efficient recursive estimators for a linear time varying gaussian model with general constraints the adaptive estimation of a time varying parameter vector in a linear gaussian model is considered where we a priori know that the parameter vector belongs to a known arbitrary subset we consider a family of efficient recursive estimators for this problem the recursive constrained maximum likelihood ml estimator the recursive affine minimax and the recursive minimum mean square error mmse estimator we show that all three estimators can be substantially simplified by using the recursive weighted least squares rwls algorithm in a first step as the rwls computes the sufficient statistic for this estimation problem the recursive constrained ml needs to solve an optimization problem in the second step for the case that the rwls solution does not fulfill the constraint in case of affine minimax we have to solve an optimization problem and to perform an affine transform the mmse estimator needs to calculate the mean of a truncated gaussian density in the second step which is done by monte carlo integration a simple rejection scheme is used to take general constraints for the parameter vector into account an example shows the superior performance of our proposed estimators in comparison to many other estimators

exploring estimator bias variance tradeoffs using the uniform cr bound we introduce a plane which we call the delta sigma plane that is indexed by the norm of the estimator bias gradient and the variance of the estimator the norm of the bias gradient is related to the maximum variation in the estimator bias function over a neighborhood of parameter space using a uniform cramer rao cr bound on estimator variance a delta sigma tradeoff curve is specified that defines an unachievable region of the delta sigma plane for a specified statistical model in order to place an estimator on this plane for comparison with the delta sigma tradeoff curve the estimator variance bias gradient and bias gradient norm must be evaluated we present a simple and accurate method for experimentally determining the bias gradient norm based on applying a bootstrap estimator to a sample mean constructed from the gradient of the log likelihood we demonstrate the methods developed in this paper for linear gaussian and nonlinear poisson inverse problems

mse bounds with affine bias dominating the cramer rao bound in continuation to an earlier work we further develop bounds on the mean squared error mse when estimating a deterministic parameter vector thetas0 in a given estimation problem as well as estimators that achieve the optimal performance the traditional cramer rao cr type bounds provide benchmarks on the variance of any estimator of thetas0 under suitable regularity conditions while requiring a priori specification of a desired bias gradient to circumvent the need to choose the bias which is impractical in many applications it was suggested in our earlier work to directly treat the mse which is the sum of the variance and the squared norm of the bias while previously we developed mse bounds assuming a linear bias vector here we study in the same spirit affine bias vectors we demonstrate through several examples that allowing for an affine transformation can often improve the performance significantly over a linear approach using convex optimization tools we show that in many cases we can choose an affine bias that results in an mse bound that is smaller than the unbiased cr bound for all values of thetas0 furthermore we explicitly construct estimators that achieve these bounds in cases where an efficient estimator exists by performing an affine transformation of the standard maximum likelihood ml estimator this leads to estimators that result in a smaller mse than ml for all possible values of thetas0

affine parameter dependent lyapunov functions and real parametric uncertainty this paper presents new tests to analyze the robust stability and or performance of linear systems with uncertain real parameters these tests are extensions of the notions of quadratic stability and performance where the fixed quadratic lyapunov function is replaced by a lyapunov function with affine dependence on the uncertain parameters admittedly with some conservatism the construction of such parameter dependent lyapunov functions can be reduced to a linear matrix inequality lmi problem and hence is numerically tractable these lmi based tests are applicable to constant or time varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations they also avoid the frequency sweep needed in real spl mu analysis and numerical experiments indicate that they often compare favorably with spl mu analysis for time invariant parameter uncertainty

is denoising dead image denoising has been a well studied problem in the field of image processing yet researchers continue to focus attention on it to better the current state of the art recently proposed methods take different approaches to the problem and yet their denoising performances are comparable a pertinent question then to ask is whether there is a theoretical limit to denoising performance and more importantly are we there yet as camera manufacturers continue to pack increasing numbers of pixels per unit area an increase in noise sensitivity manifests itself in the form of a noisier image we study the performance bounds for the image denoising problem our work in this paper estimates a lower bound on the mean squared error of the denoised result and compares the performance of current state of the art denoising methods with this bound we show that despite the phenomenal recent progress in the quality of denoising algorithms some room for improvement still remains for a wide class of general images and at certain signal to noise levels therefore image denoising is not dead yet

a less conservative lmi condition for the robust stability of discrete time uncertain systems abstract in this paper a less conservative condition for the robust stability of uncertain discrete time linear systems is proposed the uncertain parameters assumed to be time invariant are supposed to belong to convex bounded domains polytope type uncertainty the stability condition is formulated in terms of a set of linear matrix inequalities involving only the vertices of the polytope domain a simple and numerically efficient feasibility test provides a set of lyapunov matrices whose convex combination can be used to assess the stability of any dynamic matrix inside the uncertainty domain examples illustrate the results

a linear programming oriented procedure for quadratic stabilization of uncertain systems abstract this paper gives a new necessary and sufficient condition for linear quadratic stabilization of linear uncertain systems when both the dynamic as well as the control matrix are subject to uncertainty a constructive numerical procedure is defined to check the condition and it furthermore provides a stabilizing linear feedback gain some experiments are presented

