pole placement by parametric output feedback abstract this note presents a new analytical solution to the problem of pole placement via constant output feedback under the condition m p n where n m and p are the number of states inputs and outputs respectively the approach is based upon parametric eigenstructure assignment of linear time invariant multivariable systems in combination with a special explicit formulation of the pole assignment equations thus the resulting analytical solution explicitly offers all remaining m p n degrees of freedom beyond eigenvalue assignment which can be used for additional design goals such as response shaping minimizing the norm of the feedback matrix and robust control respectively

optimal systems root loci relation to the mcmillan structure of the open loop system this note shows that the orders of the butterworth patterns formed by the asymptotes of optimal closed loop poles of a time invariant linear feedback system as the weight of the input in the performance criterion approaches zero always correspond to twice the mcmillan orders of the zeros at infinity of the open loop transfer matrix furthermore bounds are given in terms of the newton polygons on the orders of the butterworth patterns formed by the root loci at finite pole zero locations finally some examples are presented to show that in the latter case the branching behavior need not always correspond to the mcmillan structure of the open loop system

an infeasible bundle method for nonsmooth convex constrained optimization without a penalty function or a filter global convergence in constrained optimization algorithms has traditionally been enforced by the use of parametrized penalty functions recently the filter strategy has been introduced as an alternative at least part of the motivation for using filter methods consists of avoiding the need for estimating a suitable penalty parameter which is often a delicate task in this paper we demonstrate that the use of a parametrized penalty function in nonsmooth convex optimization can be avoided without using the relatively complex filter methods we propose an approach which appears to be more direct and easier to implement in the sense that it is closer in spirit and structure to the well developed unconstrained bundle methods preliminary computational results are also reported

bundle level type methods uniformly optimal for smooth and nonsmooth convex optimization the main goal of this paper is to develop uniformly optimal first order methods for convex programming cp by uniform optimality we mean that the first order methods themselves do not require the input of any problem parameters but can still achieve the best possible iteration complexity bounds by incorporating a multi step acceleration scheme into the well known bundle level method we develop an accelerated bundle level method and show that it can achieve the optimal complexity for solving a general class of black box cp problems without requiring the input of any smoothness information such as whether the problem is smooth nonsmooth or weakly smooth as well as the specific values of lipschitz constant and smoothness level we then develop a more practical restricted memory version of this method namely the accelerated prox level apl method we investigate the generalization of the apl method for solving certain composite cp problems and an important class of saddle point problems recently studied by nesterov math program 103 127 152 2005 we present promising numerical results for these new bundle level methods applied to solve certain classes of semidefinite programming and stochastic programming problems

how many eigenvalues of a random matrix are real let a be an n x n matrix whose elements are independent random variables with standard normal distributions as n oo the expected number of real eigenvalues is asymptotic to v 7r we obtain a closed form expres sion for the expected number of real eigenvalues for finite n and a formula for the density of a real eigenvalue for finite n asymptotically a real normalized eigenvalue al ii of such a random matrix is uniformly distributed on the in terval 1 1 analogous but strikingly different results are presented for the real generalized eigenvalues we report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address edelmanfmath mit edu arts and sciences kapi olani community college 4303 diamond head road honolulu hawaii 98616 e mail address kostlanfkahuna math hawaii edu tj watson rsearch center 32 2 ibm yorktown heights new york 10598 0218 e mail address shubqwatson ibm com this content downloaded from 157 55 39 29 on tue 12 apr 2016 08 54 01 utc all use subject to http about jstor org terms

on root loci in several variables continuity in the high gain limit in this paper we discuss the continuity of the closed loop poles of a linear multivariable system with respect to a multidimensional polynomial family of direct output gains k 1 r this is based on and contains an exposition of the geometric formulation for including infinite gains which was developed in the lectures 2 and extended and applied in 1 to the study of output feedback systems this has been a basic tool in recent work on the classical problem of poleplacement by output feedback and in 1 the lack of continuity of the root loci in certain situations was discussed with special emphasis on the complex case here after presenting two somewhat surprising counterexamples to this continuity we give in theorem 1 and the ensuing discussion necessary and sufficient conditions for continuity of the root loci at a real infinite gain this should have significant impact on the problem of constructing graphical tests for the stability of systems subject to 2 dimensional variations in the gain parameter

grassmannian central projection and output feedback pole assignment of linear systems a sufficient rank condition for arbitrary pole assignability by static output feedback of m input p output linear systems of mcmillan degree n mp is given this condition is easy to check and is satisfied by the generic system an effective computational method to compute the feedback laws assigning any self conjugate set of closed loop poles is also given

