parameter dependent lmis in robust analysis characterization of homogeneous polynomially parameter dependent solutions via lmi relaxations this note investigates the robust stability of uncertain linear time invariant systems in polytopic domains by means of parameter dependent linear matrix inequality pd lmi conditions exploiting some algebraic properties provided by the uncertainty representation a systematic procedure to construct a family of finite dimensional lmi relaxations is provided the robust stability is assessed by means of the existence of a lyapunov function more specifically a homogeneous polynomially parameter dependent lyapunov hppdl function of arbitrary degree for a given degree if an hppdl solution exists a sequence of relaxations based on real algebraic properties provides sufficient lmi conditions of increasing precision and constant number of decision variables for the existence of an hppdl function which tend to the necessity alternatively if an hppdl solution of degree exists a sequence of relaxations which increases the number of variables and the number of lmis will provide an hppdl solution of larger degree the method proposed can be applied to determine homogeneous parameter dependent matrix solutions to a wide variety of pd lmis by transforming the infinite dimensional lmi problem described in terms of uncertain parameters belonging to the unit simplex in a sequence of finite dimensional lmi conditions which converges to the necessary conditions for the existence of a homogeneous polynomially parameter dependent solution of arbitrary degree illustrative examples show the efficacy of the proposed conditions when compared with other methods from the literature

matrix sum of squares relaxations for robust semi definite programs we consider robust semi definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description on the basis of matrix sum of squares decompositions we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart with a novel matrix version of putinar s sum of squares representation for positive polynomials on compact semi algebraic sets we prove asymptotic exactness of the relaxation family under a suitable constraint qualification if the uncertainty region is a compact polytope we provide a new duality proof for the validity of putinar s constraint qualification with an a priori degree bound on the polynomial certificates finally we point out the consequences of our results for constructing relaxations based on the so called full block s procedure which allows to apply recently developed tests in order to computationally verify the exactness of possibly small sized relaxations

a less conservative lmi condition for the robust stability of discrete time uncertain systems abstract in this paper a less conservative condition for the robust stability of uncertain discrete time linear systems is proposed the uncertain parameters assumed to be time invariant are supposed to belong to convex bounded domains polytope type uncertainty the stability condition is formulated in terms of a set of linear matrix inequalities involving only the vertices of the polytope domain a simple and numerically efficient feasibility test provides a set of lyapunov matrices whose convex combination can be used to assess the stability of any dynamic matrix inside the uncertainty domain examples illustrate the results

on 2 generator subgroups of so 3 we classify all subgroups of so 3 that are generated by two elements each a rotation of finite order about axes separated by an angle that is a rational multiple of 7r in all cases we give a presentation of the subgroup in most cases the subgroup is the free product or the amalgamated free product of cyclic groups or dihedral groups the relations between the generators are all simple consequences of standard facts about rotations by 7r and 7r 2 embedded in the subgroups are explicit free groups on 2 generators as used in the banach tarski paradox

mixed h 2 h control for discrete time systems via convex optimization abstract a mixed h 2 h control problem for discrete time systems is considered where an upper bound on the h 2 norm of a closed loop transfer matrix is minimized subject to an h constraint on another closed loop transfer matrix both state feedback and output feedback cases are considered it is shown that these problems are equivalent to finite dimensional convex programming problems in the state feedback case nearly optimal controllers can be chosen to be static gains in the output feedback case nearly optimal controllers can be chosen to have a structure similar to that of the central single objective h controller in particular the state dimension of nearly optimal output feedback controllers need not exceed the plant dimension

correspondence robust stabilization of uncertain time varying discrete systems and comments on an improved approach for constrained robust model predictive control this paper addresses the robust stabilizability problem of linear discrete time systems with time varying polytopic uncertainties a state feedback control law is proposed based on the construction of a parameter dependent lyapunov function also it points out that the main results of cuzzola geromel and morari automatica 38 2002 1183 1189 are incorrect and presents the corrected version of these results

a subgroup of so 3 r generated by rotations of orders 4 and 8 abstract we prove that a subgroup of the real 3 dimensional special orthogonal group generated by a pair of rotations of respective orders 4 and 8 belonging to a family of such groups considered by radin and sadun in c radin l sadun on 2 generator subgroups of so 3 trans amer math soc 351 1999 114469 114480 has an epimorphic image which is one of psl 2 p psl 2 p 2 pgl 2 p or pgl 2 p 2 depending on the congruence of p mod 16 for all odd primes p by considering its reduction mod p as a linear group

lmi characterization of structural and robust stability abstract this paper introduces several stability conditions for a given class of matrices expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable diagonal and simultaneous stability both characterized by polytopes of matrices are addressed using this approach a method particularly attractive to test a given matrix for d stability is proposed lyapunov parameter dependent functions are built in order to reduce conservativeness of the stability conditions the key idea is to relate hurwitz stability with a positive realness condition