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

canonical bases for the quantum group of type a_r and piecewise linear combinatorics this work was motivated by the following two problems from the classical representation theory both problems make sense for an arbitrary complex semisimple lie algebra but since we shall deal only with the ar case we formulate them in this generality 1 construct a good basis in every irreducible finite dimensional slr 1 module v which materializes the littlewood richardson rule a precise formulation of this problem was given in 3 we shall explain it in more detail a bit later 2 construct a basis in every polynomial representation of glr 1 such that the maximal element w0 of the weyl group sr 1 considered as an element of glr 1 acts on this basis by a permutation up to a sign and explicitly compute this permutation this problem is motivated by recent work by john stembridge 10 and was brought to our attention by his talk at the jerusalem combinatorics conference may 1993

maximum set estimators with bounded estimation error we consider the linear regression problem of estimating a deterministic parameter vector x from observations y hx w where h is known and w is additive noise we seek an estimator whose estimation error does not exceed a given maximum error for as wide a range of conditions as possible the maximum error is a design choice and is generally lower than the error provided by the well known least squares ls estimator we develop estimators guaranteeing the required error for as large a parameter set as possible and for as large a noise level as possible we discuss methods for finding these estimators and demonstrate that in many cases the proposed estimators outperform the ls estimator

cram acute text e r rao bound for constrained parameter estimation using lehmann unbiasedness the constrained cram acute text e r rao bound ccrb is a lower bound on the mean squared error mse of estimators that satisfy some unbiasedness conditions although the ccrb unbiasedness conditions are satisfied asymptotically by the constrained maximum likelihood cml estimator in the non asymptotic region these conditions are usually too strict and the commonly used estimators such as the cml estimator do not satisfy them therefore the ccrb may not be a lower bound on the mse matrix of such estimators in this paper we propose a new definition for unbiasedness under constraints denoted by c unbiasedness which is based on using lehmann unbiasedness with a weighted mse wmse risk and taking into account the parametric constraints in addition to c unbiasedness a cram acute text e r rao type bound on the wmse of c unbiased estimators denoted as lehmann unbiased ccrb lu ccrb is derived this bound is a scalar bound that depends on the chosen weighted combination of estimation errors it is shown that c unbiasedness is less restrictive than the ccrb unbiasedness conditions thus the set of estimators that satisfy the ccrb unbiasedness conditions is a subset of the set of c unbiased estimators and the proposed lu ccrb may be an informative lower bound in cases where the corresponding ccrb is not in the simulations we examine linear and nonlinear estimation problems under nonlinear constraints in which the cml estimator is shown to be c unbiased and the lu ccrb is an informative lower bound on the wmse while the corresponding ccrb on the wmse is not a lower bound and is not informative in the non asymptotic region

strong duality in nonconvex quadratic optimization with two quadratic constraints we consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints when the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions we then develop a connection between the image of the real and complex spaces under a quadratic mapping which together with the results in the complex case lead to a condition that ensures strong duality in the real setting preliminary numerical simulations suggest that for random instances of the extended trust region subproblem the sufficient condition is satisfied with a high probability furthermore we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems finally we discuss an application of our results to robust least squares problems

multiobjective output feedback control via lmi optimization this paper presents an overview of a linear matrix inequality lmi approach to the multiobjective synthesis of linear output feedback controllers the design objectives can be a mix of h sub spl infin performance h sub 2 performance passivity asymptotic disturbance rejection time domain constraints and constraints on the closed loop pole location in addition these objectives can be specified on different channels of the closed loop system when all objectives are formulated in terms of a common lyapunov function controller design amounts to solving a system of linear matrix inequalities the validity of this approach is illustrated by a realistic design example

pole assignment for uncertain systems in a specified disk by state feedback this paper presents a method for assigning the poles in a specified disk by state feedback for a linear discrete or continuous time uncertain system the uncertainty being norm bounded for this the quadratic d stabilizability concept which is the counterpart of quadratic stabilizability in the context of pole placement is defined and a necessary and sufficient condition for quadratic d stabilizability derived this condition expressed as a parameter dependent discrete riccati equation enables one to design the control gain matrix by solving iteratively a discrete riccati equation

pole assignment of linear uncertain systems in a sector via a lyapunov type approach the problem of designing robust control laws in performance and in stability for uncertain linear systems is considered performances are taken into account using root clustering of the closed loop dynamic matrix in a sector of the complex plane a synthesis procedure based on a sufficient condition for quadratic stabilization and root clustering such as stabilizability is given using an auxiliary convex problem the results are illustrated by a significant example from the literature

