structural properties of positive linear time invariant difference algebraic equations in this paper we analyze positive difference algebraic equations conditions regarding some of the matrices involved in the solution of this kind of systems are described geometrical conditions are used to characterize positive structural properties

adaptive pole positioning in mimo linear systems by periodic multirate input controllers abstract in this paper the certainty equivalence principle is used to combine the identification method with a control structure derived from the pole placement problem which rely on periodic multirate input controllers the proposed adaptive pole placers contain a sampling mechanism with different sampling period to each system input and rely on a periodically varying controller which suitably modulates the sampled outputs and reference signals of the plant under control such a control strategy allows us to arbitrarily assign the poles of the sampled closed loop system in desired locations and does not make assumptions on the plant other than controllability and observability of the continuous and the sampled system and the knowledge of a set of structural indices namely the locally minimum controllability indices of the continuous time plant an indirect adaptive control scheme is derived which estimates the unknown plant parameters and consequently the controller parameters on line from sequential data of the input and outputs of the plant which are recursively updated within the time limit imposed by a fundamental sampling period t0 using the proposed algorithm the controller determination is based on the transformation of the discrete analogous of the system under control to a phase variable canonical form prior to the application of the control design procedure the solution of the problem can then be obtained by a quite simple utilization of the concept of state similarity transformation known indirect adaptive pole placement schemes usually resort to the computation of dynamic controllers through the solution of a polynomial diophantine equation thus introducing high order exogenous dynamics in the control loop moreover in many cases the solution of the diophantine equation for a desired set of closed loop eigenvalues might yield an unstable controller and the overall adaptive pole placement scheme is unstable with unstable compensators because their outputs are unbounded the proposed control strategy avoids these problems since here gain controllers are essentially needed to be designed moreover persistency of excitation and therefore parameter convergence of the continuous time plant is provided without making any assumption either the richness of the reference signals or on the existence of specific convex sets in which the estimated parameters belong or finally on the coprimeness of the polynomials describing the arma model as in known adaptive pole placement schemes

time varying versus time invariant compensation for rejection of persistent bounded disturbances and robust stabilization it is shown that time varying compensation does not improve the optimal rejection of persistent bounded disturbances this result is obtained by exploiting a key observation that any time varying compensator which yields a given degree of disturbance rejection must do so uniformly over time thereby removing any advantage of time variation this key observation is exploited to show that time varying compensation does not improve the optimal rejection of disturbances regardless of the norm used to measure the disturbances thus absolutely summable finite energy or persistent bounded disturbances may be treated in the same manner it is shown that time varying compensation does not help in the bounded input bounded output robust stabilization of time invariant plants with unstructured uncertainty in doing so it is also shown that the small gain theorem is both necessary and sufficient for the bounded input bounded output stability of certain linear time varying plants subject to unstructured linear time varying perturbations

stochastic stability and robust control for sampled data systems with markovian jump parameters in this paper the problems of stochastic stability and robust control for a class of uncertain sampled data systems are studied the systems consist of random jumping parameters described by finite state semi markov process sufficient conditions for stochastic stability or exponential mean square stability of the systems are presented the conditions for the existence of a sampled data feedback control and a multirate sampled data optimal control for the continuous time uncertain markovian jump systems are also obtained the design procedure for robust multirate sampled data control is formulated as linear matrix inequalities lmis which can be solved efficiently by available software toolboxes finally a numerical example is given to demonstrate the feasibility and effectiveness of the proposed techniques

a necessary and sufficient condition for the existence of positive solutions of singular super linear m point boundary value problems abstract this paper investigates the existence of positive solutions of singular super linear m point boundary value problems of fourth order ordinary differential equations a necessary and sufficient condition for the existence of c 2 0 1 positive solutions as well as c 3 0 1 positive solutions is given by means of the fixed point theorems on cones

brief stabilization of positive systems with first integrals positive systems possessing first integrals are considered these systems frequently occur in applications this paper is devoted to two stabilization problems the first is concerned with the design of feedbacks to stabilize a given level set secondly it is shown that the same feedback allows to globally stabilize an equilibrium point if it is asymptotically stable with respect to initial conditions in its level set two examples are provided and the results are compared with those in the literature

application of material forces to hyperelastostatic fracture mechanics i continuum mechanical setting the concern of this work is a consequent exploitation of the notion of material forces for the application within hyperelastostatic fracture mechanics contrary to physical forces material forces act on the material manifold thus essentially representing the tendency of defects like cracks or inclusions to move relative to the ambient material based on the formulation of the appropriate quasi static balance laws in the material space we aim at a fresh look onto classical aspects of hyperelastostatic fracture mechanics operating throughout within the geometrically nonlinear setting we emphasize on the one hand the duality of the direct and the inverse motion description and on the other hand we re establish the classical path integrals from elementary equilibrium considerations in the material space

