system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

absolute stability of third order systems a numerical algorithm the problem of absolute stability is one of the oldest open problems in the theory of control even for the particular case of second order systems a complete solution was presented only very recently for third order systems the most general theoretical results were obtained by barabanov he derived an implicit characterization of the most destabilizing nonlinearity using a variational approach in this paper we show that his approach yields a simple and efficient numerical scheme for solving the problem in the case of third order systems this allows the determination of the critical value where stability is lost in a tractable and accurate fashion this value is important in many practical applications and we believe that it can also be used to develop a deeper theoretical understanding of this interesting problem

lie algebraic stability conditions for nonlinear switched systems and differential inclusions abstract we present a stability criterion for switched nonlinear systems which involves lie brackets of the individual vector fields but does not require that these vector fields commute a special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third order lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching this generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in d liberzon lie algebras and stability of switched nonlinear systems in v blondel a megretski eds unsolved problems in mathematical systems and control theory princeton university press nj 2004 pp 203 207 to prove the result we consider an optimal control problem which consists in finding the most unstable trajectory for an associated control system and show that there exists an optimal solution which is bang bang with a bound on the total number of switches this property is obtained as a special case of a reachability result by bang bang controls which is of independent interest by construction our criterion also automatically applies to the corresponding relaxed differential inclusion

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

on the local structure of optimal trajectories in r 3 we analyze the structure of a control function u t corresponding to an optimal trajectory for the system dot q f q u g q in a three dimensional manifold near a point where some nondegeneracy conditions are satisfied the kind of optimality which is studied includes time optimality the control turns out to be the concatenation of some bang and some singular arcs studying the index of the second variation of the switching times the number of such arcs is bounded by four

a condition for the stability of switched nonlinear systems in this paper we present a sufficient condition for the global asymptotic stability of a switched nonlinear system composed of a finite family of subsystems we show that the global asymptotic stability of each subsystem and the pairwise commutation of the vector fields that define the subsystems i e the lie bracket of any pair of them is zero are sufficient for the global asymptotic stability of the switched system we also show that these conditions are sufficient for the existence of a common lyapunov function

the problem of absolute stability a dynamic programming approach we consider the problem of absolute stability of a feedback system composed of a linear plant and a single sector bounded nonlinearity pyatnitskiy and rapoport used a variational approach and the maximum principle to derive an implicit characterization of the most destabilizing nonlinearity in this paper we address the same problem using a dynamic programming approach we show that the corresponding value function is composed of simple building blocks which are the generalized first integrals of appropriate linear systems we demonstrate how the results can be used to design stabilizing switched controllers

an optimal galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere the kinematic dynamo approximation describes the generation of magnetic field in a prescribed flow of electrically conducting liquid one of its main uses is as a proof of concept tool to test hypotheses about self exciting dynamo action indeed it provided the very first quantitative evidence for the possibility of the geodynamo despite its utility due to the requirement of resolving fine structures historically numerical work has proven difficult and reported solutions were often plagued by poor convergence in this paper we demonstrate the numerical superiority of a galerkin scheme in solving the kinematic dynamo eigenvalue problem in a full sphere after adopting a poloidal toroidal decomposition and expanding in spherical harmonics we express the radial dependence in terms of a basis of exponentially convergent orthogonal polynomials each basis function is constructed from a terse sum of one sided jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior but is everywhere infinitely differentiable including at the origin this galerkin method exhibits more rapid convergence for a given problem size than any other scheme hitherto reported as demonstrated by a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos from the literature in the axisymmetric flows we consider in this paper at a magnetic reynolds number of o 100 a convergence of 9 significant figures in the most unstable eigenvalue requires only 40 radial basis functions alternatively 4 significant figures requires 20 radial functions the terse radial discretization becomes particularly advantageous when considering flows whose associated numerical solution requires a large number of coupled spherical harmonics we exploit this new method to confirm the tentatively proposed positive growth rate of the planar flow of bachtiar et al 4 thereby verifying a counter example to the zel dovich anti dynamo theorem in a spherical geometry