structured squaring down and zero assignment the problem of zero assignment by squaring down is considered for a system of p inputs n outputs and n states m p where not all outputs are free variables for design we consider the case where a k subset of outputs is preserved in the new output set and the rest are recombined to produce a total new set of p outputs new invariants for the problem are introduced which include a new class of fixed zeros and the methodology of the global linearization developed originally for the output feedback pole assignment problem is applied to this restricted form of the squaring down problem it is shown that the problem can be solved generically if p k m p where k k p is the number of fixed outputs and is a system and compensation scheme invariant which is defined as the restricted forney degree

assigning frequencies via determinantal equations new counterexamples and invariants the problem of arbitrary pole zero assignment by fixed structure compensators has been addressed so far in terms of either necessary or sufficient conditions the strongest existing condition is based on the rank of the differential of a related map on a degenerate controller and holds true generically when the degrees of freedom of the compensator exceed the number of frequencies to be assigned mp n for the output feedback problem the complete nongeneric solvability of the problem is still open even when complex controllers are considered a simple necessary solvability condition is that the linear map appearing as a factor of the main determinantal map defining the assignment problem is onto here we examine the special problem of assignment of matrix pencil zeros via diagonal perturbations and we present a new necessary and sufficient condition for complex solvability in terms of a new invariant involving the minors of this linear map based on this result we demonstrate that for the important case where the degrees of freedom of the controller are equal to the number of frequencies to be assigned the surjectivity of this linear map although it is necessary it is not sufficient for the solvability of the problem

directed structure at infinity for infinite dimensional systems in this article the structure at infinity of infinite dimensional linear time invariant systems with finite dimensional input and output spaces is discussed it is shown that a diagonal form describing behaviour near infinity can be found this diagonal form is a generalisation of the smith mcmillan form at infinity for rational matrices it is then used to simplify certain solvability conditions of a regulation problem examples on time delay and distributed parameter systems are given

a bundle filter method for nonsmooth convex constrained optimization for solving nonsmooth convex constrained optimization problems we propose an algorithm which combines the ideas of the proximal bundle methods with the filter strategy for evaluating candidate points the resulting algorithm inherits some attractive features from both approaches on the one hand it allows effective control of the size of quadratic programming subproblems via the compression and aggregation techniques of proximal bundle methods on the other hand the filter criterion for accepting a candidate point as the new iterate is sometimes easier to satisfy than the usual descent condition in bundle methods some encouraging preliminary computational results are also reported

complex scaling circle criteria for lure systems abstractthis study reports alternative frequency domain interpretation and implementation of the circle criteria for absolute stability in lure systems by means of complex scaling and the argument principle by lure system a feedback configuration with a nominal lti model subject to sector nonlinearities is meant as usual firstly the complex scaling stability criterion is proved for asymptotic stability in lti feedback connections which dispenses open loop poles and contour locus pre orientation and possesses bounded loci without prior frequency sweeping secondly a novel frequency domain interpretation for positive realness of transfer functions is developed and employed for claiming the complex scaling circle criteria which accommodate various sector nonlinearities with unified conditions the new circle criteria are conformable in both multivariable and scalar cases implementable graphically and tractable numerically besides being a frequency complex domain analytical technique thirdly the re

riemannian symmetric superspaces and their origin in random matrix theory gaussian random matrix ensembles defined over the tangent spaces of the large families of cartan s symmetric spaces are considered such ensembles play a central role in mesoscopic physics as they describe the universal ergodic limit of disordered and chaotic single particle systems the generating function for the spectral correlations of each ensemble is reduced to an integral over a riemannian symmetric superspace in the limit of large matrix dimension such a space is defined as a pair g h m r where g h is a complex analytic graded manifold homogeneous with respect to the action of a complex lie supergroup g and m r is a maximal riemannian submanifold of the support of g h

the ginibre evolution in the large n limit we analyse statistics of the real eigenvalues of gl n r valued brownian motion the ginibre evolution in the limit of large n in particular we calculate the limiting two time correlation function of spin variables associated with real eigenvalues of the ginibre evolution we also show how the formalism of spin variables can be used to compute the fixed time correlation functions of real eigenvalues discovered originally by forrester and nagao eigenvalue statistics of the real ginibre ensemble phys rev lett 99 5 050603 2007 and borodin and sinclair the ginibre ensemble of real random matrices and its scaling limits commun math phys 291 1 177 224 2009

some remarks on real and complex output feedback we provide some new necessary and sufficient conditions which guarantee arbitrary pole placement of a particular linear system over the complex numbers we exhibit a nontrivial real linear system which is not controllable by real static output feedback and discuss a conjecture from algebraic geometry concerning the existence of real linear systems for which all static feedback laws are real

quadratically parameterized root locus analysis classical affine root locus applies to controllers that are linear in a parameter k and yield affinely parameterized closed loop denominator polynomials this paper presents root locus rules for controllers that are rational in k and yield quadratically parameterized closed loop denominator polynomials we show that the quadratic root locus has several advantages relative to the classical affine root locus specifically the quadratic root locus is high parameter stabilizing for minimum phase systems that are relative degree 1 2 or 3 moreover the quadratic root locus admits a wider variety of asymptote angles which provides more controller design flexibility the quadratic root locus is illustrated on several examples that are difficult to handle using affine root locus such as high parameter stabilization of the triple integrator

