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

geometric significance of toeplitz kernels abstract let l 2 be the lebesgue space of square integrable functions on the unit circle we show that the injectivity problem for toeplitz operators is linked to the existence of geodesics in the grassmann manifold of l 2 we also investigate this connection in the context of restricted grassmann manifolds associated to p schatten ideals and essentially commuting projections

three dimensional front tracking we describe a three dimensional front tracking algorithm discuss its numerical implementation and present studies to validate the correctness of this approach based on the results of the two dimensional computations we expect three dimensional front tracking to significantly improve computational efficiencies for problems dominated by discontinuities in some cases for which the interface computations display considerable numerical sensitivity we expect a greatly enhanced capability

pole assignment by output feedback using cylindrical algebraic decomposition recently a new polynomial method for pole assignment by static output feedback has been developed in this paper we show how this method can be generalized to partial pole clustering some poles are exactly assigned the others being clustered in a specified region of the complex plane a constructive and systematic algorithm for the numerical resolution of both problems the collins cylindrical algebraic decomposition algorithm is presented and applied to an illustrative example

robust pole assignment in singular control systems abstract necessary conditions are given for the problem of pole assignment by state feedback in singular linear systems descriptor systems to have a solution which is regular and nondefective for a robust solution such that the assigned closed loop poles are insensitive to perturbations in the system data the same conditions must hold it can be shown that these conditions are also sufficient for the existence of a feedback which assigns the maximum possible number of finite poles with regularity these results provide the basis of a procedure for constructing closed loop semistate systems with given poles guaranteed regularity and maximum robustness

assignment of the characteristic polynomial of a hessenberg matrix an algorithm is described for determining the first row of an upper hessenberg matrix associated with a completely controllable linear system so as to produce a prespecified characteristic polynomial the scheme is faster and more flexible than two existing procedures and is competitive with regard to accuracy on a variety of sample problems

algebraic characterization of controllability subspaces in the current base and its practical computation from the concepts of input chain and controllability chain introduced by heymann and in view of the generalized eigenvector assignment a new algebraic characterization of controllability subspaces css is developed compared with the previous ones based on matrix pencil theory it presents both the advantages of being stated in the current base and of being derived without resorting to use of canonical forms a practical algebraic algorithm is provided in order to compute the controllable structure of a given pair a b this approach is then applied to characterize css contained in a given subspace and span css via eigenvector assignment

metric geometry in homogeneous spaces of the unitary group of a c algebra part i minimal curves abstract we study the metric geometry of homogeneous spaces p of the unitary group of a c algebra a modulo the unitary group of a c subalgebra b where the invariant finsler metric is induced by the quotient norm of a b the main results include the following in the von neumann algebra context for each tangent vector x at a point p there is a geodesic t 0 x which is obtained by the action on of a 1 parameter group in the unitary group of a this geodesic is minimizing up to length 2

essentially commuting projections abstract let h h h be a fixed orthogonal decomposition of a hilbert space with both subspaces of infinite dimension and let e e be the projections onto h and h we study the set p c c of orthogonal projections p in h which essentially commute with e or equivalently with e i e p e p e e p is compact by means of the projection onto the calkin algebra one sees that these projections p p c c fall into nine classes four discrete classes which correspond to p being 0 1 e or e and five essential classes which we describe below the discrete classes are respectively the finite rank projections finite co rank projections the sato grassmannian of h and the sato grassmannian of h thus the connected components of each of these classes are parametrized by the integers via de rank the co rank or the fredholm index respectively the essential classes are shown to be connected we are interested in the geometric structure of p c c being the set of selfadjoint projections of the c algebra b c c of operators in b h which essentially commute with e in particular we study the problem of existence of minimal geodesics joining two given projections in the same component we show that the hopf rinow theorem holds in the discrete classes but not in the essential classes conditions for the existence and uniqueness of geodesics in these latter classes are found

a critical analysis of rayleigh taylor growth rates abstract recent simulations of rayleigh taylor instability growth rates display considerable spread we provide evidence that differences in numerical dissipation effects mass diffusion and viscosity due to algorithmic differences and differences in simulation duration are the dominant factors that produce such different results within the simulation size and durations explored here we provide evidence that the principal discrepancies are due to numerical dispersion through comparison of simulations using different algorithms we present new 3d front tracking simulations that show tentative agreement with the range of reported experimental values we begin an exploration of new physical length scales that may characterize a transition to a new rayleigh taylor mixing regime

