Strong exponential stability of switched impulsive systems with mode-constrained switching

TL;DR

The paper addresses deriving explicit strong global uniform exponential stability (S-GUES) bounds for nonlinear switched impulsive systems under mode-constrained switching. It develops a Lyapunov-based framework that combines mode-dependent average dwell-time (MDADT) and activation-time constraints, enabling computation of the maximum overshoot and decay rate through per-mode Lyapunov bounds and path-dependent weights. A constructive main theorem provides sufficient conditions for S-GUES with computable constants $K$ and $\lambda$, and a supporting lemma shows how to select per-mode coefficients to satisfy the required inequalities; the decay rate can be improved by analyzing longer mode sequences via $R(L)$. The approach handles unstable subsystems and impulses at switching, offers a path to robustness against disturbances, and is illustrated with numerical examples including perturbed nonlinear systems to demonstrate applicability and potential for controller design.

Abstract

Strong stability, defined by bounds that decay not only over time but also with the number of impulses, has been established as a requirement to ensure robustness properties for impulsive systems with respect to inputs or disturbances. Most existing results, however, only consider weak stability, where the bounds only decay with time. In this paper, we provide a method for calculating the maximum overshoot and the decay rate for strong global uniform exponential stability bounds for nonlinear switched impulsive systems. We consider the scenario of mode-constrained switching where not all transitions between subsystems are allowed, and where subsystems may exhibit unstable dynamics in the flow and/or jump maps. Based on direct and reverse mode-dependent average dwell-time and activation-time constraints, we derive stability bounds that can be improved by considering longer switching sequences for computation. We provide an example that shows how the results can be employed to ensure the stability robustness of nonlinear systems that admit a global state weak linearization.

Figures (6)

• Figure 2.1: Example of the admissible jumps for a switched impulsive system with $N=3$ subsystems, $\mathcal{J}_{_{ \neq}}=\{(1,2),(2,1),(2,3),(3,1)\}$ and $\mathcal{J}_{_{ =}} = \{(1,1),(2,2),(3,3)\}$. Switching from mode $1$ to $3$ and from $3$ to $2$ is not possible.
• Figure 3.1: Upper half-plane (dotted pattern): points where \ref{['n_imp>1']} holds. Lower half-plane: points where \ref{['n_imp<1']} holds. For different initial values $\lambda_i(0)=\bar{\lambda}(i)$ and $r_i(0)=\ln(\bar{r}(i,i))$, the values of $\lambda_i(c_i)$ and $r_i(c_i)$ change in the direction of the arrows as $c_i \geq 0$ increases. Green: points that satisfy $\lambda_i(1)<0$. These allow increasing $c_i \geq 0$ in order to make both $r_i(c_i)<0$ and $\lambda_i(c_i)<0$. Red: points for which $\lambda_i(1)\geq0$. These allow increasing $c_i\ge 0$ in order to make $r_i(c_i)<0$ at the expense of making $\lambda_i(c_i)$ positive and larger. The same value $T_J^i$ of the corresponding direct or reverse constraint of Assumption \ref{['assumption32']} is considered for each case in this example, thus the segments have the same slope.
• Figure 4.1: The jump graph of possible jumps for the system.
• Figure 4.2: Bounds from assumptions \ref{['assumption32']} and \ref{['ass:jumps']} for the current simulation with $t_0=0$. Top plot: the number of switchings $n_\nu(t,t_0)$ is bounded from above or below according to the value of $R(L)$. Middle and bottom: the impulsive jumps of subsystems $1$ and $2$ are upper and lower bounded according to \ref{['n_imp>1']} and \ref{['n_imp<1']}, respectively. All plots: bounds are plotted as dashed lines.
• Figure 4.3: Activation times (solid line) and activation time bounds (dashed line) of Assumption \ref{['ass_times']} for the current simulation. Subsystem $1$ begins active, with $t_a(1,t,t_0)$ and the lower bound \ref{['tiempo1']} in blue and $t_a(2,t,t_0)$ and the upper bound \ref{['tiempo2']} in red.

Theorems & Definitions (21)

• Remark 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.5
• Theorem 3.3
• proof
• Lemma 3.4
• proof
• Corollary 3.5: Of Theorem \ref{['thm32_strong2']}
• proof