the vinnicombe metric for nonlinear operators describes an extension of the vinnicombe metric on linear operators to a pseudometric on nonlinear operators a metric for finite dimensional time varying operators is shown to be capable of guaranteeing stability and performance robustness and reduces to the standard vinnicombe metric for the time invariant operator case which is known to be less conservative than the gap metric the analysis exploits the time varying operator equivalents of unstable poles and normalized coprime fractional descriptions in addition a time varying operator equivalent of the winding number is defined

monomorphisms between subgroups of groups of type g2 abstract abstract monomorphisms of a full subgroup of a group of type g 2 into another group of type g 2 are shown to have a standard decomposition into a field monomorphism and a special isogeny of algebraic groups

strong approximation for zariski dense subgroups of semi simple algebraic groups 1 1 theorem let k be an algebraically closed field of characteristic different from 2 and 3 and g an almost simple connected and simply connected algebraic group defined over k let f be a finitely generated zariski dense subgroup of g k and a the subring of k generated by the traces tr ad y y e p then there exist b e a a subgroup f of finite index in f and a structure gab of a group scheme over ab on g such that f c ga ab and f is dense in gala b a

stability of distributed congestion control with heterogeneous feedback delays we investigate how congestion control can achieve efficient usage of network resources in the presence of heterogeneous communication delays between network users and resources to this end we consider a fluid flow model of network behavior we study the stability of the system s behavior under small perturbations around the target equilibrium point local stability we establish several criteria for stability of certain linear delay differential equations via a technique which essentially reduces the question to studying stability of ordinary differential equations these results are then used to derive sufficient conditions for local stability of the network congestion control problem the same issue has been studied by johari et al 2001 where the authors propose a conjecture according to which local stability can be ensured in a distributed way the correctness of the conjecture was established by johari et al only in degenerate cases where feedback delays coincide our results show that a modified form of the conjecture holds true for arbitrary feedback delays

adaptive control and robustness in the gap metric we consider the construction of adaptive controllers for minimum phase linear systems that achieve nonzero robustness margins in the sense of the gap metric the gap perturbation margin may be more constrained for larger disturbances and for larger parametric uncertainties working in both and settings and within the framework of the nonlinear gap metric universal adaptive controllers are first given achieving stabilization for first order nominal plants and the results are then generalized to relative degree one nominal plants a notion of a semiuniversal control design is introduced which is the property that a bound on performance exists that is independent of the a priori known uncertainty level and a characterization is given for when semiuniversal designs outperform the class of memoryless controllers and the class of linear time invariant controllers robust semiuniversal adaptive control designs are given for nominal plants under the classical assumptions of adaptive control in both the and settings the results are applied throughout to explicit classes of unmodeled dynamics including the rohrs example

stabilization by using artificial delays an lmi approach abstract static output feedback stabilization for the n th order vector differential equations by using artificial multiple delays is considered under assumption of the stabilizability of the system by a static feedback that depends on the output and its derivatives up to the order n 1 a delayed static output feedback is found that stabilizes the system the conditions for the stability analysis of the resulting closed loop system are given in terms of simple lmis it is shown that the lmis are always feasible for appropriately chosen gains and small enough delays robust stability analysis in the presence of uncertain time varying delays and stochastic perturbation of the system coefficients is provided numerical examples including chains of three and four integrators that are stabilized by static output feedbacks with multiple delays illustrate the efficiency of the method

homomorphisms of unitary groups in problem xvii from 4 weisfeiler indicates a general formulation for homomorphisms between classical groups in terms of a congruence subgroup coming from an induced integral structure in the group as he observes certainly not all homomorphisms arise in this manner but this formulation provides a good foundation for extending the fairly well understood situation for isomorphisms we establish here weisfeiler s conjecture for unitary groups and homomorphisms which do not kill involutions for monomorphisms this problem is usually solved via group theoretical arguments using derived groups and centralizers but for homomorphisms these methods are hard to apply and it seems necessary to use a more geometrical approach in the proof here the central new idea involves moving subspaces in a controlled manner intersecting them and counting dimensions it should also be possible to use this technique in other situations let v and w be right vector spaces over the division rings k and k with involutions both denoted by where

robustness in the graph topology of a common adaptive controller for any m input m output finite dimensional linear minimum phase plant p with first markov parameter having spectrum in the open right half complex plane it is well known that the adaptive output feedback control c given by u ky dot k y 2 yields a closed loop system p c for which the state converges to zero the signal k converges to a finite limit and all other signals are of class l 2 it is first shown that these properties continue to hold in the presence of l 2 input and l 2 output disturbances working within the conceptual framework of the nonlinear gap metric approach to robust stability and by establishing gain function stability of an appropriate closed loop operator it is proved that these properties also persist when the plant p is replaced with a stabilizable and detectable linear plant p 1 within a sufficiently small neighborhood of p in the graph topology provided that the plant initial data and the l 2 magnitude of the disturbances are sufficiently small example 9 of georgiou and smith ieee trans automat control 42 1997 pp 1200 1221 is revisited unstable behavior for large initial conditions and or large l 2 disturbances is shown demonstrating that the bounds obtained from the l 2 theory are qualitatively tight this contrasts with the l infty robustness analysis of georgiou and smith which is insufficiently tight to predict the stable behavior for small initial conditions and zero disturbances

