on reachability under uncertainty the paper studies the problem of reachability for linear systems in the presence of uncertain unknown but bounded input disturbances that may also be interpreted as the action of an adversary in a game theoretic setting it defines possible notions of reachability under uncertainty emphasizing the differences between reachability under open loop and closed loop control solution schemes for calculating reachability sets are then indicated the situation when observations arrive at given isolated instances of time leads to problems of anticipative maxmin or nonanticipative minmax piecewise open loop control with corrections and to the respective notions of reachability as the number of corrections tends to infinity one comes in both cases to reachability under nonanticipative feedback control it is shown that the closed loop reach sets under uncertainty may be found through a solution of the forward hamilton jacobi bellman isaacs hjbi equation the basic relations are derived through the investigation of superpositions of value functions for appropriate sequential maxmin or minmax problems of control

brief paper reach set computation and control synthesis for discrete time dynamical systems with disturbances the paper is devoted to the formulation of the reachability problem for discrete time dynamical systems with disturbances the concept of maxmin and minmax forward and backward reach sets is addressed invariance of the backward reach set is discussed the emphasis of the paper is on discrete time linear systems for which the ellipsoidal computational method is described the synthesis of maxmin and minmax closed loop control for steering the system to a given target set using ellipsoidal backward reach set approximations is explained the ellipsoidal method covered in the paper is implemented in the ellipsoidal toolbox for matlab a popular collection of ellipsoidal calculus routines freely available online

on the structure of quasi stationary competing particle systems we study point processes on the real line whose configurations x are locally finite have a maximum and evolve through increments which are functions of correlated gaussian variables the correlations are intrinsic to the points and quantified by a matrix q q ij i j a probability measure on the pair x q is said to be quasi stationary if the joint law of the gaps of x and of q is invariant under the evolution a known class of universally quasi stationary processes is given by the ruelle probability cascades rpc which are based on hierarchically nested poisson dirichlet processes it was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasi stationary the main result of this work is a proof of this conjecture for the case where q ij assume only a finite number of values the result is of relevance for mean field spin glass models where the evolution corresponds to the cavity dynamics and where the hierarchical organization of the gibbs measure was first proposed as an ansatz

brief accuracy enhancing methods for balancing related frequency weighted model and controller reduction we consider the solution of the balancing related frequency weighted model and controller reduction problems using accuracy enhanced numerical algorithms we propose first new stability enforcing choices of the frequency weighted grammians which can guarantee the stability of reduced models for two sided frequency weights then we show that for the frequency weighted controller reduction problems with standard stability and performance enforcing frequency weights the computation of the frequency weighted grammians can be done by solving reduced order lyapunov equations for both frequency weighted model and controller reduction problems we indicate how to compute the grammians directly in terms of their cholesky factors this allows the extension of the square root and balancing free accuracy enhancing techniques to the frequency weighted case

modular centralizer algebras corresponding to p groups abstract we study the loewy structure of the centralizer algebra k p q for p a p group with subgroup q and k a field of characteristic p here k p q is a special type of hecke algebra the main tool we employ is the decomposition k p q k c p q i of k p q as a split extension of a nilpotent ideal i by the group algebra k c p q we compute the loewy structure for several classes of groups investigate the symmetry of the loewy series and give upper and lower bounds on the loewy length of k p q several of these results were discovered through the use of magma especially the general pattern for most of our computations as a final application of the decomposition we determine the representation type of k p q

balanced realizations via gradient flow techniques abstract the task of finding a class of balanced minimal realizations is shown to be equivalent to finding limiting solutions of certain gradient flow differential equations by viewing such algebraic tasks in the context of calculus they are amenable to analog computational solutions or parallel processing machines perhaps even neural networks the convergence rates of the differential equations is exponential and consequentially convergence is rapid and numerical stability properties are attractive

