hodge laplacians on graphs this is an elementary introduction to the hodge laplacian on a graph a higher order generalization of the graph laplacian we will discuss basic properties including cohomology and hodge theory t

algebraic theory for robust stability of interconnected systems necessary and sufficient conditions we consider an interconnected system s o made of linear mulrivariable subsystems which are specified by matrix fractions with elements in a ring of stable scalar transfer functions h given that the k th subsystem is perturbed from g k n rk d k 1 to tilde g k n rk delta n rk d k delta d k 1 and that the system s o is h stable we derive a computationally efficient necessary and sufficient condition for the h stability of the perturbed system these fractional perturbations are more general than the conventional additive and multiplicative perturbations the result is generalized to handle simultaneous perturbations of two or more subsystems

statistical ranking and combinatorial hodge theory we propose a technique that we call hodgerank for ranking data that may be incomplete and imbalanced characteristics common in modern datasets coming from e commerce and internet applications we are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data from raw ranking data we construct pairwise rankings represented as edge flows on an appropriate graph our statistical ranking method exploits the graph helmholtzian which is the graph theoretic analogue of the helmholtz operator or vector laplacian in much the same way the graph laplacian is an analogue of the laplace operator or scalar laplacian we shall study the graph helmholtzian using combinatorial hodge theory which provides a way to unravel ranking information from edge flows in particular we show that every edge flow representing pairwise ranking can be resolved into two orthogonal components a gradient flow that represents the l 2 optimal global ranking and a divergence free flow cyclic that measures the validity of the global ranking obtained if this is large then it indicates that the data does not have a good global ranking this divergence free flow can be further decomposed orthogonally into a curl flow locally cyclic and a harmonic flow locally acyclic but globally cyclic these provides information on whether inconsistency in the ranking data arises locally or globally when applied to statistical ranking problems hodge decomposition sheds light on whether a given dataset may be globally ranked in a meaningful way or if the data is inherently inconsistent and thus could not have any reasonable global ranking in the latter case it provides information on the nature of the inconsistencies an obvious advantage over the np hardness of kemeny optimization is that hodgerank may be easily computed via a linear least squares regression we also discuss connections with well known ordinal ranking techniques such as kemeny optimization and borda count from social choice theory

bounds on the spectrum for markov chains and markov processes a generalization of cheeger s inequality we prove a general version of cheeger s inequality for discrete time markov chains and continuous time markovian jump processes both reversible and nonreversible with general state space we also prove a version of cheeger s inequality for markov chains and processes with killing as an application we prove l 2 exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs

linear stable unity feedback system necessary and sufficient conditions for stability under nonlinear plant perturbations the authors consider a linear not necessarily time invariant stable unity feedback system where the plant and the compensator have normalized right coprime factorizations they study two cases of nonlinear plant perturbations additive and feedback with four subcases resulting from 1 allowing exogenous input to delta p or not 2 allowing the observation of the output of delta p or not the plant perturbation delta p is not required to be stable using the factorization approach the authors obtain necessary and sufficient conditions for all cases in terms of two pairs of nonlinear pseudostate maps simple physical considerations explain the form of these necessary and sufficient conditions finally the authors obtain the characterization of all perturbations delta p for which the perturbed system remain stable

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

finite element exterior calculus from hodge theory to numerical stability this article reports on the confluence of two streams of research one emanating from the fields of numerical analysis and scientific computation the other from topology and geometry in it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de rham cohomology and hodge theory are key tools for exploring the well posedness of the continuous problem the discretization methods we consider are finite element methods in which a variational or weak formulation of the pde problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace after a brief introduction to finite element methods we develop an abstract hilbert space framework for analyzing the stability and convergence of such discretizations in this framework the differential complex is represented by a complex of hilbert spaces and stability is obtained by transferring hodgetheoretic structures that ensure well posedness of the continuous problem from the continuous level to the discrete we show stable discretization is achieved if the finite element spaces satisfy two hypotheses they can be arranged into a subcomplex of this hilbert complex and there exists a bounded cochain projection from that complex to the subcomplex in the next part of the paper we consider the most canonical example of the abstract theory in which the hilbert complex is the de rham complex of a domain in euclidean space we use the koszul complex to construct two families of finite element differential forms show that these can be arranged in subcomplexes of the de rham complex in numerous ways and for each construct a bounded cochain projection the abstract theory therefore applies to give the stability and convergence of finite element approximations of the hodge laplacian other applications are considered as well especially the elasticity complex and its application to the equations of elasticity background material is included to make the presentation self contained for a variety of readers

