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

new solution method of linear static output feedback design problem for linear control systems abstract a new method of construction of the linear static output feedback for linear control systems is offered the essence of this method consists in construction of an initial approximation of the feedback matrix such that a numerical sequence which generated by arbitrary iterative method and beginning from this initial approximation converges to an exact solution of the feedback design problem examples are given

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

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

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

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

some generic invariant factor assignment results using dynamic output feedback abstract working with input output transfer functions in the frequency domain and exploiting a formulation involving generalized sylvester resultants we are able to derive necessary and sufficient conditions for generic invariant factor assignment in several cases using proper dynamic output feedback compensators

pole placement by static output feedback for generic linear systems we consider linear systems with m inputs p outputs and mcmillan degree n such that n mp if both m and p are even we show that there is a nonempty open in the usual topology subset u of such systems where the real pole placement map is not surjective it follows that for each system in u there exists an open set of pole configurations symmetric with respect to the real line which cannot be assigned by any real static output feedback

product states and c dynamical system of product type abstract a study is made of a special type of c dynamical system consisting of a class n uniformly hyperfinite c algebra a compact lie group and a natural product action of g on by automorphisms in particular a simple class of representations of the fixed point algebra g is examined those induced by restrictions of pure product states of sufficient conditions are obtained for 1 irreducibility and 2 unitary equivalence of such irreducible representations in the case of an arbitrary compact lie group g with continuous unitary representation in c n these are used to produce necessary and sufficient conditions when g u 1 or g su 2 acts by a product type conjugation on the class 2 algebra

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

regulation as an interpolation problem abstract the design of a controller such that the closed loop system will track reference signals or reject disturbance signals from a specified class is known as the servomechanism problem or the regulator problem we show here that the regulator problem can be looked at as an interpolation problem for a subspace valued function that can be viewed as a multivariable version of the nyquist curve the result is applied to obtain a simple parametrization of all solutions

regular and chaotic states in a local map description of sheared nematic liquid crystals we propose and study a local map capable of describing the full variety of dynamical states ranging from regular to chaotic obtained when a nematic liquid crystal is subjected to a steady shear flow the map is formulated in terms of a quaternion parametrization of rotations of the local frame described by the axes of the nematic director subdirector and the joint normal to these with two additional scalars describing the strength of ordering our model yields kayaking wagging tumbling aligned and coexistence states accommodated in a phase diagram which closely resembles phase diagrams obtained using representations of the dynamics which are based on ordinary differential equations we also study the behavior of the map under periodic perturbations of the shear rate such a map can serve as a building block for the construction of lattice models of the complex spatiotemporal states predicted for sheared nematics

grassmann invariants matrix pencils and linear system properties abstract the paper deals with the establishment of relationships between two different types of invariants defined on matrix pencils and polynomial matrices and highlights their significance in the context of linear systems for general singular matrix pencil sf g the grassmann invariants of the pencil are introduced as the column row grassmann representatives g c f g g r f g t and their corresponding plucker matrices p c f g p r f g the properties of these new invariants of matrix pencils are established these results provide alternative means for classifying the different families of matrix pencils which are important for the characterization of properties in linear systems finally the relationship between the grassmann plucker invariants of a general rational transfer function matrix and the system matrix pencil of a minimal realization is derived the latter results provide means for the computation of transfer function plucker matrices from state space descriptions as well as enable the linking of state space kronecker invariants to the plucker invariants of transfer functions

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

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

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

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

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

the construction of the sh lie algebra of closed bosonic strings we present a construction of the closed string algebra in terms of gaussian processes and crossed products also we give a purely functional analytical prove of the sh lie structure

