On stability and state-norm estimation of switched systems under restricted switching

TL;DR

This work addresses stability and state estimation for continuous-time switched nonlinear systems when switching is restricted by admissible transitions and dwell times. It builds a unified framework based on multiple Lyapunov-like functions to derive sufficient conditions under which $IOSS$ holds for all admissible switching signals in a restricted class, and it designs state-norm estimators that are themselves switched systems with one stable and one unstable mode. The main contributions include (i) a dwell-time based inequality that yields uniform $IOSS$ over a restricted switching set, (ii) a constructive design of state-norm estimators robust to switching instants, and (iii) a numerical example validating the theory. The results enable stable operation and state magnitude estimation for complex switched systems under practical switching constraints, with potential extensions to broader switching patterns in future work.

Abstract

This paper deals with the analysis of input/output-to-state stability (IOSS) and construction of state-norm estimators for continuous-time switched nonlinear systems under restricted switching. Our contributions are twofold. First, given a family of systems, possibly containing unstable dynamics, a set of admissible switches between the subsystems and admissible minimum and maximum dwell times on the subsystems, we identify a class of switching signals that obeys the given restrictions and preserves IOSS of the resulting switched system. Second, we design a class of state-norm estimators for switched systems under our class of stabilizing switching signals. These estimators are switched systems themselves with two subsystems -- one stable and one unstable. The key apparatus for our analysis is multiple Lyapunov-like functions. A numerical example is presented to demonstrate the results.

Figures (4)

• Figure 1: Plot for $\bigl(\left\lVert{x(t)}\right\rVert\bigr)_{t\geqslant 0}$.
• Figure 2: Plot for $\bigl(\left\lVert{x(t)}\right\rVert\bigr)_{t\geqslant 0}$ (solid line) and $\bigl(z(t)\bigr)_{t\geqslant 0}$ (dashed line).
• Figure 3: $\sigma$ and $\zeta$. $0$ and $0.5$ denote dynamics $g_0$ and $g_1$, respectively.
• Figure 4: $\overline{\beta}\bigl(\left\lVert{x(0)}\right\rVert+\left\lVert{z(0)}\right\rVert,t\bigr)$ (blue line) and $\overline{\chi}\bigl(\left\lVert{z(t)}\right\rVert\bigr)$ (black line).

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• Remark 1
• Proposition 1
• Proposition 2
• Remark 2
• Theorem 2