polynomial j spectral factorization several algorithms are presented for the j spectral factorization of a para hermitian polynomial matrix the four algorithms that are discussed are based on diagonalization successive factor extraction interpolation and the solution of an algebraic riccati equation respectively the paper includes a special algorithm for the factorization of unimodular para hermitian polynomial matrices and deals with canonical noncanonical and nearly noncanonical factorizations

partially stable lq optimal control by spectral factorization this paper presents a method for solving by spectral factorization the linear quadratic optimal control of a linear time invariant stabilizable system with an incompletely detectable state in the output in the integral of the cost function and a terminal partial stabilization constraint the solution by state feedback has been given by willems and callier 1983 the new method extends the polynomial matrix spectral factorization technique developed by kucera 1981 for the standard lq regulator with stabilization constraint an example and several comments add perspective to the results which are mainly intended to discover the correct spectral factorization equation

optimal robust stabilization and dissipativity synthesis by behavioral interconnection given a nominal plant together with a fixed neighborhood of this plant the problem of robust stabilization is to find a controller that stabilizes all plants in that neighborhood in an appropriate sense if a controller achieves this design objective we say that it robustly stabilizes the nominal plant in this paper we formulate the robust stabilization problem in a behavioral framework with control as interconnection we also formulate a relevant behavioral mathcal h infty synthesis problem which will be instrumental in solving the robust stabilization problem we use both rational and polynomial representations for the behaviors under consideration necessary and sufficient conditions for the existence of robustly stabilizing controllers are obtained using the theory of dissipative systems we will also find the optimal stability radius i e the smallest upper bound on the radii of the neighborhoods for which there exists a robustly stabilizing controller this smallest upper bound is expressed in terms of certain storage functions associated with nominal control system

discrete j spectral factorization of possibly singular polynomial matrices we develop two state space algorithms for discrete time j spectral factorization of possibly singular possibly non column reduced and with possible zeros at the origin para hermitian matrices by assignment of matrices in optimal lq return difference equality the column degrees of the spectral factor equal the column degrees of the given matrix and the dimension of a discrete time algebraic riccati system equals the sum of the column degrees necessary and sufficient condition for the least order property of our state space realizations is column reduction of the given matrix

on polynomial matrix spectral factorization by symmetric extraction we revise the 1963 davis algorithm 2 for the spectral factorization of a para hermitian nonnegative polynomial matrix phi by symmetric factor extraction this algorithm is careless about zeros at infinity by introducing the notion of diagonal reducedness of phi we obtain an easy sufficient test for the absence of zeros at infinity we show then how to get phi diagonally reduced by diagonal excess reduction steps similar to the oono and yasuura steps removing all zeros at infinity and then how to remove synunetrically finite zeros while keeping el diagonally reduced hence free of zeros at infinity this results in a revised symmetric extraction spectral factorization algorithm with monotone degree control an example shows the didactical conceptual simplicity of the method appropriate symmetric extraction is discovered by revising and discovering important particular one sided factor extraction properties of polynomial matrices

algorithm of j spectral factorization of polynomial matrices numerical algorithms for the canonical j factorization of polynomial matrices with respect to the imaginary axis are given the factorization problems for the non regular polynomial matrices and for polynomials whose determinants have zero roots are considered the constructed algorithms are not connected to the finding of roots of the polynomial the basic calculation procedure is the construction of the stabilizing solution of the matrix algebraic riccati equation

algorithms for multidimensional spectral factorization and sum of squares in this paper algorithms are developed for the problems of spectral factorization and sum of squares of polynomial matrices with n indeterminates and a natural interpretation of the tools employed in the algorithms is given using ideas from the theory of lossless and dissipative systems these algorithms are based on the calculus of 2n variable polynomial matrices and their associated quadratic differential forms and share the common feature that the problems are lifted from the original n variable polynomial context to a 2n variable polynomial context this allows to reduce the spectral factorization problem and the sum of squares problem to linear matrix inequalities lmi s to the feasibility of a semialgebraic set or to a linear eigenvalue problem

