geometric characterizations of the representation type of hereditary algebras and of canonical algebras abstract we show that a finite connected quiver q with no oriented cycles is tame if and only if for each dimension vector d and each integral weight of q the moduli space m q d s s of semi stable d dimensional representations of q is just a projective space in order to prove this we show that the tame quivers are precisely those whose weight spaces of semi invariants satisfy a certain log concavity property furthermore we characterize the tame quivers as being those quivers q with the property that for each schur root d of q the field of rational invariants k rep q d gl d is isomorphic to k or k t next we extend this latter description to canonical algebras more precisely we show that a canonical algebra is tame if and only if for each generic root d of and each indecomposable irreducible component c of rep d the field of rational invariants k c gl d is isomorphic to k or k t along the way we establish a general reduction technique for studying fields of rational invariants on schur irreducible components of representation varieties

monoids on groups with bn pairs the theory of linear algebraic monoids has been developed in the last few years by l renner and the author 21 361 as a generalization of the theory of linear algebraic groups 16 40 in applications of the theory of linear algebraic groups one is naturally led to consider groups with bnpairs or tits systems 12 4 9 42 431 this class of groups includes in particular the groups of lie type so that the abstract properties of tits systems are useful in studying the conjugacy classes and representations of groups of lie type see 4 9 our interest is in monoids so let m be a monoid such that the group of units g has a bn pair in general g has very little to do with the rest of m in fact for a given g m g could be an arbitrary semigroup so we introduce three axioms which intimately connect the idempotent structure of m with the structure of g we then say that m is a monoid on g the prime example is a connected regular linear algebraic monoid with zero but there are other natural examples for instance renner 37 is developing a theory of monoids of lie type along the lines of steinberg 41 for groups another example arises from the work of smith 39 and cabanes 3 from their work one can associate with certain irreducible modular representations 4 g gl v a set of idempotents es end v then the monoid m generated by 4 g and e is a monoid on 4 g in the examples we have worked out and very likely in general in the situation of linear algebraic monoids there is a natural type map from the finite lattice of conjugacy classes of idempotents g the lattice of f classes into the power set of the dynkin diagram of g the idea of the type map is due to lex renner and is exploited by renner and the author 31 to show in particular that the system of idempotents can be constructed from the type map in the present situation of monoids on g there is again a type map i which characterizes the system of idempotents 139 0021 8693 89 3 00

hecke algebras and semisimplicity of monoid algebras abstract we begin by constructing hecke algebras for arbitrary finite regular monoids m we then show that the semisimplicity of the complex monoid algebra c m is equivalent to the semisimplicity of the associated hecke algebras and a condition on induced group characters we apply these results to finite lie type monoids m we show that the monoid algebra fm over a field f is semisimple if and only if the characteristic of f does not divide the order of the unit group g this is accomplished by developing formulas for the unities of c j j a j class of m the unity is explicitly given when g is a simply connected chevalley group and j is associated with a borel subgroup of g

linear filtering for bilinear stochastic differential systems with unknown inputs investigates the problem of state estimation for bilinear stochastic multivariable differential systems in the presence of an additional disturbance whose statistics are completely unknown a linear filter is proposed based on a suitable decomposition of the state of the bilinear system into two components the first one is a computable function of the observations while the second component is estimated via a suitable linear filtering algorithm no a priori information on the disturbance is required for the filter implementation the proposed filter is robust with respect to the unknown input in that the covariance of the estimation error is not affected by such input numerical simulations show the effectiveness of the proposed filter

brief paper h filtering for rectangular discrete time descriptor systems this paper deals with the h recursive estimation problem for general rectangular time variant descriptor systems in discrete time riccati equation based recursions for filtered and predicted estimates are developed based on a data fitting approach and game theory in this approach the nature determines a state sequence seeking to maximize the estimation cost whereas the estimator tries to find an estimate that brings the estimation cost to a minimum a solution exists for a specified c level if the resulting cost is positive in order to present some computational alternatives to the h filters developed they are rewritten in information form along with the respective array algorithms

technical communique robust state prediction for descriptor systems this paper deals with the problem of state prediction for descriptor systems subject to bounded uncertainties the problem is stated in terms of the optimization of an appropriate quadratic functional this functional is well suited to derive not only the robust predictor for descriptor systems but also that for usual state space systems numerical examples are included in order to demonstrate the performance of this new filter

