Abstract
A linear stochastic vibration model in state-space form, x˙(t)=Ax(t)+b(t),x(0)=x0, with output equation xS(t)=Sx(t) is investigated, where A is the system matrix and b(t) is the white noise excitation. The output equation xS(t)=Sx(t) can be viewed as a transformation of the state vector x(t) that is mapped by the rectangular matrix S into the output vector x(t). It is known that, under certain conditions, the solution x(t) is a random vector that can be completely described by its mean vector, mx(t):=mx(t), and its covariance matrix, Px(t):=Px(t). If matrix A is asymptotically stable, then mx(t)→0(t→∞) and PxS(t)→PS(t→∞), where PS is a positive (semi-)definite matrix. Similar results will be derived for some output-related quantities. The obtained results are of special interest to applied mathematicians and engineers.
Highlights
In order to make the paper more readable for a large readership, we first introduce the notions of output vector and output equation common to engineers
We investigate the asymptotic behavior of mxS (t) and PxS (t) − PS
The linear stochastically excited vibration system with output equation In order to make the paper as far as possible self-contained, we summarize the known facts on linear stochastically excited systems
Summary
In order to make the paper more readable for a large readership, we first introduce the notions of output vector and output equation common to engineers. Theorem 5 (Two-sided bound on ΦS(t) = SΦ(t) by e [A]t) Let A ∈ CI n×n and Φ(t) be the fundamental matrix of A with Φ(0) = E, i.e. let Φ(t) be the solution of the initial value problem Φ (t) = AΦ(t), Φ(0) = E
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