brief papers new results in state estimation and regulation an alternative technique to design linear state estimators and regulators is presented this technique is based on transfer matrix considerations the optimal regulator or estimator gain is obtained via spectral factorization and the solution of a simple equation in polynomial matrices this approach provides further insight displays the duality of estimation and control nicely and bridges the state space and frequency domain techniques the resulting design procedure is computationally attractive and particularly simple for system matrices in the observer or controller canonical form

variational analysis for a generalized spiked harmonic oscillator a variational analysis is presented for the generalized spiked harmonic oscillator hamiltonian operator d2 dx 2 bx 2 a x 2 x where is a real positive parameter the formalism makes use of a basis provided by exact solutions of schrodinger s equation for the gol dman and krivchenkov hamiltonian and the corresponding matrix elements that were previously found for all the discrete eigenvalues the method provides bounds which improve as the dimension d of the basis set is increased extension to the n dimensional case in arbitrary angular momentum subspaces is also presented by minimizing over the free parameter a we are able to reduce substantially the number of basis functions needed for a given accuracy

new solution to the inverse regulator problem by the polynomial matrix method abstract this paper gives a new criterion for the inverse regulator problem in terms of polynomial matrix fractions the generalized kalman equation in polynomial matrix form leads to a necessary and sufficient condition for a stable feedback law to be optimal for some unknown quadratic performance index this criterion consists of i a symmetric factorizability of a polynomial matrix and ii a polynomial type criterion that guarantees detectability the relationship between the riccati and kalman equations is also established finally an example is given to illustrate the new result in contrast with the existing result

on a schur algorithm approach to spectral factorization state space formulae abstract in some earlier publications 10 13 we indicated how the classical schur algorithm can be used as a method to solve a spectral factorization problem in the present paper indicate how to obtain a state space formulation of the theory and formulae for an iterative algorithm for spectral factorization using state space representations

dissipativity of pseudorational behaviors this paper studies dissipativity for a class of infinite dimensional systems called pseudorational in the behavioral context extending the finite dimensional counterpart we show that a pseudorational behavior is dissipative if and only if it admits a storage function or a dissipation function for its proof we derive a new necessary and sufficient condition for entire functions of exponential type in the so called paley wiener class to allow a symmetric factorization characterization of dissipative behaviors and linear quadratic optimal behaviors are also given

paper efficient algorithm for matrix spectral factorization an algorithm is presented for the spectral factorization of polynomial or rational matrices arising in optimal control and filtering theory as well as in network theory there are two versions of the algorithm one applicable to continuous time problems the other to discrete time ones both versions are based on newton s method feature quadratic convergence and provide a significant improvement in efficiency over the existing methods

on linear h sup spl infin equalization of communication channels as an alternative to existing techniques and algorithms we investigate the merit of the h sup spl infin approach to the linear equalization of communication channels we first give the formulation of all causal h sup spl infin equalizers using the results of and then look at the finite delay ease we compare the risk sensitive h sup spl infin equalizer with the mmse equalizer with respect to both the average and the worst case ber performances and illustrate the improvement due to the use of the h sup spl infin equalizer

discrete j spectral factorization abstract this paper deals with the j spectral factorization for general discrete rational matrices a simple approach based on the kalman filtering in krein space is proposed the main idea is to construct a stochastic state space filtering model in krein space such that the spectral matrix of the output is equal to the rational matrix to be factorized the spectral factor is then easily derived by using the generalized kalman filtering in krein space which is similar to the h 2 spectral factorization our approach unifies the treatment of the h 2 spectral factorization and the j spectral factorization the applications of the derived results in h and risk sensitive estimation for both nonsingular and singular systems are demonstrated

the properties of reduced order minimum variance filters for systems with partially perfect measurements the problem of the finite time reduced order minimum variance full state estimation of linear continuous time invariant systems is considered in cases where the output measurement is partially free of corrupting white noise components the structure of the optimal filter is obtained and a link between this structure and the structure of the system invariant zeros is established using expressions that are derived in closed form for the invariant zeros of the system simple sufficient conditions are obtained for the existence of the optimal filter in the stationary case the structure and the transmission properties of the stationary filter for general left invertible systems are investigated a direct relation between the optimal filter and a particular minimum order left inverse of the system is obtained a simple explicit expression for the filter transfer function matrix is also derived the expression provides an insight into the mechanism of the optimal estimation