some results on the genericity of the pole placement problem abstract in this note the pole placement problem for a linear mimo systems with p outputs and m inputs is studied from the algebraic point of view a formulation is proposed that allows to analyze both theoretical and numerical aspects of the case min m p 2 with more sharpness moreover here it is shown that arbitrary pole placement by static output feedback of unitary rank is generically not possible even if m p n holds true

a new sufficient condition for arbitrary pole placement by real constant output feedback abstract a new sufficient condition for generic pole assignment is given based on the height of the first whitney class of a certain grassmannian this result contains the condition m p 1 n and is stronger than lscat p m n this test is equivalent to the factorial test in 7 but it has the advantage that it is testable for most of the cases

new variants of bundle methods in this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization convex concave games and variational inequalities we outline the ideas underlying these methods and present rate of convergence estimates

output feedback and generic stabilizability we consider questions of pole placement and stabilization for generic linear systems with prescribed state input and output dimensions where the controller must be implemented by linear memoryless output feedback we present a criterion in terms of a special pole placement property for generic stabilizability and apply this to describe constraints on the dimensions which are consistent with generic stabilizability we also discuss the rationality and solvability by radicals of stabilizing or pole positioning gains and we describe how decision algebra can theoretically handle existence questions for generic systems

the mcmillan and newton polygons of a feedback system and the construction of root loci abstract the local behaviour of root loci around zeros and poles is investigated this is done by relating the newton diagrams which arise in the local analysis to the mcmillan structure of the open loop system by means of what we shall call the mcmillan polygon this geometric construct serves to clarify the precise relationship between the mcmillan structure the principal structure and the branching patterns of the root loci in addition several rules are obtained which are useful in the construction of the root loci of multivariable control systems

grassmann invariants almost zeros and the determinantal zero pole assignment problems of linear multivariable systems the strucxpects of the rationxr spaces x c x fwhere x c is the columxof the transfer function matrix g s and x f is the space associated with right matrix fracxcriptions of g s are investigated for x f gcanonixqb s grassmgesentax g x c and g x f arxd and are shown xmplete basis free invariants for x c and x f respectively the almost zeros az and almost decoupling zeros adz of g s gdefinex local minima of a normgn defix g x c and g x f respectively the computation and certain aspects of the distribution in the complex plane of azs and adzs are examined the role of azs and adzs in the determinantal zero and pole assignment problems respectively is examined next two important families of systems are defined the strongly zero non assignable szna and the strongly pole non assignable spna systems for szna and spna systems minimal radius discs dem e z r em e z and dem f zb centred at an az and adz respectively are defined it is shown that dem e r em e z contains at least one zer

superbosonization formula and its application to random matrix theory starting from gaussian random matrix models we derive a new supermatrix field theory model in contrast to the conventional non linear sigma models the new model is applicable for any range of correlations of the elements of the random matrices we clarify the domain of integration for the supermatrices and give a demonstration of how the model works by calculating the density of states for an ensemble of almost diagonal matrices it is also shown how one can reduce the supermatrix model to the conventional sigma model

an inexact perturbed path following method for lagrangian decomposition in large scale separable convex optimization this paper studies an inexact perturbed path following algorithm in the framework of lagrangian dual decomposition for solving large scale separable convex programming problems unlike the exact versions considered in the literature we propose solving the primal subproblems inexactly up to a given accuracy this leads to an inexactness of the gradient vector and the hessian matrix of the smoothed dual function then an inexact perturbed algorithm is applied to minimize the smoothed dual function the algorithm consists of two phases and both make use of the inexact derivative information of the smoothed dual problem the convergence of the algorithm is analyzed and the worst case complexity is estimated as a special case an exact path following decomposition algorithm is obtained and its worst case complexity is given implementation details are discussed and preliminary numerical results are reported

multivariable singularly perturbed feedback systems with time delay multivariable singularly perturbed systems with both small and large time delays in the feedback loop are considered conditions for stability of the full order closed loop system are given in terms of properties of the slow and fast subsystems

computing dynamic output feedback laws the pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed loop system has a desired set of eigenvalues converting this problem into a question of enumerative geometry efficient numerical homotopy algorithms to solve this problem for general multiple input multiple output systems have been proposed recently despite the wider application range of dynamic feedback laws the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before in this note we present symbolic numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine we report on numerical experiments with our publicly available software phcpack and illustrate its application on various control problems from the literature

periodically varying compensation of time invariant systems linear time invariant discrete time scalar systems under periodically varying output feedback and the associated problems of stabilization and arbitrary eigenvalue assignment are investigated the general form of the characteristic polynomial of the closed loop periodically varying system is obtained and some detailed results are provided for the second order case interesting research questions are pointed out

