condensed cramer rule for some restricted quaternion linear equations abstract in this paper we derive an alternative more condensed cramer rule for the unique solution of some restricted left and right systems of quaternion linear equations the findings of this paper extend some known results in the literature

an iterative method for the skew symmetric solution and the optimal approximate solution of the matrix equation axb c in this paper an iterative method is constructed to solve the linear matrix equation axb c over skew symmetric matrix x by the iterative method the solvability of the equation axb c over skew symmetric matrix can be determined automatically when the equation axb c is consistent over skew symmetric matrix x for any skew symmetric initial iterative matrix x 1 the skew symmetric solution can be obtained within finite iterative steps in the absence of roundoff errors the unique least norm skew symmetric iterative solution of axb c can be derived when an appropriate initial iterative matrix is chosen a sufficient and necessary condition for whether the equation axb c is inconsistent is given furthermore the optimal approximate solution of axb c for a given matrix x 0 can be derived by finding the least norm skew symmetric solution of a new corresponding matrix equation email protected email protected finally several numerical examples are given to illustrate that our iterative method is effective

determinantal representation of the generalized inverses over the quaternion skew field with applications we first present some determinantal representations of one 1 5 inverse of a quaternion matrix within the framework of a theory of the row and column determinants as applications we show some new explicit expressions of generalized inverses a r t 1 s 1 2 a l t 2 s 2 2 and a t 1 t 2 s 1 s 2 2 over the quaternion skew field finally we give the representations of the unique solution to some restricted left and right systems of quaternionic linear equations the findings of this paper extend some known results in the literature

some comments on the behavior of sign sign adaptive identifiers abstract the sign sign variant of the lms algorithm with and without leakage is investigated three basic questions are considered through simplistic examples the need to redefine input excitations for parameter convergence the influence of designer selected variables on offset due to leakage and the effect of measurement noise in underexcited systems

the reflexive and anti reflexive solutions of the matrix equation ax b abstract an n n complex matrix p is said to be a generalized reflection matrix if p h p and p 2 i an n n complex matrix a is said to be a reflexive or anti reflexive matrix with respect to the generalized reflection matrix p if a pap or a pap this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti reflexive with respect to a generalized reflection matrix p solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm have been provided

a cramer rule for minimum norm t least squares s solution of inconsistent linear equations abstract in this paper the minimum norm t least squares s solution of inconsistent linear equations ax y a c m n r y r a is given a determinantal form and reduces to common cramer s rule if a is nonsingular

a cramer rule for least norm solutions of consistent linear equations abstract the least euclidean norm solution of a consistent linear equation ax b is given a determinantal form which reduces to cramer s rule if a is nonsingular

brief paper estimation of the disturbance structure from data using semidefinite programming and optimal weighting designing a state estimator for a linear state space model requires knowledge of the characteristics of the disturbances entering the states and the measurements in odelson b j rajamani m r rawlings j b 2006 a new autocovariance least squares method for estimating noise covariances automatica 42 2 303 308 the correlations between the innovations data were used to form a least squares problem to determine the covariances for the disturbances in this paper we present new and simpler necessary and sufficient conditions for the uniqueness of the covariance estimates we also formulate the optimal weighting to be used in the least squares objective in the covariance estimation problem to ensure minimum variance in the estimates a modification to the above technique is then presented to estimate the number of independent stochastic disturbances affecting the states this minimum number of disturbances is usually unknown and must be determined from data a semidefinite optimization problem is solved to estimate the number of independent disturbances entering the system and their covariances

a cramer rule for the least norm least squared error solution of inconsistent linear equations abstract a recent cramer rule for obtaining the least norm solution of a consistent system of linear equations ax b is shown to also provide the least squared error solution in the inconsistent case i e the solution a b where a is the moore penrose inverse of a

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

three results in connection with inverse matrices abstract first we show that the moore penrose solution of an arbitrary system of linear equations is a convex combination of the solutions of all uniquely solvable partial systems the other two results concern the elements of inverse toeplitz band matrices namely the asymptotic behavior of a determinant appearing in a formula of d s meek and a modification of a formula of w d hoskins and p j ponzo for matrices with binomial coefficients in the limit case

