j spectral factorization and equalizing vectors abstract this note presents some basic results on j spectral factorization and its connection with state space realizations the assumptions needed are mild and the proofs are given in detail

h controller synthesis by j lossless coprime factorization this paper develops a coprime factorization approach to the synthesis of internally stabilizing controllers for a given system such that the mathcal h infty norm of the closed loop is strictly less than a given bound by the use of coprime factorizations it is shown that the mathcal h infty control problem is fundamentally related to the so called analytic systems considered by helton et al regional conference series in mathematics 68 1987 such problems admit a solution if and only if a certain j lossless factorization exists interpreted in the mathcal h infty control context this means that for a controller of the requisite type to exist the plant must admit a certain j lossless coprime factorization the full mathcal h infty synthesis problem requires that two nested j lossless factorizations exist the results are independent of whether discrete time or continuous time systems are being considered it is then shown that j lossless factorization is equivalent to the existen

a behavioral approach to the h optimal control problem this paper considers the general h optimal control problem from a behavioral perspective a formalization of this problem is given that departs from the usual h control paradigm in the sense that system variables of the plant are treated in a symmetric way without distinguishing control inputs measurements exogenous inputs and to be controlled variables interconnection variables are introduced and controllers are allowed to constrain the interconnection variables of the plant necessary and sufficient conditions are given for the existence of controllers that achieve an h control objective for a given linear time invariant plant the set of all such controllers is parametrized by means of j spectral factorizations

minimal nonsquare spectral factors we consider the problem of parametrizing the set of all nonsquare minimal spectral factors of a rational matrix function taking positive semidefinite values on the imaginary axis

j inner outer factorization j spectral factorization and robust control for nonlinear systems the problem of expressing a given nonlinear state space system as the cascade connection of a lossless system and a stable minimum phase system inner outer factorization is solved for the case of a stable system having state space equations affine in the inputs the solution is given in terms of the stabilizing solution of a certain hamilton jacobi equation the stable minimum phase factor is obtained as the solution of an associated nonlinear spectral factorization problem as an application one can arrive at the solution of the nonlinear h sub spl infin control problem for the disturbance feedforward case

nonsquare spectral factors via factorizations of a unitary function abstract we consider the problem of parametrizing the set of nonsquare minimal spectral factors of a rational matrix function taking positive semidefinite values on the imaginary axis the tool here is the use of factorizations of a certain type of the function w 1w where w resp w is the minimal square spectral factor that has all its poles and zeros in the right resp left halfplane

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

inner outer factorization for nonlinear noninvertible systems this paper considers inner outer factorization of asymptotically stable nonlinear state space systems in continuous time that are noninvertible our approach will be via a nonlinear analogue of spectral factorization which concentrates on first finding the outer factor instead of the inner factor an application of the main result to control of nonminimum phase nonlinear systems is indicated

h sup infinity control for nonlinear systems with output feedback the basic question of nonlinear h sup infinity control theory is to decide for a given two port system when feedback that makes the full system dissipative and internally stable exists this problem can also be viewed as a question about circuits and after translation also has a game theoretic statement several necessary conditions for solutions to exist are presented and sufficient conditions for a certain construction to lead to a solution are given

how many hypersurfaces does it take to cut out a segre class abstract we prove an identity of segre classes for zero schemes of compatible sections of two vector bundles applications include bounds on the number of equations needed to cut out a scheme with the same segre class as a given subscheme of for example a projective variety and a segre bertini theorem controlling the behavior of segre classes of singularity subschemes of hypersurfaces under general hyperplane sections these results interpolate between an observation of samuel concerning multiplicities along components of a subscheme and facts concerning the integral closure of corresponding ideals the segre bertini theorem has applications to characteristic classes of singular varieties the main results are motivated by the problem of computing segre classes explicitly and applications of segre classes to enumerative geometry

selection of a single uniquely specifiable h controller in the chain scattering framework this paper studies the notion of central h controllers in the chain scattering framework since in the chain scattering framework the h control problem is equivalent to a factorisation problem that yields non unique factors it is important to somehow pin down these factors so that the resulting central controller is uniquely defined and corresponds to the central and minimum entropy controller frequently discussed in the literature in so doing we devise a procedure for the selection of a single uniquely specifiable h controller in the chain scattering framework the chain scattering framework is important because it has a distinct advantage over state space based methods for some applications in that it poses and solves the h control problem entirely in an operator theoretic setting thereby allowing us to contemplate direct changes of the frequency domain symbols rather than changes in the corresponding state space matrices

