topology optimization of continuum structures under uncertainty a polynomial chaos approach abstract a computational method for topology optimization in the presence of uncertainty is proposed the method combines the spectral stochastic approach for the representation and propagation of uncertainties with an existing deterministic topology optimization technique the idea in spectral stochastic formulations is to add an extra dimension a random dimension to the problem where the stochastic variability of the input parameters and outputs of interest can be modeled the resulting compact representations for the response quantities allow for efficient and accurate calculation of sensitivities of response statistics with respect to the design variables which are then fed into a gradient based optimizer that searches for the optimum design from a variety of frameworks for uncertainty informed optimization robust topology optimization is chosen to demonstrate the applicability of the method examples from continuum topology optimization under uncertainty in material properties are presented it is also shown that results obtained from the proposed method are in excellent agreement with those obtained from a monte carlo based optimization algorithm

the lqg optimal regulation problem for systems with perfect measurements explicit solution properties and application to practical designs the continuous time stationary linear quadratic gaussian lqg optimal regulation problem is considered where the measurements are free of white noise components a simple direct solution in the s domain is derived for the optimal controller for general linear time variant right invertible or left invertible systems the explicit expressions that are found for the controller transference and for the regulator return ratio matrix can be used to obtain a practical suboptimal design in the s domain these expressions are applied to derive simple conditions for the precise recovery of the return ratio matrix of the lqg regulator

the optimal projection equations for reduced order state estimation the singular measurement noise case the optimal projection equations for reduced order state estimation are generalized to allow for singular i e colored measurement noise the noisy and noise free measurements serve as inputs to dynamic and static estimators respectively the optimal solution is characterized by necessary conditions which involve a pair of oblique projections corresponding to reduced estimator order and singular measurement noise intensity

linear time variable systems balancing and model reduction a uniformly balanced realization for linear time variable systems is defined this representation is characterized by the fact that its controllability and observability gramians are equal and diagonal existence and uniqueness of the uniformly balanced realization is studied such a framework has many remarkable properties and leads to a novel method for approximating time variable systems where the subsystems of the balanced realization can be taken as a reduced model the reduced model is examined from the point of view of stability controllability and observability

structural optimization under uncertain loads and nodal locations this paper presents algorithms for solving structural topology optimization problems with uncertainty in the magnitude and location of the applied loads and with small uncertainty in the location of the structural nodes the second type of uncertainty would typically arise from fabrication errors where the tolerances for the node locations are small in relation to the length scale of the structural elements we first review the discrete form of the uncertain loads problem which has been previously solved using a weighted average of multiple load patterns with minor modifications we extend this solution to include loads described by continuous joint probability density functions we then proceed to the main contribution of this paper structural optimization under uncertainty in the nodal locations this optimization problem is computationally difficult because it involves variations of the inverse of the structural stiffness matrix it is shown however that for small uncertainties the problem can be recast into a simpler but equivalent structural optimization problem with equivalent uncertain loads by expressing these equivalent loads in terms of continuous random variables we are able to make use of the extended form of the uncertain loads problem presented in the first part of this paper the optimization algorithms are developed in the context of minimum compliance maximum stiffness design simple examples are presented the results demonstrate that load and nodal uncertainties can have dramatic impact on optimal design for structures containing thin substructures under axial loads it is shown that these uncertainties a are of first order significance influencing the linear elastic response quantities and b can affect designs by avoiding unrealistically optimistic and potentially unstable structures the additional computational cost associated with the uncertainties scales linearly with the number of uncertainties and is insignificant compared to the cost associated with solving the deterministic structural optimization problem

on the existence of an optimal observer in singular measurement systems an essential stage in each of the existing methods for singular optimal estimation is to obtain a nonsingular measurement vector which is a linear combination of the original measurement vector and a finite number of its derivatives a necessary and sufficient condition for the existence of such a nonsingular measurement is presented if this condition is satisfied then the existence of an optimal observer is guaranteed the new condition is shown to be considerably simpler and of much wider applicability than a recently published condition which is just a sufficient one the new condition requires just a rank check of a single composite matrix consisting of the system and noise matrices unlike the other condition the existence of an optimal estimator is determined here without carrying out the complex procedures for singular optimal estimation

