estimation and control of systems with unknown covariance and multiplicative noise the problem of estimation and control for systems with multiplicative noise and unknown second order statistics is considered conditions are found for the existence of a solution based on game theoretic ideas the conditions for the existence of a saddle point for the time invariant filtering problem are necessary and sufficient whereas for all other cases only necessary the central idea of the solution is to convert the stochastic problem to a deterministic optimal control problem whose minimax point is sought with respect to the control filter and unknown statistics parameters the results that are derived show that the problem of estimation for systems with unknown covariances depends on the costate matrix which in turn is a function of the performance measure thus the filter loses one of its best known properties that of independence of the performance functional this property holds not only for the classical kalman filter but also for multiplicative systems

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

detection of topological patterns in complex networks correlation profile of the internet abstract a general scheme for detecting and analyzing topological patterns in large complex networks is presented in this scheme the network in question is compared with its properly randomized version that preserves some of its low level topological properties statistically significant deviation of any topological property of a network from this null model likely reflects its design principles and or evolutionary history we illustrate this basic scheme using the example of the correlation profile of the internet quantifying correlations between degrees of its neighboring nodes this profile distinguishes the internet from previously studied molecular networks with a similar scale free degree distribution we finally demonstrate that the clustering in a network is very sensitive to both the degree distribution and its correlation profile and compare the clustering in the internet to the appropriate null model

mahler measure and entropy for commuting automorphisms of compact groups we compute the joint entropy ofd commuting automorphisms of a compact metrizable group letr d u 1 1 d 1 1 be the ring of laurent polynomials ind commuting variables andm be anr d module then the dual groupx m ofm is compact and multiplication onm by each of thed variables corresponds to an action m of d by automorphisms ofx m every action of d by automorphisms of a compact abelian group arises this way iff r d our main formula shows that the topological entropy of alpha r d left langle f right rangle is given by h left alpha r d left langle f right rangle right log m left f right int limits 0 1 ldots int limits 0 1 log left f left e 2 pi it 1 ldots e 2 pi it d right right dt 1 ldots dt d where m f is the mahler measure off this reduces to the classical result for toral automorphisms via jensen s formula while the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7 3 4 2 occur if p is a non principal prime ideal we show h left alpha r d p right 0 using an analogue of the yuzvinskii thomas addition formula we computeh m for arbitraryr d modulesm and then the joint entropy for an action of d on a not necessarily abelian compact group using a result of boyd we characterize those m which have completely positive entropy in terms of the prime ideals associated tom and show this condition implies that m is mixing of all orders we also establish an analogue of berg s theorem proving that if m has finite entropy then haar measure is the unique measure of maximal entropy if and only if m has completely positive entropy finally we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

compensatability and optimal compensation of systems with white parameters the optimal compensation problem is considered in the case of linear discrete time systems with stationary white parameters and quadratic criteria a generalization of the notion of mean square stabilizability namely mean square compensatability is introduced it is shown that suitable mean square compensatability and detectability conditions are sufficient and necessary in general for the existence of a unique optimal mean square stabilizing compensator tests are given to determine whether or not a system is mean square compensatable it is indicated how to calculate numerically the tests and the optimal mean square stabilizing compensator the results are illustrated with examples

maximum entropy type lyapunov functions for robust stability and performance analysis we present two lyapunov functions that ensure the unconditional stability and robust performance of a modal system with uncertain damped natural frequency each lyapunov function involves the sum of two matrices the first being the solution to the so called maximum entropy equation and the second being a constant auxiliary portion the significant feature of these lyapunov functions is that the guaranteed robust stability region is independent of the weighting matrix while the performance bounds are relatively tight compared to alternative approaches thus these lyapunov functions are less conservative than standard bounds that tend to be highly sensitive to the choice of state space basis kevwords maximum entropy function robust stability robust performance

a smoothing algorithm for systems with multiplicative noise a fixed point smoothing algorithm is derived for linear time varying systems with multiplicative noise in the state channels of the plant and the measurements as well as additive noise as in systems without multiplicative noise the smoothed estimates depend on the filtered estimates the steady state behavior of the algorithm is examined if the multiplicative part is absent then the results coincide with known results for systems with additive noise