robust control for linear systems with structured state space uncertainty abstract a recent sufficient condition for the stability robustness of linear systems with time varying norm bounded state space uncertainty is extended to include the structure of the uncertainty our new result requires that the nominal eigenvalues lie to the left of a vertical line in the complex plane which is determined by a norm involving the structure of the uncertainty and the nominal closed loop eigenvector matrix therefore this robustness result is especially well suited to the design of control systems using eigenstructure assignment when the uncertainty is time invariant our norm is also an upper bound on the incremental eigenvalue perturbations we also consider the use of perron weightings to reduce conservatism and the extension of the results to discrete time systems

a closed form solution to the discrete time kalman filter and its applications abstract this paper presents a closed form solution to the discrete time kalman filter and its applications we first represent the kalman filter in terms of model parameters without using the riccati equation and requiring any artificial conditions such as invertibility of a system matrix and no system noises replacing the initial time with the fixed lag time to achieve the finite memory with respect to inputs and outputs and choosing the proper initial covariances on the recent finite horizon we easily obtain the minimum variance finite memory filter and then propose its iterative computation algorithm as another application a closed form solution to the difference riccati equation on the finite horizon is utilized to obtain a stabilizing gain matrix of a luenberger type filter as in ackermann s formula

an algorithm for pole assignment in high order multivariable systems an efficient computational method is presented for solving the problem of pole assignment by state feedback in linear multivariable systems with large dimensions a given multiinput system is first transformed to an upper block hessenberg form by means of orthogonal state coordinate transformations the state feedback problem is then reformulated in terms of the sylvester equation the transformed system matrices along with certain assumed block forms for unknown matrices enable the sylvester equation to be decomposed and solved effectively a distinct point of the proposed algorithm is that the solution procedure can be tailored to parallel implementation and is therefore fast computational aspects of the algorithm are discussed and numerical examples are provided

comparison and validation of multi phase closure models the purpose of this paper is to propose a simple one or two parameter multiphase flow model suitable for the description of rayleigh taylor and richtmyer meshkov mixing layers we justify model closure assumptions in this model by comparison to rayleigh taylor and richtmyer meshkov simulation data we show that the relative errors related to model closure terms are about 10 and are about two to four times smaller than related closure models of abgrall and saurel

least squares pole assignment by memoryless output feedback in this paper a pole assignment problem for a linear time invariant control system by memoryless output feedback is posed as a least squares optimisation problem and analysed the cost functions are appropriately modified so as to obtain convergence of the corresponding gradient flow and existence of the global minimum this approach is also extended to accommodate the output pole assignment problem with insensitivity against disturbances in system parameters the relation between the modified cost functions and the original pole assignment problem is revealed the proposed approach is compared with other existing approaches and illustrated by numerical results

eigenvalue assignment in a single input linear system a procedure for eigenvalue assignment in a single input linear system in hessenberg form is described the procedure uses real arithmetic throughout and includes a facility for estimating the robustness of the assigned eigenvalues the assigned eigenvalues may be real or complex simple or multiple

hopf rinow theorem in the sato grassmannian abstract let u 2 h be the banach lie group of unitary operators in the hilbert space h which are hilbert schmidt perturbations of the identity 1 in this paper we study the geometry of the unitary orbit u p u u u 2 h of an infinite projection p in h this orbit coincides with the connected component of p in the hilbert schmidt restricted grassmannian gr res p also known in the literature as the sato grassmannian corresponding to the polarization h p h p h it is known that the components of gr res p are differentiable manifolds here we give a simple proof of the fact that gr res 0 p is a smooth submanifold of the affine hilbert space p b 2 h where b 2 h denotes the space of hilbert schmidt operators of h also we show that gr res 0 p is a homogeneous reductive space we introduce a natural metric which consists in endowing every tangent space with the trace inner product and consider its levi civita connection this connection has been considered before for instance its sectional curvature has been computed we show that the levi civita connection coincides with a linear connection induced by the reductive structure a fact which allows for the easy computation of the geodesic curves we prove that the geodesics of the connection which are of the form t e t z p e t z for z a p co diagonal anti hermitic element of b 2 h have minimal length provided that z 2 note that the condition is given in terms of the usual operator norm a fact which implies that there exist minimal geodesics of arbitrary length also we show that any two points p 1 p 2 gr res 0 p are joined by a minimal geodesic if moreover p 1 p 2 1 the minimal geodesic is unique finally we replace the 2 norm by the k schatten norm k 2 and prove that the geodesics are also minimal for these norms up to a critical value of t which is estimated also in terms of the usual operator norm in the process minimality results in the k norms are also obtained for the group u 2 h

uncertainty quantification for chaotic computational fluid dynamics we seek error models for simulations that model chaotic flow stable statistics for the solution and for the error are obtained after suitable averaging procedures