control of systems with i o delay via reduction to a one block problem in this paper the standard four block h sup spl infin control problem for systems with a single delay in the feedback loop is studied a simple procedure of the reduction of the problem to an equivalent one block problem having particularly simple structure is proposed the one block problem is then solved by the j spectral factorization approach resulting in the so called dead time compensator dtc form of the controller the advantages of the proposed procedure are its simplicity intuitively clear derivation of the dtc form of the h sup spl infin controller and extensibility to the multiple delay case

decomposition in conic optimization with partially separable structure decomposition techniques for linear programming are difficult to extend to conic op timization problems with general nonpolyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables however in many applications the convex cones have a partially separable structure that allows them to be characterized in terms of simpler lower dimensional cones the most important example is sparse semidefinite programming with a chordal sparsity pattern here partial separability derives from the clique decomposition theorems that characterize positive semidefinite and positive semidefinite completable matrices with chordal sparsity patterns the paper describes a decomposition method that exploits partial separability in conic linear optimization the method is based on spingarn s method for equality constrained convex optimization combined with a fast interior point method for evaluating proximal operators

state space approach to discrete time h control abstract state space solutions to a discrete time problem are given for a given number 0we give a characterization of all controllers such that the norm of the closed loop transfer function is less than the approach taken is based on the solution of two riccati equations using the stable deflating subspace of a symplectic pencil this ensures that no unnecessary assumptions on the a matrix on the realization of the transfer function are needed while the results are the direct analogues of those in the continuous time the state space derivation is considerably more difficult

a partial differential inequality for dissipative nonlinear systems abstract in this paper we show that storage functions for dissipative systems can be characterised by a partial differential inequality in the weak or viscosity sense the storage functions need not be differentiable this characterisation is stable with respect to perturbations we formulate a version of the positive real lemma for passive systems using the weak sense framework

state space solutions to h sampled data control problems this paper considers a general sampled data control problem with band limited continuous time exogenous inputs new results on the construction of h controllers are presented using the chain scattering matrix formalism this results in a relatively simple method in transfer function domain a generator for h controllers is obtained by solving two coprime factorizations which are required to solve a continuous time and a discrete time algebraic riccati equations in state space

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

a j spectral factorization approach to h control it is shown that necessary and sufficient conditions for the existence of sub optimal solutions to the model matching problem associated with h control are that two coupled j spectral factorizations exist the second j spectral factor provides a parameterization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of stabilizing non negative definite solutions to two algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given

chern schwartz macpherson classes and the euler characteristic of degeneracy loci and special divisors this concept overlaps a large family of interesting varieties for example the varieties of special divisors studied in section 3 several authors have worked out explicit formulas for the euler characteristic of dr 9 in terms of different cohomological and numerical invariants under the assumption that x is nonsingular and v is appropriately general for instance if 0 is a section of a vector bundle then the formulas for the euler characteristic x do fp were given by hirzebruch h and navarro aznar n if dr is a curve or a surface in a nonsingular x some explicit formulas for x dr v were given by harris and tu h t in terms of the chern classes of e f and x but under the extra assumption dr 1 v 0 which implies that dr q is nonsingular in loc cit the authors also posed the problem of finding a general formula for x dr o if such exists under the assumption dr 1 0 or even stronger without this assumption the first problem was solved positively by the second named author prl proposition 5 7 by the use of polynomials universally supported on degeneracy

corrections to on the structure of h sup spl infin control systems and related extensions the authors state that in the original paper kimura lu and kawatani ibid vol 36 p 653 67 1991 a lemma on j j losslessness is not correct its stated implications are also not fully correct they give a counterexample to these claims and show how they can be fixed by strengthening the assumption

interpolation of characteristic classes of singular hypersurfaces abstract we show that the chern schwartz macpherson class of a hypersurface x in a nonsingular variety m interpolates between two other notions of characteristic classes for singular varieties provided that the singular locus of x is smooth and that certain numerical invariants of x are constant along this locus this allows us to define a lift of the chern schwartz macpherson class of such nice hypersurfaces to intersection homology as another application the interpolation result leads to an explicit formula for the chern schwartz macpherson class of x in terms of its polar classes

