discretization and event triggered digital output feedback control of lpv systems abstract this paper investigates the problem of discretization and digital output feedback control design for continuous time linear parameter varying lpv systems subject to a time varying networked induced delay the proposed discretization procedure converts a continuous time lpv system into an equivalent discrete time lpv system based on an extension of the taylor series expansion and using an event based sampling the scheduling parameters are continuously measured and modeled as piecewise constant a new transmission of the measured output to the controller is triggered by significant changes in the parameters yielding time varying transmission intervals the obtained discretized model has matrices with polynomial dependence on the time varying parameters and an additive norm bounded term representing the discretization residual error a two step strategy based on linear matrix inequality conditions is then proposed to synthesize a digital static scheduled output feedback control law that stabilizes both the discretized and the lpv model the conditions can also be used to provide robust i e independent of the scheduling parameter static output feedback controllers the viability of the proposed design method is illustrated through numerical examples

a new class of multilevel decomposition algorithms for non monotone problems based on the quasidifferentiability concept abstract a convex multilevel decomposition approach is proposed for the solution of static analysis problems involving non monotone possibly multivalued laws the theory is developed here for model problems of structures having non monotone interface or boundary conditions first we decompose appropriately the non monotone laws writing them as a difference of monotone constituents in the general case this is related to the quasidifferentiability concept this permits us to obtain a system of convex variational inequalities the solution s of which describe the position s of static equilibrium of the considered structure then the problems are formulated as min min problems for appropriately defined lagrangian functions convex optimization algorithms of various complexity are used in a multilevel scheme for the numerical solution of the considered structural analysis problem numerical results concerning the calculation of an elastic contact problem and an elastic stamp problem illustrate the theory

open loop discretization methods for control systems design in this paper a unified theory is presented which addresses essential aspects of the open loop discretization procedure initially factors which affect the generation and propagation of discretization errors are identified by analytical heuristic and experimental arguments following this a discretization algorithm is presented which takes these factors into account the fundamental idea of the discretization method is the replacement of the analogue integrators of the prototype continuous time system by discrete time approximations this is done in such a way as to optimize a given cost function with respect to a given input unlike many discretization schemes presented in the literature the procedure developed attempts to first determine the magnitude of the expected discretization errors and then discretize with an appropriate complexity giving the designer control over the order of the discrete time system this appears to be an effective means of designing filters with low complexity which sti

asymptotic closure condition and fenchel duality for dc optimization problems in locally convex spaces a b s t r a c t we consider the dc optimization problem p inf x f1 x f2 x g1 ax g2 ax where f1 f2 g1 and g2 are proper convex functions defined on locally convex hausdorff topological vector spaces x and y respectively and a is a linear continuous operator from x to y adopting the standard convexification technique a fenchel dual problem of p is given by using properties of the epigraph of conjugate functions some sufficient and necessary conditions for the fenchel duality and for the stable fenchel duality of p are provided

sharkovsky s program for the classification of triangular maps is almost completed abstract for a continuous map of the interval there are more than 50 conditions characterizing zero topological entropy some are applicable to the class of triangular maps x y f x g x y of the square but only a few of them are equivalent in this more general setting in 1989 a n sharkovsky posed the problem of proving or disproving all possible implications between them during last 20 years 32 conditions were considered and most of the work was done only 45 relations out of 992 remained not clear in this paper we give a survey of known results provide two new examples disproving another 26 possible implications and spell out the remaining 19 open problems all but one concern distributional chaos

robust stabilisation control for discrete time networked control systems we consider the analysis and synthesis of discrete time networked control systems ncss where the plant has additive uncertainty and the controller is updated with the sensor information at stochastic time intervals it is shown that the problem is linked to h control of linear discrete time stochastic systems and a new small gain theorem is established based on this result sufficient conditions are given for ensuring mean square stability of the ncs and the genetic algorithm is utilised to design the controller of the ncs based on a linear matrix inequality technique an illustrative example is given to demonstrate the effectiveness of our proposed method

unilateral contact of two solids subject to large deformations formulation and existence results the approach that is currently the most widely used for formulating the unilateral frictionless contact between two deformable solids involves two key elements the orthogonal projection of one potential contact material surface on another and the action reaction principle of contact forces in the present paper we investigate this approach in the general case of large deformations and study unilateral frictionless boundary value problems of elasticity formulated by means of it the conditions for the orthogonal projection to be unique or further a c 1 diffeomorphism are established an extended action reaction principle is explicitly given for the forces acting on a pair of potential contact surfaces assuming the solids under consideration can be described by polyconvex hyperelastic laws we formulate the associated unilateral frictionless boundary value problem as a minimum principle and show the existence of a solution to it

