necessary and sufficient stability conditions for the case of dependent additive perturbations in this paper we examine the stability properties of linear multivariable feedback systems in the presence of structured additive perturbations characterized by dependencies occurring in the perturbation matrix such dependencies may take the form of dependent phases between elements of element by element bounded perturbations and or repeated blocks in block structured perturbations by using non diagonal and possibly complex transformation matrices it is shown how stability conditions that are both necessary and sufficient may be obtained for this class of perturbation

robust stability of interval matrices a computational approach a simple computational procedure is given to determine the robust stability of matrices containing interval parameter uncertainty the computation consists of determining the maximal phase difference of the characteristic polynomial evaluated over the vertices of the uncertain parameter set at each point of the stability boundary and verifying that this maximal phase difference is less than radians this follows as a direct application of the mapping theorem which allows us effectively to treat rank one perturbations of matrices we discuss the conservatism of the method and show how the solution may be improved to any desired level of accuracy we give several numerical examples of systems with state space perturbations and compare our results with those obtained by other methods

well posedness of feedback systems insights into exact robustness analysis and approximate computations this paper establishes a framework for robust stability analysis of linear time invariant uncertain systems the uncertainty is assumed to belong to an arbitrary subset of complex matrices the concept used here is well posedness of feedback systems leading to necessary and sufficient conditions for robust stability based on this concept some insights into exact robust stability conditions are given in particular frequency domain and state space conditions for well posedness are provided in terms of hermitian form inequalities it is shown that these inequalities can be interpreted as small gain conditions with a generalized class of scalings given by linear fractional transformations lft using the lft scaled small gain condition in the state space setting the duality is established between the h sub spl infin norm condition with frequency dependent scalings and the parameter dependent lyapunov condition connections to the existing results including the structured singular value and the integral quadratic constraints are also discussed finally we show that our well posedness conditions can be used to give a less conservative yet computable bound on the real structured singular value this result is illustrated by numerical examples

stability margin calculation for systems with structured time delay uncertainties a method of calculating a stability margin of systems with uncertain time delay elements is presented this method utilizes the bilinear transformation of time delay elements to obtain a multilinear function in order to compute the stability margin by applying the mapping theory proposed for the robust stability test

stability of polynomials under coefficient perturbation let the real polynomial a 0 x n a 1 x n 1 a n 1 x a n be stable and let the real numbers b k c k geq 0 0 leq k leq n be given we present a simple determinant criterion for finding the largest t 0 geq 0 such that the polynomial alpha 0 x n alpha 1 x n 1 alpha n 1 x alpha n is stable for all alpha k in a k b k t 0 a k c k t 0 cup a k 0 leq k leq n several further observations allow us to reduce the computational cost considerably

order 3 stability analysis of particle swarm optimization abstract particle swarm optimization pso is a stochastic optimization algorithm whose distribution is completely determined by its moments up to now most of previous theoretical researches pay attention to the first and or second moments only to give its stable condition ignoring the analysis of third moment probably due to its complex third order recurrence equation to enrich theoretical cornerstones of pso this work analyzes its first second and third moments respectively built on first order fixed point and two novel simplified strategies we present the second and third order standardized recurrence equations for pso their corresponding characteristic equations can be analyzed through the derivative functions to obtain the necessary and sufficient convergence condition besides we show that the specific third order stable region is slightly less than the second order one in illustrations

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

fast construction of robustness degradation function we develop a fast algorithm to construct the robustness degradation function which describes quantitatively the relationship between the proportion of systems guaranteeing the robustness requirement and the radius of the uncertainty set this function can be applied to predict whether a controller design based on an inexact mathematical model will perform satisfactorily when implemented on the true system

convergence and stochastic stability analysis of particle swarm optimization variants with generic parameter distributions in this paper we present the convergence and stochastic stability analysis of a set of pso variants those that differ with the classical pso in the statistical distribution of the three pso tuning parameters inertia weight local and global acceleration we provide an analytical expression for the upper limit of the second order stability regions the so called usl curves of the particle trajectories that can be applied to most of these pso algorithms thus this work generalizes to this set the result found in the literature for the classical pso we apply this analysis to some of these variants finally numerical experiments have been performed that confirm the known fact that the best algorithm performance is provided tuning the pso parameters close to the usl curve

