Introduction to mathematical systems theory. a behavioral approach [Book Review]
Introduction to mathematical systems theory. a behavioral approach [Book Review]
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数学系统理论简介。
DOI：
10.1109/tac.2002.995056
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发表时间：
2002
影响因子：
6.8
通讯作者：
J. Rosenthal
中科院分区：
文献类型：
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作者：
J. Rosenthal
传统上，数学系统理论是通过输入-输出框架来研究的。卡尔曼[8]告诉我们，如果输入输出系统还包括状态变量，状态变量与输入和输出变量一起描述系统的整个状态，那么对输入输出系统的研究最好。这个框架非常成功，迄今为止已经奠定了数学系统理论的基础。有几本优秀的教科书从数学角度涵盖了线性和非线性系统理论。其中最著名的是[1]、[4]、[6]、[7]、[10]、[14]和[18]。所有这些书都通过输入-状态-输出框架来探讨系统理论。有些人甚至可能认为，可以自由选择输入变量来影响状态，从而影响系统的输出，这是系统论的一个显着特征。正在审查的这本书做出了第二作者在过去二十年中发起的某种范式转变[15]-[17]。这种方法不再强调系统的输入输出结构，而是强调系统行为。实际上，起点比输入-状态-输出框架稍微笼统一些。基本对象是行为 B，它是某个宇宙（universe）的子集。数学模型是一种排除法则，它将所有不属于行为的结果排除在宇宙之外。输入-输出系统在该理论中作为一个特例出现。对于这种观点，时间轨迹集位于输入空间和输出空间的笛卡尔乘积内，即宇宙。然后将输入输出系统可以生成的一组轨迹定义为行为。这表明输入输出系统描述了一种行为。在行为框架中，可以自然地对没有直接输入和输出结构的情况进行建模。行星围绕太阳的运动就是一个很好的例子。作者正确地认为，数学系统理论的框架应该能够描述像开普勒行星运动定律这样基本的东西。当然，还有无数其他没有明显信号流方向的例子，因此需要一种能够处理自然界中出现的一般动态情况的理论。为此，本书发展了一种常微分方程理论，与经典情况[2]相反，常微分方程还具有“自由变量”。原则上，这样的理论可以在更一般的环境（非线性、随机和受偏微分方程控制的系统）中发展。然而，作者决定将他们的注意力限制在由具有常系数的线性常微分方程控制的系统上。这使得本书适合广大读者，并且适合作为应用数学的高级本科教材。掌握了本书内容的学生将为学习系统和控制理论中更高级的主题做好充分准备。
Traditionally, mathematical systems theory has been studied through an input–output framework. Kalman [8] taught us that an input–output system is best studied if one also includes state variables which, together with the input and output variables, describe the whole state of the system. This framework has been very successful and has built the foundation of mathematical systems theory up to this time. There are several excellent textbooks available covering linear and nonlinear systems theory from a mathematical point of view. Some of the most well known are [1], [4], [6], [7], [10], [14], and [18]. All these books approach systems theory through an input–state–output framework. Some people might even argue that input variables which can be freely chosen to influence the state and, hence, the output of a system are a distinctive feature of systems theory. The book under review makes a certain paradigm shift that has been initiated by the second author over the last two decades [15]–[17]. This approach de-emphasizes the input–output structure of a system and instead emphasizes the system behavior. The starting point is actually slightly more general than in the input–state–output framework. The basic object is a behavior B which is a subset of some universum (universe) . A mathematical model is an exclusion law that excludes from the universum all outcomes which are not part of the behavior. Input–output systems appear in this theory as a special instance. For this view the set of time trajectories lies inside the Cartesian product of the input-space and the output-space, which is the universum . Then define as the behavior the set of trajectories which can be generated by the input-output system. This shows that an input–output system describes a behavior. In the behavioral framework it is possible to naturally model situations where there is no immediate input and output structure. A good example consists of the motions of planets around the sun. The authors argue with right that a framework for mathematical systems theory should be capable of describing something as fundamental as Kepler’s law of planetary motions. Of course, there are countless other examples where there is no obvious signal flow direction and it is desirable to have a theory which can deal with general dynamical situations appearing in nature. For this purpose the book develops a theory of ordinary differential equations where in contrast to the classical situation [2] one has also ‘free variables’. In principle such a theory can be developed in a more general setting (nonlinear, stochastic and systems governed by PDE). The authors decide however to restrict their attention to systems governed by linear ordinary differential equations with constant coefficients. This makes the book accessible to a large audience and it makes it suitable as an advanced undergraduate text in applied mathematics. A student who has mastered the material of this text will be well prepared to study more advanced topics in systems and control theory.