When analyzing and designing the control system, the mathematical model of the system should be established first. The mathematical model of control system is a mathematical expression describing the relationship between physical quantities (or variables) in the system. Under static conditions (that is, the derivatives of variables are zero), the algebraic equation describing the relationship between variables is called static mathematical model. A differential equation describing the relationship between the derivatives of variables is called a mathematical model. If the initial conditions of inputs and variables are known, the expression of system output can be obtained by solving differential equations, so that the performance of the system can be analyzed. Therefore, establishing the mathematical model of the control system is the primary work of analyzing and designing the control system.

There are two methods to establish the mathematical model of control system: analytical method and experimental method. Analytical method is to analyze the motion mechanism of each part of the system and write the corresponding motion equation according to their physical or chemical laws. For example, there are Kirchhoff's law in electricity, Newton's law in mechanics, thermodynamic law in thermodynamics and so on. The experimental method is to artificially apply some test signals to the system, record its output response, and approximate it with an appropriate mathematical model. This method is called system identification. In recent years, system identification has developed into an independent discipline. This chapter focuses on the method of establishing the mathematical model of the system by analytical method.

In automatic control theory, there are many kinds of mathematical models, such as differential equation, difference equation and state equation. Complex number field has transfer function and structure diagram; Frequency domain has frequency characteristics.