method of robust stability analysis with highly structured uncertainties this paper presents a method of analyzing the stability of linear time invariant interconnected systems with uncertainties each subsystem is single input single output and its vector locus lies inside a polygon at each frequency it is shown that the convex closure of the image of the cartesian product of these polygons under the mapping phi det i fh agrees with the convex closure of the image of the vertices of these polygons under the mapping from this result the image can be estimated by a finite dumber of calculations a sufficient stability condition is obtained by applying this result to the multivariable nyquist stability criterion

structured singular values and stability analysis of uncertain polynomials part 1 the generalized m abstract one of the primary goals in this two part series is to establish a link between the structured singular value and results in stability analysis of uncertain polynomials another primary goal is to develop an improved method for accurately and efficiently computing the structured singular value for an important class of problems in robust stability analysis to achieve these goal we first introduce a generalized framework of structured singular values and next show how stability problems for uncertain polynomials may be studied in this framework part 1 of this series is devoted entirely to the generalized structured singular values specifically for the case when a certain matrix representing the nominal system is of rank one an analytical expression for the generalized notion is derived in this case which involves solving a convex optimization problem in one real variable and renders the structured singular value readily solvable in particular when the general framework is specialized to that of the standard structured singular value the expression is solved explicitly also for several additional important cases explicit solutions are obtained the framework as well as results will then be used in part 2 of this series to study stability problems for a class of polynomials whose coefficients are affine functions of real or complex uncertainties we demonstrate that the generalized structured singular value is a suitable notion for these problems and its solution unifies a number of results obtained previously via alternative approaches for several problems of interest we further demonstrate that stability conditions based upon the structured singular value and those in the spirit of kharitonov theorem can be derive from one another and hence establish a link between two drastically different type of results

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

on block limiting norm and structured singular value abstract the notion of limiting norm introduced by pokrovskii soviet math dokl 20 1979 1314 1317 is generalized to that of block limiting norm a resemblance of inequalities shared by both the block limiting norm and the structured singular value introduced by doyle proc iee 129 1982 245 250 motivates further investigation of their relationships to that effect the concept of generalized spectral radius of a set of linear operators is introduced it is then shown that for block structure of size less than 4 the block limiting norm is equal to the structured singular value and that in the general case the block limiting norm is always no less than the structured singular value finally better bounds are obtained for both the block limiting norm and the structured singular value

stability and performance robustness analysis with highly structured uncertainties necessary and sufficient test the author presents an extension of a technique of analyzing the stability of linear time invariant interconnected systems with uncertainties each element of the frequency response matrix is assumed to lie inside a polygon at each frequency it is shown that the exterior boundary of the image of the cartesian product of these polygons under the map det i m delta is contained in the boundary of the image of the cartesian product of the boundaries of these polygons under the same map thus the image can be constructed from the images of the edges the computational technique of r r e de gaston and m g safonov 1988 can then be applied to perform a nonconservative robustness analysis of the feedback system

spectral asymptotics via the semiclassical birkhoff normal form this article gives a simple treatment of the quantum birkhoff normal form for semiclassical pseudo differential operators with smooth coefficients the normal form is applied to describe the discrete spectrum in a generalised non degenerate potential well yielding uniform estimates in the energy e this permits a detailed study of the spectrum in various asymptotic regions of the parameters e h and gives improvements and new proofs for many of the results in the field in the completely resonant case we show that the pseudo differential operator can be reduced to a toeplitz operator on a reduced symplectic orbifold using this quantum reduction new spectral asymptotics concerning the fine structure of eigenvalue clusters are proved in the case of polynomial differential operators a combinatorial trace formula is obtained

robustness analysis of polynomials with polynomial parameter dependency using bernstein expansion this paper considers the robust stability verification of polynomials with coefficients depending polynomially on parameters varying in given intervals two algorithms are presented both rely on the expansion of a multivariate polynomial into bernstein polynomials the first one is an improvement of the so called bernstein algorithm and checks the hurwitz determinant for positivity over the parameter set the second one is based on the analysis of the value set of the family of polynomials and profits from the convex hull property of the bernstein polynomials numerical results to real world control problems are presented showing the efficiency of both algorithms

