max type copositive lyapunov functions for switching positive linear systems the paper aims to expand the stability analysis framework based on copositive lyapunov functions clfs for arbitrarily switching positive systems with continuous or discrete time dynamics the first part focuses on max type clfs it provides an algebraic characterization in terms of weak quasi linear inequalities as well as an existence criterion relying on a concrete construction procedure in terms of perron frobenius eigenstructure the second part explores the connections between max type and two other classes of clfs namely linear and quadratic diagonal new qualitative and quantitative features are revealed for the clfs belonging to the two mentioned classes

absolute exponential l1 gain analysis and synthesis of switched nonlinear positive systems with time varying delay this paper investigates absolute exponential stability and stabilization with l1 gain performance of switched nonlinear positive systems with time varying delay nonlinear functions considered in the paper are located in a sector field and the time varying delay is unknown but bounded the notion of absolute exponential l1 stability of switched nonlinear positive systems is first introduced a nonlinear lyapunov krasovskii functional is constructed for the underlying systems by using the lyapunov krasovskii functional a sufficient condition for absolute exponential l1 stability of the systems is established in terms of linear programming then absolute exponential l1 control synthesis of the systems is addressed a feedback control law containing nonlinear functions and a state feedback control law are designed respectively by comparing with existing results it is shown that the present design is less conservative finally an illustrative example is provided to illustrate the effectiveness of the results

hurwitz stability of metzler matrices in this note a simple method is presented that is both necessary and sufficient for determining whether a given metzler matrix a is hurwitz the method is based on the well known fact that a hurwitz metzler matrix is also diagonally stable by using this fact very simple conditions are derived for the hurwitz stability of a metzler matrix the conditions are stated in terms of the signs of the diagonal entries of a sequence of lower dimensional matrices the efficacy of the conditions is demonstrated by applying them to determine stability in several examples

a new general sufficient condition for almost sure stability of jump linear systems in this paper we study the almost sure stability of discrete time jump linear systems with a finite state markov form process a general condition for almost sure stability which is a necessary and sufficient condition for scalar one dimensional systems is derived many simpler testable sufficient conditions for almost sure stability are derived from this sufficient condition

extension of diagonal stability and stabilization for continuous time fractional positive linear systems abstract this paper considers an extension of diagonal stability for continuous fractional positive linear systems fpls with the fractional order 0 1 based on diagonal stability of metzler matrix an extension that involves the combination of a diagonal positive definite matrix and a skew symmetric anti diagonal matrix related a hurwitz and metzler matrix is established combining this extension and the linear matrix inequalities lmis criteria of stability for fractional order systems fos a necessary and sufficient condition for extension diagonal stability of fpls is presented a state feedback controller is given which ensures the stabilization and positivity of the closed loop systems numerical examples are provided to demonstrate the effectiveness and applicability of the proposed methods

on orbital variety closures in sln ii descendants of a richardson orbital variety abstract for a semisimple lie algebra g the orbit method attempts to assign representations of g to coadjoint orbits in g orbital varieties are particular lagrangian subvarieties of such orbits leading to highest weight representations of g in sl n orbital varieties are described by young tableaux in this paper we consider so called richardson orbital varieties in sl n a richardson orbital variety is an orbital variety whose closure is a standard nilradical we show that in sl n a richardson orbital variety closure is a union of orbital varieties we give a complete combinatorial description of such closures in terms of young tableaux this is the second paper in the series of three papers devoted to a combinatorial description of orbital variety closures in sl n in terms of young tableaux

diagonal lyapunov krasovskii functionals for discrete time positive systems with delay we consider the existence of diagonal lyapunov krasovskii l k functionals for positive discrete time systems subject to time delay in particular we show that the existence of a diagonal functional is necessary and sufficient for delay independent stability of a positive linear time delay system we extend this result and provide conditions for the existence of diagonal l k functionals for classes of nonlinear positive time delay systems which are not necessarily order preserving we also describe sufficient conditions for the existence of common diagonal l k functionals for switched positive systems subject to time delay

analytical and numerical solutions for the nonlinear burgers and advection diffusion equations by using a semi analytical iterative method abstract in this paper exact solutions have been obtained for 1d 2d and 3d nonlinear burgers equations and systems of equations by implementing an accurate semi analytical method this method originally proposed by temimi and ansari and herein named tam proved to be efficient and reliable for solving different types of linear and nonlinear problems this method is characterized by not requiring any restrictive assumptions for the nonlinear terms the convergence of the method is successfully presented and mathematically proved in addition the advection diffusion equation is also solved by using the tam to demonstrate the efficiency of this method several examples are solved either analytically or numerically where the accuracy of the numerical solution has been demonstrated by evaluating the absolute and relative errors to show the accuracy of the proposed method the software used in the current work is mathematica 10

the bounded real lemma for internally positive systems and h infinity structured static state feedback we consider the bounded real lemma for internally positive linear time invariant systems we show that the h norm of such systems can be evaluated by checking the existence of a certain diagonal quadratic storage function taking advantage of this fact the problem of designing a structured static state feedback controller achieving internal stability contractiveness and internal positivity in closed loop becomes convex and tractable

