on the two block h problem for a class of unstable distributed systems this paper deals with the two block h control problem for distributed plants with finitely many unstable modes we assume that weighting filters in the h mixed sensitivity problem are finite dimensional then the corresponding optimal two block problem can be solved by finding the schmidt pairs of a hankel operator whose symbol is of the form m 2 m 1u u where u r h u h and m2 rh and m1 h are inner and the suboptimal two block problem can be solved by finding the solutions of certain functional equations very similar to the ones satisfied by the schmit pairs of the above mentioned hankel operator in this paper a unified approach is proposed for solving both the optimal and suboptimal two block problems we obtain two systems of linear equations expressed in terms of state space realizations of u and m2 whose solutions give the schmidt pairs of the associated hankel operator and the functions needed for the parametrization of all the suboptimal solutions respectively

krein space approach to h sup spl infin mixed sensitivity minimization for a class of infinite dimensional systems motivated by the computation of the h sup spl infin optimal performance and optimal mixed sensitivity for a class of infinite dimensional systems the author considers the explicit computation of the norm and minimal symbols of a class of hankel plus toeplitz type operators with irrational symbols using the krein space approach the author shows how to explicitly compute the norm and parameterize all the minimal symbols the results are used to obtain a complete solution to the h sup spl infin optimal mixed sensitivity synthesis problem for a class of infinite dimensional systems an explicit formula for computing the h sup spl infin optimal performance and explicit parameterization of all optimal and suboptimal mixed sensitivities an illustrative example is given

controller reduction in the two block h optimal design for distributed plants abstract in this paper we consider a two block mixed h sensitivity minimization problem for a class of siso possibly unstable infinite dimensional systems the main issue addressed here is the problem of reducing the order of h optimal controllers it is well known that the optimal controllers for distributed plants are infinite dimensional here we obtain a finite dimensional controller from approximations of the optimal controller in the gap metric topology we show that under certain mild conditions one can obtain a finite dimensional stabilizing controller achieving a performance arbitrarily close to the optimum we also provide an explicit bound for the performance deviation this bound depends on the approximations of the infinite dimensional parts of the optimum controller an example involving an unstable delay system is given to illustrate the computational aspects of this approach

on spl hscr sub spl infin control for dead time systems a mixed sensitivity spl hscr sub spl infin problem is solved for dead time systems it is shown that for a given bound on the spl hscr sub spl infin norm causal stabilizing controllers exist that achieve this bound if and only if a related finite dimensional riccati equation has a solution with a certain nonsingularity property in the case of zero time delay the riccati equation is a standard riccati equation and the nonsingularity condition is that the solution be nonnegative definite for nonzero time delay the nonsingularity condition is more involved but still allows us to obtain controllers all suboptimal controllers are parameterized and the central controller is shown to be a feedback interconnection of a finite dimensional system and a finite memory system both of which can be implemented some spl hscr sub spl infin problems are rewritten as pure rational spl hscr sub spl infin problems using a smith predictor parameterization of the controller

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

a hamiltonian based solution to the two block h problem for general plants in h and rational weights a complete skew toeplitz type solution to the two block h 1 problem for innite dimensional stable plants with rational weights is derived with a basis free proof the solution consists of one riccati equation with a rank criterion for a transcendental function of a certain hamiltonian this gives a natural extension of the well known formula for the one block case an example is given to illustrate the result c 2000 elsevier science b v all rights reserved

mixed sensitivity optimization for a class of unstable infinite dimensional systems abstract in this paper we solve the h mixed sensitivity minimization problem for a class of unstable distributed systems the solution is based on an extension of the skew toeplitz methodology the key mathematical fact used is that the skew toeplitz operators arising in the unstable case are finite rank perturbations of the classical skew toeplitz operators obtained from compressions of rational functions a system of linear equations the singular system is derived for this class of operators from which one can compute the associated singular values and vectors an example is included to illustrate the results

a simpler formula for the singular values of a certain hankel operator this paper deals with the problem of computing the singular values and vectors of a hankel operator with symbol m w where m h is arbitrary inner and w h is rational a simplified version of the formula given in 6 is obtained for computing the singular values of the hankel operator this result is applied to the one block h optimal control of siso stable infinite dimensional plants and rational weights using this new formula a simple expression is derived for the h optimal controller whose structure was observed in 9

a note on essential spectra and norms of mixed hankel toeplitz operators in this brief hankel toeplitz operators which occur in feedback theory e g in the minimization of mixed h sensitivity and complementary sensitivity will be considered a method of computing their spectra eigenvectors and norms will be presented for infinite dimensional systems subject to continuous weightings

