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The trajectory piece-wise linear (TPWL) method is a popular technique for nonlinear model order reduction (MOR). Though widely studied, it has primarily been restricted to applications modeled by nonlinear systems with linear input operators. This paper is an effort to bridge this gap. We illustrate problems in the TPWL method in creating reduced order models for nonlinear systems with nonlinear input operators. We also propose a solution based on a quasi-linear formulation of the nonlinear system at approximation points. This results in a method for nonlinear MOR, called the trajectory piece-wise quasi-linear (TPWQ) method. TPWQ is formulated, numerically validated and a new technique to reduce the computational costs associated with simulating the quasi-linear systems is also demonstrated.