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Design of robust control systems requires efficient ways to compute variations of responses of interest with respect to a wide range of variations for a large number of initial conditions. This is necessary in order to perform engineering oriented applications such as design optimization, inverse studies, and sensitivity analysis. A reduced order model based on an adjoint approach that takes advantage of the contraction in the state rather than the response phase space is developed to calculate the variations in responses of interest with respect to input parameters. The approach is designed to combat the explosion in the state phase space often limiting the design of reduced order models. We show that the developed adjoint approach is independent of the given response, and is only dependent on the constraint equations relating initial conditions to the state variables. The mathematical framework hybridizes sampling techniques with adjoint methods to find the reduced order model. Its construction permits a general applicability to linear and nonlinear dynamical systems with general initial conditions variations. A proof of principle linear problem is demonstrated in this summary. The details of its general applicability to nonlinear models are left to a full journal article.