A two-step method for analysis of linear systems with uncertain parameters driven by Gaussian noise

• A two-step method is developed to find properties of linear systems with uncertain parameters driven by Gaussian noise.
• Step one is to determine the probability law of the system state conditional on the uncertain parameters.
• Step two is to utilize stochastic reduced-order models (SROMs) to determine the unconditional state properties.
• Bayesian methods are employed when the probability law of the uncertain parameters is unknown.

A two-step method is proposed to find state properties for linear dynamic systems driven by Gaussian noise with uncertain parameters modeled as a random vector with known probability distribution. First, equations of linear random vibration are used to find the probability law of the state of a system with uncertain parameters conditional on this vector. Second, stochastic reduced order models (SROMs) are employed to calculate properties of the unconditional system state. Bayesian methods are applied to extend the proposed approach to the case when the probability law of the random vector is not available. Various examples are provided to demonstrate the usefulness of the method, including the random vibration response of a spacecraft with uncertain damping model.