Stages and main tasks of the century-long the control theory and system identification development. Part I. State space method in the theory of linear automatic control systems

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This article opens a series of publications dedicated to the scientific and technical achievements of the century-long development of control theory and methods. Problems that could not be fully resolved based on one approach or another are also noted. The first of these articles is devoted to the state space method, which began to actively develop somewhere in the 70-80s of the last century within the framework of systems theory and general systems theory. A description of the method is given in relation to linear automatic control systems. A generally accepted description of controlled and observable dynamic processes in such systems is given. The problems of synthesis of stabilizing control in closed loop systems with feedback based on the theory of stability formulated by Lyapunov are mainly considered. In addition to stability, within the framework of the state space method, such important qualitative properties as controllability and observability are formulated. The most developed method is the synthesis of state-based linear controllers. Among them, a special pole placement by method of modal control synthesis occupies a special place which is considered within the framework of the Jordan implementation and various modifications of the method depending on the dimension of the control vector. The synthesis of an optimal controller with a quadratic criterion of a fairly general form is considered. In case of incomplete measurements, when the dimension of the output is less than the dimension of the state vector, in order to implement synthesized control laws based on the state, it becomes necessary to include a state estimator in the control system. In the deterministic case, you can use the Luenberger estimator described in the article, and in the presence of measurement errors in the form of zero-mean white noise, the widely used Kalman filter, also presented in the article. The output control method was not left unattended in the work. The conclusions give the advantages and disadvantages of control methods based on state-space descriptions.