Topological data analysis (TDA) is a powerful and promising tool for data analysis, but yet not exploited enough. It is a multidimensional method which can extract the topological features contained in a given dataset. An original TDA-based method allowing to monitor the health of structures when equipped with piezoelectric transducers (PZTs) is introduced here. Using a Lamb wave based Structural Health Monitoring (SHM) approach, it is shown that with specific pre-processing of the measured time-series data, the TDA (persistent homology) for damage detection and classification can be greatly improved. The TDA tool is first applied directly in a traditional manner in order to use homology classes to assess damage. After that, another method based on slicing the temporal data is developed to improve the persistence homology perception and to leverage topological descriptors to discriminate different damages. The dataset used to apply both methods comes from experimental campaigns performed on aeronautical composite plates with embedded PZTs where different damage types have been investigated such as delamination, impacts and stiffness reduction. The proposed approach enables to consider a priori physical information and provides a better way to classify damages than the traditional TDA approach. In summary, this article demonstrates that manipulating the topological the features of PZTs time-series signals using TDA provides an efficient mean to separate and classify the damage natures and opens the way for further developments on the use of TDA in SHM.