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

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

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

generic pole assignment using dynamic output feedback a method is presented for assigning min n q q 1 m q closed loop poles of an nth order linear time invariant system m outputs l inputs m l using a linear time invariant proper output feedback compensator of order q it is also shown how the locations of remaining unassigned poles could be controlled the compensator parameters are obtained by solving a linear matrix equation this research has been supported by nsf under grant ecs 8006896 and partially supported by afosr under grant afosr 80 0155

output feedback invariants the paper is concerned with the problem of determining a complete set of invariants for output feedback using tools from geometric invariant theory it is shown that there exists a quasi projective variety whose points parameterize the output feedback orbits in a unique way if the mcmillan degree n mp the product of the number of inputs and number of outputs then it is shown that in the closure of every feedback orbit there is exactly one nondegenerate system

characters of the infinite symplectic group a riesz ring approach we classify both the finite and infinite characters of the inductive limit symplectic group g an important feature of our technique is the systematic use of a multiplicative structure on an ordered completion of the k0 group for the group c algebra a of g we also give explicit examples of the k theory for certain primitive quotients of a

generic eigenvalue assignment by memoryless real output feedback by extensive use of methods from algebraic geometry x wang proved that arbitrary pole placement by static output feedback is generically possible for strictly proper plants with n states m inputs and p outputs if n mp here we show the same result using no more of algebraic geometry than the definition of genericity

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

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

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

reduction theory for symmetry breaking with applications to nematic systems we formulate euler poincare and lagrange poincare equations for systems with broken symmetry we specialize the general theory to present explicit equations of motion for nematic systems ranging from single nematic molecules to biaxial liquid crystals the geometric construction applies to order parameter spaces consisting of either unsigned unit vectors directors or symmetric matrices alignment tensors on the hamiltonian side we provide the corresponding poisson brackets in both lie poisson and hamilton poincare formulations the explicit form of the helicity invariant for uniaxial nematics is also presented together with a whole class of invariant quantities casimirs for two dimensional incompressible flows

generalised resultants dynamic polynomial combinants and the minimal design problem the theory of dynamic polynomial combinants is linked to the linear part of the dynamic determinantal assignment problems dap which provides the unifying description of the dynamic as well as static pole and zero dynamic assignment problems in linear systems the assignability of spectrum of polynomial combinants provides necessary conditions for solution of the original dap this paper demonstrates the origin of dynamic polynomial combinants from linear systems examines issues of their representation and the parameterisation of dynamic polynomial combinants according to the notions of order and degree and examines their spectral assignment central to this study is the link of dynamic combinants to the theory of generalised resultants which provide the matrix representation of the dynamic combinants the paper considers the case of coprime set of polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above an

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

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

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

invariant spaces at infinity of linear systems application to block decoupling the dynamic state feedback block decoupling problem in the general case i e with singular or nonsingular input transformations is solved it is proven that the problem is solvable if the number of inputs is large enough the main tools for this study are some invariant spaces associated with the columns infinite behavior of a rational matrix it turns out that when the problem is solvable it is also solvable with stability

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

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

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

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

decentralized pole assignment and product grassmannians the pole assignment problems of linear systems by decentralized static output feedback are considered in this paper a compactification of decentralized static feedback space product grassmannians is introduced in this paper its degree under plucker segre embedding is computed sufficient conditions for arbitrary and almost arbitrary pole assignability are given it is also proved that the generic m times p system of mcmillan degree n has arbitrary pole assignability by r channel decentralized static output with m i inputs and p i outputs on the i th channel if sum i 1 r m ip i geq n when the degree of the product grassmannians is odd or if sum i 1 r m ip i n and the local channels have either the same numbers of inputs or the same numbers of outputs when the degree of the product grassmannians is even

the c algebra of bosonic strings we give a rigorous definition of witten sc string algebra to this end we present a new construction ofc algebras associated to special geometric situations kahler foliations and generalize this later construction to the string case through this we get a natural geometrical interpretation of the string of semi infinite forms as well as the fermionic algebra structure using the non commutative geometric concepts for investigating the string algebra we get a natural fredholm module representation of dimension 26