admissible mimo singular observation lqg designs abstract the widely known lqg design method for linear time invariant mimo systems may lead under certain circumstances to closed loop systems with improper sensitivity matrices such systems have no stability robustness and we refer to them as inadmissible ones this phenomenon is analyzed in the paper necessary and sufficient conditions guaranteeing admissibility of lqg designs both in the frequency and the time domains are presented and proven these conditions are based on the available information in the statement of the problem thus enabling a priori detection of inadmissibility in optimal systems the key tool in the analysis is the extension of results from polynomial matrices to rational matrices a suboptimal design aimed directly at avoiding inadmissibility and which involves the minimum necessary changes in the lqg cost function is suggested illustrative examples are used to demonstrate the main results

an explicit expression for the minimum phase image of transfer function matrices a simple expression is obtained for the transfer function matrix of the minimum phase image of a left invertible continuous time invariant system with zeros in the right half plane that may be of multiplicities greater than one this expression is obtained by multiplying the system transfer function matrix from the right by a special inner matrix and it is explicitly given in terms of the input zero directions that correspond to the zeros of the system in the right half plane the result provides a simple expression for the inner outer factorization of transfer function matrices and it can thus be used in h sub infinity optimal control

minimal order wiener filter for a system with exact measurements necessary and sufficient conditions for the existence of an optimal steady state state estimator are derived under the assumption that this estimator is a linear functional of the measurements and a finite number of derivatives of the exact measurements our conditions are shown to be dual to the generalized legendre clebsch conditions of the dual optimal singular regulator a separation principle is derived and it is shown that as all process and measurement noise vanishes the error covariance of our filter converges to a null matrix

nonlinear filtering for discrete time linear systems with non gaussian initial data the optimal minimum mean square error filtering problem is discussed for a linear system with non gaussian initial distribution an input output realization is derived using an absolutely continuous change of probability measure a representation result is presented which facilitates the derivation of optimal filter realization

stress based topology optimization of frame structures under geometric uncertainty abstract probabilistic topology optimization has gained significant research attention recently this interest stems from the realization that the achieved high performance designs resulted from deterministic topology optimization algorithms may become suboptimal under real world conditions that are often accompanied with uncertainties among sources of these uncertainties the ones that define structural characteristics such as geometry are numerically challenging to treat as they lead to stochastic structural stiffness to date research on developing efficient probabilistic topology optimization under stochastic stiffness is mainly focused on displacement based objectives however in the design of structures stress is also a primary design criterion that needs to be directly controlled for a robust stress based topology optimization methodology for frame structures under geometric uncertainty is proposed in this work assuming that such uncertainties are small relative to frame member lengths the proposed methodology uses stochastic perturbation method to propagate these uncertainties up to the response level which is expressed by the maximum of expected values of von mises stresses throughout the domain sensitivities of the response with respect to design variables are derived analytically which allows using efficient gradient based optimizers the proposed algorithm is examined with stress based design of three frame structures under geometric uncertainty changes in the topology of these new designs are discussed and they are shown to outperform deterministic designs when subjected to geometric uncertainties moreover predictions and the resulting designs from the proposed methodology are found to be in excellent agreement with monte carlo simulation results

optimal design of trusses with geometric imperfections accounting for global instability abstract a topology optimization method is proposed for the design of trusses with random geometric imperfections due to fabrication errors this method is a generalization of a previously developed perturbation approach to topology optimization under geometric uncertainties the main novelty in the present paper is that the objective function includes the nonlinear effects of potential buckling due to misaligned structural members solutions are therefore dependent on the magnitude of applied loads and the direction of resulting internal member forces whether they are compression or tension direct differentiation is used in the sensitivity analysis and analytical expressions for the associated derivatives are derived in a form that is computationally efficient a series of examples illustrate how the effects of geometric imperfections and buckling may have substantial influence on truss design monte carlo simulation together with second order elastic analysis is used to verify that solutions offer improved performance in the presence of geometric uncertainties