brief overcoming the detectability obstacle in certainty equivalence adaptive control in this paper we are interested in global adaptive stabilization of nonlinear systems where we know a linearly parametrized stabilizer and a lyapunov function for the ideal closed loop system such that the standard estimator is implementable but this lyapunov function is not strict i e its derivative is only negative semi definite our motivation to consider this situation stems from the fact that for many physical systems the natural lyapunov function candidate is the total energy which is always not strict to complete the stability proof in this case it is necessary to add a rather restrictive detectability assumption our main contribution is to show that it is possible to overcome this obstacle by adding to the parameter estimator an identification error coming from an indirect identifier we thus replace the detectability assumption by a new condition that essentially requires that the negative definite terms appearing in the lyapunov function derivative dominate the uncertain terms that cannot be matched by the controller no matching or persistency of excitation assumptions are needed

immersion and invariance a new tool for stabilization and adaptive control of nonlinear systems a new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented the method relies upon the notions of system immersion and manifold invariance and in principle does not require the knowledge of a control lyapunov function the construction of the stabilizing control laws resembles the procedure used in nonlinear regulator theory to derive the invariant output zeroing manifold and its friend the method is well suited in situations where we know a stabilizing controller of a nominal reduced order model which we would like to robustify with respect to higher order dynamics this is achieved by designing a control law that asymptotically immerses the full system dynamics into the reduced order one we also show that in adaptive control problems the method yields stabilizing schemes that counter the effect of the uncertain parameters adopting a robustness perspective our construction does not invoke certainty equivalence nor requires a linear parameterization furthermore viewed from a lyapunov perspective it provides a procedure to add cross terms between the parameter estimates and the plant states finally it is shown that the proposed approach is directly applicable to systems in feedback and feedforward form yielding new stabilizing control laws we illustrate the method with several academic and practical examples including a mechanical system with flexibility modes an electromechanical system with parasitic actuator dynamics and an adaptive nonlinearly parameterized visual servoing application

brief paper on using least squares updates without regressor filtering in identification and adaptive control of nonlinear systems in continuous time system identification and adaptive control the least squares parameter estimation algorithm has always been used with regressor filtering which adds to the dynamic order of the identifier and affects its performance we present an approach for designing a least squares estimator that uses an unfiltered regressor we also consider a problem of adaptive nonlinear control and present the first least squares based adaptive nonlinear control design that yields a complete lyapunov function the design is presented for linearly parametrized nonlinear control systems in normal form a scalar linear example is included which adds insight into the key ideas of our approach and allows showing that for linear systems our lyapunov ls design with unfiltered regressor presented in the note for unnormalized least squares can also be extended to normalized least squares

a necessary and sufficient condition for stability of a perturbed system the problem of determining the stability of a feedback system in the presence of perturbation is considered a necessary and sufficient condition under which the perturbed system remains stable is obtained this condition is very general in particular it does not require the perturbed plant to have the same number of poles as the nominal plant