on a convex parameter space method for linear control design of uncertain systems this paper presents a new procedure for continuous and discrete time linear control systems design it consists of the definition of a convex programming problem in the parameter space that when solved provides the feedback gain one of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation yielding solutions to problems that have raised a great deal of interest within the last few years this is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices in this last case necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and to the authors knowledge this is one of the first numerical procedures able to handle and solve this interesting design problem for high order continuous time or discrete time linear models the theory is illustrated by examples

brief paper time varying discrete time linear systems with bounded rates of variation stability analysis and control design this paper investigates the problems of robust stability analysis and state feedback control design for discrete time linear systems with time varying parameters it is assumed that the time varying parameters lie inside a polytopic domain and have known bounds on their rate of variation a convex model is proposed to represent the parameters and their variations as a polytope and linear matrix inequality relaxations that take into account the bounds on the rates of parameter variations are proposed a feasible solution provides a parameter dependent lyapunov function with polynomial dependence on the parameters assuring the robust stability of this class of systems extensions to deal with robust control design as well as gain scheduling by state feedback are also provided in terms of linear matrix inequalities numerical examples illustrate the results

robust spl hscr sub spl infin filtering for discrete time linear systems with uncertain time varying parameters this paper deals with the problem of robust spl hscr sub spl infin filtering for linear discrete time state space models with uncertain time varying parameters the parameters enter affinely into the state space model matrices and their admissible values and variations are assumed to belong to given intervals a method is derived for designing a linear stationary asymptotically stable filter with a prescribed spl hscr sub spl infin performance in spite of large parameter uncertainty the proposed method incorporates information on available bounds on both the admissible values and variation of the uncertain parameters and is based on a lyapunov function with quadratic dependence on the parameters the filter design is given in terms of linear matrix inequalities

brief paper robust stability of time varying polytopic systems via parameter dependent homogeneous lyapunov functions this paper deals with robust stability analysis of linear state space systems affected by time varying uncertainties with bounded variation rate a new class of parameter dependent lyapunov functions is introduced whose main feature is that the dependence on the uncertain parameters and the state variables are both expressed as polynomial homogeneous forms this class of lyapunov functions generalizes those successfully employed in the special cases of unbounded variation rates and time invariant perturbations the main result of the paper is a sufficient condition to determine the sought lyapunov function which amounts to solving an lmi feasibility problem derived via a suitable parameterization of polynomial homogeneous forms moreover lower bounds on the maximum variation rate for which robust stability of the system is preserved are shown to be computable in terms of generalized eigenvalue problems numerical examples are provided to illustrate how the proposed approach compares with other techniques available in the literature

a class of robust stability conditions where linear parameter dependence of the lyapunov function is a necessary condition for arbitrary parameter dependence abstract in previous works we have proposed a robust stability condition for linear time invariant discrete time systems which makes use of a lyapunov function with linear dependence on the uncertain parameters this condition is expressed as a set of linear matrix inequalities lmi where an additional variable is kept common to all lmi these features have enabled the development of successful robust filtering and control algorithms in this short note we investigate possible extensions of this stability condition to handle lyapunov functions with arbitrary parameter dependence while keeping a variable common to all lmi by showing that feasibility of the original condition is indeed necessary for the existence of a family of robust stability conditions where the lyapunov function can have arbitrary dependence on the uncertain parameters we conclude that no such extensions are possible

uniform stabilization of discrete time switched and markovian jump linear systems the uniform stability of discrete time switched linear systems possibly with a strongly connected switching path constraint and the existence of finite path dependent dynamic output feedback controllers uniformly stabilizing such a system are both shown to be characterized by the existence of a finite dimensional feasible system of linear matrix inequalities this characterization is based on the observation that a linear time varying system is uniformly stable only if there exists a finite path dependent quadratic lyapunov function the synthesis of a uniformly stabilizing controller is done without conservatism by solving any feasible system of linear matrix inequalities among an increasing family of systems of linear matrix inequalities the result carries over to the almost sure uniform stabilization of markovian jump linear systems

brief paper stability results for linear parameter varying and switching systems in this paper stabilizability problems for discrete and continuous time linear systems with switching or polytopic uncertainties are dealt with both state feedback and full information controllers are considered possibly with an integral action several cases will be analyzed depending on the control the uncertainty and the presence of uncertainties in the input matrix a complete overview of the stabilizability implications among these different cases will be given

a convex approach to robust stability for linear systems with uncertain scalar parameters in this paper robust stability for linear systems with several uncertain complex and or real scalar parameters is studied a countable family of conditions sufficient for robust stability is given in terms of solvability of some simple linear matrix inequalities lmis these conditions are of increasing precision and it is shown conversely that robust stability implies solvability of these lmis from a certain rank and beyond this result constitutes an extension of the characterization by solvability of lyapunov inequality of the asymptotic stability for usual linear systems it is based on the search of parameter dependent quadratic lyapunov functions polynomial of increasing degree in the parameters

a robust stability problem for discrete time systems subject to an uncertain parameter abstract in this paper a linear discrete time uncertain system depending in norm bounded form on an uncertain real time varying parameter whose magnitude is between the quadratic stability radius and the real stability radius is considered in this case stability of the system depends on the time rate of variation of the parameter by utilizing parameter dependent lyapunov functions one can take the parameter rate of change into account and obtain sufficient conditions for stability which are less restrictive than those imposed by parameter independent lyapunov functions

