reliable numerical methods for polynomial matrix triangularization numerical procedures are proposed for triangularizing polynomial matrices over the field of polynomial fractions and over the ring of polynomials they are based on two standard polynomial techniques sylvester matrices and interpolation in contrast to other triangularization methods the algorithms described in this paper only rely on well worked numerically reliable tools they can also be used for greatest common divisor extraction polynomial rank evaluation or polynomial null space computation

random attractors for quasi continuous random dynamical systems and applications to stochastic reaction diffusion equations abstract this paper first introduces the so called quasi continuous random dynamical system rds on a separable banach space the quasi continuity is weaker than all the usual continuities and thus is easier to check in practice we then establish a necessary and sufficient condition for the existence of random attractors for the quasi continuous rds we also give a general method to obtain the random attractors for the rds on the banach space l q d for q 2 as an application it is shown that the rds generated by the stochastic reaction diffusion equation possesses a finite dimensional random attractor in l q d for any q 2 a comparison result of fractal dimensions under the different l q norms is also obtained

the standard ℋ2 optimal control problem a polynomial solution in this paper we present a solution to the standard 2 control problem the solution presented here is based upon the algebra of polynomials and in particular we use the polynomial equation approach in considering the scalar version of the standard 2 problem we derive a couple of linear polynomial diophantine equations which define the optimal regulator the coefficients of these equations are obtained from polynomial spectral factorization

h sub infinity optimal control with state feedback a h sub infinity optimal control problem in which the measured outputs are the states of the plant is considered the main result shows that the infimum of the norm of the closed loop transfer function using linear static state feedback equals the infimum of the norm of the closed loop transfer function over all stabilizing dynamic even nonlinear time varying state feedback controllers

polynomial lq optimization for the standard control structure scalar solution abstract the problem of lq optimization for the standard control structure is studied we obtain a solution for the scalar case of this problem using the polynomial equation approach in solving the optimal regulation problem using polynomial techniques we derive a couple of linear polynomial equations which together define the unique optimal regulator the coefficients of these equations are obtained from polynomial spectral factorization

necessary and sufficient conditions under which an h 2 optimal control problem has a unique solution a set of necessary and sufficient conditions under which a general h 2 optimal control problem has a unique solution is derived it is shown that the solution for an h 2 optimal control problem if it exists is unique if and only if i the transfer function from the control input to the controlled output is left invertible and ii the transfer function from the disturbance to the measurement output is right invertible

mixed h2 h problems approximated solutions by galerkin methods this paper presents a galerkin method to solve the mixed h2 h optimal control problem in this method the generator set is built step by step by an optimization procedure algorithm convergence is proved in the setting of weighted hardy spaces some examples are discussed showing the accelerated convergence obtained by this new algorithm and its numerical features a particular quadratic optimal control problem on finite dimensional linear systems under robustness constraints is developed as a motivation for the mixed h2 h problem

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

the attractor for a nonlinear reaction diffusion system in an unbounded domain in this paper the quasi linear second order parabolic systems of reaction diffusion type in an unbounded domain are considered our aim is to study the long time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions to this end we give a systematic study of given parabolic systems and their attractors in weighted sobolev spaces dependence of the hausdorff dimension of attractors on the weight of the sobolev spaces is considered 2001 john wiley sons inc

brief wiener hopf design of the optimal decoupling control system with state space formulas the optimal decoupling controller minimizing a given quadratic cost function is derived for the generalized plant model with two degree of freedom controller configuration a minimal set of assumptions for the existence of the optimal controller is presented in both the frequency and the state space domains the optimal controller formula is described in the frequency domain and the corresponding state space formula is derived for computational efficiency

an algebraic riccati equation approach to h optimization abstract this paper considers the general so called four block h optimal control problem with the assumption that system states are available for feedback it is shown that infimization of the h norm of the closed loop transfer function over all linear constant i e nondynamic stabilizing state feedback laws can be completely characterized via an algebraic riccati equation it is further shown that the optimal norm is not improved by allowing feedback to be dynamic thus the general state feedback h optimal control problem can be solved by iteratively solving one are and the controller can be chosen to be static gain