robust stabilization with positive real uncertainty beyond the small gain theory in many applications of feedback control phase information is available concerning the plant uncertainty for example lightly damped flexible structures with colocated rate sensors and force actuators give rise to positive real transfer functions closed loop stability is thus guaranteed by means of negative feedback with strictly positive real compensators in this paper the properties of positive real transfer functions are used to guarantee robust stability in the presence of positive real but otherwise unknown plant uncertainty these results are then used for controller synthesis to address the problem of robust stabilization in the presence of positive real uncertainty one of the principal motivations for these results is to utilize phase information in guaranteeing robust stability in this these sense results go beyond the usual limitations the small gain theorem and quadratic lyapunov functions which may be conservative when phase information is available the results of this paper are based upon a riccati equation formulation of the positive real lemma and thus are in the spirit of recent riccati based approaches to bounded real h control

controller design of systems with multiplicative noise general linear continuous stochastic systems are considered with multiplicative noise in the control and state channels and stabilizing optimal inputs are synthesized for both time varying and time invariant situations in the nonstationary case a set of nonlinear matrix differential equations has to be solved in the stationary problem a set of nonlinear matrix algebraic equations provides the solution the separation principle is not valid and both the state filtering and the control synthesis problem should be treated simultaneously

the influence of uncertainty on stability and compensatability in this paper the influence of uncertainty of linear discrete time systems modelled with white parameters on the properties of mean square stability and compensatability is considered a measure for the uncertainty in the system matrices is introduced this measure is also a measure for the uncertainty in the system in the sense that stability and compensatability decreases if this system uncertainty increases it is indicated how to calculate numerically the mean square stability and compensatability the results are illustrated with examples

the optimal projection equations with petersen hollot bounds robust stability and performance via fixed order dynamic compensation for systems with structured real valued parameter uncertainty a feedback control design problem involving structured real valued plant parameter uncertainties is considered a quadratic lyapunov bound suggested by recent work of i r petersen and c v hollot 1986 is utilized in conjunction with the guaranteed cost approach of s s l chang and t k c peng 1972 to guarantee robust stability with robust performance bound necessary conditions that generalize the optimal projection equations for fixed order dynamic compensation are used to characterize the controller that minimizes the performance bound the design equations thus effectively serve as sufficient conditions for synthesizing dynamic output feedback controllers that provide robust stability and performance

exploring the assortativity clustering space of a network s degree sequence nowadays there is a multitude of measures designed to capture different aspects of network structure to be able to say if a measured value is expected or not one needs to compare it with a refere

compact group automorphisms addition formulas and fuglede kadison determinants for a countable amenable group and an element f in the integral group ring z being invertible in the group von neumann algebra of we show that the entropy of the shift action of on the pontryagin dual of the quotient of z by its left ideal generated by f is the logarithm of the fuglede kadison determinant of f for the proof we establish an p version of rufus bowen s denition of topological entropy addition formulas for group extensions of countable amenable group actions and an approximation formula for the fuglede kadison determinant of f in terms of the determinants of perturbations of the compressions of f

brief paper optimal reduced order compensation of time varying discrete time systems with deterministic and white parameters the finite horizon optimal compensation problem is considered in the case of linear time varying discrete time systems with deterministic and white stochastic parameters and quadratic criteria the dimensions of the compensator are a priori fixed and may be time varying also the dimensions of the system may be time varying strengthened discrete time optimal projection equations sdope are developed which within the class of minimal compensators are equivalent to the first order necessary optimality conditions based on the sdope and their associated boundary conditions two numerical algorithms are presented to solve the two point boundary value problem one is a homotopy algorithm while the second iterates the sdope repeatedly forward and backward in time the latter algorithm is much more efficient and constitutes a generalization of the single iteration of the control and estimation riccati equations associated with the full order problem for systems with deterministic parameters the algorithms are illustrated with a numerical example the case of systems with deterministic parameters will be treated as a special case of systems with white parameters

robust stability and performance via fixed order dynamic compensation two robust control design problems are considered the robust stabilization problem in volves deterministically modeled bounded butunknown time varying parameter variations while the robust performance problem includes in addition a quadratic performance criterion averaged over stochastic disturbances and maximized over the admissible parameter variations for both problems the design goal is a fixed order i e reduced or full order dynamic strictly proper feedback compensator a sufficient condition for solving the robust stabilization problem is given by means of a quadratic lyapunov function parameterized by the compensator gains for the robust performance problem the lyapunov function provides an upper bound for the closed loop performance this leads to consideration of the auxiliary minimization problem minimize the performance bound over the class of fixed order controllers subject to the lyapunovofunction constraint necessary conditions for optimality in the auxiliary problem thus serve as sufficient conditions for robust stability and performance in the original problem two particular bounds are considered for constructing the quadratic lyapunov function the first corresponds to a right shift multi plicative white noise model while the second was suggested by recent work of petersen and hollot the main result is an extended version ofthe optimal projection equations for fixed order dynamic compensation whose solutions are guaranteed to provide both robust stability and robust performance