approximation metrics for discrete and continuous systems established system relationships for discrete systems such as language inclusion simulation and bisimulation require system observations to be identical when interacting with the physical world modeled by continuous or hybrid systems exact relationships are restrictive and not robust in this paper we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion approximate simulation and approximate bisimulation relations we define a hierarchy of approximation pseudo metrics between two systems that quantify the quality of the approximation and capture the established exact relationships as zero sections our approximation framework is compositional for a synchronous composition operator algorithms are developed for computing the proposed pseudo metrics both exactly and approximately the exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations or dually the solution of a static game whose cost is the so called branching distance between the systems approximations for the pseudo metrics can be obtained by considering lyapunov like functions called simulation and bisimulation functions we illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems

a jacobi type method for computing balanced realizations a new numerical scheme for computing balancing coordinate transformations in linear systems theory is presented the method is closely related to the jacobi method for diagonalizing symmetric matrices here the minimization of the sum of traces of the gramians by orthogonal and nonorthogonal jacobi type rotations is considered the algorithm is shown to be globally convergent to a balancing transformation that arranges the hankel singular values in a prescribed ordering local quadratic convergence of the algorithm is shown

computation of system balancing transformations and other applications of simultaneous diagonalization algorithms an algorithm is presented in this paper for computing state space balancing transformations directly from a state space realization the algorithm requires no squaring up or unnecessary matrix products various algorithmic aspects are discussed in detail a key feature of the algorithm is the determination of a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product other contragredient transformation applications are also described it is further shown that a similar approach may be taken involving the generalized singular value decomposition to the classical simultaneous diagonalization problem these svd based simultaneous diagonalization algorithms provide a computational alternative to existing methods for solving certain classes of symmetric positive definite generalized eigenvalue problems

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

random overlap structures properties and applications to spin glasses random overlap structures rost s are random elements on the space of probability measures on the unit ball of a hilbert space where two measures are identified if they differ by an isometry in spin glasses they arise as natural limits of gibbs measures under the appropriate algebra of functions we prove that the so called cavity mapping on the space of rost s is continuous leading to a proof of the stochastic stability conjecture for the limiting gibbs measures of a large class of spin glass models similar arguments yield the proofs of a number of other properties of rost s that may be useful in future attempts at proving the ultrametricity conjecture lastly assuming that the ultrametricity conjecture holds the setup yields a constructive proof of the parisi formula for the free energy of the sherrington kirkpatrick model by making rigorous a heuristic of aizenman sims and starr

algorithms for model reduction of large dynamical systems three algorithms for the model reduction of large scale continuous time time invariant linear dynamical systems with a sparse or structured transition matrix and a small number of inputs and outputs are described they rely on low rank approximations to the controllability and observability gramians which can efficiently be computed by adi based iterative low rank methods the first two model reduction methods are closely related to the well known square root method and schur method which are balanced truncation techniques the third method is a heuristic balancing free technique the performance of the model reduction algorithms is studied in numerical experiments

a case study in model reduction of linear time varying systems in this paper the balanced truncation procedure is applied to time varying linear systems both in continuous and in discrete time the methods are applied to a linear approximation of a diesel exhaust catalyst model the reduced order systems are obtained by using certain projections instead of direct balancing an approximate zero order hold discretization of continuous time systems is described and a new a priori approximation error bound for balanced truncation in the discrete time case is obtained the case study shows that there are several advantages to work in discrete time it gives simpler implementation with fewer computations

class expansion of some symmetric functions in jucys murphy elements abstract we present a method to compute the class expansion of a symmetric function in the jucys murphy elements of the symmetric group we apply this method to one row hall littlewood symmetric functions which interpolate between power sums and complete symmetric functions

character values and decomposition matrices of symmetric groups the relationships between the values taken by ordinary characters of symmetric groups are exploited to prove two theorems in the modular representation theory of the symmetric group 1 the decomposition matrices of symmetric groups in odd characteristic have distinct rows in characteristic 2 the rows of a decomposition matrix labelled by the different partitions and are equal if and only if and are conjugate an analogous result is proved for hecke algebras 2 a specht module for the symmetric group sn defined over an algebraically closed field of odd characteristic is decomposable on restriction to the alternating group an if and only if it is simple and the labelling partition is self conjugate this result is generalised to an arbitrary field of odd characteristic