angular embedding a robust quadratic criterion given the size and confidence of pairwise local orderings angular embedding ae finds a global ordering with a near global optimal eigensolution as a quadratic criterion in the complex domain ae is remarkably robust to outliers unlike its real domain counterpart ls the least squares embedding our comparative study of ls and ae reveals that ae s robustness is due not to the particular choice of the criterion but to the choice of representation in the complex domain when the embedding is encoded in the angular space we not only have a nonconvex error function that delivers robustness but also have a hermitian graph laplacian that completely determines the optimum and delivers efficiency the high quality of embedding by ae in the presence of outliers can hardly be matched by ls its corresponding l1 norm formulation or their bounded versions these results suggest that the key to overcoming outliers lies not with additionally imposing constraints on the embedding solution but with adaptively penalizing inconsistency between measurements themselves ae thus significantly advances statistical ranking methods by removing the impact of outliers directly without explicit inconsistency characterization and advances spectral clustering methods by covering the entire size confidence measurement space and providing an ordered cluster organization

necessary and sufficient condition for stability robustness under known additive perturbation a necessary and sufficient condition is derived for the stability robustness of a unity feedback closed loop system involving a strictly proper plant p0 and a stabilizing compensator c0 under the assumption that the plant p0 is perturbed to p0 p0 where p0 is a known strictly proper rational matrix

higher order tracking properties of nonlinear adaptive control systems abstract while the asymptotic output tracking property has been proven for many nonlinear adaptive control systems their higher order output tracking property has not been reported in this paper the higher order derivative convergence of the output tracking error e t is studied for nonlinear adaptive control systems using the feedback linearization design it is proven that not only each error component e j t from e t e 1 t e 2 t e m t t converges to zero asymptotically but also its up to k th order time derivative e j k t converges to zero for k 1 2 j where j j 1 2 m are the control relative degrees of the nonlinear system with m inputs

nonlinear adaptive control of systems in feedback form an alternative to adaptive backstepping an algorithm for the regulation via state feedback of a class of linearly parameterised systems in feedback form is presented the algorithm resembles the well known adaptive backstepping tuning functions approach but differs in the way the parameter estimator is constructed and in the way the nonlinear damping factors are assigned the prominent feature of the proposed approach is that it allows for prescribed uniformly stable dynamics to be assigned to the estimation error thus leading to a modular scheme which is easier to tune than the one obtained from lyapunov redesign and practically recovers the performance of the known parameter controller for systems with matched uncertainties an additional property is that in contrast with adaptive feedback linearisation the corresponding non adaptive controller globally stabilises the system and renders it robust with respect to the estimation error provided a structural condition on the regressor holds a comparative example is presented highlighting the advantages of the proposed methodology

global parametrization of feedback systems with nonlinear plants for unity feedback systems with a plant p and controller c we show that given a nonlinear stable p the class of all controller stabilizing the feedback system is globally parametrized by a stable map q we also show how q is useful in studying the effects of modeling errors on the stability

an exponentially stable adaptive friction compensator this note presents a novel adaptive compensation scheme for coulomb friction in a servocontrol system an adaptive observer for estimating the unknown coulomb friction coefficient is also derived on the basis of the lyapunov technique in addition a linearizing control law is developed to compensate for the friction force and obtain the tracking objective the proposed adaptive compensation guarantees an exponential convergence for state errors and parameter error and known adaptive schemes guarantee only an asymptotic or stable convergence simulation results demonstrate the effectiveness of the proposed method for a single mass servocontrol system

an adaptive compensator for a class of linearly parameterized systems a compensation design for a class of linearly parameterized systems is presented the compensator consists of a typical linearizing control and an adaptive observer for online estimation of the system s parameters the proposed method achieves the asymptotic stability of the tracking and estimation error dynamics provided the basis functions in the regressor vector are linearly independent in terms of the desired system states no persistent excitation and measurement of the highest derivatives of the system states are required a numerical example is given to demonstrate the validity of the proposed design

harmonic functions on planar and almost planar graphs and manifolds via circle packings the circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant harmonic bounded dirichlet function if p is a bounded circle packing in bbb r 2 whose contacts graph is a bounded valence triangulation of a disk then with probability 1 the simple random walk on p converges to a limit point moreover in this situation any continuous function on the limit set of p extends to a continuous harmonic function on the closure of the contacts graph of p that is this dirichlet problem is solvable we define the notions of almost planar graphs and manifolds and show that under the assumptions of transience and bounded local geometry these possess non constant harmonic bounded dirichlet functions let us stress that an almost planar graph is not necessarily roughly isometric to a planar graph