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

generalized inverses of morphisms with kernels abstract let x y be a morphism with kernel k x in an additive category with an involution then has a moore penrose inverse with respect to iff is invertible in this case 1 if x y then has a group inverse iff has a cokernel x k and 2 is invertible in this case 2 1

an iterative method for the least squares symmetric solution of the linear matrix equation axb c abstract this paper an iterative method is presented to solve the minimum frobenius norm residual problem min axb c with unknown symmetric matrix x by this iterative method for any initial symmetric matrix x0 a solution x can be obtained within finite iteration steps in the absence of roundoff errors and the solution x with least norm can be obtained by choosing a special kind of initial symmetric matrix in addition the unique optimal approximation solution x to a given matrix x in frobenius norm can be obtained by first finding the least norm solution x of the new minimum residual problem min a x b c with unknown symmetric matrix x where c c a x x t 2 b given numerical examples are show that the iterative method is quite efficient

bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations in this paper we consider bisymmetric and centrosymmetric solutions to certain matrix equations over the real quaternion algebra necessary and sufficient conditions are obtained for the matrix equation ax c and the following systemsa 1x c 1 a 1x c 1 xb 3 c 3 a 2x c 2 to have bisymmetric solutions and the systema 1x c 1 a 3xb 3 c 3 to have centrosymmetric solutions the expressions of such solutions of the matrix and the systems mentioned above are also given moreover a criterion for a quaternion matrix to be bisymmetric is established and some auxiliary results on other sets over are also mentioned

computing thecs decomposition of a partitioned orthonormal matrix this paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns

cramer s rule for some quaternion matrix equations abstract cramer s rules for some left right and two sided quaternion matrix equations are obtained within the framework of the theory of the column and row determinants

asymptotically efficient parameter estimation using quantized output observations this paper studies identification of systems in which only quantized output observations are available an identification algorithm for system gains is introduced that employs empirical measures from multiple sensor thresholds and optimizes their convex combinations strong convergence of the algorithm is first derived the algorithm is then extended to a scenario of system identification with communication constraints in which the sensor output is transmitted through a noisy communication channel and observed after transmission the main results of this paper demonstrate that these algorithms achieve the cramer rao lower bounds asymptotically and hence are asymptotically efficient algorithms furthermore under some mild regularity conditions these optimal algorithms achieve error bounds that approach optimal error bounds of linear sensors when the number of thresholds becomes large these results are further extended to finite impulse response and rational transfer function models when the inputs are designed to be periodic and full rank

completing a symmetric 2 2 block matrix and its inverse abstract we consider the following completion problems suppose n1 n2 are nonnegative integers such that n 1 n 2 n 0 let a11 a12 a21 b22 be matrices with dimensions n1 n1 n1 n2 n2 n1 and n2 n2 respectively we determine necessary and sufficient conditions so that there exists an n2 n2 matrix a22 such that a a 11 a 21 a 12 a 22 and i a is nonsingular and symmetric and b22 is the lower right block of a partitioning of a 1 ii a is symmetric positive definite and b22 is the lower right block of a partitioning of a 1

a cramer rule for solution of the general restricted matrix equation a cramer rule for finding the unique w weighted drazin inverse solution of a class of restricted matrix equationswawxwbw d r x r aw k 1 n x n wb k 2 is presented and the results in linear algebra appl 116 1989 27 appl math comput 125 2002 303 can be deduced from this paper where email protected c m x n email protected c n x m ind aw k 1 email protected c p x q email protected c q x p ind wb email protected 2 and email protected c n x p

an input output construction of finite state rho mu approximations for control design we consider discrete time plants that interact with their controllers via fixed discrete alphabets for this class of systems and in the absence of exogenous inputs we propose a general methodical approach for constructing a sequence of finite state approximate models starting from finite length sequences of input and output signal pairs we explicitly derive conditions under which the proposed construct used in conjunction with a particular generalized structure satisfies desirable properties of approximations thereby leading to nominal deterministic finite state machine models that can be used in certified by design controller synthesis we also show that the cardinality of the min imal disturbance alphabet that can be used in this setting equals that of the sensor output alphabet finally we show that the pro posed construct satisfies a relevant semi completeness property