state estimation for linear and non linear equality constrained systems this article addresses the state estimation problem for linear and non linear systems for the case in which prior knowledge is available in the form of an equality constraint the equality constrained kalman filter kf is derived as the maximum a posteriori solution to the equality constrained state estimation problem for linear and gaussian systems and is compared to alternative algorithms then four novel algorithms for non linear equality constrained state estimation based on the unscented kf are presented namely the equality constrained unscented kf the projected unscented kf the measurement augmentation unscented kf and the constrained unscented kf finally these methods are compared on linear and non linear examples

kalman filtering and riccati equations for descriptor systems a general formulation of a discrete time filtering problem for descriptor systems is considered it is shown that the nature of descriptor systems leads directly to the need to examine singular estimation problems using a dual approach to estimation the authors derive a so called 3 block form for the optimal filter and a corresponding 3 block riccati equation for a general class of time varying descriptor models which need not represent a well posed system in that the dynamics may be either over or under constrained specializing in the time invariant case they examine the asymptotic properties of the 3 block filter and in particular analyze in detail the resulting 3 block algebraic riccati equation the noncausal nature of discrete time descriptor dynamics implies that future dynamics may provide some information about the present state a modified form for the descriptor kalman filter that takes this information into account is presented

observer based controller design for stochastic descriptor systems with brownian motions in the paper a complete proof of the properness and mean square exponential stability for stochastic brownian systems is firstly addressed which lays a solid foundation for system analysis and control synthesis for this kind of stochastic systems observer based controller for the descriptor system with brownian motions is then investigated it is noticed that the system dynamics and observer dynamics are both subjected to state brownian fluctuation which makes it challenging either to design observer gain and controller gain simultaneously or design the observer and control gains by using a complete separation design technique a sequential design technique is proposed to calculate the control and observer gains by solving linear matrix inequalities the proposed design methods are readily extended to the case for stochastic descriptor systems with multiple brownian motions a numerical example is finally given to illustrate the design techniques and simulated results have demonstrated the efficiency

complex representations of matrix semigroups let m be a finite monoid of lie type these are the finite analogues of linear algebraic monoids with group of units g the multiplicative semigroup 4 f where f is a finite field is a particular example using harishchandra s theory of cuspidal representations of finite groups of lie type we show that every complex representation of m is completely reducible using this we characterize the representations of g extending to irreducible representations of m as being those induced from the irreducible representations of certain parabolic subgroups of g we go on to show that if f is any field and s any multiplicative subsemigroup of 4 f then the semigroup algebra of s over any field of characteristic zero has nilpotent jacobson radical if s 4 f then this algebra is jacobson semisimple finally we show that the semigroup algebra of 4 f over a field of characteristic zero is regular if and only if ch f p 0 and f is algebraic over its prime field

nonsymmetric generic matrix equations abstract let a i 0 i k be generic matrices over q the field of rational numbers let k q e where e denotes the entries of the a i i and let k be the algebraic closure of k we show that the generic unilateral equation a k x k a 1 x a 0 0 n has n k n solutions x m n k solving the previous equation is equivalent to solving a polynomial of degree kn with galois group s k n over k let b i i k be fixed n n matrices with entries in a field l we show that for a generic c m n l a polynomial equation g b 1 b k x c admits a finite fixed number of solutions and these solutions are simple we study when n 2 the generic non unilateral equations x 2 b x c d 0 2 and x 2 b x b c 0 2 we consider the unilateral equation x k c k 1 x k 1 c 1 x c 0 0 n when the c i i are n n generic commuting matrices we show that every solution x m n k commutes with the c i i when n 2 we prove that the generic equation a 1 x a 2 x x a 3 x x 2 a 4 a 5 x a 6 0 2 admits 16 isolated solutions in m 2 k that is according to bezout s theorem the maximum for a quadratic 2 2 matrix equation

sandwich matrices solomon algebras and kazhdan lusztig polynomials sandwich matrices have proved to be of importance in semigroup theory for the last 50 years the work of the author on algebraic monoids leads to sandwich matrices in group theory in this paper we find some connections between sandwich matrices and the hecke algebras for monoids introduced recently by louis solomon at the local level we then obtain an explicit isomorphism between solomon s hecke algebra and the complex monoid algebra of the renner monoid in the simplest case of monoids associated with a borel subgroup we find that the entries of the inverse of the sandwich matrix as well as those of the related structure matrix of solomon s hecke algebra are almost the polynomials r x y associated with the kazhdan lusztig polynomials

