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

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 h control of discrete time fuzzy systems a cone complementarity linearization approach in this paper new results are presented for h analysis and synthesis problems of discrete time takagi sugeno ts fuzzy systems by defining a multiple lyapunov function a new sufficient condition guaranteeing the h performance of the ts fuzzy systems is first derived which is expressed by a set of linear matrix inequalities lmis both theoretical analysis and numerical examples show that such a new condition is less conservative than previous results obtained within the quadratic framework based on this new condition for h performance the corresponding h controller design problem is then investigated different from the traditional quadratic framework the synthesis problem is solved by exploiting the cone complementarity linearization ccl method together with a sequential minimization problem subject to lmi constraints obtained for the existence of admissible controllers which can be readily solved by using standard numerical software

the nirenberg problem and its generalizations a unified approach making use of integral representations we develop a unified approach to establish blow up profiles compactness and existence of positive solutions of the conformally invariant equations p sigma v kv frac n 2 sigma n 2 sigma on the standard unit sphere mathbb s n for all sigma in 0 n 2 where p sigma is the intertwining operator of order 2 sigma finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures when sigma 1 it is the prescribing scalar curvature problem or the nirenberg problem and when sigma 2 it is the prescribing q curvature problem

interconnection between set and fuzzy differential equations abstract the original formulation of fuzzy differential equations suffers from the disadvantage since the solutions increases as time increases in this paper employing the results of set differential equations obtained in v laksmikantham s leela a s vatsala setvalued hybrid differential equations and stability in terms of two measures j hybrid systems 2 2 2002 169 188 a new approach is suggested in order to capture the vagueness and the rich properties of solutions without fuzziness

model approximation for discrete time state delay systems in the t s fuzzy framework this paper is concerned with the problem of h model approximation for discrete time takagi sugeno t s fuzzy time delay systems for a given stable t s fuzzy system our attention is focused on the construction of a reduced order model which not only approximates the original system well in an h performance but is also translated into a linear lower dimensional system by applying the delay partitioning approach a delay dependent sufficient condition is proposed for the asymptotic stability with an h error performance for the error system then the h model approximation problem is solved by using the projection approach which casts the model approximation into a sequential minimization problem subject to linear matrix inequality lmi constraints by employing the cone complementary linearization algorithm moreover by further extending the results h model approximation with special structures is obtained i e delay free model and zero order model finally two numerical examples are provided to illustrate the effectiveness of the proposed methods

on conformal deformations of metrics onsn abstract on s n there is a naturally metric defined n th order conformal invariant operator p n associated with this operator is a so called q curvature quantity when two metrics are pointwise conformally related their associated operators together with their q curvatures satisfy the natural differential equations this paper is devoted to the question of which function can be a q curvature candidate this is the so called prescribing q curvature problem our main result is that if q is positive nondegenerate and the naturally defined mapping associated with q has nonzero degree then our problem has a solution this is the natural generalization of prescribing gaussian curvature on s 2 into s n

q curvature flow on the standard sphere of even dimension using a gradient flow approach initiated by s brendle we generalize the existence theorem for the prescribing q curvature equation on s2 gauss curvature by m struwe 2005 14 and on s4 by malchiodi and struwe 2006 12 to sn for all even n with the similar assumption on the prescribed curvature candidate f

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

the profile of bubbling solutions of a class of fourth order geometric equations on 4 manifolds abstract we study a class of fourth order geometric equations defined on a 4 dimensional compact riemannian manifold which includes the q curvature equation we obtain sharp estimates on the difference near the blow up points between a bubbling sequence of solutions and the standard bubble

output feedback control of markovian jump repeated scalar nonlinear systems this paper is concerned with the induced l2 dynamic output feedback controller dofc design problem for discrete time markovian jump repeated scalar nonlinear systems by employing both the switching sequence dependent lyapunov function approach and the positive definite diagonally dominant lyapunov function technique a sufficient condition is first established which guarantees the underlying system to be stochastically stable with an induced l2 disturbance attenuation performance then the desired full or reduced order dofcs are designed by using projection approach cone complementarity linearization procedure is employed to cast the nonconvex feasibility problem into a sequential minimization problem finally a numerical example is presented to show the effectiveness of the proposed methods