ellipsoidal techniques for reachability analysis of discrete time linear systems this paper describes the computation of reach sets for discrete time linear control systems with time varying coefficients and ellipsoidal bounds on the controls and initial conditions the algorithms construct external and internal ellipsoidal approximations that touch the reach set boundary from outside and from inside recurrence relations describe the time evolution of these approximations an essential part of the paper deals with singular discrete time linear systems

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

equilibrium statistical mechanics of bipartite spin systems the aim of this paper is to give an extensive treatment of bipartite mean field spin systems pure and disordered at first bipartite ferromagnets are investigated and an explicit expression for the free energy is achieved through a new minimax variational principle then via the hamilton jacobi technique the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one the same is investigated for bipartite spin glasses by the borel cantelli lemma we obtain the control of the high temperature regime while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation uniquely defined by a minimax variational principle again we also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one party model counterparts for the sake of completeness we show further that at zero temperature the replica symmetric entropy becomes negative and consequently such a symmetry must be broken the treatment of the fully broken replica symmetry case is deferred to a forthcoming paper as a first step in this direction we start deriving the linear and quadratic constraints to overlap fluctuations

a time dependent hamilton jacobi formulation of reachable sets for continuous dynamic games we describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game the algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time dependent hamilton jacobi isaacs partial differential equation while alternative techniques for computing the reachable set have been proposed the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters because the time dependent equation s solution is continuous and defined throughout the state space methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions a numerical implementation of our formulation is described and has been released on the web its correctness is verified through a two vehicle three dimensional collision avoidance example for which an analytic solution is available

computing reachable states for nonlinear biological models in this paper we describe reachability computation for continuous and hybrid systems and its potential contribution to the process of building and debugging biological models we summarize the state of the art for linear systems and then develop a novel algorithm for computing reachable states for nonlinear systems we report experimental results obtained using a prototype implementation applied to several biological models we believe these results constitute a promising contribution to the analysis of complex models of biological systems

concentration for multidimensional diffusions and their boundary local times we prove that probability laws of certain multidimensional semimartingales which includes time inhomogenous diffusions under suitable assumptions satisfy quadratic transportation cost inequality under the uniform metric from this we derive concentration properties of lipschitz functions of process paths that depend on the entire history in particular we estimate concentration of boundary local time of reflected brownian motions on a polyhedral domain we work out explicit applications of consequences of measure concentration for the case of brownian motion with rank based drifts

a remark on the infinite volume gibbs measures of spin glasses in this note we point out that infinite volume gibbs measures of spin glass models on the hypercube can be identified as random probability measures on the unit ball of a hilbert space this simple observation follows from a result of dovbysh and sudakov on weakly exchangeable random matrices limiting gibbs measures can then be studied as single well defined objects this approach naturally extends the space of random overlap structures as defined by aizenman et al we discuss the ruelle probability cascades and the stochastic stability within this framework as an application we use an idea of parisi and talagrand to prove that if a sequence of finite volume gibbs measures satisfies the ghirlanda guerra identities then the infinite volume measure must be singular as a measure on a hilbert space

computational techniques for reachability analysis of max plus linear systems this work discusses a computational approach to reachability analysis of max plus linear mpl systems a class of discrete event systems widely used in synchronization and scheduling applications given a set of initial states we characterize and compute its reach tube namely the collection of set of reachable states regarded step wise as reach sets by an alternative characterization of the mpl dynamics we show that the exact computation of the reach sets can be performed quickly and compactly by manipulations of difference bound matrices and further derive worst case bounds on the complexity of these operations the approach is also extended to backward reachability analysis the concepts and results are elucidated by a running example and we further illustrate the performance of the approach by a numerical benchmark the technique comfortably handles twenty dimensional mpl systems i e with twenty continuous state variables and as such it outperforms the state of the art alternative approaches in the literature

