algebraic methods for interactive proof systems a new algebraic technique for the construction of interactive proof systems is presented our technique is used to prove that every language in the polynomial time hierarchy has an interactive proof system this technique played a pivotal role in the recent proofs that ip pspace 28 and that mip nexp 4

the largest singular value of e sup x a sub 0 e sup x is convex on convex sets of commuting matrices a short and direct proof of the convexity property is given it is shown that the theorem applies to any convex commuting set of matrices in r sup nxn where a sub 0 in r sup nxn is fixed it is also shown that the result does not hold if x is permitted to be a general square matrix a counterexample is supplied for noncommuting matrices

locating p poly optimally in the extended low hierarchy the low hierarchy in np and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes and almost all of them could be located optimally therein however until now the exact location of p poly remained open

optimal h robustness and the relationship to lqg design problems a solution to the hx optimal control problem is obtained by employing a standard lqg polynomial based solution procedure the derivation is straightforward and avoids many of the more abstract mathematical results usually employed the relationship between the lqg and hm design methods becomes transparent and the similarities and differences can be identified

relative entropy and the multivariable multidimensional moment problem entropy like functionals on operator algebras have been studied since the pioneering work of von neumann umegaki lindblad and lieb the best known are the von neumann entropy rho trace rhologrho and a generalization of the kullback leibler distance s rhoparsigma trace rhologrho rhologsigma referred to as quantum relative entropy and used to quantify distance between states of a quantum system the purpose of this paper is to explore and s as regularizing functionals in seeking solutions to multivariable and multidimensional moment problems it will be shown that extrema can be effectively constructed via a suitable homotopy the homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems this is accomplished by a renormalization of a riemannian metric induced by entropy functionals as an application we discuss the inverse problem of describing power spectra which are consistent with second order statistics which has been the main motivation behind the present work

analytic interpolation with a degree constraint for matrix valued functions we consider a nehari problem for matrix valued positive real functions and characterize the class of generically minimal degree solutions analytic interpolation problems such as the one studied herein for positive real functions arise in time series modeling and system identification the degree of positive real interpolants relates to the dimension of models and to the degree of matricial power spectra of vector valued time series the main result of the paper generalizes earlier results in scalar analytic interpolation with a degree constraint where the class of generically minimal degree solutions is characterized by an arbitrary choice of spectral zeros naturally in the current matricial setting there is freedom in assigning the jordan structure of the spectral zeros of the power spectrum i e the spectral zeros as well as their respective invariant subspaces the characterization utilizes rosenbrock s theorem on assignability of dynamics via linear state feedback

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

a complete parameterization of all positive rational extensions of a covariance sequence in this paper we formalize the observation that filtering and interpolation induce complementary or dual decompositions of the space of positive real rational functions of degree less than or equal to n from this basic result about the geometry of the space of positive real functions we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence on the one hand by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space we are able to develop estimates on the asymptotic behavior of the schur parameters 1918 of positive rational extensions on the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence indeed motivated by its application to signal processing speech processing and stochastic realization theory this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by georgiou 1983 1987 our basic result however also enables us to analyze the robustness of this parameterization with respect to variations in the problem data the methodology employed is a combination of complex analysis geometry linear systems and nonlinear dynamics

bounded queries approximations and the boolean hierarchy this paper investigates nondeterministic bounded query classes in relation to the complexity of np hard approximation problems and the boolean hierarchy nondeterministic bounded query classes turn out be rather suitable for describing the complexity of np hard approximation problems the results in this paper take advantage of this machine based model to prove that in many cases np approximation problems have the upward collapse property that is a reduction between np approximation problems of apparently different complexity at a lower level results in a similar reduction at a higher level for example if maxclique reduces to log n approximating maxclique using many one reductions then the traveling salesman problem tsp is equivalent to maxclique under many one reductions several upward collapse theorems are presented in this paper the proofs of these theorems rely heavily on the machinery provided by the nondeterministic bounded query classes in fact these results depend on a surprising connection between the boolean hierarchy and nondeterministic bounded query classes

strong and robustly strong polynomial time reducibilities to sparse sets reducibility defined by oracle strong nondeterministic machines is studied two definitions of relativized strength are presented and separated the corresponding reduction classes of the sparse sets are characterized in terms of nonuniform complexity classes an oracle restricted positive relativization of the probabilistic class zpp is developed