hankel norm approximation of linear systems with time varying delay continuous and discrete cases this paper investigates the problem of hankel norm model reduction for linear systems with time varying delay in the state for a given stable system our attention is focused on the construction of reduced order model which guarantees the corresponding error system to be asymptotically stable and has a specified hankel norm error performance two different approaches are proposed to solve this problem one casts the model reduction into a convex optimization problem by using a linearization procedure and the other is based on the cone complementarity linearization idea which casts the model reduction into a sequential minimization problem subject to linear matrix inequality constraints both continuous and discrete time cases are considered a numerical example is provided to show the effectiveness of the proposed theory

algebraic topological methods for the supercritical q curvature problem abstract we study the problem of existence of conformal metrics with prescribed q curvature on closed four dimensional riemannian manifolds this problem has a variational structure and in the case of interest here it is noncompact in the sense that accumulations points of some noncompact flow lines of a pseudogradient of the associated euler lagrange functional the so called true critical points at infinity of the associated variational problem occur using the characterization of the critical points at infinity of the associated variational problem which is established in 42 combined with some arguments from morse theory some algebraic topological methods and some tools from dynamical system originating from conley s isolated invariant sets and isolated blocks theory we derive a new kind of existence results under an algebraic topological hypothesis involving the topology of the underling manifold stable and unstable manifolds of some of the critical points at infinity of the associated euler lagrange functional

output feedback stabilization of networked control systems with random delays modeled by markov chains this note investigates the output feedback stabilization of networked control systems ncss the sensor to controller s c and controller to actuator c a random network induced delays are modeled as markov chains the focus is on the design of a two mode dependent controller that depends on not only the current s c delay but also the most recent available c a delay at the controller node the resulting closed loop system is transformed to a special discrete time jump linear system then the sufficient and necessary conditions for the stochastic stability are established further the output feedback controller is designed via the iterative linear matrix inequality lmi approach simulation examples illustrate the effectiveness of the proposed method

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

on the solutions of liouville systems for some prescribed m 0 together with the dirichlet boundary condition u 0 on 0 the liouville equation finds applications in various fields of physics and mathematics to name a few it represents the newtonian potential of a cluster of self gravitating mass distribution where a 0 1 6 36 37 the electric potential induced by charge carriers in the theory of electrolytes where a 0 31 and maximal entropy solutions of the incompressible euler equation 23 9 in the field of differential geometry it stands for the problem of finding a metric corresponding to a prescribed gaussian curvature 3 10 the system analog of l takes the form

set differential systems and vector lyapunov functions we prove a comparison result in terms of vector lyapunov like functions relative to a set differential system using this we provide sufficient conditions in terms of vector lyapunov like functions for the stability properties of the trivial solutions of set differential systems

improved results on h model reduction for markovian jump systems with partly known transition probabilities abstract this paper is concerned with the model reduction problem for markovian jump systems with partly known transition probabilities by making use of an existing result on the convex properties of the transition probabilities an improved condition is first derived for the h performance analysis of the error system two different approaches one in terms of linear matrix inequalities lmis and another in the form of an iterative algorithm subject to lmi constraints are then proposed to find a reduced order model such that the h performance of the error system is bounded by a specified level the first approach is obtained using finsler s lemma and involves fewer decision variables than the existing method it is shown that the proposed methods cover some existing results as special cases finally the advantages of our results are clearly illustrated by a numerical example

compactness of solutions to some geometric fourth order equations we prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds the proof is based on a refined bubbling analysis for which the main estimates are given in integral form our result is used in a subsequent paper to find critical points via minimax arguments of some geometric functional which give rise to conformal metrics of constant q curvature as a byproduct of our method we also obtain compactness of such metrics

