Papermaking Principles and Engineering (2nd Edition)

Author: Chief Editor: Liu Hongcai
Publisher:
Publish Date: 2004-09-01
Features:
Excerpt: System identification and parameter estimation are important branches of systems science. They not only have theoretical significance but also practical significance. They study how to establish mathematical models of controlled objects (systems or processes) through operational or experimental data. The rapid development of production practice and science and technology requires a deeper understanding of research objects, not only their static characteristics but also an in-depth analysis of their dynamic characteristics. The dynamic characteristics refer to the motion state characteristics of the working condition of a controlled object relative to a certain equilibrium point. Many industrial systems, biomedical systems, and social systems can be regarded as dynamic systems. These systems, in terms of their respective manifestations, are various, but as dynamic systems, their dynamic characteristics can all be described and analyzed by mathematical models. Therefore, how to establish a mathematical model for a controlled object has become an essential part of developing systems theory and carrying out practical applications.
Model: A model is a concise representation of essential information about an actual system in a useful descriptive form [23]. A model can simulate or imitate the behavior of an actual system without necessarily describing its actual structure. Distinguishing which parts of an actual system are essential and which are nonessential depends on the problem being studied. For practical problems, it is impossible to establish a complex model that includes all factors of the system. A model is only an approximate description of the actual system according to its purpose. Therefore, the degree of approximation between the model and the actual system, as well as the effectiveness of the model, are issues that need to be studied. When establishing a model for an actual controlled object, the accuracy of the model must be considered. If the model is required to be more accurate, it will become more complex. Conversely, if the accuracy requirement for the model is reduced, and only the essential parts of the actual system are modeled, the resulting model will be simpler. Thus, when modeling an actual system, there is a contradiction between complexity and simplicity to make the model useful. Therefore, the key to modeling lies in comprehensively considering it from an overall optimization perspective.
The representation of a model is diverse. For example, the theory of evolution is a model of the evolution of life on Earth; graphical or tabular models that represent the characteristics of controlled objects; physical models commonly used to describe the dynamic characteristics of power systems; mathematical models that describe the basic characteristics of controlled objects in the form of mathematical structures, etc.
Mathematical Model: A mathematical model is an abstract tool for studying the properties of things. Different things may be described by the same mathematical model. It is a mathematical expression that describes the relationships between various physical quantities of an actual system. Common mathematical expressions include algebraic equations, differential equations, and difference equations. The application of mathematical models is extremely broad; they are the foundation for analyzing, designing, forecasting, controlling, and diagnosing faults in an actual system. The main uses of mathematical models are as follows:
(1) Used for analyzing, designing, and controlling actual systems. In general engineering practice, when analyzing and designing a system, computer simulation experiments and physical simulation experiments are conducted first, followed by industrial experiments. Using computer digital simulation or computer-aided design (CAD) is simpler and more feasible than physical simulation, with lower investment costs, and is currently widely adopted. When using digital simulation or CAD to analyze and design a system, it is necessary to have a mathematical model that describes the actual system. The key to designing controllers using modern control theory is to have a mathematical model (state equation or input-output equation). Based on the mathematical model, various controllers are set up according to methods such as the maximum principle, dynamic programming, feedback, decoupling, pole placement, adaptive control, and intelligent control.
(2) Used for forecasting the physical quantities of an actual system. When analyzing and studying actual systems, it is often necessary to know the values of certain physical quantities. However, some of these quantities cannot be measured in advance or are not measured accurately, so it is necessary to establish a mathematical model to forecast these quantities. The future values of variables in an actual system are unmeasurable, such as future weather, power load, population, and product sales.
(3) Used for fault diagnosis in actual systems. Once a fault occurs in certain important production devices, it can cause the entire production process to shut down. Therefore, it is necessary to quickly alarm these production devices when a fault occurs, determine the fault source as soon as possible, provide a basis for further decision-making, and ensure that the production process operates reliably. A fault refers to the deterioration of the performance of a dynamic system due to the failure of some of its components. The task of fault detection and separation is to identify whether a fault has occurred and to determine the fault source. Fault detection and separation technology based on mathematical models utilizes system identification methods to determine the fault source. For the same actual system, different mathematical models can be established based on different purposes. Precise mathematical models are used for analyzing, designing actual systems, and forecasting actual physical quantities, while mathematical models used for control do not need to be very precise, especially when designing adaptive controllers, where coarser mathematical models can be used.
Mathematical models can be divided into many types. For example, static models and dynamic models, linear models and nonlinear models, constant system models and time-varying system models, single-variable system models and multi-variable system models, continuous-time models and discrete-time models, input-output models and state-space models, deterministic system models and stochastic system models, lumped-parameter system models and distributed-parameter system models, as well as parametric models and nonparametric models, etc.
Linear models are used to describe linear systems, a fundamental characteristic of which is that they satisfy the superposition principle. Nonlinear models are used to describe nonlinear systems, which do not satisfy the superposition principle. Strictly speaking, all actual systems in the real world are nonlinear, and pure linear systems do not exist. Due to the complexity of nonlinear models, a linearization is performed on a class of nonlinear models with weaker nonlinearity, treating them as linear systems. This is done by expanding the nonlinear model into a Taylor series near a working point, neglecting higher-order terms, and retaining only the first-order terms, resulting in an approximate linear model. Since linear models are simple, they are widely used.
Static models are used to describe the relationships between various physical quantities of a controlled object when it is in a steady state, considering only the relationships between the physical quantities of the controlled object at the same time, regardless of the changes in the variables over time. Static models are represented by algebraic equations. Dynamic models are used to describe the relationships between various physical quantities of a controlled object during a transient process. Generally, they are represented by differential equations or difference equations. Constant system models are used to describe controlled objects whose parameters do not change over time, generally represented by constant-coefficient linear differential equations or difference equations. Time-varying system models describe controlled objects whose parameters change over time, commonly represented by linear differential equations or difference equations with time-varying coefficients. Continuous-time models are described by differential equations, while discrete-time models are described by difference equations. Input-output models describe the dynamic characteristics of controlled objects using measurable inputs and outputs, generally used to describe the movement laws of controlled objects, revealing their internal essence through external information. It should be noted that the description of dynamic characteristics of controlled objects by such models is sometimes incomplete, potentially hiding uncontrollable and unobservable parts of the controlled object [29]. Generally, these models are represented by differential equations or difference equations. State-space models are used to describe the dynamic characteristics of controlled objects, providing a comprehensive and profound revelation of their internal essence. State-space models are represented by state equations and output equations.
A deterministic system refers to a controlled object whose characteristics and parameters change according to certain laws, and whose input variables (including controls and disturbances) also change according to certain laws. Therefore, the controlled object described by a deterministic system model has a uniquely determined output response once its state is determined. The controlled object described by a stochastic model, even if its state is determined, still has an uncertain output response. Typically, stochastic models can only be analyzed and processed using the theories and methods of probability statistics and random processes. In actual industrial systems, stochastic models are widely present.
Lumped-parameter system models are used to describe variables in controlled objects that depend only on time and not on spatial position.