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A sliding mode observer (SMO) design and convergence analysis are presented in this paper, which includes a rigorous treatment to address multiple discontinuities in the resulting estimation error dynamics. In an extension of our previous SMO results, the current work provides a non-trivial reworking of the SMO estimation error system development and stability analysis that incorporates differential inclusions. The specific contributions presented in this paper beyond the previous work include: (1) A differential inclusions-based analysis of the SMO, which incorporates the set-valued definition of the discontinuous signum function; (2) An expanded derivation of the estimation error dynamics, which emphasises advantageous properties particular to our SMO structure; (3) A Lyapunov-based stability analysis of the SMO, that rigorously incorporates the multiple discontinuities in the estimation error dynamics. The Lyapunov-based stability analysis proves that the SMO achieves finite-time estimation of the complete state vector, where the output equation is in a nonstandard mathematical form. To test the performance of the SMO, numerical simulation results are also provided, which demonstrate the capability of the SMO to estimate the state of a fluid flow dynamic system using only a single sensor measurement of the flow field velocity. [ABSTRACT FROM AUTHOR]