a novel parameter projection mechanism for smooth and stable adaptive control abstract a novel parameter projection technique for handling uncertain parameters with a priori available upper and lower bounds is the subject of this paper it is a wellknown fact that the conventional parameter projection method employed in certainty equivalence adaptive control is able to guarantee global asymptotic stability only after compromising control smoothness in contrast the technique presented in this paper preserves both stability and smoothness guarantees for the closed loop adaptive system the novelty of the proposed adaptive control formulation comes through our deliberate introduction of a nonlinear re parameterization of uncertainty thereby amounting to a significant shift from the classical certainty equivalence methodology the proposed controller is non overparameterized and has the potential to significantly improve transient performance because of full and proper utilization of all the available prior information on the unknown parameters the technical descriptions are further elaborated via the duffing nonlinear oscillator example to highlight the advantages of this new approach

integral manifolds of slow adaptation conditions are given under which an integral manifold slow manifold exists for a nonlinear system representing a broad class of adaptive algorithms the parameter update equation restricted to the manifold is an exact description of the slow adaptation process which can be approximately analyzed by averaging stability properties of the slow motion in the manifold are extended off the manifold by a set of conditions under which the manifold is shown to be attractive this two time scale analysis provides a geometrical visualization of instability phenomena observed in adaptive systems and generalizes earlier local stability results

admissible additive perturbations for the general feedback system with two input two output plant and compensator the class of all admissible additive perturbations of the plant which do not destabilize the closed loop is established

minres qlp a krylov subspace method for indefinite or singular symmetric systems cg symmlq and minres are krylov subspace methods for solving symmetric systems of linear equations when these methods are applied to an incompatible system that is a singular symmetric least squares problem cg could break down and symmlq s solution could explode while minres would give a least squares solution but not necessarily the minimum length pseudoinverse solution this understanding motivates us to design a minres like algorithm to compute minimum length solutions to singular symmetric systems minres uses qr factors of the tridiagonal matrix from the lanczos process where r is upper tridiagonal minres qlp uses a qlp decomposition where rotations on the right reduce r to lower tridiagonal form on ill conditioned systems singular or not minres qlp can give more accurate solutions than minres we derive preconditioned minres qlp new stopping rules and better estimates of the solution and residual norms the matrix norm and the condition number

an example of a globally stabilizing adaptive controller with a generically destabilizing parameter estimate in this note we consider the question of whether an adaptive controller can converge to a nonadaptive stabilizing controller specifically we show for a class of back stepping controllers with adaptive tuning functions that the set of initial conditions in a state and estimation parameter for which the estimation parameter converges to a parameter which produces a destabilizing controller can have a nonempty interior and consequently a nonzero lebesgue measure this surprising result is proved by way of a simple example with a quadratic nonlinearity

adaptive backstepping control for a class of nonlinear systems with non triangular structural uncertainties in this technical note a robust adaptive control scheme is proposed based on backstepping techniques for a class of nonlinear systems with unknown parameters a modeling error may also exist in every state equation or channel and it is bounded by a known function which is allowed to depend on all system states it is shown that the proposed adaptive control scheme can ensure all signals in the closed loop system bounded if the strength of system modeling errors is sufficiently weak transient performance is also established thus stabilizing systems in classical strict feedback forms with sufficiently small non triangular structural perturbations is successfully addressed in the case that system parameters are known a non adaptive robust controller is designed to globally exponentially stabilize such a class of nonlinear systems finally simulation studies are used to verify the effectiveness of the proposed scheme

sufficient conditions for parameter convergence in linearizable systems the major drawback of the adaptive compensation design in the previous paper 2002 in terms of parameter identification is that the criterion for parameter convergence cannot be checked in advance the purpose of this paper aims at circumventing such difficulties by providing a priori verifiable sufficient conditions that guarantee the asymptotic stability of the parametric equilibrium