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

a necessary condition for pole assignment by constant output feedback in this paper we shall investigate the pole assignment problem by constant output feedback using transfer function techniques in particular a necessary condition is given for the existence of constant output feedback which assigns the eigenvalues of closed loop system to any desired positions

poisson bracket approach to the dynamics of nematic liquid crystals we use the general poisson bracket formalism for obtaining stochastic dynamical equations for slow macroscopic fields to derive the equations that govern the dynamics of nematic liquid crystals in both their nematic and isotropic phases for uniaxial molecules we calculate the poisson bracket between the tensorial nematic order parameter q and the momentum density g as well as those between all pairs of conserved quantities we show that the full nonlinear hydrodynamical equations for the nematic phase derived in this formalism are identical to the nonlinear ericksen leslie equations we also obtain the complete dynamical equations for the slow dynamics of the tensorial nematic order parameter q valid both in the isotropic and the nematic phase they differ from those obtained by other techniques only in the values of kinetic coefficients and in the number of nonlinear terms in q which are present

stabilizability of multivariable systems and the ljusternik snirel mann category of real grassmannians abstract in this paper a numeric criterion k m p n is given for stabilizability by constant gain output feedback of the generic linear multivariable system with m inputs n states and p outputs this criterion is defined in terms of a topological invariant of the space of gains both finite and infinite arising from an interpretation of the problem of stabilizability as a problem concerning open covers of this space this invariant originally considered by ljusternik and snirel mann in the calculus of variations can be estimated from below by using a theorem of eilenberg together with the methods of the schubert calculus thus leading to some very explicit corollaries concerning generic stabilizability as far as i am aware these corollaries cannot be derived from existing results concerning pole assignability after passing to a problem on coverings of the compactified gain space the main technical problem which remains in a high gain stability analysis stated here as a high gain lemma which appears to be of independent interest that is the topological argument implies that the closed loop system is stable for some gain possibly infinite if the root locus map were defined and continuous at all infinite gains then the conclusion that the system is stabilizable by finite gain could be deduced from a simple continuity argument however it is known that if mp n then the root locus map has points of discontinuity at infinity and since mp n is now known to be necessary for generic stabilizability in most cases of interest one requires a far more subtle argument than one would give for example in the scalar case

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

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

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

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

pole placement and matrix extension problems a common point of view this paper studies a general inverse eigenvalue problem which generalizes many well studied pole placement and matrix extension problems it is shown that the problem corresponds geometrically to a so called central projection from some projective variety the degree of this variety represents the number of solutions the inverse problem has in the critical dimension we are able to compute this degree in many instances and we provide upper bounds in the general situation

solution landscapes in nematic microfluidics we study the static equilibria of a simplified leslie ericksen model for a unidirectional uniaxial nematic flow in a prototype microfluidic channel as a function of the pressure gradient g and inverse anchoring strength b we numerically find multiple static equilibria for admissible pairs g b and classify them according to their winding numbers and stability the case g 0 is analytically tractable and we numerically study how the solution landscape is transformed as g increases we study the one dimensional dynamical model the sensitivity of the dynamic solutions to initial conditions and the rate of change of g and b we provide a physically interesting example of how the time delay between the applications of g and b can determine the selection of the final steady state

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

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

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

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

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

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

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

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

fractional representation algebraic geometry and the simultaneous stabilization problem an explicit relationship between the fractional representation approach feedback system design and the algebro geometric approach to system theory is formulated and used to derive a global solution to the feedback system problem these techniques are then applied to the simultaneous stabilization problem yielding a natural geometric criterion for a set of plants to be simultaneously stabilized by a single compensator

a bilinear formulation for the output feedback problem in linear systems this note ties together a geometric theory and previous algebraic results for the closed loop eigenstructure assignment by output feedback in multivariable systems necessary and sufficient conditions for the closed loop stabilization eigenstructure assignment are presented in terms of a bilinear sylvester equation the solutions of which completely characterize the closed loop eigenstructure problem with output feedback some preliminary results for the solution of this bilinear equation are presented

output feedback pole placement with dynamic compensators derives a new rank condition which guarantees the arbitrary pole assignability of a given system by dynamic compensators of degree at most q by using this rank condition the authors establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system the authors proofs also come with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed loop poles