the design of optimal reduced order stochastic observers for discrete time linear systems the minimum variance state estimation of linear discrete time systems with random white noise input and partially noisy measurements is investigated an observer of minimal order is found which attains the minimum variance estimation error the structure of this observer is shown to depend strongly on the geometry of the system this geometry dictates the length of the delays that are applied on the measurements in order to obtain the optimal estimate the transmission properties of the observer are investigated for systems that are left invertible and free of measurement noise an explicit expression is found for the transfer function matrix of this observer from which a simple solution to the linear discrete time singular optimal filtering problem is obtained

fast significant sample detection for a pitch detector improved significant sample detection for a pitch detector for use with speech analysis and synthesis methods by performing a reverse order search and a forward order search of digitized speech samples a reverse search detector 12 is responsive to segmented digital samples for determining a set of candidate samples by initially selecting one of the digitized samples as a present candidate sample and comparing in reverse order each of the digitized samples with the present candidate sample until a digitized sample is found whose amplitude is greater than the present candidate sample or the compared sample is greater than a predefined number of samples from the present candidate sample when either of the previous conditions occurs the compared digital sample becomes the new present candidate sample and the reverse search continues during the reverse search each of the compared samples that has not replaced the present candidate sample is set equal to zero after the reverse search has been performed and a set of candidate samples has been determined a forward search detector 14 then initially determines a present significant sample the latter detector compares this significant sample with each of the candidate samples until a candidate sample is found whose amplitude is greater than the present significant sample or the compared candidate sample is more than a predefined number of samples away from the present significant sample when either of those conditions occurs the forward search detector saves the value of the amplitude and location of the candidate sample and replaces the present significant sample with that candidate sample and continues the search a single forward and reverse search determines all of the significant samples

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

component cost analysis of large scale systems abstract a brief summary of component cost analysis cca is given cca provides a means to determine the contribution of each dynamical element of the system in the overall performance metric using cca a theory of cost equivalent realizations is developed cost equivalent realizations are realizations that have the same value of the performance metric as the original system and they can be of lower order than kalman s minimal realization

lqg optimal scalar static output feedback an eigenproblem based technique is given for computing the stabilizing scalar static output feedback gain that minimizes the standard linear quadratic performance index in the presence of a gaussian white noise disturbance the method adapts recent work on eigenproblem based computation of h norms

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

a riccati equation approach to the singular lqg problem abstract the problem of optimal fixed order dynamic compensation for the singular lqg problem is considered necessary conditions characterizing the optimal compensator for the case involving both singular measurement noise and singular control weighting are given the solution consists of a set of two algebraic riccati equations and two lyapunov equations coupled by three projection matrices one projection is the standard order reduction projection while the other two projections reflect the two types of singularity that exist in the system the three projections are shown to satisfy disjointness conditions in addition to order reduction an advantage of the fixed structure approach is that differentiation which is often undersirable from a practical point of view and which may exist in the unconstrained optimal control can be avoided it is shown that the fixed order compensator agrees with the unconstrained solution when the latter possesses the same number of differentiations as are included in the prespecified controller structure and when the order is selected appropriately

