Theoretical Error Bounds for Stochastic Linearization of Feedback Systems

Stochastic model-based assessment of power systems subject to extreme wind power fluctuation

“…The obtained error bound can be understood intuitively as follows: First, we consider the case S = I corresponding to the previous result [8]. The L 1 -gain [14] of C y (sI − (A + BKC y )) −1 B is less than or equal to η y .…”

“…In addition, we can employ [8,Theorem 2] to compute η j (K) by numerical integration, which is computationally much more effective than Monte Carlo methods. Hence, it is not necessary to generate samples at all in computing the error bound in Theorem 3.1.…”

“…Differently from the previous error bounds in [8], we can relax the condition required to obtain the bounds and tighten them by choosing a suitable scaling. Especially when one wants to find a scaling that satisfies ζ η y < 1 in Theorem 3.1, it is reasonable to minimize η y with respect to S ∈ S since ζ is a constant.…”

“…To resolve these issues, a stochastic linearization method [6,7] is extended for the systems driven by stable processes based on their mathematical similarity to Gaussian distributions [3]. Moreover, our previous work [3,8] derived theoretical error bounds for stochastic linearization to guarantee the accuracy of the approximation. This makes it possible to perform approximate evaluations with guaranteed accuracy without generating any sample paths.…”

“…Actually, limits and expectation in the proof of the main result in [8] are interchanged partially carelessly. Its theoretically rigorous justification is also provided in this paper; see Appendix 2.…”