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

exact pole assignment using direct or dynamic output feedback this note addresses the problem of the assignability of the eigenvalues of the matrix a bpc by choice of the matrix p this mathematical problem corresponds to pole assignment in the direct output feedback control problem and by proper changes of variables it also represents the pole assignment problem with dynamic feedback controllers the key to our solution is the introduction of the new concept of local complete assignability which in loose terms is the arbitrary perturbability of the eigenvalues of a bpc by perturbations of p if n x is the order of the system we show that if a bp 0 c has distinct eigenvalues a necessary and sufficient condition for local complete assignability at p 0 is that the matrices c a bp 0 c i 1 b be linearly independent for 1 leq i leq n x in special cases this condition reduces to known criteria for controllability and observability although these latter properties are necessary conditions for assignability we also address the question of the assignability of uncontrollable or unobservable systems both by direct output feedback and dynamic compensation the main result of this note yields an algorithm that assigns the closed loop poles to arbitrarily chosen values in the direct and in the dynamic output feedback control problems

property t of group homomorphisms abstract in this article we define and study property t for continuous homomorphisms between topological groups if g is a locally compact group we show that g g u l 2 g has property t if and only if either g is compact or g is non amenable moreover the abelianization g ab g g g is compact if and only if every continuous homomorphism from g to any abelian topological group has property t moreover we show that g has property t fd if and only if any continuous homomorphism from g to any compact group has property t in the case when g is almost connected the above is also equivalent to the canonical map from g to its bohr compactification being a quotient map we also give some new equivalent forms of the strong property t of a locally compact group as a consequence if g is a second countable and has strong property t and h is a closed subgroup of g there exist at most one g invariant mean on g h

survey paper static output feedback a survey this paper reviews the static output feedback problem in the control of linear time invariant lti systems it includes analytical and computational methods and presents in a unified fashion the knowledge gained in the decades of research into this important open problem the paper shows that although many approaches and techniques exist to approach different versions of the problem no efficient algorithmic solutions are available

numerical algorithms for eigenvalue assignment by constant and dynamic output feedback algorithms are proposed for eigenvalue assignment eva by constant as well as dynamic output feedback the main algorithm is developed for single input multioutput systems and the results are then extended to multiinput multioutput systems in computing the feedback use is made of the fact that the closed loop eigenvalues can almost always be assigned arbitrarily close to the desired locations in the complex plane provided the system satisfies the condition m p n where m p and n are respectively the number of inputs outputs and states of the system the eva problem has been treated as a converse of the algebraic eigenvalue problem the proposed algorithms are based on the implicitly shifted qr algorithm for solving the algebraic eigenvalue problem the performance of the algorithms is illustrated by several numerical examples

error estimates for the raviart thomas interpolation under the maximum angle condition the classical error analysis for the raviart thomas interpolation on triangular elements requires the so called regularity of the elements or equivalently the minimum angle condition however in the lowest order case optimal order error estimates have been obtained in g acosta and r g duran siam j numer anal 37 2000 pp 18 36 replacing the regularity hypothesis by the maximum angle condition which was known to be sufficient to prove estimates for the standard lagrange interpolation in this paper we prove error estimates on triangular elements for the raviart thomas interpolation of any order under the maximum angle condition also we show how our arguments can be extended to the three dimensional case to obtain error estimates for tetrahedral elements under the regular vertex property introduced in g acosta and r g duran siam j numer anal 37 2000 pp 18 36

a balanced finite element method for singularly perturbed reaction diffusion problems consider the singularly perturbed linear reaction diffusion problem varepsilon 2 delta u bu f in omega subset mathbb r d u 0 on partial omega where d ge 1 the domain ome

eigenvalue generalized eigenvector assignment by output feedback this note generalizes the previous results of the closed loop eigenstructure assignment via output feedback in linear multivariable systems necessary and sufficient conditions for the closed loop eigenstructure assignment by output feedback are presented some known results on entire eigenstructure assignment are deduced from this result

on groups with property t_lp abstract let 1 p p 2 property t l p for a second countable locally compact group g is a weak version of kazhdan property t defined in terms of the orthogonal representations of g on l p property t l p is characterized by an isolation property of the trivial representation from the monomial unitary representations of g associated to open subgroups connected groups with property t l p are the connected groups with a compact abelianization in the case of a totally disconnected group isolation of the trivial representation from the quasi regular representations associated to open subgroups suffices to characterize property t l p groups with property t l p share some important properties with kazhdan groups compact generation compact abelianization simple algebraic groups over non archimedean local fields as well as automorphism groups of k regular trees for k 3 have property t l p in the case of discrete groups property t l p implies lubotzky s property and is implied by property f of glasner and monod we show that an irreducible lattice in a product g 1 g 2 of locally compact groups has property t l p whenever g 1 has property t and g 2 is connected and minimally almost periodic such a lattice does not have property t if g 2 does not have property t

