discretization of continuous linear dynamic systems analysis of the methods abstract the problem of the discretization of continuous linear systems is considered a particular case is the problem of digital filter design for a given analog prototype common and distinctive features of discrete systems which are given by a number of different methods of discretization for one and the same continuous system are analysed the sampling period is assumed to be sufficiently small

stabilization of positive switched linear systems and its application in consensus of multiagent systems in this note we study the stabilization problem of positive switched linear system psls with disturbance by introducing a class of weak common linear copositive lyapunov functions we design time dependent switching rules under which the system is asymptotically stable both continuous time and discrete time psls are taken into consideration and the obtained results are also extended to nonpositive switched systems finally the theoretical results are applied to consensus problem of multiagent systems with switching topologies and disturbance

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

improved results on stability of continuous time switched positive linear systems in this paper the problems of stability for switched positive linear systems splss under arbitrary switching are investigated in a continuous time context the so called copositive polynomial lyapunov function cplf giving a generalization of copositive types of lyapunov function is first proposed which is formulated in a higher order form of the positive states of the underlying systems it is illustrated in this paper that some classical types of lyapunov functions can be seen as special cases of the proposed cplf then new stability conditions are developed by the new lyapunov function approach it is also proved that the conservativeness of the obtained criteria can be further reduced as the degree of the lyapunov function increases a numerical example is given to demonstrate the effectiveness and less conservativeness of the developed techniques

on exponential stability of switched homogeneous positive systems of degree one in this paper we investigate the stability problem of switched homogeneous positive systems of degree one we first show that the positiveness of the considered system is guaranteed when each switched subsystem is cooperative then an integral excitation condition is formulated to ensure the exponential stability of the switched homogeneous positive system of degree one in particular the proposed condition allows that some or even all switched subsystems are only lyapunov stable rather than asymptotically stable in the second part of this paper we further consider the cases of sub homogeneous systems time varying delay systems and discrete time systems in the sub homogeneous case we first show that the previous integral excitation condition is also sufficient such that the exponential stability is achieved we also derive an exponential stability result by relaxing the cooperative and integral excitation conditions in the time varying delay case a delay independent exponential stability result is shown under the extended integral excitation condition finally in the discrete time case based on the fact that each switched subsystem is order preserving instead of cooperative the exponential stability of the considered system is shown under a weak excitation condition

asymptotic stability and stabilizability of special classes of discrete time positive switched systems abstract in this paper we consider discrete time positive switched systems switching among autonomous subsystems characterized either by monomial matrices or by circulant matrices necessary and sufficient conditions are provided guaranteeing either global uniform asymptotic stability or stabilizability i e the possibility of driving to zero the state trajectory corresponding to any initial state by resorting to some switching sequence such conditions lead to simple algorithms that allow to easily detect under suitable conditions whether a given positive switched system is not stabilizable

approximation of nonnegative systems by finite impulse response convolutions we pose the deterministic nonparametric approximation problem for scalar nonnegative input output systems via finite impulse response convolutions based on repeated observations of input output signal pairs the problem is converted into a nonnegative matrix factorization with special structure for which we use csiszar s i divergence as the criterion of optimality conditions are given on the input output data that guarantee the existence and uniqueness of the minimum we propose an algorithm of the alternating minimization type for i divergence minimization and study its asymptotic behavior for the case of noisy observations we give the large sample properties of the statistical version of the minimization problem numerical experiments confirm the asymptotic results and exhibit the fast convergence of the proposed algorithm

exponential stabilization of switched time varying systems with delays and disturbances abstract this paper deals with exponential stabilization for a class of switched time varying systems by taking time varying delays and nonlinear disturbances into consideration time dependent switching signals have been characterized in terms of metzler matrices such that the resulting system is globally exponentially stable compared with preceding works we introduce a model transformation and an approach without involving the lyapunov krasovskii functional to derive new exponential stability criteria for switched time varying systems under the average dwell time switching numerical examples show that the obtained theoretical results can be applied to some cases not covered by some existing results