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

stability of switched systems a lie algebraic condition we present a sucient condition for asymptotic stability of a switched linear system in terms of the lie algebra generated by the individual matrices namely if this lie algebra is solvable then the switched system is exponentially stable for arbitrary switching in fact we show that any family of linear systems satisfying this condition possesses a quadratic common lyapunov function we also discuss the implications of this result for switched nonlinear systems c 1999 elsevier science b v all rights reserved

stochastic stability of positive markov jump linear systems abstract this paper investigates on the stability properties of positive markov jump linear systems pmjls s i e markov jump linear systems with nonnegative state variables specific features of these systems are highlighted in particular a new notion of stability exponential mean stability is introduced and is shown to be equivalent to the standard notion of 1 moment stability moreover various sufficient conditions for exponential almost sure stability are worked out with different levels of conservatism the implications among the different stability notions are discussed it is remarkable that thanks to the positivity assumption some conditions can be checked by solving linear programming feasibility problems

dynamic output feedback control for continuous time interval positive systems under l1 performance this paper is concerned with the design of l1 induced dynamic output feedback controller for continuous time positive systems with interval uncertainties a necessary and sufficient condition for stability and l1 induced performance of positive linear systems is proposed in terms of linear inequalities based on this conditions for the existence of robust dynamic output feedback controllers are established and an iterative convex optimization approach is developed to solve the conditions one illustrative example is provided to show the effectiveness and applicability of the theoretical results

an alternative proof of the barker berman plemmons bbp result on diagonal stability and extensions abstract we revisit the theorem of barker berman and plemmons on the existence of a diagonal quadratic lyapunov function for a stable linear time invariant lti dynamical system g p barker a berman r j plemmons positive diagonal solutions to the lyapunov equations linear and multilinear algebra 5 3 1978 249 256 we use recently derived results to provide an alternative proof of this result and to derive extensions

extended h 2 and h norm characterizations and controller parametrizations for discrete time systems this paper presents new synthesis procedures for discrete time linear systems it is based on a recently developed stability condition which contains as particular cases both the celebrated lyapunov theorem for precisely known systems and the quadratic stability condition for systems with uncertain parameters these new synthesis conditions have some nice properties a they can be expressed in terms of lmi linear matrix inequalities and b the optimization variables associated with the controller parameters are independent of the symmetric matrix that defines a quadratic lyapunov function used to test stability this second feature is important for several reasons first structural constraints as those appearing in the decentralized and static output feedback control design can be addressed less conservatively second parameter dependent lyapunov function can be considered with a very positive impact on the design of robust h 2 and h x control problems third the design of controller with mixed

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

reachability properties of single input continuous time positive switched systems in this paper two reachability properties for single input positive switched systems are introduced monomial reachability and reachability monomial reachability which represents a necessary but not sufficient condition for reachability is fully characterized necessary and sufficient conditions for reachability are provided for the class of n dimensional systems switching among n subsystems several necessary or sufficient conditions are also provided moreover the definition of k switching reachability set is given and an equivalent condition for the existence of an upper bound on the number of switchings required to reach any reachable vector is given finally it is shown that for reachable systems of low dimension n 2 or n 3 each vector of the positive orthant can be reached by resorting to a switching sequence which switches no more than n 1 times

on stabilizability of switched positive linear systems under state dependent switching this paper addresses the stabilization of switched positive linear systems by state dependent switching we show that if there is a hurwitz convex or linear combination of the coefficient matrices then the switched positive linear system can be exponentially stabilized by means of a single linear co positive lyapunov function if there is not a stable combination of system matrices it is shown that the exponential stabilizability is equivalent to a completeness condition on the coefficient matrices when the switched positive systems can not be stabilized by the single lyapunov function we provide a unified criterion for piecewise exponential stabilizability in terms of multiple linear co positive lyapunov functions

geometric programming for optimal positive linear systems this article studies the parameter tuning problem of positive linear systems for optimizing their stability properties we specifically show that under certain regularity assumptions on the parameterization the problem of finding the minimum cost parameters that achieve a given requirement on a system norm reduces to a geometric program which in turn can be exactly and efficiently solved by convex optimization the flexibility of geometric programming allows the state input and output matrices of the system to simultaneously depend on the parameters to be tuned the class of system norms under consideration includes the h 2 norm h infty norm hankel norm and schatten p norm also the parameter tuning problem for ensuring the robust stability of the system under structural uncertainties is shown to be solved by geometric programming the proposed optimization framework is further extended to delayed positive linear systems where it is shown that the parameter tuning problem jointly constrained by the exponential decay rate the mathcal l 1 gain and the mathcal l infty gain can be solved by convex optimization the assumption on the system parameterization is stated in terms of posynomial functions which form a broad class of functions and thus allow us to deal with various interesting positive linear systems arising from for example dynamical buffer networks and epidemic spreading processes we present numerical examples to illustrate the effectiveness of the proposed optimization framework

