optimal h model reduction via linear matrix inequalities continuous and discrete time cases necessary and sufficient conditions are derived for the existence of a solution to the continuous time and discrete time h model reduction problems these conditions are expressed in terms of linear matrix inequalities lmis and a coupling non convex rank constraint set in addition an explicit parametrization of all reduced order models that correspond to a feasible solution is provided in terms of a contractive matrix these results follow from the recent solution of the h control design problem using lmis particularly simple conditions and a simple parametrization of all solutions are obtained for the zeroth order h approximation problem and the convexity of this problem is demonstrated computational issues are discussed and an iterative procedure is proposed to solve the h model reduction problem using alternating projections although global convergence of the algorithm is not guaranteed

frequency weighted spl lscr sub spl infin norm and optimal hankel norm model reduction a new relative error model reduction method is proposed using frequency weighted balanced realization and explicit spl lscr sub spl infin norm error bounds are also derived for the relative error and multiplicative error the method only needs to solve two lyapunov equations it is further shown that this method is equivalent to the balanced stochastic truncation if the plant is square and minimum phase this paper also gives a complete solution to the frequency weighted hankel norm approximation with antistable weighting these results are then applied to spl lscr sub spl infin norm model reduction and several numerically effective algorithms are proposed it is shown through many numerical examples that these algorithms work very well and in many cases produce almost optimal solutions

characterization of the solution to the optimal h model reduction problem abstract in this paper the h norm approximation of a given stable proper rational transfer function by a lower order stable proper rational transfer function is studied the solution method is based on the authors observation that the h model reduction problem can be converted into a hankel norm model reduction problem a simple characterization of the solution to the optimal h model reduction problem is developed

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

h _ infty filtering for discrete time state delayed systems with finite frequency specifications this article researches the problem of finite frequency ff h filtering for linear discrete time state delayed systems the disturbance is assumed to reside in low middle high frequency ranges to reduce the conservatism of the results delay partitioning idea is used to derive a new ff bounded real lemma brl by applying the generalized kalman yakubovich popov lemma two equivalent approaches to the proof of the proposed ff brl are given respectively starting from transfer function and lyapunov krasovskii functional a new ff h filter design method is proposed in terms of solving a set of linear matrix inequalities finally a numerical example clearly demonstrates the merits and effectiveness of the proposed method

weighted hankel norm approximation calculation of bounds abstract frequency domain bounds are computed for the approximation error involved in the optimum hankel norm approximation procedure the bounds differ from those arising in the unweighted optimal hankel norm approximation procedure by a multiplicative factor depending solely on the weighting function this factor becomes unity when the weighting function is constant

a finite frequency approach to control of markov jump linear systems with incomplete transition probabilities this paper is concerned with the state feedback control of continuous markov jump linear systems with incomplete transition probabilities in finite frequency domain by developing a new technique to handle the coupling among lyapunov variable system matrix and controller parameter new sufficient conditions for the closed loop system to be stochastically stable with the required finite frequency performance are established in terms of linear matrix inequalities meanwhile the finite frequency state feedback controller is also obtained by the proposed conditions directly the validity of the proposed method is demonstrated by a numerical example

internal positivity preserved model reduction this article studies model reduction of continuous time stable positive linear systems under the hankel norm h norm and h 2 norm performance the reduced order systems preserve the stability as well as the positivity of the original systems this is achieved by developing new necessary and sufficient conditions of the model reduction performances in which the lyapunov matrices are decoupled with the system matrices in this way the positivity constraints in the reduced order model can be imposed in a natural way as the model reduction performances are expressed in linear matrix inequalities with equality constraints the desired reduced order positive models can be obtained by using the cone complementarity linearisation iterative algorithm a numerical example is presented to illustrate the effectiveness of the given methods

l1 induced norm and controller synthesis of positive systems in this paper the problem of 1 1 induced controller design for discrete time positive systems is investigated with the use of linear lyapunov function an analytical method to compute the exact value of 1 1 induced norm is first presented then a novel characterization for stability and 1 1 induced performance is proposed based on the characterization a necessary and sufficient condition for the existence of desired controllers is derived and an iterative convex optimization approach is developed to solve the condition in addition the synthesis of the state feedback controller for single input multiple output simo positive systems is investigated for this special case an analytic solution is established to show how the optimal 1 1 induced controller can be designed and some links to the spectral radius of the closed loop systems are provided finally the theoretical results are illustrated through a numerical example

families of rational maps and iterative root finding algorithms in this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root finding algorithms we answer a question of smale s by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more we settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics and we give a classification of all such algorithms in the context of conformal dynamics our main result is the following a stable algebraic family of rational maps is either trivial all its members are conjugate by mobius transformations or affine its members are obtained as quotients of iterated addition on a family of complex tori our classification of generally convergent algorithms follows easily from this result as another consequence of rigidity we observe that the eigenvalues of a nonaffine rational map at its periodic points determine the map up to finitely many choices this implies that bounded analytic functions nearly separate points on the moduli space of a rational map