new strategy for robust stability analysis of discrete time uncertain systems in this paper a new robust stability analysis approach is developed for uncertain discrete time linear time invariant systems with polytopic or affine parameter dependent uncertainty models the proposed approach is based on a combination of a branch and bound like strategy with linear matrix inequality lmi based analysis formulations two sufficient conditions are considered one for the robust stability of the uncertain system and other one for the contrary situation if both sufficient conditions fail to characterize the polytope then it is iteratively subdivided into subpolytopes until some one proves to be unstable or all ones are verified to be robustly stable the polytope subdivision is implemented by means of a specially developed simplex subdivision algorithm exhaustive numerical tests prove the efficiency of the proposed approach when compared with the most recent lmi based formulations

controller design with regional pole constraints a design procedure is developed that combines linear quadratic optimal control with regional pole placement specifically a static and dynamic output feedback control problem is addressed in which the poles of the closed loop system are constrained to lie in specified regions of the complex plane these regional pole constraints are embedded within the optimization process by replacing the covariance lyapunov equation by a modified lyapunov equation whose solution in certain cases leads to an upper bound on the quadratic cost functional the results include necessary and sufficient conditions for characterizing static output feedback controllers with bounded performance and regional pole constraints sufficient conditions are also presented for the fixed order i e full and reduced order dynamic output feedback problem with regional pole constraints circular elliptical vertical strip parabolic and section regions are considered

lmi characterization of structural and robust stability the discrete time case abstract this paper extends to the discrete time case some robust stability conditions recently obtained for continuous time systems those conditions are expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable as in the continuous time case parameter dependent lyapunov functions can be constructed and consequently the new approach can yield much sharper and less conservative results than the simultaneous stability approach in particular well known stability problems namely d stability and robust stability in the presence of diagonally structured uncertainty can be more efficiently addressed numerical examples are included to illustrate the advantages of the new stability conditions

a neural network for linear matrix inequality problems gradient type hopfield networks have been widely used in optimization problems solving the paper presents a novel application by developing a matrix oriented gradient approach to solve a class of linear matrix inequalities lmis which are commonly encountered in the robust control system analysis and design the solution process is parallel and distributed in neural computation the proposed networks are proven to be stable in the large representative lmis such as generalized lyapunov matrix inequalities simultaneous lyapunov matrix inequalities and algebraic riccati matrix inequalities are considered several examples are provided to demonstrate the proposed results to verify the proposed control scheme in real time applications a high speed digital signal processor is used to emulate the neural net based control scheme

linear quadratic control with stability degree constraint abstract the problem of static state feedback linear quadratic lq optimal control subject to a prescribed degree of stability for the closed loop system is considered in this paper a necessary optimality condition is given via the lagrange multiplier method a globally convergent algorithm is provided to solve the optimization problem it is shown that the algorithm recovers the standard lq feedback gain provided the desired stability degree is small enough to be within the range by the standard lq design as for other cases the optimum occurs on the boundary of the region a numerical example shows that the proposed algorithm provides a better design compared to the existing methods

prox method with rate of convergence o 1 t for variational inequalities with lipschitz continuous monotone operators and smooth convex concave saddle point problems we propose a prox type method with efficiency estimate o epsilon 1 for approximating saddle points of convex concave c 1 1 functions and solutions of variational inequalities with monotone lipschitz continuous operators application examples include matrix games eigenvalue minimization and computing the lovasz capacity number of a graph and these are illustrated by numerical experiments with large scale matrix games and lovasz capacity problems

an improved lmi condition for robust d stability of uncertain polytopic systems a sufficient condition for robust d stability of linear systems with polytope type uncertainties is proposed the result is based on a linear parameter dependent lyapunov function obtained from the feasibility test of a set of linear matrix inequalities lmis defined at the vertices of the polytope this improved lmi condition encompasses previous results based on additional inequalities as well as results based on extra variables

geometric lifting of the canonical basis and semitoric degenerations of richardson varieties in the sl n case a berenstein and a zelevinsky studied the schutzenberger involution in terms of lusztig s canonical basis 3 we generalize their construction and formulas for any semisimple lie algebra we use for this the geometric lifting of the canonical basis on which an analogue of the schutzenberger involution can be given as an application we construct semitoric degenerations of richardson varieties following a method of p caldero 6

