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　　控制问题的科学表述必须基于两种信息： (A) 系统的行为必须以数学上的方式描述；(B) 控制的目的（标准）和环境（干扰）必须以数学上的方式。

　　A 类信息意味着应用于系统的任何潜在控制动作的效果在所有可能的环境情况下都是已知的。然后在可能可用的许多可能性中选择一个或几个适当的控制动作，然后基于类型 B 的信息。这种选择称为优化。

　　控制理论的任务是研究这两个基本问题的数学量化，然后推导出应用的数学方法，从而获得优化的具体答案。控制理论不直接处理物理现实，而是处理数学模型。因此，该理论的局限性仅取决于可用模型与要控制的系统的实际行为之间的一致性。可以对标准和干扰的数学表示做出类似的评论。

　　一旦从上述信息中通过数学方法推导出适当的控制动作，控制的实施就成为一项技术任务，在工程的各个领域进行处理。化工厂的具体控制方式可能与汽车厂的控制方式大不相同，但基本原理是相同的。因此，这里对控制问题解决方案的进一步讨论将限于数学层面。

　　为了得到这个意义上的解，描述要控制的系统很方便，称为植物，就其内部动态而言。这意味着一个表（称为状态向量) 以定量的形式表示当前时刻之前所有外部影响对植物的影响，因此可以从当前状态和未来输入的知识中准确地给出植物的未来进化。这种情况意味着给定时间的控制动作可以为当时状态的某个函数。这种状态的函数，它决定在任何时刻要采取的控制动作，称为控制规律。这是一个比早期的反馈概念更普遍的概念；事实上，控制律可以同时包含反馈和前馈控制方法。

　　在开发模型来表示控制问题时，假设状态向量的每个分量都可以和瞬时地测量是不现实的。因此，在大多数情况下，控制问题必须扩大到包括进一步的问题：状态确定，这可以被视为统计预测和过滤理论的中心任务。原则上，任何控制问题都可以分两步解决：（1）建立一个滤波器（所谓的曼滤波器）来确定当前状态向量的估计；(2) 确定一个控制律并通过将步骤 1 中获得的状态向量的估计值代入其中来使其机械化。

　　在实践中，这两个步骤由单个硬件单元实现，称为控制器，可以看作是一台专用计算机。这里给出的理论公式可以被证明包括所有其他以前的方法作为一个特例；的区别在于控制器的工程细节。

　　控制问题的数学解可能并不总是存在。从 1950 年代后期开始，严格存在条件的确定对现代控制的演变产生了重要影响，无论是从理论的角度还是从应用的角度来看。最重要的是可控性；它表达了某种控制是可能的事实。如果满足这个条件，优化方法可以使用 B 类信息挑选出正确的控制类型。

　　可控性条件具有重要的现实意义和哲学意义。因为状态向量方程准确地代表了大多数物理系统，它们的稳态行为只有很小的偏差，所以在自然界中，小规模控制几乎总是可能的，至少在原则上是这样。这一自然事实实际上是目前所有现有控制技术的理论基础。另一方面，人们对最终的限制知之甚少当所讨论的模型不是线性的时，控制的控制，在这种情况下，输入的微小变化可能会导致大的偏差。特别是，不知道在什么条件下可以进行大规模控制，即对于与现有条件的任意偏差。在经济学家和社会学家关于控制可能改善人类社会的经常夸大的主张时，应牢记缺乏科学知识。

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　　The scientific formulation of a control problem must be based on two kinds of information: (A) the behaviour of the system must be described in a mathematically precise way; (B) the purpose of control (criterion) and the environment (disturbances) must be specified, again in a mathematically precise way.

　　Information of type A means that the effect of any potential control action applied to the system is precisely known under all possible environmental circumstances. The choice of one or a few appropriate control actions, among the many possibilities that may be available, is then based on information of type B. This choice is called optimization.

　　The task of control theory is to study the mathematical quantification of these two basic problems and then to deduce applied mathematical methods whereby a concrete answer to optimization can be obtained. Control theory does not deal directly with physical reality but with mathematical models. Thus, the limitations of the theory depend only on the agreement between available models and the actual behaviour of the system to be controlled. Similar comments can be made about the mathematical representation of the criteria and disturbances.

　　Once the appropriate control action has been deduced by mathematical methods from the information mentioned above, the implementation of control becomes a technological task, which is best treated under the various specialized fields of engineering. The detailed manner in which a chemical plant is controlled may be quite different from that of an automobile factory, but the essential principles will be the same. Hence further discussion of the solution of the control problem will be limited here to the mathematical level.

　　To obtain a solution in this sense, it is convenient to describe the system to be controlled, which is called the plant, in terms of its internal dynamical state. By this is meant a list of numbers (called the state vector) that expresses in quantitative form the effect of all external influences on the plant before the present moment, so that the future evolution of the plant can be exactly given from the knowledge of the present state and the future inputs. This situation implies that the control action at a given time can be specified as some function of the state at that time. Such a function of the state, which determines the control action that is to be taken at any instant, is called a control law. This is a more general concept than the earlier idea of feedback; in fact, a control law can incorporate both the feedback and feedforward methods of control.

　　In developing models to represent the control problem, it is unrealistic to assume that every component of the state vector can be measured exactly and instantaneously. Consequently, in most cases the control problem has to be broadened to include the further problem of state determination, which may be viewed as the central task in statistical prediction and filtering theory. In principle, any control problem can be solved in two steps: (1) building an optimal filter (a so-called Kalman filter) to determine the best estimate of the present state vector; (2) determining an optimal control law and mechanizing it by substituting into it the estimate of the state vector obtained in step 1.

　　In practice, the two steps are implemented by a single unit of hardware, called the controller, which may be viewed as a special-purpose computer. The theoretical formulation given here can be shown to include all other previous methods as a special case; the only difference is in the engineering details of the controller.

　　The mathematical solution of a control problem may not always exist. The determination of rigorous existence conditions, beginning in the late 1950s, has had an important effect on the evolution of modern control, equally from the theoretical and the applied point of view. Most important is controllability; it expresses the fact that some kind of control is possible. If this condition is satisfied, methods of optimization can pick out the right kind of control using information of type B.

　　The controllability condition is of great practical and philosophical importance. Because the state-vector equations accurately represent most physical systems, which only have small deviations about their steady-state behaviour, it follows that in the natural world small-scale control is almost always possible, at least in principle. This fact of nature is the theoretical basis of practically all the presently existing control technology. On the other hand, little is known about the ultimate limitations of control when the models in question are not linear, in which case small changes in input can result in large deviations. In particular, it is not known under what conditions control is possible in the large, that is, for arbitrary deviations from existing conditions. This lack of scientific knowledge should be kept in mind in assessing often-exaggerated claims by economists and sociologists in regard to a possible improvement in human society by governmental control.