minimal strongly eventually positive realization for a class of externally positive systems the minimal positive realization of externally positive systems is still an open problem due to various restrictions caused by the nonnegativity constraint on the state space matrices in this paper the minimal strongly eventually positive realization which relaxes the nonnegativity constraint on the state space model and only requires that the state trajectory is nonnegative after a certain number of steps is proposed for an externally positive system it is shown that a discrete time transfer function with a simple strictly dominant root is externally positive if and only if it has a minimal strongly eventually positive realization a constructive proof is provided the continuous time counterpart is also addressed by transforming it into a corresponding discrete time case

on discrete time linear systems over cones abstract the subject of this paper is the theory of linear discrete time systems with input constrained to belong to a closed convex cone after discussing the generality of this class of systems a number of invariant cones of states are introduced in the last section these latter are shown to allow the extension of some classical results of the theory of linear unconstrained systems

on the reachability in any fixed time for positive continuous time linear systems this paper deals with the reachability of continuous time linear positive systems the reachability of such systems which we will call here the strong reachability amounts to the possibility of steering the state in any fixed time to any point of the positive orthant by using nonnegative control functions the main result of this paper essentially says that the only strongly reachable positive systems are those made of decoupled scalar subsystems moreover the strongly reachable set is also characterized

minimal positive realizations of transfer functions with nonnegative multiple poles this note concerns a particular case of the minimality problem in positive system theory a standard result in linear system theory states that any nth order rational transfer function of a discrete time invariant linear single input single output siso system admits a realization of order n in some applications however one is restricted to realizations with nonnegative entries i e a positive system and it is known that this restriction may force the order n of realizations to be strictly larger than n a general solution to the minimality problem i e determining the smallest possible value of n is not known in this note we consider the case of transfer functions with nonnegative multiple poles and give sufficient conditions for the existence of positive realizations of order n n with the help of our results we also give an improvement of an existing result in positive system theory

reachability observability and realizability of continuous time positive systems this paper discusses reachability observability and realizability of single input single output linear time invariant systems in which state variables and or input output functions are restricted to be nonnegative to reflect physical constraints frequently encountered in real systems we define a set reachable from the origin with nonnegative inputs and also a set observable with nonnegative outputs we investigate geometrical structures of the sets through convex analysis and a duality relation between them is established next we consider positive realization of a given transfer function using the reachable set and the observable set we give a necessary and sufficient condition for positive realizability an example is given to demonstrate that a positive realizable transfer function does not in general have a jointly controllable and observable positive realization

an example of how positivity may force realizations of large dimension abstract a standard result of linear system theory states that an siso proper rational transfer function of degree n always has a realization of dimension n in some applications one is interested in having a realization with nonnegative entries and it is known that when the dominant poles display a specific pattern forcing nonnegativity leads to a system which is not jointly reachable and observable in this paper we show that the minimal dimension of a positive realization may be large even in the case of a single dominant pole more precisely we provide a family of transfer functions each of which is of degree n 3 such that for any integer n 3 the corresponding member of the family admits a minimal positive realization of state space dimension not smaller than n

advances on the reachability index of positive 2 d systems this work was supported by dgi grant mtm2010 18228 recommended by associate editor z wang

parameter estimation of continuous time bilinear systems based on numerical integration and separable non linear least squares this paper considers the problem of estimating the parameters in continuous time bilinear systems the system identification approach is based on numerical integration and separable non linear least squares the situation where the output signal is contaminated with noise is also discussed the suggested estimation method uses a bias compensating approach and the model parameters together with parameters associated with the noise are estimated by solving an overdetermined system of equation in a least squares sense

realization of continuous time positive linear systems positive linear systems are used in biomathematics economics and other research areas for discrete time positive linear systems part of the realization problem has been solved in this paper the solution of the corresponding problem for continuous time positive linear systems will be presented which can be deduced from that of the discrete time case by a transformation sufficient and necessary conditions for the existence of a positive realization are presented to solve the problem of minimality the solution of the factorization of positive matrices is needed

on the nonergodic convergence rate of an inexact augmented lagrangian framework for composite convex programming in this paper we consider the linearly constrained composite convex optimization problem whose objective is a sum of a smooth function and a possibly nonsmooth function we propose an inexact aug

