on the optimal tuning of a robust controller for parabolic distributed parameter systems abstract optimal tuning of robust multivariable controllers for stable parabolic distributed parameter systems is discussed in order to evaluate the behaviour of the closed loop system a quadratic criterion is selected then it is proved that the system may be decomposed into slow and fast subsystems and that the value of the criterion will mainly depend on the slow subsystem tuning of the controller for this subsystem turns out to be a standard finite dimensional quadratic regulator problem two examples are given to show the improvement due to the new selection of the control parameters

survey paper transfer functions of distributed parameter systems a tutorial in this tutorial article the rich variety of transfer functions for systems described by partial differential equations is illustrated by means of several examples under various boundary conditions an important feature is the strong influence of the choice of boundary conditions on the dynamics and on system theoretic properties such as pole and zero locations properness relative degree and minimum phase it is sometimes possible to design a controller using the irrational transfer function and several such techniques are outlined more often the irrational transfer function is approximated by a rational one for the purpose of controller design various approximation techniques and their underlying theory are briefly discussed

maximal cohen macaulay modules over the cone of an elliptic curve let r k y1 y2 y3 f f y13 y23 y33 where k is an algebraically closed field with chark 3 using atiyah bundle classification over elliptic curves we describe the matrix factorizations of the graded indecomposable reflexive r modules equivalently we describe explicitly the indecomposable bundles over the projective curve v f pk2 using the fact that over the completion r of r every reflexive module is gradable we obtain a description of the maximal cohen macaulay modules over r k y1 y2 y3 f

robust controller for systems with exponentially stable strongly continuous semigroups abstract the theory of robust controllers is extended to the case where we have boundary and or distributed control and the system operator is an infinitesimal generator of a strongly continuous exponentially stable semigroup an example on hyperbolic systems is presented

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

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

what periodic signals can an exponentially stabilizable linear feedforward control system asymptotically track we study asymptotic tracking and rejection of continuous periodic signals in the context of exponentially stabilizable linear infinite dimensional systems our reference signals are in sobolev type spaces h omega n f n and they as well as the disturbance signals are generated by an infinite dimensional exogenous system we show that there exists a feedforward controller which achieves output regulation if and only if the so called regulator equations are satisfied and a decomposability condition holds for siso systems this result allows us to completely answer the question posed in the title we show that if the stabilized plant does not have transmission zeros at the frequencies i omega n of the reference signals then all reference signals in h omega n f n can be asymptotically tracked in the presence of disturbances if and only if bigl h k i omega n 1 1 h d n f n 1 bigr n in i in ell 2 here h k i omega n n in i is the transfer function of the stabilized plant evaluated at i omega n and h d n n in i is a sequence of disturbance coefficients for the stabilized plant moreover the sequence f n n in i consists of weights for the fourier coefficients of the reference signals we give four examples to illustrate the theory

obstructions to determinantal representability there has recently been ample interest in the question of which sets can be represented by linear matrix inequalities lmis a necessary condition is that the set is rigidly convex and it has been conjectured that rigid convexity is also sufficient to this end helton and vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices we disprove this conjecture by relating the question of finding lmi representations to the problem of determining whether a polymatroid is representable over the complex numbers we find a real zero polynomial such that no power of it admits a determinantal representation the proof uses recent results of wagner and wei on matroids with the half plane property and the polymatroids associated to hyperbolic polynomials introduced by gurvits

hyperbolic polynomials and interior point methods for convex programming hyperbolic polynomials have their origins in partial differential equations we show in this paper that they have applications in interior point methods for convex programming each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone we give an explicit representation of this cone in terms of polynomial inequalities the function fx log px is a logarithmically homogeneous self concordant barrier function for the hyperbolicity cone with barrier parameter equal to the degree of p the function fx possesses striking additional properties that are useful in designing long step interior point methods for example we show that the long step primal potential reduction methods of nesterov and todd and the surface following methods of nesterov and nemirovskii extend to hyperbolic barrier functions we also show that there exists a hyperbolic barrier function on every homogeneous cone

determinantal representations of singular hypersurfaces in pn abstract a global determinantal representation of projective hypersurface x p n is a matrix whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface we study the properties of such representations for singular possibly reducible or non reduced hypersurfaces in particular we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces further we classify the determinantal representations in terms of the corresponding kernel sheaves on x finally we extend the results to the case of symmetric self adjoint representations with implications to hyperbolic polynomials and the generalized lax conjecture

the cayley hamilton theorem and inverse problems for multiparameter systems abstract we review some of the current research in multiparameter spectral theory we prove a version of the cayley hamilton theorem for multiparameter systems and list a few inverse problems for such systems some consequences of results on determinantal representations proved by dixon dickson and vinnikov for the inverse problems are discussed