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

on orbital variety closures in sln i induced duflo order abstract for a semisimple lie algebra g the orbit method attempts to assign representations of g to coadjoint orbits in g orbital varieties are particular lagrangian subvarieties of such orbits leading to highest weight representations of g in sl n orbital varieties are described by young tableaux inclusion relation on orbital variety closures defines a partial order on young tableaux our aim is to describe this order the paper is devoted to the combinatorial description of induced duflo order on young tableaux the order generated by inclusion of generating subspaces of orbital varieties this is a very interesting and complex combinatorial question this is the first paper in the series in parts ii and iii we use repeatedly the results of this paper as a basis for further study of orbital variety closures

survey paper structured low rank approximation and its applications fitting data by a bounded complexity linear model is equivalent to low rank approximation of a matrix constructed from the data the data matrix being hankel structured is equivalent to the existence of a linear time invariant system that fits the data and the rank constraint is related to a bound on the model complexity in the special case of fitting by a static model the data matrix and its low rank approximation are unstructured we outline applications in system theory approximate realization model reduction output error and errors in variables identification signal processing harmonic retrieval sum of damped exponentials and finite impulse response modeling and computer algebra approximate common divisor algorithms based on heuristics and local optimization methods are presented generalizations of the low rank approximation problem result from different approximation criteria e g weighted norm and constraints on the data matrix e g nonnegativity related problems are rank minimization and structured pseudospectra

stabilization of continuous time jump linear systems we investigate almost sure and moment stabilization of continuous time jump linear systems with a finite state markov jump form process we first clarify the concepts of spl delta moment stabilizability exponential spl delta moment stabilizability and stochastic spl delta moment stabilizability we then present results on the relationships among these concepts coupled riccati equations that provide necessary and sufficient conditions for mean square stabilization are given in detail and an algorithm for solving the coupled riccati equations is proposed moreover we show that individual mode controllability implies almost sure stabilizability which is not true for other types of stabilizability finally we present some testable sufficient conditions for spl delta moment stabilizability and almost sure stabilizability

stability analysis and l1 gain characterization for switched positive systems under dwell time constraint abstract this paper deals with the problems of stability analysis and l 1 gain characterization for continuous time switched systems consisting of positive subsystems with the aid of a discretized copositive lyapunov function a sufficient condition ensuring the asymptotic stability of continuous time switched positive systems under dwell time constraint is obtained which can be checked via linear programming furthermore the conservatism of the proposed approach is studied and the result with least conservatism in the framework of discretized copositive lyapunov function is obtained the result is then extended to l 1 gain characterization for switched positive systems with a prescribed dwell time an unweighted l 1 gain can be computed via solving a linear programming problem a numerical example and a practical traffic example are given to illustrate our results

a semi analytical iterative technique for solving nonlinear problems we present a semi analytical iterative method for solving nonlinear differential equations to demonstrate the working of the method we consider some nonlinear ordinary differential equations with appropriate initial boundary conditions in each of the examples we demonstrate the accuracy and convergence of the method to the solution we demonstrate clearly that the method is accurate fast and has a high order of convergence

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

brief paper on almost sure stability of continuous time markov jump linear systems in this paper we study the almost sure stability of continuous time jump linear systems with a finite state markov form process a sufficient condition for almost sure stability is derived that refers to the statistics of the transition matrix over m switches it is shown that if the system is exponentially almost sure stable there exists a finite m such that the criterion is satisfied in order to evaluate the expected value appearing in the condition an efficient monte carlo algorithm is worked out

on the variety of a highest weight module 1 2 let u g be the enveloping algebra of g detailed information on the primitive spectrum prim u g of u g was summarized by conjecture 7 4 of 181 here a main question was to relate prim u g with the springer correspondence parts i ii of this conjecture were established in i ii and a partial solution to iii was given in 21 sufficient e g to treat g of type a meanwhile borho and brylinski 5 have established iii for induced ideals while barbasch and vogan 1 2 have established a version of iii in which gr i i e prim u g is replaced by a certain wavefront set known to be contained in y gr 1 this last work involves some case by case analysis and gives little indication as to why the springer correspondence should arise in the study of prim u g