coprime fractional representations and stability of non linear feedback systems fractional representation methods for finite dimensional linear time invariant fdlti systems have been developed in an algebraic framework of principal ideal domains the concept of coprime fractional representations has been generalized to discrete time non linear systems by hammer 1985 1987 by viewing the bezout identity as the basis of the definition of coprimeness in this paper we view the role of right coprime fractional representation in representing the set of all stable input output pairs of a possibly unstable non linear system this notion of coprimeness is shown to be equivalent to that of fdlti systems using coprime fractional representations of non linear systems we obtain a characterization of the stability of non linear feedback systems this characterization is similar to that obtained in the case of fdlti systems we also obtain a parameterization of all non linear and time varying stabilizing controllers for linear possibly time varying plants which has the same form as t

the class of stabilizing nonlinear plant controller pairs in this paper a general approach is taken to yield a characterization of the class of stable plant controller pairs which is a generalization of the youla parameterization for linear systems this is based on the idea of representing the input output pairs of the plant and controller as elements of the kernel of some related operator which is denoted the kernel representation of the system it is demonstrated that in some sense the kernel representation is a generalization of the left coprime factorization of a general nonlinear system results giving one method of deriving a kernel representation for a nonlinear plant with a general state space description are presented

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

robust stabilization by linear output delay feedback the main result of this paper establishes that if a controller c comprising of a linear feedback of the output and its derivatives globally stabilizes a nonlinear plant p then global stabilization of p can also be achieved by an output feedback controller c h where the output derivatives in c are replaced by an euler approximation with sufficiently small delay h 0 this is proved within the conceptual framework of the nonlinear gap metric approach to robust stability the main result is then applied to finite dimensional linear minimum phase systems with unknown coefficients but known relative degree and known sign of the high frequency gain results are also given for systems with nonzero initial conditions

on the gap metric and coprime factor perturbations abstract new conditions are derived for when the distance between two linear systems in the gap metric is less than one by including a coprimeness assumption in the coprime factor uncertainty description it is shown that an open ball in the gap metric is equivalent to an open ball phrased in terms of coprime factor perturbations a new criterion is given for robust stabilization

berry s phase and space symmetries we study the role played by space symmetries in the context of the berry phase it is found that the parameter space may be expanded to include non compact spaces that minimal surfaces may be associated with the berry magnetic field and that in s 3 the hopf invariant may be used to classify these surfaces and fields up to a gauge transformation a number of examples are discussed and some evidence is offered for the existence of berry phases that are not of the monopole type

graph topologies gap metrics and robust stability for nonlinear systems graph topologies for nonlinear operators which admit coprime factorizations are defined w r t a gain function notion of stability in a general normed signal space setting several metrics are also defined and their relationship to the graph topologies are examined in particular relationships between nonlinear generalizations of the gap and graph metrics georgiou type formulae and the graph topologies are established closed loop robustness results are given w r t the graph topology where the role of a coercivity condition on the nominal plant is emphasized

abstract isomorphisms of simple algebraic groups split by quadratic extensions abstract abstract isomorphisms of a big subgroup of an anisotropic absolutely almost simple algebraic group split by a separable quadratic extension of the base field with big subgroups of other semi simple algebraic groups are shown to have a standard decomposition into a field isomorphism a special isogeny of algebraic groups and a radial isomorphism

stability for network flow control against disturbances and time delays new stability results are presented for kelly s algorithm for network traffic control these are applicable to a broad class of algorithms including most of those previously discussed in the literature three types of results are derived an asymptotic stability result for networks with external disturbances but without time delays an asymptotic stability result for a class of algorithms with a linear cost with both external disturbances and time delays and a gain stability result for more general algorithms with both disturbances and time delays the time delay results are obtained using a gap metric approach

on the computation of the gap metric abstract this paper deals with the computation of the gap metric introduced by zames and el sakkary 17 it is shown that the gap between two systems p1 p2 is precisely the maximum of the two expressions inf q h d i n i d j n j q for i j equal to 1 2 and 2 1 and ni di being normalized right coprine factorizations of pi i 1 2 in the sense of vidyasagar 12 this expression is computable using well known techniques from interpolation theory