temperley lieb algebra and a new integrable electronic model a new correlated electron model is presented which is derived using the quantum inverse scattering method and is thus integrable in one dimension on a periodic lattice the r matrix is constructed from a representation of the temperley lieb algebra the associated hamiltonian describes correlated hopping electron pair hopping and a generalized spin interaction

stabilization of discrete time switched positive linear systems via weak switched linear copositive lyapunov function abstract by introducing a class of weak switched linear copositive lyapunov functions this paper designs new time dependent switching signals guaranteeing asymptotic stability of discrete time switched positive linear systems whose subsystems are only marginally stable our results have advantages over some existing ones in the literature moreover the main results have been extended to the general switched linear time varying system with disturbance numerical examples well demonstrate the superiority of the obtained results

h_2 analysis of lti systems via conversion to externally positive systems motivated by recent advances in the study of linear time invariant lti positive systems we explore analysis techniques of general not necessarily positive lti systems using positive system theory even though a positive system is characterized by its peculiar property that its impulse response is nonnegative we often deal with nonnegative impulse responses even in general lti system analysis a typical example is the computation of the h 2 norm where we focus on squared impulse responses to deal with such products of impulse responses in a systematic fashion in this paper we first establish a construction technique of an lti system whose impulse response is given by the product of impulse responses of two different lti systems then as the main result we reduce the h 2 norm computation problem of a general lti system into the l infty induced norm computation problem or l 1 problem in short of a positive system by which we can derive various formulas for the h 2 norm computation

minimum time control of boolean networks boolean networks bns are discrete time dynamical systems with boolean state variables bns have recently been attracting considerable interest as computational models for biological systems and in particular gene regulating networks boolean control networks bcns are boolean networks with boolean inputs we consider the problem of steering a bcn from a given state to a desired state in minimal time using the algebraic state space representation assr of bcns we derive several necessary conditions stated in the form of maximum principles mps for a control to be time optimal in the assr every state and input vector is a canonical vector using this special structure yields an explicit state feedback formula for all time optimal controls to demonstrate the theoretical results we develop a bcn model for the genetic switch controlling the lambda phage development upon infection of a bacteria our results suggest that this biological switch is designed in a way that guarantees minimal time respon

stability analysis and delay control for switched positive linear systems in this paper the problems of stability analysis and delay control for switched positive linear systems splss are studied by using average dwell time adt switching by developing a new class of multiple piecewise continuous linear copositive lyapunov function mplclf approach an improved stability condition for splss is obtained which can obtain a tighter bound on the adt guaranteeing the stability the obtained stability condition covers the existing results on stability analysis for splss since the multiple linear copositive lyapunov function used in the literature can be viewed as a special case of our proposed mplclf and our result can obtain a tighter bound of the adt then a more efficient controller design method over the existing ones is also established for splss with switching delay to guarantee the underlying system to be exponentially stable to illustrate the advantages of our proposed results two numerical examples are finally given

stability of positive linear switched systems on ordered banach spaces abstract we provide sufficient criteria for the stability of positive linear switched systems on ordered banach spaces the switched systems can be generated by finitely many bounded operators in infinite dimensional spaces with a general class of order inducing cones in the discrete time case we assume an appropriate interior point of the cone whereas in the continuous time case an appropriate interior point of the dual cone is sufficient for stability this is an extension of the concept of linear lyapunov functions for positive systems to the setting of infinite dimensional partially ordered spaces we illustrate our results with examples

on easily verifiable conditions for the existence of common linear copositive lyapunov functions this technical note addresses the existence of a common linear copositive lyapunov function clclf for continuous time switched positive linear systems two necessary and sufficient conditions for the existence of a clclf are provided one condition shows that the existence of a clclf for a family of n n dimensional metzler matrices can only be determined by n 1 lower dimensional hurwitz matrices the other condition further shows that the existence of a clclf is also determined by n 1 characteristic constants which gives an easily verifiable computation algorithm without checking a large amount of metzler hurwitz matrices