on eigenstructure assignment by gain output feedback in this paper the eigenstructure assignment of linear multivariable control systems is studied from a geometric point of view for the class of systems in which the number of outputs plus the number of inputs exceeds the number of states genericity properties relative to this problem are derived it is shown without any assumption on the genericity of the system that the pole assignment can be carried out by choosing some closed loop eigenvectors almost freely the crucial point is that all the expected degrees of freedom in pole assignment are so described without redundancy which fully justifies the practical interest of such techniques despite the technicality required for the derivation of intermediate results the main result which is an eigenstructure assignment algorithm is very easy to implement because it is only based on the computation of certain sums and intersections of characteristic subspaces furthermore it is shown how the basic tools developed here can be used to tackle the prob

pole assignment of strictly proper and proper linear systems by constant output feedback a new unifying approach for the study of pole assignment by constant output feedback cpap for strictly proper and proper linear systems is presented the multilinear nature of cpap is reduced to a linear problem and to the standard multilinear problem of decomposability of multivectors the solvability of cpap thus becomes a problem of finding real intersections of a linear variety with the grassmann variety of a projective space an alternative proof to the ml n m i n are the numbers of outputs inputs states necessary and sufficient condition for generic pole assignability of strictly proper systems by complex output feedback is given the above result is extended to the ml n 1 condition for the case of proper systems stronger necessary conditions for generic pole assignment by a real output feedback are obtained by using the recently introduced invariant the plucker matrix pn such conditions are that pn must have full rank and ml n strictly proper case or ml n 1 proper case it is sh

paper low order control design for lmi problems using alternating projection methods computational techniques based on alternating projections are proposed to solve control design problems described by linear matrix inequalities lmis in particular we concentrate on the stabilization and the suboptimal h output feedback control design problems these problems can be described by a pair of lmis and an additional coupling condition this coupling condition is convex for the full order control design problem but convexity is lost for the control problem of order strictly less than the plant order we formulate these problems as feasibility problems with matrix constraint sets of simple geometry and we utilize this geometry to obtain analytical expressions for the orthogonal projection operators onto these sets full order and low order controllers are designed using alternating projection methods for the full order controller case global convergence of the alternating projection methods to a feasible solution is guaranteed however for the low order control case only local convergence is guaranteed an example is provided to illustrate the use of these methods for the full order and the low order controller design

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

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

quadrilateral h div finite elements we consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the piola transform extending results previously obtained for scalar approximation the finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element which is then transformed to a space of vector fields on each convex quadrilateral element via the piola transform associated to a bilinear isomorphism of the square onto the element for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms the situation is more complicated and we give a precise characterization of what is needed for optimal order l2 approximation of the function and of its divergence as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of h div we also derive new estimates for approximation by quadrilateral raviart thomas elements requiring less regularity and propose a new quadrilateral finite element space which provides optimal order approximation in h div finally we demonstrate the theory with numerical computations of mixed and least squares finite element approximations of the solution of poisson s equation

a geometric framework for pole assignment algorithms the problem of pole assignment by gain output feedback or by low order dynamical compensator is considered from a geometric point of view this makes it possible to unify in a general framework most of the existing pole assignment methods formulated in a state space context such as the minimal order observers the brasch pearson compensator and the methods proposed by h kimura and to simplify their presentation moreover new pole assignment algorithms may be derived from this general formulation

the maximum angle condition for mixed and nonconforming elements application to the stokes equations for the lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition the objective of this paper is to obtain similar results for the raviart thomas interpolation arising in the analysis of mixed methods we prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and moreover that this condition is indeed necessary to have these estimates error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the stokes equations are derived from our results

property t and strong property t for unital c algebras in this paper we will give a thorough study of the notion of property t for c algebras as introduced by m b bekka as well as a slightly stronger version of it called strong property t which is also an analogue of the corresponding concept in the case of discrete groups and type ii1 factors more precisely we will give some interesting equivalent formulations as well as some permanence properties for both property t and strong property t we will also relate them to certain t type properties of the unitary group of the underlying c algebra

a parameter robust numerical method for a two dimensional reaction diffusion problem in this paper a singularly perturbed reaction diffusion partial differential equation in two space dimensions is examined by means of an appropriate decomposition we describe the asymptotic behaviour of the solution of problems of this kind a central finite difference scheme is constructed for this problem which involves an appropriate shishkin mesh we prove that the numerical approximations are almost second order uniformly convergent in the maximum norm with respect to the singular perturbation parameter some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method

amenable unitary representations of locally compact groups a notion of amenability for an arbitrary unitary group representation is introduced this unifies and generalizes the notions of amenable homogeneous spaces and of inner amenable groups amenable locally compact groups are characterized by the amenability of all their unitary representations amenable representations are characterized by several properties which are operator theoretic analogues of properties characterizing amenable groups we give a generalization to arbitrary representations of hulanicki reiter theorem this is used in order to describe the amenable representations of the groups with kazhdan property t

