a polynomial approach to robust estimation of uncertain discrete time systems the robust estimation problem for uncertain discrete time systems is treated in this paper considering parametric uncertainties in the polynomial framework the estimator is the minimiser of a cost function written in a epsilon contaminated form uncertainties are not formally modelled it is shown that in the present context the proposed approach can be connected to stochastic modelling of uncertainties the optimal robust estimator is obtained by computing a spectral factorisation and by solving a single diophantine equation an example shows the efficiency of the proposed method

quantum risk sensitive estimation and robustness this paper studies a quantum risk sensitive estimation problem and investigates robustness properties of the filter this is a direct extension to the quantum case of analogous classical results all investigations are based on a discrete approximation model of the quantum system under consideration this allows us to study the problem in a simple mathematical setting we close the paper with some examples that demonstrate the robustness of the risk sensitive estimator

polynomial matrix solution of the h infinity filtering problem and the relationship to riccati equation state space results a solution to the standard suboptimal h filtering problem is presented using a new polynomial systems ap proach the polynomial matrix solution is closely related to the recent h state space based riccati equation results and the links between the polynomial and state equation methodologies are demonstrated the h filter is derived by solving a special type of minimum variance filtering problem using a technique based on game theory the solution of this game problem in a stochastic setting by polynomial methods is novel the cal culation of the h filter involves a j spectral factorization and the solution of two coupled diophantine equations there are several computational advantages over previous polynomial methods for calculating h filters

game theory approach to h sub spl infin optimal discrete time fixed point and fixed lag smoothing optimal h sub spl infin fixed point and fixed lag discrete time smoothing estimators are developed by applying a game theory approach a deterministic discrete time game is defined where the estimator plays against nature nature determines the system initial condition the driving input and the measurement noise whereas the estimator tries to find an estimate that brings a prescribed cost function that is based on the error of the estimation at a fixed time instant to a saddle point equilibrium the latter estimate yields the h sub spl infin optimal fixed point smoothing differently from the usual case in h sub spl infin optimal estimation and control the critical value of the scalar design parameter of the smoothing game is obtained in closed form explicitly in the terms of the corresponding h sub 2 solution unlike the h sub 2 case the recursive application of the h sub spl infin fixed point smoothing algorithm does not lead to fixed lag smoothing in the h sub spl infin norm sense the h sub spl infin fixed lag smoothing filter is derived by augmenting the state vector of the system with additional delayed states

global optimal attitude estimation using uncertainty ellipsoids a deterministic attitude estimation problem for a rigid body in a potential field with bounded attitude and angular velocity measurement errors is considered an attitude estimation algorithm that globally minimizes the attitude estimation error is obtained assuming that the initial attitude the initial angular velocity and measurement noise lie within given ellipsoidal bounds an uncertainty ellipsoid that bounds the attitude and the angular velocity of the rigid body is obtained the center of the uncertainty ellipsoid provides point estimates and the size of the uncertainty ellipsoid measures the accuracy of the estimates the point estimates and the uncertainty ellipsoids are propagated using a lie group variational integrator and its linearization respectively the attitude and angular velocity estimates are optimal in the sense that the sizes of the uncertainty ellipsoids are minimized

robust finite horizon minimax filtering for discrete time stochastic uncertain systems we study a finite horizon robust minimax filtering problem for time varying discrete time stochastic uncertain systems the uncertainty in the system is characterized by a set of probability measures under which the stochastic noises driving the system are defined the optimal minimax filter has been found by applying techniques of risk sensitive lqg control the structure and properties of resulting filter are analyzed and compared to h and kalman filters

a game theory approach to robust discrete time h sub spl infin estimation the deterministic robust estimation problem is formulated in the discrete time case as a zero sum two person difference game in which a statistician plays against nature the statistician tries to estimate a linear combination of the states of an uncertain linear system which is driven by nature saddle point strategies for the statistician in the above game are difficult to find a suboptimal strategy is therefore considered which guarantees an h sub spl infin performance bound on the estimation error different patterns of the information that is available to the statistician are treated and the corresponding fixed point fixed lag and fixed interval smoothing strategies are derived

