julia a fresh approach to numerical computing bridging cultures that have often been distant julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing julia is designed to be easy and fast and questions notions generally held to be laws of nature by practitioners of numerical computing beginlist item high level dynamic programs have to be slow item one must prototype in one language and then rewrite in another language for speed or deployment item there are parts of a system appropriate for the programmer and other parts that are best left untouched as they have been built by the experts endlist we introduce the julia programming language and its design a dance between specialization and abstraction specialization allows for custom treatment multiple dispatch a technique from computer science picks the right algorithm for the right circumstance abstraction which is what good computation is really about recognizes what remains the same after dif

non reciprocity in nonlinear elastodynamics abstract reciprocity is a fundamental property of linear time invariant lti acoustic waveguides governed by self adjoint operators with symmetric green s functions the break of reciprocity in lti elastodynamics is only possible through the break of time reversal symmetry on the micro level and this can be achieved by imposing external biases adding nonlinearities or allowing for time varying system properties we present a volterra series based asymptotic analysis for studying spatial non reciprocity in a class of one dimensional 1d time invariant elastic systems with weak stiffness nonlinearities we show that nonlinearity is neither necessary nor sufficient for breaking reciprocity in this class of systems rather it depends on the boundary conditions the symmetries of the governing linear and nonlinear operators and the choice of the spatial points where the non reciprocity criterion is tested extension of the analysis to higher dimensions and time varying systems is straightforward from a mathematical point of view but not in terms of new non reciprocal physical phenomena whereas the connection of non reciprocity and time irreversibility can be studied as well finally we show that suitably defined non reciprocity measures enable optimization and can provide physical understanding of the nonlinear effects in the dynamics enabling one to establish regimes of maximum nonlinearity we highlight the theoretical developments by means of a numerical example

a companion matrix approach to the study of zeros and critical points of a polynomial in this paper we introduce a new type of companion matrices namely d companion matrices by using these d companion matrices we are able to apply matrix theory directly to study the geometrical relation between the zeros and critical points of a polynomial in fact this new approach will allow us to prove quite a number of new as well as known results on this topic for example we prove some results on the majorization of the critical points of a polynomial by its zeros in particular we give a different proof of a recent result of gerhard schmeisser on this topic the same method allows us to prove a higher order schoenberg type conjecture proposed by m g de bruin and a sharma

a closed formula for the product in simple integral extensions abstract let be an algebraic number and let q a closed formula for the coordinates of the product is given in terms of the coordinates of and and the companion matrix of the minimal polynomial of the formula as well as its proof extend to fairly general simple integral extensions

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

subalgebras of matrix algebras generated by companion matrices abstract let f g z x be monic polynomials of degree n and let c d m n z be the corresponding companion matrices we find necessary and sufficient conditions for the subalgebra z c d to be a sublattice of finite index in the full integral lattice m n z in which case we compute the exact value of this index in terms of the resultant of f and g if r is a commutative ring with identity we determine when r c d m n r in which case a presentation for m n r in terms of c and d is given

parameterized convergence bounds for volterra series expansion of narx models evaluation of the convergence bound in the frequency domain for volterra series expansion of nonlinear systems described by narx models is studied this provides new convergence criteria under which the nonlinear system of interest has a convergent volterra series expansion and the new criteria are expressed explicitly in terms of the input magnitude model parameters and frequency variable the new convergence criteria are firstly developed for harmonic inputs and then extended to multi tone and general input cases based on the theoretical analysis a general procedure for calculating the convergence bound is provided the results provide a fundamental basis for nonlinear signal processing using the volterra series theory

discrete time subharmonic modelling and analysis traditionally the volterra time and frequency domain analysis tools cannot be applied to severely non linear systems in this paper a new method of building a time domain narx miso model for a class of severely siso non linear systems that exhibit subharmonics is introduced and it is shown how this allows the volterra time and frequency domain analysis to be extended to this class of non linear systems the new approach is based on decomposing the original single input based on a fourier analysis to provide a set of modified input signals which have the same period as the output signal a miso narx model can then be constructed from the decomposed multiple inputs and the single output signal the resulting miso model is shown to meet the basic requirement for the existence of a volterra series representation from which important frequency domain properties can be derived explained and discussed this is done by first introducing the derivation of generalized frequency response functions gfrfs from tim

