SIAM Journal on Control and Optimization, Volume 58, Issue 4, Page 2588-2615, January 2020.
We propose an extremum seeking control law for a driftless control-affine system with a state-dependent real-valued output function. The purpose of the control law is to asymptotically stabilize the closed-loop system around states at which the output function attains a local minimum. An implementation of the control law only requires measurements of the output values. The approach employs highly oscillatory inputs with suitably chosen frequencies. A detailed averaging analysis reveals that the closed-loop system is driven approximately into descent directions of the output function along Lie brackets of the control vector fields. Those descent directions also originate from an approximation of suitably chosen Lie brackets. The approximation properties are ensured if the amplitudes and frequencies of the oscillatory inputs are sufficiently large. The proposed method can lead to practical asymptotic stability even if the control vector fields do not span the entire tangent. It suffices instead that the tangent space is spanned by the elements in the Lie algebra generated by the control vector fields. This novel feature extends extremum seeking by Lie bracket approximations from the class of fully actuated systems to a larger class of nonholonomic systems.

中文翻译：

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一类非完整系统的极值搜索控制

SIAM控制与优化杂志，第58卷，第4期，第2588-2615页，2020年1月。
我们提出了一个极值寻求具有状态依赖实值输出函数的无漂移仿射系统的控制律。控制定律的目的是在输出函数达到局部最小值的状态附近渐近稳定闭环系统。控制律的实施仅需要测量输出值。该方法采用具有适当选择的频率高振荡输入。详细的平均分析表明，闭环系统大约沿着控制矢量场的李括号进入输出函数的下降方向。那些下降方向也从适当选择的李氏括号的近似起源。如果振荡输入的幅度和频率足够大，则可以确保近似特性。即使控制矢量场不跨越整个切线，所提出的方法也可以导致实用的渐近稳定性。相反，切线空间被控制矢量场生成的李代数中的元素所覆盖即可。这种新颖的功能将李氏逼近法的极值搜索范围从全驱动系统类别扩展到更大的非完整系统类别。