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

variational data assimilation for the initial value dynamo problem the secular variation of the geomagnetic field as observed at the earth s surface results from the complex magnetohydrodynamics taking place in the fluid core of the earth one way to analyze this system is to use the data in concert with an underlying dynamical model of the system through the technique of variational data assimilation in much the same way as is employed in meteorology and oceanography the aim is to discover an optimal initial condition that leads to a trajectory of the system in agreement with observations taking the earth s core to be an electrically conducting fluid sphere in which convection takes place we develop the continuous adjoint forms of the magnetohydrodynamic equations that govern the dynamical system together with the corresponding numerical algorithms appropriate for a fully spectral method these adjoint equations enable a computationally fast iterative improvement of the initial condition that determines the system evolution the initial condition depends on the three dimensional form of quantities such as the magnetic field in the entire sphere for the magnetic field conservation of the divergence free condition for the adjoint magnetic field requires the introduction of an adjoint pressure term satisfying a zero boundary condition we thus find that solving the forward and adjoint dynamo system requires different numerical algorithms in this paper an efficient algorithm for numerically solving this problem is developed and tested for two illustrative problems in a whole sphere one is a kinematic problem with prescribed velocity field and the second is associated with the hall effect dynamo exhibiting considerable nonlinearity the algorithm exhibits reliable numerical accuracy and stability using both the analytical and the numerical techniques of this paper the adjoint dynamo system can be solved directly with the same order of computational complexity as that required to solve the forward problem these numerical techniques form a foundation for ultimate application to observations of the geomagnetic field over the time scale of centuries

stability analysis of switched systems using variational principles an introduction many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode switched systems provide a suitable mathematical model for such processes and their stability analysis is important for both theoretical and practical reasons we review a specific approach for stability analysis based on using variational principles to characterize the most unstable solution of the switched system we also discuss a link between the variational approach and the stability analysis of switched systems using lie algebraic considerations both approaches require the use of sophisticated tools from many different fields of applied mathematics the purpose of this paper is to provide an accessible and self contained review of these topics emphasizing the intuitive and geometric underlying ideas

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

a numerical technique for the stability analysis of linear switched systems in this paper the ray gridding approach a new numerical technique for the stability analysis of linear switched systems is presented it is based on uniform partitions of the state space in terms of ray directions which allow refinable families of polytopes of adjustable complexity to be examined for invariance in this framework the existence of a polyhedral lyapunov function that is common to a family of asymptotically stable subsystems can be checked efficiently via simple iterative algorithms the technique can be used to prove the stability of switched linear systems classes of linear time varying systems and linear differential lnclusions we also present preliminary results on another related problem namely the construction of multiple polyhedral lyapunov functions for the specification of stabilizing switching sequences for a switched system constructed from a family of stable linear subsystems

periodicity and chaos from switched flow systems contrasting examples of discretely controlled continuous systems two examples of the discrete control of a continuous variable system are analyzed these examples exhibit what may be regarded as the two extremes of complexity of the closed loop behavior one is eventually periodic the other is chaotic the examples are derived from sampled deterministic flow models these are of interest in their own right but have also been used as models for certain aspects of manufacturing systems in each case the authors give a precise characterization of the closed loop behavior

determination of the instantaneous geostrophic flow within the three dimensional magnetostrophic regime in his seminal work taylor 1963 proc r soc lond a 274 274 283 doi 10 1098 rspa 1963 0130 argued that the geophysically relevant limit for dynamo action within the outer core is one of negligibly small inertia and viscosity in the magnetohydrodynamic equations within this limit he showed the existence of a necessary condition now well known as taylor9s constraint which requires that the cylindrically averaged lorentz torque must everywhere vanish magnetic fields that satisfy this condition are termed taylor states taylor further showed that the requirement of this constraint being continuously satisfied through time prescribes the evolution of the geostrophic flow the cylindrically averaged azimuthal flow we show that taylor9s original prescription for the geostrophic flow as satisfying a given second order ordinary differential equation is only valid for a small subset of taylor states an incomplete treatment of the boundary conditions renders his equation generally incorrect here by taking proper account of the boundaries we describe a generalization of taylor9s method that enables correct evaluation of the instantaneous geostrophic flow for any three dimensional taylor state we present the first full sphere examples of geostrophic flows driven by non axisymmetric taylor states although in axisymmetry the geostrophic flow admits a mild logarithmic singularity on the rotation axis in the fully three dimensional case we show that this is absent and indeed the geostrophic flow appears to be everywhere regular