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

the optimal projection equations for finite dimensional fixed order dynamic compensation of infinite dimensional systems one of the major difficulties in designing implementable finite dimensional controllers for distributed parameter systems is that such systems are inherently infinite dimensional while controller dimension is severely constrained by on line computing capability while some approaches to this problem initially seek a correspondingly infinite dimensional control law whose finite dimensional approximation may be of impractically high order the usual engineering approach involves first approximating the distributed parameter system with a high order discretized model followed by design ofa relatively low order dynamic controller among the numerous approaches suggested for the latter step are model controller reduction techniques used in conjunction with the standard lqg result an alternative approach developed in 36 relies upon the discovery in 31 that the necessary conditions for optimal fixed order dynamic compensation can be transformed into a set of equations possessing remarkable structural coherence the present paper generalizes this result to apply directly to the distributed parameter system itself in contrast to the pair of operator riccati equations for the full order lqg case the optimal finite dimensional fixed orderdynamic compensator is characterized byfour operator equations two modified riccati equations and two modified lyapunov equations coupled by an oblique projection whose rank is precisely equal to the order ofthe compensatorand which determines the optimal compensator gains this optimal projection is obtained by a full rank factorization of the product of the finite rank nonnegative definite hilbert space operators which satisfy the pair ofmodified lyapunov equations the coupling represents a graphic portrayal of the demise of the classical separation principle for the finite dimensional reduced order controller case the results obtained apply to a semigroup formulation in hilbert space and thus are applicable to control problems involving a broad range of specific partial and functional differential equations

design of reduced order optimal estimators directly in the frequency domain the frequency domain design of optimal reduced order estimators caiman filters for continuous time linear time invariant systems containing 0 k m noise free output measurements is considered in a direct frequency domain approach to state feedback and estimation the states of a system do not appear explicitly and therefore we here focus our attention mainly on the frequency domain characteristics of the optimal filter by suitably modifying known time domain results a simple polynomial matrix equation can be derived from which the optimal filter parameters are obtained by spectral factorization the equivalence between the time and the frequency domain results is established with the aid of a non minimal representation of reduced order observers

level set topology optimization of fluids in stokes flow we propose the level set method of topology optimization as a viable robust and efficient alternative to density based approaches in the setting of fluid flow the proposed algorithm maintains the discrete nature of the optimization problem throughout the optimization process leading to significant advantages over density based topology optimization algorithms specifically the no slip boundary condition is implemented directly this is accurate removes the need for interpolation schemes and continuation methods and gives significant computational savings by only requiring flow to be modeled in fluid regions topological sensitivity information is utilized to give a robust algorithm in two dimensions and familiar two dimensional power dissipation minimization problems are solved successfully computational efficiency of the algorithm is also clearly demonstrated on large scale three dimensional problems

topology optimization of creeping fluid flows using a darcy stokes finite element a new methodology is proposed for the topology optimization of fluid in stokes flow the binary design variable and no slip condition along the solid fluid interface are regularized to allow for the use of continuous mathematical programming techniques the regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by darcy s law fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the stokes and darcy equations the mixed formulation of the new darcy stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems convergence to the no slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid fluid interface are not needed the optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature the advantages of the darcy stokes approach include that it uses existing stabilization techniques to solve the finite element problem it produces 0 1 void solid topologies i e there are no regions of artificial material and that it can potentially be used to optimize the layout of a microscopically porous material copyright 2005 john wiley sons ltd

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

explicit expressions for cascade factorization of general nonminimum phase systems explicit expressions for two different cascade factorizations of any detectable left invertible nonminimum phase systems are given the first one is a well known minimum phase all pass factorization by which all nonminimum phase zeros of a transfer function g s are collected into an all pass factor v s and g s is written g sub m s v where g sub m s is considered as a minimum phase image of g s the second one is a new cascade factorization by which g s is rewritten as g sub m s u s where u s collects all awkward zeros including all nonminimum phase zeros of g s both g sub m s and g sub m s retain the given infinite zero structure of g s further properties of g sub m s g sub m s and u s are discussed these factorizations are useful in several applications including loop transfer recovery

properness of feedback transfer matrices this article presents some sufficient conditions for the transfer matrix of a state estimator type feedback system to be proper or improper these conditions are needed in the discussion of the well posedness of the feedback system