the four block adamjan arov krein problem abstract this paper is concerned with the mixed sensitivity h design in the most general i e the four block case this problem involves in a crucial manner the so called four block operator the norm of which is the achievable feedback tolerance our objective in this paper is to provide an interpretation for all singular values of these singular values are given an adamjan arov krein interpretation in terms of the l distance of an l function to h l intuitively the singular values of are the various tolerance levels that can be achieved if we allow a various number of unstable poles in the closed loop we finally provide an upper bound on the number of singular values

linear control theory with an h 0e optimality criterion this expository paper sets out the principal results in bf h infty control theory in the context of continuous time linear systems the focus is on the mathematical theory rather than computational methods

on the computation of induced norms for non compact hankel operators arising from distributed control problems abstract this paper considers the problem of computing the induced l 2 norm for hankel operators which are not necessarily compact in the literature such a norm has traditionally been computed by solving an infinite dimensional eigenvalue eigenfunction problem in this paper it is shown that a sequence of finite dimensional eigenvalue eigenvector problems can be solved to estimate the desired induced norm

singular values and vectors of a class of hankel operators abstract this paper considers the problem of determining the singular values and vectors of the class of hankel operators whose symbol can be written in the form w where is inner and w is rational we will obtain a new and simplified proof of the fact conjectured in 17 and first proved in 9 that the zeros of a certain determinantal expression coincide with the hankel singular values we will also show how this expression can be used to generate the singular vectors

weighted sensitivity minimization for delay systems in this note we discuss the h infty sensitivity minimization problem for linear time invariant delay systems while the unweighted case reduces to simple nevanlinna pick interpolation the weighted case turns out to be much more complicated and demands certain functional analytic techniques for its solution

on stable orthogonal decompositions of idempotents in fact the two relative projectivities of a pointed group are equivalentin some sense and it is the same for the two relative freeness see 2 2 and2 29 below it is clear that the relative freeness implies the relative projectivity theconverse is certainly false to some extent our aim is just to look for theconditions which make the converse true if

tutorial review h optimal controller design for a class of distributed parameter systems this paper is a tutorial on a frequency domain design method for the two block h optimal control of a class of siso distributed parameter systems uncertainties in the system are assumed to be unmodelled dynamics represented in the form of either multiplicative or additive or coprime factor perturbations of a nominal infinite dimensional plant performance of the closed loop system is measured in terms of the energy amplification from external input signals to the outputs of interest in this paper transfer functions of the nominal plant and the weights are used in order to derive a solution to two block h control problems the solution presented here uses results from operator theory the h optimal performance and controller in this method are computed from a finite set of linear equations

a skew toeplitz approach to the h optimal control of multivariable distributed systems in this paper the problem of the h infty optimization of multivariable distributed systems in the four block setting is studied this work is based on several previous papers and employs the skew toeplitz framework developed in operator theory adv appl 32 1988 pp 21 43 operator theory adv appl 32 1988 pp 93 112 operator theory and integral equations 11 1988 pp 726 767 j functional anal 74 1987 pp 146 159 siam j math anal 19 1988 pp 1081 1091

g algebras clifford theory and the green correspondence let g be a finite group with subgroup h loosely speaking clifford theory as developed by e c dade and others is a tool for constructing reduction arguments in the case where h is normal in g the green correspondence is a tool for constructing reduction arguments in the case where h contains the normaliser in g of a given p subgroup or more generally where h contains the normaliser of a given local pointed group here we cover both of these situations by considering the case where no h contains the normaliser in g of a given local pointed group a brief discussion in 1 indicates how puig s theory subsumes the green correspondence and aspects of clifford theory here this theme is given a more systematic treatment we shall see in particular that defect pointed subgroups the green correspondence and brauer s extended first main theorem are all manifestations of a more general mechanism let be a complete local commutative noetherian ring with an algebraically closed residue field k of prime characteristic p throughout we fix a g algebra a all algebras are deemed to possess a unity element all algebras and modules are to be free over or over k and of finite rank for the rest of this section let us fix points 3 a of h g respectively on a and suppose that ht3 g have a common defect pointed subgroup pr such that nc p ng h it is easy to show see section 4 t h a t if p is another defect pointed subgroup of ht3 then p is also a defect pointed subgroup of g and ng p ng h it t ranspires again see section 4 t h a t ht3 g if and only if a c trgh ah 3ah when these equivalent

