This paper proposes an estimator-based tracking control for a thermoelectric cooling system with Peltier cells. It is intended for use in a novel manufacturing system leveraging ice clamping. Starting from physical principles and conservation equations, a 4th-order state space model is developed and exploited within an input-output linearization, where the internal dynamics can be shown to be asymptotically stable. A backstepping-based tracking control is designed to accurately track the desired cold side temperature even in the presence of disturbances. Unknown states and disturbances are observed using an Unscented Kalman Filter, which outperforms previous results with alternative estimators. Conclusive simulation results are discussed and demonstrate the performance of the combined control and estimation structure.