a lagrange duality characterisation for stability under arbitrary switching in switched positive linear systems abstractthe present communication is concerned with uniform exponential stability under arbitrary switching in discrete time switched positive linear systems lagrange duality is used in order to obtain a new characterisation for uniform exponential stability which is in terms of sets of inequalities involving each of the matrices that represent the modes of the system these sets of inequalities are shown to generalise the classical linear lyapunov inequality that characterises in positive matrices the property of being schur each solution to these sets of inequalities is shown to provide a representation in terms of a number of linear functionals for a common lyapunov function for the switched positive linear system a result is further presented which conveys to a conservative upper bound on the minimum required number of linear functionals in the above mentioned representation and also to a method for computing them our proof for the aforementioned characterisation is based on another equ

state unilateral tracking control of positive switched systems via designing a switching law abstractthis paper focuses on the model reference state unilateral tracking control problem of positive switched systems the state unilateral tracking means that the state always keeps in the same direction of the reference objective either above or below this feature makes unilateral tracking play a role of avoiding the overshoot via the dual design of both the controller and the switching signal the state unilateral tracking control problem with l1 gain performance for positive switched systems is solved through adopting multiple linear co positive lyapunov functions technique the design method is applicable to the case that the unilateral tracking control problem is unsolvable for any subsystem of a positive switched system moreover two techniques are presented in order to deal with the state unilateral tracking problem finally a practical example demonstrates the validity of the presented result

sign properties of metzler matrices with applications abstract several results about sign properties of metzler matrices are obtained it is first established that checking the sign stability of a metzler sign matrix can be either characterized in terms of the hurwitz stability of the unit sign matrix in the corresponding qualitative class or in terms the negativity of the diagonal elements of the metzler sign matrix and the acyclicity of the associated directed graph similar results are obtained for the case of metzler block matrices and metzler mixed matrices the latter being a class of metzler matrices containing both sign and real type entries the problem of assessing the sign stability of the convex hull of a finite and summable family of metzler matrices is also solved and a necessary and sufficient condition for the existence of common lyapunov functions for all the matrices in the convex hull is obtained the concept of sign stability is then generalized to the concept of ker b sign stability a problem that arises in the analysis of certain jump markov processes a sufficient condition for the ker b sign stability of metzler sign matrices is obtained and formulated using inverses of sign matrices and the concept of l matrices several applications of the results are discussed in the last section

brief paper probabilistic sorting and stabilization of switched systems in this paper we consider lyapunov stability of switched linear systems whose switching signal is constrained to a subset of indices we propose a switching rule that chooses the most stable subsystem among those belonging to the subset this rule is based on an ordering of the subsystems using a common lyapunov function we develop randomized algorithms for finding the ordering as well as for finding a subset of systems for which a common lyapunov function exists it is shown that the class of randomized algorithms known as the las vegas type is useful in the design procedure a third order example illustrating the efficacy of the approach is presented

stability of discrete time positive switched linear systems with stable and marginally stable subsystems abstract in this paper we investigate the stability of discrete time positive switched linear systems the highlight of this work is that each subsystem is only assumed to be stable or marginally stable some sufficient conditions imposed on the subsystem matrices and switching signals are established to guarantee exponential stability of the considered positive switched linear systems an extension to the case of time varying and unbounded delay is also given compared with the continuous time counterparts in the literature our discrete time case is much more complicated involving the transitive connectivity between the subsystems specified by the switching signals

on common quadratic lyapunov functions for pairs of stable lti systems whose system matrices are in companion form in this note the problem of determining necessary and sufficient conditions for the existence of a common quadratic lyapunov function for a pair of stable linear time invariant systems whose system matrices a sub 1 and a sub 2 are in companion form is considered it is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product a sub 1 a sub 2 does not have an eigenvalue that is real and negative examples are presented to illustrate the result