model based control and stability analysis of discrete time polynomial fuzzy systems with time delay and positivity constraints this paper proposes a novel lyapunov stabilization analysis of discrete time polynomial fuzzy model based control systems with time delay under positivity constraint the polynomial fuzzy model is constructed to describe the dynamics of a nonlinear discrete time system with time delay a model based polynomial fuzzy controller is designed using nonparallel distributed compensation technique to stabilize the system while driving the system states to positive using the positivity constraints the lyapunov stability and positivity conditions are formulated as sum of squares to relax the conservativeness of the obtained stability results two main methods are proposed in this paper first the piecewise linear membership functions plmfs are used to introduce the approximate error between piecewise and the original membership functions into the stability analysis and second introduce the boundary information of the premise variables into the stability analysis since the premise variables hold rich nonlinearity information a numerical example is given to demonstrate the effectiveness of the proposed approach

on orbital variety closures in sln iii geometric properties abstract this is the third paper in the series here we define a few combinatorial orders on young tableaux the first order is obtained from the induced duflo order by the extension with the help of vogan t procedure we call it the duflo vogan order the second order is obtained from the generalization of spaltenstein s construction by consideration of an orbital variety as a double chain of nilpotent orbits we call it the chain order again we use vogan s t procedure however this time to restrict the chain order we call it the vogan chain order the order on young tableaux defined by the inclusion of orbital variety closures is called the geometric order and the order on young tableaux defined by inverse inclusion of primitive ideals is called the algebraic order we get the following relations between the orders the duflo vogan order is an extension of the induced duflo order the algebraic order is an extension of the duflo vogan order the geometric order is an extension of the algebraic order the vogan chain order is an extension of the geometric order and finally the chain order is an extension of the vogan chain order the computations show that the duflo vogan and the vogan chain orders coincide on sl n for n 9 and in n 10 there is one case up to t procedure and transposition where the chain vogan order is a proper extension of the duflo vogan order in this only case the algebraic order coincides with the vogan chain order these computations permit us to conjecture that in sl n the algebraic order coincides with the geometric order as well we conjecture that the combinatorics of both the inclusions on primitive ideals and on orbital variety closures is defined by the vogan chain order on young tableaux

analysis and synthesis of interconnected positive systems this paper is concerned with the analysis and synthesis of interconnected systems constructed from heterogeneous positive subsystems and a nonnegative interconnection matrix we first show that admissibility to be defined in this paper is an essential requirement in constructing such interconnected systems then we clarify that the interconnected system is admissible and stable if and only if a metzler matrix which is built from the coefficient matrices of positive subsystems and the nonnegative interconnection matrix is hurwitz stable by means of this key result we further provide several results that characterize the admissibility and stability of the interconnected system in terms of the frobenius eigenvalue of the interconnection matrix and the weighted l1 induced norm of the positive subsystems again to be defined in this paper moreover in the case where every subsystem is siso we provide explicit conditions under which the interconnected system has the property of persistence i e its state converges to a unique strictly positive vector that is known in advance up to a strictly positive constant multiplicative factor for any nonnegative and nonzero initial state as an important consequence of this property we show that the output of the interconnected system converges to a scalar multiple of the right eigenvector of a nonnegative matrix associated with its frobenius eigenvalue where the nonnegative matrix is nothing but the interconnection matrix scaled by the steady stage gains of the positive subsystems this result is then naturally and effectively applied to formation control of multiagent systems with positive dynamics this result can be seen as a generalization of a well known consensus algorithm that has been basically applied to interconnected systems constructed from integrators

on a theorem of redheffer concerning diagonal stability abstract an important problem in system theory concerns determining whether or not a given lti system x ax is diagonally stable more precisely this problem is concerned with determining conditions on a matrix a such that there exists a diagonal matrix d with positive diagonal entries i e a positive diagonal matrix satisfying a t d da q 0 while this problem has attracted much attention over the past half century two results of note stand out i a result based on theorems of the alternative derived by barker berman and plemmons and ii algebraic conditions derived by redheffer this paper is concerned with the second of these conditions our principal contribution is to show that redheffer s result can be obtained from the kalman yacubovich popov lemma we then show that this method of proof leads to natural generalisations of redheffer s result and we use these results to derive new conditions for diagonal and hurwitz stability for special classes of matrices

stability and l1 gain analysis of linear periodic piecewise positive systems this paper investigates the stability stabilization and l1 gain of linear periodic piecewise positive systems the monotonicity of linear periodic piecewise positive systems is first studied then a time varying co positive lyapunov function for periodic piecewise positive systems is employed and a sufficient condition for the asymptotic stability of the system is established based on the provided co positive lyapunov function and the sufficient stability condition a state feedback periodic piecewise controller to stabilize the system is formed and an upper bound of l1 gain of the system is given finally numerical examples are given to illustrate the theoretical results