general eigenvalue correlations for the real ginibre ensemble we rederive in a simplified version the lehmann sommers eigenvalue distribution for the gaussian ensemble of asymmetric real matrices invariant under real orthogonal transformations as a basis for a detailed derivation of a pfaffian generating functional for n point densities this produces a simple free fermion diagram expansion for the correlations leading to quaternion determinants in each order n all will explicitly be given with the help of a very simple symplectic kernel for even dimension n the kernel is valid for both complex and real eigenvalues and describes a deep connection between both a slight modification by an artificial additional grassmannian also solves the more complicated odd n case as illustration we present some numerical results in the space of complex eigenvalue n tuples

composite self concordant minimization we propose a variable metric framework for minimizing the sum of a self concordant function and a possibly non smooth convex function endowed with an easily computable proximal operator we theoretically establish the convergence of our framework without relying on the usual lipschitz gradient assumption on the smooth part an important highlight of our work is a new set of analytic step size selection and correction procedures based on the structure of the problem we describe concrete algorithmic instances of our framework for several interesting applications and demonstrate them numerically on both synthetic and real data

on decentralized dynamic pole placement and feedback stabilization in this paper we study the feedback control problem using an r channel decentralized dynamic feedback control scheme we will develop the theory in the behavioral framework using this framework we introduce an algebraic parameterization of the space of all possible feedback compensators having a bounded mcmillan degree and we show that this parameterization has the structure of an algebraic variety we define the pole placement map for this problem and we give exact conditions when this map is onto and almost onto finally we provide new necessary and sufficient conditions which guarantee that the set of stabilizable plants is a generic set

a nyquist type stability test for multivariable distributed systems a multivariable nyquist criterion is given that applies to a wide class of open loop transfer matrices including some that are improper or have infinitely many poles in the right half plane application of the test to specific examples requires only elementary knowledge of complex variable theory the actual test requires operations similar to those used in the standard nyquist test the simplicity of the frequency domain description of the class t analyticity and polynomial boundedness makes it relatively easy to verify if the test is applicable to a given transfer matrix

complex static skew symmetric output feedback control we study the problem of feedback control for skew symmetric and skew hamiltonian transfer functions using skew symmetric controllers this extends work of helmke et al who studied static symmetric feedback control of symmetric and hamiltonian linear systems we identify spaces of linear systems with symmetry as natural subvarieties of the moduli space of rational curves in a grassmannian give necessary and sufficient conditions for pole placement by static skew symmetric complex feedback and use schubert calculus for the orthogonal grassmannian to count the number of complex feedback laws when there are finitely many of them finally we also construct a real skew symmetric linear system with only real feedback for any set of real poles

eigenvalue statistics of the real ginibre ensemble the real ginibre ensemble consists of random n ifmmode times else texttimes fi n matrices formed from independent and identically distributed standard gaussian entries by using the method of skew orthogonal polynomials the general n point correlations for the real eigenvalues and for the complex eigenvalues are given as n ifmmode times else texttimes fi n pfaffians with explicit entries a computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented this is relevant to may s stability analysis of biological webs

decentralized feedback pole placement of linear systems a projective product spaces model is used to analyse decentralized systems the degree of the pole placement map is computed the conditions under which the degree is odd are also given twin lift systems are studied it is proved that the poles of a twin lift system can be assigned to any values by local static and local dynamic feedback laws if and only if the system is jointly controllable communicated by professor a isidori

fixed point continuation for ell_1 minimization methodology and convergence we present a framework for solving the large scale ell 1 regularized convex minimization problem min x 1 mu f x our approach is based on two powerful algorithmic ideas operator splitting and continuation operator splitting results in a fixed point algorithm for any given scalar mu continuation refers to approximately following the path traced by the optimal value of x as mu increases in this paper we study the structure of optimal solution sets prove finite convergence for important quantities and establish q linear convergence rates for the fixed point algorithm applied to problems with f x convex but not necessarily strictly convex the continuation framework motivated by our convergence results is demonstrated to facilitate the construction of practical algorithms

on the solvability of polynomial systems arising in control we consider a compactification of a general system of polynomial equations in weighted projective space and give some sufficient conditions about the solvability of a polynomial system over c and r we also prove that for generic linear subspaces the inverse eigenvalue problem with perturbations in the linear subspace always has n solutions

output feedback pole assignment for transfer functions with symmetries this paper studies the problem of pole assignment for symmetric and hamiltonian transfer functions a necessary and sufficient condition for pole assignment by complex symmetric output feedback transformations is given moreover in the case where the mcmillan degree coincides with the number of parameters appearing in the symmetric feedback transformations we derive an explicit combinatorial formula for the number of pole assigning symmetric feedback gains the proof uses intersection theory in projective space as well as a formula for the degree of the complex lagrangian grassmann manifold