homogeneous quantization and multiplicities of group representations let b be a compact manifold a cone over b is a principal r bundle x with base b let a x a x be the mapping associated with the action of a r on x x is called a symplectic cone if it possesses a symplectic form such that a a a compact lie group g is said to act in a homogeneous fashion on x if it acts on x in such a way that both and the principal bundle structure are preserved it is known that to such an action one can associate in a fairly canonical way a representation of g on a hilbert space h see 3 in this paper we propose a symplectic recipe for the multiplicities with which h decomposes into g irreducibles and show that this recipe is correct generically

robust stability of polynomials with multilinear parameter dependence abstract the problem is studied of testing for stability a class of real polynomials in which the coefficients depend on a number of variable parameters in a multilinear way we show that the testing for real unstable roots can be achieved by examining the stability of a finite number of corner polynomials obtained by setting parameters at their extreme values while checking for unstable complex roots normally involves examining the real solutions of up to m 1 simultaneous polynomial equations where m is the number of parameters when m 2 this is an especially simple task

robust synthesis in sub 1 a globally optimal solution in this paper a method to synthesize controllers that achieve globally optimal robust performance within any prespecified tolerance against structured norm bounded time varying and or nonlinear uncertainty is developed the performance measure considered is the infinity to infinity induced norm of a system s transfer function the method developed utilizes linear relaxation techniques to solve the infinite dimensional nonconvex problem via finite dimensional linear programming problems

robust stability of state space models with structured uncertainties a method for robust eigenvalue location analysis of linear state space models affected by structured real parametric perturbations is proposed the approach based on algebraic matrix properties deals with state space models in which system matrix entries are perturbed by polynomial functions of a set of uncertain physical parameters a method converting the robust stability problem into nonsingularity analysis of a suitable matrix is proposed the method requires a check of the positivity of a multinomial form over a hyperrectangular domain in parameter space this problem which can be reduced to finding the real solutions of a system of polynomial equations simplifies considerably when cases with one or two uncertain parameters are considered for these cases necessary and sufficient conditions for stability are given in terms of the solution of suitable real eigenvalue problems

strong stabilizability of systems with multiaffine uncertainties and numerator denominator coupling this paper considers a set of proper transfer functions whose numerator and denominator polynomial coefficients display dependent multiaffine parametric uncertainties it is shown that provided no member transfer function has positive real pole zero cancellations all members satisfy the parity interlacing property iff all corner members do the same notice that while this implies that each member is strongly stabilizable it does not imply the existence of a single stable controller that stabilizes the whole set the paper also shows that whenever the numerator and denominator polynomials lie in dependent polytopes the task of verifying the absence of positive real pole zero cancellations can be accomplished by checking the edges

fast computation of the multivariable stability margin for real interrelated uncertain parameters an alternative implementation of an algorithm for the computation of the multivariable stability margin to check the robust stability of feedback systems with real parametric uncertainty is proposed this method eliminates the need for the frequency search by reducing it to the testing of a finite number of conditions these conditions have a special structure that allows a significant improvement in the speed of computation

asymptotic euler maclaurin formula over lattice polytopes abstract formulas for the riemann sums over lattice polytopes determined by the lattice points in the polytopes are often called euler maclaurin formulas an asymptotic euler maclaurin formula by which we mean an asymptotic expansion formula for riemann sums over lattice polytopes was first obtained by guillemin and sternberg 2007 11 then the problem is to find a concrete formula for each term of the expansion in this paper an asymptotic euler maclaurin formula of the riemann sums over general lattice polytopes is given the formula given here is an asymptotic form of the so called local euler maclaurin formula of berline and vergne 2007 3 for delzant polytopes our proof given here is independent of the local euler maclaurin formula furthermore a concrete description of differential operators which appear in each term of the asymptotic expansion for delzant lattice polytopes is given by using this description when the polytopes are delzant lattice a concrete formula for each term of the expansion in two dimension and a formula for the third term of the expansion in arbitrary dimension are given