the multivariable servomechanism problem for positive lti systems in this technical note we study the servomechanism problem for positive open loop stable lti systems under constant tracking signals and constant disturbances in particular we show that in general the mimo servomechanism problem for positive systems is not solvable but is solvable for various classes of mimo systems and for the class of siso systems in which the disturbances are small compared to the tracking signal in particular we give existence results as to what subclass of reference and disturbance signals can be considered for the case when either the mathematical model of the plant is known or unknown and we provide the controller structure that solves the problem for the case of measurable or unmeasurable disturbance signals

computational complexity certification for dual gradient method application to embedded mpc abstract in this paper we analyze the computational complexity of the dual gradient method for solving linearly constrained convex problems when it is difficult to project on the primal feasible set described by linear constraints we use the lagrangian relaxation to handle the complicated constraints and then we apply the dual gradient algorithm for solving the corresponding dual we give a unified convergence rate analysis for the dual gradient algorithm we provide sublinear or linear estimates on the primal suboptimality and feasibility violation of the generated approximate primal solutions our analysis relies on the lipschitz property of the dual function or an error bound property furthermore the iteration complexity analysis is based on two types of approximate primal solutions an average primal sequence or the last primal iterate sequence we also discuss complexity certifications and implementation aspects of the dual gradient algorithm on constrained mpc problems for embedded linear systems

positive nonlinear realizations of response maps abstract the problem of realization of a positive response map as a positive initialized system on a finite dimensional euclidean state space is investigated the dynamical part of the system is described by a delta differential equation on an arbitrary time scale this incorporates continuous and discrete time systems the main result states necessary and sufficient conditions for existence of positive realization of a particular class the criterion is expressed in the language of skew differential global universes

canonical forms for positive discrete time linear control systems abstract in this paper the properties of reachability controllability and essential reachability of positive discrete time linear control systems are studied these properties are characterized in terms of the directed graph of the state matrix from these characterizations canonical forms of those properties are deduced

stability analysis of positive descriptor systems abstract in this paper we discuss stability properties of positive descriptor systems in the continuous time as well as in the discrete time case we present characterisations of positivity and establish generalised stability criteria for the case of positive descriptor systems we show that if the spectral projector onto the finite deflating subspace of the matrix pair e a is nonnegative then all stability criteria for standard positive systems take a comparably simple form in the positive descriptor case furthermore we provide sufficient conditions that guarantee entry wise nonnegativity along with positive semidefiniteness of solutions of generalised projected lyapunov equations as an application of the framework established throughout this paper we exemplarily generalise two criteria for the stability of two switched standard positive systems under arbitrary switching to the descriptor case

brief identification of positive linear systems with poisson output transformation positive systems are systems in which the input state output variables are always positive since they represent quantities we propose an identification procedure for a class of positive linear systems

positive realizations of nonlinear input output discrete time systems abstract the problem of realization of a positive nonlinear input output discrete time system as a positive state space system is investigated the input output system is given by a difference equation of a higher order a necessary and sufficient condition for existence of positive and regularly positively observable realization is provided the state space variables of the realization are found as independent generators of some positive universe associated to the input output system only the single input single output case is studied and all functions describing the systems and their properties are assumed to be continuous

a tutorial on the positive realization problem this paper is a tutorial on the positive realization problem that is the problem of finding a positive state space representation of a given transfer function and characterizing existence and minimality of such representation this problem goes back to the 1950s and was first related to the identifiability problem for hidden markov models then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory within this framework developing some ideas sprang in the 1960s during the 1980s the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem in this paper the reader is carried through the key ideas which have proved to be useful in order to tackle this problem in order to illustrate the main results contributions and open problems several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided

positive realizations of linear systems abstract this paper deals with the realization problem of linear systems in which a sign pattern is imposed on the realization a b c t this problem arises in the context of positive linear systems i e systems in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative using a unitary framework for continuous time and discrete time systems we derive necessary and sufficient conditions for positive realizability by means of convex analysis moreover the complete solution in the case of a transfer function of mcmillan degree n 3 is obtained

positive and compartmental systems when dealing with compartmental systems an important question is given an experiment i e an input output sequence and supposing there is no error in the data is the sequence compatible with the compartmental assumption if the process under analysis is linear then the previous question is obviously equivalent to asking whether a given transfer function is that of a compartmental system in this note we provide an answer to the latter question giving necessary and sufficient conditions for a transfer function to be that of a compartmental system of some finite order i e number of compartments another problem tackled in this note originates from the observation that in many cases one wants to determine the number of compartments involved in the process in this note we report a step toward the solution of this fundamental problem by proving necessary and sufficient conditions for a given third order transfer function with real poles to be that of a compartmental system with three compartments