brief paper robust filtering for uncertain linear discrete time descriptor systems this paper is concerned with the problem of robust filtering for uncertain linear discrete time descriptor systems the matrices of the system state space model are uncertain belonging to a given polytope a linear matrix inequality based method is proposed for designing a linear stationary filter that guarantees the asymptotic stability of the estimation error and gives an optimized upper bound on the asymptotic error variance irrespective of the parameter uncertainty the proposed robust filter design is based on a parameter dependent lyapunov function which is shown to outperform parameter independent ones

representations of monoids arising from finite groups of lie type a class of finite monoids m constructed from a group g of lie type is considered we describe the irreducible complex representations and prove the complete reducibility of the representations of m the sandwich matrix of m is decomposed into a product of matrices corresponding to maximal parabolic subgroups of g

h_ infty unbiased filtering for linear descriptor systems via lmi this note concerns the optimal h infin unbiased filtering problem for linear descriptor systems it unifies the design of reduced order minimal order and full order filters for continuous and discrete time systems necessary and sufficient conditions for the solvability of the problem are obtained in terms of a set of linear matrix inequalities lmis the parametrization of all h infin unbiased filters is presented application to standard systems with or without unknown inputs is given a numerical example is given to show the applicability of the presented approach

on the graded identities for elementary gradings in matrix algebras over infinite fields we find a basis for the g graded identities of the n n matrix algebra mn k over an infinite field k of characteristic p 0 with an elementary grading such that the neutral component corresponds to the diagonal of mn k

high level primitives for recursive maximum likelihood estimation this paper proposes a high level language constituted of a small number of primitives and macros for describing recursive maximum likelihood ml estimation algorithms this language is applicable to estimation problems involving linear gaussian models or processes taking values in a finite set the use of high level primitives allows the development of highly modular ml estimation algorithms based on simple numerical building blocks the primitives which correspond to the combination of different measurements the extraction of sufficient statistics and the conversion of the status of a variable from unknown to observed or vice versa are first defined for linear gaussian relations specifying mixed deterministic stochastic information about the system variables these primitives are used to define other macros and are illustrated by deriving new filtering and smoothing algorithms for linear descriptor systems the primitives are then extended to finite state processes and used to implement the viterbi ml state sequence estimator for a hidden markov model

poincare series of some pure and mixed trace algebras of two generic matrices we work over a field k of characteristic zero the poincare series for the algebra cn 2 of gln invariants and the algebra tn 2 of gln concomitants of two generic n n matrices x and y are computed for n 6 both simply graded and bigraded cases are included the cases n 4 were known previously for c4 2 and c5 2 we construct a minimal set of generators and give an application to specht s theorem on unitary similarity of matrices by identifying the space mn2 of pairs of n n matrices with mn k2 we extend the action of gln to gln gl2 for n 5 we compute the poincare series for the polynomial invariants of this action when restricted to the subgroups gln sl2 and gln 1 where 1 is the maximal torus of sl2 consisting of diagonal matrices five conjectures are proposed concerning the numerators and denominators of various poincare series mentioned above some heuristic formulas and open problems are stated

h filter design for a class of nonlinear discrete time singular systems this paper deals with the h filter design problem for a class of discrete time lipschitz nonlinear singular systems the approach is based on the parameterisation of the obtained algebraic constraints from the estimation errors sufficient conditions for the existence of the filters which guarantee stability and the worst case filter error energy over all bounded energy disturbances is minimised are given the method also unifies the design for the full order reduced order minimal order filters for discrete time systems the presented results are less conservative than those existing in the literature this is due to the introduction of new slack variables application to systems with unknown input is presented via a numerical example

the ring of generic matrices in chapter 3 we defined a single n x n generic matrix we now introduce the ring generated by a set of n x n generic matrices

observer design for general linear time invariant systems it is noted that for both explicit and implicit linear systems several observer design problems including system inversion for systems with or without unknown inputs can all be formulated as a single problem involving descriptor systems the method used for solving this problem is based on recursively applying the reduction operation which consists of identifying and discarding equations that do not provide useful information for the estimation problem the reduction algorithm produces a reduced dimension system for which standard observers can be constructed the method presented can be implemented using numerically robust algorithms one such algorithm is presented

information filtering and array algorithms for descriptor systems subject to parameter uncertainties this paper presents information filters in riccati recursions and in array algorithms for descriptor systems subject to parameters uncertainties the filters are developed in filtered and predicted forms the inversion of the state matrix is avoided in the new information recursive formulas therefore it turns out clear that the invertibility of the state matrix usually considered in the state space information recursions is not necessary a numerical example is provided to illustrate the performance of the proposed robust array algorithms