structured noncommutative multidimensional linear systems we introduce a class of multidimensional linear systems with evolution along a free semigroup the transfer function for such a system is a formal power series in noncommuting indeterminates standard system theoretic properties the operations of cascade parallel connection and inversion controllability observability kalman decomposition state space similarity theorem minimal state space realizations hankel operators realization theory are developed for this class of systems we also draw out the connections with the much earlier studied theory of rational and recognizable formal power series applications include linear fractional models for classical discrete time systems with structured time varying uncertainty dimensionless formulas in robust control multiscale systems and automata theory and the theory of formal languages

nonuniform proof systems a new framework to describe nonuniform and probabilistic complexity classes the concept of non uniform proof systems is introduced this notion allows a uniform description of non uniform complexity classes 10 probabilistic classes e g bpp 8 15 27 32 am 2 3 and language classes defined by simultaneous non uniform and nondeterministic time bounds non uniform proof systems provide a better understanding of many results concerning these classes particularly their connections to uniform complexity measures we give an uniform approach to lowness results 19 20 for various complexity classes for instance we show that co np poly np is contained in the third level of the low hierarchy and that np c np co np poly implies that the polynomial time hierarchy collapses to its second level see also 1 finally some evidence is given that the low hierarchy cannot be extended beyond its third level by the current techniques

complexity classes without machines on complete languages for up this paper develops techniques for studying complexity classes that are not covered by known recursive enumerations of machines often counting classes probabilistic classes and intersection classes lack such enumerations concentrating on the counting class up we show that there are relativizations for which upa has no complete languages and other relativizations for which pb upb npb and upb has complete languages among other results we show that p up if and only if there exists a set s in p of boolean formulas with at most one satisfying assignment such that s sat is not in p p up coup if and only if there exists a set s in p of uniquely satisfiable boolean formulas such that no polynomial time machine can compute the solutions for the formulas in s if up has complete languages then there exists a set r in p of boolean formulas with at most one satisfying assignment so that sat r is complete for up finally we indicate the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class bpp there is a relativized world where bppa has no complete languages if bpp has complete languages then it has a complete language of the form b majority where b p and majority f f is true for at least half of all assignments is the canonical pp complete set

on the uniqueness result of theorem 6 in relative entropy and the multivariable multidimensional moment problem matrix valued covariance extension and multivariate spectral estimation are formulated as generalized moment problems in the three approach and its extensions under this context we discuss theorem 6 in the title paper concerning the bijectivity of a moment map defined over a parametric family of spectral densities in particular we provide a counterexample in which the moment map under consideration is shown to have a critical point namely a point at which the jacobian loses rank then with standard techniques in bifurcation theory we conclude further that the computed critical point is a bifurcation point which means that the moment map is not injective

new collapse consequences of np having small circuits we show that if a self reducible set has polynomial size circuits then it is low for the probabilistic class zpp np as a consequence we get a deeper collapse of the polynomial time hierarchy ph to zpp np under the assumption that np has polynomial size circuits this improves on the well known result in karp and lipton proceedings of the 12th acm symposium on theory of computing acm press new york 1980 pp 302 309 stating a collapse of ph to its second level sigmap 2 under the same assumption furthermore we derive new collapse consequences under the assumption that complexity classes like up fewp and c p have polynomial size circuits finally we investigate the circuit size complexity of several language classes in particular we show that for every fixed polynomial s there is a set in zpp np which does not have o s n size circuits

optimal model matching control for time delay systems model matching control for time delay systems is formulated as an optimization problem in h2 norm space as well as h norm space wiener hopf techniques and the theory of optimal interpolation are successfully employed to treat this model matching problem including perfect and imperfect model matching cases prior to this work for time delay systems only the problem of perfect model matching control had been considered and no attempt had been made to examine that of imperfect model matching control with the optimization approach presented here an optimal model matching control scheme is constructed for the imperfect model matching case to produce perfect model matching results whenever possible