numerical ranges of companion matrices we show that an n by n companion matrix a can have at most n line segments on the boundary w a of its numerical range w a and it has exactly n line segments on w a if and only if for n odd a is unitary and for n even a is unitarily equivalent to the direct sum a1 a2 of two n 2 by n 2 companion matrices a1

application of higher order spectral analysis to cubically nonlinear system identification in this study digital higher order spectral analysis and frequency domain volterra system models are utilized to yield a practical methodology for the identification of weakly nonlinear time invariant systems up to third order the primary focus is on consideration of random excitation of nonlinear systems and thus the approach makes extensive use of higher order spectral analysis to determine the frequency domain volterra kernels which correspond to linear quadratic and cubic transfer functions although the volterra model is nonlinear in terms of its input it is linear in terms of its unknown transfer functions thus a least squares approach is used to determine the optimal in a least squares sense set of linear quadratic and cubic transfer functions of particular practical note is the fact that the approach of this paper is valid for non gaussian as well as gaussian random excitation it may also be utilized for multitone inputs the complexity of the problem addressed in this paper arises from two principal causes 1 the necessity to work in a 3d frequency space to describe cubically nonlinear systems and 2 the necessity to characterize the non gaussian random excitation by computing higher order spectral moments up to sixth order a detailed description of the approach used to determine the nonlinear transfer functions including considerations necessary for digital implementation is presented

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

a constraint satisfaction approach for enclosing solutions to parametric ordinary differential equations this paper considers initial value problems for ordinary differential equations odes where some of the data is uncertain and given by intervals as is the case in many areas of science and engineering interval methods provide a way to approach these problems but they raise fundamental challenges in obtaining high accuracy and low computation costs this work introduces a constraint satisfaction approach to these problems which enhances traditional interval methods with a pruning step based on a global relaxation of the ode the relaxation uses hermite interpolation polynomials and enclosures of their error terms to approximate the ode our work also shows how to find an evaluation time for the relaxation that minimizes its local error theoretical and experimental results show that the approach produces significant improvements in accuracy over the best interval methods for the same computation costs the results also indicate that the new algorithm should be significantly faster when the ode contains many operations

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

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

a new algorithm for factoring polynomials over finite fields in f t there are two standard methods one is due to e berlekamp the other appears to be a folk method both are described by d knuth in 3 pp 381 397 further improvements for special values of q are given by r moenck in 5j see also e berlekamp 1 note that both methods as described in the references apply only when d 1 however straightforward modifications described below allow d 1 both methods use the calculation of resultants or equivalently the solution of linear equations to reduce the problem to finding the roots of a polynomial which has all of its roots in f when p is very small probabilistic methods are used an improvement to the folk method method along with a more explicit calculation of the work required has recently been given by m rabin 6 we present here a new probabilistic method which when combined with the above algorithms avoids the need for both resultants and linear equations it leads to algorithms which are conceptually simpler than previous methods moreover it works equally well for all finite fields f regardless of the magnitude of q when used for factoring a quadratic x2 a it reduces to berlekamp s algorithm other standard algorithms for factoring quadratics are due to d h lehmer 4 and d shanks 7 our algorithm is also suitable for finding solutions of polynomial equations over finite fields