optimal time weighted h 2 model reduction for markovian jump systems this article addresses the optimal time weighted h 2 model reduction problem for markovian jump linear systems that is for a given mean square stable markovian jump system our aim is to find a mean square stable jump system of lower order such that the time weighted h 2 norm of the corresponding error system is minimised the time weighted h 2 norm of the system is first defined and then a computational method is constructed the computation requires the solution of two sets of recursive lyapunov type linear matrix equations associated with the markovian jump system to solve the optimal time weighted h 2 model reduction problem we propose a gradient flow method for its solution a necessary condition for minimality is derived and a computational procedure is provided to obtain the minimising reduced order model the necessary condition generalises the standard result for systems when markov jumps and the time weighting term do not appear finally two numerical examples are given to demonstrate the

the q curvature on a 4 dimensional riemannian manifold m g with mqdvg 8π2 abstract we deal with the q curvature problem on a 4 dimensional compact riemannian manifold m g with m q g d v g 8 2 and positive paneitz operator p g let q be a positive smooth function the question we consider is when can we find a metric g which is conformal to g such that q is just the q curvature of g a sufficient condition to this question is given in this paper

controllability observability and discrete time markovian jump linear quadratic control this paper is concerned with the controllability and observability of discrete time linear systems that possess randomly jumping parameters described by finite stale markov processes and the relationship between these properties and the solution of the infinite time jump linear quadratic jlq optimal control problem the solution of the markovian jlq problem with finite or infinite time horizons is known necessary and sufficient conditions for the existence of optimal constant control laws that lead to finite optimal expected costs as the time horizon becomes infinite are also known sufficient conditions for these steady state control laws to stabilize the controlled system are also available chizeck et al 1986 these conditions are not easy to test however various definitions of controllability and observability for stochastic systems exist in the literature these definitions are unfortunately not related to the steady state jlq control problem in a manner that is analogous to the role of deter

fixed order robust h sub spl infin filter design for markovian jump systems with uncertain switching probabilities this paper discusses the fixed order robust h sub spl infin filtering problem for a class of markovian jump linear systems with uncertain switching probabilities the uncertainties under consideration are assumed to be norm bounded in the system matrices and to be elementwise bounded in the mode transition rate matrix respectively first a criterion based on linear matrix inequalities is provided for testing the h sub spl infin filtering level of a filter over all the admissible uncertainties then a sufficient condition for the existence of the fixed order robust h sub spl infin filters is established in terms of the solvability of a set of linear matrix inequalities with equality constraints to determine the filter a globally convergent algorithm involving convex optimization is suggested finally a numerical example is used to illustrate that the developed theory is more effective than the existing results

brief paper stabilization of markovian jump linear system over networks with random communication delay this paper is concerned with the stabilization problem for a networked control system with markovian characterization we consider the case that the random communication delays exist both in the system state and in the mode signal which are modeled as a markov chain the resulting closed loop system is modeled as a markovian jump linear system with two jumping parameters and a necessary and sufficient condition on the existence of stabilizing controllers is established an iterative linear matrix inequality lmi approach is employed to calculate a mode dependent solution finally a numerical example is given to illustrate the effectiveness of the proposed design method

brief on robust stabilization of markovian jump systems with uncertain switching probabilities this brief paper is concerned with the robust stabilization problem for a class of markovian jump linear systems with uncertain switching probabilities the uncertain markovian jump system under consideration involves parameter uncertainties both in the system matrices and in the mode transition rate matrix first a new criterion for testing the robust stability of such systems is established in terms of linear matrix inequalities then a sufficient condition is proposed for the design of robust state feedback controllers a globally convergent algorithm involving convex optimization is also presented to help construct such controllers effectively finally a numerical simulation is used to illustrate the developed theory

improved batch fuzzy learning vector quantization for image compression in this paper we develop a batch fuzzy learning vector quantization algorithm that attempts to solve certain problems related to the implementation of fuzzy clustering in image compression the algorithm s structure encompasses two basic components first a modified objective function of the fuzzy c means method is reformulated and then is minimized by means of an iterative gradient descent procedure second the overall training procedure is equipped with a systematic strategy for the transition from fuzzy mode where each training vector is assigned to more than one codebook vectors to crisp mode where each training vector is assigned to only one codebook vector the algorithm is fast and easy to implement finally the simulation results show that the method is efficient and appears to be insensitive to the selection of the fuzziness parameter