extremal norms for positive linear inclusions abstract for finite dimensional linear semigroups which leave a proper cone invariant it is shown that irreducibility with respect to the cone implies the existence of an extremal norm in case the cone is simplicial a similar statement applies to absolute norms the semigroups under consideration may be generated by discrete time systems continuous time systems or continuous time systems with jumps the existence of extremal norms is used to extend results on the lipschitz continuity of the joint spectral radius beyond the known case of semigroups that are irreducible in the representation theory interpretation of the word

nice reachability for planar bilinear control systems with applications to planar linear switched systems corrected we consider planar bilinear control systems with measurable controls we show that any point in the reachable set can be reached by a ldquonicerdquo control specifically a control that is a concatenation of a bang arc with either 1 a bang bang control that is periodic after the third switch or 2 a piecewise constant control with no more than two discontinuities under the additional assumption that the bilinear system is positive or invariant for any proper cone we show that the reachable set is spanned by a concatenation of a bang arc with either 1 a bang bang control with no more than two discontinuities or 2 a piecewise constant control with no more than two discontinuities in particular any point in the reachable set can be reached using a piecewise constant control with no more than three discontinuities several known results on the stability of planar linear switched systems under arbitrary switching follow as corollaries of our result we demonstrate this with an example

stability and stabilizability criteria for discrete time positive switched systems in this paper we consider the class of discrete time switched systems switching between p autonomous positive subsystems first sufficient conditions for testing stability based on the existence of special classes of common lyapunov functions are investigated and these conditions are mutually related thus proving that if a linear copositive common lyapunov function can be found then a quadratic positive definite common function can be found too and this latter in turn ensures the existence of a quadratic copositive common function secondly stabilizability is introduced and characterized it is shown that if these systems are stabilizable they can be stabilized by means of a periodic switching sequence which asymptotically drives to zero every positive initial state conditions for the existence of state dependent stabilizing switching laws based on the values of a copositive linear quadratic lyapunov function are investigated and mutually related too finally some properties of the patterns of the stabilizing switching sequences are investigated and the relationship between a sufficient condition for stabilizability the existence of a schur convex combination of the subsystem matrices and an equivalent condition for stabilizability the existence of a schur matrix product of the subsystem matrices is explored

conditions on the stability of a class of second order switched systems for a special class of systems it is shown that the existence of a common quadratic lyapunov function cqlf is necessary and sufficient for the stability of an associated switched system under arbitrary switching furthermore it is shown that the existence of a cqlf for n n 2 subsystems is equivalent to the existence of a cqlf for every pair of subsystems an algorithm is proposed to compute a cqlf for the subsystems when it exists using the left and right eigenvectors of a critical matrix obtained from a matrix pencil

stability properties of a class of positive switched systems with rank one difference abstract given a single input continuous time positive system described by a pair a b with a a diagonal matrix we investigate under what conditions there exists a state feedback law u t c x t that makes the resulting controlled system positive and asymptotically stable by this meaning that a b c is metzler and hurwitz in the second part of this note we assume that the state space model switches among different state feedback laws c i i 1 2 p each of them ensuring the positivity and show that the asymptotic stability of this type of switched system is equivalent to the asymptotic stability of all its subsystems while its stabilizability is equivalent to the existence of an asymptotically stable subsystem

on the dynamic instability of a class of switching system in this note we use floquet theory to establish that a sufficient condition for the existence of a periodic destabilizing switching sequence for the system x a t x a t a1 a2 am where the ai are hurwitz matrices is given by the existence of non negative real constants 1 2 m i 0 i 0 such that the matrix pencil has at least one eigenvalue with a positive real part we use this result to provide insights into the relationship between the non existence of a common quadratic lyapunov function and the existence of a destabilizing switching sequence for low order systems as well as the robustness of a class of switching system that is known to be exponentially stable