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

polynomial solution of the standard multivariable spl hscr sub 2 optimal control problem in this work the authors solve the standard multivariable spl hscr sup 2 optimal control problem using polynomial matrix techniques

riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices until recently it was believed that a necessary and sufficient condition for convergence of the riccati difference equation of optimal filtering was that the system be both delectable and stabilizable recently it has been shown that the stabilizability condition can be removed but convergence has only established under restrictive assumptions including the requirement that the state transition matrix be nonsingular the present paper generalizes these results in several directions first properties of the algebraic riccati equation are established for the case of singular state transition matrix second several assumptions previously imposed in establishing convergence of the riccati difference equation for systems with unreachable modes on the unit circle are relaxed including replacing observability by detectability weakening the conditions on the initial covariance and allowing the state transition matrix to be singular third results on the convergence and properties of the riccati equations are expressed as both necessary and sufficient conditions whereas previous results were only sufficient these extensions mean that the results have wider applicability including fixed lag smoothing problems and filtering for systems with time delays the implications of the results in the dual problem of optimal control are also studied

the attractors for the nonhomogeneous nonautonomous navier stokes equations abstract in this paper we consider the attractors for the two dimensional nonautonomous navier stokes equations in nonsmooth bounded domain with nonhomogeneous boundary condition u on assuming f f x t l loc 2 0 t d a 4 which is translation compact and l we establish the existence of the uniform attractor in l 2 and d a 1 4

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

the singular h 2 control problem in this paper we discuss the standard h2 control problem for linear finite dimensional time invariant systems without any assumptions on the system parameters we give an explicit formula for the infimum over all internally stabilizing strictly proper compensators and give a characterization when the infimum is attained

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

optimal boundary interpolation and one block optimal control problem with invariant zeros on the imaginary axis and infinity we formulate a matrix interpolation problem with existing interpolation points on the imaginary axis and infinity and existing equal left and right interpolation points using the concept of parametrisation of stabilising controllers then we solve the problem of obtaining all its solutions if interpolation points at infinity are absent we show that the introduced problem is equivalent to the existing one we apply this result to solve the problem of optimal interpolation with existent interpolation points on the imaginary axis and infinity we show by an example that the solution of optimal interpolation is directly applicable to the one block optimal control with existent invariant zeros on the imaginary axis and infinity it is seen from the example that not only the transfer matrix of the closed loop system is constrained on the extended imaginary axis but also its derivatives

polynomial design of stochastic tracking systems a new technique to design optimal stochastic tracking systems is presented for single input output plants described by rational transfer functions the optimal controller is shown to consist of feedback and feedforward parts and it is designed by solving two couples of linear polynomial equations whose coefficients are obtained by spectral factorization

on general ℋ2 control from frequency to time domain this article presents a general formula for discrete time 2 control it works with the regular and singular case of 2 control i e in the case of possibly non left invertible matrices g 12 and non right invertible matrices g 21 with possible unit circle invariant zeros in the generic case it can be simplified and adapted to work with the plant transfer matrix directly without invoking the matrices of the parameterisation of stabilising controllers a further result of this article is the presented necessary and sufficient conditions for state space 2 control under only stabilisability and detectability assumptions if the conditions are satisfied an observer based 2 controller is constructed the corresponding numerical algorithm consists of solving two discrete time algebraic riccati systems darss and two eigen problems

wiener hopf design of optimal decoupling controllers for plants with non square transfer matrices an important consideration in the design of many control systems is decoupling each command input controlling only one output the two degree of freedom or three degree of freedom controller structure is ideally suited to this task since only one of the degrees of freedom is affected by the decoupling requirement the decoupling constraint on this degree of freedom however can lead to some deterioration in achievable system performance a wiener hopf design procedure which minimizes this performance deterioration is described here for plants with non square transfer matrices moreover the parametrization of the class of all decoupled systems is given for which the cost functional representing system performance remains finite

technical communique wiener hopf approach to derivation of rational doubly coprime factorizations this paper presents a method to construct a stable rational doubly coprime factorization dcf of a transfer matrix proper or improper from a given polynomial dcf wiener hopf factor polynomials are used to rationalize the polynomial dcf matrices it is shown that the suggested construction procedure results in the stable rational dcf description that includes the normalized matrix fractional description mfd and inner coprime mfd as special cases

phaseless inverse scattering problems in three dimensions three dimensional inverse scattering problems in the frequency domain are considered in the case when only the modulus of the scattered field is given while the phase is unknown uniqueness theorems are proved