on the convergence of conditional e subgradient methods for convex programs and convex concave saddle point problems abstract the paper provides two contributions first we present new convergence results for conditional e subgradient algorithms for general convex programs the results obtained here extend the classical ones by polyak sov math doklady 8 1967 593 ussr comput math math phys 9 1969 14 introduction to optimization optimization software new york 1987 as well as the recent ones in math program 62 1993 261 eur j oper res 88 1996 382 math program 81 1998 23 to a broader framework secondly we establish the application of this technique to solve non strictly convex concave saddle point problems such as primal dual formulations of linear programs contrary to several previous solution algorithms for such problems a saddle point is generated by a very simple scheme in which one component is constructed by means of a conditional e subgradient algorithm while the other is constructed by means of a weighted average of the inexact subproblem solutions generated within the subgradient method the convergence result extends those of minimization methods for non differentiable functions springer verlag berlin 1985 oper res lett 19 1996 105 math program 86 1999 283 for lagrangian saddle point problems in linear and convex programming and of int j numer meth eng 40 1997 1295 for a linear quadratic saddle point problem arising in topology optimization in contact mechanics

improved robust ℋ control for neutral systems via discretised lyapunov krasovskii functional this paper deals with the robust stability analysis and robust control synthesis with guaranteed performance index for time invariant uncertain neutral systems based on linear matrix inequalities lmis the novelty is to extend the discretisation technique introduced by k gu for an appropriate parameter dependent lyapunov krasovskii functional then the approach proposed improves the previous criteria from the literature that treats robust stability and control for uncertain neutral systems the main strategy used to derive the obtained conditions is to introduce slack variables allowing one to decouple the system matrices from the lyapunov functional matrices without including additional dynamics two numerical examples are performed to support the theoretical predictions the first example deals with stability analysis and the other illustrates how the proposed approach may provide a better solution for the guaranteed cost control problem when compared with others available in the literature

weyl group multiple dirichlet series eisenstein series and crystal bases we show that the whittaker coecients of borel eisenstein series on the metaplectic covers of glr 1 can be described as multiple dirichlet series in r complex variables whose coecients are computed by attaching a number theoretic quantity a product of gauss sums to each vertex in a crystal graph these gauss sums depend on string data previously introduced in work of lusztig berenstein and zelevinsky and littelmann these data are the lengths of segments in a path from the given vertex to the vertex of lowest weight depending on a factorization of the long weyl group element into simple reections the coecients may also be described as sums over strict gelfand tsetlin patterns the description is uniform in the degree of the metaplectic cover

continuous time analysis eigenstructure assignment and h sub 2 synthesis with enhanced linear matrix inequalities lmi characterizations this note describes a new framework for the analysis and synthesis of control systems which constitutes a genuine continuous time extension of results that are only available in discrete time in contrast to earlier results the proposed methods involve a specific transformation on the lyapunov variables and a reciprocal variant of the projection lemma in addition to the classical linearizing transformations on the controller data for a wide range of problems including robust analysis and synthesis multichannel h sub 2 stateand output feedback syntheses the approach leads to potentially less conservative linear matrix inequality lmi characterizations this comes from the fact that the technical restriction of using a single lyapunov function is to some extent ruled out in this new approach moreover the approach offers new potentials for problems that cannot be handled using earlier techniques an important instance is the eigenstructure assignment problem blended with lyapunov type constraints which is given a simple and tractable formulation

deepest minimum criterion for biased affine estimation a new strategy called the deepest minimum criterion dmc is presented for optimally obtaining an affine transformation of a given unbiased estimator when a priori information on the parameters is known here it is considered that the samples are drawn from a distribution parametrized by an unknown deterministic vector parameter the a priori information on the true parameter vector is available in the form of a known subset of the parameter space to which the true parameter vector belongs a closed form exact solution is given for the non linear dmc problem in which it is known that the true parameter vector belongs to an ellipsoidal ball and the covariance matrix of the unbiased estimator does not depend on the parameters a closed form exact solution is also given for the min max strategy for this same case

stability of a polytope of matrices counterexamples while there have been significant breakthroughs for the stability of a polytope of polynomials since v l kharitonov s 1978 seminal result on interval polynomials for a polytope of matrices the stability problem is considered far from completely resolved counterexamples are provided for three conjectures that are directly motivated by the results in the polynomial case these counterexamples illustrate the fundamental differences between polynomial stability and matrix stability problems and indicate that some obvious lines of attack on the matrix polytope stability problem will fail

gain scheduling via linear fractional transformations abstract a linear finite dimensional plant with rational state space parameter dependence is controlled using a parameter dependent controller the parameters are known to take on values in a unit ball and are known in real time the goal of control is to stabilize the parameter dependent closed loop system and provide disturbance error attenuation as measured in induced l 2 norms the approach taken uses the optimally scaled small gain theorem and solves the control synthesis problem by reformulating the existence conditions into a finite dimensional convex optimization

