## 系统

System (from Latin systēma, in turn from Greek σύστημα systēma) is a set of entities, real or abstract, comprising a whole where each component interacts with or is related to at least one other component and they all serve a common objective. Any object which has no relation with any other element of the system is not part of that system but rather of the system environment. A subsystem then is a set of elements, which is a system itself, and a part of the whole system.

## 动力学系统

The dynamical system concept is a mathematical formalization for any fixed “rule” which describes the time dependence of a point’s position in its ambient space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems.

A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic: for a given time interval only one future state follows from the current state.

## 系统动力学

System Dynamics is an approach to understanding the behaviour of complex systems over time. It deals with internal feedback loops and time delays that affect the behaviour of the entire system.

## 控制理论

In engineering and mathematics, control theory deals with the behavior of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system.

## 一个例子

$$I\frac{d^2\theta}{dt^2}+mlgsin\theta=\tau$$

$e = q-R \tag{2}$

$S = \dot e+ke \tag{3}$

$V = (1/2)*SQ(S)\tag{4}$

SQ表示平方运算square

$\dot V = S( (\frac{u - mgLsin(q)}{I} - \ddot R + k\dot e) \tag{6}$

(6) 式的右边出现了u,这正是我们想要的，因为u是控制信号，是需要我们去产生的，于是我们可以设计一个控制器，产生控制信号u，使得$\dot V < 0$

$0 < I_1 < I < I_2$ 且 $0 < m_1 < m < m_2$

$u = - I_2Sign(S)(|\ddot R+k|\dot e||+mgL/I_1)\tag{7}$

Sign 函数表示取符号. 如果 Sign(5) =1, sign(-7.2) = -1…