on cancellation in h optimal controllers abstract this note clarifies the mechanism of pole zero cancellations that appeared in the derivation of the h optimal controller via a solution of the corresponding model matching problem the cancellable factor is explicitly expressed using chain scattering matrix descriptions a simple state space formula of the sub optimal controller is also presented based on the chain scattering forms and we show that there is a mixed sensitivity problem that can be solved by calculating a solution of only one riccati equation in each iteration

h sub spl infin control in a behavioral context the full information case the authors formulate the h sub spl infin control problem in a behavioral setting given a mathematical model say a set of higher order differential equations together with some static equations the vector of manifest variables is partitioned into yet to be controlled variables unknown exogenous variables and interconnection variables the interconnection variables are available for interconnection in the sense that they can be made to obey certain differential or static equations to be specified by the designer such a system of differential equations and static equations is called a controller the design problem that we consider is to find controllers such that the size of the to be controlled variables is less than a given tolerance for all disturbances in the unit ball and such that the interconnection is a stable system we find necessary and sufficient conditions for the existence of suitable controllers under the hypothesis that we have a full information problem these conditions involve indefinite factorizations of polynomial matrices and a test on a given pick matrix

minimal square spectral factors abstract recently there has been renewed interest in the problem of spectral factorization and in particular the problem of parametrizing all square minimal spectral factors of a given spectrum for instance let us mention two recent papers in this journal one of which dealing with more computational aspects of this problem clements 1993 the other giving a parametrization under additional constraints ferrante et al 1993 this paper is motivated in a large part by ferrante et al 1993 in fact we shall show that a parametrization similar to the one given there can be achieved without one of the additional constraints imposed in ferrante et al 1993 a secondary aim of this paper is to give an overview of several parametrizations available in the literature

convergence of the dre solution to the are strong solution in this paper we use the boundary solutions to a linear matrix inequality lmi associated with kalman filtering to investigate the convergence of the solution of a differential riccati equation dre to the so called strong solution of an algebraic riccati equation are we furthermore extend our results to kalman filtering with indefinite input noise covariances

the segre zeta function of an ideal abstract we define a power series associated with a homogeneous ideal in a polynomial ring encoding information on the segre classes defined by extensions of the ideal in projective spaces of arbitrarily high dimension we prove that this power series is rational with poles corresponding to generators of the ideal and with numerator of bounded degree and with nonnegative coefficients we also prove that this segre zeta function only depends on the integral closure of the ideal the results follow from good functoriality properties of the shadows of rational equivalence classes of projective bundles more precise results can be given if all homogeneous generators have the same degree and for monomial ideals in certain cases the general description of the segre zeta function given here leads to substantial improvements in the speed of algorithms for the computation of segre classes we also compute the projective ranks of a nonsingular variety in terms of the corresponding zeta function and we discuss the segre zeta function of a local complete intersection of low codimension in projective space

on the structure of a two degree of freedom h loop shaping controller explicit state space formulae are presented for a two degree of freedom 2 dof controller that stabilizes an augmented plant with respect to unstructured additive coprime factor perturbations whilst simultaneously providing robust model following the 2 dof problem was known to inherit certain features of the mcfarlane glover loop shaping design method from which it is derived in this paper the precise structure of one admissible controller is derived that structure is shown to comprise a state estimator an on line step response model and an augmented state estimate feedback law the formulae derived herein are applied to a classical nonlinear stabilization problem

exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion a basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities the sparsity characterized with a chordal graph structure can be detected in the variable matrix or in a linear or nonlinear matrix inequality constraint of the problem we classify the sparsity in two types the domain space sparsity d space sparsity for the symmetric matrix variable in the objective and or constraint functions of the problem which is required to be positive semidefinite and the range space sparsity r space sparsity for a linear or nonlinear matrix inequality constraint of the problem four conversion methods are proposed in this framework two for exploiting the d space sparsity and the other two for exploiting the r space sparsity when applied to a polynomial semidefinite program sdp these conversion methods enhance the structured sparsity of the problem called the correlative sparsity as a result the resulting polynomial sdp can be solved more effectively by applying the sparse sdp relaxation preliminary numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems

inner outer factorization of nonlinear operators abstract a fundamental result in the theory of hardy spaces of analytic matrix and operator valued functions on the unit disk is inner outer factorization for the case of rational matrix functions explicit formulas for the inner and outer factors have recently been obtained in these formulas the functions are presented in terms of a collection of finite matrices as the transfer function of a linear system the key ingredient in the construction is the solution of a linear matrix equation called a stein equation in this paper we obtain extensions of these results to the setting of input output maps of nonlinear discrete time systems a particaular computable class of examples is presented for illustration

j spectral factorization and equalizing vectors for the wiener class of matrix valued functions we provide necessary and sufficient conditions for the existence of a j spectral factorization one of these conditions is in terms of equalizing vectors the second one states that the existence of a j spectral factorization is equivalent to the invertibility of the toeplitz operator associated to the matrix to be factorized our proofs are simple and only use standard results of general factorization theory clancey and gohberg factorization of matrix functions and singular integral operators operator theory advances and applications vol 3 birkhauser basel 1981 note that we do not use a state space representation of the system however we make the connection with the known results for the pritchard salamon class of systems where an equivalent condition with the solvability of an algebraic riccati equation is given

the four block adamjan arov krein problem for discrete time systems abstract we consider the problem of approximating in h norm a given discrete inverse time stable system possibly improper t t 11 t 12 t 21 t 22 by a discrete time system s with no more than r poles outside the unit disk possibly at infinity such that t 11 s t 12 t 21 t 22 here r 0 is an integer and 0 is a prescribed tolerance if r 0 this is the four block nehari problem while if we consider the one block case where t t11 we obtain the well known hankel norm approximation problem the theoretical developments are based on a frequency domain signature condition while the class of solutions is constructed in state space in terms of the solutions to two riccati equations

j lossless and extended j lossless factorizations approach for domain h control the paper addresses problems of numerical conditioning of the discrete time h x control design based on j lossless factorizations of the chain scattering representation of a plant it is demonstrated that for a sufficiently small sampling period forward shift operator techniques may become ill conditioned and numerical robustness and reliability of design procedures can be significantly improved by utilizing a so called i operator form of the source problem state space models of all i domain controllers are given the solutions are obtained by considering two coupled i domain algebraic riccati equations that can be solved by applying a generalized eigenproblem formulation numerical conditioning of these solutions can effectively be evaluated by using a suitable relative condition number an extended j lossless factorization for generalized plants with zeros on the stability boundary circle is also given a so called zero compensator technique adapted to the i domain models is utilized to cancel suc

a derivation of the glover doyle algorithms for general h control problems abstract we show that the glover doyle algorithm can be formulated simply by using the j j lossless factorization method and chain scattering matrix description this algorithm was first stated by glover and doyle in 1988 because the corresponding diagonal block of the j j lossless matrix in the general 4 block h control problem of the glover doyle algorithm is not square a new type of chain scattering matrix description is developed with this description in hand we obtain two types of state space solution which are similar to each other thus a similarity transformation between these solutions in the 4 block h control problem can also be obtained the main idea of the solution is illustrated by means of block diagrams

rational spectral factorization using state space methods abstract we describe a state space based algorithm to find the minimum phase spectral factor of any rational spectral density whether it be proper improper or polynomial

on a state space approach to nonlinear h control we study the standard h optimal control problem using state feedback for smooth nonlinear control systems the main theorem obtained roughly states that the l2 induced norm from disturbances to inputs and outputs can be made smaller than a constant 0 if the corresponding h norm for the system linearized at the equilibrium can be made smaller than by linear state feedback necessary and sufficient conditions for the latter problem are by now well known e g from the state space approach to linear h optimal control our approach to the nonlinear h optimal control problem generalizes the state space approach to the linear h problem by replacing the hamiltonian matrix and corresponding riccati equation as used in the linear context by a hamiltonian vector field together with a hamiltonian jacobi equation corresponding to its stable invariant manifold

on a hardy space approach to the analysis of spectral factors the paper presents a study of several problems related to spectral factorizations we assume only a very weak form of coercivity for the p x p spectral function phi and look at the set wm of all rectangular p x m spectral factors the main object is the arithmetization of the geometry of the set of minimal stable spectral factors by employing hardy space techniques and the arithmetic of inner functions particular attention is paid to the study of various partial orders associated with the set of spectral factors