output feedback stabilisation of minimum phase systems by delays for linear square multi input multi output minimum phase relative degree one or two systems proportional output feedback controllers are studied in particular we show that derivative feedback can be replaced by delay feedback essentially an euler approximation ofthe derivative ifthe delay is su3ciently small stability regions are determined and related c

static output feedback stabilization necessary conditions for multiple delay controllers this note focuses on the static output feedback stabilization problem for a class of single input single output systems when the control law includes multiple distinct delays we are interested in giving necessary conditions for the existence of such stabilizing controllers illustrative examples second order system chain of integrators or chain of oscillators are presented and discussed

general fast sampling theorems for nonlinear systems abstract this paper is concerned with the gap metric approach to controller discretisation problems for continuous time nonlinear systems with disturbances in both input and output channels the principal idea is to construct a discrete controller based on a given stabilising continuous time controller via a fast sampling and hold procedure and to calculate the gap between the two controllers it is expected that under general conditions the computed gap depends on the discrete sample size and the faster the sample rate the smaller the gap and therefore existing gap metric robust stability theorems can be applied to obtain both stability and performance results for the appropriately discretised controller this is shown for the case of memoryless controllers and for a more general class of controllers specified by stable causal operators in both cases both regional and global results are obtained under respective local and global incremental stability assumptions on the controllers

robustness of network flow control against disturbances and time delay this paper studies robustness of kelly s source and link control laws in j oper res soc 49 1998 237 with respect to disturbances and time delays this problem is of practical importance because of unmodelled ows and propagation and queueing delays which are ubiquitous in networks we rst show lp stability for p 1 with respect to additive disturbances we pursue l stability within the input to state stability iss framework of sontag ieee trans automat control 34 1989 435 which makes explicit the vanishing eaect of initial conditions next using this iss property and a loop transformation we prove that global asymptotic stability is preserved for subciently small time delays in forward and return channels for larger delays we achieve global asymptotic stability by scaling down the control gains as in paganini et al proceedings of 2001 conference on decision and control orlando fl december 2001 pp 185 190 c

gap metrics representations and nonlinear robust stability various alternative definitions for the nonlinear h2 l2 and nu gap metrics are studied the concept of beta conjugacy and multiplicative homogeneity are introduced to relate the metrics to each other and to compare the stability margins of nonlinear feedback loops expressed in terms of the norms of complementary parallel projections left and right representations for the graph of a nonlinear system are studied a new definition of normalized is introduced for left representations formulas for the gap metrics as the norm of the product of left and right representations are derived the problem of controller synthesis for input affine nonlinear systems to achieve norm bounds on the parallel projection operators is studied for input affine nonlinear systems the duality between the optimization of the two parallel projections is highlighted state space realizations for the normalized left and right representations are derived using nonlinear h infty synthesis methods

representations of anisotropic unitary groups ii abstract let su f be the special unitary group of an anisotropic hermitian form f over a field k assume f represents only one norm class in k the representations su f sl n r are characterized when r is a commutative local ring with 3 a unit and n dim f 4

right coprime factorizations and stabilization for nonlinear systems the problem of constructing right coprime factorizations which are based on the graph of an input output map instead of the bezout identity of nonlinear input output maps is considered the map is assumed to arise from a state variable realization with a fixed initial state the main result is that the existence of a stabilizing state feedback implies the existence of a right coprime factorization for the map the technique is illustrated by application to nonlinear systems which are affine in the control and have a controllable linear past and nonlinear systems which are feedback linearizable a notion of input output stability that requires a bound on the magnitude of the input signals is introduced methods of constructing such bounds are developed for a locally feedback linearizable system the problem of input bounds is transferred to the equivalent linear system this leads to a technique that allows state and input constraints for a feedback linearizable system to be mapped to the equivalent linear system

robust stabilization of nonlinear systems via stable kernel representations with l 2 gain bounded uncertainty the approach to robust stabilization of linear systems using normalized left coprime factorizations with h bounded uncertainty is generalized to nonlinear systems a nonlinear perturbation model is derived based on the concept of a stable kernel representation of nonlinear systems the robust stabilization problem is then translated into a nonlinear disturbance feedforward h optimal control problem whose solution depends on the solvability of a single hamilton jacobi equation

brief paper robust stabilization of nonlinear systems via normalized coprime factor representations a reasonably complete theory for the synthesis of robust controllers for a broad class of nonlinear systems is now available we use this theory to generalize the linear theory of normalized coprime factor robustness optimization to the case of affine input nonlinear systems in particular we show that the equilibrium controller may be characterized in terms of the stabilizing and destabilizing solutions of the hamilton jacobi equation used to calculate the normalized right coprime factors of the plant we also show that the optimal robustness margin email protected 1 mn 2 hgeneralizes to the nonlinear case in preparation for the nonlinear analysis we review the linear case in a way which motivates our approach to the nonlinear case and highlights the parallels with it