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

static output feedback pole placement via a trust region approach this technical note presents two closely related algorithms for the problem of pole placement via static output feedback the algorithms are based on two different trust region methods and utilize the derivatives of the closed loop poles extensive numerical experiments show the effectiveness of the algorithms in practice though convergence to a solution is not guaranteed for either algorithm while desired poles must be distinct strategies for dealing with repeated poles are also presented

a necessary and sufficient condition for output feedback stabilizability a necessary and sufficient condition for a linear system to be stabilizable via static output feedback is presented it makes an appeal to the linear quadratic regulator theory

exact pole assignment by output feedback part 1 abstract we shall prove that exact assignment of distinct poles using output feedback is possible in any controllable and observable linear time invariant multivariable system in which the number of inputs plus the number of outputs exceeds the number of states except in clearly identified circumstances furthermore we give a construction for the entire class of pathological cases the main part of our argument is concerned with the assigning of complex conjugate eigenvalues by means of a real feedback matrix indeed we begin by giving a simple algorithm which is effective over any field although this problem has been discussed by numerous authors in the past they have usually presented only generic results we examine the complete class of systems satisfying the hypotheses mentioned

geometric methods for the classification of linear feedback systems abstract the precise determination of the nature and form of system invariants under output feedback is of fundamental interest in classical and modern control theory an algebraic solution of this problem would entail for example a description of the generators and relations of the ring of output feedback invariants geometrically we seek to describe the basic properties of the quotient space whose points consist of feedback equivalence classes of the system recovering the algebraic theory in terms of the function theory on this space naturally any description should be made in system theoretic terms in this paper we construct and study in the scalar case the quotient manifold for linear systems modulo the full output feedback group using classical algebraic geometry as a corollary we obtain explicit generators for the associated ring of invariants which we then interpret in terms of classical control theory e g in terms of root locus plots turning to the question of the existence of continuous canonical forms we show that these exist globally only when the system order is odd in the even case the obstruction to the existence of globally defined continuous forms is described in terms of the values of the system cauchy index this is illustrated for low order examples and certain questions concerning relations among the generators is treated using methods of the schubert calculus the complete set of invariants for the full output feedback group is obtained and our results concerning the existence and nonexistence of canonical forms is also compared to related results contained in the literature

multistage parametric eigenstructure assignment by output feedback control this paper develops an efficient multistage parametric approach for eigenstructure assignment in linear multivariable systems by output feedback control the sets of closed loop eigenvalues and associated eigenvectors are suitably divided into subsets and the entire eigenstructure is constructed by parts in two or more consecutive stages this allows the subsets of right and left eigenvectors to be assigned in separate stages and thus the computational algorithm is relaxed from the orthogonality conditions the number of effective free parameters beyond eigenvalue assignment is determined and the notion of redistributing these parameters among the assignable right and left eigenvectors is introduced numerical examples are included to demonstrate the design procedure

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

output feedback eigenstructure assignment using two sylvester equations the eigenstructure assignment problem with output feedback is studied for systems satisfying the condition p m n the main tool used is the concept of c a b invariance and two coupled sylvester equations the solution of which leads to the computation of an output stabilizing feedback a computationally efficient algorithm for the solution of these two coupled equations which leads to the computation of a desired output feedback is presented

quadratic characterization and use of output stabilizable subspaces this note treats the problem of stabilization of linear systems by static output feedback using the concept of c a b invariant subspaces the work provides a new characterization of output stabilizable c a b invariant subspaces through two coupled quadratic stabilization conditions an equivalence is shown between the existence of a solution to this set of conditions and the possibility to stabilize the system by static output feedback an algorithm is provided and numerical examples are reported to illustrate the approach

error estimates for raviart thomas interpolation of any order on anisotropic tetrahedra we prove optimal order error estimates for the raviart thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition namely the so called three dimensional maximum angle condition and the regular vertex property for tetrahedra our techniques are different from those used in previous papers on the subject and the results obtained are more general in several aspects first intermediate regularity is allowed that is for the raviart thomas interpolation of degree k 0 we prove error estimates of order j 1 when the vector field being approximated has components in w j 1 p for triangles or tetrahedra where 0 j k and 1 p these results are new even in the two dimensional case indeed the estimate was known only in the case j k on the other hand in the three dimensional case results under the maximum angle condition were known only for k 0

