Type-2 Fuzzy Logic and Systems by Robert John Hani Hagras & Oscar Castillo

Author:Robert John, Hani Hagras & Oscar Castillo
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

2.3 Accurate Nonlinear System Models Are Hard and Expansive to Obtain in Practice

Conventional nonlinear control theory is powerful and effective if a nonlinear system model is mathematically available. In order to design a conventional controller for controlling a physical system, the mathematical model of the system is needed. A common form of the system model is differential equation for a continuous-time system or difference equation for a discrete-time system. Strictly speaking, all physical systems are nonlinear. Unless physical insight and the laws of physics can be applied, establishing an accurate nonlinear model using measurement data and system identification methods is difficult in practice.

For any dynamic system modeling problems, linear or nonlinear, two tasks need to be accomplished. The first task is model structure identification, and the second model parameter identification. These tasks are relatively easier for linear system modeling as there have already existed a set of popular linear model structures to choose from, which include AR (Auto Regressive), ARX (Auto Regressive with eXtra input) and ARMA (Auto Regressive Moving Average). They are different types of difference equations and are black-box models. Strictly speaking, a linear system does not exist—a linear model is an approximate model of the nonlinear system valid for a region around one of the system operation points.

Nonlinear system modeling, however, is far more complicated because there exist an infinitive number of possible model structures. Correctly assuming a nonlinear model structure is a hard problem in nonlinear system modeling theory and no general theory exists. Though difficult, different nonlinear system modeling techniques have still been developed, including the Volterra and Wiener theories of nonlinear systems. Such nonlinear system models are black-box models because they only attempt to mimic system’s input-output relationship with system measurement data and hence cannot provide any insight on internal structure of the system. Another option is to model a nonlinear system as a (piecewise) linear system. This approach can be over-simplistic in nature and fails to capture diverse and peculiar nonlinear system behaviors, such as limit circles, chaos and bifurcation.

Once the model structure is selected/determined, parameters in the model can be found using system’s input-output data and some system optimization procedures (e.g., the least-squares methods), which is the second task.

A linear system model is often adequate for control system development. The whole knowledge base of linear control theory, from linear PID control to modern linear robust control, has been developed based on the notation of linear system models. Once designed, control performance and system stability as well as other properties of the linear control system can usually be examined mathematically. This is because these linear models are difference equations and thus can be analytically analyzed. Whether this linear controller development approach will succeed in practice depends highly on whether the linear model captures the essence of the nonlinear physical system and whether it is a reasonable representation and approximation of the physical system.

In contrast, accurately establishing a nonlinear system model is generally difficult, which significantly limits the application scope of nonlinear control theory.

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