robustness analysis of nonlinear feedback systems an input output approach this paper presents an approach to robustness analysis for nonlinear feedback systems we pursue a notion of model uncertainty based on the closeness of input output trajectories which is not tied to a particular uncertainty representation such as additive parametric structured etc the basic viewpoint is to regard systems as operators on signal spaces we present two versions of a global theory where stability is captured by induced norms or by gain functions we also develop local approaches over bounded signal sets and give a treatment for systems with potential for finite time escape we compute the relevant stability margin for several examples and demonstrate robustness of stability for some specific perturbations e g small time delays we also present examples of nonlinear control systems which have zero robustness margin and are destabilized by arbitrarily small gap perturbations the paper considers the case where uncertainty is present in the controller as well as the plant and the generalization of the approach to the case where uncertainty occurs in several subsystems in an arbitrary interconnection

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

w stability of feedback systems abstract this paper introduces the notion of w stability of feedback systems this is a generalization of a concept of well posedness discussed by willems w stability requires that a feedback system should remain stable under a certain class of perturbations which may not be vanishingly small when measured by the gap metric the paper derives necessary and sufficient conditions for a feedback system to be w stable further an investigation of well posedness is given for feedback systems where high frequency roll off has been suppressed for modelling simplicity this question is pertinent in standard pid control

finite time global stabilization by means of time varying distributed delay feedback the paper contains certain results concerning the finite time global stabilization for triangular control systems described by retarded functional differential equations by means of time varying distributed delay feedback these results enable us to present solutions to feedback stabilization problems for systems with delayed input the results are obtained by using the backstepping technique

stabilizing a chain of integrators using multiple delays this note addresses the output feedback stabilization problem of a chain of integrators using multiple delays we shall prove that either n distinct delays or a proportional delay compensator with n 1 distinct delays are sufficient to stabilize a chain including n integrators we present two different approaches both are constructive and rely on frequency domain techniques on a derivative feedback approximation idea and a pole placement idea respectively an illustrative example triple integrator is presented

linear feedback systems and the graph topology it is established for general linear systems that the gap metric induces the coarsest topology with respect to which both closed loop stability and closed loop performance are robust properties in earlier works similar topological results were obtained by exploiting the existence of particular coprime factor system representations not known to exist in general by contrast the development here does not rely on any specific system representations systems are simply characterized as subspaces of norm bounded input output pairs and the analysis hinges on the underlying geometric structure of the feedback stabilization problem unlike other work developed within such a framework fundamental issues concerning the causality of feedback interconnections are discussed explicitly the key result of this paper concerns the difference between linear feedback interconnections with identical controllers in terms of a performance robustness related closed loop mapping upper and lower bounds on the induced norm of this difference are derived allowing for possibly infinite dimensional input output spaces and time varying behavior the bounds are both proportional to the gap metric distance between the plants which clearly demonstrates the gap to be an appropriate measure of the difference between open loop systems from the perspective of closed loop behavior to conclude an example is presented to show that bounds of the form derived here for linear systems do not hold in a general nonlinear setting

normalized coprime factorizations and balancing for unstable nonlinear systems in this paper we first study the normalized right and left coprime factorizations of a nonlinear system from a state space point of view in order to do so we introduce the notion of inner and co inner nonlinear systems secondly we deal with balancing for unstable nonlinear systems for linear systems there are several ways to approach this problem and we generalize the ideas of balancing the normalized coprime factorization and of lqg balancing to the nonlinear case lqg balancing can be seen as measuring the influence of a state component on a certain future and past energy function we extend this interpretation to the nonlinear case furthermore we use the normalized coprime factorization to give another way of balancing the unstable nonlinear system we give the relation between these two methods of balancing which is similar to the relation in the linear case

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

a generalization of berry s connection a generalization of berry s connection is proposed for quantum mechanical models whose wavefunctions are sections of a vector bundle over a bundle space with fibre the configuration space of the model and base space a space of parameters the transformation properties of the generalized berry s connection under diffeomorphisms of the parameter space are also studied

a biased approach to nonlinear robust stability and performance with applications to adaptive control the nonlinear robust stability theory of georgiou and smith ieee trans automat control 42 1997 pp 1200 1229 is generalized to the case of notions of stability with bias terms an example from adaptive control illustrates nontrivial robust stability certificates for systems which the previous unbiased theory could not establish a nonzero robust stability margin this treatment also shows that the bounded input bounded output robust stability results for adaptive controllers in french ieee trans automat control 53 2008 pp 461 478 can be refined to show preservation of biased forms of stability under gap perturbations in the nonlinear setting it also is shown that in contrast to linear time invariant systems the problem of optimizing nominal performance is not equivalent to maximizing the robust stability margin