the caratheodory fejer pisarenko decomposition and its multivariable counterpart when a covariance matrix with a toeplitz structure is written as the sum of a singular one and a positive scalar multiple of the identity the singular summand corresponds to the covariance of a purely deterministic component of a time series whereas the identity corresponds to white noise this is the caratheacuteodory fejeacuter pisarenko cfp decomposition in the present paper we study multivariable analogs for block toeplitz matrices as well as for matrices with the structure of state covariances of finite dimensional linear systems which include block toeplitz ones to this end we develop theory which addresses questions of existence uniqueness and realization of multivariable power spectra possibly having deterministic components we characterize state covariances which admit only a deterministic input power spectrum and we explain how to realize multivariable power spectra which are consistent with singular state covariances via decomposing the contribution of the singular part we then show that multivariable decomposition of a state covariance in accordance with a deterministic component white noise hypothesis for the input does not exist in general we finally reinterpret the cfp dictum and consider replacing the scalar multiple of the identity by a covariance of maximal trace which is admissible as a summand the summand can be either block diagonal corresponding to white noise or have a short range correlation structure corresponding to a moving average component the trace represents the maximal variance energy that can be accounted for by a process at the input e g noise with the aforementioned structure and this maximal solution can be computed via convex optimization the decomposition of covariances and spectra according to the range of their time domain correlations is an alternative to the cfp dictum with potentially great practical significance

sampled data control of continuous time systems with an h optimality criterion abstract the synthesis of sampled data controllers for continuous time systems is studied in the context of h control theory the sampled data h optimal control problem consists of finding a sampled data controller which minimizes the l2 induced norm from the continuous time disturbance to the continuous time output the problem is here solved via a discrete system representation this is achieved by describing the closed loop system as a discrete system with finite state dimension and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite dimensional spaces via finite rank approximations of a compact integral operator associated with the system the sampled data h control problem can be solved to any degree of accuracy from the structure of the problem it follows that irrespective of the order of the approximation a feasible sampled data controller can be found which has the same order as the system

paper adaptive controllers for discrete time systems with arbitrary zeros an overview the paper presents a comprehensive review of the problem of designing discrete time adaptive controllers for systems with arbitrary zeros minimum or non minimum phase systems being particular cases the authors have attempted to give a unified presentation of the various existing schemes by stripping away the technical details and highlighting their close interrelation through the use of a common framework both algorithmic and theoretical aspects are discussed with particular emphasis on the stability of the schemes

likelihood analysis of power spectra and generalized moment problems we develop an approach to the spectral estimation that has been advocated by a ferrante et al time and spectral domain relative entropy a new approach to multivariate spectral estimation ieee trans autom control vol 57 no 10 pp 2561 2575 oct 2012 and in the context of the scalar valued covariance extension problem by p enqvist and j karlsson minimal itakura saito distance and covariance interpolation in proc 47th ieee conf decision control 2008 pp 137 142 the aim is to determine the power spectrum that is consistent with given moments and minimizes the relative entropy between the probability law of the underlying gaussian stochastic process to that of a prior the approach is analogous to the framework of earlier work by byrnes georgiou and lindquist and can also be viewed as a generalization of the classical work by burg and jaynes on the maximum entropy method in this paper we present a new fast algorithm in the general case i e for general gaussian priors and show that for priors with a specific structure the solution can be given in closed form

logarithmic advice classes abstract karp and lipton 1980 introduced the notion of nonuniform complexity classes where a certain amount of additional information the advice is given for free the advice only depends on the length of the input karp and lipton initiated the study of classes with either logarithmic or polynomial advice however later yap 1983 schoning 1984 balcazar 1987 and ko and schoning 1985 concentrated on the study of classes of the form c poly where c is p np or pspace and poly denotes a polynomial size advice this paper considers classes of the form c log as a main result it is shown that in the context of an np log computation log bounded advice is equivalent to a sparse oracle in np in contrast it has been shown that a poly bounded advice to an arbitrary sparse oracle set furthermore a general theorem is presented that generalizes karp and lipton s round robin tournament method

how much uncertainty can be dealt with by feedback feedback is used primarily for reducing the effects of the plant uncertainty on the performance of control systems and as such understanding the following questions is of fundamental importance how much uncertainty can be dealt with by feedback what are the limitations of feedback how does the feedback performance depend quantitatively on the system uncertainty how can the capability of feedback be enhanced if a priori information about the system structure is available as a starting point toward answering these questions a typical class of first order discrete time dynamical control systems with both unknown nonlinear structure and unknown disturbances is selected for our investigation and some concrete answers are obtained in the paper in particular we find that in the space of unknown nonlinear functions the generalized lipschitz norm is a suitable measure for characterizing the size of the structure uncertainty and that the maximum uncertainty that can be dealt with by the feedback mechanism is described by a ball with radius 3 2 spl radic 2 in this normed function space

if np has polynomial size circuits then ma am abstract it is shown that the assumption of np having polynomial size circuits implies apart from a collapse of the polynomial time hierarchy as shown by karp and lipton that the classes am and ma of babai s arthur merlin hierarchy coincide this means that also a certain inner collapse of the remaining classes of the polynomial time hierarchy occurs