a spectral characterization of h optimal feedback performance and its efficient computation abstract we study the spectral properties of a toeplitz hankel operator which arises in the context of the mixed sensitivity h optimization problem and whose largest eigenvalue characterizes the optimal achievable performance 0 the existence of such an operator was first shown by verma and jonckheere 26 who also noted the potential numerical advantage of computing eo through its eigenvalue characterization rather than through the iteration here we investigate this operator in detail with the objective of efficiency computing its spectrum we define an adjoint linear quadratic problem that involves the same toeplitz hankel operator as shown by jonckheere and silverman 13 16 consequently a finite polynomial algorithm allows 0 to be characterized as simply as the largest root of a polynomial finally a computationally more attractive state space algorithm emerges from the h t 8 lq relationship this algorithm yields a very good accuracy evaluation of the performance 0 by solving just one algebraic riccati equation thorough exploitation of this algorithm results in a drastic computation reduction with respect to the standard e iteration

robust stabilization in the gap metric controller design for distributed plants the problem of robustness in the gap metric for infinite dimensional systems is considered the problem of computing the optimal controller and the optimal robustness radius for a class of systems whose normalized coprime factors have elements which are h sub infinity functions with continuous boundary values is studied the underlying hankel and related operators which are important in the gap optimization problem are studied and relations between their singular values and vectors are established a computational approach to the optimal robustness problem is developed for single input single output systems whose transfer function is an inner function in h sub infinity times a rational function the procedure is applied to a general first order delay system and a closed form formula is obtained for the optimal controller the frequency response plots of the compensated system for various values of time delay are examined

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

ordering of the trees with a perfect matching by minimal energies the ordering of the trees with a perfect matching according to minimal energies is investigated using a new method proposed here we obtain the preceding trees in the increasing order of their energies within the class under consideration for n 3 where 2n is the vertex number of the tree we deduce the first 11 12 13 14 12 9 16 and 21 trees for n 21 15 n 20 n 14 10 n 13 n 9 n 8 n 7 and n 6 respectively for n 5 4 3 we list all the trees in the increasing order of their energies

ordering of huckel trees according to minimal energies abstract the tree with a perfect matching having degrees not greater than three is referred to as the huckel tree the ordering of huckel trees according to their minimal energies is investigated by means of a quasi ordering relation using a simpler method than that of li h li on minimal energy ordering of acyclic conjugated molecules j math chem 25 1999 145 169 we obtain the first 2 n 10 1 1 n 2 trees in the increasing order of their energies within the class under consideration for n 13 where 2 n is the vertex number of the tree the number of the trees obtained here exceeds the reported result li 1999 by n 10 we also get a lot of preceding trees in the increasing order of their energies within the class for 5 n 12

on the h optimal sensitivity problem for systems with delays in this paper we extend some of the results of ieee trans automat control ac 31 1986 pp 763 766 to more general delay systems in particular we analyze the effect of the interaction of delays and nonminimum phase zeros on the h infty optimal weighted sensitivity

minimal energies of trees with given parameters abstract the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix let t k be the set of trees with given order k suppose that t t k and v 1 v 2 v k be the ordering vertex set of t we denote by t n 1 n 2 n k the graph obtained by attaching n i pendent vertices to vertex v i i 1 2 k of t respectively let t n k t n 1 n 2 n k t t k n 1 n 2 n k n k n i 1 i 1 2 k in this paper we determine the trees in t n k with the first and the second minimal energies as applications we can characterize the trees with the first and the second minimal energies among the set of trees with given domination number matching number independence number respectively

on the mixed sensitivity minimization for systems with infinitely many unstable modes in this note we consider a class of linear time invariant systems with infinitely many unstable modes by using the parameterization of all stabilizing controllers and a data transformation we show that h controllers for such systems can be computed using the techniques developed earlier for infinite dimensional plants with finitely many unstable modes

duality in g algebras keywords finite group complete discrete valuation ring symmetric g algebra symmetric bilinear form projective ideal symmetric algebra rg lattice unimodular symmetric g algebra almost projective ideal almost split sequences block defect groups virtually irreducible lattices splitting trace lattices reference ctg article 1988 005doi 10 1007 bf01161746 record created on 2008 12 16 modified on 2017 05 12

computation of the singular values of toeplitz operators and the gap metric it is widely recognized that the computation of gap metric is equivalent to a certain two block h problem i e the gap is equal to the norm of a certain two block operator however it can also be characterized as the smallest singular value of a certain toeplitz operator this paper derives a simple computational method for finding such singular values and the gap between two plants by using a state space approach

