LMI Based Gain-Scheduled Sliding Mode Control for Nonlinear and Time Varying Systems

变结构系统或滑模控制采用不连续控制结构来控制非线性系统。它是一种强有力的控制方法，在控制工程的许多领域得到了越来越广泛的应用。增益调度是一种特殊类型的非线性反馈，用于各种控制。鲁棒控制理论的最新进展提供了一个新的理论框架和系统的增益调度。线性变参数(LPV)对象的定义可以描述为线性时变对象或非线性对象的模型，这些对象使用时变参数的向量进行线性化，如θ(T)。利用LPV对象定义，以线性矩阵不等式(LMI‘s)的形式广泛地给出了增益调度H_*控制设计，其解在有效的凸优化技术的范围内。这里，我们的目标是将这种方法扩展到设计一个滑模超平面，并将其应用到一个实际的系统中。本文研究了一类线性变参数(LPV)对象的滑模超平面设计，其状态空间矩阵是时变物理参数的仿射函数。所提出的超平面涉及线性矩阵不等式(LMI)方法，由于调度参数具有连续的动力学特性，并且对参数不确定性具有稳定性和鲁棒性。针对具有陀螺效应的转子-磁轴承系统，我们设计了一种时变超平面，由于参数依赖于转速，可以将其视为LPV对象。所获得的超平面相对于旋转速度被连续调度。我们利用商品化的涡轮分子泵系统成功地进行了实验，结果是合理的和良好的。

A variable structure system or sliding mode control uses a discontinuous control structure to control nonlinear systems. It is a powerful control method with increasing application in many areas of control engineering. Gain scheduling is a special type of nonlinear feedback used in a variety of control. Recent advances in robust control theory have offered a new theoretical framework and systematic gain scheduling.A linerar parameter-varying (LPV) plant definition can be described as model of linear time-varying plants or nonlinear plants which are linearized using a vector of a time-varying parameter, such as theta(t). Using LPV plant definition, the gain scheduled H_* contol design is extensively presented in terms of linear matrix inequalitiees (LMI's), the solution of which are within the scope of efficient convex optimization techniques. Here, our aims are to extend this approach to designing a sliding mode hyperplane and applying it to a practical system. All definitions and theorems given for LPV plants are also valid in our study and will not be repeated here.This reserach project deals with slidiing mode hyperplane design for a class of linearr parameter-varying (LPV) plants, the state-space matrices' of which are an affine function of time varying physical parameters. The proposed hyperplane, involving a linear matrix inequality (LMI) approach, has continuous dynamics due to scheduling parameters and provides stability and robustness against parametric uncertainties, We have designcd a time-varying hyperplane for a rotor-magnetic bearing system with a gyroscopic effect, which can be considered an LPV plant due to parameter dependence on rotational speed. The obtained hyperplane is continuously scheduled with respect to rotational speed. We successfully carried out experiments using a commercially available turbomolecular pump system and results were reasonable and good.