generalized interpolation in h infty with a complexity constraint in a seminal paper sarason generalized some classical interpolation problems for h infinity functions on the unit disc to problems concerning lifting onto h 2 of an operator t that is defined on k

tunable line spectral estimators based on state covariance subspace analysis subspace methods for spectral analysis can be adapted to the case where state covariance of a linear filter replaces the traditional toeplitz matrix formed out of a partial autocorrelation sequence of a time series this observation forms the basis of a new framework for spectral analysis the goal of this paper is to quantify potential advantages in working with state covariance data instead of the autocorrelation sequence to this end we identify tradeoffs between resolution and robustness in spectral estimates and how these are affected by the filter dynamics the approach leads to a novel tunable high resolution frequency estimator

a convex optimization approach to the rational covariance extension problem in this paper we present a convex optimization problem for solving the rational covariance extension problem given a partial covariance sequence and the desired zeros of the modeling filter the poles are uniquely determined from the unique minimum of the corresponding optimization problem in this way we obtain an algorithm for solving the covariance extension problem as well as a constructive proof of georgiou s seminal existence result and his conjecture a stronger version of which we have resolved in byrnes et al ieee trans automat control ac 40 1995 pp 1841 1857

a new approach to spectral estimation a tunable high resolution spectral estimator traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates here we present a new approach to spectral estimation which is based on the use of filter banks as a means of obtaining spectral interpolation data such data replaces standard covariance estimates a computational procedure for obtaining suitable pole zero arma models from such data is presented the choice of the zeros ma part of the model is completely arbitrary by suitable choices of filter bank poles and spectral zeros the estimator can be tuned to exhibit high resolution in targeted regions of the spectrum

on the well posedness of a parametric spectral estimation problem and its numerical solution this paper concerns a spectral estimation problem in which we want to find a spectral density function that is consistent with estimated second order statistics it is an inverse problem admitting multiple solutions and selection of a solution can be based on prior functions we show that the problem is well posed when formulated in a parametric fashion and that the solution parameter depends continuously on the prior function in this way we are able to obtain a smooth parametrization of admissible spectral densities based on this result the problem is reparametrized via a bijective change of variables out of a numerical consideration and then a continuation method is used to compute the unique solution parameter numerical aspects such as convergence of the proposed algorithm and certain computational procedures are addressed a simple example is provided to show the effectiveness of the algorithm

competing provers yield improved karp lipton collapse results via competing provers we show that if a language a is self reducible and has polynomial size circuits then s2a s2 building on this we strengthen the kamper afk theorem namely we prove that if np np conp poly then the polynomial hierarchy collapses to s2np conp we also strengthen yap s theorem namely we prove that if np conp poly then the polynomial hierarchy collapses to s2np under the same assumptions the best previously known collapses were to zppnp and zppnpnp respectively siam journal on computing 28 1 1998 311 journal of computer and system sciences 52 3 1996 421 building on proceedings of the 12th acm symposium on theory of computing acm press new york 1980 pp 302 309 journal of computer and system sciences 39 1989 21 theoretical computer science 85 2 1991 305 theoretical computer science 26 3 1983 287 it is known that s2 zppnp proceedings of the 42nd ieee symposium on foundations of computer science ieee computer society press silver spring md 2001 pp 620 629 that result and its relativized version show that our new collapses indeed improve the previously known results the kamper afk theorem and yap s theorem are used in the literature as bridges in a variety of results ranging from the study of unique solutions to issues of approximation and so our results implicitly strengthen those results

robustness of probabilistic computational complexity classes under definitional perturbations various types of probabilistic algorithms play an increasingly important role in computer science especially in computational complexity theory probabilistic polynomial time complexity classes are defined and compared to each other emphasizing some structural relationships to the known complexity classes p np pspace the classes r and zpp corresponding to the so called las vegas polynomial time bounded algorithms are given special attention it is shown that many definitions that arise naturally from different types of algorithms are equivalent in defining the same class r or zpp resp these robustness results justify finally the tractability of the above probabilistic classes