robust state space filtering under incremental model perturbations subject to a relative entropy tolerance this paper considers robust filtering for a nominal gaussian state space model when a relative entropy tolerance is applied to each time increment of a dynamical model the problem is formulated as a dynamic minimax game where the maximizer adopts a myopic strategy this game is shown to admit a saddle point whose structure is characterized by applying and extending results presented earlier in robust least squares estimation with a relative entropy constraint b c levy and r nikoukhah ieee trans inf theory vol 50 no 1 89 104 jan 2004 for static least squares estimation the resulting minimax filter takes the form of a risk sensitive filter with a time varying risk sensitivity parameter which depends on the tolerance bound applied to the model dynamics and observations at the corresponding time index the least favorable model is constructed and used to evaluate the performance of alternative filters simulations comparing the proposed risk sensitive filter to a standard kalman filter show a significant performance advantage when applied to the least favorable model and only a small performance loss for the nominal model

probability guaranteed h finite horizon filtering for a class of nonlinear time varying systems with sensor saturations in this paper the probability guaranteed h finite horizon filtering problem is investigated for a class of nonlinear time varying systems with uncertain parameters and sensor saturations the system matrices are functions of mutually independent stochastic variables that obey uniform distributions over known finite ranges attention is focused on the construction of a time varying filter such that the prescribed h performance requirement can be guaranteed with probability constraint by using the difference linear matrix inequalities dlmis approach sufficient conditions are established to guarantee the desired performance of the designed finite horizon filter the time varying filter gains can be obtained in terms of the feasible solutions of a set of dlmis that can be recursively solved by using the semi definite programming method a computational algorithm is specifically developed for the addressed probability guaranteed h finite horizon filtering problem finally a simulation example is given to illustrate the effectiveness of the proposed filtering scheme 2012 elsevier b v all rights reserved

robust h sub 2 filtering for uncertain systems with measurable inputs this paper deals with the robust minimum variance filtering problem for linear time varying systems subject to a measurable input and to norm bounded parameter uncertainty in the state and or the output matrices of the state space model the problem addressed is the design of linear filters having an error variance with a guaranteed upper bound for any allowed uncertainty and any input of bounded energy three types of input signals are considered a signal that is a priori known for the whole time interval an unknown signal of very large bandwidth that is perfectly measured on line and a large bandwidth signal that is measured ahead of time in a fixed preview time interval both the time varying finite horizon and stationary infinite horizon cases are treated

robust output feedback stabilization via risk sensitive control we consider a problem of robust linear quadratic gaussian lqg control for discrete time stochastic uncertain systems with partial state measurements for a finite horizon case the problem was recently introduced by petersen et al ieee trans automat control 45 2000 398 in this paper an infinite horizon extension of the results of petersen et al ieee trans automat control 45 2000 398 is discussed we show that for a broad class of uncertain systems under consideration a controller constructed in terms of the solution to a specially parameterized risk sensitive stochastic control problem absolutely stabilizes the stochastic uncertain system

minimax lqg control of stochastic partially observed uncertain systems we consider an infinite horizon linear quadratic minimax optimal control problem for stochastic uncertain systems with output measurement a new description of stochastic uncertainty is introduced using a relative entropy constraint for the stochastic uncertain system under consideration a connection between the worst case control design problem and a specially parametrized risk sensitive stochastic control problem is established using this connection a minimax optimal lqg controller is constructed which is based on a pair of algebraic matrix riccati equations arising in risk sensitive control it is shown that this minimax optimal controller absolutely stabilizes the stochastic uncertain system

real time calculation of a limiting form of the renyi entropy applied to detection of subtle changes in scattering architecture previously a new method for ultrasound signal characterization using entropy hf was reported and it was demonstrated that in certain settings further improvements in signal characterization could be obtained by generalizing to renyi entropy based signal characterization if r with values of r near 2 specifically r 1 99 m s hughes et al j acoust soc am 125 3141 3145 2009 it was speculated that further improvements in sensitivity might be realized at the limit r 2 at that time such investigation was not feasible due to excessive computational time required to calculate if r near this limit in this paper an asymptotic expression for the limiting behavior of if r as r 2 is derived and used to present results analogous to those obtained with if 1 99 moreover the limiting form if is computable directly from the experimentally measured waveform f t by an algorithm that is suitable for real time calculation and implementation