mean dimension mean rank and von neumann luck rank we introduce an invariant called mean rank for any module m of the integral group ring of a discrete amenable group as an analogue of the rank of an abelian group it is shown that the mean dimension of the induced action on the pontryagin dual of m the mean rank of m and the von neumann luck rank of m all coincide as applications we establish an addition formula for mean dimension of algebraic actions prove the analogue of the pontryagin schnirelmnn theorem for algebraic actions and show that for elementary amenable groups with an upper bound on the orders of finite subgroups algebraic actions with zero mean dimension are inverse limits of finite entropy actions

the optimal projection equations for reduced order discrete time state estimation for linear systems with multiplicative white noise the optimal projection equations obtained in 2 3 for reduced order discrete time state estimation are generalized to include the effects of state and measurement dependent noise to provide a model of parameter uncertainty in contrast to the single matrix riccati equation arising in the full order kalman filter case the optimal steady state reduced order discrete time estimator is characterized by three matrix equations one modified riccati equation and two modified lyapunov equations coupled by both an oblique projection and stochastic effects

matrix criterion robust linear quadratic control problem this paper studies a robust linear quadratic lq control problem with multiple quadratic performance indices specifically a linear model is assumed to be available for each of a prespecified set of operating points of a system a vector valued quadratic control performance index is given at each operating point it is then desired to find the constant feedback gains of a linear controller to obtain satisfactory control performance over all the operating points this is a realistic case involving the minimization of a vector criterion of vector criteria i e a matrix valued criterion the matrix valued criterion gives insight into standard robust lq design and leads to several related vector minimization problems and to natural procedures for computing a satisfactory non inferior solution to the matrix valued minimization problem

some remarks on the entropy for algebraic actions of amenable groups in this short note we study the entropy for algebraic actions of certain amenable groups the possible values for this entropy are studied various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant we provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to these results extend previous work by lind and ward on p adic entropy

sofic mean dimension abstract we introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces these generalize the gromov lindenstrauss weiss mean dimensions for actions of countable amenable groups and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy

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

on turing dynamical systems and the atiyah problem main theorems of the article concern the problem of atiyah on possible values of l 2 betti numbers it is shown that all non negative real numbers are l 2 betti numbers and that many for example all non negative algebraic real numbers are l 2 betti numbers of simply connected manifolds with respect to a free cocompact action also an explicit example is constructed which leads to a simply connected manifold with a transcendental l 2 betti number with respect to an action of the threefold direct product of the lamplighter group mathbb z 2 mathbb z wr mathbb z the main new idea is embedding turing machines into integral group rings the main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids

free lamplighter groups and a question of atiyah we compute the von neumann dimensions of the kernels of adjacency operators on free lamplighter groups and show that they are irrational thus providing an elementary constructive answer to a question of atiyah

robust optimal parametric lq control with a guaranteed cost bound and applications for an optimal parametric linear quadratic lq control problem a design objective is to determine a controller of constrained structure such that the closed loop system is asymptotically stable and an associated performance measure is optimized in the presence of system uncertainty the system via a parametric lq design is further required to be robust in terms of maintaining the closed loop stability with a guaranteed cost bound this problem is referred to as robust optimal parametric lq control with a guaranteed cost bound and is addressed in this work a new design method is proposed to find an optimal controller for simultaneously guaranteeing robust stability and performance over a specified range of parameter variations the results presented generalize some previous work in this area a versatile numerical algorithm is also given for computing the robust optimal gains the usefulness of the design method is demonstrated by numerical examples and a design of the robust control of a vtol helico

entropy determinants and l 2 torsion we show that for any amenable group and any z module m of type fl with vanishing euler characteristic the entropy of the natural action on the pontryagin dual of m is equal to the l 2 torsion of m as a particular case the entropy of the principal algebraic action associated with the module z z f is equal to the logarithm of the fuglede kadison determinant of f whenever f is a non zero divisor in z this conrms a conjecture of deninger as a key step in the proof we provide a general szeg o type approximation theorem for the fuglede kadison determinant on the group von neumann algebra of an amenable group as a consequence of the equality between l 2 torsion and entropy we show that the l 2 torsion of a non trivial amenable group with nite classifying space vanishes this was conjectured by luck finally we establish a milnor turaev formula for the l 2 torsion of a nite acyclic chain complex

