## Abstract

This paper investigates the problem of observer-based finite time sliding mode control (SMC) for a class of one-sided Lipschitz (OSL) systems with uncertainties. The parameter uncertainties are assumed to be time-varying norm-bounded appearing not only in both the state and output matrices but also in the nonlinear function. For a time interval $[0,T]$, we divide it into two parts: one part is the reaching phase within $[0,T*]$ and another part is the sliding motion phase within $[T*,T]$. First, the reachability of the sliding mode surface with $T*≤T$ is proved. Next, several conditions are proposed which ensure robust finite time boundedness (FTB) of the corresponding closed-loop systems in the interval $[0,T*]$ and $[T*,T]$, respectively. Then, the sufficient conditions, which guarantee robust finite time boundedness of the closed-loop system in whole time interval $[0,T]$, are given in terms of linear matrix inequalities (LMIs), and further the robust observer and controller can be designed in an LMI frame. A convex optimization problem subject to LMIs is formulated to optimize the desired performance indices of interest to us. Finally, a practical example is given to demonstrate the effectiveness of the proposed methods.

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