robust discrete time minimum variance filtering the bounded variance filtered estimation of the state of an uncertain linear discrete time system with an unknown norm bounded parameter matrix is considered an upper bound on the variance of the estimation error is found for all admissible systems and estimators are derived that minimize the latter bound we treat the finite horizon time varying case and the infinite time case where the nominal system model is time invariant in the special stationary case where it is known that the uncertain system is time invariant we provide a robust filter for all uncertainties that still keep the system asymptotically stable

brief paper adaptive divided difference filtering for simultaneous state and parameter estimation a novel adaptive version of the divided difference filter ddf applicable to non linear systems with a linear output equation is presented in this work in order to make the filter robust to modeling errors upper bounds on the state covariance matrix are derived the parameters of this upper bound are then estimated using a combination of offline tuning and online optimization with a linear matrix inequality lmi constraint which ensures that the predicted output error covariance is larger than the observed output error covariance the resulting sub optimal high gain filter is applied to the problem of joint state and parameter estimation simulation results demonstrate the superior performance of the proposed filter as compared to the standard ddf

analysis of digitized waveforms using shannon entropy a novel symbol oriented information theoretic approach to signal analysis is described based on application of shannon entropy it is shown that this can be a preferred alternate to conventional digital signal energy analysis by making an experimental comparison of scanned ultrasonic images formed from shannon entropy signal energy and log signal energy measurements in these images which were made by c scanning a plexiglas bar with drill hole defects the shannon entropy exhibits approximately two to three times the sensitivity of the two energy measures however the shannon entropy analysis of waveforms suffers from a severe fundamental defect it depends on the properties of the analog to digital converter adc used to acquire the waveform as well as the properties of the waveform itself as digitizer dynamic range is increased the shannon entropy ultimately saturates at a level governed by the properties of the digitizer alone this effect arises from the necessary limited bandwidth of a real adc if adc bandwidth were not limited the shannon entropy would grow without bound with increasing dynamic range to overcome this difficulty a variant of the shannon entropy is defined that depends only on the properties of the waveform it is then shown that this variant is sensitive to the same scattering features as is the shannon entropy for data acquired from the plexiglas specimen discussed initially furthermore the variant offers the same sensitivity improvements obtained from the shannon entropy analysis finally computation of this variant for waveforms having noise is described derivation of the variant is given in appendix a of this work several of its more interesting features are also discussed there derivation of the procedure required to compute the variant is given in appendix b

mixed norm h 2 h regulation and estimation the discrete time case abstract a discrete time h2 static output feedback design problem involving a constraint on h disturbance attenuation is addressed and state space formulae are derived the dual problem of discrete time dynamic estimation with an h error bound is also addressed these results are analogous to results obtained previously for the continuous time problem

a new approach to the word and conjugacy problems in the braid groups abstract a new presentation of the n string braid group b n is studied using it a new solution to the word problem in b n is obtained which retains most of the desirable features of the garside thurston solution and at the same time makes possible certain computational improvements we also give a related solution to the conjugacy problem but the improvements in its complexity are not clear at this writing

h sub infinity fixed lag smoothing filter for scalar systems the solution of the h sub infinity optimal linear fixed lag smoothing problem is considered using a polynomial matrix description for the discrete system the smoother is obtained from the solution of a linear equation and a spectral factorization calculation the pole zero properties of the optimal smoother are obvious in the polynomial representation and insights into the measurement noise rejection properties of the smoother are obtained allowance is made for both dynamic cost weighting and colored measurement noise the polynomial form of the smoothing filter may be incorporated into a self tuning algorithm and a simple adaptive smoother is discussed

finite horizon robust state estimation for uncertain finite alphabet hidden markov models with conditional relative entropy constraints we consider a robust state estimation problem for time varying uncertain discrete time homogeneous first order finite state finite alphabet hidden markov models hmms a class of time varying uncertain hmms is considered in which the uncertainty is sequentially described by a conditional relative entropy constraint on perturbed conditional probability measures given a realized observation sequence for this class of uncertain hmms the robust state estimation problem is formulated as a constrained optimization problem using a lagrange multiplier technique and a variational formula for conditional relative entropy the above problem is converted into an unconstrained optimization problem and a problem related to partial information risk sensitive filtering a measure transformation technique and an information state method are employed to solve this equivalent problem related to risk sensitive filtering a characterization of the solution to the robust state estimation problem is also presented