approximation of non archimedean lyapunov exponents and applications over global fields let k be an algebraically closed field of characteristic 0 that is complete with respect to a non archimedean absolute value we establish a locally uniform approximation formula of the lyapunov exponent of a rational map f of mathbb p 1 of degree d 1 over k in terms of the multipliers of n periodic points of f with an explicit control in terms of n f and k as an immediate consequence we obtain an estimate for the blow up of the lyapunov exponent near a pole in one dimensional families of rational maps over k combined with our former archimedean version this non archimedean quantitative approximation allows us to show a quantified version of silverman s and ingram s recent comparison between the critical height and any ample height on the moduli space mathcal m d bar mathbb q two improvements of mcmullen s finiteness of the multiplier maps reduction to multipliers of cycles of exact given period and an effective bound from below on the period a characterization of non affine isotrivial rational maps defined over the function field mathbb c x of a normal projective variety x in terms of the growth of the degree of the multipliers

brief paper positivity preserving h model reduction for positive systems this paper is concerned with the model reduction of positive systems for a given stable positive system our attention is focused on the construction of a reduced order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed h performance based upon a system augmentation approach a novel characterization on the stability with h performance of the error system is first obtained in terms of linear matrix inequality lmi then a necessary and sufficient condition for the existence of a desired reduced order model is derived accordingly furthermore iterative lmi approaches with primal and dual forms are developed to solve the positivity preserving h model reduction problem finally a compartmental network is provided to show the effectiveness of the proposed techniques

brief paper robust finite frequency h filtering for uncertain 2 d roesser systems this paper investigates the problem of robust finite frequency ff h filtering for two dimensional 2 d roesser models with polytopic uncertainties our attention is focused on designing filters guaranteeing the robustly asymptotic stability and ff h disturbance attenuation level of the filtering error system by the parameter dependent idea and the generalized kalman yakubovich popov lemma for 2 d roesser models the existence conditions of robust ff h filters are obtained in terms of solving an optimization problem which is more general than the existing results an example is given to validate the proposed methods the contribution of the paper is twofold 1 systematic methods are proposed for designing ff h filters for roesser models 2 an improved strategy has been presented to deal with the robust h filter design for roesser models

positive filtering for positive takagi sugeno fuzzy systems under l 1 performance in this paper the positive filtering problem is addressed for positive takagi sugeno t s fuzzy systems under the 1 induced performance to estimate the output of positive t s fuzzy systems error bounding positive filters are constructed a new performance characterization is first established to guarantee the asymptotic stability of the filtering error system with the 1 induced performance moreover sufficient conditions expressed by linear programming problems are derived to design the required filters finally a numerical example is presented to show the effectiveness of the derived theoretical results

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

finite time balanced truncation for linear systems via shifted legendre polynomials abstract in this paper we present a finite time model order reduction method for linear systems via shifted legendre polynomials the main idea of the approach is to use finite time empirical gramians which are constructed from impulse responses by solving block tridiagonal linear systems to generate approximate balanced system for the large scale system the balancing transformation is directly computed from the expansion coefficients of impulse responses in the space spanned by shifted legendre polynomials without individual reduction of the gramians and a separate eigenvector solve then the reduced order model is constructed by truncating the states corresponding to the small approximate hankel singular values hsvs the stability preservation of the reduced model is briefly discussed and in combination with the dominant subspace projection method we modify the reduction procedure to alleviate the shortcomings of the above method which may unexpectedly lead to unstable systems even though the original one is stable furthermore the properties of the resulting reduced models are considered finally numerical experiments are given to demonstrate the effectiveness of the proposed methods

generalized kyp lemma with real data a recent generalization of the kalman yakubovich popov kyp lemma establishes the equivalence between a semi infinite inequality on a segment of a line or circle in the complex plane and a linear matrix inequality lmi in this technical note we show that when the data are real the matrix variables in the lmi can be restricted to be real even when the frequency range is asymmetric with respect to the real axis

model simplification for switched hybrid systems abstract this paper is concerned with the problem of h model reduction for switched system which is an important class of hybrid systems frequently encountered in practical situations two sharply different approaches are proposed to solve this problem the first approach casts the model reduction into a convex optimization problem which is the first attempt to solve the model reduction problem by using linearization procedure the second one based on the cone complementarity linearization idea casts the model reduction problem into a sequential minimization problem subject to linear matrix inequality constraints both approaches have their own advantages and disadvantages concerning conservatism and computational complexity a numerical example illustrates the effectiveness of the proposed theories