h 2 design of one degree of freedom decoupling controllers for square plants decoupling design of one degree of freedom controllers is treated for the generalised plant model within the h 2 framework the class of all realisable closed loop transfer matrices is parametrised under a mild assumption on non unity feedback sensors a set of compact assumptions is presented to guarantee the existence of the optimal h 2 controller together with the optimal one the class of all h 2 controllers that yield finite h 2 cost is obtained the optimal closed transfer matrix and its associated controller are shown to be strictly proper under the reasonable order assumptions on the generalised plant

h2 optimal controllers with measurement feedback for continuous time systems flexibility in closed loop pole placement for a general h 2 optimal control problem at first all h 2 optimal measurement feedback controllers are characterized and parameterized and then attention is focused on controllers with observer based architecture both full order as well as reduced order observer based h 2 optimal controllers are characterized and parameterized also systematic methods of designing them are presented an important problem coined as an h 2 optimal control problem with simultaneous pole placement is formulated and solved that is since in general there exist many h 2 optimal measurement feedback controllers utilizing such flexibility and freedom we can solve the problem of simultaneously placing the closed loop poles at desirable locations whenever possible while still preserving h 2 optimality all the design algorithms developed here are easily computer implementable

the h2 control problem a general transfer function solution a general solution of the h2 control problem is presented for linear systems described by rational transfer matrices not necessarily proper or stable the control system is considered in the standard configuration which includes the synthesis model of the plant and the controller the h2 control problem consists of internally stabilizing the control system while minimizing the h2 norm of its transfer function the notion of internal stability is based on bounded input bounded output stability and means that subsystems defined by any pair of input and output signals within the control system all are bounded input bounded output stable in this manner the optimal control system is devoid of impulsive as well as non decaying exponential modes the solution proceeds in three steps first the set of all controllers that internally stabilize the control system is parameterized then the subset of the stabilizing controllers that achieve a finite value of the h2 norm of the control system transfer matrix is

a general theory for the wiener hopf design of multivariable control systems multivariable control system design in a quadratic cost setting is treated for a completely general configuration the class of all controllers is determined for which the general configuration is internally asymptotically stable and a quadratic cost functional is finite this controller class is parameterized in terms of an arbitrary real rational strictly proper matrix z s which is analytic in the closed right hand side of the s plane the choice z identical to 0 gives the controller which minimizes the quadratic cost functional

fifo stable control systems abstract motivated by the dead beat and shortest correlation control strategies the paper introduces the notion of finite input finite output fifo stability for linear discrete time control systems a simple parametrization of all controllers that fifo stabilize a given plant is obtained the best controller is then chosen for various applications including h 2 optimization disturbance rejection and transfer function shaping

simplified formula for a controller in optimal control problems we derive a simplified frequency domain formula for a controller transfer matrix for lq optimal output feedback control of stochastic systems for this purpose we apply a generalization of the wiener hopf method the generalization is characterized by the following three properties 1 we generalize the usual operator cdot on rational matrices in the traditional wiener hopf approach the same algorithms as for computation of cdot are applicable to compute the new operator 2 the matrices of the youla kuccera parametrization do not appear in the optimal controller transfer matrix bf c they appear only in its derivation even if the plant is unstable 3 unlike the traditional wiener hopf method d c youla h a jabr and j j bongiorno jr ieee trans automat control ac 21 1976 pp 319 338 where the spectral factors in the two spectral factorizations are both stable and minimum phase our spectral factors need to be minimum phase only finally three state space applications of the formula are presented

algorithms for optimal control with invariant zeros on the extended imaginary axis abstract we transform an control problem with invariant zeros on the extended imaginary axis into a regular problem then using the nevanlinna pick theorem we find necessary and sufficient existence conditions for the original control problem an algorithm and examples with invariant zeros on the extended imaginary axis are given by the same approach we also solve the singular 2 control problem

two single shot methods for locating multiple electromagnetic scatterers we develop two inverse scattering schemes for locating multiple electromagnetic em scatterers by the electric far field measurement corresponding to a single incident detecting plane wave the first scheme is for locating scatterers of small size compared to the wavelength of the detecting plane wave the multiple scatterers could be extremely general with an unknown number of components and each scatterer component could be either an impenetrable perfectly conducting obstacle or a penetrable inhomogeneous medium with an unknown content the second scheme is for locating multiple perfectly conducting obstacles of regular size compared to the detecting em wavelength the number of the obstacle components is not required to be known in advance but the shape of each component must be from a certain known admissible class the admissible class may consist of multiple different reference obstacles the second scheme could also be extended to include the medium components if a certain generic condition is s