nonconvex optimization problem the infinite horizon linear quadratic control problem with quadratic constraints abstract we present a general method for the solution of the problem of minimization of a quadratic functional in a hilbert space over the intersection of some subspace and a set defined by special quadratic constraints many optimal control problems can be formulated in this abstract form as an example a stationary infinite horizon linear quadratic optimization problem with quadratic constraints is posed and solved

joint entropy of continuously differentiable ultrasonic waveforms this study is based on an extension of the concept of joint entropy of two random variables to continuous functions such as backscattered ultrasound for two continuous random variables x and y the joint probability density p x y is ordinarily a continuous function of x and y that takes on values in a two dimensional region of the real plane however in the case where x f t and y g t are both continuously differentiable functions x and y are concentrated exclusively on a curve t f t g t in the x y plane this concentration can only be represented using a mathematically singular object such as a schwartz distribution its use for imaging requires a coarse graining operation which is described in this study subsequently removal of the coarse graining parameter is accomplished using the ergodic theorem the resulting expression for joint entropy is applied to several data sets showing the utility of the concept for both materials characterization and detection of targeted liquid nanoparticle ultrasonic contrast agents in all cases the sensitivity of these techniques matches or exceeds sometimes by a factor of two that demonstrated in previous studies that employed signal energy or alternate entropic quantities

a comparison of shannon entropy versus signal energy for acoustic detection of artificially induced defects in plexiglas an experimental comparison of scanned images formed from shannon entropy total energy and log energy measurements of acoustic wave propagation in a plexiglas bar with drill hole defects shows that shannon entropy in the instance investigated is approximately two to three times as sensitive to the presence of small defects than are the two energy measures this suggests that the shannon entropy technique may be useful for flaw detection in materials that are too thick or lossy for the conventional use of acoustic energy for flaw detection

the rotating normal form of braids is regular abstract defined on birman ko lee monoids the rotating normal form has strong connections with the dehornoy s braid ordering it can be seen as a process for selecting between all the representative words of a birman ko lee braid a particular one called rotating word in this paper we construct for all n 2 a finite state automaton which recognizes rotating words on n strands proving that the rotating normal form is regular as a consequence we obtain the regularity of a definite normal form defined on the whole braid group

filtering and smoothing in an h sup infinity setting the problems of filtering and smoothing are considered for linear systems in an h sup infinity setting i e the plant and measurement noises have bounded energies are in l sub 2 but are otherwise arbitrary two distinct situations for the initial condition of the system are considered the initial condition is assumed known in one case while in the other the initial condition is not known but the initial condition the plant and measurement noise are in some weighted ball of r sup n xl sub 2 finite horizon and infinite horizon cases are considered necessary and sufficient conditions are presented for the existence of estimators both filters and smoothers that achieve a prescribed performance bound and algorithms that result in performance within the bounds are developed in case of smoothers the optimal smoother is also presented the approach uses basic quadratic optimization theory in a time domain setting as a consequence of which both linear time varying and time invariant systems can be considered with equal ease in the smoothing problem for linear time varying systems one considers only the finite horizon case

a simple algorithm for finding short sigma definite representatives abstract we describe a new algorithm which for each braid returns a quasi geodesic definite word representative defined as a braid word in which the generator i with maximal index i appears either only positively or only negatively

a frequency domain approach to the problems of h sub infinity minimum error state estimation and deconvolution the properties of the minimum h sub infinity norm filtering estimation error are investigated and the relation between the optimal estimator and the equalizing solution to the standard h sub infinity minimization problem is discussed the optimal estimation method is applied in the multivariable deconvolution problem a simple deconvolution filter of minimum order which minimizes the h sub infinity norm of the deconvolution error is obtained the proposed methods of optimal estimation and deconvolution are useful in cases where the statistics of the disturbance and the noise signals is not completely known or in cases where it is required to minimize the maximum singular value of the estimation or the deconvolution error problem