higher asymptotics of unitarity in quantization commutes with reduction let m be a compact kahler manifold equipped with a hamiltonian action of a compact lie group g guillemin and sternberg invent math 67 515 538 1982 no 3 showed that there is a geometrically natural isomorphism between the g invariant quantum hilbert space over m and the quantum hilbert space over the symplectic quotient m g this map though is not in general unitary even to leading order in hslash hall and kirwin commun math phys 275 401 422 2007 no 2 showed that when the metaplectic correction is included one does obtain a map which while not in general unitary for any fixed hslash becomes unitary in the semiclassical limit hslash rightarrow0 cf the work of ma and zhang c r math acad sci paris 341 297 302 2005 no 5 and asterisque no 318 viii 154 2008 the unitarity of the classical guillemin sternberg map and the metaplectically corrected analogue is measured by certain functions on the symplectic quotient m g in this paper we give precise expressions for these functions and compute complete asymptotic expansions for them as hslash rightarrow0

fedosov products and quantum momentum maps the purpose of this paper is to study various aspects of star products on a symplectic manifold related to the fedosov method by introducing the notion of quantum exponential maps we give a characterization of fedosov connections as an application a geometric realization is obtained for the equivalence between an arbitrary product and a fedosov one every fedosov product is shown to be a vey product consequently we find that every product is equivalent to a vey product a classical result of lichnerowicz quantization of a hamiltonian g space and in particular quantum momentum maps are studied lagrangian submanifolds are also studied under a deformation quantization

brief positive polynomial matrices and improved lmi robustness conditions recently several new lmi conditions for stability of linear systems have been proposed introducing additional slack variables to reduce the gap between conservative convex quadratic stability conditions and intractable non convex robust stability conditions in this paper we show that these improved lmi conditions can be derived with the help of some basic results on positive polynomial matrices the approach allows us to derive in a unifying way results in the state space and polynomial frameworks applications to robust stability analysis and robust stabilization of systems with multi linear parametric uncertainty are fully described

unitarity in quantization commutes with reduction let m be a compact kahler manifold equipped with a hamiltonian action of a compact lie group g in this paper we study the geometric quantization of the symplectic quotient m g guillemin and sternberg invent math 67 515 538 1982 have shown under suitable regularity assumptions that there is a natural invertible map between the quantum hilbert space over m g and the g invariant subspace of the quantum hilbert space over m

a study of particle swarm optimization particle trajectories particle swarm optimization pso has shown to be an efficient robust and simple optimization algorithm most of the pso studies are empirical with only a few theoretical analyses that concentrate on understanding particle trajectories these theoretical studies concentrate mainly on simplified pso systems this paper overviews current theoretical studies and extend these studies to investigate particle trajectories for general swarms to include the influence of the inertia term the paper also provides a formal proof that each particle converges to a stable point an empirical analysis of multi dimensional stochastic particles is also presented experimental results are provided to support the conclusions drawn from the theoretical findings

a generalization of kharitonov s four polynomial concept for robust stability problems with linearly dependent coefficient perturbations kharitonov s four polynomial concept is generalized to the case of linearly dependent coefficient perturbations and more general zero location regions to this end a specially constructed scalar function of a scalar variable is instrumental to the robustness analysis the present work is motivated by two fundamental limitations of kharitonov s theorem namely 1 the theorem only applies to polynomials with independent coefficient perturbations and 2 it only applies to zeros in the left hand plane

semiclassical quantization and spectral limits of ħ pseudodifferential and berezin toeplitz operators we introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of commuting operators converges to the convex hull of the spectrum of the associated classical system this gives a quick alternative solution to the isospectrality problem for quantum toric systems if the operators are uniformly bounded the convergence is uniform analogous results hold for non commuting operators

robustness analysis with correlated highly structured complex uncertainties it is shown that the results reported by the author ibid vol 35 no 3 pp 350 352 mar 1990 on robustness analysis with highly structured independent perturbations is directly extendable to the case of correlated perturbations a direct consequence of this extension is that the phase sensitive structured singular value and the multiloop phase margin can be computed exactly

inverse spectral theory for semiclassical jaynes cummings systems quantum semitoric systems form a large class of quantum hamiltonian integrable systems with circular symmetry which has received great attention in the past decade they include systems of high interest to physicists and mathematicians such as the jaynes cummings model 1963 which describes a two level atom interacting with a quantized mode of an optical cavity and more generally the so called systems of jaynes cummings type in this paper we consider the joint spectrum of a pair of commuting semiclassical operators forming a quantum integrable system of jaynes cummings type we prove assuming the bohr sommerfeld rules hold that if the joint spectrum of two of these systems coincide up to mathcal o hbar 2 then the systems are isomorphic