lq control of descriptor systems a spectral factorisation approach a spectral factorisation algorithm of improper matrices and existence results under controllability conditions are presented based on these results an algorithm for linear quadratic lq control of rectangular descriptor systems without the detectability and infinite observability conditions is presented the matrix pencil associated with the euler lagrange differential equations can be singular can have finite generalised eigenvalues on the imaginary axis and infinite generalised eigenvalues of any multiplicity in the class of optimal state feedback controls we find a control that renders the closed loop system marginally stable and impulse free also we find conditions on x 0 that guarantee the optimality it is shown how the disturbance attenuation problem of descriptor systems can be re formulated as an optimal state feedback feedforward control problem for which the results of this article can be applied when applied to state space systems the algorithm of this article which is based on

construction and parameterization of all static and dynamic h sub 2 optimal state feedback solutions optimal fixed modes and fixed decoupling zeros a problem of h sub 2 optimization via state feedback is considered the problems dealt with are of the general singular type with a left invertible transfer matrix function from the control input to the controlled output all the static and dynamic h sub 2 optimal state feedback solutions are constructed and parameterized and all the eigenvalues of an optimal closed loop system are characterized all optimal closed loop systems share a set of eigenvalues which are termed the optimal fixed modes which must be assigned among the closed loop eigenvalues this set includes a set of optimal fixed decoupling zeros which shows the minimum absolutely necessary number and location of pole zero cancellations present in any h sub 2 optimal design it is shown that both the sets of optimal fixed modes and optimal fixed decoupling zeros do not vary

disturbance attenuation and h optimization a design method based on the algebraic riccati equation this note presents a method for designing a state feedback control law to reduce the effect of disturbances on the output of a given linear system the problem under consideration involves attenuating the disturbances to a prespecified level the construction of the state feedback control law requires the solution of a certain algebraic riccati equation by applying the procedure with successively smaller values of the prespecifled disturbance level the result also provides an approach to the h infty optimization problem for the state feedback case

parameterization of decoupling controllers in the generalized plant model a diagonal decoupling problem in linear multi variable systems is formulated for the most general setting a necessary and sufficient condition for the existence of a decoupling controller is presented and when a solution exists the class of all decoupling controllers is characterized in terms of free parameters of stable rational vectors

wiener hopf design of optimal decoupled multivariable feedback control system an important consideration in the design of many control systems is decoupling each command input controlling only one output the 2dof two degree of freedom or 3dof three degree of freedom structure is ideally suited to this task since only one of the degrees of freedom is affected by the decoupling requirement the decoupling constraint on this degree of freedom however can lead to some deterioration in achievable system performance a design procedure which minimizes this performance deterioration is established the parameterization of the class of all decoupled systems is given for which the cost functional representing system performance remains finite

a feedback theory of two degree of freedom optimal wiener hopf design the design of linear two degree of freedom stabilizing controllers is treated in a quadratic cost setting the class of all such controllers which give finite cost is established and the tradeoff possible between optimum performance tracking cost sensitivity and stability margins is discussed

persistent inputs and the standard h 2 multivariable control problem the h 2 control problem is formulated with exogenous inputs having unstable shape deterministic components such inputs are called persistent some issues in connection with past solutions of the h 2 control problem in this case are carefully described and addressed in particular it is established that two and three degree of freedom 2dof and 3dof systems with persistent inputs cannot be treated within the framework of the standard configuration in addition past treatments of persistent inputs within the framework of the generalised 2dof configuration focused on solutions which yielded stable error transforms without explicitly requiring the same for controller outputs in this article a more general configuration is treated and physical considerations are invoked to justify imposition of the requirement that both the regulated variable z s and the controller output u s be stable a persistent input model for which there exists a stabilising controller that makes both z s and u s stable is cal

multiple objective optimal control of linear systems the quadratic norm case control of a linear system to meet requirements embodied in multiple performance criteria is recognized as the typical design problem the authors study the design of control when every criterion of interest takes the form of an h sub 2 norm defined over certain exogenous inputs and controlled outputs of the system the objective is to find all stabilizing controllers that are noninferior with respect to the defined objectives that is to find the set of pareto optimal controllers the novel feature is the use of the youla parametrization to represent the admissible controllers this representation enables the authors to show that each criterion and hence their weighted sum is convex over the class of admissible controllers since the set of admissible controllers is itself convex the pareto optimal controllers can be determined by the standard scalarization technique it is concluded that in general the controller order is higher than the order of the plant

the general distance problem in h optimal control theory abstract the general distance problem that arises in h optimal control theory is considered transformations that reduce the general h problem to the general distance problem are reviewed and an iterative scheme called iteration is considered in which the general distance problem is reduced to a standard best approximation problem the iteration is viewed as a problem of finding the zero crossing of a function this function is shown to be continuous monotonically decreasing convex and bounded by some very simple functions these properties make it possible to obtain very rapid convergence of the iterative process the issues of model reduction and approximation in h synthesis will also be addressed

