2,141 research outputs found
Transverse Contraction Criteria for Existence, Stability, and Robustness of a Limit Cycle
This paper derives a differential contraction condition for the existence of
an orbitally-stable limit cycle in an autonomous system. This transverse
contraction condition can be represented as a pointwise linear matrix
inequality (LMI), thus allowing convex optimization tools such as
sum-of-squares programming to be used to search for certificates of the
existence of a stable limit cycle. Many desirable properties of contracting
dynamics are extended to this context, including preservation of contraction
under a broad class of interconnections. In addition, by introducing the
concepts of differential dissipativity and transverse differential
dissipativity, contraction and transverse contraction can be established for
large scale systems via LMI conditions on component subsystems.Comment: 6 pages, 1 figure. Conference submissio
Regulation and robust stabilization: a behavioral approach
In this thesis we consider a number of control synthesis problems within the behavioral approach to systems and control. In particular, we consider the problem of regulation, the H! control problem, and the robust stabilization problem. We also study the problems of regular implementability and stabilization with constraints on the input/output structure of the admissible controllers. The systems in this thesis are assumed to be open dynamical systems governed by linear constant coefficient ordinary differential equations. The behavior of such system is the set of all solutions to the differential equations. Given a plant with its to-be-controlled variable and interconnection variable, control of the plant is nothing but restricting the behavior of the to-be-controlled plant variable to a desired subbehavior. This restriction is brought about by interconnecting the plant with a controller (that we design) through the plant interconnection variable. In the interconnected system the plant interconnection variable has to obey the laws of both the plant and the controller. The interconnected system is also called the controlled system, in which the controller is an embedded subsystem. The interconnection of the plant and the controller is said to be regular if the laws governing the interconnection variable are independent from the laws governing the plant. We call a specification regularly implementable if there exists a controller acting on the plant interconnection variable, such that, in the interconnected system, the behavior of the to-becontrolled variable coincides with the specification and the interconnection is regular. Within the framework of regular interconnection we solve the control problems listed in the first paragraph. Solvability conditions for these problems are independent of the particular representations of the plant and the desired behavior.
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
On a small-gain approach to distributed event-triggered control
In this paper the problem of stabilizing large-scale systems by distributed
controllers, where the controllers exchange information via a shared limited
communication medium is addressed. Event-triggered sampling schemes are
proposed, where each system decides when to transmit new information across the
network based on the crossing of some error thresholds. Stability of the
interconnected large-scale system is inferred by applying a generalized
small-gain theorem. Two variations of the event-triggered controllers which
prevent the occurrence of the Zeno phenomenon are also discussed.Comment: 30 pages, 9 figure
Linear Hamiltonian behaviors and bilinear differential forms
We study linear Hamiltonian systems using bilinear and quadratic differential forms. Such a representation-free approach allows us to use the same concepts and techniques to deal with systems isolated from their environment and with systems subject to external influences and allows us to study systems described by higher-order differential equations, thus dispensing with the usual point of view in classical mechanics of considering first- and second-order differential equations only
Kuhn-Tucker-based stability conditions for systems with saturation
This paper presents a new approach to deriving stability conditions for continuous-time linear systems interconnected with a saturation. The method presented can be extended to handle a dead-zone, or in general, nonlinearities in the form of piecewise linear functions. By representing the saturation as a constrained optimization problem, the necessary (Kuhn-Tucker) conditions for optimality are used to derive linear and quadratic constraints which characterize the saturation. After selecting a candidate Lyapunov function, we pose the question of whether the Lyapunov function is decreasing along trajectories of the system as an implication between the necessary conditions derived from the saturation optimization, and the time derivative of the Lyapunov function. This leads to stability conditions in terms of linear matrix inequalities, which are obtained by an application of the S-procedure to the implication. An example is provided where the proposed technique is compared and contrasted with previous analysis methods
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