parametrizing projections with selfadjoint operators abstract let h h h be an orthogonal decomposition of a hilbert space with e e the corresponding projections let a be a selfadjoint operator in h which is codiagonal with respect to this decomposition i e a h h and a h h we consider three maps which assign a selfadjoint projection to a 1 the graph map a projection onto the graph of a h 2 the exponential map of the grassmann manifold p of h the space of selfadjoint projections in h at e exp a e i 2 a e e i 2 a 3 the map p called here the davis map based on a result by chandler davis characterizing the selfadjoint contractions which are the difference of two projections the ranges of these maps are studied and compared using davis map one can solve the following operator matrix completion problem given a contraction a h h complete the matrix a 2 a 2 to a projection p in order that p e is minimal

pairs of orthogonal projections with a fixed difference abstract let a be a self adjoint contraction on a hilbert space h in 2014 e andruchow proved that if a is a difference of two orthogonal projections with non trivial generic part then there exist infinitely many pairs p q of orthogonal projections such that a p q in the present paper we give a sufficient and necessary condition for a to be a difference of a pair of orthogonal projections then we give a characterization of all pairs p q of orthogonal projections such that a p q moreover we characterize the von neumann algebra generated by such pairs p q and consider the connected components of the set p q a p q

an algorithm to assign eigenvalues in a hessenberg matrix single input case we propose a conceptually simple algorithm to assign eigenvalues in a hessenberg matrix the method is based on the evaluation of a simple recursive relation and it is extremely easy to program on a computer numerical experiments compare favorably to those of other methods recently proposed for the problem

duality in reconstruction systems we consider the notion of finite dimensional reconstructions systems rs s which includes the fusion frames as projective rs s we study erasures some geometrical properties of these spaces the spectral picture of the set of all dual systems of a fixed rs the spectral picture of the set of rs operators for the projective systems with fixed weights and the structure of the minimizers of the joint potential in this setting we give several examples

stability robustness of almost linear state equations sufficient conditions for stability robustness of finite dimensional autonomous systems are discussed the system comprises a stable linear nominal part and different types of unstructured norm bounded perturbations quantitative sufficient conditions for stability robustness are introduced for the case in which the perturbations are almost linear in the sense that both the size and the derivative of the perturbations are bounded these conditions describe a tradeoff between the size of the perturbations and their distance from linearity the arbitrary nonlinear case and a special linear case are shown to be limiting cases of the almost linear type as an illustration the same quantitative sufficient conditions for stability are applied to a system with slowly varying linear perturbations

maximal stability robustness for state equations a measure for stability robustness of a linear time invariant finite dimensional system state equations is introduced an upper bound for the measure is derived using the characteristic values of the system it is shown that the set of optimal systems namely systems for which the stability robustness measure attains the bound contains the normal set which has been considered as the set of optimal robustness

output deadbeat control discrete time multivariate systems an algorithm is presented to compute output deadbeat controls for linear multivariable systems the algorithm based on quadratic optimization with no cost on control cheap control is numerically stable and covers the most general cases the resulting controls are internally stable and are shown to drive the outputs of the system to zero in minimum time using state feedback the central part of the algorithm is the construction of the nonunique state cost matrix two alternatives are presented having different complexities and resulting in solutions having different properties

discrete equations for physical and numerical compressible multiphase mixtures we have recently proposed in 21 a compressible two phase unconditionally hyperbolic model able to deal with a wide range of applications interfaces between compressible materials shock waves in condensed multiphase mixtures homogeneous two phase flows bubbly and droplet flows and cavitation in liquids one of the difficulties of the model as always in this type of physical problems was the occurrence of non conservative products in 21 we have proposed a discretisation technique that was without any ambiguity only in the case of material interfaces not in the case of shock waves this model was extended to several space dimensions in 24 in this paper thanks to a deeper analysis of the model we propose a class of schemes that are able to converge to the correct solution even when shock waves interact with volume fraction discontinuities this analysis provides a more accurate estimate of closure terms but also an accurate resolution method for the conservative fluxes as well as non conservative terms even for situations involving discontinuous solutions the accuracy of the model and method is clearly demonstrated on a sequence of difficult test problems

the iterated aluthge transforms of a matrix converge abstract given an r r complex matrix t if t u t is the polar decomposition of t then the aluthge transform is defined by t t 1 2 u t 1 2 let n t denote the n times iterated aluthge transform of t i e 0 t t and n t n 1 t n n we prove that the sequence n t n n converges for every r r matrix t this result was conjectured by jung ko and pearcy in 2003 we also analyze the regularity of the limit function