a smooth primal dual optimization framework for nonsmooth composite convex minimization we propose a new and low per iteration complexity first order primal dual optimization framework for a convex optimization template with broad applications our analysis relies on a novel combination of three classic ideas applied to the primal dual gap function smoothing acceleration and homotopy the algorithms due to the new approach achieve the best known convergence rate results in particular when the template consists of only nonsmooth functions we also outline a restart strategy for the acceleration to significantly enhance the practical performance we demonstrate relations with the augmented lagrangian method and show how to exploit the strongly convex objectives with rigorous convergence rate guarantees we provide representative examples to illustrate that the new methods can outperform the state of the art including chambolle pock and the alternating direction method of multipliers algorithms we also compare our algorithms with the well known nesterov smoothing method

on the existence of a positive realization the positive realization problem for linear systems is to find conditions for a given transfer function with nonnegative impulse response to have a realization such that the resulting system is a positive system recently it has been shown that under a mild assumption on the long term behaviour of the impulse response this problem is related to the maximum modulus poles only in this paper necessary and sufficient conditions for positive realizability of discrete time systems are given they show that also nondominant poles play a role in the most general case positive realizability conditions for the continuous time case are also given

convergence rate analysis for the alternating direction method of multipliers with a substitution procedure for separable convex programming recently in he et al he bs tao m yuan xm 2012 alternating direction method with gaussian back substitution for separable convex programming siam j optim 22 2 313 340 we have showed the first possibility of combining the douglas rachford alternating direction method of multipliers admm with a gaussian back substitution procedure for solving a convex minimization model with a general separable structure this paper is a further study on this theme we first derive a general algorithmic framework to combine admm with either a forward or backward substitution procedure then we show that convergence of this framework can be easily proved from the contraction perspective and its local linear convergence rate is provable if certain error bound condition is assumed without such an error bound assumption we can estimate its worst case convergence rate measured by the iteration complexity

periodic signal analysis by maximum likelihood modeling of orbits of nonlinear odes this paper treats a new approach to the problem of periodic signal estimation the idea is to model the periodic signal as a function of the state of a second order nonlinear ordinary differential equation ode this is motivated by poincare theory which is useful for proving the existence of periodic orbits for second order odes the functions of the right hand side of the nonlinear ode are then parameterized by a multivariate polynomial in the states where each term is multiplied by an unknown parameter a maximum likelihood algorithm is developed for estimation of the unknown parameters from the measured periodic signal the approach is analyzed by derivation and solution of a system of odes that describes the evolution of the cramer rao bound over time this allows the theoretically achievable accuracy of the proposed method to be assessed in the ideal case where the signals can be exactly described by the imposed model the proposed methodology reduces the number of estimated unknowns at least in cases where the actual signal generation resembles that of the imposed model this in turn is expected to result in an improved accuracy of the estimated parameters

metric selection in fast dual forward backward splitting the performance of fast forward backward splitting or equivalently fast proximal gradient methods depends on the conditioning of the optimization problem data this conditioning is related to a metric that is defined by the space on which the optimization problem is stated selecting a space on which the optimization data is better conditioned improves the performance of the algorithm in this paper we propose several methods with different computational complexity to find a space on which the algorithm performs well we evaluate the proposed metric selection procedures by comparing the performance to the case when the euclidean space is used for the most ill conditioned problem we consider the computational complexity is improved by two to three orders of magnitude we also report comparable to superior performance compared to state of the art optimization software

realization of positive linear systems abstract positive linear systems are frequently used as mathematical models in research areas like biology and economics the problem of classifying all minimal realizations of these systems is treated in this paper extensive use is made of the theory of polyhedral cones sufficient and necessary conditions for the existence of a positive realization are given but the problem of minimality leads to the factorization problem for positive matrices ideas and results are given to come towards a solution of this factorization problem