symmetry groupoids and higher dimensional analogues abstract we give some history and motivation for the notion of groupoid as a powerful and flexible generalization of the notion of group stressing the links with symmetry and with paths in dynamical systems two dimensional groupoids are explained and their possibilities for new notions of symmetry are explored

on kublanovskaya s approach to the solution of the generalized latent value problem for functional lambda matrices in siam j numer anal 7 1970 pp 532 537 kublanovskaya proposed a quadratically convergent algorithm for computing the eigenvalues of the generalized latent value problem for lambda matrices in this contribution we show that kublanovskaya s algorithm does not constitute a newton iteration though it can be justified through its equivalence with the generalized rayleigh quotient an alternative iteration scheme that can provide a higher order of convergence is proposed

rational spectral factorization using state space methods abstract we describe a state space based algorithm to find the minimum phase spectral factor of any rational spectral density whether it be proper improper or polynomial

on a hardy space approach to the analysis of spectral factors the paper presents a study of several problems related to spectral factorizations we assume only a very weak form of coercivity for the p x p spectral function phi and look at the set wm of all rectangular p x m spectral factors the main object is the arithmetization of the geometry of the set of minimal stable spectral factors by employing hardy space techniques and the arithmetic of inner functions particular attention is paid to the study of various partial orders associated with the set of spectral factors

a parametrisation for dissipative behaviours the matrix case we study linear time invariant dynamical systems that are dissipative with respect to a generalised power defined by a quadratic differential form we address several cases in an increasing order of complexity and show how dissipative systems can be parametrised in this process we also establish a number of results for polynomial matrices that are of independent interest the present article is a generalisation of our earlier work that dealt with single input single output dissipative systems pendharkar i and pillai h k 2004 a parametrization for dissipative behaviours systems and control letters 51 123 132

least squares approximation of perfect reconstruction filter banks designing good causal filters for perfect reconstruction pr filter banks is a challenging task due to the unusual nature of the design constraints we present a new least squares ls design methodology for approximating prfbs that avoids most of these difficult constraints the designer first selects a set of subband analysis filters from an almost unrestricted class of rational filters then given some desired reconstruction delay this design procedure produces the causal and rational synthesis filters that result in the best least squares approximation to a prfb this technique is built on a multi input multi output mimo system model for filter banks derived from the filter bank polyphase representation using this model we frame the ls approximation problem for prfbs as a causal ls equalization problem for mimo systems we derive the causal ls solution to this design problem and present an algorithm for computing this solution the resulting algorithm includes a mimo spectral factorization that accounts for most of the complexity and computational cost for this design technique finally we consider some design examples and evaluate their performance

brief positive polynomial matrices and improved lmi robustness conditions recently several new lmi conditions for stability of linear systems have been proposed introducing additional slack variables to reduce the gap between conservative convex quadratic stability conditions and intractable non convex robust stability conditions in this paper we show that these improved lmi conditions can be derived with the help of some basic results on positive polynomial matrices the approach allows us to derive in a unifying way results in the state space and polynomial frameworks applications to robust stability analysis and robust stabilization of systems with multi linear parametric uncertainty are fully described

overcomplete discrete wavelet transforms with rational dilation factors this paper develops an overcomplete discrete wavelet transform dwt based on rational dilation factors for discrete time signals the proposed overcomplete rational dwt is implemented using self inverting fir filter banks is approximately shift invariant and can provide a dense sampling of the time frequency plane a straightforward algorithm is described for the construction of minimal length perfect reconstruction filters with a specified number of vanishing moments whereas in the nonredundant rational case no such algorithm is available the algorithm is based on matrix spectral factorization the analysis synthesis functions discrete time wavelets can be very smooth and can be designed to closely approximate the derivatives of the gaussian function

on the factorization of rational discrete time spectral densities in this paper we consider an arbitrary matrix valued rational spectral density phi z we show with a constructive proof that phi z admits a factorization of the form phi z w top z 1 w z where w z is stochastically minimal moreover w z and its right inverse are analytic in regions that may be selected with the only constraint that they satisfy some symplectic type conditions by suitably selecting the analyticity regions this extremely general result particularizes into a corollary that may be viewed as the discrete time counterpart of the matrix factorization method devised by youla in his celebrated work 48