geometric design techniques for observers in singular systems abstract we present geometric design techniques for observer design in generalized state space or singular systems the unknown input conditioned invariant subspaces are defined for singular systems and are related to the solution of a generalized lyapunov equation the approach emphasizes solution procedures based on the lyapunov equation and a singular system structure algorithm this paper extends the singular system observer theory both by providing a geometric basis for it and showing the best that may be accomplished if the system is not observable

brief paper h observers design for a class of nonlinear singular systems the present paper is devoted to the h observer or filter design problem for a class of lipschitz nonlinear singular systems the approach is based on the parameterization of the obtained algebraic constraints from the estimation errors sufficient conditions for the existence of the observers which guarantee stability and the worst case observers error energy over all bounded energy disturbances is minimized are given the method also unifies the design for the full order reduced order minimal order observers for continuous time systems it can be seen that the obtained conditions when applied to linear systems are also necessary a numerical example is given to show the applicability of our results

induction theorems on the stable rationality of the center of the ring of generic matrices following procesi and formanek the center of the division ring of rn x n generic matrices over the complex numbers c is stably equivalent to the fixed field under the action of sn of the function field of the group algebra of a zsn lattice denoted by gn we study the question of the stable rationality of the center cn over the complex numbers when rn is a prime in this module theoretic setting let n be the normalizer of an rn sylow subgroup of sn let m be a zsn lattice we show that under certain conditions on m inductionrestriction from n to sn does not affect the stable type of the corresponding field in particular c gn and c zg0zn gn are stably isomorphic and the isomorphism preserves the sn action we further reduce the problem to the study of the localization of gn at the prime n all other primes behave well we also present new simple proofs for the stable rationality of cn over c in the cases r 5 and rn 7

optimal filtering for systems with unknown inputs via the descriptor kalman filtering method this paper addresses globally optimal unbiased minimum variance state estimation for systems with unknown inputs that affect both the system and the output with the descriptor kalman filtering method it is shown that directly applying the conventional descriptor kalman filter dkf to the considered problem may not yield the globally optimal solution because the unknown input vector may not be estimable to remedy this problem three approaches are proposed to facilitate optimal filter design the transformed approach uses some input and output transformations the untrammeled approach does not require any transformations and the augmented approach reconstructs the unknown input dynamics then three 5 block forms of the extended dkf 5 block edkf are derived as globally optimal state estimators in the sense that the first two filters are equivalent to the recently developed extended recursive three step filter and the third is equivalent to the conventional augmented state kalman filter the relationship between the proposed edkfs and the existing results in the literature is addressed simulation results are given to illustrate the usefulness of the proposed filters

kalman filtering for general discrete time linear systems recursive state estimation problems for explicit and implicit time invariant linear systems both for systems with and without unknown inputs can be formulated as a single problem usually referred to as descriptor kalman filtering solutions to this problem have been proposed in the literature however these solutions either neglect possible contributions of future dynamics to the current estimate or make unnecessary assumptions on the structure of the system in this paper the authors propose a solution to this problem which leads to a constructive method lifting these unnecessary assumptions this method uses a generalization of the shuffle algorithm

h infinity kalman estimation for rectangular descriptor systems with unknown inputs this note considers h filtering and prediction for rectangular descriptor systems with unknown inputs that affect both the system and the output an optimal h descriptor kalman estimator hdke which can simultaneously solve the h filtering and prediction problems for rectangular descriptor systems with unknown inputs is developed based on the maximum likelihood descriptor kalman filtering method the hdke serves as a unified solution to solve h and kalman filtering for descriptor systems and standard systems with or without unknown inputs to reduce the computational complexity problem some efficient alternatives to the developed hdke are further proposed the relationship between the hdke and the existing literature results is also addressed an illustrative example is given to show the usefulness of the proposed results

brief risk sensitive filtering prediction and smoothing for discrete time singular systems this paper is concerned with steady state risk sensitive filtering prediction and smoothing problems for discrete time singular systems it is shown that a risk sensitive estimator can be obtained by ensuring the minimum of an indefinite quadratic form to be maximum minimum when the risk sensitivity parameter q is negative positive an auxiliary state space signal model and an innovation sequence in krein space are introduced to simplify the derivation of the estimator the estimator is calculated based on one j spectral factorization for risk seeking q 0 a numerical example is given to demonstrate the applicability of the result

