32,487 research outputs found
Input-to-state stability of infinite-dimensional control systems
We develop tools for investigation of input-to-state stability (ISS) of
infinite-dimensional control systems. We show that for certain classes of
admissible inputs the existence of an ISS-Lyapunov function implies the
input-to-state stability of a system. Then for the case of systems described by
abstract equations in Banach spaces we develop two methods of construction of
local and global ISS-Lyapunov functions. We prove a linearization principle
that allows a construction of a local ISS-Lyapunov function for a system which
linear approximation is ISS. In order to study interconnections of nonlinear
infinite-dimensional systems, we generalize the small-gain theorem to the case
of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov
function for an entire interconnection, if ISS-Lyapunov functions for
subsystems are known and the small-gain condition is satisfied. We illustrate
the theory on examples of linear and semilinear reaction-diffusion equations.Comment: 33 page
The converse of the passivity and small-gain theorems for input-output maps
We prove the following converse of the passivity theorem. Consider a causal
system given by a sum of a linear time-invariant and a passive linear
time-varying input-output map. Then, in order to guarantee stability (in the
sense of finite L2-gain) of the feedback interconnection of the system with an
arbitrary nonlinear output strictly passive system, the given system must
itself be output strictly passive. The proof is based on the S-procedure
lossless theorem. We discuss the importance of this result for the control of
systems interacting with an output strictly passive, but otherwise completely
unknown, environment. Similarly, we prove the necessity of the small-gain
condition for closed-loop stability of certain time-varying systems, extending
the well-known necessity result in linear robust control.Comment: 15 pages, 3 figure
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