certainty equivalence implies detectability abstract it is shown that any stabilizing certainty equivalence control used within an adaptive control system causes the familiar interconnection of a controlled process and associated output estimator to be detectable through the estimator s output error e p for every frozen value of the index or parameter vector p upon which both the estimator and controller dynamics depend the fact that certainty equivalence implies detectability has been known for some time this has been shown to be so whenever the process model is linear and the controller and estimator models are also linear for every frozen value of p in this paper use is made of recently introduced concepts of input to state stability and detectability for nonlinear systems to prove that the same implication is valid in a more general nonlinear setting

a relationship between sensitivity and stability of multivariable feedback systems both comparison sensitivity and robust stability analysis attempt to extend concepts which are well understood in scalar feedback systems to multivariable control system design although the two bodies of work have developed about a decade apart the basic philosophies are similar both attempt to maintain a system property sensitivity reduction versus stability in the presence of plant modeling errors this paper demonstrates how the similarity in philosophy translates into a similarity in results and interpretations

stability and robustness properties of a simple adaptive controller a low order adaptive tracking controller is proposed for linear time invariant plants with a relative degree not exceeding two the order of the plant is not required to be known a priori the algorithm is robustly stable with respect to small linear time invariant plant perturbations and bounded disturbances the robustness is achieved by using a projection of the parameter estimates in the control law an additional a priori information needed to design the controller is bounds on the plant parameters in the absence of plant perturbations and disturbances perfect tracking is obtained

stabilization of a class of nonlinear systems by a smooth feedback control we consider the stabilizability problem for systems of the form x f x ub in the neighborhood of an equilibrium point of f x first by means of center manifold theory a lower order system is introduced if this system is stabilizable then so is the original system results on the stabilization of the low order system are presented no resort is taken to liapunov theory the relation between stabilizability and controllability is discussed

immersion and invariance adaptive control for discrete time systems in strict feedback form abstract this paper presents a new design procedure for the adaptive regulation tracking via state feedback for discrete time nonlinear systems in a parametric strict feedback form the algorithm utilizes discrete time adaptive backstepping together with a new parameter estimator design this approach provides an adaptive control law which guarantees boundedness of the closed loop trajectories global convergence to the origin tracking of the state of the closed loop system and local stability the performance of the proposed method is illustrated by simulations

switched adaptive control of switched nonlinearly parameterized systems with unstable subsystems this paper investigates the problem of adaptive stabilization for a class of switched nonlinearly parameterized systems where the solvability of the adaptive stabilization problem for subsystems is unnecessary a new switched adaptive control technique for the problem studied is set up by exploiting the generalized multiple lyapunov functions method and the parameter separation technique different update laws are designed to reduce the conservativeness caused by adoption of a common update law for all subsystems also the proposed technique permits removal of a common restriction in which the parameterization in the switched systems is restricted to a linear parameterization as an application of the proposed design technique stabilization for a class of non triangular switched systems with nonlinear parameterization is achieved by design of adaptive controllers since no subsystem of such a switched system is assumed to be adaptively stabilizable the problem under study cannot be handled by the existing methods a two inverted pendulums as a practical example is also provided to demonstrate the effectiveness of the proposed design method

flows and decompositions of games harmonic and potential games in this paper we introduce a novel flow representation for finite games in strategic form this representation allows us to develop a canonical direct sum decomposition of an arbitrary game into three components which we refer to as the potential harmonic and nonstrategic components we analyze natural classes of games that are induced by this decomposition and in particular focus on games with no harmonic component and games with no potential component we show that the first class corresponds to the well known potential games we refer to the second class of games as harmonic games and demonstrate that this new class has interesting properties which contrast with properties of potential games exploiting the decomposition framework we obtain explicit expressions for the projections of games onto the subspaces of potential and harmonic games this enables an extension of the equilibrium properties of potential and harmonic games to nearby games