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

the algorithmic analysis of hybrid systems we present a general framework for the formal specification and algorithmic analysis of hybrid systems a hybrid system consists of a discrete program with an analog environment we model hybrid systems as finite automata equipped with variables that evolve continuously with time according to dynamical laws for verification purposes we restrict ourselves to linear hybrid systems where all variables follow piecewise linear trajectories we provide decidability and undecidability results for classes of linear hybrid systems and we show that standard program analysis techniques can be adapted to linear hybrid systems in particular we consider symbolic model checking and minimization procedures that are based on the reachability analysis of an infinite state space the procedures iteratively compute state sets that are definable as unions of convex polyhedra in multidimensional real space we also present approximation techniques for dealing with systems for which the iterative procedures do not converge

on the convergence of the volterra series representation of the duffing s oscillators subjected to harmonic excitations abstract this paper is concerned with the investigation of the convergence issue of the volterra series representations of the duffing s oscillator subjected to harmonic inputs a simple criterion is proposed to determine the upper limit of the magnitude of the harmonic inputs a comparison between the new proposed criterion and a criterion suggested by tomlinson et al g r tomlinson g manson g m lee a simple criterion for establishing an upper limit to the harmonic excitation level of the duffing oscillator using the volterra series journal of sound and vibration 190 1996 751 762 was carried out the results show that the new criterion can provide a more accurate prediction about the convergence of the volterra series representation of the duffing s oscillator

continuous time non linear system identification in the frequency domain a new algorithm for identifying non linear differential equation models is introduced the algorithm avoids the computation of derivatives by using the generalized frequency response functions to reconstruct the model it is shown that the model can be constructed sequentially by building in first the linear terms then the quadratic terms and so on and provides unbiased estimates in the presence of noise a simulated example and the identification of a model relating the action of waveforces on an offshore structure are included to demonstrate the procedure

mapping non linear integro differential equations into the frequency domain a recursive algorithm is derived to compute the generalized frequency response functions for a large class of non linear integro differential equations applications to duffing s equation and a modified van der pol model are discussed

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

recursive algorithm for computing the frequency response of a class of non linear difference equation models abstract a recursive algorithm is developed which is both computationally compact and enables the nth order frequency response functions of a large class of non linear difference equation models to be found without restriction on the order n

analysis of nonlinear oscillators using volterra series in the frequency domain the volterra series representation is a direct generalisation of the linear convolution integral and has been widely applied in the analysis and design of nonlinear systems both in the time and the frequency domain the volterra series is associated with the so called weakly nonlinear systems but even within the framework of weak nonlinearity there is a convergence limit for the existence of a valid volterra series representation for a given nonlinear differential equation barrett 1965 1 proposed a time domain criterion to prove that the volterra series converges within a given region for a class of nonlinear systems with cubic stiffness nonlinearity in this paper this time domain criterion is extended to the frequency domain to accommodate the analysis of nonlinear oscillators subject to harmonic excitation a common and severe nonlinear phenomenon called jump a behavior associated with the duffing oscillator and the multi valued properties of the response solution is investigated using the new frequency domain criterion of establishing the upper limits of the nonlinear oscillators to predict the onset point of the jump and the volterra time and frequency domain analysis of this phenomenon are carried out based on graphical and numerical techniques

reachability analysis of discrete time systems with disturbances this paper presents new results that allow one to compute the set of states that can be robustly steered in a finite number of steps via state feedback control to a given target set the assumptions that are made in this paper are that the system is discrete time nonlinear and time invariant and subject to mixed constraints on the state and input a persistent disturbance dependent on the current state and input acts on the system existing results are not able to address state and input dependent disturbances and the results in this paper are therefore a generalization of previously published results one of the key aims of this paper is to present results such that one can perform the relevant set computations using polyhedral algebra and computational geometry software provided the system is piecewise affine and the constraints are polygonal existing methods are only applicable to piecewise affine systems that either have no control inputs or no disturbances whereas the results in this paper remove this limitation some simple examples are also given that show that even if all the relevant sets are convex and the system is linear convexity of the set of controllable states cannot be guaranteed