necessary and sufficient conditions for zero assignment by constant squaring down the problem of zero assignment by constant squaring down czap is studied for minimal systems described by a transfer function g s rm l s m l using tools from exterior algebra and algebraic geometry conditions for its solvability are given in terms of the numbers m l the forney dynamical order and the rank of the plucker matrix p for the cases l 1 or l m 1 it is shown that g s is completely zero assignable cza if and only if 1 for the cases l 1 m 1 it is proved that g s is cza or generically zero assignable gza under a complex squaring down if and only if l m l 1 and 1 the latter conditions also provide necessary conditions for the existence of a real solution for a generic g s sufficient conditions for solvability of czap by a real squaring down are that l m l 1 and that the number g a0 al 1 1 a0 al 1 i j ai aj is odd for some set a0 al 1 satisfying 1 l 1i 0ai l l 1 2 and 0 a0 a1 al 1 m 1 apart from the existence results the present approach also allows the computation of the solutions whenever they exist finally it is proved that the only fixed zeros of czap are the zeros of g s and a procedure for computing the solutions of czap whenever such solutions exist based on an optimization problem is given

output feedback eigenstructure assignment using a new reduced orthogonality condition this paper presents a new method of output feedback eigenstructure assignment a new reduced orthogonality condition is derived which is less restrictive on the design degrees of freedom than others in the literature from this a novel general formulation for the gain matrix is admitted which utilizes a two stage design process in the first stage a subset of desired eigenvectors is assigned either left or right in the second stage a dual set is assigned which adheres to the necessary and sufficient conditions for output feedback eigenstructure assignment proved earlier in the paper a numerical example is presented to illustrate the whole procedure

primal dual subgradient methods for convex problems in this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure our methods are primal dual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem besides other advantages this useful feature provides the methods with a reliable stopping criterion the proposed schemes differ from the classical approaches divergent series methods mirror descent methods by presence of two control sequences the first sequence is responsible for aggregating the support functions in the dual space and the second one establishes a dynamically updated scale between the primal and dual spaces this additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set however we always assume the uniform boundedness of subgradients we present the variants of subgradient schemes for nonsmooth convex minimization minimax problems saddle point problems variational inequalities and stochastic optimization in all situations our methods are proved to be optimal from the view point of worst case black box lower complexity bounds

conditions for scale based decompositions in singularly perturbed systems abstract singularly perturbed models of the form x t a x t with a analytic at 0 nonsingular for 0 0 and singular at 0 arise naturally in various problems of systems and control theory under a so called multiple semisimple null structure mssns condition on a the eigenstructure of this matrix has a multiple scale property that allows the asymptotic eigenstructure of the matrix to be studied via reduced order matrices associated with the separate scales under a stronger multiple semistability or msst condition this eigenstructure decomposition translates into a time scale decomposition of the solution x t of the system this paper is aimed at illuminating the mssns and msst conditions using ideas from an algebraic approach that we have developed for the study of singularly perturbed systems we show among other results that the smith decomposition of a permits transformation to a form in which the scales become explicit and the computations become transparent and simple that this form allows us to identify perturbations of a that all have the same scale based decompositions that the eigenstructure of a does indeed display multiple scales under mssns and that the eigenvalues in this case can be approximated using reduced order calculations that a has mssns if and only if the orders of its invariant factors equal the orders of its eigenvalues that this happens if and only if the orders of its invariant factors and principal minors are related in a specified way and that a has msst if and only if it is hurwitz for 0 0 and has mssns and satisfies the requirement that the order in of the real part of every eigenvalue is not greater than the order of its imaginary part

this is spiral tap sparse poisson intensity reconstruction algorithms theory and practice observations in many applications consist of counts of discrete events such as photons hitting a detector which cannot be effectively modeled using an additive bounded or gaussian noise model and instead require a poisson noise model as a result accurate reconstruction of a spatially or temporally distributed phenomenon f from poisson data y cannot be effectively accomplished by minimizing a conventional penalized least squares objective function the problem addressed in this paper is the estimation of f from y in an inverse problem setting where the number of unknowns may potentially be larger than the number of observations and f admits sparse approximation the optimization formulation considered in this paper uses a penalized negative poisson log likelihood objective function with nonnegativity constraints since poisson intensities are naturally nonnegative in particular the proposed approach incorporates key ideas of using separable quadratic approximations to the objective function at each iteration and penalization terms related to l1 norms of coefficient vectors total variation seminorms and partition based multiscale estimation methods