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

supervisory control of hybrid systems within a behavioural framework abstract this contribution addresses the synthesis of supervisory control for hybrid systems with discrete external signals such systems are in general neither l complete nor can they be represented by finite state machines we find an l complete approximation abstraction l for represent it by a finite state machine and investigate the control problem for the approximation if a solution exists we synthesize the maximally permissive supervisor for l we show that it also solves the control problem for the hybrid system if no solution exists approximation accuracy can be increased by computing a k complete abstraction k k l this paper is entirely set within the framework on willems behavioural systems theory

convergence analysis of self adaptive equalizers a theoretical analysis of self adaptive equalization for data transmission is carried out starting from known convergence results for the corresponding trained adaptive filter the development relies on a suitable ergodicity model for the sequence of observations at the output of the transmission channel thanks to the boundedness of the decision function used for data recovery it can be proved that the algorithm is bounded strong convergence results can be reached when a perfect noiseless equalizer exists the algorithm will converge to it if the eye pattern is initially open otherwise convergence may take place towards certain other stationary points of the algorithm for which domains of attraction have been defined some of them will result in a poor error rate the case of a noisy channel exhibits limit points for the algorithm that differ from those of the classical trained algorithm the stronger the noise the greater the difference is one of the principal results of this study is the proof of the stability of the usual decision feedback algorithms once the learning period is over

iterative methods for solving restricted linear equations in this paper the iterative methods for computing some solution of the general restricted linear equations ax b x et are established and the iterative methods for finding the solution of several special linear equations considered are constructed

the general solution to a system of real quaternion matrix equations in this paper we consider the system of matrix equations a 1x c 1 a 2x c 2 a 3xb 3 c 3 and a 4xb 4 c 4 over the real quaternion algebra a necessary and sufficient condition for the existence and the expression of the general solution to the system are given as particular cases the corresponding results on other systems over are also obtained

the number of connected components in a graph associated with a rectangular 0 1 matrix abstract with a nonzero rectangular 0 1 matrix a we associate an undirected graph g a that corresponds to the linear transformation x a x t a we use the generalized singular value decomopsition of incidence matrices to count the number of connected components and the number of bipartite connected components of g a we show that g a is connected if and only if g a has no bipartite connected component or isolated vertex we also show that if a has no row or column of 0 s then the number of connected components is 1 2 s s 1 and the number of bipartite connected components is 1 2 s s 1 where s is the number of chainable components of a that is the number of connected components in the undirected graph with adjacency matrix o a a t o

minimum norm hamiltonian solutions of a class of generalized sylvester conjugate matrix equations abstract in this study we consider the iteration solutions of the generalized sylvester conjugate matrix equation a x b c x d e by a modified conjugate gradient method when the system is consistent the convergence theorem shows that a solution can be obtained within finite iterative steps in the absence of round off error for any initial value given hamiltonian matrix furthermore we can get the minimum norm solution x by choosing a special kind of initial matrix finally some numerical examples are given to demonstrate the algorithm considered is quite effective in actual computation

cramer rule for the unique solution of restricted matrix equations over the quaternion skew field in this paper we establish the determinantal representations of the generalized inverses a r t 1 s 1 2 a l t 2 s 2 2 and a t 1 t 2 s 1 s 2 2 over the quaternion skew field by the theory of the column and row determinants in addition we derive some generalized cramer rules for the unique solution of some restricted quaternion matrix equations the findings of this paper extend some known results in the literature

an iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation axb c an iteration method is constructed to solve the linear matrix equation axb c over symmetric x by this iteration method the solvability of the equation axb c over symmetric x can be determined automatically when the equation axb c is consistent over symmetric x its solution can be obtained within finite iteration steps and its least norm symmetric solution can be obtained by choosing a special kind of initial iteration matrix furthermore its optimal approximation solution to a given matrix can be derived by finding the least norm symmetric solution of a new matrix equation axb c finally numerical examples are given for finding the symmetric solution and the optimal approximation symmetric solution of the matrix equation axb c

transient and tracking performance bounds of the sign sign algorithm the paper provides a rigorous tracking analysis of the sign sign algorithm when used in the identification of a time varying plant with a white gaussian input the plant parameters vary according to a random walk model the assumptions allow nonstationarity of the plant input plant noise and increments of the plant parameters upper bounds are derived for the long term averages of the mean of the weight misalignment norm mean absolute error mean square weight misalignment and mean square error these bounds hold for all values of the algorithm step size all initial filter weight settings and all degrees of nonstationarity of the plant input plant noise and plant parameter increments lower bounds of the mean square weight misalignment and mean square error are also derived the step sizes that minimize the above bounds are derived a transient analysis of the algorithm is done in the case of a time invariant plant a tight lower bound of the convergence time is derived the above analytical results are supported by computer simulations