duality for multiobjective optimization problems with convex objective functions and d c constraints in this paper we provide a duality theory for multiobjective optimization problems with convex objective functions and finitely many d c constraints in order to do this we study first the duality for a scalar convex optimization problem with inequality constraints defined by extended real valued convex functions for a family of multiobjective problems associated to the initial one we determine then by means of the scalar duality results their multiobjective dual problems finally we consider as a special case the duality for the convex multiobjective optimization problem with convex constraints

semidefinite representation of convex sets let s x in mathbb r n g 1 x geq 0 ldots g m x geq 0 be a semialgebraic set defined by multivariate polynomials g i x assume s is convex compact and has nonempty interior let s i x in mathbb r n g i x geq 0 and s resp s i be the boundary of s resp s i this paper as does the subject of semidefinite programming sdp concerns linear matrix inequalities lmis the set s is said to have an lmi representation if it equals the set of solutions to some lmi and it is known that some convex s may not be lmi representable helton and vinnikov in commun pure appl math 60 5 654 674 2007 a question arising from nesterov and nemirovski siam studies in applied mathematics society for industrial and applied mathematics siam philadelphia 1994 see helton and vinnikov in commun pure appl math 60 5 654 674 2007 and nemirovski in plenary lecture international congress of mathematicians icm madrid spain 2006 is given a subset s of mathbb r n does there exist an lmi representable set in some higher dimensional space mathbb r n n whose projection down onto mathbb r n equals s such s is called semidefinite representable or sdp representable this paper addresses the sdp representability problem the following are the main contributions of this paper i assume g i x are all concave on s if the positive definite lagrange hessian condition holds i e the hessian of the lagrange function for optimization problem of minimizing any nonzero linear function l t x on s is positive definite at the minimizer then s is sdp representable ii if each g i x is either sos concave 2 g i x w x t w x for some possibly nonsquare matrix polynomial w x or strictly quasi concave on s then s is sdp representable iii if each s i is either sos convex or poscurv convex s i is compact convex whose boundary has positive curvature and is nonsingular i e g i x 0 on s i s then s is sdp representable this also holds for s i for which s i s extends smoothly to the boundary of a poscurv convex set containing s iv we give the complexity of schmudgen and putinar s matrix positivstellensatz which are critical to the proofs of i iii

design of sampled data critical control systems based on the fast discretization technique the fast discretization is known as an approximate but efficient technique for design and analysis of sampled data systems in this paper we propose a fast discretization based design for sampled data critical control systems supposing a tracking problem or a slow changing disturbance rejection problem we assume that an exogenous input is a persistent and or transient input with bound on the rate of change it is shown that the critical constraint for such exogenous inputs can be given in the form of the inequality constraint on the unit step response the design parameters are determined by a numerical search method subject to this constraint however instead of evaluating it strictly we check the corresponding constraint which is obtained from the fast discretized system although this approach is approximate it provides an efficient numerical procedure for a computer aided design to show the validity of the proposed method an example of a multi objective critical control system design is presented

on li yorke pairs the li yorke definition of chaos proved its value for interval maps in this paper it is considered in the setting of general topological dynamics we adopt two opposite points of view on the one hand sufficient conditions for li yorke chaos in a topological dynamical system are given we solve a long standing open question by proving that positive entropy implies li yorke chaos on the other hand properties of dynamical systems without li yorke pairs are investigated in addition to having entropy 0 they are minimal when transitive and the property is stable under factor maps arbitrary products and inverse limits finally it is proved that minimal systems without li yorke pairs are disjoint from scattering systems

convexity in semialgebraic geometry and polynomial optimization we review several and provide new results on the theory of moments sums of squares and basic semialgebraic sets when convexity is present in particular we show that under convexity the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence a highly desirable feature as convex problems are in principle easier to solve in addition if a basic semialgebraic set mathbf k is convex but its defining polynomials are not we provide two algebraic certificates of convexity which can be checked numerically the second is simpler and holds if a sufficient and almost necessary condition is satisfied it also provides a new condition for mathbf k to have semidefinite representation for this we use and extend some of the recent results from the author and helton and nie math program to appear finally we show that when restricting to a certain class of convex polynomials the celebrated jensen s inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

a new variational principle and duality for periodic solutions of hamilton s equations we are dealing with a classical boundary value problem concerning periodic solutions of hamiltonian equations p f k f 4th p f i f h x t p t 1 where h r x r x r r is a given function called the hamiltonian in mechanics 1 and also in mathematics 171 with a given hamiltonian the lagrangian l defined by the legendre transform l 4 x a sup k p wc x pi d 2