interface tracking for axisymmetric flows a front tracking method for inviscid gas dynamics is presented the key constructions and algorithms used in the code are described and the interrelations between shock capturing interface dynamics computational geometry grid construction and parallelism are discussed for the code as a whole validation is carried out by comparing the single mode bubble velocity for rayleigh taylor instability with theoretical models and experimental results the calculations are validated by mesh refinement studies and by the comparison of the asymptotic limit of the minimum radius r rm min rightarrow infty to a pure planar computation in two dimensions

an algorithm for the assignment of system zeros abstract this paper presents an algorithm for assigning the locations and number of zeros to a linear system through synthesis of the system input and output space couplings the approach imposes no restrictions on the state space model can assign both real and complex zeros using only real arithmetic and is computationally efficient numerical properties of the algorithm are discussed and examples are given to illustrate its performance

on computational algorithms for pole assignment a number of computationally reliable direct methods for pole assignment by feedback have recently been developed these direct procedures do not necessarily produce robust solutions to the problem however in the sense that the assigned poles are insensitive to perturbalions in the closed loop system this difficulty is illustrated here with results from a recent algorithm presented in this transactions and its causes are examined a measure of robustness is described and techniques for testing and improving robustness are indicated

dynamical evolution of rayleigh taylor and richtmyer meshkov mixing fronts dynamic behavior of mixing fronts plays a crucial role in multifluid turbulent mixing in this paper we derive an analytic solution for the dynamic evolution of mixing fronts driven by constant acceleration rayleigh taylor rt and impulsive acceleration richtmyer meshkov instabilities from a simple physics model expressed as a pair of ordinary differential equations an approximate closed form asymptotic evaluation of the rt solution is obtained through terms of order o 1 as t infinity this three term expansion including lower order terms is used to interpret experimental and simulation data our solutions improve on previous analyses in their agreement with experimental data in that we can fit both the slope and the intercept of the z b vs agt 2 experimental plots by adjusting parameters in our model since the experimental data are close to self similar the improvement due to the lower order contributions in the asymptotic expansion is modest we also apply this analysis to simulation data for which preasymptotic data exist we reexamine previous simulation data and determine an improved growth rate alpha b 0 0625 the present paper provides concepts and tools to explore the preasymptotic aspects of these data

verification and validation for turbulent mixing abstract verification and validation constitute a relatively new requirement for numerical analysis it is one which results from the growing practical use of computation as a basis for engineering decisions while a number of procedures have been proposed for these tasks their application to the more difficult cases remains a research issue here we explain methods developed by ourselves and collaborators for verification and validation of simulations for chaotic multiscale flows specifically for turbulent mixing simulations

approximate pole assignment a novel approach and some new algorithms for the pole assignment problem of a controllable time invariant linear multivariable system with linear state or output feedback are presented the main theme of the approach and the algorithms is to satisfy some constraints in the problem approximately various rank 1 update algorithms can then be derived depending on different extents to which the constraints are approximated functionals have to be minimized when eigenvectors of the closed loop system are chosen a perturbation analysis on the effect of the approximation and several numerical examples will be presented

robust pole placement in lmi regions discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality lmi regions a class of convex regions of the complex plane that embraces most practically useful stability regions the focus is on linear systems with static uncertainty on the state matrix for this class of uncertain systems the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary lmi regions the resulting tests for robust pole clustering are all numerically tractable because they involve solving linear matrix inequalities lmis and cover both unstructured and parameter uncertainty these analysis results are then applied to the synthesis of dynamic output feedback controllers that robustly assign the closed loop poles in a prescribed lmi region with some conservatism this problem is again tractable via lmi optimization in addition robust pole placement can be combined with other control objectives such as h sub 2 or h sub spl infin performance to capture realistic sets of design specifications physically motivated examples demonstrate the effectiveness of this robust pole clustering technique

sensitivity properties of multirate feedback control systems based on eigenstructure assignment some sampled data systems e g fly by wire control schemes have a necessarily multirate structure various input and or outputs sampled at different rates when considering a multirate system which has parameter uncertainty it is important to examine ways in which the full freedom of the multivariable design can be utilized to minimize the sensitivity to parameter variations given the accompanying problems induced by intersample ripple disturbance this note examines the design capabilities of a class of multirate systems with multiple input and fixed state sampling rates mifs based on eigenstructure assignment although the use of eigenstructure assignment for continuous and single rate discrete systems is well understood the eigenstructure assignment for the design of multirate feedback systems is an open topic of research accepting that the problems of intersample ripple are often magnified through multirate control there are advantages in terms of increased freedom for minimizing sensitivity and optimizing robustness to parameter variations a special feature of the mifs class of multirate systems is the ability to introduce extra design freedom in the eigenproblem by a suitable choice of eigenstructure assignment and sample rates the criteria for the selection of minimum sample rates to produce this extra freedom and the implication that this has on the eigenstructure assignment problem are outlined the improved insensitivity properties are demonstrated using an example comparing the performance of multirate and corresponding single rate designs