robust pole placement in lmi regions discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality lmi regions a class of convex regions of the complex plane that embraces most practically useful stability regions the focus is on linear systems with static uncertainty on the state matrix for this class of uncertain systems the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary lmi regions the resulting tests for robust pole clustering are all numerically tractable because they involve solving linear matrix inequalities lmis and cover both unstructured and parameter uncertainty these analysis results are then applied to the synthesis of dynamic output feedback controllers that robustly assign the closed loop poles in a prescribed lmi region with some conservatism this problem is again tractable via lmi optimization in addition robust pole placement can be combined with other control objectives such as h sub 2 or h sub spl infin performance to capture realistic sets of design specifications physically motivated examples demonstrate the effectiveness of this robust pole clustering technique

a new lmi condition for robust stability of discrete time uncertain systems in this paper a new robust stability condition is derived for uncertain discrete time linear systems with convex polytopic uncertainties the condition is expressed in terms of a set of linear matrix inequalities lmis involving only the vertices of the polytope domain it enables us to determine robust stability of uncertain systems easily by solving some lmis a rigorous proof is given to show that an interesting result appeared recently is a special case of the proposed condition numerical examples also demonstrate the merit of the present condition in the aspect of conservativeness over other results in the literature

stability of polytopes of matrices via affine parameter dependent lyapunov functions asymptotically exact lmi conditions abstract this paper investigates necessary and sufficient conditions for the existence of an affine parameter dependent lyapunov function assuring the hurwitz or schur stability of a polytope of matrices a systematic procedure for constructing a family of linear matrix inequalities conditions of increasing precision is given at each step a set of linear matrix inequalities provides sufficient conditions for the existence of the affine parameter dependent lyapunov function necessity is asymptotically attained through a relaxation based on a generalization of polya s theorem to the case of matrix valued functions numerical experiments illustrate the results

a lower bound on the bayesian mse based on the optimal bias function a lower bound on the minimum mean squared error mse in a bayesian estimation problem is proposed in this paper this bound utilizes a well known connection to the deterministic estimation setting using the prior distribution the bias function which minimizes the cramer rao bound can be determined resulting in a lower bound on the bayesian mse the bound is developed for the general case of a vector parameter with an arbitrary probability distribution and is shown to be asymptotically tight in both the high and low signal to noise ratio snr regimes a numerical study demonstrates several cases in which the proposed technique is both simpler to compute and tighter than alternative methods

uniformly improving maximum likelihood snr estimation of known signals in gaussian channels the signal to noise ratio snr estimation problem is considered for an amplitude modulated known signal in gaussian noise the benchmark method is the maximum likelihood estimator mle whose merits are well documented in the literature in this work an affinely modified version of the mle ammle that uniformly outperforms over all snr values the traditional mle in terms of the mean square error mse is obtained in closed form however construction of an ammle whose mse is lower at every snr than the unbiased cramer rao bound ucrb is shown to be infeasible in light of this result the ammle construction rule is modified to provision for an a priori known set s where the snr lies and the mse enhancement target is pursued within s the latter is realized through proper extension of an existing framework due to eldar which settles the design problem by solving a semidefinite program the analysis is further extended to the general case of vector signal models numerical results show that the proposed design demonstrates enhancement of the mse for all the considered cases

stability analysis of linear time varying systems improving conditions by adding more information about parameter variation abstract in this paper a new lyapunov function is proposed for stability analysis of linear time varying systems this new function carries more information regarding parameter variation leading to less conservative conditions using finsler s lemma and a suitable form to describe the high order time derivatives of the parameters finite sets of lmis are obtained which are progressively less conservative as a pair of parameters grow previous results can be seen as a special case and numerical examples are carried out for the sake of illustration

simplicial grid refinement on freudenthal s algorithm and the optimal number of congruence classes in the present paper we investigate freudenthal s simplex refinement algorithm which can be considered to be the canonical generalization of bank s well known red refinement strategy for triangles freudenthal s algorithm subdivides any given n simplex into 2 n subsimplices in such a way that recursive application results in a stable hierarchy of consistent triangulations our investigations concentrate in particular on the number of congruence classes generated by recursive refinements after presentation of the method and the basic ideas behind it we will show that freudenthal s algorithm produces at most n 2 congruence classes for any initial n simplex no matter how many subsequent refinements are performed moreover we will show that this number is optimal in the sense that recursive application of any affine invariant refinement strategy with 2 n sons per element results in at least n 2 congruence classes for almost all n simplices

robust model reduction of uncertain systems maintaining uncertainty structure this article presents a new approach for or robust model reduction for uncertain systems based on an optimisation algorithm applied directly in the space of reduced order model parameters the proposed approach can be applied to both polytopic or affine parameter dependent models as well as for continuous or discrete time systems the developed procedure is capable of computing both fixed reduced order model as well as uncertain reduced order model keeping the original uncertainty structure the resulting reduced order model is guaranteed to be robust stable