robust stabilization of delay systems by approximation of coprime factors given a delay system with transfer function g s h2 s h1 s where h1 s 01npi s e is and h2 s 02nqi s e is with 0 0 1 n1 0 0 n2 the pi being polynomials of degree i and i 0for i 0 and the qi polynomials of degree di 0 for each i the robust stabilization of a class of perturbed coprime factors of this system is considered asymptotic estimates are obtained based on recent results on the approximation and stabilization of normalized coprime factors an explicit formula is given for the normalized coprime factor stability margin for the case of a multivariable transfer function of the form g s e str s with r rational

paper uniformly optimal control of linear feedback systems the problem of uniformly optimal control is posed in the context of linear time varying discrete time systems the controller is optimal uniformly over a pre specified set of exogenous signals existence of an optimal controller is proved and a formula for the minimum cost is derived the time invariant case is treated in the frequency domain it is shown that for time invariant systems an optimal time varying controller is no better than an optimal time invariant controller

the parametrization of interior algebras keywords finite group indecomposable og lattice invariants vertex source multiplicity module indecomposable projective lattice twisted group algebra primitive interior g algebras reference epfl article 130446doi 10 1007 bf02571667 record created on 2008 12 16 modified on 2017 05 12

stable controller design for mixed sensitivity reduction of infinite dimensional systems abstract in this paper we study the design of stable controllers for mixed sensitivity reduction for systems that have input output delays and infinitely many unstable zeros the proposed method is based on interpolation with a unit in h which gives a necessary and sufficient condition for stabilization by stable controllers using the youla parameterization we introduce a two block problem for the stable h controller design the two block problem is then transformed to a one block problem solvable by the skew toeplitz approach

defect theory for maximal ideals and simple functors keywords green functor source green correspondence maximal ideal finite group fixed point algebra defect theory for g algebras defect group restriction transfer conjugation defect pair functorial ideal simple g functors reference ctg article 1991 002doi 10 1016 0021 8693 91 90168 8 record created on 2008 12 16 modified on 2017 05 12

ordering trees and graphs with few cycles by algebraic connectivity abstract several approaches for ordering graphs by spectral parameters are presented in the literature we can find graph orderings either by the greatest eigenvalue spectral radius or index or by the sum of the absolute values of the eigenvalues the energy of a graph or by the second smallest eigenvalue of the laplacian matrix the algebraic connectivity among others by considering the fact that the algebraic connectivity is related to the connectivity and shape of the graphs several structural properties of graphs relative to this parameter have been studied hence a large number of papers about ordering graphs by algebraic connectivity mainly about trees and graphs with few cycles have been published this paper surveys the significant results concerning these topics trying to focus on possible points to be investigated in order to understand the difficulties to obtain partial orderings via algebraic connectivity

on computing the spectral radius of the hankel plus toeplitz operator the square root of the spectral radius of the hankel plus toeplitz operator has been shown to be the achievable performance of the mixed sensitivity h sup infinity design the computation of the spectral radius is the bottleneck in the synthesis of the h sup infinity controller in this paper the spectral properties of the hankel plus toeplitz operator are investigated a finite procedure for computing the spectral radius of the hankel plus toeplitz operator is proposed

explicit formulas for optimally robust controllers for delay systems this paper considers single input single output systems whose transfer functions take the form of a strictly proper rational function times a delay a closed form expression is presented for the controller which is optimally robust with respect to perturbations measured in the gap metric the formula allows the h sub spl infin loop shaping procedure of glover mcfarlane to be carried out explicitly for this class of systems without the need to first find a rational approximation of the plant the form of the controller involves a certain algebra of pseudo derivation operators these operators and their matrix generalizations play a central role in the derivation of the controller a discussion of the main properties of these operators is given an example is presented of a controller design to achieve disturbance attenuation and robust set point following for a plant with two lightly damped poles and a nontrivial time delay the performance is compared and shown to be superior to that of a smith predictor

h sup spl infin optimal and suboptimal controllers for infinite dimensional siso plants this paper presents a simple formula for h sup spl infin optimal and suboptimal controllers for unstable siso distributed plants with rational weighting functions the controller is expressed in terms of i inner and outer parts of the plant ii a finite dimensional spectral factor obtained from the weighting functions and iii a rational function satisfying certain interpolation conditions under certain genericity assumptions this rational function is of dimension less than or equal to n sub 1 l 1 n sub 1 l in the suboptimal case where l is the number of unstable poles of the plant and n sub 1 is the order of the sensitivity weighting function there are 2 n sub 1 l 2 n sub 1 l 1 in the suboptimal case linear equations which determine this rational function these linear equations can be written directly from the structure of the controller

