Modern Control Theory

Chapter 0 Backgrounds 0.1 Development of Control Theory The history of Automatic control technology， which is utilized by human， can be traced back to thousands of years ago. However， it was until the middle of the 20th century that Automatic control theory had been formed， and developed as a separate discipline. In the 1930-1940s， H. Nyquist， H. W. Bode， N. Wiener and many others had made outstanding contributions to the formation of the Automatic control theory. After World War II， through the effort of many scholars， a more perfect frequency method theory was presented， which depends on practical experience and knowledge of the feedback and frequency response theories. In 1948， the root-locus method was introduced， and the first stage of automatic control theory was laid at this time. This theory， based on the frequency-response and root-locus methods， is often called classical control theory. The classical control theory takes Laplace transform as the mathematical tools， considers single-input-single-output (SISO) linear time-invariant systems as the main research object， transforms differential equations or difference equations describing physical systems to the complex field， and uses transfer functions to design and analyze systems， and to determine the structure and parameters of controllers in the frequency domain. This design approach suffers from certain drawbacks， since it is restricted to SISO systems and difficult to reveal the internal behavior. In the 1960s， the development of the aeronautics and aerospace industry stimulated the field of feedback control. Significant progress had been made. In the meantime， R. Bellman proposed the dynamic programming method for optimal control. Pontryagin proved Maximum principle and developed further the optimal control theory. R. E. Kalman systematically introduced the state-space method， including the concepts of controllability and observerability and the filtering theory. These work， which used the ordinary differential equation (ODE) as a model for control systems， laid the foundations of modern control theory and this approach relying on ODEs， is now often called modern control to distinguish it from classical control， which uses the complex variable methods of Bode and others. In contrast to frequency domain analysis of the classical control theory， the mo- dern control theory relies on first-order ordinary differential equations and utilizes the time-domain state-space representation. To abstract from the number of inputs， outputs and states， the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear). The state space representation (also known as the “time-domain approach”) provides a convenient and compact way to model and ana- lyze systems with multiple inputs and outputs. Given inputs and outputs， we would otherwise have to write down Laplace transforms to encode all the information on a system. Unlike the frequency domain approach， the use of the state space represen- tation is not limited to systems with linear components and zero initial conditions. “State space” refers to the space whose axes are the state variables. The state of the system can be represented as a vector within that space. In the late 1970s， the control theory under development had entered the period of diversified development. The large scale system theory and intelligent control theory were established. Afterwards， some new ideas and new theoretical control， like， multi- variable frequency domain theory by H. H. Rosenbroek， fuzzy control theory by L. A. Zadeh formed the new control concept. In recent years， with the economy and the rapid development of science and tech- nology， automatic control theory and its applications continue to deepen and ex- pand. An enormous impulse was given to the field of automatic control theory. New problems， new ideas and new methods are proposed to meet the need of practical engineering problems. 0.2 Main Contents of Modern Control Theory In summary， there mainly exist the following branches in the field of modern control theory. 1. Linear system theory It is the basis of modern control theory， based on linear systems， aiming at study- ing the motion rule of the system states and the possibilities and implementation methods to change them， establishing and explaining the system structure， para- meters， behaviors and the relationship between them. Linear system theory includes not only the system controllability， observability， stability analysis， but also the state feedback， state estimation compensator theory and design methods， etc. 2. Optimal filtering theory The research object focuses on stochastic systems which are described by stochastic difference equations or differential equations. It focuses on obtaining the desired signals by applying s