minimal eventually positive realizations of externally positive systems it is a well known fact that externally positive linear systems may fail to have a minimal positive realization in order to investigate these cases we introduce the notion of minimal eventually positive realization for which the state update matrix becomes positive after a certain power eventually positive realizations capture the idea that in the impulse response of an externally positive system the state of a minimal realization may fail to be positive but only transiently as a consequence we show that in discrete time it is possible to use downsampling to obtain minimal positive realizations matching decimated sequences of markov coefficients of the impulse response in continuous time instead if the sampling time is chosen sufficiently long a minimal eventually positive realization leads always to a sampled realization which is minimal and positive

bilinear continuous time systems identification via hartley based modulating functions abstract this paper highlights the relevance and merits of the hartley modulating functions hmf method for the identification of bilinear continuous time bct systems from recorded input and noise contaminated output data and it provides an insight into parameter estimation of a wider range of nonlinear systems in practice the methodology replaces the i o differential equation representing the dynamic system behavior by the hartley spectrum equation as a result it involves the known derivatives of the modulating function instead of the derivatives of the input and noisy output data by applying integral transformation to signals a frequency weighted least squares algorithm is also applied in the identification and a normalized root mean square criterion is used to investigate some computational considerations and the bias of the estimates results of the simulation studies demonstrate the appropriateness of the approach and its efficiency

lyapunov functions and duality for convex processes the paper studies convex lyapunov functions for differential and difference inclusions with right hand sides given by convex processes that is by set valued mappings the graphs of which are convex cones convex conjugacy between weak lyapunov functions for such inclusions and lyapunov functions for adjoint inclusions is established asymptotic stability concepts are compared and the existence of convex lyapunov functions for classes of convex processes is shown the relevance of the results for the study of asymptotic controllability or stabilizability and of detectability for linear control systems with conical control and state constraints is underlined

bounds on the size of minimal nonnegative realizations for discrete time lti systems nonnegative linear time invariant lti state space models are characterized by the property that their state variables and outputs remain nonnegative whenever the initial state and the inputs are nonnegative necessary and sufficient conditions for the existence of a nonnegative state space realization of a given discrete time dt impulse response were obtained recently in this letter we look at minimal nonnegative realizations of dt impulse responses and develop bounds on the number of state variables that are necessary

linear convergence rate of a class of distributed augmented lagrangian algorithms we study distributed optimization where nodes cooperatively minimize the sum of their individual locally known convex costs f i x s x in bbr d is global distributed augmented lagrangian al methods have good empirical performance on several signal processing and learning applications but there is limited understanding of their convergence rates and how it depends on the underlying network this paper establishes globally linear geometric convergence rates of a class of deterministic and randomized distributed al methods when the f i s are twice continuously differentiable and have a bounded hessian we give explicit dependence of the convergence rates on the underlying network parameters simulations illustrate our analytical findings

on the reachable set for third order linear discrete time systems with positive control the case of complex eigenvalues abstract in this paper we study the geometrical properties of the set of reachable states of a single input third order discrete time linear system with positive controls this set is a cone and we give a complete geometrical characterization of this set when the system has a complex conjugate pair of eigenvalues more in detail we give necessary and sufficient conditions for properness and polyhedrality of the cone and provide the number of its edges in terms of eigenvalue locations

analysis of systems with slope restricted nonlinearities using externally positive zames falb multipliers this paper proposes an approach for assessing the stability of feedback interconnections where one element is a static slope restricted nonlinearity and the other element is a linear system the approach is based on the use of zames falb multipliers where the dynamic portion of the multiplier is chosen as an externally positive noncausal transfer function by restricting attention to a subset of these multipliers a set of pure linear matrix inequality conditions is obtained which requires no initial parameterization by the user a useful byproduct of using externally positive systems is that the results are applicable to nonodd slope restricted nonlinearities which is not the case for all classes of zames falb multipliers

structural properties of discrete time linear positive periodic systems abstract we study reachability and controllability properties for discrete time positive periodic systems the equivalence between n periodic systems with n states and m inputs and invariant systems with n states and nm inputs preserves the positivity of such systems and in addition closely relates their reachability cones this equivalence permits us to show that the complete contrallability reachability of a positive n periodic system is equivalent to the complete contrallability reachability of n associated positive invariant systems

minimal positive realizations of transfer functions with real poles a standard result of linear system theory states that a single input single output siso rational n th order transfer function always has a state space realization of the same order in some applications one is interested in having a realization with nonnegative entries i e a positive system and it is known that such constraints may lead to a minimal order positive realization of order much greater than the transfer function order n in this technical note necessary and sufficient conditions for a third order transfer function with distinct real poles to have a third order positive realization are given these conditions are expressed in terms of lower bounds for the first three samples of the impulse response and therefore are very easy to check this result is an extension of a previous result for transfer functions with distinct real positive poles