non minimal factorization of nonlinear systems the observable case minimal factorizations of nonlinear systems have been studied earlier here non minimal factorizations of nonlinear systems are studied for the case where the product contains no extra non observable states observable factorizations a description for all such factorizations is given in terms of invariant foliations of the system and its inverse the results are applied to the special cases of stable factorizations coprime

unstable and nonproper weights in h control an h control problem is examined in which the controller can and need only stabilize a part of the generalized closed loop due to unstable weights the procedure is a simple extension of known results and is computationally the easiest method available

positive semidefinite matrices with a given sparsity pattern abstract this paper concerns the cone of positive semidefinite matrices which have zeros in prescribed entries one of the main purposes is to obtain information and if possible describe the ranks of external matrices in such cones in terms of the pattern of prescribed zeros in particular we study those patterns for which the ranks of extremal matrices are less than a given number

on extended j j lossless factorization abstract the extended j j lossless factorization for discrete time rational matrix functions with zeros on the unit circle is considered a necessary and sufficient condition for the existence of the extended j j lossless factorization is obtained the condition is given in terms of the original system parameters in the form of a generalized eigenvalue problem and a discrete time algebraic riccati equation state space representations for the extended j j lossless factorization are provided

stable invariant lagrangian subspaces factorization of symmetric rational matrix functions and other applications several applications of earlier results by the authors concerning various notions of stability of invariant lagrangian subspaces are studied the applications include stability of symmetric minimal factorizations of real symmetric rational matrix functions stability of factorizations of certain classes of symmetric matrix polynomials and stably well posed matricial boundary value problems with symmetries an interpretation of many stability results in terms of equilibria of lie group actions on subspaces is given as well

parametrization of all minimal square spectral factors abstract two different parametrizations of minimal square spectral factors are given these parametrizations extend recent results by picci and pinzoni since they characterize spectral factors without specification on the zero or pole structure

necessary and sufficient conditions for j spectral factorizations with a j lossless property for infinite dimensional systems in continuous and discrete time necessary and sufficient conditions for j spectral factorizations are given in terms of the existence of a self adjoint stabilizing solution of an appropriate riccati equation in infinite dimensions furthermore it is shown that a certain j lossless property holds if and only if the stabilizing solution to the riccati equation is nonnegative definite we prove these results first for the continuous time case and thereby derive analogous results for the discrete time case by means of a bilinear transformation of the complex plane

technical communique j spectral factorization of regular para hermitian transfer matrices this paper characterizes a class of regular para hermitian transfer matrices and then reveals the elementary characteristics of j spectral factorization for this class a transfer matrix l in this class admits a j spectral factorization if and only if there exists a common nonsingular matrix to similarly transform the a matrices of l and l 1 resp into 2x2 lower upper resp triangular block matrices with the 1 1 block including all the stable modes of l l 1 resp for a transfer matrix in a smaller subset this nonsingular matrix is formulated in terms of the stabilizing solutions of two algebraic riccati equations the j spectral factor is formulated in terms of the original realization of the transfer matrix

segre classes on smooth projective toric varieties we provide a generalization of the algorithm of eklund jost and peterson for computing segre classes of closed subschemes of projective k space the algorithm is here generalized to computing the segre classes of closed subschemes of smooth projective toric varieties

quadratic optimal control of stable well posed linear systems we consider the infinite horizon quadratic cost minimization problem for a stable time invariant well posed linear system in the sense of salamon and weiss and show that it can be reduced to a spectral factorization problem in the control space more precisely we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution if both the system and the spectral factor are regular then the feedback operator can be expressed in terms of the riccati operator and the riccati operator is a positive self adjoint solution of an algebraic riccati equation this riccati equation is similar to the usual algebraic riccati equation but one of its coefficients varies depending on the subspace in which the equation is posed similar results are true for unstable systems as we have proved elsewhere

a chain scattering matrix approach to the h output feedback control for state delayed systems a state space coprime factorization approach to the h controllers design for state delayed systems is developed the coprime factors that are generated by the present factorization possess the same delayed state as that of the original system this property together with the well known youla parameterization and the chain scattering matrix description provides a relatively simple method to obtain a family of the internally stabilizing h output feedback controllers a numerical example is given to demonstrate the validity and the applicability of the proposed approach