amalgams blocks weights fusion systems and finite simple groups abstract we associate an iterated amalgam of finite groups to a certain class of fusion systems on finite p groups p a prime in such a way that the p group of the fusion system is a maximal finite p subgroup of the resulting group unique up to conjugacy and furthermore the conjugation action of the resulting usually infinite group on p subgroups induces the original fusion system on the p group 1 in view of earlier work of puig and of broto castellana grodal levi and oliver c broto n castellana j grodal r levi r oliver subgroup families controlling p local finite groups proc london math soc 3 91 2005 325 354 the fusion systems we deal with include all saturated fusion systems the resulting amalgam has free normal subgroups of finite index and we examine the images of the group by its maximal free normal subgroups of finite index these images all contain isomorphic copies of the original p group and are generated by the images of the finite groups used in the amalgamation if there is no non trivial normal p subgroup of the fusion system equivalently if the iterated amalgam constructed has no non trivial normal p subgroup then the generalised fitting subgroup of each of these homomorphic images is a direct product of non abelian simple groups each of order divisible by p and if there is no proper non trivial strongly closed p subgroup in the fusion system then the generalised fitting subgroup of any of these finite groups is characteristically simple we note that the finite dimensional representation theory of this amalgam is almost by construction p locally determined in the case of the fusion system associated to a p block of a finite group we suggest strong links between block theoretic invariants of the above finite epimorphic images of the associated amalgam and of the original block we believe that the results of this paper offer the prospect of at least the beginnings of a structural explanation for some of the current numerical conjectures in block theory e g j l alperin weights for finite groups in proc sympos pure math vol 47 amer math soc providence 1987 pp 369 379 e c dade counting characters in blocks i invent math 109 1 1992 187 210

rational functions with identical measure of maximal entropy abstract we discuss when two rational functions f and g can have the same measure of maximal entropy the polynomial case was completed by beardon levin baker eremenko schmidt steinmetz etc 1980s 1990s and we address the rational case following levin and przytycki 1997 we show f g implies that f and g share an iterate f n g m for some n and m for general f with degree d 3 and for generic f rat d 3 f g implies g f n for some n 1 for generic f rat 2 f g implies that g f n or f f n for some n 1 where f psl 2 c permutes two points in each fiber of f finally we construct examples of f and g with f g such that f n g m for any psl 2 c and m n 1

technical communique on the non conservatism of a novel lmi relaxation for robust analysis of polytopic systems it is known that robust analysis of polytopic systems i e linear system with polytopic state space uncertainty is a difficult problem recently we have proposed for this problem in both continuous time and discrete time cases the use of homogeneous polynomially parameter dependent quadratic lyapunov functions hpd qlfs by introducing a novel lmi relaxation which provides a sufficient condition via a convex optimization without requiring the use of auxiliary multipliers of unknown degree in this paper we show that this condition is not only sufficient but it is also necessary

a new discrete time robust stability condition a new robust stability condition for uncertain discrete time systems with convex polytopic uncertainty is given it enables to check stability using parameter dependent lyapunov functions which are derived from lmi conditions it is shown that this new condition provides better results than the classical quadratic stability besides the use of a parameter dependent lyapunov function this condition exhibits a kind of decoupling between the lyapunov and the system matrices which may be explored for control synthesis purposes a numerical example illustrates the results

brief an improved approach for constrained robust model predictive control in this paper we present a new technique to address constrained robust model predictive control the main advantage of this new approach with respect to other well known techniques is the reduced conservativeness specifically the technique described in this paper can be applied to polytopic uncertain systems and is based on the use of several lyapunov functions each one corresponding to a different vertex of the uncertainty s polytope

robust stability of time varying polytopic systems abstract this paper provides global asymptotic stability conditions for time varying continuous time polytopic systems using a parameter dependent lyapunov function the time varying parameter uncertainty as well as its time derivative are modelled as belonging to polytopic convex sets and their dependence is made explicit to get less conservative results a particular case characterized by the parameter uncertainty satisfying a linear differential equation is analyzed and a simpler version of the aforementioned stability conditions is presented the results are expressed in terms of linear matrix inequalities being thus numerically solvable with no big difficulty the theory is illustrated by the determination of the asymptotic stability region of a mathieu s type equation with an uncertain time varying parameter

parameter dependent lyapunov functions for discrete time systems with time varying parametric uncertainties abstract in this paper we consider discrete time systems with polytopic time varying uncertainty we look for a class of parameter dependent lyapunov functions which are quadratic on the system state and depend in a polytopic way on the uncertain parameter we show that extending the new discrete time stability condition proposed by de oliveira et al systems control lett 36 1999 135 to the case of time varying uncertainty leads to a necessary and sufficient condition for the computation of such a lyapunov function this allows to check asymptotic stability of the system under study the obtained linear matrix inequalities condition can also be used to cope with the control synthesis problem