technical communique preservation of lyapunov functions under bilinear mapping bilinear mapping preserves quadratic lyapunov functions between continuous time and discrete time lti systems this note studies if this property holds for other types of lyapunov functions specified by vector norms of the system state some implications of the obtained results in the common lyapunov function problems are discussed

brief paper markov jump linear systems with switching transition rates mean square stability with dwell time the stability of a class of markov jump linear systems characterized by piecewise constant transition rates and system dynamics is investigated for these switching markov jump linear systems mean square stability is analyzed through the time evolution of the second order moment of the state the main result is a sufficient condition that guarantees mean square stability under constraints on the dwell time between switching instants an alternative condition based on kronecker calculus is worked out it is shown that both the stability criteria admit an lmi implementation

on linear copositive lyapunov functions and the stability of switched positive linear systems we consider the problem of common linear copositive lyapunov function existence for positive switched linear systems in particular we present a necessary and sufficient condition for the existence of such a function for switched systems with two constituent linear time invariant systems several applications of this result are also given

convexity of the cost functional in an optimal control problem for a class of positive switched systems abstract this paper deals with the optimal control of a class of positive switched systems the main feature of this class is that switching alters only the diagonal entries of the dynamic matrix the control input is represented by the switching signal itself and the optimal control problem is that of minimizing a positive linear combination of the final state variable first the switched system is embedded in the class of bilinear systems with control variables living in a simplex for each time point the main result is that the cost is convex with respect to the control variables this ensures that any pontryagin solution is optimal algorithms to find the optimal solution are then presented and an example taken from a simplified model for hiv mutation mitigation is discussed

a note on recursive schur complements block hurwitz stability of metzler matrices and related results it is known that the stability of a metzler matrix can be characterized in a routh hurwitz like fashion based on a recursive application of scalar schur complements 1 our objective in this brief note is to show that recently obtained stability conditions are equivalent statements of this result and can be deduced directly therefrom using only elementary results from linear algebra implications of this equivalence are also discussed and several examples are given to illustrate potentially interesting system theoretic applications of this observation

a new iterative technique for solving nonlinear second order multi point boundary value problems abstract we present a semi analytical iterative method for solving nonlinear second order multi point boundary value problems to demonstrate the working of the method we consider a particular example of this class of problems in this example we demonstrate the accuracy and convergence of the method to the solution we demonstrate clearly that the method is accurate fast and has a reasonable order of convergence

a constructive approach to linear lyapunov functions for positive switched systems using collatz wielandt sets we establish the link between linear lyapunov functions for positive switched systems and corresponding collatz wielandt sets this leads to an algorithm to compute a linear lyapunov function whenever a lyapunov function exists

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

input output gain analysis for linear systems on cones abstract this paper is concerned with the input output gain analysis for dynamic systems that are defined on cones for such monotone systems we characterize their input output gain induced by the so called cone linear absolute norm in terms of generalized inequalities it is further shown that for a linear time invariant system represented by a convolution operator that maps one proper cone to another in the l 1 space its operator norm is governed by the static gain of the system as a byproduct the spectral radius of a cone preserving convolution operator turns out to be the same as that of the static gain matrix moreover dual results based on the cone max norm are presented finally the theoretical results are illustrated via linear systems over polyhedral cones or second order cones as well as a class of stochastic systems

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

generalized joint spectral radius and stability of switching systems abstract this paper extends the notion of generalized joint spectral radius with exponents originally defined for a finite set of matrices to probability distributions we show that under a certain invariance condition the radius is calculated as the spectral radius of a matrix that can be easily computed extending the classical counterpart using this result we investigate the mean stability of switching systems in particular we establish the equivalence of mean square stability simultaneous contractibility in square mean and the existence of a quadratic lyapunov function also the stabilization of positive switching systems is studied numerical examples are given to illustrate the results

on the stability of positive linear switched systems under arbitrary switching laws we consider n dimensional positive linear switched systems a necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is hurwitz several researchers conjectured that for positive linear switched systems this condition is also sufficient recently gurvits shorten and mason showed that this conjecture is true for the case n 2 but is not true in general their results imply that there exists some minimal integer np such that the conjecture is true for all n np but is not true for n np we show that np 3

stabilizability of switched linear systems does not imply the existence of convex lyapunov functions counterexamples are given which show that a linear switched system with controlled switching that can be stabilized by means of a suitable switching law does not necessarily admit a convex lyapunov function both continuous and discrete time cases are considered this fact contributes in focusing the difficulties encountered so far in the theory of stabilization of switched system in particular the result is in contrast with the case of uncontrolled switching in which it is known that if a system is stable under arbitrary switching then admits a polyhedral norm as a lyapunov function