set membership filtering with incomplete observations abstract this study addresses the set membership estimation problem for a class of discrete time varying systems with incomplete observations a set membership filter is developed and a recursive algorithm is proposed to calculate the state estimate ellipsoid which contains the true value to solve the problem that the conventional set membership filter cannot guarantee the stability when applied to discrete time varying systems with incomplete observations a quantitative analysis method about incomplete observations is developed and a tight bound of the estimation error is found based on interval analysis and some bounded noise assumptions in terms of bounded assumptions the relationship between the bound of estimated error and the data dropout rate is obtained if the data dropout rate is less than a maximal value the set membership filter is asymptotically stable the proposed filter is applied to a two state example to demonstrate the effectiveness of theoretical results

balancing related methods for minimal realization of periodic systems we propose balancing related numerically reliable methods to compute minimal realizations of linear periodic systems with time varying dimensions the first method belongs to the family of square root methods with guaranteed enhanced computational accuracy and can be used to compute balanced minimal order realizations an alternative balancing free square root method has the advantage of a potentially better numerical accuracy in case of poorly scaled original systems the key numerical computation in both methods is the solution of nonnegative periodic lyapunov equations directly for the cholesky factors of the solutions for this purpose a numerically reliable computational algorithm is proposed to solve nonnegative periodic lyapunov equations with time varying dimensions

eigenspace stability conditions in mean field replica theories i consider the eigenspace form of the saddle point conditions that apply for mean field replica models that are characterized by a single overlap matrix order parameter these allow for a characterization of the full set of saddle point solutions further they motivate consideration of a particular class of solutions which i call the principally commuting set these are the solutions that simultaneously satisfy the replica space rotational stability constraints for all physical models i show that this set satisfies the replica equivalence symmetry as well as other more stringent conditions and that it also contains the set of parisi type ultrametric solutions as a subset

reachability analysis using dissipation inequalities for uncertain nonlinear systems abstract we propose a method to outer bound forward reachable sets on finite horizons for uncertain nonlinear systems with polynomial dynamics this method makes use of time dependent polynomial storage functions that satisfy appropriate dissipation inequalities that account for time varying uncertain parameters l 2 disturbances and perturbations characterized by integral quadratic constraints iqcs with both hard and soft factorizations the use of iqcs in forward reachability analysis allows for various types of uncertainty including unmodeled dynamics the generalized s procedure and sum of squares techniques are used to derive algorithms with the goal of finding the tightest outer bound with a desired shape both pedagogical and practically motivated examples are presented including a 7 state f 18 aircraft model

construction of pure states in mean field models for spin glasses if a mean field model for spin glasses is generic in the sense that it satisfies the extended ghirlanda guerra identities and if the law of the overlaps has a point mass at the largest point q of its support we prove that one can decompose the configuration space into a sequence of sets ak such that generically the overlap of two configurations is equal to q if and only if they belong to the same set ak for the study of the overlaps each set ak can be replaced by a single point combining this with a recent result of panchenko a connection between ghirlanda guerra identities and ultrametricity ann probab 2008 to appear this proves that if the overlaps take only finitely many values ultrametricity occurs we give an elementary self contained proof of this result based on simple inequalities and an averaging argument

a note on closed loop balanced truncation closed loop balanced truncation is a model reduction method that aims at preserving the closed loop properties of the original controller we show that this method is equivalent to frequency weighted balanced truncation with certain weighting and that for observer based controllers a significant amount of computation can be saved because of the observer state feedback structure of the controller