minimax optimal control of stochastic uncertain systems with relative entropy constraints this paper considers a new class of discrete time stochastic uncertain systems in which the uncertainty is described by a constraint on the relative entropy between a nominal noise distribution and the perturbed noise distribution this uncertainty description is a natural extension to the case of stochastic uncertain systems of the sum quadratic constraint uncertainty description this paper solves problems of worst case robust performance analysis and output feedback minimax optimal controller synthesis in a general nonlinear setting specializing these results to the linear case leads to a minimax linear quadratic gaussian lqg optimal controller this controller is defined by riccati difference equations and a kalman filter like state equation the paper also shows that the minimax lqg problem will have a solution if and only if a corresponding h sup spl infin control problem has a solution a linear example is presented to illustrate the minimax lqg methodology

brief robust h filtering with error variance constraints for discrete time varying systems with uncertainty the robust h filtering problem with error variance constraints is considered for discrete time varying systems subject to norm bounded parameter uncertainties in both the state and the output matrices of the state space model sufficient conditions for a finite horizon filter to satisfy state estimation error variance constraints as well as prescribed h performance for all admissible perturbations are given in terms of two discrete riccati difference equations which are of a form suitable for recursive computation the results are extended to cover the case of stationary filtering over an infinite horizon for uncertain time invariant systems

a connection between h control and the absolute stabilizability of uncertain systems abstract this paper considers a stabilization problem for a class of uncertain systems in which the uncertainty satisfies a certain integral quadratic constraint such uncertain systems have been previously studied in the theory of absolute stability with most publications in this area being in the russian literature the paper introduces a notion of absolute stabilizability and shows that an uncertain system is absolutely stabilizable if and only if there exists a solution to a corresponding h control problem the paper also shows that if an uncertain system can be absolutely stabilized via nonlinear output feedback control then it can be absolutely stabilized via linear output feedback control

braid groups and left distributive operations the decidability of the word problem for the free left distributive law is proved by introducing a structure group which describes the underlying identities this group is closely connected with artin s braid group b braid colourings associated with free left distributive structures are used to show the existence of a unique ordering on the braids which is compatible with left translation and such that every generator ai is preponderant over all ak with k i this ordering is a linear ordering the first goal of the present paper is to give a proof of the following result which had been conjectured for several years theorem there is an effective algorithm for deciding whether a given identity is or is not a consequence of the left distributivity identity x yz xy xz until recently this question has had a rather unusual status conditional solutions were given independently in 6 and 24 where the decision problem was reduced to a specific algebraic hypothesis one which had been shown by richard laver to be a consequence of a very strong set theoretical axiom of a type which certainly cannot be derived from the usual axioms of set theory namely a large cardinal axiom the question as to whether a strong axiom of this type was actually needed remained open opinions were in fact divided a connection between large cardinals and a purely finitistic problem of this caliber would seem paradoxical but it is well known that problems of a combinatorial type can embody surprisingly strong proof principles see for instance 28 and some work on free distributive structures has shown that they do give rise to intrinsically complex objects typically nonprimitive recursive ones cf 12 13 we will show that in the present case a solution which is purely algebraic in terms of the methods employed and the spirit of the argument can in fact be given in particular no unusual set theoretical axioms are required for this argument the decision method described in 6 was fully effective but the proof of the correctness of the algorithm amounted to a direct invocation of this specific algebraic hypothesis which followed from a large cardinal axiom with no hint of a direct proof we shall refine this decision method below introducing uniqueness at each step of the process and the correctness will then be seen to follow very naturally received by the editors february 5 1993 1991 mathematics subject classification primary 20f36 1 7a30 1 7a50