semidefinite programming duality and linear time invariant systems several important problems in control theory can be reformulated as semidefinite programming problems i e minimization of a linear objective subject to linear matrix inequality lmi constraints from convex optimization duality theory conditions for infeasibility of the lmis as well as dual optimization problems can be formulated these can in turn be reinterpreted in control or system theoretic terms often yielding new results or new proofs for existing results from control theory we explore such connections for a few problems associated with linear time invariant systems

on the simultaneous diagonal stability of a pair of positive linear systems in this paper we derive a necessary and sufficient condition for the existence of a diagonal common quadratic lyapunov function cqlf for a pair of positive linear time invariant lti systems

differential operators on homogeneous spaces iii characteristic varieties of harish chandra modules and of primitive ideals letg be a semi simple complex algebraic group with lie algebra and flag varietyx g b for each primitive idealj with trivial central character in the enveloping algebrau we define a characteristic variety in the cotangent bundle ofx which projects under the springer resolution mapt x onto the closure of a nilpotent orbit we prove that this characteristic variety is theg saturation of the characteristic variety of a highest weight module with annihilatorj we conjecture that it is irreducible forg sl n our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of weyl group representations as achieved by a joseph the presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space and to study more generally characteristic varieties of harish chandra modules

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

cooperative distributed model predictive control abstract in this paper we propose a cooperative distributed linear model predictive control strategy applicable to any finite number of subsystems satisfying a stabilizability condition the control strategy has the following features hard input constraints are satisfied terminating the iteration of the distributed controllers prior to convergence retains closed loop stability in the limit of iterating to convergence the control feedback is plantwide pareto optimal and equivalent to the centralized control solution no coordination layer is employed we provide guidance in how to partition the subsystems within the plant we first establish exponential stability of suboptimal model predictive control and show that the proposed cooperative control strategy is in this class we also establish that under perturbation from a stable state estimator the origin remains exponentially stable for plants with sparsely coupled input constraints we provide an extension in which the decision variable space of each suboptimization is augmented to achieve pareto optimality we conclude with a simple example showing the performance advantage of cooperative control compared to noncooperative and decentralized control strategies

stability and boldsymbol l_ 1 gain analysis for positive takagi sugeno fuzzy systems with impulse this paper is concerned with the stability analysis and l 1 gain characterization for positive takagi sugeno fuzzy systems subject to impulse effects the main tool utilized is the construction of an impulse time dependent discretized copositive lyapunov function where the impulse interval partitioning idea is exploited to derive less conservative results for the disturbance free case stability criteria are developed to guarantee global exponential stability of the concerned systems under ranged and constant dwell time constraints the analysis of the influence of the discretizing parameter on the reduction of conservativeness is performed we further employ the proposed impulse time dependent discretized copositive lyapunov function to analyze the l 1 gain performance for the case where external disturbances are considered by performing a grid search for a scalar in an interval all the proposed criteria can be checked via linear programming finally numerical simulations are discussed to illustrate the effectiveness of the theoretical results

exponential stability and interval stability of a class of stochastic hybrid systems driven by both brownian motion and poisson jumps abstract a class of stochastic singular hybrid systems driven by both brownian motion and poisson jumps are studied this paper is devoted to discussing the exponential stability and interval stability of such stochastic singular hybrid systems the concept of interval admissibility is proposed sufficient conditions are given for exponential mean square admissibility and interval admissibility by using ito s formula h representation and spectrum technique finally two simulation cases are presented to demonstrate the theoretical results

almost sure and δmoment stability of jump linear systems abstract in this paper we study the almost sure stability and moment stability of discrete time jump linear systems with a finite state form process criteria for almost sure and moment stability which are necessary and sufficient conditions for scalar one dimensional systems are derived the relationship between almost sure and 6 moment stability for scalar one dimensional systems is also presented if the system matrices commute necessary and sufficient conditions for almost sure stability are obtained

stability and stabilization of switched linear time invariant systems with saddle points and switching delays this paper addresses global asymptotic stability gas and stabilization issues of switched linear time invariant lti systems with saddle points and time varying switching delays firstly two criteria of gas are proposed for such switched systems with all subsystems sharing a common unique saddle point under arbitrary periodic quasi periodic switching paths psp qsp secondly by using our stability results we design global asymptotic stabilizing controls gasc periodic quasi periodic stabilizing switching paths pssp qssp and an algorithm of gasc and pssp qssp for the switched systems finally we present a numerical example of a switching multi agent system to illustrate the effectiveness and practicality of our new results

discretisation of sparse linear systems an optimisation approach abstract this paper addresses the discretisation problem for sparse linear systems classical methods usually destroy sparsity patterns of continuous time systems we develop an optimisation procedure that yields the best approximation to the discrete time dynamical matrix with a prescribed sparsity pattern and subject to stability and other constraints by formulating this problem in an adequate manner tools from convex optimisation can be then applied error bounds for the approximation are provided for special classes of matrices numerical examples are included