convergence rates for rank based models with applications to portfolio theory we determine rates of convergence of rank based interacting diffusions and semimartingale reflecting brownian motions to equilibrium bounds on fluctuations of additive functionals are obtained using transportation cost information inequalities for markov processes we work out various applications to the rank based abstract equity markets used in stochastic portfolio theory for example we produce quantitative bounds including constants for fluctuations of market weights and occupation times of various ranks for individual coordinates another important application is the comparison of performance between symmetric functionally generated portfolios and the market portfolio this produces estimates of probabilities of beating the market

the extremal process of branching brownian motion we prove that the extremal process of branching brownian motion in the limit of large times converges weakly to a cluster point process the limiting process is a randomly shifted poisson cluster process where the positions of the clusters is a poisson process with intensity measure with exponential density the law of the individual clusters is characterized as branching brownian motions conditioned to perform unusually large displacements and its existence is proved the proof combines three main ingredients first the results of bramson on the convergence of solutions of the kolmogorov petrovsky piscounov equation with general initial conditions to standing waves second the integral representations of such waves as first obtained by lalley and sellke in the case of heaviside initial conditions third a proper identification of the tail of the extremal process with an auxiliary process based on the work of chauvin and rouault which fully captures the large time asymptotics of the extremal process the analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses

hierarchical exchangeability of pure states in mean field spin glass models the main result in this paper is motivated by the mezard parisi ansatz which predicts a very special structure for the distribution of spins in diluted mean field spin glass models such as the random k sat model at positive temperature using the fact that one can safely assume the validity of the ghirlanda guerra identities in these models we prove hierarchical exchangeability of pure states for the asymptotic gibbs measures which allows us to apply a representation result for hierarchically exchangeable arrays recently proved in austin and panchenko in probab theory relat fields 2013 comparing this representation with the predictions of the mezard parisi ansatz one can see that the key property still missing is that the multi overlaps between pure states depend only on their overlaps

the extremal process of critical points of the pure p spin spherical spin glass model recently sharp results concerning the critical points of the hamiltonian of the p spin spherical spin glass model have been obtained by means of moments computations in particular these moments computations allow for the evaluation of the leading term of the ground state i e of the global minimum in this paper we study the extremal point process of critical points that is the point process associated to all critical values in the vicinity of the ground state we show that the latter converges in distribution to a poisson point process of exponential intensity in particular we identify the correct centering of the ground state and prove the convergence in distribution of the centered minimum to a minus gumbel variable these results are identical to what one obtains for a sequence of i i d variables correctly normalized namely we show that the model is in the universality class of rem

the parisi ultrametricity conjecture in this paper we prove that the support of a random measure on the unit ball of a separable hilbert space that satises the ghirlanda guerra identities must be ultrametric with probability one this implies the parisi ultrametricity conjecture in mean eld spin glass models such as the sherrington kirkpatrick and mixed p spin models for which gibbs measures are known to satisfy the ghirlanda guerra identities in the thermodynamic limit

complexity reduction through a schur based decomposition for reachability analysis of linear time invariant systems this article presents a method for complexity reduction in reachability analysis and safety preserving controller synthesis via schur based decomposition the decomposition results in either decoupled or weakly coupled lower dimensional subsystems reachable sets computed independently for each subsystem are back projected and intersected to yield an overapproximation of the actual reachable set moreover applying this technique to a class of unstable lti systems we show that when certain eigenvalue and state constraint conditions are satisfied further reduction of complexity is possible evaluating our method for a variety of examples we demonstrate that significant reduction in the computational costs can be achieved this technique has considerable potential utility for use in conjunction with computationally intensive reachability tools

structure of finite rsb asymptotic gibbs measures in the diluted spin glass models we suggest a possible approach to proving the mezard parisi formula for the free energy in the diluted spin glass models such as diluted k spin or random k sat model at any positive temperature in the main contribution of the paper we show that a certain small modification of the hamiltonian in any of these models forces all finite rsb asymptotic gibbs measures in the sense of the overlaps to satisfy the mezard parisi ansatz for the distribution of spins unfortunately what is still missing is a description of the general full rsb asymptotic gibbs measures if one could show that the general case can be approximated by finite rsb case in the right sense then one could a posteriori remove the small modification of the hamiltonian to recover the mezard parisi formula for the original model