toeplitz operators and hamiltonian torus actions this paper is devoted to semi classical aspects of symplectic reduction consider a compact prequantizable kahler manifold m with a hamiltonian torus action in the seminal paper v guillemin s sternberg geometric quantization and multiplicities of group representations invent math 67 3 1982 515 538 guillemin and sternberg introduced an isomorphism between the invariant part of the quantum space associated to m and the quantum space associated to the symplectic quotient of m provided this quotient is non singular we prove that this isomorphism is a fourier integral operator and that the toeplitz operators of m descend to toeplitz operators of the reduced phase space we also extend these results to the case where the symplectic quotient is an orbifold and estimate the spectral density of a reduced toeplitz operator a result related to the riemann roch kawasaki theorem

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

berezin toeplitz operators a semi classical approach this article is devoted to the toeplitz operators 4 in the context of the geometric quantization 11 15 we propose an ansatz for their schwartz kernel from this we deduce the main known properties of the principal symbol of these operators and obtain new results we define their covariant and contravariant symbols which are full symbols and compute the product of these symbols in terms of the kahler metric this gives canonical star products on the kahlerian manifolds this ansatz is also useful to introduce the notion of microsupport

when is the multiaffine image of a cube a convex polygon abstract we give two simple sufficient conditions under which the multiaffine image in the complex plane of an m dimensional cube is a convex polygon a third condition which in some generic sense is necessary and sufficient is then obtained our conditions involve checking the locations of the image of the vertices of the cube these results help determine whether a parameterized family of polynomials th stable

fast computation of the largest stability radius for a two parameter linear system the author describes a fast algorithm to compute the minimum distance to instability or parameter margin for a linear time invariant system subject to rational variations in two real parameters both the l sub infinity and the euclidean norms are considered for measuring parameter deviations

a gain based lower bound algorithm for real and mixed µ problems a new lower bound algorithm for real and mixed m problems is presented the basic idea of this algorithm is to use a related worst case gain problem to compute the real blocks and if the block structure is mixed the standard power iteration to compute the complex blocks numerical tests indicate that the algorithm is fast and provides good lower bounds for both real and mixed m problems of small to moderate size

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

necessary and sufficient stability criterion for systems with structured uncertainties the major principal direction alignment principle a necessary and sufficient stability criterion is proved for multivariable systems in the presence of element by element bounded perturbations the result constitutes a solution to the problem of computing the function for this class of perturbations and the paper also provides an explicit approximation to the exact criterion

robust stability for time delay systems the edge theorem and graphical tests the robust stability problem is discussed for a class of uncertain delay systems where the characteristic equations involve a polytope p of quasi polynomials i e polynomials in one complex variable and exponential powers of the variable given a set d in the complex plane the goal is to find a constructive technique to verify whether all roots of every quasi polynomial in p belong to d that is to verify the d stability of p first it is demonstrated by counterexample that kharitonov s theorem does not hold for general delay systems next it is shown that under a mild assumption on the set d a polytope of quasi polynomials is d stable if and only if the edges of the polytope are d stable this extends the edge theorem for the d stability of a polytope of polynomials the third result gives a constructive graphical test for checking the d stability of a polytope of quasi polynomials which is especially simple when the set d is the open left half plane an application is given to demonstrate the power of the results

a routh like test for the stability of families of polynomials with linear uncertainty abstract the stability of a polynomial can be deduced from a routh table in this paper we show that such tabular tests can also be developed for the robust stability of polynomials whose coefficients depend linearly on real parameters

computation of algebraic combinations of uncertainty value sets a robust analysis approach based on the computation of boundaries of uncertainty value sets is presented the idea is to use the strongly connected subsystem separation and the tree structured decomposition tsd concept to extend the class of uncertainty structures to be considered to this end we compute the exact boundaries of regions produced by operations spl times spl divide among boundaries of simply connected value sets in polynomial time and without incrementing the total amount of input data parametric dynamic or general multiply connected value set descriptions may be accommodated