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

counting classes are at least as hard as the polynomial time hierarchy in this paper it is shown that many natural counting classes such as pp c p and text mod k text p are at least as computationally hard as ph the polynomial time hierarchy in the following sense for each bf k of the counting classes above every set in bf k ph is polynomial time randomized many one reducible to a set in bf k with two sided exponentially small error probability as a consequence of the result it is seen that all the counting classes above are computationally harder than ph unless ph collapses to a finite level some other consequences are also shown

weight selection in feedback design with degree constraints we present an approach for feedback design which is based on recent developments in analytic interpolation with a degree constraint performance is cast as an interpolation problem with bounded analytic functions minimizers of a certain weighted entropy functional provide interpolants having degree less than the number of constraints the choice of weight parameterizes all such bounded degree solutions however the relationship between the weights and the shape of corresponding transfer functions is not direct thus in this paper we develop a formalism that guides weight selection

matrix valued nevanlinna pick interpolation with complexity constraint an optimization approach over the last several years a new theory of nevanlinna pick interpolation with complexity constraint has been developed for scalar interpolants in this paper we generalize this theory to the matrix valued case also allowing for multiple interpolation points we parameterize a class of interpolants consisting of most interpolants of no higher degree than the central solution in terms of spectral zeros this is a complete parameterization and for each choice of interpolant we provide a convex optimization problem for determining it this is derived in the context of duality theory of mathematical programming to solve the convex optimization problem we employ a homotopy continuation technique previously developed for the scalar case these results can be applied to many classes of engineering problems and to illustrate this we provide some examples in particular we apply our method to a benchmark problem in multivariate robust control by constructing a controller satisfying all design specifications but having only half the mcmillan degree of conventional h sup spl infin controllers we demonstrate the advantage of the proposed method

output feedback decentralized control of large scale systems using weighted sensitivity functions minimization this paper considers the problem of achieving stability and desired dynamical transient behavior for linear large scale systems by decentralized control it can be done by making the effects of the interconnections between the subsystems arbitrarily small sufficient conditions for stability and diagonal dominance of the closed loop system are introduced these conditions are in terms of decentralized subsystems and directly make a constructive h control design possible a mixed h pole region placement is suggested such that by assigning the closed loop eigenvalues of the isolated subsystems appropriately the eigenvalues of the overall closed loop system are assigned in desirable range the designs are illustrated by an example

h control with transients in h infty or uniformly optimal control problems it is usually assumed that the system initial conditions are zero in this paper an h infty like control problem that incorporates uncertainty in initial conditions is formulated this is done by defining a worst case performance measure both finite and infinite horizon problems are considered necessary and sufficient conditions are derived for the existence of controllers that yield a closed loop system for which the above mentioned performance measure is less than a prespecified value state space formulae for the controllers are also presented

computing solutions uniquely collapses the polynomial hierarchy is there an np function that when given a satisfiable formula as input outputs one satisfying assignment uniquely that is can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments we show that if there is such a nondeterministic function then the polynomial hierarchy collapses to zppnp and thus in particular to npnp as the existence of such a function is known to be equivalent to the statement every np function has an np refinement with unique outputs our result provides the strongest evidence yet that np functions cannot be refined we prove our result via a result of independent interest we say that a set a is npsv selective npmv selective if there is a 2 ary partial np function with unique values a 2 ary partial np function that decides which of its inputs if any is more likely to belong to a this is a nondeterministic analog of the recursion theoretic notion of the semi recursive sets and the extant complexity theoretic notion of p selectivity our hierarchy collapse result follows by combining foo the easy observation that every set in np is npmv selective with the following result if a in np is npsv selective then a in np cap conp poly relatedly we prove that if a in np is npsv selective then a is low 2 we prove that the polynomial hierarchy collapses even further namely to np if all conp sets are npmv selective this follows from a more general result we prove every self reducible npmv selective set is in np

rational hyperholomorphic functions in r4 we introduce the notion of rationality for hyperholomorphic functions functions in the kernel of the cauchy fueter operator following the case of one complex variable we give three equivalent definitions the first in terms of cauchy kovalevskaya quotients of polynomials the second in terms of realizations and the third in terms of backward shift invariance also introduced and studied are the counterparts of the arveson space and blaschke factors

brief paper reliable stabilization using a multi controller configuration given a plant p we consider the problem of designing a pair of controllers c 1 and c 2 such that their sum stabilizes p and in addition each of them also stabilizes p should the other one fail this is referred to as the reliable stabilization problem it is shown that every strongly stabilizable plant can be reliably stabilized moreover one of the two controllers can be specified arbitrarily subject only to the constraint that it should be stable the stabilization technique is extended to reliable regulation

a generalized entropy criterion for nevanlinna pick interpolation with degree constraint we present a generalized entropy criterion for solving the rational nevanlinna pick problem for n 1 interpolating conditions and the degree of interpolants bounded by n the primal problem of maximizing this entropy gain has a very well behaved dual problem this dual is a convex optimization problem in a finite dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n the criterion requires a selection of a monic schur polynomial of degree n it follows that this class of monic polynomials completely parameterizes all such rational interpolants and it therefore provides a set of design parameters for specifying such interpolants the algorithm is implemented in a state space form and applied to several illustrative problems in systems and control namely sensitivity minimization maximal power transfer and spectral estimation