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

lmi based reset mathcal h _ infty design for linear continuous time plants in this technical note an optimization based synthesis of a multi objective reset controller for linear plants is presented the result allows designing via convex tools with a line search a multi objective reset controller optimizing both the exponential converging rate and the mathcal l 2 gain a novel peculiarity of our scheme is that the underlying linear flow dynamics is not necessarily stabilizing this last property is the consequence of taking into account where the flowing trajectories lay and lead to improved performance especially in terms of decay rate some simulations illustrate the usefulness of our techniques

lmi conditions for robust stability analysis based on polynomially parameter dependent lyapunov functions abstract the robust stability of uncertain linear systems in polytopic domains is investigated in this paper the main contribution is to provide a systematic procedure for generating sufficient robust stability linear matrix inequality conditions based on homogeneous polynomially parameter dependent lyapunov matrix functions of arbitrary degree on the uncertain parameters the conditions exploit the positivity of the uncertain parameters being constructed in such a way that as the degree of the polynomial increases the number of linear matrix inequalities and free variables increases and the test becomes less conservative if a feasible solution exists for a certain degree the conditions will also be verified for larger degrees for any given degree the feasibility of a set of linear matrix inequalities defined at the vertices of the polytope assures the robust stability both continuous and discrete time uncertain systems are addressed as illustrated by numerical examples

h sub spl infin design with pole placement constraints an lmi approach this paper addresses the design of state or output feedback h sub spl infin controllers that satisfy additional constraints on the closed loop pole location sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane these conditions are expressed in terms of linear matrix inequalities lmis and the authors formulation is therefore numerically tractable via lmi optimization in the state feedback case mixed h sub 2 h sub spl infin synthesis with regional pole placement is also discussed finally the validity and applicability of this approach are illustrated by a benchmark example

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

output feedback decentralized control of large scale systems using weighted sensitivity functions minimization this paper considers the problem of achieving stability and desired dynamical transient behavior for linear large scale systems by decentralized control it can be done by making the effects of the interconnections between the subsystems arbitrarily small sufficient conditions for stability and diagonal dominance of the closed loop system are introduced these conditions are in terms of decentralized subsystems and directly make a constructive h control design possible a mixed h pole region placement is suggested such that by assigning the closed loop eigenvalues of the isolated subsystems appropriately the eigenvalues of the overall closed loop system are assigned in desirable range the designs are illustrated by an example

explicit controller formulas for lmi based h synthesis the set of h controllers with closed loop performance c can be implicitly parametrized by the solutions r s of a system of linear matrix inequalities lmi the matrices r and s play a role analogous to that of the riccati solutions x and y in classical riccati based h control useful applications include lmi based h synthesis mixed h 2h design and h design with a pole placement constraint this paper is concerned with the reliable computation of h controllers given a solution r s of the characteristic system of lmis explicit formulas are derived for both the regular and singular cases remarkably these formulas are extensions of the usual central controller formulas where the lmi solutions r and s replace the riccati solutions x and y simple and numerically appealing new formulas for discrete time h controllers are also derived

performance predictions for parameter estimators that minimize cost functions using wirtinger calculus with application to cm blind equalization we present calculations of the first order bias and first order variance of parameters which are estimated batch wise from the global minimum of a cost function of complex valued signals embedded in zero mean gaussian noise the derivation involves the calculation of the multidimensional taylor series of the non analytic cost function up to third order using the elegant wirtinger calculus whereas closed form expressions for the variance can be obtained straightforwardly from a second order taylor series and have been presented in various other contexts an exact expression for the bias cannot be derived in general in this paper we propose approximate expressions for the first order bias and confirm them in a comparison of results from extensive analytical calculations with results from monte carlo mc simulations for the statistical efficiency of a batch processing blind equalizer using the constant modulus cm criterion we study the equalization of independent and identically distributed i i d random symbols and obtain asymptotic for large batch size expressions for the averages of the bias and variance over zero mean random real valued signals of binary phase shift keying bpsk and complex valued signals of m ary psk modulation m 2 finally we compare the statistical efficiency of the cm estimator with the one of the maximum likelihood ml blind estimation of the path parameters and equalized symbols with cm constraint

minimum variance in biased estimation bounds and asymptotically optimal estimators we develop a uniform cramer rao lower bound ucrlb on the total variance of any estimator of an unknown vector of parameters with bias gradient matrix whose norm is bounded by a constant we consider both the frobenius norm and the spectral norm of the bias gradient matrix leading to two corresponding lower bounds we then develop optimal estimators that achieve these lower bounds in the case in which the measurements are related to the unknown parameters through a linear gaussian model tikhonov regularization is shown to achieve the ucrlb when the frobenius norm is considered and the shrunken estimator is shown to achieve the ucrlb when the spectral norm is considered for more general models the penalized maximum likelihood pml estimator with a suitable penalizing function is shown to asymptotically achieve the ucrlb to establish the asymptotic optimality of the pml estimator we first develop the asymptotic mean and variance of the pml estimator for any choice of penalizing function satisfying certain regularity constraints and then derive a general condition on the penalizing function under which the resulting pml estimator asymptotically achieves the ucrlb this then implies that from all linear and nonlinear estimators with bias gradient whose norm is bounded by a constant the proposed pml estimator asymptotically results in the smallest possible variance