multivariate root locus behaviour and the relationship to transfer function pole zero structure this paper is concerned with the qualitative behaviour of the multivariable root locus for high and low gains for a non singular m m rational transfer function the butterworth approach and departure patterns are shown to be determined in all cases by the zeros and poles of certain algebraic functions these are in turn characterized at each point of the extended complex plane by a set of m rational numbers called the root locus indices in addition the smith mcmillan pole zero structure of the transfer function is represented at each point of the extended complex plane by a sequence of m integers called the mcmillan indices an algebraic condition for the coincidence of the two sets of indices is established using the algebraic function theory underlying the newton diagram method together with a valuation characterization of the smith mcmillan form it is shown directly without simple null structure assumptions or equivalent conditions on constant matrices that the condition is generically met

solving semidefinite quadratic linear programs using sdpt3 this paper discusses computational experiments with linear optimization problems involving semidefinite quadratic and linear cone constraints sqlps many test problems of this type are solved using a new release of sdpt3 a matlab implementation of infeasible primal dual path following algorithms the software developed by the authors uses mehrotra type predictor corrector variants of interior point methods and two types of search directions the hkm and nt directions a discussion of implementation details is provided and computational results on problems from the sdplib and dimacs challenge collections are reported

a pole placement design approach for systems with multiple operating conditions the author proposes design procedures based on state space pole placement techniques for systems with multiple operating conditions this is the so called simultaneous pole placement problem first the full state feedback problem is studied in which a nonlinear local pole placement solution is proposed the design condition is formulated in terms of the rank condition of a multimode controllability matrix then the output feedback problem is approached using a multimodel controller design which is an extension of the observer design to multimode systems the design is decomposed into separated global pole placement subproblems and a local pole placement subproblem for a system with some operating conditions having modes on the j omega axis but no modes at the origin in the open right half of the complex plane stabilizability and detectability conditions for the design of an asymptotically stabilizing control are established without any restriction on the number of inputs or outputs relations of this approach to other simultaneous control design approaches are pointed out

characteristic polynomials in real ginibre ensembles we calculate the average of two characteristic polynomials for the real ginibre ensemble of asymmetric random matrices and its chiral counterpart considered as quadratic forms they determine a skew symmetric kernel from which all complex eigenvalue correlations can be derived our results are obtained in a very simple fashion without going to an eigenvalue representation and are completely new in the chiral case they hold for gaussian ensembles which are partly symmetric with kernels given in terms of hermite and laguerre polynomials respectively depending on an asymmetry parameter this allows us to interpolate between the maximally asymmetric real ginibre and the gaussian orthogonal ensemble as well as their chiral counterparts

on some gaussian ensembles of hermitian matrices a gaussian ensemble of hermitian matrices with an arbitrary ratio of their symmetric and anti symmetric parts is studied similarly a gaussian ensemble of hermitian quaternion matrices with an arbitrary ratio of their self dual and anti self dual parts is studied analytical expressions for the correlation and cluster functions as well as their limits when the order of the matrices is large are derived for both ensembles

special loop connections of linear systems in this paper we completely determine the possible eigenvalues of a matrix of a system obtained as a result of special loop connections of arbitrary many linear systems

multivariable nyquist criteria root loci and pole placement a geometric viewpoint classical control theory is concerned with the topics in our title in the context of single input single output systems there is now a large and growing literature on the extension of these ideas to the multiinput multioutput case this development has posed certain difficulties some due to the intrinsic nature of the problem and some we would argue due to an inadequate reflection on what the multivariable problem calls for in this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory a rather satisfactory generalization of the nyquist criterion will be described and a clear analog of the asymptotic properties of the root locus will be obtained in the multiparameter case however an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains this calls for a new ingredient a compactification of the space of gains and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement this problem is difficult in the multivariable case but by establishing a correspondence with a classical set of problems in geometry we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field

the approximate determinantal assignment problem the determinantal assignment problem dap has been introduced as the unifying description of all frequency assignment problems in linear systems and it is studied in a projective space setting this is a multi linear nature problem and its solution is equivalent to finding real intersections between a linear space associated with the polynomials to be assigned and the grassmann variety of the projective space this paper introduces a new relaxed version of the problem where the computation of the approximate solution referred to as the approximate dap is reduced to a distance problem between a point in the projective space from the grassmann variety gm rn the cases g2 rn and its hodge dual gn 2 rn are examined and a closed form solution to the distance problem is given based on the skew symmetric matrix description of multivectors via the gap metric a new algorithm for the calculation of the approximate solution is given and stability radius results are used to investigate the acceptability of the resulting perturbed solutions

the ginibre ensemble of real random matrices and its scaling limits we give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the pfaffian of an antisymmetric matrix formed from a 2 2 matrix kernel associated to the ensemble we apply this result to the real ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large

cascade equivalence of linear systems abstract this paper concerns the characterization of a linear system with respect to output feedback as well as to slightly more general transformations called cascade transformations with a transfer function t s we associate a finite tuplee of numbers that is invariant under cascade transformation