on the robustness of meop design versus asymptotic lqg synthesis an analysis is presented of generalized lqg linear quadratic gaussian type algorithms with regard to robustness characteristics comparisons are made of the maximum entropy optimal projection meop methodology versus the linear quadratic gaussian with loop transfer recovery lqg ltr technique and the parametric robust linear quadratic gaussian synthesis procedure the performance of a reduced order direct controller design is compared with the full order designs on a example representative of typical flexible structure dynamics

robust controller synthesis using the maximum entropy design equations this note presents an application of the optimality conditions obtained in 1 for dynamic compensation in the presence of state control and measurement dependent noise by solving these equations which represent a fundamental generalization of standard steady state lqg theory a series of increasingly robust control designs is obtained for the example considered in 2

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

optimal projection equations for discrete time fixed order dynamic compensation of linear systems with multiplicative white noise the optimal projection equations for discrete time reduced order dynamic compensation are generalized to include the effects of state control and measurement dependent noise in addition the discrete time static output feedback problem with multiplicative disturbances is considered for both problems the design equations are presented in a concise unified manner to facilitate their accessibility for developing numerical algorithms for practical applications

degree landscapes in scale free networks we generalize the degree organizational view of real world networks with broad degree distributions in a landscape analog with mountains high degree nodes and valleys low degree nodes for exam

homoclinic groups ie groups and expansive algebraic actions we give algebraic characterizations for expansiveness of algebraic actions of countable groups the notion of p expansiveness is introduced for algebraic actions and we show that for countable amenable groups a finitely presented algebraic action is 1 expansive exactly when it has finite entropy we also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms and show that the ie group determines the pinsker factor for such actions for an expansive algebraic action of a polycyclic by finite group on x it is shown that the entropy of the action is equal to the entropy of the induced action on the pontryagin dual of the homoclinic group the homoclinic group is a dense subgroup of the ie group the homoclinic group is nontrivial exactly when the action has positive entropy and the homoclinic group is dense in x exactly when the action has completely positive entropy

computational methods for parametric lq problems a survey iterative methods for finding the optimal constant feedback gains for parametric lq problems notably for optimal constant output feedback problems are surveyed the connections of several methods to loss function expansions are discussed with important implications to the understanding of their convergence properties especially the descent anderson moore method levine athans like methods and the newton method are considered convergence results are also included the initialization problem and the output feedback stabilization problem are also discussed furthermore it is shown that the concepts and methods surveyed in this paper are useful in solving many realistic generalized parametric lq problems as well notably so called robust parametric lq problems

why social networks are different from other types of networks we argue that social networks differ from most other types of networks including technological and biological networks in two important ways first they have nontrivial clustering or network transitivity and second they show positive correlations also called assortative mixing between the degrees of adjacent vertices social networks are often divided into groups or communities and it has recently been suggested that this division could account for the observed clustering we demonstrate that group structure in networks can also account for degree correlations we show using a simple model that we should expect assortative mixing in such networks whenever there is variation in the sizes of the groups and that the predicted level of assortative mixing compares well with that observed in real world networks

multiple models multiplicative noise and linear quadratic control algorithmic aspects numerical solution of certain linear quadratic lq control problems for robust design is addressed an iterative method is suggested and analysed for solving a minimax multiple model lq control problem a convergent iterative method is studied for finding the constant feedback gains of a given linear controller so as to solve a spectral radius functional minimization problem for multiple plants furthermore a convergent iterative method is proposed for solving the maximum entropy parametric lq control problem with multiplicative noise

robust static and dynamic output feedback stabilization deterministic and stochastic perspectives three parallel gaps in robust feedback control theory are examined sufficiency versus necessity deterministic versus stochastic uncertainty modeling and stability versus performance deterministic and stochastic output feedback control problems are considered with both static and dynamic controllers the static and dynamic robust stabilization problems involve deterministically modeled bounded but unknown measurable time varying parameter variations while the static and dynamic stochastic optimal control problems feature state control and measurement dependent white noise general sufficiency conditions for the deterministic problems are obtained using lyapunov s direct method while necessary conditions for the stochastic problems are derived as a consequence of minimizing a quadratic performance criterion the sufficiency tests are then applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems as an additional application of the deterministic result the modified riccati equation approach of petersen and hollot is generalized in the static case and extended to dynamic compensation