complete description of determinantal representations of smooth irreducible curves abstract determinantal representations of algebraic curves are interesting in themselves and their classification is equivalent to the simultaneous classification of triples of matrices we present a complete description of determinantal representations of smooth irreducible curves over any algebraically closed field we used the notion of the class of divisors of the vector bundle corresponding to a determinantal representation we prove that two determinantal representations of a smooth curve f are equivalent if and only if the classes of divisors of the corresponding vector bundles coincide we give a precise characterization of those classes of divisors that arise from vector bundles corresponding to determinantal representations of f then we obtain a parametrization of determinantal representations of f up to equivalence by the points of the jacobian variety of f not on some exceptional subvariety in particular it follows that any smooth curve of order 3 or greater possesses an infinite number of nonequivalent determinantal representations we also specialize our results to symmetrical and self adjoint representations

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

co evolving prisoner s dilemma performance indicators and analytic approaches understanding the intrinsic relation between the dynamical processes in a co evolving network and the necessary ingredients in formulating a reliable theory is an important question and a challenging task using two slightly different definitions of performance indicator in the context of a co evolving prisoner s dilemma game it is shown that very different cooperative levels result and theories of different complexity are required to understand the key features when the payoff per opponent is used as the indicator case a non cooperative strategy has an edge and dominates in a large part of the parameter space formed by the cutting and rewiring probability and the strategy imitation probability when the payoff from all opponents is used case b cooperative strategy has an edge and dominates the parameter space two distinct phases one homogeneous and dynamical and another inhomogeneous and static emerge and the phase boundary in the parameter space is studied in detail a simple theory assuming an average competing environment for cooperative agents and another for non cooperative agents is shown to perform well in case a the same theory however fails badly for case b it is necessary to include more spatial correlation into a theory for case b we show that the local configuration approximation which takes into account of the different competing environments for agents with different strategies and degrees is needed to give reliable results for case b the results illustrate that formulating a proper theory requires both a conceptual understanding of the effects of the adaptive processes in the problem and a delicate balance between simplicity and accuracy

livsic type determinantal representations and hyperbolicity abstract hyperbolic homogeneous polynomials with real coefficients i e hyperbolic real projective hypersurfaces and their determinantal representations play a key role in the emerging field of convex algebraic geometry in this paper we consider a natural notion of hyperbolicity for a real subvariety x p d of an arbitrary codimension l with respect to a real l 1 dimensional linear subspace v p d and study its basic properties we also consider a class of determinantal representations that we call livsic type and a nice subclass of these that we call very reasonable much like in the case of hypersurfaces l 1 the existence of a definite hermitian very reasonable livsic type determinantal representation implies hyperbolicity we show that every curve admits a very reasonable livsic type determinantal representation our basic tools are cauchy kernels for line bundles and the notion of the bezoutian for two meromorphic functions on a compact riemann surface that we introduce we then proceed to show that every real curve in p d hyperbolic with respect to some real d 2 dimensional linear subspace admits a definite hermitian or even definite real symmetric very reasonable livsic type determinantal representation

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

on the structure at infinity of linear square decoupled systems the purpose of this note is to demonstrate that a square invertible linear system c a b is decouplable by static state feedback if and only if its infinite zero orders are respectively equal to the infinite zero orders of the subsystems c i a b where c i denotes the i th row of c until now only the necessity of this condition has been established 1

robust multivariable pi controller for infinite dimensional systems a robust multivariable controller is introduced for a class of distributed parameter systems the system to be controlled is given as dot x ax bu y cx in a banach space the purpose of the control which is based on the measurement y is to stabilize and regulate the system so that y t rightarrow y r as t rightarrow infty where y r is a constant reference vector under the assumptions that operator a generates a holomorphic stable semigroup b is linear and bounded c is linear and a bounded and the input and output spaces are of the same dimension a necessary and sufficient condition is found for the existence of a robust multivariable controller this controller appears to be a multivariable pi controller also a simple necessary criterion for the existence of a decentralized controller is derived the tuning of the controller is discussed and it is shown that the i part of the controller can be tuned on the basis of step responses without exact knowledge of the system s parameters the presented theory is then used as an example to control the temperature profile of a bar with the dirichlet boundary conditions

robust regulation of stable systems in the h algebra in this paper robust regulation of stable infinite dimensional plants with transfer functions in the h algebra is considered the reference and disturbance signals are allowed to have an infinite number of poles on the imaginary axis which makes it possible to consider e g arbitrary periodic signals a controller depending on a positive scalar e and design parameters kk is proposed the design parameters are matrices over h and by the main result of the paper if the design parameters satisfy a certain condition the proposed controller is a robust regulator for sufficiently small values of e the only knowledge needed to verify this condition is the so called stability of the plant and values of the plant transfer function at the poles of the reference and disturbance signals the analysis in this paper is in the frequency domain and it is based on the fractional representation approach

