119,335 research outputs found
A characterization of integral input-to-state stability
Abstract-The notion of input-to-state stability (ISS) is now recognized as a central concept in nonlinear systems analysis. It provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite 2 gains. It plays a central role in recursive design, coprime factorizations, controllers for nonminimum phase systems, and many other areas. In this paper, a newer notion, that of integral input-to-state stability (iISS), is studied. The notion of iISS generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states, in that sense generalizing the linear " 2 " theory. It allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance. We obtain here several necessary and sufficient characterizations of the iISS property, expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations. These characterizations serve to show that integral input-to-state stability is a most natural concept, one that might eventually play a role at least comparable to, if not even more important than, ISS
Semi-uniform Input-to-state Stability of Infinite-dimensional Systems
We introduce the notions of semi-uniform input-to-state stability and its
subclass, polynomial input-to-state stability, for infinite-dimensional
systems. We establish a characterization of semi-uniform input-to-state
stability based on attractivity properties as in the uniform case. Sufficient
conditions for linear systems to be polynomially input-to-state stable are
provided, which restrict the range of the input operator depending on the rate
of polynomial decay of the product of the semigroup and the resolvent of its
generator. We also show that a class of bilinear systems are polynomially
integral input-to-state stable under a certain smoothness assumption on
nonlinear operators.Comment: 28 page
On an integral variant of incremental input/output-to-state stability and its use as a notion of nonlinear detectability
We propose a time-discounted integral variant of incremental
input/output-to-state stability (i-iIOSS) together with an equivalent Lyapunov
function characterization. Continuity of the i-iIOSS Lyapunov function is
ensured if the system satisfies a certain continuity assumption involving the
Osgood condition. We show that the proposed i-iIOSS notion is a necessary
condition for the existence of a robustly globally asymptotically stable
observer mapping in a time-discounted ``-to-'' sense. In
combination, our results provide a general framework for a Lyapunov-based
robust stability analysis of observers for continuous-time systems, which in
particular is crucial for the use of optimization-based state estimators (such
as moving horizon estimation).Comment: replaced with accepted versio
On Relaxed Conditions of Integral ISS for Multistable Periodic Systems
International audienceA novel characterization of the integral Inputto-State Stability (iISS) property is introduced for multistable systems whose dynamics are periodic with respect to a part of the state. First, the concepts of iISS-Leonov functions and output smooth dissipativity are introduced, then their equivalence to the properties of bounded-energy-bounded-state and global attractiveness of solutions in the absence of disturbances are proven. The proposed approach permits to relax the usual requirements of positive definiteness and periodicity of the iISS-Lyapunov functions. Moreover, the usefulness of the theoretical results is illustrated by a robustness analysis of a nonlinear pendulum with a constant bias input and an unbounded statedependent input coefficient
Online Optimization of Dynamical Systems with Deep Learning Perception
This paper considers the problem of controlling a dynamical system when the
state cannot be directly measured and the control performance metrics are
unknown or partially known. In particular, we focus on the design of
data-driven controllers to regulate a dynamical system to the solution of a
constrained convex optimization problem where: i) the state must be estimated
from nonlinear and possibly high-dimensional data; and, ii) the cost of the
optimization problem -- which models control objectives associated with inputs
and states of the system -- is not available and must be learned from data. We
propose a data-driven feedback controller that is based on adaptations of a
projected gradient-flow method; the controller includes neural networks as
integral components for the estimation of the unknown functions. Leveraging
stability theory for perturbed systems, we derive sufficient conditions to
guarantee exponential input-to-state stability (ISS) of the control loop. In
particular, we show that the interconnected system is ISS with respect to the
approximation errors of the neural network and unknown disturbances affecting
the system. The transient bounds combine the universal approximation property
of deep neural networks with the ISS characterization. Illustrative numerical
results are presented in the context of control of robotics and epidemics.Comment: This is an extended version of the paper submitted to the IEEE Open
Journal of Control Systems - Special Section on Machine Learning with
Control, containing proof
Lyapunov Conditions for Input-to-State Stability of Impulsive Systems
This paper introduces appropriate concepts of input-to-state stability (ISS) and integral-ISS for impulsive systems, i.e., dynamical systems that evolve according to ordinary differential equations most of the time, but occasionally exhibit discontinuities (or impulses). We provide a set of Lyapunov-based sufficient conditions for establishing these ISS properties. When the continuous dynamics are ISS but the discrete dynamics that govern the impulses are not, the impulses should not occur too frequently, which is formalized in terms of an average dwell-time (ADT) condition. Conversely, when the impulse dynamics are ISS but the continuous dynamics are not, there must not be overly long intervals between impulses, which is formalized in terms of a novel reverse ADT condition. We also investigate the cases where (i) both the continuous and discrete dynamics are ISS and (ii) one of these is ISS and the other only marginally stable for the zero input, while sharing a common Lyapunov function. In the former case we obtain a stronger notion of ISS, for which a necessary and sufficient Lyapunov characterization is available. The use of the tools developed herein is illustrated through examples from a Micro-Electro-Mechanical System (MEMS) oscillator and a problem of remote estimation over a communication network
Feedforward/feedback control of multivariable nonlinear processes
This paper concerns general MIMO nonlinear processes, whose dynamic behavior is described by a standard state-space model of arbitrary order, including measurable disturbances. The concept of relative order of an output with respect to an input, extended to include disturbance as well as manipulated inputs, is generalized in a MIMO context and it is used to obtain a characterization of the dynamic interactions among the input and the output variables. A synthesis formula is calculated for a feedforward/state feedback control law that completely eliminates the effect of the measurable disturbances on the process outputs and induces a linear behavior in the closed-loop system between the outputs and a set of reference inputs. The input/output stability and the degree of coupling in the closed-loop system are determined by appropriate choice of adjustable parameters. A MIMO linear controller with integral action completes the feedforward/feedback control structure. The developed control methodology is applied to a continuous polymerization reactor and its performance is evaluated through simulations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/37411/1/690361003_ftp.pd
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