a j lossless coprime factorisation approach to h control in delta domain the paper addresses problems of numerical conditioning of the discrete time h control design based on a j lossless coprime factorisations of a chain scattering description of a controlled plant it is demonstrated that forward shift operator techniques for solving the design problem may become ill conditioned for a sufficiently small sampling period it is shown that numerical robustness and reliability of computations can be significantly improved via utilising a so called delta operator form of the origin problem state space formulae for all delta domain controllers are given the solution is obtained via solving two coupled riccati equations leading to j lossless coprime factorisations of a chain scattering description of a controlled plant in order to evaluate a measure of numerical conditioning of these solutions the relative condition number of the delta domain algebraic riccati equation is introduced an example dealing with a problem of mixed sensitivity h design is given to illustrate the mthod

coprime factorization and dynamic stabilization of transfer functions it is known that a matrix valued transfer function p has a stabilizing dynamic controller q i e begin smallmatrix i q p i end smallmatrix 1 in rm h infty iff p has a right or left coprime factorization we show that the same result is true in the operator valued case thus the standard youla bongiorno parameterization applies to every dynamically stabilizable function we then derive further equivalent conditions one of them being that p has a stabilizing controller with internal loop this and some others are new even in the scalar valued case we also establish certain related results for example we extend the classical results on coprime factorization and partial feedback measurement feedback stabilization to nonrational transfer functions all our results apply in both discrete and continuous time settings except that in the latter it is not clear whether the controller q can always be chosen so that it is continuous time proper holomorphic and bounded on a right half plane unless e g p z rightarrow0 as operatorname re z rightarrow infty

matrices attaining the minimum semidefinite rank of a chordal graph every positive semidefinite matrix whose graph g is chordal may be represented as a sum of rank 1 positive semidefinite matrices whose graphs are subgraphs of g we show that if the matrix is of minimum rank then this representation is unique this result is used to give a full characterization of the completely positive matrices of minimum rank with a given chordal graph every such matrix is shown to have a unique easily computable minimal rank 1 representation and cp rank equal to its rank we also characterize all chordal graphs with the property that every minimum rank doubly nonnegative matrix realization of the graph is completely positive

nonsquare spectral factorization for nonlinear control systems this paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time more specifically we obtain a parametrization of nonsquare spectral factors in terms of invariant lagrangian submanifolds and associated solutions of hamilton jacobi inequalities this inequality is a nonlinear analogue of the bounded real lemma and the control algebraic riccati inequality by way of an application we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system

cuts matrix completions and graph rigidity this paper brings together several topics arising in distinct areas polyhedral combinatorics in particular cut and metric polyhedra matrix theory and semidefinite programming in particular completion problems for positive semidefinite matrices and euclidean distance matrices distance geometry and structural topology in particular graph realization and rigidity problems cuts and metrics provide the unifying theme indeed cuts can be encoded as positive semidefinite matrices this fact underlies the approximative algorithm for max cut of goemans and williamson and both positive semidefinite and euclidean distance matrices yield points of the cut polytope or cone after applying the functions 1 arccos or when fixing the dimension in the euclidean distance matrix completion problem we find the graph realization problem and the related question of unicity of realization which leads to the question of graph rigidity our main objective here is to present in a unified setting a number of results and questions concerning matrix completion graph realization and rigidity problems these problems contain indeed very interesting questions relevant to mathematical programming and we believe that research in this area could yield to cross fertilization between the various fields involved

chordal decomposition in operator splitting methods for sparse semidefinite programs we employ chordal decomposition to reformulate a large and sparse semidefinite program sdp either in primal or dual standard form into an equivalent sdp with smaller positive semidefinite psd constraints in contrast to previous approaches the decomposed sdp is suitable for the application of first order operator splitting methods enabling the development of efficient and scalable algorithms in particular we apply the alternating direction method of multipliers admm to solve decomposed primal and dual standard form sdps each iteration of such admm algorithms requires a projection onto an affine subspace and a set of projections onto small psd cones that can be computed in parallel we also formulate the homogeneous self dual embedding hsde of a primal dual pair of decomposed sdps and extend a recent admm based algorithm to exploit the structure of our hsde the resulting hsde algorithm has the same leading order computational cost as those for the primal or dual problems only with the advantage of being able to identify infeasible problems and produce an infeasibility certificate all algorithms are implemented in the open source matlab solver cdcs numerical experiments on a range of large scale sdps demonstrate the computational advantages of the proposed methods compared to common state of the art solvers