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

right half plane poles and zeros and design tradeoffs in feedback systems this paper expresses limitations imposed by right half plane poles and zeros of the open loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed loop system the limitations are determined by integral relationships which must be satisfied by these functions the integral relationships are interpreted in the context of feedback design

the structure of state covariances and its relation to the power spectrum of the input we study the relationship between power spectra of stationary stochastic inputs to a linear filter and the corresponding state covariances and identify the structure of positive semidefinite matrices that qualify as state covariances of the filter this structure is best revealed by a rank condition pertaining to the solvability of a linear equation involving the state covariance and the system matrices we then characterize all input power spectra consistent with any specific state covariance the parametrization of input spectra is achieved through a relation to solutions of an analytic interpolation problem which is analogous but not equivalent to a matricial nehari problem

relativizing complexity classes with sparse oracles baker gill and solovay constructed sparse sets a and b such that p a np a and np b co np b in contrast to their results we prove that p np if and only if for every tally language t p t np t and that np co np if and only if for every tally language t np t co np t we show that the polynomial hierarchy collapses if and only if there is a sparse set s such that the polynomial hierarchy relative to s collapses similar results are obtained for several other complexity classes

h2 optimal control of linear systems with tracking disturbance rejection constraints this paper considers the design of two degree of freedom linear multivariable controllers for four block matrix transfer functions with objectives pertaining to both random and deterministic signals the former are taken into account in the cost functional whereas the latter entail that the controller is constrained to solve a servomechanism problem involving a class of reference signals and two classes of disturbances the cost functional is set up by adding quadratic terms to the usual lqg wiener hopf cost in such a way as to ensure the existence of unique solutions even when the signals to be tracked rejected as well as models have poles on the imaginary axis the solutions of the servomechanism problem are explicitly parametrized by free stable rational and proper parameters under two sets of assumptions one which is necessary and sufficient for the existence of solutions and another stronger set which avoids kronecker products and the associated higher dimensional factorizations involved in

two properties of l sub 1 optimal controllers the solution of the l sub 1 sensitivity minimization problem is shown to have two properties which contrast markedly with properties of the solutions to the better known h infinity sensitivity minimization problem or the lqg linear quadratic gaussian problem is given of a one parameter family of first order plants where the order of the l sub 1 optimal compensator can be arbitrarily large and thus it is impossible to bound the order of an l sub 1 optimal compensator in terms of the order of the plant a plant is considered which has a continuous one parameter family of l sub 1 optimal compensators and thus l sub 1 optimal compensators need not be unique the author s two examples are considered to answer two questions left open by m a daleh and j b pearson ibid vol 32 p 314 23 1987

some consequences of non uniform conditions on uniform classes we obtain some results of the form if certain complexity classes satisfy a non uniform condition then some unlikely consequences follow more precisely 1 1 if the non uniform polynomial time hierarchy collapses at level i 0 i e ipoly ipoly then the meyer stockmeyer hierarchy collapses at level i 2 i e i 2 i 2 this strengthens a generalization of a result of karp and lipton 1980 2 2 if co np is conjunctively reducible to a sparse set then p np this generalizes a theorem of fortune 1979 3 3 if np is conjunctively and disjunctively reducible to a sparse np complete set then p np this is a partial generalization of a result of mahaney 1980 conjuctive and disjunctive reducibility were introduced by ladner lynch and selman 1975 4 4 if co np is reducible to a sparse set then np co np reducibility was introduced 5 by adleman and manders 1977