sliding mode design via quadratic performance optimization with pole clustering constraint we consider a novel method for designing the sliding mode that minimizes the quadratic performance while keeping a pole clustering constraint our approach is based on the manipulations of linear matrix inequalities lmis imposed by the design objectives for this purpose we newly propose lmi conditions for the quadratic performance optimization and the pole clustering problem respectively in a full order state then they are combined in the lmi framework that is typically devised for the sliding mode design in the convex form an effort is made to reduce the generic conservatism by allowing different lyapunov matrices in addition a class of polytope uncertain systems is addressed to illustrate the advantages of the proposed method

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

mixed h sub 2 h sub infinity control a convex optimization approach the problem of finding an internally stabilizing controller that minimizes a mixed h sub 2 h sub infinity performance measure subject to an inequality constraint on the h sub infinity norm of another closed loop transfer function is considered this problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint both the state feedback and output feedback problems are considered it is shown that in the state feedback case one can come arbitrarily close to the optimal even over full information controllers mixed h sub 2 h sub infinity performance measure using constant gain state feedback moreover the state feedback problem can be converted into a convex optimization problem over a bounded subset of n n and n q where n and q are respectively the state and input dimensions real matrices using the central h sub infinity estimator it is shown that the output feedback problem can be reduced to a state feedback problem in this case the dimension of the resulting controller does not exceed the dimension of the generalized plant

a new robust d stability condition for real convex polytopic uncertainty abstract the problem of robust d stability analysis with respect to real convex polytopic uncertainties is tackled a new lmi based sufficient condition for the existence of parameter dependent lyapunov functions is proposed this condition generalises previously published conditions numerical comparisons with quadratic stability results as well as previous results based on parameter dependent lyapunov functions illustrate the relevance of this new condition finally this result appears to be promising for robust multi objective performance analysis and control synthesis purposes

pole assignment in a specified disk the problem of assigning all poles of a closed loop system in a specified disk by state feedback is considered for both continuous and discrete systems a state feedback control law is determined by using a discrete riccati equation this kind of pole assignment problem is named d pole assignment and its relation to the optimal control problem and its robustness properties are discussed the gain and phase margins for all closed loop poles to stay inside the specified disk d are determined for the proposed control

a general theory for matrix root clustering in subregions of the complex plane we consider the general problem of root clustering of a matrix in the complex plane let a in c n times n and s subset c find the largest class of s and an algebraic criterion which is necessary and sufficient for lambda i a in s i 1 2 n we introduce two types of regions which constitute the largest class of s known to date the criterion is presented both for open regions and closed ones the results are used to define a design methodology for control systems moreover all classical results are shown to be special cases of the present theory

a new discrete time robust stability condition a new robust stability condition for uncertain discrete time systems with convex polytopic uncertainty is given it enables to check stability using parameter dependent lyapunov functions which are derived from lmi conditions it is shown that this new condition provides better results than the classical quadratic stability besides the use of a parameter dependent lyapunov function this condition exhibits a kind of decoupling between the lyapunov and the system matrices which may be explored for control synthesis purposes a numerical example illustrates the results

analysis and comparison of biased affine estimators affine biased estimation is particularly useful when there is some a priori knowledge on the parameters that can be exploited in adverse situations when the number of samples is low or the noise is high three different affine estimation strategies are discussed namely the deepest minimum criterion dmc the min max mm and the linear matrix inequality lmi strategies and closed form expressions are obtained for all of them for the case when the a priori knowledge is given in the form of ellipsoidal constraints on the parameter space and when the covariance matrix of the unbiased estimator is constant a relationship between affine estimation and bayesian estimation of the mean of a multivariate gaussian distribution with gaussian prior is established and it is shown how affine estimation theory can help in the choice of the gaussian prior distribution

establishing robust stability of discrete time systems with time varying uncertainty this paper addresses the problem of establishing robust asymptotical stability of discrete time linear systems polynomially affected by time varying uncertainty confined into a polytope a linear matrix inequality lmi condition for establishing robust asymptotical stability is proposed by introducing a novel approach for establishing the existence of a common homogeneous polynomial lyapunov function hplf this approach consists firstly of introducing a gram matrix built with respect to the state and parametrized by an arbitrary vector function of the uncertainty and secondly of requiring that a transformation of the introduced gram matrix is a sum of squares sos of matrix polynomials the approach hence is referred to as a gram sos approach it is shown that the proposed lmi condition is sufficient for any degree of the hplf candidate that includes quadratic robust stability as a special case and that is also necessary for a sufficiently large degree of the hplf candidate numerical examples also show that the proposed lmi condition can outperform alternative ones in terms of conservatism and computational burden

