Background of Control TheorySystem and Control TheoryAccording to the Encyclopedia Americana, a system is “an aggregation or assemblage of things so combined by nature or man as to form an integral and complex whole ”.Mathematical systems theory is the study ,of the interruptions and behavior of such an assemblage of “things ”when subjected to certain conditions or inputs ,The abstract of systems theory is due to the fact that it is concerned with mathematical properties rather than the physical form of the constituent parts.Control theory is more often concerned with physical applications.A control system is considered to be any system which exists for the purpose of regulating or controlling or controlling the flow of energy, information, money, or other puantities in some desired fashion. In more general terms, a control system is an interconnection of many components or functional units in such a way as to produce a desired result . Fig1 is a general representation of an open-loop control system, The input control u(t) is selected based on goals for the system and all available a priori knowledge about the system. The input is in no way influenced by the output of the system, represented by y(t).If unexpected disturbances act upon an open-loop system, or if its behavior is not completely understood, then the output will not completely understood, then the output will not behave precisely as expected.Another general class of control systems is the closed-loop or feedback system, as illustrated in Fig2. In the closed-loop system, the control u(t) is modified in some way by information about the behavior of the system output. A feedback system is often better able to cope with unexpected disturbances and uncertainties about the system’s dynamic behavior. However, it need not be true that closed-loop control is always superior to open-loop control. When the measured output measured output has errors which are sufficiently large, and when unexpected disturbances are relatively unimportant, closed-loop control can have a performance which is inferior to open-loop control.Fig1 An Open-loop Control SystemIntroduction to Modern Control TheorySeveral factors provided the stimulus for the development of modern control theory:(a)The necessity of dealing with more realistic models of systems.(b)The shift in emphasis towards optimal control and optimal systemdesign.(c)The continuing developments in digital computer technology.(d)The shortcomings of previous approaches.(e) A recognition of the applicability of well-known methods in other fieldsof knowledge.The transition from simple approximate models, which are easy to work with, to more realistic models produces two effects. First, a larger number of variables must be included in the model. Second, a more realistic model is more likely to contain nonlinearities and time-varying parameters. Previously ignored aspects of the system, such as interactions and feedback through the environment, are more likely to be included.With an advancing technological society, there is a trend towards more ambitious goals. This also means dealing with complex systems with a larger number of interacting components. The need for greater accuracy and efficiency has changed the emphasis on control system performance. The classical specifications in terms of percent overshoot, settling tome, bandwidth, etc. , have in many cases given way to optimal criteria such as minimum energy, minimum cost , and minimum timeoperation. Optimization of these criteria makes it even more difficult to avoid dealing with unpleasant nonlinearities. Optimal control theory often dictates that nonlinear time0-varying control laws be used, even if the basic system is linear and time-invariant.The continuing advances in computer technology have had three principal effects on the controls field. One of these relates to the gigantic supercomputers. The size and class of problems that can now be modeled, analyzed, and controlled are considerably larger than they were when the fist of this book was written.The second impact of computer technology has to do with the proliferation and wide availability of microcomputers in homes and in the work place. Classical control theory was dominated by graphical methods because at the time that was the only way to solve certain problems. Now every control designer has easy access to powerful computer packages for system analysis and designer. The old graphic methods have not yet disappeared, but have been automated. They survive because of the insight and intuition that they can provide. However, some different techniques are often better suited to a computer. Although a computer can be used to carry out the classical transform-inverse transform methods, it is usually more efficient for a computer to integrate differential equations directly.The third major impact of computer is that they are now so commonly used as just another component in the control system. Their cost, size and reliability make it possible to use them routinely in many systems. This means that the discrete-time and digital system control now deserves much more attention than it did in the past. Modern control theory is well suited to the above trends because its time-domain techniques and its mathematical language (matrices, linear vector space, etc.) are ideal when dealing with a computer. Computers are a major reason for the existence of state variable methods.Most classical control techniques were developed for linear constant coefficient systems with one input and one output (perhaps a few inputs and outputs). The language of classical techniques is the Laplace or z-transform and transfer functions. When nonlinearities and time variations are present, the very basic for these classical techniques is removed. Some successful techniques such as phase-plane methods,describing functions, and other ad hoc methods, have been developed to alleviate this shortcoming. However, the greatest success has been limited to low-order systems. The state variable approach of modern control theory provides a uniform and powerful method of representing systems of arbitrary order, linear or .nonlinear, with time-varying or constant coefficients. It provides an ideal formulation for computer implementation and is responsible for much of the progress in optimization theory. Modern control theory is a recent development in the field of control. Therefore, the name is justified at least as a descriptive title. However, the foundation of modern control theory is to be found in other well-established fields. Representing a system in terms of state variables is equivalent to the approach of Hamiltonian mechanics, using generalized coordinates and generalized momenta. The advantages of this approach have been well-known in classical physics for many years. The advantages of using matrices when deal with simultaneous equations of various kinds have long been appreciated in applied mathematics. The field of linear algebra also contributes heavily to modern control theory. This due to the concise notation, the generality of the results, and the economy of thought that linear algebra provides.Fig2 A Closed-Loop Errors控制理论基础系统及控制理论按照美国大百科全书的解释，所谓系统就是指“一个各种物体的集合，根据其性质或人的愿望而结合起来的以至形成一个集中、复杂的整体”。