on the minimization of sensitivity to additive disturbances for linear distributed parameter mimo feedback systems the problem of minimizing the sensitivity of the output or input of a plant under a feedback control with respect to external additive disturbances is considered the mathematical model assumed for the plant is a rectangular matrix of transfer functions which can arise from both lumped parameter and distributed parameter systems including time delays the feedback controller is assumed to guarantee the closed loop stability it is shown that an arbitrarily small sensitivity function on a compact set of complex frequencies can be obtained by a suitable choice of feedback transfer function provided no zero of the plant transfer function is in this set additionally the sensitivity function can be designed to be bounded on an unbounded region containing the imaginary axis if there are no zeros of the plant in the right half plane and a certain hypothesis holds multiplicative delays are not excluded if there are zeros in the right half plane the latter design is not realizable since minimizing the sen

strong nondeterministic polynomial time reducibilities abstract building on the work of adleman and manders 1 we define a new class of reducibilities the strong nondeterministic polynomial time reducibilities we show that several reducibilities in this class differ over the recursive sets and assuming np conp that they each determine different classes of np hard sets comparisons in strength are also made between the strong nondeterministic polynomial time reducibilities and the classes of deterministic and non deterministic polynomial time reducibilities in ladner lynch and selman 6

linear feedback systems and the graph topology it is established for general linear systems that the gap metric induces the coarsest topology with respect to which both closed loop stability and closed loop performance are robust properties in earlier works similar topological results were obtained by exploiting the existence of particular coprime factor system representations not known to exist in general by contrast the development here does not rely on any specific system representations systems are simply characterized as subspaces of norm bounded input output pairs and the analysis hinges on the underlying geometric structure of the feedback stabilization problem unlike other work developed within such a framework fundamental issues concerning the causality of feedback interconnections are discussed explicitly the key result of this paper concerns the difference between linear feedback interconnections with identical controllers in terms of a performance robustness related closed loop mapping upper and lower bounds on the induced norm of this difference are derived allowing for possibly infinite dimensional input output spaces and time varying behavior the bounds are both proportional to the gap metric distance between the plants which clearly demonstrates the gap to be an appropriate measure of the difference between open loop systems from the perspective of closed loop behavior to conclude an example is presented to show that bounds of the form derived here for linear systems do not hold in a general nonlinear setting

realization of power spectra from partial covariance sequences in this paper we address the problem of realization of a spectral density function from incomplete information about the underlying stochastic process the standing assumption is the availability of an incomplete partial sequence of covariance samples of the process we study the set of rational extensions of this finite sequence to an infinite covariance function that agrees with the available samples the classical theory of orthogonal polynomials with respect to the unit circle and the theory of moments have been utilized extensively in a variety of engineering problems including the one we are dealing with these have been known to provide a unifying framework for a variety of current spectral estimation techniques maximum entropy method pisarenko s harmonic decomposition etc in this work we consider and study the set of all covariance realizations of dimension lower than or equal to the length of the partial sequence and equal to the dimension of the maximum entropy realization the me solution is a point in this set other points correspond to pole zero models a general formula is obtained for recursively updated pole zero models of dimension increasing with the data record information about the zeros is obtained from the asymptotic behavior of the partial autocorrelation coefficients our approach combines techniques from the theory of orthogonal polynomials on the unit circle the theory of moments and also techniques from degree theory topology our objective is to develop a theory that will provide a frame for constructing recursively pole zero realizations of increasing dimension

further results on limitations to the capability of feedback in this paper we study the limitations of the capability of the feedback mechanism in dealing with structural uncertainty for a class of pth order non linear autoregressive control systems the size of the structural uncertainty is characterized by multiple lipschitz constants say l 1 l 2 lp of the uncertain non linear function it is shown that if l 1 l 2 lp satisfy certain algebraic conditions then there exists no globally stabilizing feedback control law for the corresponding uncertain systems the result here is an extension of the previous result obtained by zhang and guo 2002

relativizing relativized computations abstract this paper introduces a technique of relativizing already relativized computations and gives two interesting applications the techniques developed here are simpler than the usual methods for constructing oracles that satisfy several requirements simultaneously the first application shows that a result of karp and lipton if sets in np are decidable with polynomial size circuits then p 2 p 2 cannot be strengthened in the presence of certain oracles this means that relativizable proof techniques cannot strengthen the conclusion to say p np such a stronger conclusion would be desirable as it would establish the equivalence of polynomial time programs and polynomial size circuits for solving np complete problems and would extend the known equivalence of polynomial time programs and programs that are allowed a single query to a polynomial size table the second application gives an oracle c for which p c np c conp c np c and np c conp c has complete sets under polynomial time many one reductions this complements a result of sipser in which an oracle b is constructed for which np b conp b has no complete sets these results suggest that current proof methods will not settle whether np conp has complete sets

