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Some notes on the control problems for general nonlinear

已有 1411 次阅读 2017-11-4 08:52 |个人分类:动力学模型化简与逼近|系统分类:科研笔记

Some notes on the control problems for general nonlinear systems


In the control theory field, I think, the biggest open problem is the system analysis and control design problems for the general nonlinear systems(GNLS). Seeking for the analytic methods for these problems is an unremitting pursuit of the researchers in the control field.


For some special nonlinear systems, many methods of these analysis and design problems are proposed. For example,


1) the affine nonlinear systems. For the the affine nonlinear systems satisfied a series of strictly Lie-derivative conditions, the analytic system analysis and control design methods are proposed based on the modern differential geometry analysis.


2) some chained-form (series connection) systems. For some special nonlinear systems with the chained-form (series connection) structures in the state variable space, the analytic back-steping methods for the controller designing are got.


3) systems with some nonlinear feedback. For the linear controlled systems with the nonlinear feedback loop, based on the Liapunov's stability theory, Popov's absolute stability theory and Popov's hyperstability stability theory, some special controller design methods are proposed.


Unfortunately, the above control methods are only for the special nonlinear systems with some strict conditions about the system dynamics, system structures, and the feedback loop, and not for the GNLS. For the analytic methods of these analysis and design problems of the GNLS, a little progress has been gotten.


In fact, in many pure mathematical fields, except for some special cases, the analytic mathematical methods haven't been made, such as, integrating, finding roots, and optimizing for the general functions, solving the general ordinary differential equations(ODE) and partial differential equations(PDE), etc. For the general functions and equations, there isn't exist the analytic solutions with the universal significance. Therefore, with the developments of the computer and the computational mathematics, solving numerical solutions for the general functions and equations becomes an almost unique choice. The continuous-time nonlinear systems can be describe by ODE, and then the systems analysis and control design problems are essentially solving ODE, optimizing on the functions and functionals. Hence, maybe there isn't exist the analytic systems analysis and control design methods for the GNLS.


In essence, the current fuzzy control, neural network control and intelligent control methods are belong to a class of system analysis and control design methods for the GNLS. But, so far, what are lacking in these methods are the relatively rigorous theory basis and analytic processing for these analysis and design, such as, the dynamical analysis and optimization, the stability analysis and design, the control performance design for the closed-loop systems, etc. In nonlinear conbtrol field, the improtant problem is how to abstract and construct an universally analytic analysis and design method for the GNLS from the fuzzy control, neural network control, and intelligent control methods, and so on. Based on the multivariate function approximation and numerical analysis, the universal methods will be dealed with the analysis and design problems in three variable spaces, such as, real variable space, symbolic variable space (e.g., fuzzy domain, hidden unit, feature variable for the three methods), approximation variable space in between the above two variable spaces, and then the rigorous theoretical analysis and design will be carried out on among of the three variable spaces. The new analysis and design method maybe is an universal and effective way to deal with the GNLS.




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