fundamental connections among the stability conditions using higher order time derivatives of lyapunov functions for the case of linear time invariant systems abstract it has already been recognized that looking for a positive definite lyapunov function such that a high order linear differential inequality with respect to the lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard lyapunov approach examining only the first derivative fails in this context the main purpose of this paper is on one hand to theoretically unveil deeper connections among existing stability conditions especially for linear time invariant lti systems and from the other hand to examine the effect of the higher order time derivatives approach on the stability results for uncertain polytopic lti systems in terms of conservativeness to this end new linear matrix inequality lmi stability conditions are derived by generalizing the concept mentioned above and through the development relations among some existing stability conditions are revealed examples illustrate the improvement over the quadratic approach

comparing between estimation approaches admissible and dominating linear estimators we treat the problem of evaluating the performance of linear estimators for estimating a deterministic parameter vector x in a linear regression model with the mean squared error mse as the performance measure since the mse depends on the unknown vector x a direct comparison between estimators is a difficult problem here we consider a framework for examining the mse of different linear estimation approaches based on the concepts of admissible and dominating estimators we develop a general procedure for determining whether or not a linear estimator is mse admissible and for constructing an estimator strictly dominating a given inadmissible method so that its mse is smaller for all x in particular we show that both problems can be addressed in a unified manner for arbitrary constraint sets on x by considering a certain convex optimization problem we then demonstrate the details of our method for the case in which x is constrained to an ellipsoidal set and for unrestricted choices of x as a by product of our results we derive a closed form solution for the minimax mse estimator on an ellipsoid which is valid for arbitrary model parameters as long as the signal to noise ratio exceeds a certain threshold

robust mean squared error estimation in the presence of model uncertainties we consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties where the vector x is known to satisfy a weighted norm constraint we first assume that the model is known exactly and seek the linear estimator that minimizes the worst case mean squared error mse across all possible values of x we show that for an arbitrary choice of weighting the optimal minimax mse estimator can be formulated as a solution to a semidefinite programming problem sdp which can be solved very efficiently we then develop a closed form expression for the minimax mse estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of mayer and willke with a specific choice of shrinkage factor that explicitly takes the prior information into account next we consider the case in which the model matrix is subject to uncertainties and seek the robust linear estimator that minimizes the worst case mse across all possible values of x and all possible values of the model matrix as we show the robust minimax mse estimator can also be formulated as a solution to an sdp finally we demonstrate through several examples that the minimax mse estimator can significantly increase the performance over the conventional least squares estimator and when the model matrix is subject to uncertainties the robust minimax mse estimator can lead to a considerable improvement in performance over the minimax mse estimator

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

semigroups of sl_3 mathbb c tensor product invariants abstract we find presentations for a family of semigroup algebras related to the problem of decomposing sl 3 c tensor products along the way we find new toric degenerations of the grassmannian variety gr 3 c n which are t invariant for t gl n c the diagonal torus

technical communique time invariant uncertain systems a necessary and sufficient condition for stability and instability via homogeneous parameter dependent quadratic lyapunov functions this paper investigates linear systems with polynomial dependence on time invariant uncertainties constrained in the simplex via homogeneous parameter dependent quadratic lyapunov functions hpd qlfs it is shown that a sufficient condition for establishing whether the system is either stable or unstable can be obtained by solving a generalized eigenvalue problem moreover this condition is also necessary by using a sufficiently large degree of the hpd qlf

uniformly improving the cramer rao bound and maximum likelihood estimation an important aspect of estimation theory is characterizing the best achievable performance in a given estimation problem as well as determining estimators that achieve the optimal performance the traditional cramer rao type bounds provide benchmarks on the variance of any estimator of a deterministic parameter vector under suitable regularity conditions while requiring a priori specification of a desired bias gradient in applications it is often not clear how to choose the required bias a direct measure of the estimation error that takes both the variance and the bias into account is the mean squared error mse which is the sum of the variance and the squared norm of the bias here we develop bounds on the mse in estimating a deterministic parameter vector x0 over all bias vectors that are linear in x0 which includes the traditional unbiased estimation as a special case in some settings it is possible to minimize the mse over all linear bias vectors more generally direct minimization is not possible since the optimal solution depends on the unknown x0 nonetheless we show that in many cases we can find bias vectors that result in an mse bound that is smaller than the cramer rao lower bound crlb for all values of x0 furthermore we explicitly construct estimators that achieve these bounds in cases where an efficient estimator exists by performing a simple linear transformation on the standard maximum likelihood ml estimator this leads to estimators that result in a smaller mse than the ml approach for all possible values of x0