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

hermitian pencils and output feedback stabilization of scalar systems abstract necessary and sufficient semi algebraic conditions for a the stabilization of scalar transfer functions and b the assignability of real poles by static output feedback are given in terms of the weierstrass invariants of an associated hermitian matrix pencil an explicit graphical test for output feedback stabilizability is derived which is equivalent to the nyquist criterion

properties of numerical algorithms related to computing controllability the numerical properties of some methods for computing controllability are used in an expository way to motivate a wider understanding of numerical computations in particular the numerical rank of a matrix the numerical stability of algorithms the sensitivity of problems and the scaling of problems are discussed a numerically stable algorithm is given for computing controllability but it is pointed out that a measure of the distance of the given system from the nearest uncontrollable system would be more useful and this appears to be an open computational problem

mixed finite element based fully conservative methods for simulating wormhole propagation abstract wormhole propagation during reactive dissolution of carbonates plays a very important role in the product enhancement of oil and gas reservoir because of high velocity and nonuniform porosity the darcy forchheimer model is applicable for this problem instead of conventional darcy framework we develop a mixed finite element scheme for numerical simulation of this problem in which mixed finite element methods are used not only for the darcy forchheimer flow equations but also for the solute transport equation by introducing an auxiliary flux variable to guarantee full mass conservation in theoretical analysis aspects based on the cut off operator of solute concentration we construct an analytical function to control and handle the change of porosity with time we treat the auxiliary flux variable as a function of velocity and establish its properties we employ the coupled analysis approach to deal with the fully coupling relation of multivariables from this the stability analysis and a priori error estimates for velocity pressure concentration and porosity are established in different norms numerical results are also given to verify theoretical analysis and effectiveness of the proposed scheme

eigenstructure assignment by output feedback and residue analysis this note deals with the use of feedback to approximate the closed loop eigenstructure of a system to a prescribed set of values a new method is proposed to reduce the controller complexity based on eigenvalue sensitivity concepts a flight control example is presented as an application

output feedback pole assignment procedure a new algorithm based on pole assignment considerations for linear time invariant m input p output systems by means of static output feedback is presented for the case in which the number of inputs m and the number of outputs p are less than the order of the system n this algorithm determines the output feedback matrix required to assign max p m 1 poles and simultaneously calculates the coefficients of the residual polynomial the roots of which are the remaining unassigned poles the root locus method is used to find an appropriate feedback gain to provide acceptable locations for the unassigned poles some numerical examples are provided to illustrate this method

perturbation theory of output feedback pole assignment perturbation theory for output feedback eigenstructure assignment is considered the analysis is based on general conditions for eigenstructure assignment together with the well known perturbation theory for the pseudo inverse of a matrix this enables us to present a variety of sensitivity analyses for this problem and to indicate some precautions when choosing eigenvectors in the pole assignment problem

a new approach to eigenstructure assignment by output feedback the eigenstructure assignment problem for a linear time invariant multi input multi output system using output feedback is considered a new approach is developed which identifies the eigenspaces for the desired set of all the closed loop eigenvalues for the assignment of this desired set necessary and sufficient conditions are established these conditions contain two coupled sylvester matrix equations one of which is proven to be a reduced order square sylvester matrix equation this results in an efficient analytical procedure numerically superior to known techniques for the determination of the output feedback gain matrix

solutions of the equation av bw vf and their application to eigenstructure assignment in linear systems two new simple complete analytical and restriction free solutions with complete and explicit freedom of the matrix equation av bw vf are proposed here ab is known and is controllable and f is in the jordan form with arbitrary given eigenvalues based on the proposed solutions of this matrix equation a complete parametric approach for eigenstructure assignment in linear systems via state feedback is proposed and two new algorithms are presented the proposed solutions of the matrix equation and the eigenstructure assignment result are generalizations of some previous results and are simpler and more effective

analysis and synthesis of robust control systems via parameter dependent lyapunov functions in this paper the problem of robust stability of systems subject to parametric uncertainties is considered sufficient conditions for the existence of parameter dependent lyapunov functions are given in terms of a criterion which is reminiscent of but less conservative than popov s stability criterion an equivalent frequency domain criterion is demonstrated the relative sharpness of the proposed test and existing stability criteria is then discussed the use of parameter dependent lyapunov functions for robust controller synthesis is then considered it is shown that the search for robustly stabilizing controllers may be limited to controllers with the same order as the original plant a possible synthesis procedure and a numerical example are then discussed

invariant measures for algebraic actions zariski dense subgroups and kazhdan s property t let k be any locally compact non discrete field we show that finite invariant measures for k algebraic actions are obtained only via actions of compact groups this extends both borel s density and fixed point theorems over local fields for semisimple solvable groups resp we then prove that for k algebraic actions finitely additive finite invariant measures are obtained only via actions of amenable groups this gives a new criterion for zariski density of subgroups and is shown to have representation theoretic applications the main one is to kazhdan s property t for algebraic groups which we investigate and strengthen