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royAdaptiveSlidingMode2020

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文献标题

On adaptive sliding mode control without a priori bounded uncertainty

Abstract

Adaptive Sliding Mode Control (ASMC) aims to adapt the switching gain in such a way to cope with possibly unknown uncertainty. In state-of-the-art ASMC methods, a priori boundedness of the uncertainty is crucial to ensure boundedness for the switching gain and uniformly ultimately boundedness. A priori bounded uncertainty might impose a priori bounds on the system state before obtaining closed-loop stability. A design removing this assumption is still missing in literature. A positive answer to this quest is given by this note where a novel ASMC methodology is proposed which does not require a priori bounded uncertainty. An illustrative example is presented to highlight the main features of the approach, after which a general class of Euler–Lagrange systems is taken as a case study to show the applicability of the proposed design.

Contents

问题描述

问题背景

• A design challenge in sliding mode control is to tackle uncertainties in the system to be controlled without prior knowledge about them
  • Based on this, lead to #ASMC, the switching-gain is adapted to cope with possibly unknown uncertainty.

前人工作

• increase monotonically the switching-gain
  • lead to high gain
• Another way
  • increasing-decreasing ASMC
    • @plestanNewMethodologiesAdaptive2010
  • equivalent control ASMC

前人缺陷

• Most work assume the uncertainty or its time-derivative to be upper bounded by a constant
  • such prior constant upper bound might be restrictive hence it requires the states have a upper bound before obtaining system stability
  • Illustrative example
    • shows that if initial K0 is not high enough, the instablity might arise
    • 

本文工作

本文意义

实验方法

控制方法

在 @plestanNewMethodologiesAdaptive2010 提出的方法基础上：

$$\begin{array}{l}\tau \left(t\right)=-K\mathrm{sgn}\left(s\left(t\right)\right)\\ \dot{K}\left(t\right)=\{\begin{array}{l}\bar{K}\left|s\left(t\right)\right|\mathrm{sgn}\left(\left|s\left(t\right)\right|-ϵ\right)\\ \mu \end{array}\begin{array}{c}\mathrm{if}K\left(t\right)⩾\mu \\ \mathrm{if}K\left(t\right)<\mu \end{array}\end{array}$$

^eqn-raw-sm

提出新的方法：

考虑这样一个 scalar system：

$$\dot{q}(t)=-cq(t)+\tau (t)+d(q,t)$$

其中 $c>0$ 是未知的，$\tau$ 是控制输入，$d$ 代表建模误差和与时间相关的扰动

根据选择的滑模变量，有：

$$\dot{s}=\frac{\partial s}{\partial t}+\frac{\partial s}{\partial q}\dot{q}=\frac{\partial s}{\partial t}+\underset{a\left(q,t\right)}{\underbrace{\frac{\partial s}{\partial q}\left(-cq+d\right)}}+\underset{b\left(q,t\right)}{\underbrace{\frac{\partial s}{\partial q}}}\tau$$

假定

$$\left|a\left(q,t\right)\right|⩽K_0^{\ast }+K_1^{\ast }\left|q\right|,\forall q,\forall t⩾0$$

note

这里我个人觉得其实不是很好的假设，还是没有脱离上界限制，无非是修改成了可变大小的上界，这个可变大小如何界定是一个问题

从而提出：

$$\begin{array}{l}\tau \left(t\right)=-\Lambda s\left(t\right)-\rho \left(t\right)\mathrm{sgn}\left(s\left(t\right)\right)\\ \rho \left(t\right)={\hat{K}}_0\left(t\right)+{\hat{K}}_1\left(t\right)\left|q\left(t\right)\right|\\ {\dot{\hat{K}}}_0\left(t\right)=\left|s\left(t\right)\right|-{\alpha }_0{\hat{K}}_0\left(t\right)\\ {\dot{\hat{K}}}_1\left(t\right)=\left|s\left(t\right)\right|\left|q\left(t\right)\right|-{\alpha }_1{\hat{K}}_1\left(t\right)\end{array}$$

#todo 进一步分析可以以后再做，看这篇文章主要是为了找一下 #ASMC 相关研究的论文

优点缺点

优点

缺点

个人评价