## Abstract

This paper applies the $H2$ norm along time and parameter domains. The norm is related to the probabilistic $H2$ problem. It is calculated using polynomial chaos to handle uncertainty in the plant model. The structure of expanded states resulting from Galerkin projections of a state space model with uncertain parameters is used to formulate cost functions in terms of mean performances of the states, as well as covariances. Also, bounds on the norm are described in terms of linear matrix inequalitys. The form of the gradient of the norm, which can be used in optimization, is given as a Lyapunov equation. Additionally, this approach can be used to solve the related probabilistic LQR problem. The legitimacy of the concept is demonstrated through two mechanical oscillator examples. These controllers could be easily implemented on physical systems without observing uncertain parameters.

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