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1、Introduction to Modern Control TheoryWhen differential equations are encountered, they are linearized and subjected to whatever constraints are necessary to establish useful input-output relationships. A recognition of the applicability of well-known methods in other fields of knowledge. Optimal con
2、trol theory often dictates that nonlinear time varying control law be used, even if the basic system is linear and time-invariant. When nonlinearities and time variation are present, the very basis for classical techniques is removed. Some successful techniques such as phase-plane, describing functi
3、on, and ad hoc methods, have been developed to alleviate this shortcoming .With an advancing technological society, there is a trend towards more ambitious goals. This also means dealing with complex system with a larger number of interesting components. The need for greater accuracy and efficiency
4、has changed the emphasis on control system performance. The classical specifications in terms of percent overshoot, settling time, bandwidth, etc., have in many cases given way to optimal criteria such as minimum energy, minimum cost, and minimum time operation. Optimization of these criteria makes
5、it even more difficult to avoid dealing with unpleasant nonlinearities. Optimal control theory often dictates that nonlinear time varying control law be used, even if the basic system is linear and time-invariantThe concept of state occupies a central position in modern control theory. However, it a
6、ppear in many other technical and non-technical context as well. In thermodynamics the equations of state are prominently used. Binary sequential networks are normally analyzed in term of their state. In everyday life, monthly financial statements are commonplace. The President state of the Union me
7、esage is another familiar example.In all of these examples the concept of state is essentially the same. It is a complete summary of the status of the system at a particular point in time. Knowledge of the state at some initial time t0 plus knowledge of the system inputs after t0, allows the determi
8、nation of the state at a later time t1. As far as the state at t1 is concerned, it makes no difference how the initial state was attained. Thus the state at t0 constitutes a complete history of the system behavior prior to t0, insofar as that history affects future behavior. Knowledge of the present
9、 state allows a sharp separation between the past and the future.At any fixed time the state of a system can be described by the values of a set of variables xi called state variables. One of the state variables of a thermodynamic system is temperature and its value can arrange over the continuum of
10、 real number. In a binary network state variable can take on only two discrete values, 0 or 1. Note that the state of your cheking account at the end of the month can be represented by single number, the balance. The state of the Union can be represented by such things as gross national product, per
11、cent unemployment, the balance of trade deficit, ect. For the systems considered in this article the state variables may take on any scalar value, real or complex. That is . Although some systems require an infinite number of state variables, only system which can be described by a finite number n o
12、f state variables will be considered here. Then the state can be represented by an n component state vector It belongs to an n- dimensional vector space defined over the field C.For Continuous-time system, the state is defined for all time in some interval, for example, a continually varying tempera
13、ture or voltage. Discrete-time system have their state defined only at discrete times, as with the monthly financial statement or the annual State of the Union message. Continuous-time and discrete-time systems can be discussed simultaneously by defining the times of interest as T. For continuous-ti
14、me system T consists of the set of times . In either case the initial time could be and the final time could be in some circumstance The state vector x(t) is defined only for those . At any given t, it is simply an ordered set of n numbers. However, the character of a system could change with time,
15、causing the number of required state variables (and not just the values) to change. If the dimension of the state space varies with time, the notation could be used. It is assumed here that is the same n-dimensional state space at all. 现代控制理论简述当使用微分方程时，要对其进行线性化并受限于一定的约束条件才能建立有用的输入-输出关系。认识到其他领域的一些有名的
16、方法的适用性。即使系统是线性定常的，最优控制理论通常给出非线性时变控制律。当系统存在非线性和时变特性时，经典方法赖以存在的基础就不存在了。一些成功的方法，如相平面法、描述函数法以及一些特定的方法可以改进经典控制理论。随着社会技术的进步，人们总是选择更高的目标。这就意味着要处理复杂的具有更多相互作用的部件的系统。由于需要更高的精度和效率控制系统的性能指标已经发生变化。经典的指标如超调量、调节时间、带宽等已经让位于最优化指标如最小能量、最小成本已经最小时间等。即使系统是线性定常的，最优控制理论通常给出非线性时变控制律。状态的概念在现代控制理论中占据中心位置。然而其也出现在其他技术和非技术领域。在热
17、力学中状态方程的概念被突出地使用。二进制序列网络通常使用状态的术语进行分析。在日常生活中每月的也使用财政（财务）状况。美国总统的国情咨文也是一个熟悉的例子。在上述所有的例子中，“状态”的概念是基本相同的。“状态”完全就是系统在某个特殊时刻的“状况”的一个总结。状态在某个时刻t0的值再加上t0时刻的输入的知识可以确定以后时刻t1的状态。就t1时刻的状态而言，它与初始状态是如何实现的无关。因此，t0时刻的状态就构成了t0以前行为的历史，这个历史状态在一定程度上影响系统未来的行为。当前状态就将过去与未来作了一个截然的划分。待添加的隐藏文字内容2在任何一个固定的时刻，系统的状态可以用变量集合的值xi来
18、描述，称为状态变量。热力学系统的一个状态变量是温度，其值是在一个实数连续区间R变化。对于一个二进制网络状态变量可以仅仅有两个离散的值，0和1。你在月底帐目的平衡的状态可以用一个数来表示。国情咨文中的状态可以用国民生产总值、失业率、贸易赤字等来表示。对于本文所考虑的系统，状态变量可以用任何一个标量值（实数或复数）来表示。即。虽然有的系统需要用无穷多个状态变量来表示，但是在这里我们仅仅考虑有限个数目状态变量的系统。因此，状态可以表示为n个分量的状态向量。状态向量属于某个域C上的状态空间。对于连续时间系统，状态可以定义某个区间上的所有时间。例如，连续变化的温度或者电压。离散时间系统的状态只定义在离散时刻。例如，每月财务状况或者年度国情咨文。连续时间系统和离散时间系统可以通过定义时间域T来统一讨论。对于连续时间系统，T由的所有实数构成。对于离散时间系统，T由离散时刻集合构成。在任何一种情形，有时，初始时刻可以为，最终时刻可以是。.状态向量x(t)仅仅是在上有定义。对于任意给定的t ，x(t)仅仅是一个有序的n个数的集合。然而系统的特性可以随时间变化，会引起系统状态变量个数（不是变量的值）的变化。如果状态空间的维数发生变化需要使用符号。这里假设这里表示，对于系统的维数都是n维。