a new robust stability measure for state feedback systems abstract this brief paper proposes the use of sensitivities of each individual system pole to measure robust stability of state feedback systems compared with the existing measures of this kind m 1 and m 2 this measure is much easier to work with than m 1 while at least as accurate and is definitely more accurate than m 2 the upper and lower bounds of this measure are also derived and proved to be the same as those of m 1 this measure can be maximized numerically using existing explicit and systematic procedures some analytical guidelines which relate explicitly and directly the state feedback system s individual pole selection performance and decoupling to this new measure are also available and presented

natural variational problems in the grassmann manifold of a c algebra with trace we consider a family of natural variational problems in the grassmannian of a c algebra with trace which can be considered as slightly degenerate finsler metrics we show that all these problems have as solutions the standard geodesics and that the short standard geodesics are absolute minima of the functionals restricted to a special class of curves

a front tracking algorithm for limited mass diffusion the advantage of front tracking over capturing methods for the numerical simulation of discontinuity surfaces in fluid flow is to eliminate the numerical diffusion of mass across a fluid interface for accurate modeling of an interface between miscible fluids however physical mass diffusion may be significant and must be added to the calculation the main purpose of this paper is to introduce a new front tracking algorithm in which the physical diffusion of mass across a tracked interface is included the accuracy and convergence properties of this algorithm are discussed the new algorithm is a subgrid algorithm in the sense that the asymptotic fine grid behavior is identical to conventional untracked capturing methods while the coarse grid behavior is improved it is thus most suitable for modeling small values of physical mass diffusion for which adequate numerical resolution is computationally demanding the mixing rates for the 3d rayleigh taylor instability of miscible fluids based on this algorithm agree with experimental values

robust control with pole shifting via performance index modification robust controllers that ensure closed loop eigenvalue placement are designed for linear systems with parametric uncertainties by using an appropriate state weighting matrix choice and a relative stability degree an lmi based algorithm is proposed to provide guaranteed cost controllers that ensure i the closed loop nominal eigenvalues shifting by a pre specified amount ii a prescribed degree of stability for all system uncertainties of a given class a certain level of closed loop performance is also ensured by minimizing the associated guaranteed cost

spaces of nonpositive curvature arising from a finite algebra fil conde cristian marcelo universidad nacional de general sarmiento instituto de ciencias argentina consejo nacional de investigaciones cientificas y tecnicas oficina de coordinacion administrativa saavedra 15 instituto argentino de matematica alberto calderon argentina

multiscale models for fluid mixing recent work of the authors and colleagues on the turbulent mixing of compressible fluids is developed and extended with an emphasis on the multiscale aspects of this work specifically we study an interplay between micro and macro aspects of mixing

a robust periodic pole assignment algorithm in this note a robust periodic pole assignment algorithm is proposed for linear time invariant discrete time systems the condition numbers of the eigenvector matrices of the closed loop system are assumed as a robustness measure and a periodic state feedback law is deduced by the minimization of the condition numbers associated to the eigenvectors of the monodromy matrix of the closed loop system the proposed periodic pole assignment algorithm has been tested on a number of examples giving satisfactory results

metric geometry of partial isometries in a finite von neumann algebra abstract we study the geometry of the set i p v m v v p of partial isometries of a finite von neumann algebra m with initial space p p is a projection of the algebra this set is a c submanifold of m in the norm topology of m however we study it in the strong operator topology in which it does not have a smooth structure this topology allows for the introduction of inner products on the tangent spaces by means of a fixed trace in m the quadratic norms do not define a hilbert riemann metric for they are not complete nevertheless certain facts can be established a restricted result on minimality of geodesics of the levi civita connection and uniqueness of these as the only possible minimal curves we prove also that i p d g is a complete metric space where d g is the geodesic distance of the manifold or the metric given by the infima of lengths of piecewise smooth curves

brief augmented gradient flows for on line robust pole assignment via state and output feedback this paper is concerned with robust pole assignment in synthesis of linear control systems via state and output feedbacks first both the pole assignment and robustness requirements are appropriately formulated as two optimization problems then gradient flow models are developed for the on line computation of feedback gain matrices that result in robust pole assignment by solving these two optimization problems a technique is introduced to facilitate the real time matrix inverse involved for realizing the gradient flow models the resulting augmented gradient flows have desired convergence properties simulation results are included to show the effectiveness of the proposed approach