copositive lyapunov functions for switched systems over cones abstract we answer two open questions on copositive lyapunov functions which were recently posed by m k camlibel and j m schumacher in the book unsolved problems in mathematical systems and control theory edited by v d blondel and a megretski m k camlibel j m schumacher copositive lyapunov functions in v d blondel a megretski eds unsolved problems in mathematical systems and control theory princeton university press 2004 pp 189 193 available online at http press princeton edu math blondel these questions are what are necessary and sufficient conditions for the existence of a lyapunov function for a linear system which is defined over a cone how can this be extended to switched linear systems where the system matrix varies over time we present conditions answering these questions our conditions amount to checking feasibility or infeasibility of a system of linear inequalities

limiting zeros of decouplable mimo systems for sufficiently rapid zero order hold zoh sampling it is known that the zeros of single input single output discrete time systems expressed in the forward shift operator converge to values determined only by the degrees of the finite and infinite zeros of the underlying continuous time system in this paper it is shown how this result can be generalized to multi input multi output mimo systems decouplable by static state feedback equivalently having a diagonal interactor matrix in the fast sampling limit the authors show how invariant and infinite zero structure is mapped under zoh sampling for discrete time systems expressed in either the delta forward difference or forward shift operators

stabilisation of second order lti switched positive systems the stabilisation problem of second order switched positive systems consisting of two unstable subsystems is considered in this article by considering the vector fields and geometric characteristics a necessary and sufficient condition for the stabilisability of second order switched positive systems with two unstable subsystems is provided furthermore it is shown via this condition that neither second order switched positive systems consisting of two subsystems with unstable nodes nor second order switched positive systems consisting of one subsystem with unstable nodes and the other with a saddle point can be stabilised via any switching law

lmi approach to linear positive system analysis and synthesis abstract this paper is concerned with the analysis and synthesis of linear positive systems based on linear matrix inequalities lmis we first show that the celebrated perron frobenius theorem can be proved concisely by a duality based argument again by duality we next clarify a necessary and sufficient condition under which a hurwitz stable metzler matrix admits a diagonal lyapunov matrix with some identical diagonal entries as the solution of the lyapunov inequality this new result leads to an alternative proof of the recent result by tanaka and langbort on the existence of a diagonal lyapunov matrix for the lmi characterizing the h performance of continuous time positive systems in addition we further derive a new lmi for the h performance analysis where the variable corresponding to the lyapunov matrix is allowed to be non symmetric we readily extend these results to discrete time positive systems and derive new lmis for the h performance analysis and synthesis we finally illustrate their effectiveness by numerical examples on robust state feedback h controller synthesis for discrete time positive systems affected by parametric uncertainties

sampling zeros and the euler frobenius polynomials we show that the zeros of sampled data systems resulting from rapid sampling of continuous time systems preceded by a zero order hold zoh are the roots of the euler frobenius polynomials using known properties of these polynomials we prove two conjectures of hagiwara et al 1993 the first of which concerns the simplicity negative realness and interlacing properties of the sampling zeros of zoh and first order hold foh sampled systems to prove the second conjecture we show that in the fast sampling limit and as the continuous time relative degree increases the largest sampling zero for foh sampled systems approaches 1 e where e is the base of the natural logarithm

a converse lyapunov theorem for a class of dynamical systems which undergo switching the authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields they allow switching between members of the family of vector fields to be completely random the focus of this paper is to prove a converse lyapunov theorem for this class of systems

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

on the stability of switched positive linear systems it was recently conjectured that the hurwitz stability of the convex hull of a set of metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching in this note we show that 1 this conjecture is true for systems constructed from a pair of second order metzler matrices 2 the conjecture is true for systems constructed from an arbitrary finite number of second order metzler matrices and 3 the conjecture is in general false for higher order systems the implications of our results both for the design of switched positive linear systems and for research directions that arise as a result of our work are discussed toward the end of the note