Estimation and Control of Dynamical Systems
Estimation and Control of Dynamical Systems
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DOI：
10.1007/978-3-319-75456-7
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发表时间：
2018-01-01
期刊：
影响因子：
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通讯作者：
Bensoussan, A.
中科院分区：
文献类型：
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作者：
Bensoussan, A.
动力系统是相对于时间演化的系统。它们代表了绝大多数系统。科学家的目标首先是尽可能准确地对它们进行建模，以便对它们的进化有最佳的了解，并设计进化的控制，以实现某些目标。动力系统的建模和理论在20世纪50年代末和60年代初发生了重大的演变，导致了所谓的现代动力系统理论和控制理论。一个关键要素是动力系统的状态表示，也称为内部表示。以前，动态系统是由外部表示建模的，也称为输入输出关系。这就是黑盒子的概念。在这样的框架中，人们只能看到系统如何对输入做出反应。当没有可用的知识模型时，这种方法特别有意义，证明术语黑盒是合理的。对于一个工程师来说，他对系统的全部知识不感兴趣，而是对系统在接受刺激时的行为感兴趣，外部表征是一种有用的方法。我们在这里不展开它，因为它显然已被现代理论所取代。让我们简单回顾几个基本要素。假设一个系统可以接收m个输入，用一个时间依赖向量v（t）∈ Rm表示。假设我们观察到输出z（t）∈ Rp。在系统线性反应的最简单情况下，输入-输出关系可以写为z（t）= t t0 h（t，τ）v（τ）dτ，其中t0是初始时间，矩阵函数h（t，τ）表征反应函数。工程师感兴趣的所有信息都包含在该功能中。稳态情况对应于t0= 0和h（t，τ）= h（t-τ），
Dynamical systems are systems that evolve with respect to time. They represent the overwhelming majority of systems. The objective of scientists is first to model them as accurately as possible in order to have optimal knowledge of their evolution and to design controls of the evolution in order to accomplish certain objectives. A significant evolution in the modeling and theory of dynamical systems occurred in the late 1950s and early 1960s, resulting in what is called the modern theory of dynamical systems and control theory. A key element is the state representation of dynamical systems, also called the internal representation. Previously, dynamical systems were modeled by an external representation, also called an input–output relation. It is the idea of a black box. In such a framework, one can see only how the system reacts to inputs. This approach is particularly meaningful when there is no knowledge model available, justifying the terminology black box. For an engineer, who is not so interested in a full knowledge of a system but rather in its behavior when it receives stimuli, the external representation is a useful approach. We shall not develop it here, since it has clearly been superseded by the modern theory. Let us simply recall a few basic elements. Suppose a system can receive m inputs, represented by a time-dependent vector v (t)∈ Rm. Suppose we observe outputs z (t)∈ Rp. In the simplest case, in which the system reacts linearly, the input–output relation can be written as z (t)=∫ t t0 h (t, τ) v (τ) dτ, in which t0 is an initial time, and the matrix function h (t, τ) characterizes the reaction function. All the information of interest for the engineer is included in that function. The stationary case corresponds to t0= 0 and h (t, τ)= h (t− τ),