classical results on the stability of linear time invariant systems and the schwarz form the paper deals with the classical results of routh and hurwitz biehler and kharitonov concerning the stability of a linear time invariant differential equation it is shown that these results follow directly from a schwarz matrix representation of stable systems

on the decay rate of discrete time linear delay systems with cone invariance this technical note is concerned with the decay rate constrained stability analysis for linear delay systems that possess cone invariant property in order to capture the decay rate of such systems we introduce a nondecreasing positive function whose reciprocal represents the decay rate under mild assumptions on the growth rate of this function an explicit condition is given to ensure that a cone preserving linear system with unbounded time varying delays is asymptotically stable with a given decay rate as typical cases necessary and sufficient conditions are given to characterize the decay rate when the delay is restricted by a linear sublinear or logarithmic growth rate finally some numerical examples are given to illustrate the effectiveness of the theoretical results

co positive lyapunov functions for the stabilization of positive switched systems in this paper exponential stabilizability of continuous time positive switched systems is investigated for two dimensional systems exponential stabilizability by means of a switching control law can be achieved if and only if there exists a hurwitz convex combination of the metzler system matrices in the higher dimensional case it is shown by means of an example that the existence of a hurwitz convex combination is only sufficient for exponential stabilizability and that such a combination can be found if and only if there exists a smooth positively homogeneous and co positive control lyapunov function for the system in the general case exponential stabilizability ensures the existence of a concave positively homogeneous and co positive control lyapunov function but this is not always smooth the results obtained in the first part of the paper are exploited to characterize exponential stabilizability of positive switched systems with delays and to provide a description of all the switched equilibrium points of an affine positive switched system

stability of continuous time positive switched linear systems a weak common copositive lyapunov functions approach in this paper we study the problem of asymptotic stability of continuous time positive switched linear systems under both arbitrary and restricted switchings it is well known that asymptotic stability under arbitrary switching can be implied by several classes of strong common copositive lyapunov functions clfs i e functions whose derivative along the nontrivial system trajectories is negative however asymptotically stable positive switched systems may not admit strong common clfs the main contribution of this paper is to study the stability problem by requiring only weak common clfs firstly necessary and sufficient conditions are established for asymptotic stability under arbitrary switching among them an easily verifiable graphical stability criterion based on the connectivity of the digraphs associated with the subsystem matrices is proposed secondly we further relax the obtained graphical condition to derive a relaxed weak excitation condition for asymptotic stability under dwell time switching finally two examples are provided to illustrate the effectiveness of our theoretical results

primitive ideals and orbital integrals in complex exceptional groups in 2 two related problems were studied for complex classical groups determination of the primitive spectrum of the enveloping algebra as a set and fourier inversion of unipotent orbital integrals the first of these was solved completely extending work of joseph and others in sl n c and small groups although our techniques presumably solve the second as well up to determination of normalizing constants we carried out the calculations only for special unipotent classes see 181 we had of course tried to apply the same methods to the exceptional groups but they seem to be inadequate since that work was done however brylinski and kashiwara and beilinson and bernstein have established the kazhdan lusztig conjecture giving character formulas for irreducible highest weight modules 3 71 because of a conjecture of joseph proved in 24 this determines in principle the primitive ideals with a fixed regular integral infinitesimal character they are in one to one correspondence with what kazhdan and lusztig call left cells in the weyl group however the algorithm given by kazhdan and lusztig to compute these cells is enormously complicated requiring one to compute roughly 1 i polynomials of degrees on the order of half the number of positive roots this is very unsatisfactory however it

on delay independent stability of a class of nonlinear positive time delay systems we present a condition for delay independent stability of a class of nonlinear positive systems this result applies to systems that are not necessarily monotone and extends recent work on cooperative nonlinear systems

block wise discretization accounting for structural constraints this paper presents an approximate discretization method named mixed euler zoh me zoh which can be applied to any continuous time linear system this method has been explicitly developed to preserve the system sparsity a property that is particularly important when dealing with the analysis and design of distributed controllers for large scale systems in terms of stability preservation as a function of the sampling interval we show that me zoh outperforms the classical forward euler fe approach which is the only known discretization method guaranteeing the preservation of sparsity for all possible sampling times it is then shown that this new discretization method is capable of preserving stability for all sampling times for a wide classes of dynamical systems including the important class of positive systems besides stability also positivity of the resulting discrete time system is preserved contrarily to what happens for the fe approach a couple of examples are reported to illustrate the main theoretical results of the paper

on the kalman yakubovich popov lemma for discrete time positive linear systems a novel simple proof and some related results theorems of alternatives on the feasibility of linear matrix inequalities lmis are used in order to provide novel simple proofs for two considered versions of the kalman yakubovich popov kyp lemma for discrete time positive linear systems two different and novel recursive methods to determine whether a positive matrix is or is not schur are obtained as an application of an existing connection between the strict inequality version of the kyp lemma for single input single output siso discrete time positive linear systems and a schur matrix condition examples are included which provide illustration on these recursive methods