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

sampled data and discrete time h_2 optimal control this paper deals with the sampled data h 2 optimal control problem given a linear time invariant continuous time system the problem of minimizing the h 2 performance over all sampled data controllers with a fixed sampling period can be reduced to a pure discrete time h 2 optimal control problem this discrete time h 2 problem is always singular motivated by this in this paper we give a treatment of the discrete time h 2 optimal control problem in its full generality the results we obtain are then applied to the singular discrete time h 2 problem arising from the sampled data h 2 problem in particular we give conditions for the existence of optimal sampled data controllers we also show that the h 2 performance of a continuous time controller can always be recovered asymptotically by choosing the sampling period sufficiently small finally we show that the optimal sampled data h 2 performance converges to the continuous time optimal h 2 performance as the sampling period converges to zero

a coprime factorisation and its application in ℋ2 control given a plant transfer matrix as a fraction of two polynomial matrices we present a doubly coprime factorisation identity over the proper stable rational matrices of the plant transfer matrix and

near peak scattering and the corrected kirchhoff approximation for a convex obstacle abstract the uniform asympotic behavior of the scattering amplitude near the forward peak in the case of classical scattering of waves by a convex obstacle is derived a microlocal model is obtained for the scattering operator this is achieved by use of a parametrix for diffractive boundary problems and by a new study of a class of fourier integral operators those with folding canonical relations a crucial ingredient consists of putting a fourier integral operator with folding canonical relation into a normal form the analysis also gives the asymptotic behavior of the normal derivative of the scattered wave on a neighbourhood of the shadow boundary thus providing a corrected version of the kirchhoff approximation

the lqg ltr procedure for multivariable feedback control design this paper provides a tutorial overview of the lqg ltr design procedure for linear multivariable feedback systems lqg ltr is interpreted as the solution of a specific weighted h2 tradeoff between transfer functions in the frequency domain properties of this solution are examined for both minimum phase and nonminimum phase systems this leads to a formal weight augmentation procedure for the minimum phase case which permits essentially arbitrary specification of system sensitivity functions in terms of the weights while such arbitrary specifications are not possible for nonminimum phase problems a direct relationship between weights and sensitivities is developed for nonminimum phase siso and certain nonminimum phase mimo cases which guides the weight selection process

recovering scattering obstacles by multi frequency phaseless far field data abstract it is well known that the modulus of the far field pattern or phaseless far field pattern is invariant under translations of the scattering obstacle if only one plane wave is used as the incident field so the shape but not the location of the obstacle can be recovered from the phaseless far field data this paper aims to devise an approach to break the translation invariance property of the phaseless far field pattern to this end we make use of the superposition of two plane waves rather than one plane wave as the incident field in this paper it is mathematically proved that the translation invariance property of the phaseless far field pattern can indeed be broken if superpositions of two plane waves are used as the incident fields for all wave numbers in a finite interval furthermore a recursive newton type iteration algorithm in frequencies is also developed to numerically recover both the location and the shape of the obstacle simultaneously from multi frequency phaseless far field data numerical examples are also carried out to illustrate the validity of the approach and the effectiveness of the inversion algorithm

h2 output feedback control for descriptor systems this paper solves the h 2 output feedback control problem for continuous time descriptor systems an existence condition of the h 2 optimal controller is derived based on the standard model matching technique and a characterization of the optimal controller is given in terms of the solutions of two generalized algebraic riccati equations it turns out that the optimal controller has an observer based structure well known in the state space case so that the results in this paper are natural extensions of the state space results to the descriptor systems a numerical example is also included to show the applicability of the proposed results

derivation of the maximum entropy h controller and a state space formula for its entropy the problem of maximizing the entropy of a stabilized closed loop system is solved the solution exploits the parametrization of all closed loop systems that meet an h norm bound the central solution of this set is shown to maximize the entropy at infinity and for this case a particularly simple formula is derived for the maximum value of the entropy

analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators we deal with an inverse obstacle problem for general second order scalar elliptic operators with real principal part and analytic coefficients near the obstacle we assume that the boundary of the obstacle is a non analytic hypersurface we show that when we put dirichlet boundary conditions one measurement is enough to reconstruct the obstacle in the neumann case we have results only for n 2 3 in general more precisely we show that one measurement is enough for n 2 and we need 3 linearly independent inputs for n 3 however in the case for the helmholtz equation we only need n 1 linearly independent inputs for any n 2 here n is the dimension of the space containing the obstacle these are justified by investigating the analyticity properties of the zero set of a real analytic function in addition we give a reconstruction procedure for each case to recover the shape of obstacle although we state the results for the scattering problems similar results are true for the associated boundary value problems

existence of a global attractor for semilinear dissipative wave equations on rn abstract we consider the semilinear hyperbolic problem utt ut x u f u x x r n t 0 with the initial conditions u x 0 u0 x and ut x 0 u1 x in the case where n 3 and x 1 g x lies in ln 2 r n the energy space x 0 d 1 2 r n l2g r n is introduced to overcome the difficulties related with the noncompactness of operators which arise in unbounded domains we derive various estimates to show local existence of solutions and existence of a global attractor in x 0 the compactness of the embedding d 1 2 r n l2g r n is widely applied

sensitivity minimization in an h norm parametrization of all suboptimal solutions a number of problems in the control of linear feedback systems can be reduced to the following we are given three stable rational matrix functions k of sizes p 1 x q 1 p 1 x q 2 and p 2 x q 1 respectively and seek a stable rational q 2 x p 2 matrix function s so as to minimize k s we assume that p 1 q 2 p 2 q 1 and that and have maximal rank q 2 and p 2 respectively on the j axis given a tolerance level sufficiently large we obtain a linear fractional map g f 0 11 g 0 12 0 13 0 21 g 0 22 0 23 1 such that f k s with s stable and f if and only if g is a stable q 2 x p 2 matrix function with g 1 the computation of 0 0 ij 1 i 2 1 j 3 reduces to solving a pair of symmetric wiener hopf factorization problems for the special case where i q2 0 t i p2 0 and k not necessarily stable to which the general case can be reduced we provide explicit state space formulae for 0 in terms of a state space realization of k and the

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

wiener hopf design of the general two degree of freedom controllers and the connection to state space formulas h 2 design of the two degree of freedom controller configuration is treated for the generalized plant in both the frequency and state space domains a frequency domain wiener hopf formula is derived first under a set of compact assumptions that gives the entire class of controllers for which the feedback control system is stable and a quadratic cost functional is finite the h 2 design method suggested in this paper allows the reference inputs to include the persistent signals such as step or ramp functions and the class of persistent reference inputs that can be tracked with zero steady state error is characterized explicitly to circumvent the computational difficulties of the wiener hopf formulas the corresponding state space domain formulas are derived by rationalizing the polynomial fractional descriptions it is shown that the well known standard time domain assumptions for the h 2 design problem have the corresponding equivalent frequency domain conditions and this equivalence relationship is show

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

polynomial lqg regulator design for general system configurations a polynomial equation approach to the linear quadratic gaussian lqg regulation problem for a general system configuration is presented the solution is given in terms of a left spectral factorization plus a pair of bilateral diophantine equations the resulting control design procedure is based on an innovations representation of the system this can be obtained from a physical description by solving via polynomial equations a minimum mean square error filtering problem the use in cascade of the above two procedures allows one to generalize previous polynomial design results to general system configurations

mathscr h _ infty control of descriptor systems possessing invariant zeros on the imaginary axis and infinity we present an algorithm for suboptimal mathscr h infty control of descriptor systems possessing invariant zeros on the imaginary axis and infinity it is proved that the algorithm works also in an optimality case the algorithm is illustrated by examples

extension of linear quadratic optimal control theory for mixed backgrounds the problem of finding an optimal controller while both deterministic and stochastic signals are present in the feedback system is addressed a general framework is developed to solve the above problem and to unify conventional frequency domain lq and lqg optimal control theories both the system transient and steady state behaviours are considered and the exact tracking of the arbitrary reference signal is guaranteed the decomposition of a mixed signal into the deterministic and stochastic parts gives a chance to investigate the effect of the weighting factors which were specified in the cost function on the compromise between transient and steady state system performance the solution which characterizes the structure of the optimal controller results in a set of independent diophantine equations instead of a set of coupled ones moreover the number of equations can be reduced under weak conditions finally both transient command tracking and steady state noise rejection capabilities can be op