stable interval observers in bbc for linear systems with time varying input bounds this technical note deals with the design of stable interval observers and estimators for continuous time linear dynamic systems under uncertain initial states and uncertain inputs enclosed within time varying zonotopic bounds no monotony assumption such as cooperativity is required in the vector field the interval observer stability directly derives from the stability of the observer state matrix where any poles real or complex single or multiple are handled in the same way

time interval model reduction of bilinear systems in this paper a new method for model reduction of bilinear systems is presented the proposed technique is from the family of gramian based model reduction methods the method uses time interval generalised gramians in the reduction procedure rather than the ordinary generalised gramians and in such a way that it improves the accuracy of the approximation within the time interval for which the method is applied the time interval generalised gramians are the solutions to the generalised time interval lyapunov equations the conditions for these equations to be solvable are derived and an algorithm is proposed to solve these equations iteratively the method is further illustrated with the help of two illustrative examples the numerical results show that the method is more accurate than its previous counterpart which is based on the ordinary gramians

model reduction via schur decomposition a simple procedure for model reduction based upon a schur decomposition of a given system is presented the obtained partitioned system allows the combination of singular perturbation and aggregation approaches

a new method of reachable set estimation for time delay systems with polytopic uncertainties we address the reachable set bounding problem of linear systems subject to both time varying delay and polytopic uncertainties a novel criterion is established to give a more accurate description of the set which bounds the states starting from the origin by inputs with peak values by combining the maximal lyapunov functional and the reciprocally convex approach we obtain a tighter reachable set estimation with lower computational complexity the corresponding reachable set condition for the common lyapunov functional can be obtained naturally as a special case of the above result finally two examples illustrate the advantages of our proposed method

a schur method for balanced truncation model reduction it is shown that a not necessarily balanced state space realization of the moore reduced model can be computed directly without balancing via projections defined in terms of arbitrary bases for the left and right eigenspaces associated with the large eigenvalues of the product pq of the reachability and controllability grammians two specific methods for computing these bases are proposed one based on the ordered schur decomposition of pq and the other based on the cholesky factors of p and q the algorithms perform reliably even for nonminimal models

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

the modelling and vibration control of beams with active constrained layer damping abstract the finite element method fem is combined with the golla hughes mctavish ghm model of viscoelastic materials vem to model a cantilever beam with active constrained layer damping treatments this approach avoids time consuming iteration in solving modal frequencies modal damping ratios and responses but the resultant finite element fe model has too many degrees of freedom d o f s from the point of view of control nor is it observable and controllable a new model reduction procedure is proposed an iterative dynamic condensation is performed in the physical space and guyan condensation is taken as an initial iteration approximation a reduced order model rom of suitable size emerges but it is still not observable and controllable accordingly a robust model reduction method is then employed in the state space a numerical example proves that this procedure reduces the model and assures the stability controllability and observability of the final reduced order model from finally a controller is designed by linear quadratic gaussian lqg method based on the from the vibration attenuation is evident

rational matrix gcds and the design of squaring down compensators a state space theory a state space construction for rational matrix greatest common divisors gcds of rational transfer matrices is given it is shown how the gcd results can be used to solve the problem of designing stable minimum phase squaring down compensators for multi variable plants one application is a direct state space construction for such compensators and a state space solution to fat plant h infinity control problems the results make use of the concepts of strongly observable systems and maximally observable systems and build upon the concepts introduced in the state space gcd extraction results for polynomial matrices of l m silverman and p van dooren 1979