discretisation and control of polytopic systems with uncertain sampling rates and network induced delays this paper investigates the problems of uncertain sampling rate discretisation and the networked control of uncertain time invariant continuous time linear systems in polytopic domains the sampling period is assumed to be unknown but belonging to a given interval to avoid the difficulty of dealing with the exponential of uncertain matrices a discrete time model is obtained by applying a taylor series expansion of degree l to the original system the resulting discrete time model is composed of homogeneous polynomial matrices with parameters lying in the cartesian product of simplexes called a multi simplex plus an additive norm bounded term representing the discretisation residual error the original continuous time system is controlled through a communication network that introduces a time delay in the process linear matrix inequality relaxations that include a scalar parameter search are proposed for the design of a digital robust state feedback controller that guarantees the closed loop stability

a triangular map of type 2 with positive topological entropy on a minimal set we provide a class of triangular maps of the square x y f x gx y of type 2 i e such that the periods of periodic points are the powers of 2 which has a minimal set supporting positive topological entropy this improves the famous example by s kolyada from 1992 and contributes to the solution of an old problem by a n sharkovsky

on minimizing difference of a sos convex polynomial and a support function over a sos concave matrix polynomial constraint in this paper we establish tractable sum of squares characterizations of the containment of a convex set defined by a sos concave matrix inequality in a non convex set defined by difference of a sos convex polynomial and a support function with slater s condition using our set containment characterization we derive a zero duality gap result for a dc optimization problem with a sos convex polynomial and a support function its sum of squares polynomial relaxation dual problem the semidefinite representation of this dual problem and the dual problem of the semidefinite programs also we present the relations of their solutions finally through a simple numerical example we illustrate our results particularly in this example we find the optimal solution of the original problem by calculating the optimal solution of its associated semidefinite problem

critical point approximation through exact regularization construction d algorithmes iteratifs pour la resolution des problemes extremaux non differentiables regularisation d une fonctionnelle application aux problemes differentiels semi elliptiques en physique des plasmas et en mecanique des fluides

farkas type results for fractional programming problems abstract considering a constrained fractional programming problem within the present paper we present some necessary and sufficient conditions which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant the desired results are obtained using the fenchel lagrange duality approach applied to an optimization problem with convex or difference of convex dc objective functions and finitely many convex constraints these are obtained from the initial fractional programming problem using an idea due to dinkelbach we also show that our general results encompass as special cases some recently obtained farkas type results

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

stability of model based networked control systems with time varying transmission times in model based networked control systems mb ncss an explicit model of the plant is used to produce an estimate of the plant state behavior between transmission times in this paper the stability of mb ncss is studied when the controller actuator is updated with the sensor information at nonconstant time intervals networked control systems with transmission times that are varying either within a time interval or are driven by a stochastic process with identically independently distributed and markov chain driven transmission times are studied sufficient conditions for lyapunov stability are derived for stochastically modeled transmission times almost sure stability and mean square sufficient conditions for stability are introduced

convex multilevel decomposition algorithms for non monotone problems a convex multilevel decomposition algorithm is proposed in this paper for the solution of static analysis problems involving non monotone possibly multivalued laws the theory is developed here for a model structure with non monotone interface or boundary conditions first the non monotone laws are written in the form of a difference of two monotone functions under this decomposition the non linear elastostatic analysis problem is equivalent to a system of convex variational inequalities and to non convex min min problems for appropriately defined lagrangian functions the solution s of each one of the aforementioned problems describe the position s of static equilibrium of the considered structure in this paper a multilevel optimization scheme due to auchmuty 1 is used for the numerical solution of the problem the most interesting feature of this method from the computational mechanics standpoint is the fact that each one of the subproblems involved in the multilevel algorithm is a convex optimization problem or in terms of mechanics an appropriately modified monotone unilateral problem thus existing algorithms and software can be used for the numerical solution with minor modifications numerical results concerning the calculation of elastic and rigid stamp problems and of material inclusion problems with delamination and non monotone stick slip frictional effects illustrate the theory

pareto optimality and walrasian equilibria abstract the paper concerns the study of a class of convex constrained multiobjective optimization problems from the viewpoint of the existence issues the main feature of the presented approach is that the classical qualification condition requiring the existence of interior points in the effective domains of functions under consideration does not hold a variant of duality theory for multiobjective optimization problems based on the fenchel theorem is formulated next by using very recent results on the walrasian general equilibrium model of economy obtained in naniewicz z naniewicz pseudo monotonicity and economic equilibrium problem in reflexive banach space math oper res 32 2007 436 466 the conditions ensuring the existence of pareto optimal solutions for the class of multiobjective optimization problems are established the concept of the proper efficiency is used as the solution notion finally a new version of the second fundamental theorem of welfare economics is presented