symplectic structure of the real ginibre ensemble we give a simple derivation of all n point densities for the eigenvalues of the real ginibre ensemble with even dimension n as quaternion determinants a very simple symplectic kernel governs both the real and complex correlations one and two point correlations are discussed in more detail scaling forms for large dimension n are derived

the ordered subsets mirror descent optimization method with applications to tomography we describe an optimization problem arising in reconstructing three dimensional medical images from positron emission tomography pet a mathematical model of the problem based on the maximum likelihood principle is posed as a problem of minimizing a convex function of several million variables over the standard simplex to solve a problem of these characteristics we develop and implement a new algorithm ordered subsets mirror descent and demonstrate theoretically and computationally that it is well suited for solving the pet reconstruction problem

compressed sensing off the grid this paper investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples unlike previous work in compressed sensing the frequencies are not assumed to lie on a grid but can assume any values in the normalized frequency domain 0 1 an atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies which is then reformulated as an exact semidefinite program even with this continuous dictionary it is shown that o slog s log n random samples are sufficient to guarantee exact frequency localization with high probability provided the frequencies are well separated extensive numerical experiments are performed to illustrate the effectiveness of the proposed method

self concordant inclusions a unified framework for path following generalized newton type algorithms we study a class of monotone inclusions called self concordant inclusion which covers three fundamental convex optimization formulations as special cases we develop a new generalized newton type framework to solve this inclusion our framework subsumes three schemes full step damped step and path following methods as specific instances while allows one to use inexact computation to form generalized newton directions we prove the local quadratic convergence of both full step and damped step algorithms then we propose a new two phase inexact path following scheme for solving this monotone inclusion which possesses an mathcal o sqrt nu log 1 varepsilon worst case iteration complexity to achieve an varepsilon solution where nu is the barrier parameter and varepsilon is a desired accuracy as byproducts we customize our scheme to solve three convex problems the convex concave saddle point problem the nonsmooth constrained convex program and the nonsmooth convex program with linear constraints we also provide three numerical examples to illustrate our theory and compare with existing methods

an inexact proximal path following algorithm for constrained convex minimization many scientific and engineering applications feature nonsmooth convex minimization problems over convex sets in this paper we address an important instance of this broad class where we assume that the nonsmooth objective is equipped with a tractable proximity operator and that the convex constraint set affords a self concordant barrier we provide a new joint treatment of proximal and self concordant barrier concepts and illustrate that such problems can be efficiently solved without the need of lifting the problem dimensions as in disciplined convex optimization approach we propose an inexact path following algorithmic framework and theoretically characterize the worst case analytical complexity of this framework when the proximal subproblems are solved inexactly to show the merits of our framework we apply its instances to both synthetic and real world applications where it shows advantages over standard interior point methods as a byproduct we describe how our framework can obtain points on t

on the linear static output feedback problem the annihilating polynomial approach abstract one of the fundamental open problems in control theory is that of the stabilization of a linear time invariant dynamical system through static output feedback we are given a linear dynamical system defined through w a w b u y c w the problem is to find if it exists a feedback u k y such that the matrix a b k c has all its eigenvalues in the complex left half plane and if such a feedback does not exist to prove that it does not substantial progress has not been made on the computational aspect of the solution to this problem in this paper we consider instead which annihilating polynomials can a matrix of the form a b k c possess we give a simple solution to this problem when the system has either a single input or a single output for the multi input multi output case we use these ideas to characterize the annihilating polynomials when k has rank one and suggest possible computational solutions for general k we also present some numerical evidence for the plausibility of this approach for the general case as well as for the problem of shifting the eigenvalues of the system

gaussian ensembles of random hermitian matrices intermediate between orthogonal and unitary ones a gaussian ensemble of hermitian matrices depending on a parameter is considered when 0 the ensemble is gaussian orthogonal and when 1 it is gaussian unitary an analytic expression for then level correlation and cluster functions is given for anyn and 0 1 this ensemble is of relevance in the study of time reversal symmetry breaking of nuclear interactions

squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem we extend the so called single ring theorem feinberg and zee 1997 nucl phys b 504 579 also known as the haagerup larsen theorem haagerup and larsen 2000 j funct anal 176 331 we do this by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non hermitian matrix x being the spectral density weighted by the squared eigenvalue condition number is given by a simple formula involving only the radial spectral cumulative distribution function of x we show that this object allows the calculation of the conditional expectation of the squared eigenvalue condition number we give examples and provide a cross check of the analytic prediction by the large scale numerics

augmented self concordant barriers and nonlinear optimization problems with finite complexity in this paper we study special barrier functions for convex cones which are the sum of a self concordant barrier for the cone and a positive semidefinite quadratic form we show that the central path of these augmented barrier functions can be traced with linear speed we also study the complexity of finding the analytic center of the augmented barrier a problem that has some interesting applications we show that for some special classes of quadratic forms and some convex cones the computation of the analytic center requires an amount of operations independent of the particular data set we argue that these problems form a class that is endowed with a property which we call finite polynomial complexity