factorization and feedback stabilization for nonlinear systems we establish the equivalence of internal input out stability for two feedback configurations of a nonlinear time varying plant p for which a related plant g is assumed to have a factorization g r with both r and r 1 incrementally stable this extends a factorization principle for stabilizability previously given only for the linear time invariant case as an application of a special case we recover a version of the youla parametrization of stabilizing compensators for the nonlinear case previously presented in the literature we use degree theory to parametrize a collection of solutions of the h control problem for the case of a 1 gain stable or lossless plant in the case of a plant g having a j inner outer factorization this last result combined with the above mentioned factorization principle leads to results on the h control problem for p

a direct proof for the matrix decomposition of chordal structured positive semidefinite matrices agler helton mccullough and rodman proved that a graph is chordal if and only if any positive semidefinite psd symmetric matrix whose nonzero entries are specified by a given graph can be decomposed as a sum of psd matrices corresponding to the maximal cliques this decomposition is recently exploited to solve positive semidefinite programming efficiently their proof is based on a characterization for psd matrix completion of a chordal structured matrix due to grone johnson sa and wolkowicz this note gives a direct and simpler proof for the result of agler et al which leads to an alternative proof of grone et al

analytic system problems and j lossless coprime factorization for infinite dimensional linear systems abstract this paper extends the coprime factorization approach to the synthesis of internally stabilizing controllers satisfying an h norm bound to a class of systems with irrational transfer matrices using the coprime factorization description the h control problem can be reduced to two stable analytic system problems such problems have solutions if and only if a certain j lossless factorization exists the full h synthesis problem is shown to be equivalent to the solution of two nested j lossless factorizations if the irrational transfer matrix has a state space realization then the known state space formulas for the h control problem may be recovered using the relationship between j lossless factorizations and solutions of riccati equations however the results derived here are valid for a larger class of infinite dimensional systems

a characterization of minimal square spectral factors two different well known approaches to the spectral factorization problem phi s w s w s are connected together by relating the geometric properties of the solution set of the underlying algebraic riccati equation to the structure of the all pass factor of each minimal solution w s

synthesis of dissipative systems using quadratic differential forms part ii the problem discussed is that of designing a controller for a linear system that renders a quadratic functional nonnegative our formulation and solution of this problem is completely representation free the system dynamics are specified by a differential behavior and the performance is specified through a quadratic differential form we view control as interconnection a controller constrains a distinguished set of system variables the control variables the resulting behavior of the to be controlled variables is called the controlled behavior the constraint that the controller acts through the control variables only can be succinctly expressed by requiring that the controlled behavior should be wedged in between the hidden behavior obtained by setting the control variables equal to zero and the plant behavior obtained by leaving the control variables unconstrained the main result is a set of necessary and sufficient conditions for the existence of a controlled behavior that meets the performance specifications the essential requirement is a coupling condition an inequality that combines the storage functions of the hidden behavior and the orthogonal complement of the plant behavior

non linear minimal square spectral factorization recent work has identified a correspondence between minimal square spectral factors of a given spectral density invariant subspaces of an associated hamiltonian matrix solutions of an appropriate algebraic riccati equation and minimal unitary left divisors of a certain unitary function on the imaginary line the purpose of this report is to obtain a natural extension of these ideas to a non linear setting at least for the special case where only minimal square stable spectral factors are considered

discrete time h algebraic riccati equation and parametrization of all h filters this paper is concerned with the algebraic riccati equations ares related to the h filtering problem a necessary and sufficient condition for the h problem to be solvable is that the h are has a positive semidefinite stabilizing solution with an additional condition that a certain matrix is positive definite it is shown that such a stabilizing solution is a monotonically non increasing convex function of the prescribed h norm bound this property of the h are is very important for the analysis of the performance of the h filter in this paper the size of the set of all h filters is considered on the basis of the monotonicity of the above riccati solution it turns out that under a certain condition the degree of freedom of the h filter reduces a1 the optimal h norm bound these results provide a guideline for selecting the value of some numerical examples are included