on the symmetric solutions of linear matrix equations abstract the symmetric solutions of linear matrix equations ax c and axb c are considered the necessary and sufficient conditions for the consistency of the equations with a symmetric condition on solutions are derived using the singular value decomposition and the generalized singular value decomposition numerical algorithms for solving the equations are suggested

a characterization and determinantal formula for the generalized inverse at s 2 and its applications in this paper we present a characterization and determinantal formula for the generalized inverse a t s 2 as applications on the one hand we get the corresponding results for many common important generalized inverses such as a a m n a d a and so on on the other hand we apply these results to solve restricted linear equations ax b email protected t

condensed cramer rule for solving restricted matrix equations a cramer rule for solving a kind of restricted matrix equationswawxw bw d r x r aw k 1 n x n w b k 2 is presented in g wang jie sun a cramer rule for solution of the general restricted matrix equation appl math comput 154 2004 415 422 this paper gives a more condensed cramer rule for the solution of the matrix equations the results in g wang z l xu solving a kind of restricted matrix equations and cramer rule appl math comput 162 2005 329 338 chao gu g r wang z l xu pcr algorithm for the parallel computation of the solution of a class of singular linear systems appl math comput 176 2006 237 244 are partially the special cases in our paper

convergence analysis of an adaptive filter equipped with the sign sign algorithm convergence with probability 1 of the sign sign algorithm with a decreasing step size is proven when it is used in adaptive plant identification the plant input and plant noise are assumed stationary and m dependent the assumptions do not pose any upper bound on the memory m of the plant input and plant noise and hence they allow arbitrarily strong correlation in these signals a main interest of the paper is that it provides a condition of persistent excitation of the algorithm examples satisfying this condition are provided

symmetric and skew antisymmetric solutions to systems of real quaternion matrix equations in this paper we consider symmetric and skew antisymmetric solutions to certain matrix equations over the real quaternion algebra h first a criterion for a quaternion matrix to be symmetric and skew antisymmetric is given then necessary and sufficient conditions are obtained for the matrix equation ax c and the following system a 1x c 1xb 3 c 3 to have symmetric and skew antisymmetric solutions the expressions of such solutions of the matrix equation and the system mentioned above are also given

quiver diagrams and signed adaptive filters a graphical technique for exploring the behavior of deterministic discrete time adaptive algorithms excited by periodic inputs is examined the constraint to periodic inputs is convenient the underlying concepts used also apply to stationary inputs drawn from finite alphabets the resulting graph is based on plotting single period parameter trajectories in the parameter error space and is essentially a discrete version of flow diagrams associated with continuous time systems for the least mean square lms adaptive fir filter with the algorithm step size sufficiently small the change of adapted parameters from a particular initial value is given by the total update over the input period with the parameterization frozen at its initial value this results in the familiar elliptical hyperparaboloid surface steepest descent interpretation this method can be extended to the lms adaptive infinite input response iir filter with suitable approximations based on a small step size

determinantal representations of the drazin inverse over the quaternion skew field with applications to some matrix equations abstract within the framework of the theory of the column and row determinants we obtain determinantal representations of the drazin inverse both for hermitian and arbitrary matrices over the quaternion skew field using the obtained determinantal representations of the drazin inverse we get explicit representation formulas analogs of cramer s rule for the drazin inverse solutions of a quaternion matrix equation axb d and consequently ax d and xb d in two cases if a b are hermitian or arbitrary

solving a kind of restricted matrix equations and cramer rule the problem of finding determinantal formula for solutions of some restricted linear systems ax b has been discussed in linear multilinear algebra 15 1984 319 linear algebra appl 116 1989 27 linear multilinear algebra 34 1993 177 appl math comput 125 2002 303 this paper deals with some more extensive cases of this kind of problem and establishes determinantal formulas for solutions of the restricted matrix equationsax d r x r a k 1 xb d n x n b k 2 axb d r x r a k 1 n x n b k 2 where a c n x n with ind a k 1 b c m x m with ind b k 2 and d c n x m the results in linear multilinear algebra 15 1984 319 linear algebra appl 74 1986 213 linear multilinear algebra 34 1993 177 appl math comput 125 2002 303 are partially the special cases in our paper the classic cramer rule is also a special case of our results