robustness in partial pole placement the robustness of state feedback solutions to the problem of partial pole placement obtained by a new projection procedure is examined the projection procedure gives a reduced order pole assignment problem it is shown that the sensitivities of the assigned poles in the complete closed loop system are bounded in terms of the sensitivities of the assigned reduced order poles and the sensitivities of the unaltered poles are bounded in terms of the sensitivities of the corresponding open loop poles if the assigned poles are well separated from the unaltered poles these bounds are expected to be tight the projection procedure is described in 3 and techniques for finding robust or insensitive solutions to the reduced order problem are given in 1 2

the minimal and maximal symmetries for j contractive projections abstract in this paper we first characterize the structure of symmetries j such that a projection p is j contractive then the minimal and maximal elements of the symmetries j with p j p j or j p 0 are given moreover some formulas between p 2 i p p p 2 i p p and p p p p p p are established

new approach to robust observer design this paper shows that based on the recent development of observer design solution an observer pole selection method can be formulated to minimize the observer gain to the system input it is proved that this method is a deterministic approach to the recovery of the loop transfer function and robustness of direct state feedback systems

a multiphase godunov method for compressible multifluid and multiphase flows we propose a new model and a solution method for two phase compressible flows the model involves six equations obtained from conservation principles applied to each phase completed by a seventh equation for the evolution of the volume fraction this equation is necessary to close the overall system the model is valid for fluid mixtures as well as for pure fluids the system of partial differential equations is hyperbolic hyperbolicity is obtained because each phase is considered to be compressible two difficulties arise for the solution one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations we propose robust and accurate discretisation of these terms the method solves the same system at each mesh point with the same algorithm it allows the simulation of interface problems between pure fluids as well as multiphase mixtures several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible the method provides reliable results is able to compute strong shock waves and deals with complex equations of state

projection and deflation method for partial pole assignment in linear state feedback two projection methods are proposed for partial pole placement in linear control systems these methods are of interest when the system is very large and only a few of its poles must be assigned the first method is based on computing an orthonormal basis of the left invariant subspace associated with the eigenvalues to be assigned and then solving a small inverse eigenvalue problem resulting from projecting the initial problem into that subspace the second method can be regarded as a variant of the weilandt deflation technique used in eigenvalue methods

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

neurodynamics based robust pole assignment for high order descriptor systems in this paper a neurodynamic optimization approach is proposed for synthesizing high order descriptor linear systems with state feedback control via robust pole assignment with a new robustness measure serving as the objective function the robust eigenstructure assignment problem is formulated as a pseudoconvex optimization problem a neurodynamic optimization approach is applied and shown to be capable of maximizing the robust stability margin for high order singular systems with guaranteed optimality and exact pole assignment two numerical examples and vehicle vibration control application are discussed to substantiate the efficacy of the proposed approach

a new design approach to unknown input observers this paper proposes a fundamentally new unknown input observer uio which has uniquely r decoupled modes provides r linearly independent combinations of plant system states and has zero gain to all p unknown inputs the parameter r is adjustable if the plant system satisfies the conditions of the existing uio then r n plant system order and there is no difference between our uio and the existing regular uio however if the plant system satisfies either of the following two significantly more general conditions m p m is the number of plant output measurements and at least one stable transmission zero then 1 n and our uio can still generate some constrained state feedback control signals hence this new uio can completely unify the regular uio r n and the static output feedback r m the two basic feedback structures of modern control theory far from being unified before

shock wave interactions in spherical and perturbed spherical geometries the interaction of shock waves with spherical and perturbed spherical layers provides an interesting and important class of problems we present recent results for these and related planar geometry problems we address the following issues 1 the accuracy of different numerical solution methods 2 the magnitude of numerical solution errors and their causes 3 the development of the instabilities 4 reduced descriptions for chaotic flows

some necessary and sufficient conditions for eigenstructure assignment abstract some necessary and sufficient conditions for closed loop eigenstructure assignment by output feedback in time invariant linear multivariable control systems are presented a simple condition on a square matrix necessary and sufficient for it to be the closed loop plant matrix of a given system with some output feedback is the basis of the paper some known results on entire eigenstructure assignment are deduced from this the concept of an inner inverse of a matrix is employed to obtain a condition concerning the assignment of an eigenstructure consisting of the eigenvalues and a mixture of left and right eigenvectors

a new approach for computing the state feedback gains of multivariable systems this note presents some new results in linear systems first the relationship between the polynomial matrix description and the state space representation of multivariable systems is clarified then we show that once such a relationship is determined the coprime matrix fraction description can be easily computed and we can further develop a closed form formula to solve the pole assignment problem of multivariable systems such formula can be thought of as an extension of the ackermann s formula for mimo systems thus this note potentially gives us a clearer insight into linear systems from the theoretical viewpoint

