Mathematical model is a new discipline developed in recent years, and it is a science that combines mathematical theory with practical problems. It reduces practical problems to corresponding mathematical problems, and on this basis, it uses mathematical concepts, methods and theories to conduct in-depth analysis and research, thus depicting practical problems from a qualitative or quantitative perspective and providing accurate data or reliable guidance for solving practical problems. According to the research purpose, a structure (called real prototype or prototype) that generalizes and approximately expresses the main characteristics and relations of the processes and phenomena studied by formal mathematical language. The so-called "mathematization" refers to the construction of mathematical models. The method of understanding things by studying their mathematical models is called mathematical model method, or MM method for short. Mathematical model is the product of mathematical abstraction, and its prototype can be concrete objects and their properties and relationships, or mathematical objects and their properties and relationships. Mathematical models can be divided into broad sense and narrow sense. In a broad sense, mathematical concepts, numbers, sets, vectors and equations can all be called mathematical models. In a narrow sense, only the mathematical relationship structure models that reflect specific problems and specific things systems can be roughly divided into two categories: (1) deterministic models that describe the inevitable phenomena of objects, and their mathematical tools are generally substitution equations, differential equations, integral equations and difference equations. The mathematical model of excellent athletes is often mentioned in sports practice. According to investigation and statistics, the model of modern world-class sprinters is about 1.80m in height and 70kg in weight, 100 Mi Yue 100 seconds or better. An equation or inequality composed of letters, numbers and other mathematical symbols, or a model that describes the characteristics of the system and its internal connection or connection with the outside world with charts, images, block diagrams and mathematical logic. It is an abstraction of a real system. Mathematical model is a powerful tool to study and master the law of system motion, and it is the basis for analyzing, designing, predicting or controlling the actual system. There are many mathematical models and different classification methods. Static model and dynamic model Static model means that the relationship between variables of the system to be described does not change with time, and is generally expressed by algebraic equations. Dynamic model refers to a mathematical expression that describes the laws of system variables changing with time, and is generally expressed by differential equations or difference equations. The transfer function of the system commonly used in classical control theory is also a dynamic model, because it is transformed from the differential equation describing the system (see Laplace transform). Distributed parameter model and lumped parameter model describe the dynamic characteristics of the system with various partial differential equations, while lumped parameter model describes the dynamic characteristics of the system with linear or nonlinear ordinary differential equations. In many cases, the distributed parameter model can be simplified to a lumped parameter model with low complexity through spatial discretization. Continuous-time models and discrete-time models with time variables varying in a certain interval are called continuous-time models, and the above models described by differential equations are all continuous-time models. When dealing with lumped parameter model, time variables can also be discretized, and the obtained model is called discrete time model. The discrete-time model is described by the difference equation. The relationship between variables in stochastic model and deterministic model is given in the form of statistical value or probability distribution, while the relationship between variables in deterministic model is deterministic. Parametric and nonparametric models The models described by algebraic equations, differential equations, differential equations and transfer functions are all parametric models. The establishment of parametric model is to determine the parameters in the known model structure. Parametric models are always obtained through theoretical analysis. The nonparametric model is the response obtained directly or indirectly from the experimental analysis of the actual system. For example, the impulse response or step response of the system recorded by experiments is a nonparametric model. Using various system identification methods, parametric models can be obtained from nonparametric models. If the structure of the system can be determined before the experiment, the parameter model can be obtained directly through experimental identification. The relationship between variables in linear and nonlinear models is linear, and the superposition principle can be applied, that is, several different inputs act on the response of the system at the same time, which is equal to the sum of the responses of several inputs acting alone. The linear model is simple and widely used. The relationship between quantities in the nonlinear model is not linear and does not satisfy the superposition principle. When allowed, nonlinear models can usually be linearized into linear models. The method is to expand the nonlinear model into Taylor series near the working point, keep the first-order term and omit the higher-order term, and then the approximate linear model can be obtained.