74 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
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
Uniform bounded input bounded output stability of fractional‐order delay nonlinear systems with input
The bounded input bounded output (BIBO) stability for a nonlinear Caputo
fractional system with time-varying bounded delay and nonlinear output is
studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input
bounded output stability criteria are derived. Also, explicit and independent on
the initial time bounds of the output are provided. Uniform BIBO stability and
uniform BIBO stability with input threshold are studied. A numerical simulation is carried out to show the system’s dynamic response, and demonstrate the
effectiveness of our theoretical results.publishe
Well-posedness and robust stability of a nonlinear ODE-PDE system
This work studies stability and robustness of a nonlinear system given as an
interconnection of an ODE and a parabolic PDE subjected to external
disturbances entering through the boundary conditions of the parabolic
equation. To this end we develop an approach for a construction of a suitable
coercive Lyapunov function as one of the main results. Based on this Lyapunov
function we establish the well-posedness of the considered system and establish
conditions that guarantee the ISS property. ISS estimates are derived
explicitly for the particular case of globally Lipschitz nonlinearities.Comment: 41 page
Roughness of dichotomy for the interconnected system of operator-differential equations
The paper is devoted to obtaining conditions for the roughness of dichotomy
in the Banach spaces. Deep analysis of the well known papers was considered.
The main results also works for the case of unbounded operators
Robustness of Delayed Multistable Systems with Application to Droop-Controlled Inverter-Based Microgrids
Motivated by the problem of phase-locking in droop-controlled inverter-based microgrids with delays, the recently developed theory of input-to-state stability (ISS) for multistable systems is extended to the case of multistable systems with delayed dynamics. Sufficient conditions for ISS of delayed systems are presented using Lyapunov-Razumikhin functions. It is shown that ISS multistable systems are robust with respect to delays in a feedback. The derived theory is applied to two examples. First, the ISS property is established for the model of a nonlinear pendulum and delay-dependent robustness conditions are derived. Second, it is shown that, under certain assumptions, the problem of phase-locking analysis in droop-controlled inverter-based microgrids with delays can be reduced to the stability investigation of the nonlinear pendulum. For this case, corresponding delay-dependent conditions for asymptotic phase-locking are given
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