mathcal h _2 state feedback control design of continuous time positive linear systems this technical note focuses on state feedback control design of positive linear systems in continuous time domain the main goal is to analyze stability and performance preserving the closed loop system positivity in some sense the provided results fill the literature gap by considering the mathcal h 2 performance index indeed although mathcal h infty state feedback control design of positive systems has received remarkable attention in the recent years we believe that for the first time a numerical method able to provide a local optimal solution to this class of nonconvex control problems is addressed by means of a simple since no line search is required convergent sequence of convex programming problems the theory and the algorithm behavior are illustrated and evaluated by numerical examples borrowed from the recent literature

on varieties in an orbital variety closure in semisimple lie algebras abstract in this note we discuss the closure of an orbital variety as a union of varieties we show that if semisimple lie algebra g contains factors not of type a n then there are orbital varieties whose closure contains components which are not lagrangian we show that the argument does not work if all the factors are of type a n and provide the facts supporting the conjecture claiming that if all the factors of g are of type a n then the closure of an orbital variety is a union of orbital varieties

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

brief paper on linear co positive lyapunov functions for sets of linear positive systems in this paper we derive necessary and sufficient conditions for the existence of a common linear co positive lyapunov function for a finite set of linear positive systems both the state dependent and arbitrary switching cases are considered our results reveal an interesting characterisation of linear stability for the arbitrary switching case namely the existence of such a linear lyapunov function can be related to the requirement that a number of extreme systems are metzler and hurwitz stable examples are given to illustrate the implications of our results

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

stochastic stability properties of jump linear systems jump linear systems are defined as a family of linear systems with randomly jumping parameters usually governed by a markov jump process and are used to model systems subject to failures or changes in structure the authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties it is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability and a testable necessary and sufficient condition for second moment stability is derived the lyapunov exponent method for the study of almost sure sample stability is discussed and a theorem which characterizes the lyapunov exponents of jump linear systems is presented finally for one dimensional jump linear system it is proved that the region for delta moment stability is monotonically converging to the region for almost sure stability at delta down arrow 0 sup

almost sure stabilization of uncertain continuous time markov jump linear systems this work deals with the robust almost sure as stabilization problem for continuous time markov jump linear systems mjls norm bounded uncertainties affecting the system state and input matrices are considered a deterministically testable sufficient condition for robust as stability is provided which relies on a bound on the 2 norm of the system transition matrix such a condition can be profitably employed also to design a robust feedback stabilization strategy such feedback design is based on a formulation of the sufficient condition for robust as stability in terms of an equivalent lmi problem

stability and stabilizability of continuous time linear compartmental switched systems in this paper we introduce continuous time linear compartmental switched systems and investigate their stability and stabilizability properties by their nature these systems are always stable necessary and sufficient conditions for asymptotic stability for arbitrary switching functions and sufficient conditions for asymptotic stability under certain dwell time conditions on the switching functions are proposed finally stabilizability is thoroughly investigated and proved to be equivalent to the existence of a hurwitz convex combination of the subsystem matrices a condition that for positive switched systems is only sufficient for stabilizability

some extensions on the bounded real lemma for positive systems recent advances have shown that the l 2 gain of positive systems is fully determined by the static gain matrix and hence the linear matrix inequality lmi characterizing their h infty performance admits a diagonal lyapunov matrix variable in this technical note we aim to extend this property for systems with input state and output defined on general cones although it is known that the l 2 gain is no longer governed by the static gain matrix for systems defined on general proper cones we show that this property is still valid when the cone in the state space is symmetric and the cones in the input and output spaces are self dual meanwhile it is proved that the lyapunov matrix appearing in the lmi characterization for the h infty performance can be constructed by the quadratic representation of the jordan algebra asso ciated with the symmetric cone in the state space finally the established results are illustrated via a linear system defined on second order cones

on pade approximations quadratic stability and discretization of switched linear systems in this note we consider the stability preserving propertie s of diagonal pade approximations to the matrix exponential we show that while diagonal pade approximations preserve quadratic stability when going from continuous time to discrete time the converse is not true we discuss the implications of this result for discretizing switched linear systems we also show that for continuous time switched systems which are exponentially stable but not quadratically stable a pade approximation may not preserve stability

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

linear copositive lyapunov functions for continuous time positive switched systems continuous time positive systems switching among p subsystems are introduced and a complete characterization for the existence of a common linear copositive lyapunov function for all the subsystems is provided when the subsystems are obtained by applying different feedback control laws to the same continuous time single input positive system the above characterization leads to a very easy checking procedure

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