robust fuzzy design for nonlinear uncertain stochastic systems via sliding mode control this paper deals with the sliding mode control smc problem for nonlinear stochastic time delay systems by means of fuzzy approach the takagi sugeno t s fuzzy stochastic time delay model with parametric uncertainties and unknown nonlinearities is presented a sufficient condition for the exponential stability in mean square of the sliding motion is also derived moreover it is shown that when the linear matrix inequalities lmis with equality constraint are feasible the designs of both sliding surface and sliding mode controller can be easily obtained via convex optimization a simulation example illustrating the proposed method is given

robust l2 l control of uncertain differential linear repetitive processes abstract for two dimensional 2 d systems information propagates in two independent directions 2 d systems are known to have both system theoretical and applications interest and the so called linear repetitive processes lrps are a distinct class of 2 d discrete linear systems this paper is concerned with the problem of l 2 l energy to peak control for uncertain differential lrps where the parameter uncertainties are assumed to be norm bounded for an unstable lrp our attention is focused on the design of an l 2 l static state feedback controller and an l 2 l dynamic output feedback controller both of which guarantee the corresponding closed loop lrps to be stable along the pass and have a prescribed l 2 l performance sufficient conditions for the existence of such l 2 l controllers are proposed in terms of linear matrix inequalities lmis the desired l 2 l dynamic output feedback controller can be found by solving a convex optimization problem a numerical example is provided to demonstrate the effectiveness of the proposed controller design procedures

an lmi based algorithm for designing suboptimal static cal h_2 cal h_ infty output feedback controllers we consider the problem of designing a suboptimal cal h 2 cal h infty feedback control law for a linear time invariant control system when a complete set of state variables is not available this problem can be necessarily restated as a nonconvex optimization problem with a bilinear multiobjective functional under suitably chosen linear matrix inequality lmi constraints to solve such a problem we propose an lmi based procedure which is a sequential linearization programming approach the properties and the convergence of the algorithm are discussed in detail finally several numerical examples for static cal h 2 cal h infty output feedback problems demonstrate the applicability of the considered algorithm and also verify the theoretical results numerically

prescribing q curvature problem on sn let pn be the n th order paneitz operator on sn n 3 we consider the following prescribing q curvature problem on sn pnu n 1 q x enuon sn where q is a smooth positive function on sn satisfying the following non degeneracy condition q 2 q 2 0 let g sn rn 1 be defined by g x q x q x we show that if q 0 is non degenerate and deg g g sn 0 then the above equation has a solution when n is even this has been established in our earlier work j wei x xu on conformal deformation of metrics on sn j funct anal 157 1998 292 325 when n is odd pn becomes a pseudo differential operator here we develop a unified approach to treat both even and odd cases the key idea is to write it as an integral equation and use liapunov schmidt reduction method

model reduction of a class of discrete time nonlinear systems abstract this paper considers the problem of model reduction of a class of discrete time systems subject to lipschitzian nonlinearities it is shown that under some conditions the nonlinear system can be either approximated by a discrete time linear time invariant system or a nonlinear system of reduced order the computation of the matrices of the reduced order system is carried out through the solutions of a set of linear matrix inequalities the proposed design is approved by the simulation of reduced order dynamics of a mass spring system subject to a nonlinear friction and a linear electric circuit with uncertain parameters

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

on the prescribed q curvature problem on sn in this paper we prescribe a fourth order curvature the q curvature on the standard n sphere n 5 under flatness condition of order n 4 n near each critical point of the prescribed q curvature function we characterize the critical points at infinity of the associated variational problem and we prove new existence results through euler hopf formulae type our argument gives an upper bound on the morse index of the obtained solutions we also give a lower bound on the number of conformal metrics having the same q curvature

on the perturbed q curvature problem on s4 abstract let g 0 denote the standard metric on s 4 and p g 0 g 0 2 2 g 0 denote the corresponding paneitz operator in this work we study the following fourth order elliptic problem with exponential nonlinearity p g 0 u 6 2 q x e 4 u on s 4 here q is a prescribed smooth function on s 4 which is assumed to be a perturbation of a constant we prove existence results to the above problem under assumptions only on the shape of q near its critical points these are more general than the non degeneracy conditions assumed so far we also show local uniqueness and exact multiplicity results for this problem the main tool used is the lyapunov schmidt reduction

