Discrete-time Lyapunov-based small-gain theorem for parameterized interconnected ISS systems

Abstract

Input-to-state stability (ISS) of a feedback interconnection of two discrete-time ISS systems satisfying an appropriate small gain condition is investigated via the Lyapunov method. In particular, an ISS Lyapunov function for the overall system is constructed from the ISS Lyapunov functions of the two subsystems. We consider parameterized families of discrete-time systems that naturally arise when an approximate discrete-time model is used in controller design for a sampled-data system.

Stability Analysis of Hybrid Systems Via Small-Gain Theorems

“…Theorem 1 is still valid if we let z 1 and z 2 include only the states of interest for each subsystem. 5 Small-gain theorems have been widely used for analysis of continuous-time as well as discrete-time systems with feedback interconnection structure. The discussion of Section 2.2 suggests that it is also very natural to use this idea to analyze (internal or external) stability of hybrid systems.…”

“…The discussion of Section 2.2 suggests that it is also very natural to use this idea to analyze (internal or external) stability of hybrid systems. Of course, we will need to be able to prove that the subsystems in a given feedback decomposition satisfy suitable ISS properties, and calculate the ISS gains in order to check (5). There exist efficient tools for doing this, as exemplified in the next section.…”

“…The question naturally arises whether Theorem 1 can be formulated and proved entirely in terms of such Lyapunov functions. It is known that such an alternative formulation is possible for continuous-time as well as discretetime small-gain theorems [4,5], but this issue has not been pursued for hybrid systems.…”

“…As an alternative to time-domain proofs, Lyapunov function constructions for interconnected systems under small-gain conditions were suggested for continuoustime systems in [4] and for discrete-time systems in [5]. It is generally well known that having a Lyapunov function provides additional insight into the behavior of a stable system and is important for tasks such as perturbation analysis and estimating the region of attraction.…”

Towards -stability of discrete-time reset control systems via dissipativity theory

“…First, characterisation of passivity, dissipativity, and feedback passivity and dissipativity by using zero dynamics and relative degree properties [11,27,28,31,30,35]. Second, preservation of dissipativity and stability properties under sampling [26,24,25,32].…”