on stochastic linear systems with zonotopic support sets abstract this paper analyzes stochastic linear discrete time processes whose process noise sequence consists of independent and uniformly distributed random variables on given zonotopes we propose a cumulant based approach for approximating both the transient and limit distributions of the associated state sequence the method relies on a novel class of k symmetric lyapunov equations which are used to construct explicit expressions for the cumulants the state distribution is recovered via a generalized gram charlier expansion with respect to products of a multivariate variant of wigner s semicircle distribution using chebyshev polynomials of the second kind this expansion converges uniformly under surprisingly mild conditions to the exact state distribution of the system a robust feedback control synthesis problem is used to illustrate the proposed approach

a short note on constrained linear control systems with multiplicative ellipsoidal uncertainty this technical note revisits the classical robust control question of how to design linear control laws for uncertain linear dynamic systems we formulate this robust control design problem from a modern optimal control perspective which allows us to take into account control and state constraints that have to be satisfied by the closed loop system for all possible uncertainty scenarios the contribution of this technical note is the derivation of a conservative but computationally tractable robust control design problem formulation for the case that multiplicative uncertainties are present in the linear dynamic systems which render the robust control problem non convex in general here our approximation technique relies on propagating ellipsoidal bounds on the reachable states of the closed loop system the methods developed in this technical note can also be used as a building block for tube based model predictive control schemes where robustly designed linear control laws can help to reduce the conservatism of open loop predictions we illustrate the proposed techniques with a numerical case study

generators for the centre of the group algebra of a symmetric group abstract the ring of symmetric functions is used to obtain an explicit set of generators for the centre of the integral group algebra of a symmetric group different to those given by h k farahat and g higman this generating set is used to shows that the centre of the 2 modular group algebra is generated by certain sums of 2 classes proper subalgebras of the centre of the 2 modular group algebra are studied in the context of symmetric functions these include the algebra that is the span of the 2 regular class sums and the algebra that is generated by the involution class sums various related subalgebras of the modular ring of symmetric functions are shown to be polynomial algebras and furnished with explicit sets of algebraically independent generators

a comprehensive method for reachability analysis of uncertain nonlinear hybrid systems reachability analysis of nonlinear uncertain hybrid systems i e continuous discrete dynamical systems whose continuous dynamics guard sets and reset functions are defined by nonlinear functions can be decomposed in three algorithmic steps computing the reachable set when the system is in a given operation mode computing the discrete transitions i e detecting and localizing when and where the continuous flowpipe intersects the guard sets and aggregating the multiple trajectories that result from an uncertain transition once the whole flow pipe has transitioned so that the algorithm can resume this paper proposes a comprehensive method that provides a nicely integrated solution to the hybrid reachability problem at the core of the method is the concept of mspb i e geometrical object obtained as the minkowski sum of a parallelotope and an axes aligned box mspb are a way to control the over approximation of the taylor s interval integration method as they happen to be a specific type of zonotope they articulate perfectly with the zonotope bounding method that we propose to enclose in an optimal way the set of flowpipe trajectories generated by the transition process the method is evaluated both theoretically by analyzing its complexity and empirically by applying it to well chosen hybrid nonlinear examples

model reduction for discrete unstable systems based on generalized normal representations the input and output normal representations of stable systems are generalized for unstable systems these are then used to reduce the order of unstable discrete time linear multivariable systems resulting in reduced order models with the same number of unstable poles as the full order model and with an a priori upper bound on the reduction error the generalized normal representations are then modified to include a frequency weighting resulting in frequency weighted generalized normal representations these are used to derive frequency weighted reduced order models for unstable discrete systems with the same number of unstable poles as the full order model and with an a priori upper bound on the frequency weighted reduction error

reachability analysis of linear dynamic systems with constant arbitrary and lipschitz continuous inputs abstract this paper sets forth a method for reachability analysis of linear dynamic systems in continuous time that can be used to compute time domain bounds of states and outputs with floating point precision the focus is on the particular initial conditions and inputs that cause state or output trajectories to attain their extreme values in time inputs can be constant arbitrary or lipschitz continuous waveforms uncertainties in initial conditions and inputs are modeled using zonotopes the calculations exploit the definition of zonotopes and the modal information of the linear system dynamics