balanced truncation for discrete time markov jump linear systems this technical note investigates the model reduction problem for mean square stable discrete time markov jump linear systems for this class of systems a balanced truncation algorithm is developed the reduced order model is suboptimal however the approximation error which is captured by means of the stochastic gain is bounded from above by twice the sum of singular numbers associated to the truncated states of each mode such a result allows rigorous simplification of the dynamics of each mode in an independent manner with respect to a metric which is relevant from a robust control point of view

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

mixed h sub 2 h sub spl infin control of discrete time markovian jump linear systems in this paper we consider the mixed h sub 2 h sub spl infin control problem for the class of discrete time linear systems with parameters subject to markovian jump linear systems it is assumed that both the state variable and the jump variable are available to the controller the transition probability matrix may not be exactly known but belongs to an appropriate convex set for this controlled discrete time markovian jump linear system the problem of interest can be stated in the following way find a robust with respect to the uncertainty on the transition markov probability matrix mean square stabilizing state and jump feedback controller that minimizes an upper bound for the h sub spl infin norm under the restriction that the h sub spl infin norm is less than a prespecified value spl delta the problem of the determination of the smallest h sub spl infin norm is also addressed we present an approximate version of these problems via linear matrix inequality optimization

existence of conformal metrics with constant q curvature given a compact four dimensional manifold we prove existence of conformal metrics with constant q curvature under generic assumptions the problem amounts to solving a fourth order nonlinear elliptic equation with variational structure since the corresponding euler functional is in general unbounded from above and from below we employ topological methods and min max schemes jointly with the compactness result of 35

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

group representations arising from lorentz conformal geometry abstract it is shown that there exist conformally covariant differential operators d 2 l k of all even orders 2 l on differential forms of all orders k in the double cover n of the n dimensional compactified minkowski space n these act as intertwining differential operators for natural representations of o 2 n the conformal group of n for even n the resulting decompositions of differential form representations of o 2 n the orthochronous conformal group produce infinite families of unitary representations the most interesting of which are carried by positive mass squared positive frequency quotients for 2l n 2k physically these generalize unitary representations of the conformal group associated with the modified wave operator d 2 0 n 2 2 2 and the maxwell operator on vector potentials d 2 n 2 2 d all the representation spaces produced unitary and nonunitary may be viewed as infinite systems of harmonic oscillators as a by product of the spectral resolution of the d 2 l k one gets some striking wave propagative properties for all of the equations d 2 l k 0 including huygens principle in the curved spacetime n the operators d 2 l k have not been seen before except in the special cases k 0 or n and k n 2 2 l 1 the maxwell operator thus much new information is obtained even in the physical case n 4

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

sliding mode control of mimo markovian jump systems this paper addresses the sliding mode control problem for uncertain mimo linear markovian jump systems firstly by using the linear matrix inequality approach sufficient conditions are proposed to guarantee the stochastically asymptotical stability of the system on the sliding surfaces secondly an equivalent control based sliding mode control is proposed such that the closed loop system can be driven onto the desired sliding surfaces in a finite time finally combining with multi step state transition probability the reaching and sliding probabilities are derived for situations where the control force may not be strong enough to ensure the fully asymptotical stability simulation results are presented to illustrate the effectiveness of the proposed design method

convergence of the q curvature flow on s4 in this paper we study the q curvature flow on the standard sphere s4 we show that the flow converges exponentially for all initial data

h model reduction of markovian jump linear systems abstract in this paper the h model reduction problem for linear systems that possess randomly jumping parameters is studied the development includes both the continuous and discrete cases it is shown that the reduced order models exist if a set of matrix inequalities is feasible an effective iterative algorithm involving linear matrix inequalities is suggested to solve the matrix inequalities characterizing the model reduction solutions using the numerical solutions of the matrix inequalities the reduced order models can be obtained an example is given to illustrate the proposed model reduction method