on pole placement via eigenstructure assignment approach this note comes back to the hard problem of pole placement by static output feedback let a triplet of matrices a b c be given with n state variables m inputs and p ouputs find a matrix k such that the spectrum of a bkc equals a specified set when mp m p a simple noniterative technique based upon the notion of eigenstructure that in most cases assigns m p roots is proposed it therefore enables the designer to assign the whole of the desired spectrum when m p n mp

schur function averages for the real ginibre ensemble we derive an explicit simple formula for expectations of all schur functions in the real ginibre ensemble it is a positive integer for all entries of the partition even and zero otherwise the result can be used to determine the average of any analytic series of elementary symmetric functions by schur function expansion

exact pole assignment with regularity by output feedback in descriptor systems part 2 in this paper the first of two we examine the problem of modal control of linear time invariant multivariable descriptor systems in particular for systems satisfying a certain inequality we will give necessary and sufficient conditions for the existence of an output feedback matrix possibly complex which exactly assigns an arbitrary set of r the number of independent semi state variables of the system distinct complex numbers to a generalized eigenspectrum of the closed loop system while ensuring that the resulting system has a unique smooth solution for each admissible initial condition furthermore it is shown that if the set of proposed poles contains only real elements the assignment can be carried out using real feedback next we present conditions that are necessary and sufficient for pole assignability by real feedback of a self conjugate set of complex numbers provided that the system satisfies a further inequality

a sufficient condition for arbitrary eigenvalue assignment in linear descriptor systems by output feedback the problem of eigenvalue assignment in the linear descriptor system e mathdotx ax bu y cx via output feedback is considered it is shown that mp rank e m and p are respectively the numbers of inputs and outputs of the system is a sufficient condition for generic real output feedback eigenvalue assignability the result is the best possible for general m p rank e in general the assignment is achieved not in an exact sense but in arbitrarily small neighborhoods around a given set of eigenvalues

rational functions with real critical points and the b and m shapiro conjecture in real enumerative geometry suppose that 2d 2 tangent lines to the rational normal curve z 1 z z d in d dimensional complex projective space are given it was known that the number of codimension 2 subspaces intersecting all these lines is always finite for a generic configuration it is equal to the d th catalan number we prove that for real tangent lines all these codimension 2 subspaces are also real thus confirming a special case of a general conjecture of b and m shapiro this is equivalent to the following result if all critical points of a rational function lie on a circle in the riemann sphere for example on the real line then the function maps this circle into a circle

identification and estimation of dynamic errors in variables models abstract we consider here the factor analysis approach to the analysis of linear dynamic errors in variables models identification of the most general models and the question of nonuniqueness are discussed we show how these models are generalizations of the total least squares model and how the grassmann manifold provides a natural setting for their analysis under certain assumptions we analyze the problem of maximum likelihood estimation and indicate how this involves maximization of a functional on a grassmann manifold

eigenvalue assignment by static output feedback on a new solvability condition and the computation of low gain feedback matrices in this article the static output feedback problem for linear time invariant systems is considered for arbitrary assignability of the roots of the characteristic polynomial by static output feedback a new necessary and sufficient condition is derived although the proof is based on simple analysis the known sufficient conditions derived by techniques of algebraic geometry are directly covered furthermore an algorithm for the calculation of feedback matrices assigning a desired set of eigenvalues is proposed this algorithm does not require the desired eigenvalues to be distinct and it explicitly exploits the available degrees of freedom for reducing the feedback gain the presented approach is illustrated on computational examples

a new approach to derive pfaffian structures for random matrix ensembles correlation functions for matrix ensembles with orthogonal and unitary symplectic rotation symmetry are more complicated to calculate than in the unitary case the supersymmetry method and the orthogonal polynomials are two techniques to tackle this task recently we presented a new method to average ratios of characteristic polynomials over matrix ensembles invariant under the unitary group here we extend this approach to ensembles with orthogonal and unitary symplectic rotation symmetry we show that pfaffian structures can be derived for a wide class of orthogonal and unitary symplectic rotation invariant ensembles in a unifying way this also includes those for which this structure was not known previously as the real ginibre ensemble and the gaussian real chiral ensemble with two independent matrices as well

model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data we consider the problem of estimating the parameters of a gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse our approach is to solve a maximum likelihood problem with an added l1 norm penalty term the problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes we present two new algorithms for solving problems with at least a thousand nodes in the gaussian case our first algorithm uses block coordinate descent and can be interpreted as recursive l1 norm penalized regression our second algorithm based on nesterov s first order method yields a complexity estimate with a better dependence on problem size than existing interior point methods using a log determinant relaxation of the log partition function wainwright and jordan 2006 we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case we test our algorithms on synthetic data as well as on gene expression and senate voting records data