the p q generalized reflexive and anti reflexive solutions of the matrix equation ax b abstract an n n complex matrix p is said to be a generalized reflection matrix if p h p and p 2 i an n n complex matrix a is said to be a p q generalized reflexive or anti reflexive matrix with respect to the generalized reflection matrix dual p q if a paq or a paq this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the p q generalized reflexive and anti reflexive solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm has been provided

the hermitian p k 1 anti reflexive solutions of a linear matrix equation in this paper we first establish the necessary and sufficient conditions for the existence and the explicit expressions of the hermitian p k 1 anti reflexive solutions of the matrix equation a x b and meanwhile the best approximation solution is considered then if the solvability conditions are not satisfied the least squares hermitian p k 1 anti reflexive solutions and the least squares hermitian p k 1 anti reflexive solutions with the minimum norm of the above matrix equation are respectively derived in addition two algorithms are shown to compute the least squares hermitian p k 1 anti reflexive solutions and the corresponding numerical examples are also given to illustrate the feasibility of the algorithms

new matrix iterative methods for constraint solutions of the matrix equation axb c in this paper two new matrix iterative methods are presented to solve the matrix equation axb c the minimum residual problem min x email protected email protected and the matrix nearness problem min x s email protected x x where s is the set of constraint matrices such as symmetric symmetric r symmetric and r s symmetric and s e is the solution set of above matrix equation or minimum residual problem these matrix iterative methods have faster convergence rate and higher accuracy than the matrix iterative methods proposed in deng et al 2006 13 huang et al 2008 15 peng 2005 16 and lei and liao 2007 17 paige s algorithms are used as the frame method for deriving these matrix iterative methods numerical examples are used to illustrate the efficiency of these new methods

asynchronous l complete approximations abstract this paper extends the l complete approximation method developed for time invariant systems to a larger system class ensuring that the resulting approximation can be realized by a finite state machine to derive the new abstraction method called asynchronous l complete approximation an asynchronous version of the well known concepts of state property memory span and l completeness is introduced extending the behavioral systems theory in a consistent way

convergence of the signed output error adaptive identifier this paper considers an adaptive output error identifier with a signum function in its update kernel it is shown that the classical strict positive real spr condition required for the convergence of traditional adaptive identifiers does not suffice for the convergence of this signed identifier instead what is needed is a stronger operator condition called the strict dominant passive sdp condition we give an analog of the kalman yakubovic popov lemma for the sdp conditions and using it give a convergence proof under the assumptions of persistent excitation and sdp

ranks of least squares solutions of the matrix equation axb c for a complex matrix equation axb c we solve the following two problems 1 the maximal and minimal ranks of least square solution x to axb c and 2 the maximal and minimal ranks of two real matrices x 0 and x 1 in least square solution x x 0 ix 1 to axb c we also give a necessary and sufficient condition for matrix equations a ix ib i c i i 1 2 to have a common least square solution

analogs of cramer s rule for the minimum norm least squares solutions of some matrix equations abstract the least squares solutions with the minimum norm of the matrix equations ax b xa b and axb d are considered in this paper we use the determinantal representations of the moore penrose inverse obtained earlier by the author and get analogs of the cramer rule for the minimum norm least squares solutions of these matrix equations

pcr algorithm for parallel computing minimum norm t least squares s solution of inconsistent linear equations this paper presents a new highly parallel algorithm for computing the minimum norm t least squares s solution of inconsistent linear equations ax b email protected r m x n r email protected r a by this algorithm the solution x a s t b is obtained in t 1 m 1 log 2m n 6 log 2 n r 1 log 2m log 2n r 1 log 2n steps with p mn processors when m 2 n 1 and with p 2n n 1 processors otherwise

cramer s rule for a system of quaternion matrix equations with applications abstract in this paper we investigate cramer s rule for the general solution to the system of quaternion matrix equations a 1 x b 1 c 1 a 2 x b 2 c 2 and cramer s rule for the general solution to the generalized sylvester quaternion matrix equation a x b c y d e respectively as applications we derive the determinantal expressions for the hermitian solutions to some quaternion matrix equations the findings of this paper extend some known results in the literature