approaches to robust pole assignment abstract robust pole assignment is a non linear optimization problem in many variables we describe numerical methods for determining robust or well conditioned so lutions to the problem of pole assignment by state feedback the solutions are chosen to minimize various objective functions based on the condition number of the eigenvector matrix careful choice of parametrization and objective function avoids singularities and artificial variable constraints explicit formulae for the gradient and hessian permit rigorous stopping criteria and rapid local convergence several computational examples are included

finsler geometry and actions of the p schatten unitary groups let p be an even positive integer and u p h the banach lie group of unitary operators u which verify that u 1 belongs to the p schatten ideal b p h let o be a smooth manifold on which u p h acts transitively and smoothly then one can endow o with a natural finsler metric in terms of the p schatten norm and the action of u p h our main result establishes that for any pair of given initial conditions x o and x to x there exists a curve 6 t e t z x in o with z a skew hermitian element in the p schatten class such that 0 x and 0 x which remains minimal as long as t z p 4 moreover is unique with these properties we also show that the metric space o d where d is the rectifiable distance is complete in the process we establish minimality results in the groups u p h and a convexity property for the rectifiable distance as an example of these spaces we treat the case of the unitary orbit o uau u u p h of a self adjoint operator a b h

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

robust pole assignment for synthesizing feedback control systems using recurrent neural networks this paper presents a neurodynamic optimization approach to robust pole assignment for synthesizing linear control systems via state and output feedback the problem is formulated as a pseudoconvex optimization problem with robustness measure i e the spectral condition number as the objective function and linear matrix equality constraints for exact pole assignment two coupled recurrent neural networks are applied for solving the formulated problem in real time in contrast to existing approaches the exponential convergence of the proposed neurodynamics to global optimal solutions can be guaranteed even with lower model complexity in terms of the number of variables simulation results of the proposed neurodynamic approach for 11 benchmark problems are reported to demonstrate its superiority

an algorithm for computing state feedback in multiinput linear systems a new algorithm is proposed in this note to compute the state feedback for arbitrary pole placement in multiinput linear systems the algorithm has good numerical behavior it also allows the user to adjust the eigenstructure of the resulting feedback system and therefore improve the condition of the problem and computation

the decay of stabilizability with reynolds number in a linear model of spatially developing flows this article characterizes the gradual decay of stabilizability with reynolds number in the linear complex ginzburg landau model of spatially developing ow systems when a single spatially localized actuator is used to apply the control forcing it is shown that technically the system considered is linearly stabilizable for all actuator locations at any reynolds number however as the reynolds number is increased and an increasing number of modes of the open loop system become unstable the control authority on some of these open loop unstable modes is found to be exponentially small using nite precision arithmetic and any given numerical method for computing the feedback gains an e ective upper bound on the reynolds number is reached above which it is not possible to compute a linearly stabilizing control

robust eigenvalue assignment for generalized systems abstract in this paper we examine the problem of robust pole placement using state feedback in generalized systems we develop a robustness theory for the finite generalized spectrum of the system as a partial problem and for the infinite pole placement problem as a second partial problem where perfect conditioning is always achievable we also give bounds between the complete closed loop robust eigenvalue problem and the two partial ones the basic tool that is exploited in the presented theory is the concept of chordal metric in this paper we take advantage of this notion and present a compact theory for the robust eigenvalue assignment problem in generalized systems the proposed theory is easy to implement retrieves the results for the state variable case as a special case and takes advantage of well known computational results

effects of using observers on stabilization of uncertain linear systems the effects of observers on robust linear feedback controllers are studied sufficient conditions are obtained that guarantee full recovery of the allowable uncertainty bounds attainable by full state feedback the effects of the resulting high gain observers on the disturbance rejection bounds are also studied it is shown that full recovery of the uncertainty bound leads to possible large degradation in disturbance rejection however if there is only additive plant disturbance and no measurement disturbance this degradation can be prevented

geometry of unitary orbits of pinching operators abstract let i be a symmetrically normed ideal of the space of bounded operators acting on a hilbert space h let p i 1 w 1 w be a family of mutually orthogonal projections on h the pinching operator associated with the former family of projections is given by p i i p x i 1 w p i x p i let u i denote the banach lie group of the unitary operators whose difference with the identity belongs to i we study geometric properties of the orbit u i p l u p l u u u i where l u is the left representation of u i on the algebra b i of bounded operators acting on i the results include necessary and sufficient conditions for u i p to be a submanifold of b i special features arise in the case of the ideal k of compact operators in general u k p turns out to be a non complemented submanifold of b k we find a necessary and sufficient condition for u k p to have complemented tangent spaces in b k we also show that u i p is a covering space of another orbit of pinching operators