model reduction of markovian jump systems with uncertain probabilities this paper studies the problem of model reduction for nonhomogeneous markovian jump systems the transition probability matrix of the nonhomogeneous markovian chain has the characteristic of a polytopic structure an asynchronous reduced order model is considered and the asynchronization is modeled by a hidden markov model with a partially unknown conditional probability matrix under this framework a new sufficient condition is proposed to ensure that the augmented system is stochastically mean square stable with a specified level of h infty performance finally a numerical example is provided to show the effectiveness and advantages of the theoretic results obtained

attracting cycles in p adic dynamics and height bounds for postcritically finite maps a rational function of degree at least 2 with coefficients in an algebraically closed field is postcritically finite pcf if and only if all of its critical points have finite forward orbit under iteration we show that the collection of pcf rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers once the well understood family known as flexible lattes maps is excluded as a consequence there are only finitely many conjugacy classes of non lattes pcf rational maps of a given degree defined over any given number field the key ingredient of the proof is a nonarchimedean version of fatou s classical result that every attracting cycle of a rational function over c attracts a critical point

generalized kalman yakubovich popov lemma for 2 d fm lss model this paper is concerned with the generalized kalman yakubovich popov kyp lemma for 2 d fornasini marchesini local state space fm lss systems by carefully analyzing the feature of the states in 2 d fm lss models a linear matrix inequality lmi characterization for a rectangular finite frequency region is constructed and then by combining this characterization with s procedure a generalized kyp lemma is proposed for 2 d fm lss models as applications of the developed kyp lemma new conditions are further derived for designing controllers guaranteeing finite frequency positive realness of the closed loop systems where both state feedback and dynamic output feedback control laws are considered finally two illustrative examples clearly show the effectiveness of the proposed results the main contribution of the paper is trifold 1 a novel characterization for a finite frequency region has been developed 2 a generalized kyp lemma has been proposed including the existing bounded realness and positive realness results as special cases and 3 systematic methods for finite frequency positive real control of 2 d systems have been presented

bifurcation measures and quadratic rational maps we study critical orbits and bifurcations in the curves per1 within the moduli space m2 of quadratic rational maps f p 1 p 1 for each 2 c the curve per1 is dened by the condition that the map has a xed point of multiplier we prove that the curve per1 contains innitely many postcritically n ite maps if and only if 0 addressing a special case of bd2 conjecture 1 4 we also show that the two critical points of f dene distinct bifurcation measures along per 1

technical communique h model reduction for uncertain switched linear discrete time systems in this paper the problem of h model reduction for switched linear discrete time systems with polytopic uncertainties is investigated a reduced order switched model is constructed for a given robustly stable switched system which has the same structural polytopic uncertainties as the original system such that the resulting error system is robustly asymptotically stable and an h error performance is guaranteed a sufficient condition for the existence of the desired reduced order model is derived and formulated in terms of a set of linear matrix inequalities by solving the corresponding convex optimization problem in such existence condition the vertex system of reduced order model can be obtained which also provides an h gain for the error system between the original system and the reduced order model a numerical example is given to show the effectiveness and the potential of the proposed techniques

technical communique static output feedback stabilization an ilmi approach in this note the static output feedback stabilization problem is addressed using the linear matrix inequality technique a necessary and sufficient condition for static output feedback stabilizability for linear time invariant systems is derived in the form of a matrix inequality the extension of the result to h control is studied an iterative lmi ilmi algorithm is proposed to compute the feedback gain numerical examples are employed to demonstrate the effectiveness and the convergence of the algorithm

a finiteness theorem for canonical heights attached to rational maps over function fields let k be a function field let k t be a rational map of degree d 2 defined over k and suppose that is not isotrivial in this paper we show that a point p p k has canonical height zero if and only if p is preperiodic for this answers affirmatively a question of szpiro and tucker and generalizes a recent result of benedetto from polynomials to rational functions we actually prove the following stronger result which is a variant of the northcott finiteness principle there exists e 0 such that the set of points p p k with canonical height at most e is finite our proof is essentially analytic making use of potential theory on berkovich spaces to study the dynamical green s functions g v x y attached to at each place v of k for example we show that every conjugate of has bad reduction at v if and only if g v x x 0 for all x p1berk v where p 1 berk v denotes the berkovich projective line over the completion of kv in an appendix we use a similar method to give a new proof of the mordell weil theorem for elliptic curves over k

l induced output feedback controller synthesis for positive nonlinear systems via t s fuzzy model approach abstract this paper investigates the l induced output feedback controller synthesis problem for positive takagi sugeno t s fuzzy systems for positive t s fuzzy systems a sufficient condition for stability and l induced performance is first put forward using a group of linear inequalities then on the basis of these inequalities the conditions on which static output feedback controllers exist are set up and to solve these conditions a method of iterative convex optimization is provided moreover a complete solution to the synthesis problem of the controller is proposed based on linear programming for positive t s fuzzy systems with single output or input finally two examples are presented to illustrate and verify the effective and applicable methods derived in this paper