generalized lqr control and kalman filtering with relations to computations of inner outer and spectral factorizations we investigate the generalized linear quadratic regulator lqr control where the dimension of the control input is strictly greater than the dimension of the controlled output and the weighting matrix on the control signal is singular the dual problem is the generalized kalman filtering where the dimension of the input noise process is strictly smaller than the dimension of the output measurement and the covariance of the observation noise is singular these two problems are intimately related to inner outer factorizations for nonsquare stable transfer matrices with square inners of the smaller size such inner outer factorizations are in turn related to spectral factorizations for power spectral density psd matrices whose normal ranks are not full we propose iterative algorithms and establish their convergence for inner outer and spectral factorizations which in turn solve the generalized lqr control and kalman filtering

on the lyapunov equation coinvariant subspaces and some problems related to spectral factorizations a geometric approach to stochastic realization theory and hence to spectral factorization problems has been developed by lindquist and picci 1985 1991 and lindquist et al 1995 most of this work was done abstractly fuhrmann and gombani 1998 adopted an entirely hardy space approach to this set of problems studying the set of rectangular spectral factors of given size for a weakly coercive spectral function the parametrization of spectral factors in terms of factorizations of related inner functions as developed in fuhrmann 1995 had to be generalized this led to a further understanding of the partial order introduced by lindquist and picci in the set of stable spectral factors in the present paper we study the geometry of finite dimensional coinvariant subspaces of a vectorial hardy space h 2 via realization theory emphasizing the role of the lyapunov equation in lifting the hardy space metric to the statespace domain we follow this by deriving state space formulas for rectangular spect

lossless and dissipative distributed systems this paper deals with linear shift invariant distributed systems by this we mean systems described by constant coefficient linear partial differential equations we define dissipativity with respect to a quadratic differential form i e a quadratic functional in the system variables and their partial derivatives the main result states the equivalence of dissipativity and the existence of a storage function or a dissipation rate the proof of this result involves the construction of the dissipation rate we show that this problem can be reduced to hilbert s 17th problem on the representation of a nonnegative rational function as a sum of squares of rational functions

spectral factorization by symmetric extraction for distributed parameter systems the spectral factorization problem of a scalar coercive spectral density is considered in the framework of the callier desoer algebra of distributed parameter system transfer functions criteria are given for the infinite product representation of a meromorphic coercive spectral density of finite order and for the convergence of infinite product representations of spectral factors i e for the convergence of the symmetric extraction method for solving the spectral factorization problem of such spectral density these convergence criteria are applied to the solution of the linear quadratic optimal control problem by spectral factorization for a specific class of semigroup hilbert state space systems with a riesz spectral generator the speed of convergence of the symmetric extraction method is also considered as an example a damped vibrating string model is handled

a toeplitz algorithm for polynomial j spectral factorization a block toeplitz algorithm is proposed to perform the j spectral factorization of a para hermitian polynomial matrix the input matrix can be singular or indefinite and it can have zeros along the imaginary axis the key assumption is that the finite zeros of the input polynomial matrix are given as input data the algorithm is based on numerically reliable operations only namely computation of the null spaces of related block toeplitz matrices polynomial matrix factor extraction and linear polynomial matrix equations solving

a polynomial approach to minimax frequency domain optimization of multivariable feedback systems the properties of a feedback system where the plant has rational transfer matrix h and the compensator has transfer matrix g can be characterized through the system functions s i hg and t g i hg 1 good disturbance attenuation robustness limited bandwidth and compensator roll off may be obtained by minimizing a criterion of the form z where z v s t w1 wls t w2 w2t v with respect to the compensator transfer matrix g here v w1 and w2 are suitable rational weighting matrices the solution of the problem can be reduced to a pair of matrix polynomial equations since the optimal solution is highly non unique special solutions with additional optimality properties are considered as well the paper includes a discussion of the numerical solution of the polynomial equations and of the question how to choose the weighting matrices to ensure required feedback system properties an example illustrates the results