infinite structure and exact model matching problem a geometric approach we first generalize to a b c d quadruples the geometric characterization of the infinite zero structure we then give a new geometric proof for a solvability condition of the exact model matching problem expressed in terms of this concept and previously established in the polynomial context by morse verghese and pernebo this emphasizes the approach independent character of the infinite zero structure

structure and smith macmillan form of a rational matrix at infinity the use of special row and column operations in the reduction of a rational matrix to its smith macmillan form at infinity is investigated the connections between this reduction procedure and the valuation approach are established a graphical method for finding the smith macmillan form of a rational matrix at infinity from its bode magnitude array and some new results on realization theory for polynomial matrices are presented

on the characteristic polynomials of multiparameter pencils abstract in this paper we study the characteristic polynomial of multiparameter pencil z 1 a 1 z 2 a 2 z s a s the main theorem states that a unitary representation of a finitely generated group contains a one dimensional representation if and only if the characteristic polynomial of its generators contains a linear factor it follows that a two or three dimensional unitary representation of a finitely generated group is irreducible if and only if the characteristic polynomial of the pencil of its generators is irreducible the result is of kin to the dedekind and frobenius theorem on finite group determinant

internal model based tracking and disturbance rejection for stable well posed systems in this paper we solve the tracking and disturbance rejection problem for infinite dimensional linear systems with reference and disturbance signals that are finite superpositions of sinusoids we explore two approaches both based on the internal model principle in the first approach we use a low gain controller and here our results are a partial extension of results by hamalainen and pohjolainen in their papers the plant is required to have an exponentially stable transfer function in the callier desoer algebra while in this paper we only require the plant to be well posed and exponentially stable these conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable our second approach concerns the case when the second component of the plant transfer function from control input to tracking error is positive in this case we identify a very simple stabilizing controller which is again an internal model but which does not require low gain we apply our results to two problems involving systems modeled by partial differential equations the problem of rejecting external noise in a model for structure acoustics interactions and a similar problem for two coupled beams

de branges spaces on compact riemann surfaces and a beurling lax type theorem abstract using the notion of commutative operator vessels this work investigates de branges rovnyak spaces whose elements are sections of a line bundle of multiplicative half order differentials on a compact real riemann surface as a special case we obtain a beurling lax type theorem in the setting of the corresponding hardy space on a finite bordered riemann surface

theta characteristics on algebraic curves the theory of theta characteristics is developed algebraically so that it may be applied to possibly singular and or reducible algebraic curves the configuration of theta characteristics on a curve is described in terms of its singularities with applications to the geometry of plane quartic curves and the problem of appolonius some results on gorenstein local rings are appended 0 introduction the theory of theta characteristics has been studied from different points of view and often inadvertently cf for example the 19th century literature on bitangents to smooth plane quartic curves e g 4 and 10 until recently however virtually all treatments of theta characteristics per se involved the transcendental theory of the theta function associated to a compact riemann surface then in 1971 mumford in 8 gave a completely algebraic treatment of the subject formulating the main results and proving them without reference to the transcendental theory this in turn suggested that theta characteristics could be studied in a more general context for example on possibly singular and or reducible algebraic curves in this paper we undertake such an analysis in 1 following mumford we establish some of the basic facts about theta characteristics algebraically so as to apply to more general curves in fact 1 10 i and 1 13 in their present generality are due to mumford while he assumes throughout 8 that his curve x is smooth his proof seems to apply as given the argument for 1 13 here is somewhat different in order to make the proof more elementary and selfcontained one bonus is the result 1 11 on the subvariety of the moduli space 9thg consisting of curves with large semicanonical linear series this fact seemingly can be proved only via the algebraic set up in 2 we investigate the number and configuration of odd and even theta characteristics on a singular curve the results are expressed in the two main theorems 2 12 and 2 22 giving a more or less complete answer to these questions in terms of the singularities of the curve interestingly a recurrent motif in these arguments is one introduced by mumford in 8 the behavior of the maximal isotropic subspaces for a nondegenerate quadratic form on an even dimensional vector space here the mainspring of theorems 1 13 2 12 and 2 22 is assertion 1 4 below that the sum of the dimensions of the pairwise intersections of any three such subspaces is congruent to n mod 2 received by the editors july 11 1980 and in revised form october 22 1981 1980 mathematics subject classification primary 14h99 14h30 secondary 14h20 14h40 14k25 research partially supported by nsf grant mcs78 04008 aol dl 982 american mathematical society 0002 9947 8 1 0000 0371 10 25