l1 gain analysis and model reduction problem for boolean control networks in this paper the weighted l1 gain analysis and l1 model reduction problem for boolean control networks are proposed and investigated via semi tensor product method first the input energy and output energy are described by pseudo boolean functions based on which the definition of the weighted l1 gain is introduced by constructing a co positive lyapunov function a sufficient condition is established to ensure that a boolean control network is not only internally asymptotically stable but also has an l1 gain no more than a given scalar along this line by virtue of the properties of semi tensor product the l1 model reduction problem of a boolean control network is defined and converted to the l1 gain problem of another boolean control network with more nodes a sufficient condition for the l1 model reduction problem is then derived immediately and an algorithm is presented to compute the matrices in the reduced order model finally two examples including the boolean model for biochemical oscillators in the cell cycle are displayed to show the feasibility of the theoretical results

generalized kyp lemma unified frequency domain inequalities with design applications the celebrated kalman yakubovic spl caron popov kyp lemma establishes the equivalence between a frequency domain inequality fdi and a linear matrix inequality and has played one of the most fundamental roles in systems and control theory this paper first develops a necessary and sufficient condition for an s procedure to be lossless and uses the result to generalize the kyp lemma in two aspects the frequency range and the class of systems and to unify various existing versions by a single theorem in particular our result covers fdis in finite frequency intervals for both continuous discrete time settings as opposed to the standard infinite frequency range the class of systems for which fdis are considered is no longer constrained to be proper and nonproper transfer functions including polynomials can also be treated we study implications of this generalization and develop a proper interface between the basic result and various engineering applications specifically it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain phase properties in several frequency ranges the method is illustrated by numerical design examples of digital filters and proportional integral derivative controllers

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

fault detection for discrete time markov jump linear systems with partially known transition probabilities in this article the fault detection fd problem for a class of discrete time markov jump linear system mjls with partially known transition probabilities is investigated the proposed systems are more general which relax the traditional assumption in markov jump systems that all the transition probabilities must be completely known a residual generator is constructed and the corresponding fd is formulated as an h filtering problem by which the error between residual and fault are minimised in the h sense the linear matrix inequality based sufficient conditions for the existence of fd filter are derived a numerical example on a multiplier accelerator model economic system is given to illustrate the potential of the developed theoretical results

improved results on h model reduction for continuous time linear systems over finite frequency ranges this paper revisits the h model reduction problem for continuous time linear systems over finite frequency ranges given an asymptotically stable system our goal is to find a stable reduced order system in such a way that the error of the transfer functions between the original system and the reduced order one is bounded over a finite frequency range by virtue of the generalized kalman yakubovich popov gkyp lemma we first establish necessary and sufficient characterizations for this problem in terms of linear matrix inequalities lmis for the low and mid frequency cases through introducing a non conservative multiplier and resorting to the projection lemma the reduced order system matrices are decoupled with the matrix variables from the gkyp lemma then by introducing a new diagonal matrix variable and based on congruence transformation the reduced order system matrices are further decoupled with the matrix variable induced by the projection lemma and can be parameterized by a new matrix variable the results are extended to the high frequency case without the use of projection lemma to reduce the conservatism moreover an iterative convex optimization algorithm is developed to solve the conditions finally we demonstrate via numerical examples that our method can achieve much smaller approximation error than existing results

controllability of multi input positive discrete time systems linear discrete time positive systems arise in various fields of science such as economic modelling behavioural science stochastic processes etc recently some authors have examined the problem of the controllability of such systems for the single input case this paper provides necessary and sufficient conditions for characterizing the controllability properties of multi input systems the results here developed show that only a narrow class of these systems enjoys the properties of reachability and controllability

critical orbits of polynomials with a periodic point of specified multiplier answering a question posed by adam epstein we show that the collection of conjugacy classes of polynomials admitting a parabolic fixed point and at most one infinite critical orbit is a set of bounded height in the relevant moduli space we also apply the methods over function fields to draw conclusions about algebraically parametrized families and prove an analogous result for quadratic rational maps

brief paper weighted h model reduction for linear switched systems with time varying delay this paper is concerned with model reduction for continuous time linear switched systems with time varying delay for a given stable switched system our attention is focused on construction of a reduced order model such that the error system is exponentially stable with a prescribed weighted performance by applying the average dwell time approach and the piecewise lyapunov function technique delay dependent deley independent sufficient conditions are proposed in terms of linear matrix inequality lmi to guarantee the exponential stability and the weighted performance for the error system the model reduction problem is solved by using the projection approach which casts the model reduction problem into a sequential minimization problem subject to lmi constraints by employing the cone complementary linearization algorithm a numerical example is provided to illustrate the effectiveness of the proposed theory