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利用经典李对称理论,研究一类抛物型分布参数系统的边界控制问题,分别设计开环和闭环形式的边界控制律,实现系统状态的定态控制。借助于无穷小生成元作为分析工具,应用微分方程的不变性条件,确定系统经典李对称的具体表示形式,即其所对应的无穷小生成元表达式。之后,分别针对开环和闭环控制结构,设计出系统解析形式的边界控制条件。通过设定系统参数、初始条件和控制目标,开环和闭环边界控制都能实现设定的控制要求。相比较而言,开环控制的输出误差收敛速度较慢;闭环控制收敛速度较快,不过入口附近有无法完全避免的超调现象。提供的研究结果,对于一类包含传导和对流特性的温度或浓度模型的定态控制问题有一定指导意义。
The classical Li Symmetry theory is used to study the boundary control problems of a parabolic distributed parameter system. The boundary control law of open-loop and closed-loop is designed respectively to realize the steady state control of the system state. With the help of infinitesimal generator as analysis tool, the invariant condition of differential equation is used to determine the concrete representation of system symmetry Li symmetry, that is, its corresponding infinitesimal generator expression. Then, the boundary conditions of system analysis are designed for the open-loop and closed-loop control structures respectively. By setting system parameters, initial conditions and control targets, both open-loop and closed-loop boundary control can achieve the set control requirements. In contrast, the output error of the open-loop control converges slowly; the closed-loop control has a faster convergence rate, but overshoot can not be completely avoided near the entrance. The results provided are of guiding significance for the steady state control of a class of temperature or concentration models that include conduction and convection characteristics.