iterative solutions of coupled discrete markovian jump lyapunov equations in this paper two iterative methods are given to solve coupled discrete markovian jump lyapunov equations arising from the stability analysis of discrete time markovian jump systems when such equations have unique positive definite solutions a sufficient condition for the convergence of iterates for one iterative method is derived when such equations only have unique solutions but not necessarily positive definite the above iterative method may fail and an alterative iterative method is provided by using the solutions of discrete time lyapunov equations as its intermediate steps a sufficient condition is also presented for the convergence of the iterates

technical communique robust h2 control of continuous time markov jump linear systems this paper is concerned with the problem of designing robust h 2 state feedback controllers for continuous time markov jump linear systems subject to polytopic type parameter uncertainty based on the parameter dependent lyapunov function approach a new method for designing robust h 2 controllers is presented in terms of solutions to a set of linear matrix inequalities a numerical example is given to illustrate the effectiveness of the proposed design method

controllability stabilizability and continuous time markovian jump linear quadratic control consideration is given to the control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes the relationship between appropriately defined controllability stabilizability properties and the solution of the infinite time jump linear quadratic jlq optimal control problems is also examined although the solution of the continuous time markov jlq problem with finite or infinite time horizons is known only sufficient conditions for the existence of finite cost constant stabilizing controls for the infinite time problem appear in the literature in this paper necessary and sufficient conditions are established these conditions are based on new definitions of controllability observability stabilizability and detectability that are appropriate for continuous time markovian jump linear systems these definitions play the same role for the jlq problem as the deterministic properties do for the linear quadratic regulator lqr problem

h model reduction for discrete time delay systems delay independent and dependent approaches this paper investigates the problem of h model reduction for linear discrete time state delay systems for a given stable system our attention is focused on the construction of reduced order models which guarantee the corresponding error system to be asymptotically stable and have a prescribed h error performance both delay independent and dependent approaches are developed with sufficient conditions obtained for the existence of admissible reduced order solutions since these obtained conditions are not expressed as strict linear matrix inequalities lmis the cone complementary linearization method is exploited to cast them into sequential minimization problems subject to lmi constraints which can be readily solved in standard numerical software in addition the development of reduced order models with special structures such as delay free models and zeroth order models is also addressed the approximation methods presented in this paper can be further extended to cope with systems with unc

robust decentralized stabilization of markovian jump large scale systems a neighboring mode dependent control approach abstract this paper is concerned with the decentralized stabilization problem for a class of uncertain large scale systems with markovian jump parameters the controllers use local subsystem states and neighboring mode information to generate local control inputs a sufficient condition involving rank constrained linear matrix inequalities is proposed for the design of such controllers a numerical example is given to illustrate the developed theory

the principal eigenvalue of a conformally invariant differential operator with an application to semilinear elliptic pde in this paper we give conditions under which the paneitz operator a conformally invariant fourth order differential operator is positive we also prove a sharp estimate for a related conformal invariant these results are combined to study the existence of a semilinear pde with critical sobolev growth

ℋ model reduction for linear parameter varying systems with distributed delay this paper is concerned with the model reduction for linear parameter varying lpv systems with both discrete and distributed delays for a given stable system our attention is focused on the construction of reduced order models which approximate the original system well in an norm sense first a sufficient condition is proposed for the asymptotic stability with an performance of the error system by using the parameter dependent lyapunov functional method then the decoupling technique is applied such that there does not exist any product term between the lyapunov matrices and the system matrices in the parametrised linear matrix inequality plmi constraints thus a new sufficient condition is obtained based on the new condition two different approaches are developed to solve the model reduction problem one is the convex linearisation approach and the other is the projection approach finally a numerical example is provided to demonstrate the effectiveness of the proposed design method

