. This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach. Keywords. Piecewise linear systems, Lyapunov stability, linear matrix inequalities. 1. Introduction Construction of Lyapunov functions is one of the most fundamental problems in systems theory. The most direct application is stability analysis, but analogous problems appear more or less implicitly also in performance analysis, controller synthesis and system identification. Consequently, methods for constructing Lyapunov functions for general nonlinear systems is of great theoretical and practical interest. The objective of this paper is to develop a uniform and compu...

In order to cope with the large amounts of data that have become available in genomics, mathematical tools for the analysis of networks of interactions between genes, proteins, and other molecules are indispensable. We present a method for the qualitative simulation of genetic regulatory networks, based on a class of piecewise-linear (PL) differential equations that has been well-studied in mathematical biology. The simulation method is well-adapted to state-of-the-art measurement techniques in genomics, which often provide qualitative and coarsegrained descriptions of genetic regulatory networks. Given a qualitative model of a genetic regulatory network, consisting of a system of PL differential equations and inequality constraints on the parameter values, the method produces a graph of qualitative states and transitions between qualitative states, summarizing the qualitative dynamics of the system. The qualitative simulation method has been implemented in Java in the computer tool Genetic Network Analyzer.

this paper is to give a brief tutorial review of four different classes of hybrid systems of this type - each consists of a continuous-time process to be controlled, a parameterized family of candidate controllers, and an event driven switching logic. Three of the logics, called prerouted switching, hysteresis switching and dwelltime switching respectively, are simple strategies capable of determining in real time which candidate controller should be put in feedback with a process in order to achieve desired closed-loop performance. The fourth, called cyclic switching, has been devised to solve the long-standing stabilizability problem which arises in the synthesis of identifier-based adaptive controllers because of the existence of points in parameter space where the estimated model upon which certainty equivalence synthesis is based, loses stabilizability

In this work we suggest a novel methodology for synthesizing switching controllers for continuous and hybrid systems whose dynamics are defined by linear differential equations. We formulate the synthesis problem as finding the conditions upon which a controller should switch the behavior of the system from one "mode" to another in order to avoid a set of bad states, and propose an abstract algorithm which solves the problem by an iterative computation of reachable states. We have implemented a concrete version of the algorithm, which uses a new approximation scheme for reachability analysis of linear systems.

Stability and robustness issues for hybrid systems are considered in this paper. General stability results that are extensions of classical Lyapunov theory have recently been formulated. However, these results are in general not straightforward to apply due to the following reasons. First, a search for multiple Lyapunov functions must be performed. However, existing theory does not unveil how to find such functions. Secondly, if the most general stability result is applied, knowledge about the continuous trajectory is required, at least at some time instants. Because of these drawbacks stronger conditions for stability are suggested, in which case it is shown that the search for Lyapunov functions can be formulated as a linear matrix inequality (LMI) problem for hybrid systems consisting of linear subsystems. Additionally, it is shown how robustness properties can be achieved when the Lyapunov functions are given. Specifically, it is described how to determine permitted switch regions ...

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The work of this paper is inspired by the flocking phenomenon observed in Reynolds (1987). We introduce a class of local control laws for a group of mobile agents that result in: (i) global alignment of their velocity vectors, (ii) convergence of their speeds to a common one, (iii) collision avoidance, and (iv) minimization of the agents artificial potential energy. These are made possible through local control action by exploiting the algebraic graph theoretic properties of the underlying interconnection graph. Algebraic connectivity a#ects the performance and robustness properties of the overall closed loop system. We show how the stability of the flocking motion of the group is directly associated with the connectivity properties of the interconnection network and is robust to arbitrary switching of the network topology.

Network flow control regulates the traffic between sources and links based on congestion, and plays a critical role in ensuring satisfactory performance. In recent studies, global stability has been shown for several flow control schemes. By using a passivity approach, this paper presents a unifying framework which encompasses these stability results as special cases. In addition, the new approach significantly expands the current classes of stable flow controllers by augmenting the source and link update laws with passive dynamic systems. This generality offers the possibility of optimizing the controllers, for example, to improve robustness and performance with respect to time delay, unmodeled flows, and capacity variation.

It is shown that every asymptotically controllable system can be globally stabilized by means of some (discontinuous) feedback law. The stabilizing strategy is based on pointwise optimization of a smoothed version of a control-Lyapunov function, iteratively sending trajectories into smaller and smaller neighborgoods of a desired equilibrium. A major technical problem, and one of contributions of the present paper, concerns the precise meaning of "solution" when using a discontinuous controller. I. Introduction A longstanding open question in nonlinear control theory concerns the relationship between asymptotic controllability to the origin in R n of a nonlinear system x = f(x; u) (1) by an "open loop" control u : [0; +1) ! U and the existence of a feedback control k : R n ! U which stabilizes trajectories of the system x = f(x; k(x)) (2) with respect to the origin. For the special case of linear control systems x = Ax + Bu, this relationship is well understood: asymptotic cont...

Fish in a school efficiently find the densest source of food by individually responding not only to local environmental stimuli but also to the behavior of nearest neighbors. It is of great interest to enable a network of autonomous vehicles to function similarly as an intelligent sensor array capable of climbing or descending gradients of some spatially distributed signal. We formulate and study a coordinated control strategy for a group of autonomous vehicles to descend or climb an environmental gradient using measurements of the environment together with relative position measurements of nearest neighbors. Each vehicle is driven by an estimate of the local environmental gradient together with control forces, derived from artificial potentials, that maintain uniformity in group geometry.