discrete approximation and supervisory control of continuous systems addresses the following hybrid control problem a continuous plant its state evolving in euclidean space is to be controlled via symbolic output feedback both measurement and control signal live on finite sets of symbols we adopt the following approach the hybrid problem is first translated into a purely discrete problem by approximating the continuous plant model by a nondeterministic finite state machine by taking into account past measurement and control symbols approximation accuracy can be improved and adjusted to the specification requirements supervisory control theory for discrete event systems des is then applied to find the optimal controller which enforces the specifications as the behavior of the approximating automaton is guaranteed to contain the behavior of the underlying continuous plant model the controller also forces the latter to obey the specifications

a condensed cramer s rule for the minimum norm least squares solution of linear equations abstract in this paper we will derive the condensed cramer s rule of werner for minimal norm least squares solution of linear equations ax b from the cramer s rule of ben israel and verghese in addition a new condensed cramer s rule will be obtained in this paper

persistence of excitation conditions and the convergence of adaptive schemes in the study of the behavior of adaptive filtering algorithms persistence of excitation of the input process arises as a sufficient condition for convergence and perhaps more importantly for convergence rate of the parameter estimates in this paper the underlying nature of the persistence requirement is presented and discussed relating its normal specification in terms of moment conditions with covariance decays etc to sample path properties deterministic and stochastic persistence conditions and persistence measures are treated as well as persistence conditions for output error equation error and adaptive control schemes

optimal control of discrete time hybrid automata under safety and liveness constraints abstract in this contribution we address an optimal control problem for a class of discrete time hybrid automata under safety and liveness constraints the solution is based on a hierarchical decomposition of the problem where the low level controller enforces safety and liveness constraints while the high level controller exploits the remaining degrees of freedom for performance optimization lower level control is based on a discrete abstraction of the continuous dynamics the action of low level control can be interpreted as restricting invariants in the hybrid automaton representing the plant model a state feedback solution for the high level control is provided based on the off line construction of an appropriate partition of the state space

stabilization of nonlinear systems by qualitative feedback controllers the paper considers the problem of stabilizing a non linear plant by means of a qualitative controller u k k x k that has access only to the qualitative value x k of the system state x k and can only determine one of a set of qualitative values u k of the command input u k necessary and sufficient conditions for the existence of a stabilizing qualitative controller and a procedure for determining such a controller are given these conditions and the design algorithm refer merely to a qualitative model of the system rather than the precise state space description the results are illustrated by a numerical example

a new iteration method for the matrix equation ax b an iteration method for the matrix equation ax b is constructed by this iteration method the least norm solution for the matrix equation can be obtained when the matrix equation is consistent and the least norm least squares solutions can be obtained when the matrix equation is not consistent the related optimal approximation solution is also obtained by this iteration method a preconditioned method for improving the iteration rate is put forward finally some numerical examples are given

adaptive filtering with binary reinforcement recently there has been increased interest in high speed adaptive filtering where the usual stochastic gradient or least mean square lms algorithm is replaced with the simpler algorithm where adaptation is guided only by the polarity of the error signal in this paper the convergence of this binary reinforcement br algorithm is proved under the usual independence assumption and the surprising observation is made that unlike the lms algorithm convergence occurs for any positive value of the step size parameter while the stochastic gradient algorithm attempts to minimize a mean square error cost function the binary reinforcement algorithm in fact attempts to minimize a mean absolute error cost function it is proved for the binary reinforcement algorithm that the tap weight vector converges in distribution to a random vector that is suitably concentrated about the optimal value based on a least mean absolute error cost function for a sufficiently small step size the expected cost of the asymptotic weight vector can be made as close as desired to thc minimum cost attain ed by the optimum weight vector

the hermitian reflexive solutions to the matrix inverse problem ax b in this paper the necessary and sufficient conditions for the solvability of the matrix inverse problem ax b in hermitian reflexive matrix set hj n x n are given and the solution expression of this matrix inverse problem is also presented moreover an optimal approximation to any matrix in the above solution set in frobenius norm is provided

linear matrix equations from an inverse problem of vibration theory abstract the symmetric positive semidefinite and positive definite real solutions of matrix equations a t xa d and a t xa xa y ad d 0 are considered necessary and sufficient conditions for the existence of such solutions and their general forms are derived using the singular value decomposition the theory is motivated and illustrated with a problem of vibration theory

characterizing persistent excitation for the sign sign equation error identifier abstract the sign sign ss algorithm is a computationally efficient adaptive identifier often used in signal processing tasks it is obtained by introducing signum functions on both the regressor and the prediction error multiplicands in the update kernel of the well known lms algorithm this paper gives a deterministic persistent excitation condition on the regressor sequence which guarantees ss convergence it also gives conditions under which ss may diverge and discusses how these persistent excitation conditions may be verified through a finite amount of computations

