Switching ℒ₂ gain for evaluating the fluctuations in transient responses after an unpredictable system switch

ABSTRACT

We propose a new switching ${\mathcal{L}}_2$ gain for evaluating the fluctuations in transient responses after an unpredictable switch, such as a switch caused by a failure in a control device occurring in the normal operation, between two linear time-invariant systems. We also present ${\mathcal{L}}_2$ gain conditions that enable us to compute the value of the proposed switching ${\mathcal{L}}_2$ gain. We then theoretically establish its applicability as an analysis index. Furthermore, by applying the proposed gain to switching matrix design, we show its potential applicability as a design index.

KEYWORDS:

• Control system analysis
• ${\mathcal{L}}_2$ gain
• system switches
• linear systems
• fault-tolerant systems

Acknowledgments

This research was supported by KAKENHI Grant #17K06488, Japan Society for the Promotion of Science.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. This result plays an important role in resetting the internal state x _c (t) of a controller so that x _c (t _{0 +}) = αx _c (t ₀) (Suyama & Sebe, 2017a).

2. The input giving $\widehat{\gamma }$ obtained from (22(22) $$\begin{array}{c}\hfill \widehat{w}\left(t\right)=\left\{\begin{array}{cc}\hfill B_p^{\mathrm{T}}{e}^{-A_p^{\mathrm{T}}(t-t_0)}{X}_p^{-1}v,& \hfill t\le t_0\\ \hfill K_f{e}^{(A_f+B_f{K}_f)(t-t_0)}Sv,& \hfill t>t_0,\end{array}\right.\end{array}$$(22) ) can be normalised by adjusting |v| so that ‖w(t)‖_{2 ( − ∞, ∞)} = 1. The same holds true for ${\widehat{\gamma }}_{\mathrm{dit}}$ and ${\widehat{\gamma }}_{\mathrm{H}}$.

Additional information

Funding

Japan Society for the Promotion of Science: KAKENHI [grant number 17K06488].