14 research outputs found
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. The results are obtained in a general formulation of ISS, the
cases of summation, maximization and separation with respect to external gains
are obtained as corollaries.Comment: provisionally accepted by SIAM Journal on Control and Optimizatio
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 (ISS) 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 two subsystems,
linear gains, linear systems and an associated artificial dynamical system.Comment: 26 pages, 3 figures, submitted to Mathematics of Control, Signals,
and Systems (MCSS
Applications of the general Lyapunov ISS small-gain theorem for networks
In the framework of the ISS Lyapunov formulation a small gain theorem has recently been proved which allows the explicit construction of Lyapunov functions for interconnected systems. In this note we recall the definitions of ISS Lyapunov functions and the corresponding general small gain theorems. These are then exemplarily used to prove input-to-state stability of and to construct ISS Lyapunov functions for four areas of applications: Linear systems, a Cohen-Grossberg neuronal network, error dynamics in formation control, as well as nonlinear transistor-linear resistor circuits