on the symmetric solutions of a linear matrix equation a formula for the partitioned minimum norm reflexive generalized inverse is applied to find the general symmetric solution x to the matrix equation ax b also the dimension of the space of symmetric solutions is established

a cramer rule for finding the solution of a class of singular equations a cramer rule for finding the unique solution x r ak of singular equations ax b ind a k b r ak is given and reduces to the common cramer rule if a is a nonsingular

a new method for computing moore penrose inverse through gauss jordan elimination performing elementary row operations on a i we can calculate matrices whose columns form bases for n a and n a easily these matrices are then used to construct a bordered matrix through which the moore penrose inverse a of a general matrix a can be obtained through further elementary row operations our method is reduced to the classical gauss jordan elimination procedure for the regular inverse when applied to a nonsingular matrix an example is included to illustrate the new method

on the perturbation and subproper splittings for the generalized inverse at s 2 of rectangular matrix a abstract in this paper the perturbation and subproper splittings for the generalized inverse at s 2 the unique matrix x such that xax x r x t and n x s are considered we present lower and upper bounds for the perturbation of at s 2 convergence of subproper splittings for computing the special solution at s 2 b of restricted rectangular linear system ax b x t are studied for the solution at s 2 b we develop a characterization therefore we give a unified treatment of the related problems considered in literature by ben israel berman hanke neumann plemmons etc

the modified conjugate gradient methods for solving a class of generalized coupled sylvester transpose matrix equations in this paper we consider the iteration solutions of generalized coupled sylvester transpose matrix equations a1xb1 c1ytd1 f1 a2yb2 c2xtd2 f2 when the coupled matrix equations are consistent we propose a modified conjugate gradient method to solve the equations and prove that a solution x y can be obtained within finite iterative steps in the absence of roundoff error for any initial value furthermore we show that the minimum norm solution can be got by choosing a special kind of initial matrices when the coupled matrix equations are inconsistent we present another modified conjugate gradient method to find the least squares solution with the minimum norm finally some numerical examples are given to show the behavior of the considered algorithms

a characterization for the w weighted drazin inverse and a cramer rule for the w weighted drazin inverse solution we establish characterization for the w weighted drazin inverse of an arbitrary rectangular matrix which reduces to the well known result if the matrix is nonsingular also a cramer rule for finding the unique w weighted drazin inverse solution x r aw k 1 of special restricted linear equationswawx b b r wa k 2 is presented and reduces to the classical cramer rule if a is invertible

tracking analysis of the sign sign algorithm for nonstationary adaptive filtering with gaussian data this correspondence is concerned with the analysis of the sign sign algorithm ssa when used to track a plant with randomly time varying parameters the input of the plant and the plant noise are assumed stationary and gaussian the paper derives expressions of the steady state excess mean square error spl xi the steady state mean square weight misalignment spl eta and the step sizes that minimize each one of them the paper also presents a comparison among the tracking properties of the ssa the sign algorithm sa and the signed regressor algorithm sra it is found that the three algorithms share the features that 1 spl xi does not depend on the eigenvalue spread of the input covariance matrix 2 spl eta depends on the eigenvalue spread and 3 the step size that minimizes spl xi is different from the one that minimizes spl eta the minimum values of spl xi attained by the sa and the sra are equal to each other and they are 1 db less than the one attained by the ssa the ratio of the minimum value of spl eta of the ssa to the one of the sa is found to be dependent on the input eigenvalue spread for equal eigenvalues this ratio is equal to 1 db the minimum value of spl eta of the ssa is found to be 1 db higher than the one of the sra independently of the input eigenvalue spread it is found that an advantage of the ssa with respect to both the sa and the sra is that the two optimum step sizes of the ssa are independent of the mean square plant input and the mean square plant noise

sharf convergence properties a class of stable algorithms for adapting infinite impulse response iir digital filters based on the concepts of nonlinear stability theory prominent in the control literature is emerging while this class of adaptive filters offers much promise in practical applications little has been done toward providing a characterization that would guide selection of design parameters such as adaptation constants and error smoothing coefficients this paper focuses on the simplest well behaved member of this class of adaptive recursive filters sharf progression from a local linearization of the nonlinear parameter estimate convergence behavior through an idealized eigenvalue eigenvector analysis of the parameter estimate time varying recursion to lyapunov function establishment for the full output and parameter error system reveals the exponential local nongradient descent convergence character of sharf and provides initial insight into the effects of adaptation constants and error smoothing coefficients on these characteristics

