Robust Stability Analysis of Nonlinear Hybrid Systems

We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems

Distributed H-infinity filtering for polynomial nonlinear stochastic systems in sensor networks

Copyright [2010] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this paper, the distributed H1 filtering problem is addressed for a class of polynomial nonlinear stochastic systems in sensor networks. For a Lyapunov function candidate whose entries are polynomials, we calculate its first- and second-order derivatives in order to facilitate the use of Itos differential role. Then, a sufficient condition for the existence of a feasible solution to the addressed distributed H1 filtering problem is derived in terms of parameter-dependent linear matrix inequalities (PDLMIs). For computational convenience, these PDLMIs are further converted into a set of sums of squares (SOSs) that can be solved effectively by using the semidefinite programming technique. Finally, a numerical simulation example is provided to demonstrate the effectiveness and applicability of the proposed design approach.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the U.K. under Grant GR/S27658/01, the Royal Society of the U.K., the National 973 Program of China under Grant 2009CB320600, the National Natural Science Foundation of China under Grant 60974030 and the Alexander von Humboldt Foundation of Germany

Stability and Control of Power Systems using Vector Lyapunov Functions and Sum-of-Squares Methods

Recently sum-of-squares (SOS) based methods have been used for the stability analysis and control synthesis of polynomial dynamical systems. This analysis framework was also extended to non-polynomial dynamical systems, including power systems, using an algebraic reformulation technique that recasts the system's dynamics into a set of polynomial differential algebraic equations. Nevertheless, for large scale dynamical systems this method becomes inapplicable due to its computational complexity. For this reason we develop a subsystem based stability analysis approach using vector Lyapunov functions and introduce a parallel and scalable algorithm to infer the stability of the interconnected system with the help of the subsystem Lyapunov functions. Furthermore, we design adaptive and distributed control laws that guarantee asymptotic stability under a given external disturbance. Finally, we apply this algorithm for the stability analysis and control synthesis of a network preserving power system.Comment: to appear at the 14th annual European Control Conferenc

An Overview of Integral Quadratic Constraints for Delayed Nonlinear and Parameter-Varying Systems

A general framework is presented for analyzing the stability and performance of nonlinear and linear parameter varying (LPV) time delayed systems. First, the input/output behavior of the time delay operator is bounded in the frequency domain by integral quadratic constraints (IQCs). A constant delay is a linear, time-invariant system and this leads to a simple, intuitive interpretation for these frequency domain constraints. This simple interpretation is used to derive new IQCs for both constant and varying delays. Second, the performance of nonlinear and LPV delayed systems is bounded using dissipation inequalities that incorporate IQCs. This step makes use of recent results that show, under mild technical conditions, that an IQC has an equivalent representation as a finite-horizon time-domain constraint. Numerical examples are provided to demonstrate the effectiveness of the method for both class of systems

Backstepping PDE Design: A Convex Optimization Approach

Abstract\u2014Backstepping design for boundary linear PDE is formulated as a convex optimization problem. Some classes of parabolic PDEs and a first-order hyperbolic PDE are studied, with particular attention to non-strict feedback structures. Based on the compactness of the Volterra and Fredholm-type operators involved, their Kernels are approximated via polynomial functions. The resulting Kernel-PDEs are optimized using Sumof- Squares (SOS) decomposition and solved via semidefinite programming, with sufficient precision to guarantee the stability of the system in the L2-norm. This formulation allows optimizing extra degrees of freedom where the Kernel-PDEs are included as constraints. Uniqueness and invertibility of the Fredholm-type transformation are proved for polynomial Kernels in the space of continuous functions. The effectiveness and limitations of the approach proposed are illustrated by numerical solutions of some Kernel-PDEs

Positive trigonometric polynomials for strong stability of difference equations

We follow a polynomial approach to analyse strong stability of linear difference equations with rationally independent delays. Upon application of the Hermite stability criterion on the discrete-time homogeneous characteristic polynomial, assessing strong stability amounts to deciding positive definiteness of a multivariate trigonometric polynomial matrix. This latter problem is addressed with a converging hierarchy of linear matrix inequalities (LMIs). Numerical experiments indicate that certificates of strong stability can be obtained at a reasonable computational cost for state dimension and number of delays not exceeding 4 or 5