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

some decomposition results for companion matrices let c p denote the frobenius companion matrix of the monic polynomial p with complex coefficients the singular value decomposition and the qr decomposition of c p are computed

a simple criterion for establishing an upper limit to the harmonic excitation level of the duffing oscillator using the volterra series abstract high order frequency response functions based on the volterra series are employed to represent the input output characteristics of the duffing oscillator subject to sinusoidal excitation from these a series representation of a first order frequency response function of the non linear system is obtained and used to derive a simple criterion which provides an estimate of the magnitude of the input sine wave above which the predicted response begins to diverge the criterion results in a small overestimate of the magnitude of the input force excitation at the undamped natural frequency condition although the derived criterion strictly applies only to the duffing oscillator the procedure can be applied generally and is extended to a two degree of freedom system with a non linear cubic stiffness producing a similar criterion to that of the single degree of freedom system

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

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

a hybrid bounding method for computing an over approximation for the reachable set of uncertain nonlinear systems in this paper we show how to compute an over approximation for the reachable set of uncertain nonlinear continuous dynamical systems by using guaranteed set integration we introduce two ways to do so the first one is a full interval method which handles whole domains for set computation and relies on state of the art validated numerical integration methods the second one relies on comparison theorems for differential inequalities in order to bracket the uncertain dynamics between two dynamical systems where there is no uncertainty since the derived bracketing systems are piecewise ck differentiable functions validated numerical integration methods cannot be used directly hence our contribution resides in the use of hybrid automata to model the bounding systems we give a rule for building these automata and we show how to run them and address mode switching in a guaranteed way in order to compute the over approximation for the reachable set the computational cost of our method is also analyzed and shown to be smaller that the one of classical interval techniques sufficient conditions are given which ensure the epsiv practical stability of the enclosures given by our hybrid bounding method two examples are also given which show that the performance of our method is very promising

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

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

convergence analysis of volterra series response of nonlinear systems subjected to harmonic excitation volterra series provides a strong platform for non linear analysis and higher order frequency response functions however limited convergence is an inherent difficulty associated with the series and needs to be addressed rigorously prior to its application to a physical system the power series representation of the response of non linear systems subjected to harmonic excitation is investigated in this study the problem of convergence is addressed in terms of the convergence of individual frequency harmonics of the non linear response though the procedure is applicable to general polynomial form non linearity it is illustrated for a duffing oscillator subjected to harmonic excitation a general and structured series expression is obtained for amplitudes of all the response harmonics and convergence is investigated in terms of a non dimensional non linear parameter critical values of this parameter representing the upper limit of excitation level for the convergence are defined for a wide range of excitation frequencies zones of convergence and divergence of the response series are presented graphically for a range of the non dimensional non linear parameter and the number of terms included in the approximation of a response harmonic an algorithm based on ratio test is presented to compute the critical value of the non dimensional non linear parameter results obtained from the suggested algorithm are found to be in close agreement with the exact values the method gives better results compared to previous methods and has wider application in terms of excitation frequency the procedure is also investigated for a two degree of freedom system

is every matrix similar to a polynomial in a companion matrix abstract given a field f an integer n 1 and a matrix a m n f are there polynomials f g f x with f monic of degree n such that a is similar to g c f where c f is the companion matrix of f for infinite fields the answer is easily seen to positive so we concentrate on finite fields in this case we give an affirmative answer provided f n 2 moreover for any finite field f with f m we construct a matrix a m m 3 f that is not similar to any matrix of the form g c f of use above but also of independent interest is a constructive procedure to determine the similarity type of any given matrix g c f purely in terms of f and g without resorting to polynomial roots in f or in any extension thereof this in turn yields an algorithm that given g and the invariant factors of any a returns the elementary divisors of g a it is a rational procedure as opposed to the classical method that uses the jordan decomposition of a to find that of g a finally extending prior results by the authors we show that for an integrally closed ring r with field of fractions f and companion matrices c d the subalgebra r c d of m n r is a free r module of rank n n m n 1 where m is the degree of gcd f g f x and a presentation for r c d is given in terms of c and d a counterexample is furnished to show that r c d need not be a free r module if r is not integrally closed the preceding information is used to study m n r and others as r x modules

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

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