nearly singular filtering for uniform and non uniform rank linear continuous systems abstract the problem of the minimum variance stationary estimation of linear time invariant continuous systems is considered in the case where the norm of tho measurement white noise intensity approaches zero solutions in closed form are obtained for the kalman gain and the corresponding minimum error covariance matrix of the filtered estimate in terms of the noise intensity norm and the system transfer function matrix both uniform and non uniform rank systems are considered and for the first time a solution is obtained for the case where the infinite zeros of the system tond to infinity with multiple rates

the optimal projection equations for reduced order state estimation first order necessary conditions for optimal steady state reduced order state estimation for a linear time invariant plant in the presence of correlated disturbance and nonsingular measurement noise are derived in a new and highly simplified form in contrast to the lone matrix riccati equation arising in the full order kalman filter case the optimal steady state reduced order estimator is characterized by three matrix equations one modified riccati equation and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the estimator and which determines the optimal estimator gains this coupling is a graphic reminder of the suboptimality of proposed approaches involving either model reduction followed by full order estimator design or full order estimator design followed by estimator reduction techniques the results given here complement recently obtained results which characterize the optimal reduced order model by means of a pair of coupled modified lyapunov equations 7 and the optimal fixed order dynamic compensator by means of a coupled system of two modified riceati equations and two modified lyapunov equations 6

on the recovery procedure for lqg systems abstract the loop transfer function recovery ltr procedure of doyle and stein 1979 ieee trans aut control ac 24 607 611 is re examined in the context of singular measurement lqg systems using existing closed form expressions for singular optimal estimation an alternative proof of the ltr is presented some of the structural properties of the ltr are obtained naturally using this approach it is shown that in the limit the control effort becomes infinite and therefore the critical tradeoff is between control effort and robustness

return difference matrix properties of optimal linear stationary estimation and control in singular case a characterization of the return difference matrix properties of optimal linear stationary estimation in singular mode is presented in which the non singular kalman problem and the coloured measurement noise problem occur as special cases linear least squares estimators of minimal order are treated for both known state space models and known separable output covariance models return difference properties of the optimal linear regulator where the control weighting matrix is singular are also shown to be the dual of those of the singular estimation solutions

on the properties of reduced order kalman filters several known results are unified by considering properties of reduced order kalman filters for the case in which the number of noise sources equals the number of observations it is shown that the reduced order kalman filter achieves zero steady state variance of the estimation error if and only if the plant has no transmission zeros in the right half plane since these would be among the poles of the kalman filter the reduced order kalman filter cannot achieve zero variance of the estimation error if the number of independent noise sources exceeds the number of observations it is also shown that the reduced order kalman filter achieves the generalized doyle stein condition for robustness when the noise sources are colocated with the control inputs when there are more observations than noise sources additional noise sources can be postulated to improve the observer frequency response without diminishing robustness

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

on partially augmented observers for systems with coloured noises the problem of optimal fixed order observers for continuous time linear systems with deterministic input and coloured process and measurement noises is considered optimal estimation of the state involves augmentation of the system and as a result the observer has the augmented dimension in this paper the order of the observer is prefixed to a value which ranges between the dimension of the original system and that of the augmented system the presence of a deterministic input imposes the structure of an observer on the estimator the optimal observer estimator under these dimensional and structural constraints is then derived solution is given for the general case where the intensity of the white noise component of the measurement noise may be singular necessary conditions fully characterizing the optimal observer estimate are given the solution consists of a modified riccati and three modified lyapunov equations coupled by three projection matrices the three projections correspond to the order

state estimation for linear systems with observations partially corrupted by noise abstract this paper exploits the fact that any row vector of the observability matrix applied for transforming the state converts the latter to the new state component in the form of some derivative of the output component using the same but appropriately chosen vectors for transforming the system with the observation not fully corrupted by white noise we can accurately determine some state components these vectors create the basis for the l dimensional subspace of transformation vectors to the new accurately determinable state components using this basis the state transformation is constructed which in one step converts the singular linear filtering problem to a nonsingular one with state dimension decreased by l