gaussian groups and garside groups two generalisations of artin groups it is known that a number of algebraic properties of the braid groups extend to arbitrary finite coxeter type artin groups here we show how to extend the results to more general groups that we call garside groups define a gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths and where left and right lowest common multiples exist a garside monoid is a gaussian monoid in which the left and right lowest common multiples satisfy an additional symmetry condition a gaussian group is the group of fractions of a gaussian monoid and a garside group is the group of fractions of a garside monoid braid groups and more generally finite coxeter type artin groups are garside groups we determine algorithmic criteria in terms of presentations for recognizing gaussian and garside monoids and groups and exhibit infinite families of such groups we describe simple algorithms that solve the word problem in a gaussian group show that these algorithms have a quadratic complexity if the group is a garside group and prove that garside groups have quadratic isoperimetric inequalities we construct normal forms for gaussian groups and prove that in the case of a garside group the language of normal forms is regular symmetric and geodesic has the 5 fellow traveller property and has the uniqueness property this shows in particular that garside groups are geodesically fully biautomatic finally we consider an automorphism of a finite coxeter type artin group derived from an automorphism of its defining coxeter graph and prove that the subgroup of elements fixed by this automorphism is also a finite coxeter type artin group that can be explicitly determined 1991 mathematics subject classification primary 20f05 20f36 secondary 20b40 20m05

analysis of digitized waveforms using shannon entropy ii high speed algorithms based on green s functions an improved approach for calculating continuous waveform entropy hc is described this approach is based on the use of the green s function for the one dimensional differential operator l u u with boundary conditions u 1 u 0 0 the method is applied to data acquired from a plexiglas block having artificial defects in order to produce an entropy image the defect contrast obtained with the new green s function computation is then compared with that obtained from a previously described fourier series approach it is found that the new approach produces the same or higher image contrast in a time that is roughly three orders of magnitude smaller than that required by the fourier series method this speedup arises from two sources the green s function approach has greater inherent immunity to noise and the number of calculations required in the green s function approach is much smaller than the number required in the fourier series approach

robust h filtering for discrete time varying uncertain systems with a known deterministic input in this paper a new discrete time dynamic game where two players seek to reinforce and suppress the output of a system under the influence of a known deterministic input is considered a necessary and sufficient condition for the existence of a saddle point solution to this game is derived in terms of a discrete time riccati difference equation the result of the game is used to solve the robust h x filtering problem for discrete time varying systems subject to a known deterministic input and norm bounded parameter uncertainties occurring in both the state and the output matrices of the state space model there is no restriction on the form of the filter this allows the effect of the known input on the estimation error due to system uncertainties to be optimally reduced without any prior assumption on the filter structure

robust kalman filtering under model perturbations we consider a family of divergence based minimax approaches to perform robust filtering the mismodeling budget or tolerance is specified at each time increment of the model more precisely all possible model increments belong to a ball which is formed by placing a bound on the tau divergence family between the actual and the nominal model increment then the robust filter is obtained by minimizing the mean square error according to the least favorable model in that ball it turns out that the solution is a family of kalman like filters their gain matrix is updated according to a risk sensitive like iteration where the risk sensitivity parameter is now time varying as a consequence we also extend the risk sensitive filter to a family of risk sensitive like filters according to the tau divergence family

properties of an entropy based signal receiver with an application to ultrasonic molecular imaging qualitative and quantitative properties of the finite part hf of the shannon entropy of a continuous waveform f t in the continuum limit are derived in order to illuminate its use for waveform characterization simple upper and lower bounds on hf based on features of f t are defined quantitative criteria for a priori estimation of the average case variation of hf and logef where ef is the signal energy of f t are also derived these provide relative sensitivity estimates that could be used to prospectively choose optimal imaging strategies in real time ultrasonic imaging machines where system bandwidth is often pushed to its limits to demonstrate the utility of these sensitivity relations for this application a study designed to assess the feasibility of identification of angiogenic neovasculature targeted with perfluorocarbon nanoparticles that specifically bind to v 3 integrin expression in tumors was performed the outcome of this study agrees with the prospective sensitivity estimates tha

gain scheduled filtering for time varying discrete systems this paper deals with the design of gain scheduled filters whose state space realization depends on real time parameters of plants similar to well recognized advantages of gain scheduled controllers in control theory gain scheduled filters are expected to provide enhanced performance in comparison with customary nonadjustable filters our construction technique is based on nonlinear fractional transformation nft representations of systems that are a generalization of widely used linear fractional transformation lft representations both generalized h sub 2 and h sub spl infin discrete time filter design problems are investigated together with their extension to mixed designs this study leads to new linear matrix inequality lmi formulations which in turn provide an effective and reliable design tool the proposed design technique is finally evaluated in the light of simulation examples