concave duality application to problems dealing with difference of functions in this paper a duality is introduced in a concave sense and its relationship with toland s duality is studied along with several formulas dealing with the conjugates of differences of convex functions

stable and total fenchel duality for convex optimization problems in locally convex spaces we consider the optimization problem p a inf x in x f x g ax where f and g are proper convex functions defined on locally convex hausdorff topological vector spaces x and y respectively and a is a linear operator from x to y by using the properties of the epigraph of the conjugated functions some sufficient and necessary conditions for the strong fenchel duality and the strong converse fenchel duality of p a are provided sufficient and necessary conditions for the stable fenchel duality and for the total fenchel duality are also derived

on open problems concerning distributional chaos for triangular maps abstract we show that in the class t of the triangular maps x y f x g x y of the square there is a map of type 2 with non minimal recurrent points which is not dc3 we also show that every dc1 continuous map of a compact metric space has a trajectory which cannot be weakly approximated by trajectories of compact periodic sets these two results make possible to answer some open questions concerning classification of maps in t with zero topological entropy and contribute to an old problem formulated by a n sharkovsky

h sub spl infin control for networked systems with random communication delays this note is concerned with a new controller design problem for networked systems with random communication delays two kinds of random delays are simultaneously considered i from the controller to the plant and ii from the sensor to the controller via a limited bandwidth communication channel the random delays are modeled as a linear function of the stochastic variable satisfying bernoulli random binary distribution the observer based controller is designed to exponentially stabilize the networked system in the sense of mean square and also achieve the prescribed h sub spl infin disturbance attenuation level the addressed controller design problem is transformed to an auxiliary convex optimization problem which can be solved by a linear matrix inequality lmi approach an illustrative example is provided to show the applicability of the proposed method

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

duality for non convex variational principles abstract this paper treats the construction of dual variational principles for non convex problems using lagrangian methods besides the well known saddle point theory there are dualities involving minimizing the lagrangian in both variables and dualities where one first maximizes and then minimizes the lagrangian in each variable when the primal problem has certain structure an associated canonical lagrangian is defined which leads to the appropriate dual variational principle and to a hamiltonian description the properties of and correspondence between the critical points are analyzed the last five sections are devoted to examples of these dual principles

mpc for lpv systems with bounded parameter variations this article deals with the model predictive control mpc design problem for systems which are linearly dependent on a time varying parameter the main novelty is that motivated by practical issues the bounds on the rate of variation of the parameters are taken into account in the control design moreover a lyapunov function depending multiaffinely on the parameters computed at a set of instants of time and a parameter dependent control gain are used to provide an upper bound to a quadratic performance index the solution is obtained by means of a semidefinite programming algorithm examples illustrate the efficiency of the proposed approach

periodic solutions to hamiltonian inclusions abstract we prove the existence of periodic trajectories of hamiltonian inclusions which reduce to the usual hamiltonian equations in the presence of smoothness this is accomplished via a direct variational principle involving a new action integral in an extended sense

duality in d c programming the case of several d c constraints abstract we introduce a duality scheme for the problem of minimizing the difference of two extended real valued convex functions d c functions under finitely many d c constraints in terms of the legendre fenchel conjugates of the involved convex functions

variational principles for operator equations and initial value problems on decrit et on etudie des principes variationnels pour certaines classes d equations operateur et de problemes aux valeurs initiales

an auxiliary system discretization approach to takagi sugeno fuzzy models abstract this paper proposes a new procedure for discretizing nonlinear systems described by takagi sugeno fuzzy models the discretization procedure consists of obtaining a linear auxiliary system that approximates the takagi sugeno model over a sampling instant by discretizing this auxiliary system a norm bounded uncertain linear discrete time system is found which is capable of representing the fuzzy model this auxiliary system as well as the norm bounded uncertainty is found by solving an optimization problem with linear matrix inequality lmi constraints to illustrate the discretization procedure a constant state observer is synthesized based on simple lmi conditions and then applied to a real nonlinear chua s circuit additionally a state feedback controller based on our discretization approach is synthesized and we obtain larger maximum sampling periods than other tested strategies from the literature

duality in reverse convex optimization a duality theorem for the general problem of minimizing an extended real valued convex function on a locally convex linear space under a reverse convex constraint is considered in the particular case of the distance to a reverse convex subset in a normed linear space we recover as a corollary a duality theorem due to c franchetti and i singer boll un mat ital b 5 17 1980 pp 33 43 similar to the one known for the distance to a convex subset the general theorem also contains the duality principle of toland singer for d c optimization