robust h2 control for discrete time markovian jump linear systems this paper deals with the robust h2 control of discrete time markovian jump linear systems it is assumed that both the state and jump variables are available to the controller uncertainties satisfying some norm bounded conditions are considered on the parameters of the system an upper bound for the h2 control problem is derived in terms of a linear matrix inequality lmi optimization problem for the case in which there are no uncertainties we show that the convex formulation is equivalent to the existence of the mean square stabilizing solution for the set of coupled algebraic riccati equations arising on the quadratic optimal control problem of discrete time markovian jump linear systems therefore for the case with no uncertainties the convex formulation considered in this paper imposes no extra conditions than those in the usual dynamic programming approach finally some numerical examples are presented to illustrate the technique

stability results for discrete time linear systems with markovian jumping parameters stability results for discrete time linear systems subject to random jumps in the parameters are obtained first necessary and sufficient conditions for mean square stability mss including the case in which the system is driven by an independent wide sense stationary random sequence are derived it is shown that mss is equivalent to the spectral radius of an augmented matrix being less than one or to the existence of a solution of a certain lyapunov equation in addition it is shown that the lyapunov equation can be derived in four equivalent forms and as a by product each one gives risc to an easier to check sufficient condition these results give inter alia a unified and rather complete picture for mss of markovian jump linear systems next we derive sufficient conditions for almost sure stability of the noiseless ease finally we conclude by presenting an illustrative application to adaptive filtering where a mild condition for almost sure convergence is provided

jump linear quadratic gaussian control in continuous time the optimal quadratic control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes is addressed the systems are also subject to gaussian input and measurement noise the optimal solution for the jump linear quadratic gaussian jlqc problem is given this solution is based on a separation theorem the optimal state estimator is sample path dependent if the plant parameters are constant in each value of the underlying jumping process then the controller portion of the compensator converges to a time invariant control law however the filter portion of the optimal infinite time horizon jlqc compensator is not time invariant thus a suboptimal filter which does converge to a steady state solution under certain conditions is derived and a time invariant compensator is obtained

a new method for stabilization of networked control systems with random delays we consider the stabilization problem for a class of networked control systems in the discrete time domain with random delays the sensor to controller and controller to actuator delays are modeled as two markov chains and the resulting closed loop systems are jump linear systems with two modes the necessary and sufficient conditions on the existence of stabilizing controllers are established it is shown that state feedback gains are mode dependent an iterative linear matrix inequality lmi approach is employed to calculate the state feedback gains

the xy centring algorithm for the dual lmi problem a new approach to fixed order control design many fixed order suboptimal control problems with stability performance and robustness specifications can be reduced to a search for a matrix x 0 satisfying a linear matrix inequality lmi while x 1 satisfies another lmi this paper defines a certain class of these problems we shall call the dual lmi problem and a computational algorithm to solve our dual lmi problem is given properties and limitations of the algorithm are discussed in comparison with the existing algorithm the min max algorithm an extension to optimal control problems is provided numerical examples for the fixed order stabilization problem and the static output feedback linear quadratic optimal control problem demonstrate the applicability of the proposed algorithm

convergence of gradient based iterative solution of coupled markovian jump lyapunov equations the solution of coupled discrete time markovian jump lyapunov matrix equations cdmjlmes is important in stability analysis and controller design for markovian jump linear systems this paper presents a simple and effective iterative method to produce numerical solutions to this class of matrix equations the gradient based algorithm is developed from an optimization point of view a necessary and sufficient condition guaranteeing the convergence of the algorithm is established this condition shows that the algorithm always converges provided the cdmjlmes have unique solutions which is evidently different from the existing results that converge conditionally a simple sufficient condition which is easy to test is also provided the optimal step size in the algorithm such that the convergence rate of the algorithm is maximized is given explicitly it turns out that an upper bound of the convergence rate is bounded by a function of the condition number of the augmented coefficient matrix of the cdmjlmes some parameters are introduced to the algorithm that will potentially reduce the condition number and thus increase the convergence rate of the algorithm a numerical example is used to illustrate the efficiency of the proposed approach

the mean square stability probability of h_ infty control of continuous markovian jump systems in this technical note h infty control problem of continuous markovian jump systems is investigated a linear feedback control scheme combined with a state observer design is proposed in the form of linear matrix inequalities which can ensure the systems mean square stability with h infty performance then a multi step state transition conditional probability function is introduced for the continuous markovian process which is used to define the system s mean square stability probability furthermore the formulas for calculating the mean square stability probability are derived for situations where the control force may not be strong enough to ensure the full stability simulation results are presented to show the effectiveness of the theoretical results

