure imaginary roots by using the Dixon's resultant
for polynomials. Then to check that the cases corresponding to the parameters
obtained in the first step are in fact impossible by using the generalized Sturm
criterion and some stability criteria.
The
time delays are usually short in many practical applications, so it is natural
to use some approximate methods to study the stability of time delay systems. By
means of the Pade approximation, a stability analysis for a controlled vibration
system with a short time delay in the feedback paths of displacement and
velocity, taken as an example, is given in chapter 4. By approximating the
exponential function in the characteristic equation by proper Pade formula, the
stability is completed by analyzing the root location of a polynomial.
Estimation on effective ranges of the system parameter is given. Then, the test
to interval stability (robust stability) of the approximated system is completed
on the basis of the Kharitonov theorem. Numerical examples indicates that the
approach give excellent accuracy for the test of stability of linear dynamic
systems with short time delays.
In
engineering, a great number of mechanical systems are composed of two kinds of
substructures, one is relatively stiff and has high nature frequency, while the
other is relatively soft and has low natural frequency. This kind of systems is
usually referred to as stiff-soft systems. A dimensional reduction for
stiff-soft systems with time delays is proposed in chapter 5 by means of center
manifold reduction. By introducing properly a small parameter, time variables,
and transformations between the state variables, the equation of motion can be
changed into critically stable functional differential equations. Then, the
center manifold theorem can be used to reduce the original system of infinite
dimensional solution space to a low system described by ordinary differential
equations whose dimension equals to the order of the slow variable. With this
reduction, one can complete stability analysis and other local dynamics.
KEY WORDS: time delay, delay-independent stability, stability switches, interval stability, system reduction, polynomial theory.
