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Input-to-state stability of nonlinear impulsive systems
Sergey Dashkovskiy, Andrii Mironchenko
TL;DR
The paper addresses how to establish input-to-state stability for impulsive systems when unrestricted impulse sequences may not suffice. It develops dwell-time, small-gain, and linearization-based Lyapunov tools, proving ISS results for single systems and interconnections, including in Banach spaces.
Problem
When continuous or discrete dynamics is not ISS, ISS cannot generally be achieved for all impulse-time sequences, motivating restrictions on admissible sequences.
Method
The paper develops Lyapunov sufficient conditions, dwell-time analyses, small-gain constructions for interconnections, and linearization methods for local ISS-Lyapunov functions.
Results
The paper proves small-gain results for impulsive interconnections and establishes ISS under fixed dwell-time, with uniform ISS under generalized average dwell-time for exponential ISS Lyapunov functions.
Takeaways & Limitations
The results broaden ISS analysis to non-exponential Lyapunov functions, interconnected impulsive systems, and abstract systems based on equations in Banach spaces.
Takeaways & Limitations
For interconnected systems with subsystem-dependent impulse sequences, the resulting hybrid system is not covered by the theory developed in the paper.
Abstract
from arXiv · showhide
We prove that impulsive systems, which possess an ISS Lyapunov function, are ISS for time sequences satisfying the fixed dwell-time condition. If an ISS Lyapunov function is the exponential one, we provide a stronger result, which guarantees uniform ISS of the whole system over sequences satisfying the generalized average dwell-time condition. Then we prove two small-gain theorems that provide a construction of an ISS Lyapunov function for an interconnection of impulsive systems, if the ISS-Lyapunov functions for subsystems are known. The construction of local ISS Lyapunov functions via linearization method is provided. Relations between small-gain and dwell-time conditions as well as between different types of dwell-time conditions are also investigated. Although our results are novel already in the context of finite-dimensional systems, we prove them for systems based on differential equations in Banach spaces that makes obtained results considerably more general.
1. Introduction.
The introduction identifies dwell-time restrictions as necessary when continuous or discrete dynamics are not ISS, then presents Lyapunov, small-gain, and linearization tools for broader impulsive-system analysis.
- When either continuous or discrete dynamics is not ISS, dwell-time conditions restrict impulse sequences to recover ISS.
- The paper proves ISS under nonlinear fixed dwell-time conditions using ISS Lyapunov functions, including uniform stability under a slightly weaker condition.
- Exponential ISS Lyapunov functions yield uniform ISS over impulse sequences satisfying the generalized average dwell-time condition.
- The paper also develops global and local ISS-Lyapunov constructions, including local exponential functions obtained through linearization.
- Small-gain theorems construct ISS Lyapunov functions for interconnected impulsive systems, including exponential functions when subsystem gains are power functions.
- Relations among dwell-time conditions and their interplay with small-gain conditions are investigated for interconnected systems.
- The results are proved for abstract systems based on equations in Banach spaces, extending their scope beyond finite-dimensional systems.
2. Preliminaries.
The preliminaries formulate impulsive systems on Banach state and input spaces, define their trajectories and stability properties, and introduce the Lyapunov framework used later.
- An impulsive system combines continuous evolution with state jumps at a strictly increasing sequence of impulse times.
- The state belongs to a Banach space X, inputs belong to a Banach space U, and A generates a C0-semigroup while f and g specify system nonlinearities.
- The model assumes existence and uniqueness of solutions, with piecewise-continuous state trajectories and well-defined left limits at impulse times.
- The transition map φ records the state at time t for a specified initial condition, initial time, impulse sequence, and admissible input.
- Because impulse sequences affect time shifts, the system is generally not time-invariant, although shifted trajectories satisfy the stated relation.
- The unforced system is assumed to have the equilibrium x ≡ 0, and subsequent ISS analysis exploits ISS-Lyapunov functions.
- ISS requires the stability estimate globally, whereas LISS restricts initial states and inputs to a neighborhood; uniform ISS or GS uses sequence-independent comparison functions.
3. Lyapunov ISS theory for an impulsive system.
The section develops Lyapunov-based ISS conditions for impulsive systems, linking stability to dwell-time restrictions and extending results to local, exponential, and interconnected settings.
- Dwell-time conditions: An ISS-Lyapunov function does not ensure ISS for arbitrary impulse sequences; additional dwell-time restrictions are required.The impulse sequence affects stability even though the continuous-flow Lie derivative and Lyapunov function do not depend on it.
- Fixed dwell-time: An ISS-Lyapunov function guarantees ISS over sequences satisfying a nonlinear fixed dwell-time condition, while a slightly weaker condition guarantees uniform global stability.The corresponding theorem establishes ISS for all sequences in the admissible fixed-dwell-time class.
- Relations between dwell-time conditions: Dwell-time conditions expose a trade-off between allowable impulse density and gain size, and gADT can be weaker than classical ADT over long intervals.For h(x)=(x+1)e^(µ−λx), the resulting condition resembles ADT locally but imposes a considerably weaker restriction for large time intervals.
- Generalized average dwell-time: If the Lyapunov function is exponential, the generalized average dwell-time condition yields uniform ISS over its admissible sequence class.The gADT condition is described as providing, in a certain sense, tight estimates of the sequences for which ISS holds.
- Local construction and interconnections: A linearization method constructs local exponential ISS-Lyapunov functions, while small-gain results construct Lyapunov functions for interconnected impulsive systems.The construction extends to systems with power-function gains and uses bounded coercive operators in the linearization result.
4. ISS of interconnected impulsive systems.
The section formulates interconnected impulsive subsystems on Banach spaces and develops small-gain constructions for ISS Lyapunov functions, including exponential cases. It also relates gain choices to dwell-time requirements while identifying a class of interconnections beyond the paper’s theory.
- System formulation: The interconnected system is represented by subsystem dynamics on Banach spaces, with continuous evolution and jumps occurring at a common impulse sequence.Each subsystem has its own state space, input coupling, and jump map; the shared-sequence formulation can be written in vector form.
- Scope boundary: The paper’s results cover shared impulse sequences, but not general interconnections whose subsystems have different impulse-time sequences.Aggregating the sequences does not remove their dependence from the whole-system jump function, so such systems remain an open research topic.
- Small-gain construction: Theorem 8 constructs an ISS-Lyapunov function for the whole system when the subsystem gains satisfy the small-gain condition.The construction applies to subsystem ISS-Lyapunov functions and their corresponding gains.
- Small-gain construction: Theorem 9 extends the construction to exponential ISS Lyapunov functions when the gains are power functions and the small-gain condition holds.The resulting function is an exponential ISS Lyapunov function for the whole shared-impulse-time system.
- Small-gain construction: The interconnected example requires Lyapunov functions with one negative rate coefficient because positive-rate subsystem functions do not satisfy the small-gain condition.The constructed interconnection function has d = −1 and c = 1.672, after choosing b ≈ 0.612; Theorem 5 then gives impulse-sequence classes for GAS.
- Gain–dwell-time relations: The allowed impulse frequency depends on the difference c − d: smaller gains allow a smaller impulse frequency under the dwell-time condition.For the stated gain family, the admissible parameter interval is (ρ, ˜c), with limiting behavior described by Proposition 4.2.
5. Concluding remarks and open questions.
A future research direction is developing a theory for interconnected impulsive systems whose subsystems use different impulse-time sequences.
- Future work should address interconnected impulsive systems whose subsystems have different impulse-time sequences.