Björn S. Rüffer

An ISS small gain theorem for general networks

We provide a generalized version of the nonlinear small gain theorem for the case of more than two coupled input-to-state stable systems. For this result the interconnection gains are described in a nonlinear gain matrix, and the small gain condition requires bounds on the image of this gain matrix. The condition may be interpreted as a nonlinear generalization of the requirement that the spectral radius of the gain matrix is less than 1. We give some interpretations of the condition in special cases covering two subsystems, linear gains, linear systems and an associated lower-dimensional discrete time dynamical system.

Small Gain Theorems for Large Scale Systems and Construction of ISS Lyapunov Functions

We consider interconnections of n nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems.

Robust Stability of Networks of iISS Systems: Construction of Sum-Type Lyapunov Functions

This paper gives a solution to the problem of verifying stability of networks consisting of integral input-to-state stable (iISS) subsystems. The iISS small-gain theorem developed recently has been restricted to interconnections of two subsystems. For large-scale systems, stability criteria relying only on gain-type information that were previously developed address only input-to-state stable (ISS) subsystems. To address the stability problem involving iISS subsystems interconnected in general structure, this paper shows how to construct Lyapunov functions of the network by means of a sum of nonlinearly rescaled individual Lyapunov functions of subsystems under an appropriate small-gain condition.

Local ISS of large-scale interconnections and estimates for stability regions

We consider interconnections of locally input-to-state stable (LISS) systems. The class of LISS systems is quite large, in particular it contains input-to-state stable (ISS) and integral input-to-state stable (iISS) systems. Local small-gain conditions both for LISS trajectory and Lyapunov formulations guaranteeing LISS of the composite system are provided in this paper. Notably, estimates for the resulting stability region of the composite system are also given. This in particular provides an advantage over the linearization approach, as will be discussed.

A small-gain type stability criterion for large scale networks of ISS systems

We provide a generalized version of the nonlinear small-gain theorem for the case of more than two coupled input-to-state stable systems. For this result the interconnection gains are described in a nonlinear gain matrix and the small gain condition requires bounds on the image of this gain matrix. The condition may be interpreted as a nonlinear generalization of the requirement that the spectral radius of the gain matrix is less than one. We give some interpretations of the condition in special cases covering linear gains and linear systems.

Numerical verification of local input-to-state stability for large networks

We consider networks of locally input-to-state stable (LISS) systems. Under a small gain condition the entire network is again LISS. An efficient numerical test to check the small gain condition is presented in this paper. An example from applications serves as a demonstration for quantitative results.

A LYAPUNOV ISS SMALL-GAIN THEOREM FOR STRONGLY CONNECTED NETWORKS

Abstract We consider strongly connected networks of input-to-state stable (ISS) systems. Provided a small gain condition holds it is shown how to construct an ISS Lyapunov function using ISS Lyapunov functions of the subsystems. The construction relies on two steps: The construction of a strictly increasing path in a region defined on the positive orthant in ℝ n by the gain matrix and the combination of the given ISS Lyapunov functions of the subsystems to a ISS Lyapunov function for the composite system. Novelties are the explicit path construction and that all the involved Lyapunov functions are nonsmooth, i.e., they are only required to be locally Lipschitz continuous. The existence of a nonsmooth ISS Lyapunov function is qualitatively equivalent to ISS.

A small-gain theorem and construction of sum-type Lyapunov functions for networks of iISS systems

This paper gives a solution to the problem of verifying stability of networks consisting of integral input-to state stable (iISS) subsystems. The iISS small-gain theorem developed recently has been restricted to interconnection of two subsystems. For large-scale systems, stability criteria re lying only on gain-type information have been successful only in dealing with input-to-state stable stable (ISS) subsystems. To address the stability problem involving iISS subsystems interconnected in general structure, this paper shows how to construct Lyapunov functions of the network by means of nonlinear sum of individual Lyapunov functions of subsystems given in a dissipation formulation under an appropriate small gain condition.

Separable Lyapunov functions for monotone systems

Separable Lyapunov functions play vital roles, for example, in stability analysis of large-scale systems. A Lyapunov function is called max-separable if it can be decomposed into a maximum of functions with one-dimensional arguments. Similarly, it is called sum-separable if it is a sum of such functions. In this paper it is shown that for a monotone system on a compact state space, asymptotic stability implies existence of a max-separable Lyapunov function. We also construct two systems on a non-compact state space, for which a max-separable Lyapunov function does not exist. One of them has a sum-separable Lyapunov function. The other does not.

Max- and sum-separable Lyapunov functions for monotone systems and their level sets

For interconnected systems and systems of large size, aggregating information of subsystems studied individually is useful for addressing the overall stability. In the Lyapunov-based analysis, summation and maximization of separately constructed functions are two typical approaches in such a philosophy. This paper focuses on monotone systems which are common in control applications and elucidates some fundamental limitations of max-separable Lyapunov functions in estimating domains of attractions. This paper presents several methods of constructing sum- and max-separable Lyapunov functions for second order monotone systems, and some comparative discussions are given through illustrative examples.