the optimal projection equations for static and dynamic output feedback the singular case oblique projections have been shown to arise naturally in both static and dynamic optimal design problems for static controllers an oblique projection was inherent in the early work of levine and athans while for dynamic controllers an oblique projection was developed by hyland and bernstein this note is motivated by the following natural question what is the relationship between the oblique projection arising in optimal static output feedback and the oblique projection arising in optimal fixed order dynamic compensation we show that in nonstrictly proper optimal output feedback there are indeed three distinct oblique projections corresponding to singular measurement noise singular control weighting and reduced compensator order moreover we unify the levine athans and hyland bernstein approaches by rederiving the optimal projection equations for combined static dynamic nonstrictly proper output feedback in a form which clearly illustrates the role of the three projections in characterizing the optimal feedback gains even when the dynamic component of the nonstrictly proper controller is of full order the controller is characterized by four matrix equations which generalize the standard lqg result

extended limiting forms of optimum observers and lqg regulators the general problem of optimal estimation in singular observation systems is considered by generalizing the limiting procedure of friedland explicit and closed expressions for the transfer matrices of the optimum observer as well as for the optimal control law in the singular measurement lqg problem are derived the number of integrators that are required for the implementation of the singular optimum observer is shown for square systems to be equal to the number of the system s zeros the closed form of the optimum observer given here is advantageous as compared with existing procedures structural properties of the singular observation problem are discussed and the role of the zeros of the system in such problems is clarified the dual results for the case of stochastic cheap control are also given

optimal observers for systems with colored noises the problem of optimal full order observers for continuous time linear systems with colored process and measurement noises is considered in such cases optimal estimation of the state involves augmenting the system thus a higher order observer is required the structure of a full order observer is assumed and necessary conditions for the optimal observer are derived the conditions are given for the general case where the intensity of the white noise component of the measurement noise may be singular the solution consists of a modified riccati equation and a lyapunov equation coupled by two projection matrices in the singular case and one projection matrix in the nonsingular case

pitch detector for speech analysis speech signal pitch detection uses the residual signal output of an lpc linear prediction coder filter the residual signal or its hilbert transform is divided into frames of 20 milliseconds and each frame searched for signal peak periodicity by first detecting the highest peak then detecting a set of equally spaced selected samples from which the pitch period is determined if no periodicity is found over three frames an unvoiced decision is made

direct solution to the general reduced order stochastic observation problem the stochastic optimal state observation problem is considered for a general linear continuous time invariant system with unmeasurable stationary inputs and measurement outputs that may be at least in part perfect a general solution to the problem is obtained by processing the perfect measurements through a specific differentiation transformation scheme in order to extract the maximum accurate information on the system states using this information the original system is transformed to a new reduced order model whose measurements are corrupted by a white noise of non singular intensity matrix a minimum order full state estimator to the original system is then constructed by combining the outputs of any full order observer to the reduced order model and the perfect combinations of the system states that were derived by the differentiation transformation scheme a solution to the general singular kalman filtering problem is then obtained by minimizing the variance of the estimation error of the obser

method of and device for quantizing excitation gains in speech coders based on analysis by synthesis techniques an optimum excitation signal for each subframe is determined in a speech coder based on analysis by synthesis techniques and operating on frames of samples divided into a number of subframes the excitation signal includes a shape contribution innovation and an amplitude contribution gain which are quantized separately a circuit it for gain quantization includes means qu for determining a gain index for each subframe a comparison logic network cfr for detecting the maximum value taken by the gain index in the frame and means for computing a normalized index for each subframe as a difference between the maximum index and the gain index relevant to that subframe the coded signal includes the coded values of the maximum index and of the normalized indexes as information on the gain relevant to a frame

a simple solution to the singular linear minimum variance estimation problem a simple expression is obtained for the transfer function matrix of the minimum variance estimator for the states of a linear stationary continuous time right invertible system whose output is measured perfectly using the s domain approach it is shown that the optimal estimator first finds the driving noise input that achieves when applied on the minimum phase image model of the system an output spectrum which is identical to the measurement spectrum this input is then applied on a state space representation of the minimum phase image to produce the optimal estimate