recovering a polyhedral obstacle by a few backscattering measurements abstract we propose an inverse scattering scheme of recovering a polyhedral obstacle in r n n 2 3 by only a few high frequency acoustic backscattering measurements the obstacle could be sound soft or sound hard it is shown that the modulus of the far field pattern in the backscattering aperture possesses a certain local maximum behavior from which one can determine the exterior normal directions of the front sides faces then by using the phaseless backscattering data corresponding to a few incident plane waves with suitably chosen incident directions one can determine the exterior unit normal vector of each side face of the obstacle after the determination of the exterior unit normals the recovery is reduced to a finite dimensional problem of determining a location point of the obstacle and the distance of each side face away from the location point for the latter reconstruction we need to make use of the far field data with phases numerical experiments are also presented to illustrate the effectiveness of the proposed scheme

an algorithm for polynomial matrix factor extraction an algorithm is described for extracting a polynomial matrix factor featuring any subset of the zeros of a given non singular polynomial matrix it is assumed that the zeros to be extracted are given as input data complex or repeated zeros are allowed the algorithm is based on interpolation and relies upon numerically reliable subroutines only it makes use of a procedure that computes the generalized characteristic vectors of a polynomial matrix at a given point the extracted factor is provided in column and row reduced popov form applications of the algorithm include polynomial matrix interpolation plus minus factorization column and row reduction or computation of the smith form of a polynomial matrix the numerical routines described in this paper are implemented in the new release 2 0 of the polynomial toolbox for matlab

a high frequency hp boundary element method for scattering by convex polygons in this paper we propose and analyze a hybrid hp boundary element method for the solution of problems of high frequency acoustic scattering by sound soft convex polygons in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution we demonstrate both theoretically and via numerical examples exponential convergence with respect to the order of the polynomials moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern in which the dependence on the frequency of all constants is explicit importantly these estimates prove that to achieve any desired accuracy in the computation of these quantities it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases in contrast to the at least linear growth required by conventional methods

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

a polynomial approach to the mimo lq servo and disturbance rejection problems abstract a novel matrix fraction solution to the discrete time lq stochastic tracking and servo problems is extended to the disturbance rejection problem unlike recent contributions here the resulting control law at time t is given in terms of an expectation conditioned to the accessible disturbance given up to time t k if k is positive and large enough the optimal feedforward input can be tightly approximated without requiring any stochastic model for the disturbance if the latter is known also for recurrent or predictable disturbance the optimal feedforward input is obtained by solving a single bilateral diophatine equation

comments on optimizing simultaneously over the numerator and denominator polynomials in the youla kucera parameterization it is noted that the parameterization of the set of stabilizing regulators was first presented in a monograph by larin v b naumenko k i and suntsev v n

general optimal attenuation of harmonic disturbance with unknown frequencies we present algorithms for optimal harmonic disturbance attenuation in standard discrete time control structure based on a parametrisation of marginally stabilising controllers the frobenius norm and the spectral norm of the closed loop transfer matrix at the disturbance frequencies are minimised if there is only one frequency of the disturbance the controller has an observer based form which we obtain by solving a static output feedback sof stabilisation control problem although the sof stabilisation problem is hard the generical case of nonsquare matrix g 22 is solved by linear algebra methods numerical simulation results are presented as a corollary we transform the control problem with unit circle invariant zeros into a control problem without such zeros the elimination of the unit circle invariant zeros is based on the fact that matrix y zi a bf 1 is stable where y f with y 0 is a solution of a discrete time algebraic riccati system

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft

on the wiener hopf approach to optimal feedback design abstract a simple solution is given to the problem of designing a proper feedback which minimizes the h 2 norm of the sensitivity function multiplied by a suitable weighting function

synthesis of linear multivariable regulators this paper generalizes the results obtained by the authors in 1 and presents new results on the construction of proper controllers for linear multivariable regulators these results may lead to efficient computation methods that do not involve smith forms or the order relations used in 1 to construct proper controllers the methods are based on the recently developed theory of generalized polynomials 2 we discuss several numerical algorithms and how they can be applied to the regulator problem we also point out an important computational problem in need of an efficient numerical algorithm