用集值KKL观测器重构不可区分解
Reconstructing Indistinguishable Solutions Via Set-Valued KKL Observer
摘要：
KKL观测器设计在于寻找坐标的平滑变化，将系统动态特性转换为输出的线性滤波。然后，通过从任何初始条件实现该滤波器并在\textit{backward-distinguishability}属性下对变换进行左反转来重构原始系统的状态。在本文中，我们考虑的情况下，后者的假设并不成立，即当不同的解决方案可能会产生相同的输出，从而无法区分。KKL变换不再是单射的，因此它的“左逆”被允许是集值的，产生一个集值KKL观测器。假设变换是满秩的，其原像具有常数基数，我们证明了存在一个全局定义的集值左逆，即在Hausdorff意义下的Lipschitz。利用最近的结果连接这个左逆与\textit{backward-indistinguishable sets}，我们表明，集值KKL观测器收敛在Hausdorff意义上的向后不可区分的系统解决方案。此外，当一个给定的输出所产生的特定数量的解决方案不收敛到对方，我们表明，所设计的观察器渐近重建这些解决方案。最后，不同的假设进行了讨论，并通过例子说明。
英文摘要：
KKL observer design consists in finding a smooth change of coordinates transforming the system dynamics into a linear filter of the output. The state of the original system is then reconstructed by implementing this filter from any initial condition and left-inverting the transformation, under a \textit{backward-distinguishability} property. In this paper, we consider the case where the latter assumption does not hold, namely when distinct solutions may generate the same output, and thus be indistinguishable. The KKL transformation is no longer injective and its ``left-inverse'' is thus allowed to be set-valued, yielding a set-valued KKL observer. Assuming the transformation is full-rank and its preimage has constant cardinality, we show the existence of a globally defined set-valued left-inverse that is Lipschitz in the Hausdorff sense. Leveraging on recent results linking this left-inverse with the \textit{backward-indistinguishable sets}, we show that the set-valued KKL observer converges in the Hausdorff sense to the backward-indistinguishable set of the system solution. When, additionally, a given output is generated by a specific number of solutions not converging to each other, we show that the designed observer asymptotically reconstructs each of those solutions. Finally, the different assumptions are discussed and illustrated via examples.
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