computational complexity of a problem arising in fixed order output feedback design this paper is concerned with a matrix inequality problem which arises in fixed order output feedback control design this problem involves finding two symmetric and positive definitive matrices x and y such that each satisfies a linear matrix inequality and that xy i it is well known that many control problems such as fixed order output feedback stabilization h control guaranteed h2 control and mixed h2 h control can all be converted into the matrix inequality problem above including static output feedback problems as a special case we show however that this matrix inequality problem is np hard

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

existence and interrelation between set and fuzzy differential equations abstract an existence result is proved for set differential equation when the function involved is upper semicontinuous and the result is very general then the connection between the solutions of fuzzy differential equation and the set differential equation that is generated from it is studied

brief paper h filtering for 2d markovian jump systems this paper is concerned with the problem of h filtering for 2d discrete markovian jump systems the mathematical model of 2d jump systems is established upon the well known roesser model our attention is focused on the design of a full order filter which guarantees the filtering error system to be mean square asymptotically stable and has a prescribed h disturbance attenuation performance sufficient conditions for the existence of a desired filter are established in terms of linear matrix inequalities lmis and the corresponding filter design is cast into a convex optimization problem which can be efficiently solved by using commercially available numerical software a numerical example is provided to illustrate the effectiveness of the proposed design method

remarks on prescribing gauss curvature we study the nonlinear partial differential equation for the problem of prescribing gauss curvature k on s 2 we give an example of a rotationally symmetric k for which the kazdan warner obstruction is satisfied but the equation has no rotationally symmetric solution on the other hand we give a simple sufficient condition for solvability of the equation when k is rotationally symmetric finally we give a sufficient condition for solvability when k is not necessarily rotationally symmetric

brief continuous time state feedback h2 control of markovian jump linear systems via convex analysis continuous time h 2 control problem for the class of linear systems with markovian jumps mjls using convex analysis is considered in this paper the definition of the h 2 norm for continuous time mjls is presented and related to the appropriate observability and controllability gramians a convex programming formulation for the h 2 control problem of mjls is developed that enables us to tackle the optimization problem of mjls under the assumption that the transition rate matrix p p i j for the markov chain may not be exactly known but belongs to an appropriate convex set an equivalence between the convex formulation when p is exactly known and the usual dynamic programming approach of quadratic optimal control of mjls is established it is shown that there exists a solution for the convex programming problem if and only if there exists the mean square stabilizing solution for a set of coupled algebraic riccati equations these results are compared with other related works in the current literature

full informationh control for discrete time infinite markov jump parameter systems abstract in this paper we consider the full information discrete timeh control problem for the class of linear systems with markovian jumping parameters the state space of the markov chain is assumed to take values in a countably infinite set full information here means that the controller has access to both the state variables and jump variables a necessary and sufficient condition for the existence of a feedback controller that makes the l 2 induced norm of the system less than a prespecified bound is obtained this condition is written in terms of a set of infinite coupled algebraic riccati equations

output feedback control of markov jump linear systems in continuous time this paper addresses the dynamic output feedback control problem of continuous time markovian jump linear systems the fundamental point in the analysis is an lmi characterization comprising all dynamical compensators that stabilize the closed loop system in the mean square sense the h sub 2 and h sub spl infin norm control problems are studied and the h sub 2 and h sub spl infin filtering problems are solved as a by product

robust h sub spl infin control of uncertain markovian jump systems with time delay this paper is concerned with the robust stochastic stabilizability and robust h sub spl infin disturbance attenuation for a class of uncertain linear systems with time delay and randomly jumping parameters the transition of the jumping parameters is governed by a finite state markov process sufficient conditions on the existence of a robust stochastic stabilizing and spl gamma suboptimal h sub spl infin state feedback controller are presented using the lyapunov functional approach it is shown that a robust stochastically stabilizing h sub spl infin state feedback controller can be constructed through the numerical solution of a set of coupled linear matrix inequalities