brief paper recursive linear estimation for general discrete time descriptor systems this paper considers the optimal linear estimates recursion problem for discrete time linear systems in its more general formulation the system is allowed to be in descriptor form rectangular time variant and with the dynamical and measurement noises correlated we propose a new expression for the filter recursive equations which presents an interesting simple and symmetric structure convergence of the associated riccati recursion and stability properties of the steady state filter are provided

stable rationality of certain pgln quotients stable rationality of the field of matrix invariants m n m n pgl n is proved forn 5 andn 7 in combination with existing results this shows that v pgl n is stably rational wheneverv is an almost free representation ofpgl n andn divides 420 22 3 5 7

multiple window moving horizon estimation long horizon lengths in moving horizon estimation are desirable to reach the performance limits of the full information estimator however the conventional mhe technique suffers from a number of deficiencies in this respect first the problem complexity scales at least linearly with the horizon length selected which restrains from selecting long horizons if computational limitations are present second there is no monitoring of constraint activity inactivity which results in conducting redundant constrained minimizations even when no constraints are active in this study we develop a multiple window moving horizon estimation strategy mw mhe that exploits constraint inactivity to reduce the problem size in long horizon estimation problems the arrival cost is approximated using the unconstrained full information estimator arrival cost to guarantee stability of the technique a new horizon length selection criterion is developed based on the sensitivity between remote states in time the development will be in terms of general causal descriptor systems which includes the standard state space representation as a special case the potential of the new estimation algorithm will be demonstrated with an example showing a significant reduction in both computation time and numerical errors compared to conventional mhe

filtering and lqg problems for discrete time stochastic singular systems discrete time stochastic singular systems are discussed filtering and linear quadratic gaussian lqg problems are considered the problems are first transformed into the normal form via state augmentation and then solved by utilizing standard results for nonsingular systems the linear unbiased least squares state estimation algorithm and the optimal control law for the lqg problem are given the order of the filtering algorithm obtained in this way is not much increased moreover this algorithm allows the presence of some kinds of control inputs

optimal unbiased reduced order filtering for discrete time descriptor systems via lmi abstract in this paper the reduced order unbiased optimal h filtering problem for discrete time descriptor linear systems is considered necessary and sufficient conditions for the existence of unbiased causal time invariant filters are obtained solutions of the filtering problems are given in terms of linear matrix inequalities lmis parametrization of all optimal causal unbiased filters is provided the proposed method generalizes the existing results in the standard case

brief paper minimax state estimation for linear discrete time differential algebraic equations this paper presents a state estimation approach for an uncertain linear equation with a non invertible operator in hilbert space the approach addresses linear equations with uncertain deterministic input and noise in the measurements which belong to a given convex closed bounded set a new notion of a minimax observable subspace is introduced by means of the presented approach new equations describing the dynamics of a minimax recursive estimator for discrete time non causal differential algebraic equations daes are presented for the case of regular daes it is proved that the estimator s equation coincides with the equation describing the seminal kalman filter the properties of the estimator are illustrated by a numerical example

probability guaranteed set membership filtering for systems with incomplete measurements in this note the probability guaranteed set membership filtering problem is investigated for a class of time varying systems with incomplete measurements the uniform distribution over known finite ranges is assumed for mutually independent stochastic variables existing in the functions of system matrices the process noise and measurement noise are unknown bounded and confined to a specified ellipsoidal set respectively which is handled by transformation to the inequality constraints the purpose is to design a set membership filter for all the admissible unknown but bounded noises known deterministic input and incomplete measurements such that the ellipsoidal set including all possible states can be determined by a convex optimization approach with probability constraint a recursive linear matrix inequality rlmi approach is proposed to address the probability guaranteed set membership filtering problem finally a simulation example demonstrates the effectiveness of the proposed filtering scheme

h sub spl infin filtering for singular systems this note considers the h sub spl infin filtering problem for linear continuous singular systems the purpose is the design of a linear filter such that the resulting error system is regular impulse free and stable while the closed loop transfer function from the disturbance to the filtering error output satisfies a prescribed h sub spl infin norm bound constraint without decomposing the original system matrices a necessary and sufficient condition for the solvability of this problem is obtained in terms of a set of linear matrix inequalities lmis when these lmis are feasible an explicit expression of a desired filter is given finally an illustrative example is presented to demonstrate the applicability of the proposed approach