on the optimal and suboptimal solutions of the mixed sensitivity problem abstract in this note using elementary operator theoretic methods we give a method for the explicit computation of the suboptimal and optimal solutions to the mixed sensitivity problem for stable single input single output distributed systems

operator theoretic approach to the optimal two disk problem the nonstandard two disk problem plays a fundamental role in robust feedback optimization here it is shown via banach space duality theory that its solutions satisfy an extremal identity and may be viewed as a dual extremal kernel of a particular l sup 1 optimization problem a novel operator theoretic framework to characterize explicitly its solutions is developed in particular the two disk optimization is shown to be equal to the induced norm of a specific operator defined on a projective tensor product space involving a non hilbert version of a vector valued h sup 2 space moreover this operator is shown to be a combination of multiplication and toeplitz operators under certain conditions existence of maximal vectors is established leading to an explicit formula for the optimal controller an infinite matrix representation with respect to a canonical basis is derived together with an algorithm to compute it the norm of the relevant operator is approximated by special finite dimensional optimizations whose solutions lead to solving semi definite programming problems involving the computation of a matrix projective tensor norm

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

characterizing trees with large laplacian energy we investigate the problem of ordering trees according to their laplacian energy more precisely given a positive integer n we find a class of cardinality approximately p n whose elements are the n vertex trees with largest laplacian energy the main tool for establishing this result is a new upper bound on the sum sk t of the k largest laplacian eigenvalues of an n vertex tree t with diameter at least four where k 2 f1 ng

g algebras jacobson radical and almost split sequences keywords finite group indecomposable kg module sas sequence almost split exact sequences restriction relative jacobson radicals local criterion multiplicity module vertex source indecomposable projective module twisted group algebra induction reference ctg article 1988 006doi 10 1007 bf01393690 record created on 2008 12 16 modified on 2017 05 12

an operator theoretic approach to the mixed sensitivity minimization problem in this paper we consider the mixed sensitivity minimization problem scalar case lt gives rise to the so called two block problem on the algebra he we analyze this problem from an operator point of view using krein space theory we obtain a necessary and sufficient condition for the uniqueness of the solution and a parameterization of all solutions in the non uniqueness case moreover an interpolation interpretation is given for the finite dimensional case

on stabilization and the existence of coprime factorizations let k be an integral domain and f its quotient field we show there are plants over f which have no stable coprime factorizations but can be stabilized in a stable closed loop this answers a question posed by vidyasagar schneider and francis

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

finite rank criteria for h infty control of infinite dimensional systems this paper studies the hinfin control of infinite dimensional systems whose transfer matrices are expressible as a series connection of a rational transfer matrix and a scalar possibly irrational inner function this class of systems is adequate for describing many control problems in practice when weighting functions are rational and plants have at most finitely many unstable modes we show that this problem can be reduced to solving two matrix valued riccati equations and an additional rank condition furthermore the obtained controller structure is characterized by the inner function and controllers for the finite dimensional part this result provides us with a solution that is easily implementable without much prior knowledge on infinite dimensional control theory a numerical example is given to illustrate the result

minimax frequency domain performance and robustness optimization of linear feedback systems it is shown that feedback system design objectives such as disturbance attenuation and rejection power and bandwidth limitation and robustness may be expressed in terms of required bounds of the sensitivity function and its complement on the imaginary axis this leads to a minimax frequency domain optimization problem whose solution is reduced to the solution of a polynomial equation

optimal mixed sensitivity for siso distributed plants the authors extend previous results for the single input single output siso mixed sensitivity problem for distributed parameter systems to a more general setting in which the transfer function of the plant to be controlled has both irrational stable and irrational unstable parts these developments were originally motivated by the computation of the infimal h sup spl infin norm of the weighted mixed sensitivity for the irrational transfer function model of a damped flexible beam

2dof controller parametrization for systems with a single i o delay this note puts forward a parametrization of all stabilizing two degrees of freedom controllers for possibly unstable processes with dead time the proposed parametrization is based on a doubly coprime factorization of the plant and takes the form of a generalized smith predictor dead time compensator feedback part and a finite dimensional feedforward part prefilter some alternative dead time compensation schemes and disturbance attenuation limitations are also discussed