galerkin orthogonal polynomials abstract the galerkin method offers a powerful tool in the solution of differential equations and function approximation on the real interval 1 1 by expanding the unknown function in appropriately chosen global basis functions each of which explicitly satisfies the given boundary conditions in general this scheme converges exponentially fast and almost always supplies the most terse representation of a smooth solution to date typical schemes have been defined in terms of a linear combination of two jacobi polynomials however the resulting functions do not inherit the expedient properties of the jacobi polynomials themselves and the basis set will not only be non orthogonal but may in fact be poorly conditioned using a gram schmidt procedure it is possible to construct in an incremental fashion polynomial basis sets that not only satisfy any linear homogeneous boundary conditions but are also orthogonal with respect to the general weighting function 1 x 1 x however as it stands this method is not only cumbersome but does not provide the structure for general index n of the functions and obscures their dependence on the parameters in this paper it is shown that each of these galerkin basis functions as calculated by the gram schmidt procedure may be written as a linear combination of a small number of jacobi polynomials with coefficients that can be determined moreover this terse analytic representation reveals that for large index the basis functions behave asymptotically like the single jacobi polynomial p n x this new result shows that such galerkin bases not only retain exponential convergence but expedient function fitting properties too in much the same way as the jacobi polynomials themselves this powerful methodology of constructing galerkin basis sets is illustrated by many examples and it is shown how the results extend to polar geometries in exploring more generalised definitions of orthogonality involving derivatives we discuss how a large class of differential operators may be discretised by galerkin schemes and represented in a sparse fashion by the inverse of band limited matrices

technical communique analysis and design of aimd congestion control algorithms in communication networks we present a model of a network of synchronised sources operating additive increase multiplicative decrease aimd congestion control algorithms we show i that networks of such devices in the presence of a drop tail bottleneck buffer may be modelled as a positive linear system ii that such networks possess a unique stationary point and iii that this stationary point is globally exponentially stable we use these results to establish conditions for the fair co existence of traffic in networks employing heterogeneous aimd algorithms and to design a new protocol for operation over high speed and long distance links

criteria of asymptotic stability of differential and difference inclusions encountered in control theory abstract for a class of differential inclusions to which many of the practically important control systems can be reduced necessary and sufficient conditions for asymptotic stability of the zero solution are established by the method of lyapunov functions similar results are obtained for the corresponding class of difference inclusions

lie algebraic stability criteria for switched systems it was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable lie algebra possesses a quadratic common lyapunov function which implies that the corresponding switched linear system is exponentially stable for arbitrary switching in this paper we prove that the same properties hold under the weaker condition that the lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact lie group the corresponding local stability result for nonlinear switched systems is also established moreover we demonstrate that if a lie algebra fails to satisfy the above condition then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable relevant facts from the theory of lie algebras are collected at the end of the paper for easy reference

brief on absolute stability analysis by polyhedral lyapunov functions this paper addresses the problem of computing a polyhedral lyapunov function plf for a linear differential inclusion ldi a numerically efficient algorithm is presented to compute a plf for an ldi it is also pointed out that for the case of a polyhedron defined by vertices rather than by faces one can state algebraic stability conditions analogous to those given by molchanov and pyatnitskiy 1986

numerical methods of lyapunov function construction and their application to the absolute stability problem the minimax approach is used to study numerical selection of a lyapunov function from a multiparameter class of functions a numerical method is suggested for plotting an absolute stability region in the space of system gains for a control system with several nonstationary nonlinearities

on the computation of switching surfaces in optimal control a grobner basis approach a number of problems in control can be reduced to finding suitable real solutions of algebraic equations in particular such a problem arises in the context of switching surfaces in optimal control a powerful new methodology for doing symbolic manipulations with polynomial data has been developed and tested namely the use of grobner bases we apply the grobner basis technique to find effective solutions to the classical problem of time optimal control

control systems which generate decomposable lie algebras let x y be real analytic vector fields on an n manifold m ad x y x y denote the lie product inductively ad k 1 x y x ad k x y and 1 ad v x y v 0 the attainable set at time t denoted download full size image t for the control system 1 x x x u t y x x 0 p u t 1 is contained in the integral manifold through p of the lie algebra l 1 generated by 1 l 1 is said to be decomposable at p if essentially when dim l 1 p n the solution of 1 can be written as exp tx exp f 1 t u v 1 exp f n t u v n p with v 1 v n l 1 this is true for example at any p if l 1 is nilpotent the f i t u are shown to satisfy a system on n of the form 2 x i u t g i t x 1 x i 1 x i 0 0 i 1 n where if l 1 is nilpotent each g i is a polynomial in x 1 x i 1 in this case reachability properties of 2 for controls being countably additive measures impluses reduce to the study of algebraic equations homogeneity allows these to be considered on projective space n giving local results by bezout s theorem similar results hold for controls in the unit ball of krener s results on approximating systems of the form 1 by systems which generate nilpotent lie algebras can then be used to remove the condition that l 1 be nilpotent the method applies to show that if m denotes the set of all products of k elements of 1 with k m and dim span m 1 p dim span m for all odd m then exp tx p int download full size image t for all t 0