optimal sensitivity theory for multivariate distributed plants in this note we generalized the results of foias and tannenbaum 1986 a and foias et al 1987 on optimal sensitivity minimization to a much broader class of distributed plants we relate our results to certain kinds of hankel norm approximation problems that frequently appear in control engineering applications

delay dependent stability and h control for a class of fuzzy descriptor systems with time delay this paper addresses issues of stability analysis and h control for a class of fuzzy time delay descriptor systems described by an extended takagi sugeno t s fuzzy model a delay dependent stability criterion is first presented and a h control synthesis approach is then developed both the stability criterion and the h controller synthesis are based on the solutions of linear matrix inequalities lmis finally two examples are given to illustrate the effectiveness of the approaches proposed in this paper

infinite horizon h control for nonlinear stochastic markov jump systems with x u v dependent noise via fuzzy approach in this paper the h control problem is studied for nonlinear stochastic markov jump systems with state control and external disturbance dependent noise x u v dependent noise for short a sufficient condition is derived for the infinite horizon h control of such systems in terms of a set of coupled second order hamilton jacobi inequalities hjis in general it is difficult to solve these coupled hjis by using fuzzy approach the infinite horizon h control design for nonlinear stochastic markov jump systems is developed via solving a set of linear matrix inequalities lmis instead of hjis two numerical examples are presented to illustrate the effectiveness of the proposed design methods

minimal state space realization for a class of linear discrete nd siso systems minimal realizations are a key feature in systems theory applications in the case of nd linear systems i e systems where transfer function descriptions are functions of n 1 indeterminates however the problem of the existence and construction of such a realization is still an open problem with solutions available in only a few special cases this paper adds one more case to this list in the form of a single input single output linear system where denominator and denominator polynomials of the transfer function description are multi linear polynomials i e they are of degree one in each indeterminate also in the case when an absolutely minimum realization does not exist the use of the elementary operations algorithm eoa to produce non minimal realizations with a moderate degree of redundancy or even of least order is advocated

brief the weighted incremental norm approach from linear to nonlinear h control weighted induced norms can be used to handle robustness and sensitivity requirements in the linear context as shown by zames in its seminal work this paper deals with an attempt to extend linear h control concepts to the nonlinear context making use of a weighted incremental norm it is shown how these concepts make it possible to handle basic requirements such as robust stability disturbance attenuation and steady state behaviors

an approach to h control of a class of nonlinear systems this paper considers robust h control of a class of nonlinear systems that can be represented by a heterogeneous model a new controller design method using both state feedback is proposed based on quadratic stability theory a number of necessary and sufficient conditions are derived for quadratic stabilizability with disturbance attenuation of the control system concerned an example is also given to demonstrate the performance of the proposed controller design methods

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part i weakly doubly coprime factorizations in this paper we show how to reformulate the fractional representation approach to analysis and synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions so that a system admits weakly left right doubly coprime factorizations moreover we explicitly characterize the integral domains a such that every plant defined by means of a transfer matrix whose entries belong to the quotient field of a admits weakly doubly coprime factorizations finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 v r sule siam j control optim 32 1994 pp 1675 1695 and 36 1998 pp 2194 2195 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

entropy minimizing and risk sensitive control rules abstract glover and doyle 4 have shown that the entropy minimising criterion is identical with the infinite horizon form of the leqg optimality criterion leqg linear exponential of quadratic gaussian whittle and kuhn 11 have given a hamiltonian formulation which provides a natural treatment of leqg optimisation this paper contains a direct proof that the control yielded by these hamiltonian methods is entropy minimising the conclusion would follow from the two assertions above but a simple adjoining of the proofs of these two assertions yields an argument so circuitous that a direct proof is of interest

complexity classes of equivalence problems revisited to determine if two lists of numbers are the same set we sort both lists and see if we get the same result the sorted list is a canonical form for the equivalence relation of set equality other canonical forms arise in graph isomorphism algorithms to determine if two graphs are cospectral have the same eigenvalues we compute their characteristic polynomials and see if they are equal the characteristic polynomial is a complete invariant for cospectrality finally an equivalence relation may be decidable in p without either a complete invariant or canonical form blass and gurevich 1984 asked whether these conditions on equivalence relations having an fp canonical form having an fp complete invariant and being in p are distinct they showed that this question requires non relativizing techniques to resolve we extend their results and give new connections to probabilistic and quantum computation