on the reachable set for third order linear discrete time systems with positive control abstract in this paper we study the geometrical properties of the set of reachable states of a single input third order discrete time linear system with positive controls this set is a cone and we give a complete geometrical characterization of this set when the system has all real eigenvalues more in detail we give necessary and sufficient conditions for properness and polyhedrality of the cone and provide the number of its edges in terms of eigenvalue locations moreover we provide necessary and sufficient conditions for finite time reachability of every reachable state and characterize the minimum number of steps needed to reach every state in terms of eigenvalue locations

positivity preserving balanced truncation for descriptor systems we propose a model reduction method for positive systems that ensures the positivity of the reduced order model in the standard as well as in the descriptor case for continuous time and discrete time systems our approach is based on constructing diagonal solutions of lyapunov inequalities these are linear matrix inequalities lmis which are shown to be feasible positivity and stability are preserved and an error bound in the mathcal h infty norm is provided

computational complexity of inexact gradient augmented lagrangian methods application to constrained mpc we study the computational complexity certification of inexact gradient augmented lagrangian methods for solving convex optimization problems with complicated constraints we solve the augmented lagrangian dual problem that arises from the relaxation of complicating constraints with gradient and fast gradient methods based on inexact first order information moreover since the exact solution of the augmented lagrangian primal problem is hard to compute in practice we solve this problem up to some given inner accuracy we derive relations between the inner and the outer accuracy of the primal and dual problems and we give a full convergence rate analysis for both gradient and fast gradient algorithms we provide estimates on the primal and dual suboptimality and on primal feasibility violation of the generated approximate primal and dual solutions our analysis relies on the lipschitz property of the dual function and on inexact dual gradients we also discuss implementation aspects of the proposed algor

the niep for four dimensional leslie and doubly stochastic matrices with zero trace from the coefficients of the characteristic polynomial abstract the constraints on the coefficients of the characteristic polynomials of four dimensional doubly stochastic matrices with zero trace and leslie stochastic matrices with zero trace are determined then on the basis of these constraints the regions of the complex plane consisting of those points which can serve as characteristic roots of four dimensional doubly stochastic matrices with zero trace and leslie stochastic matrices with zero trace are characterized

qualitative theory of compartmental systems dynamic models of many processes in the biological and physical sciences which depend on local mass balance conditions give rise to systems of ordinary differential equations many nonlinear that are called compartmental systems in this paper the authors define compartmental systems specify their relations to other nonnegative systems and discuss examples of applications the authors review the qualitative results on linear and nonlinear compartmental systems including their relation to cooperative systems they review the results for linear compartmental systems and then integrate and expand the results on nonlinear compartmental systems providing a framework for unifying them under a few general theorems in the course of that they complete the solution of a problem posed by bellman and show that closed nonlinear autonomous n compartment systems can show the full gamut of possible behaviors of systems of odes finally to provide additional structure to this study the authors show how to partiti

optimal move blocking strategies for model predictive control this paper presents a systematic methodology for designing move blocking strategies to reduce the complexity of a model predictive controller for linear systems with explicit optimisation of the blocking structure using mixed integer programming given a move blocked predictive controller with a terminal invariant set constraint for stability combined with an input parameterisation to preserve recursive feasibility two different optimisation problems are formulated for blocking structure selection the first problem calculates the maximum achievable reduction in the number of input decision variables and prediction horizon length subject to the controller s region of attraction containing a specified subset of the state space then for a given fixed horizon length and block count determined by hardware capabilities the second problem seeks to maximise the volume of an inner approximation to the region of attraction numerical examples show that the resulting blocking structures are able to optimally reduce controller complexity and improve region of attraction volume

controllability and reachability criteria for discrete time positive systems discrete time positive systems have been the object of widespread interest in the literature in the last decade particular attention has been devoted to the analysis and characterization of the different notions of reachability and controllability which in this context make their appearance in this paper the properties of ordinary and essential controllability and reachability are analysed based on a graph theoretic approach we provide practical criteria to verify whether a given positive system f g is endowed with these properties together with canonical forms for describing reachable controllable pairs