explicit expressions of the generalized inverses and condensed cramer rules in this paper we obtain an explicit representation of the 2 inverse a 2 t s of a matrix a c m n with the prescribed range t and null space s as special cases new expressions for the moore penrose inverse a and drazin inverse a d are derived through explicit expressions we re derive the condensed cramer rules of werner for minimal norm least squares solution of linear equations ax b and propose two new condensed cramer rules for the unique solution of a class of singular system ax b x r a k b r a k k ind a finally condensed determinantal expressions for a a d a a a a a ndaa d are also presented

joint identification of plant rational models and noise distribution functions using binary valued observations system identification of plants with binary valued output observations is of importance in understanding modeling capability and limitations for systems with limited sensor information establishing relationships between communication resource limitations and identification complexity and studying sensor networks this paper resolves two issues arising in such system identification problems first regression structures for identifying a rational model contain non smooth nonlinearities leading to a difficult nonlinear filtering problem by introducing a two step identification procedure that employs periodic signals empirical measures and identifiability features rational models can be identified without resorting to complicated nonlinear searching algorithms second by formulating a joint identification problem we are able to accommodate scenarios in which noise distribution functions are unknown convergence of parameter estimates is established recursive algorithms for joint identification and their key properties are further developed

synthesis of supervisory controllers for hybrid systems based on approximating automata the paper concerns the synthesis of supervisory controllers for a class of continuous time hybrid systems with discrete valued input signals that select differential inclusions for continuous valued state trajectories and event valued output signals generated by threshold crossings in the continuous state space the supervisor is allowed to switch the input signal value when threshold events are observed the objective is to synthesize a nonblocking supervisor such that the set of possible sequences of control and threshold event pairs for the closed loop system lies between given upper and lower bounds in the sense of set containment we show how this problem can be converted into a supervisor synthesis problem for a standard controlled discrete event system des a finite representation may not exist for the exact des model of the hybrid system however to circumvent this difficulty we present an algorithm for constructing finite state muller automata that accept outer approximations to the exact controlled threshold event language and we demonstrate that supervisors that solve the synthesis problem for the approximating automata achieve the control specifications when applied to the original hybrid system

explicit formulas for determinantal representations of the drazin inverse solutions of some matrix and differential matrix equations the drazin inverse solutions of the matrix equations ax b xa b and axb d are considered in this paper we use both the determinantal representations of the drazin inverse obtained earlier by the author and in the paper we get analogs of the cramer rule for the drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations x ax b and x xa b where the matrix a is singular

identification input design for consistent parameter estimation of linear systems with binary valued output observations input design is of essential importance in system identification for providing sufficient probing capabilities to guarantee convergence of parameter estimates to their true values this paper presents conditions on input signals that characterize their probing richness for strongly consistent parameter estimation of linear systems with binary valued output observations necessary and sufficient conditions on periodic signals are derived for sufficient richness these conditions are further studied under different system configurations including open loop and feedback systems and different scenarios of noises including actuator noise input measurement noise and output measurement noise in addition to system parameter estimation essential properties of identifiability and input conditions are also derived when sensor thresholds or noise distribution functions are unknown the findings of this paper provide a foundation to study identification of systems that either use binary valued or quantized sensors or involve communication channels which mandate quantization of signals

solutions with special structure to the linear matrix equation ax b in this article we solve the following three kinds of problems first some formulas for the minimal rank of the submatrices in a solution x of matrix equation ax b and the minimal and maximal rank of x itself are derived by using the matrix rank method from these formulas necessary and sufficient conditions are given for x to be nonsingular or the submatrices to be zero respectively second some formulas for the minimal rank of x x x x and the corresponding expressions of submatrices of x are investigated combined with matrix decomposition necessary and sufficient conditions are given for the existence of solutions to be hermitian local hermitian skew hermitian and local skew hermitian respectively third necessary and sufficient conditions are given for the existence of solutions to be local positive negative semidefinite and for some hermitian solution with zero submatrix the structure form of a positive negative semidefinite solution are obtained using results of inertia