numerically robust pole assignment for second order systems we propose two new methods for solution of the eigenvalue assignment problem associated with the second order control system global hsize 30pc specifically the methods construct feedback matrices f 1 and f 2 such that the closed loop quadratic pencil has a desired set of eigenvalues and the associated eigenvectors are well conditioned method 1 is a modification of the singular value decomposition based method proposed by juang and maghami which is a second order adaptation of the well known robust eigenvalue assignment method by kautsky et al for first order systems method 2 is an extension of the recent non modal approach of datta and rincon for feedback stabilization of second order systems robustness to numerical round off errors is achieved by minimizing the condition numbers of the eigenvectors of the closed loop second order pencil control robustness to large plant uncertainty will not be explicitly considered in this paper numerical results for both the two methods are favourable a compara

a schur method for pole assignment in this paper a new numerical algorithm for pole assignment of linear time invariant systems is presented the proposed algorithm uses reliable numerical techniques based on the schur form of the state matrix and on the use of orthogonal transformations only the bad eigenvalues of the system are modified by a sequential pole shifting procedure and the resulted gains in the feedback matrix are minimized

operators which are the difference of two projections abstract we study the set d of differences d a p q p q p where p denotes the set of orthogonal projections in h we describe models and factorizations for elements in d which are related to the geometry of p the study of d throws new light on the geodesic structure of p we show that two projections in generic position are joined by a unique minimal geodesic the topology of d is examined particularly its connected components are studied also we study the subsets d c d f where d c are the compact elements in d and d f are the differences a p q such that the pair p q is a fredholm pair p q is a fredholm pair if q p r p r p r q is a fredholm operator

a crisis for the verification and validation of turbulence simulations abstract three algorithms have been proposed for solution of the rayleigh taylor turbulent mixing problem they are based upon three different physical principles governing the euler equations for fluid flow the principles serve to select the physically relevant solution from among many nonunique solutions the admissibility principle is in dispute the three different algorithms expressing the three physical admissibility principles can be formulated in terms of the three energy dissipation rates or the entropy production rates as selected by the size of the sub grid scale coefficients these have maximal values or less than maximal values the resulting solutions are markedly different we find strong validation evidence that supports the maximum rate principle based on a review of prior results and on new results presented here we review experimental data used for validation and sufficient to discriminate among the three we present a new analysis of this data we show that the hypothesized long wave length perturbations in the initial conditions are not significant so that validation can be based on this data in a straight forward manner one of the algorithms is labeled direct numerical simulation but is not and as a consequence the two algorithms with less than maximal sgs coefficients are variants of one another recommendations for the numerical modeling of the deflagration to detonation transition in type ia supernova are discussed

the rectifiable metric on the set of closed subspaces of hilbert space consider the set of selfadjoint projections on a fixed hilbert space it is well known that the connected components under the norm topology are the sets p rank p rank 1 p where and are appropriate cardinal numbers on a given component instead of using the metric induced by the norm we can use the rectifiable metric d r which is defined in terms of the lengths of rectifiable paths or equivalently in this case the lengths of e chains if p q 1 then d r p q sin 1 p q but if p q 1 d r p q can have any value in 2 assuming and are infinite

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

geometric operator inequalities abstract the geometrical meaning of several well known inequalities is discussed they include the so called loewner heinz mcintosh and segal inequalities it is shown that some of them can be deduced from the others even for unitarily invariant forms some spectral properties of the elementary operators associated to the inequalities are studied

a unified closed loop stability measure for finite precision digital controller realizations implemented in different representation schemes a computationally tractable unified finite word length closed loop stability measure is derived which is applicable to fixed point floating point and block floating point representation schemes both the dynamic range and precision of an arithmetic scheme are considered in this new unified measure for each arithmetic scheme the optimal controller realization problem is defined and a numerical optimization approach is adopted to solve it numerical examples are used to illustrate the design procedure and to compare the optimal controller realizations in different representation schemes

eigenvalue assignment in an upper hessenberg matrix a simple algorithm is described for the determination of the elements in the first row of a real unreduced upper hessenberg matrix so that this matrix has a specified set of eigenvalues the method is extended for application to descriptor systems

an algorithm for pole assignment of time invariant linear systems an algorithm is suggested for the computation of a linear state feedback for a single input system such that the resultant closed loop system has specified eigenvalues the algorithm is more efficient than many comparable techniques and has some desirable numerical properties it is closely related to the qr algorithm for the eigenproblem and extends to descriptor systems