the optimal reduced order estimator for systems with singular measurement noise the optimal reduced order estimator is completely characterized by necessary conditions resulting from the optimal projection equations the solution consists of one riccati equation and two lyapunov equations coupled by two projections explicit expressions for all of the estimator parameters are given the relation between the reduced order singular estimator and the full order optimal singular estimator which is of reduced order itself is investigated it is shown that under certain conditions the optimal estimator is recovered from the reduced order estimator

a new interpretation of the fault detection filter part 2 the optimal detection filter park and rizzoni 1993 obtained closed form expressions for detection filters i e the structure of all detection filters for a given fault direction was defined an important consequence of these results is that they permit the formation of the optimal detection filter problem for optimization with respect to process and measurement noises the necessary conditions for the existence of the optimal detection filter are obtained and a numerical solution technique is shown to be feasible by virtue of the uniqueness of the detection filter gains from an optimization point of view the problem can be regarded as optimal estimation with some structural constraints on the observer gain this problem is solved for both the continuous time and the discrete time cases

the optimal projection equations for fixed order dynamic compensation first order necessary conditions for quadratically optimal steady state fixed order dynamic compensation of a linear time invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form in contrast to the pair of matrix riccati equations for the full order lqg case the optimal steady state fixed order dynamic compensator is characterized by four matrix equations two modified riccati equations and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains the coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced order controller case

solution in the z domain to the discrete time stationary kalman filtering problem for systems with perfect measurements the discrete time stationary kalman filtering problem is solved in the z domain for general right invertible linear systems whose measurements are all noise free a simple expression in closed form is obtained for the transfer function of the optimal filter both for the uniform and the general nonuniform rank cases it provides an optimal solution also for systems with uniform and nonuniform time delays the minimum variance error of the estimate is analyzed and the independent contributions to this error of the system zeros outside the unit circle finite and infinite and the excess of the number of inputs over the number of measurements are investigated

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

limiting forms of optimal observers halevi and palmor 1986 presented closed form expressions for optimal observers for singular measurement systems two assumptions were made there i the measurement is totally singular and ii the number of outputs is less than or equal to the number of stochastic inputs these two assumptions are relaxed in this paper and the optimal observer for the most general singular measurement case is given

materials with prescribed constitutive parameters an inverse homogenization problem abstract this paper deals with the construction of materials with arbitrary prescribed positive semi definite constitutive tensors the construction problem can be called an inverse problem of finding a material with given homogenized coefficients the inverse problem is formulated as a topology optimization problem i e finding the interior topology of a base cell such that cost is minimized and the constraints are defined by the prescribed constitutive parameters numerical values of the constitutive parameters of a given material are found using a numerical homogenization method expressed in terms of element mutual energies numerical results show that arbitrary materials including materials with poisson s ratio 1 0 and other extreme materials can be obtained by modelling the base cell as a truss structure furthermore a wide spectrum of materials can be constructed from base cells modelled as continuous discs of varying thickness only the two dimensional case is considered in this paper but formulation and numerical procedures can easily be extended to the three dimensional case

the hankel operator and its induced norms the hankel operator of a linear stable time invariant system is usually considered as a mapping of square integrable past inputs into square integrable future outputs and as such has an induced norm which can be calculated from the controllability and observability gramains of the system in this paper the hankel operator is considered as a mapping between some other spaces and expressions are derived for its induced norm

hankel singular values and lqg characteristic values of discrete time linear systems in cascade with inner systems recent results have shown that for continuous time systems obtained by cascading an asymptotically stable system with an inner system the first n hankel singular values are greater than or equal to those of the original system similarly cascading a system in minimal form with an inner system the same property holds for the linear quadratic gaussian lqg characteristic values in this article we consider the discrete time case and demonstrate that the property also holds for these systems a very important consequence stemming from these results is that the hankel singular values and the lqg characteristic values of input delayed discrete systems are greater than or equal to those of their zero delay counterpart