linear time varying h 2 optimal control of sampled data systems abstract the goal in this paper is twofold first it is shown that linear time varying control is in general superior to linear time invariant lti control for lti plants with an h 2 criterion an expression for the optimal performance using time varying control is derived and a condition under which there is no advantage in using linear time varying control over lti control is determined second the h 2 optimal control of continuous time systems is studied via dual rate sampled data controllers it is shown that such a problem can be recast via the lifting technique as a standard lti h 2 problem in discrete time a design procedure for the dual rate h 2 optimal controller is proposed

a constructive algorithm for sensitivity optimization of periodic systems in a recent paper a feintuch p p khargonekar and a tannenbaum on the sensitivity minimization problem for linear time varying periodic systems this journal 24 1986 pp 1076 1085 the problem of weighted sensitivity optimization was considered for linear discrete time periodic time varying systems here we present a constructive algorithm for solving this problem

optimum h designs under sampled state measurements abstract we present the complete solution to the h optimal control problem when only sampled values of the state are available for linear time varying systems the optimum controller is characterized in terms of the solution of a particular generalized riccati differential equation with the optimum performance determined by the conjugate point conditions associated with a family of generalized riccati differential equations for the infinite horizon time invariant problem however the optimum controller is characterized in terms of the solution of a particular generalized algebraic riccati equation and the performance is determined in terms of the conjugate point conditions of a single generalized riccati equation defined on the longest sampling interval if the distribution of the sampling times is also taken as part of the general design uniform sampling turns out to be optimal for the infinite horizon case while for the finite horizon problem a nonuniform sampling generally leads to a better performance

h and h 2 optimal controllers for periodic and multirate systems in this paper we present the solutions to the optimal l2 to l2 disturbance rejection problem h as well as to the lqg h2 problem in periodic systems using the lifting technique both problems involve a causality condition on the optimal lti compensator when viewed in the lifted domain the h problem is solved using nehari s theorem whereas in the h2 problem the solution is obtained using the projection theorem exactly the same methods of solution can be applied in the case of multirate sampled data systems finally stability robustness issues in periodic and multirate systems are analyzed

multivariable multirate sampled data systems state space description transfer characteristics and nyquist criterion first a discrete time state space description is derived for multivariable multirate sampled data systems then the multirate impulse modulation mim and the symmetric coordinate multirate impulse modulation smim of a transfer matrix are defined and their relation to the discrete time state space description is clarified based on the above results four stability criteria are derived the contents of the paper will be useful to extend the modern results about continuous time and single rate sampled data control systems to the multirate sampled data case especially extensions of the ina method and of the robust stability conditions will be straightforward

a lifting technique for linear periodic systems with applications to sampled data control abstract a lifting technique is developed for periodic linear systems and applied to the h and h 2 sampled data control problems

eigenvalue regions for positive systems positive systems are systems in which the state variables take only nonnegative values for all times in this paper we derive eigenvalue regions for discrete and continuous time positive linear systems by resorting to the immediately available information on the values of the main diagonal entries of the system matrix

parameter dependent lyapunov functions for robust stability of time varying linear systems under point delays abstract this paper investigates the stabilization dependent on the delays of in general time varying linear systems with multiple constant point time delays the matrices describing the state space dynamics are parameterized by in general time varying function matrices governed by coefficient functions in compact sets the closed loop sufficiency type stability conditions are obtained from lyapunov s stability theory by constructing parameter dependent candidates which satisfy in the most general case a riccati matrix differential inequality

a new representation of the parameters of lifted systems lifting i e discretization with built in intersample behavior is an emerging technique for the analysis and design of sampled data systems the applicability of the lifting technique however is severely limited owing to difficulties in dealing with the parameters of the lifted systems which are operators over infinite dimensional spaces rather than finite dimensional matrices in this paper a new representation for the parameters of the lifted systems is proposed the technical machinery developed in the paper based on this representation simplifies considerably algebraic manipulation over parameters of the lifted systems thus extending the scope of applicability of the lifting technique to illustrate the advantages of the proposed approach the computational issues in the sampled data h sup spl infin problem are considered

further study of a fourth order elliptic equation with negative exponent we continue to study the nonlinear fourth order problem t u d 2 u l u 2 l u u 0 u 0 on where n is a bounded smooth domain and 0 is a parameter when n 2 and is a convex domain we know that there is c 0 such that for 0 c the problem possesses at least two regular solutions we will see that the convexity assumption on can be removed i e the main results are still true for a general bounded smooth domain the main technique in the proofs of this paper is the blow up argument and the main difficulty is the analysis of touch down behaviour

multiplicity of solutions for a fourth order equation with power type nonlinearity let b be the unit ball in mathbb r n n 3 and n be the exterior unit normal vector on the boundary we consider radial solutions to delta 2 u lambda 1 rm sign p u p quad rm in b quad u 0 quad frac partial u partial n 0 quad rm on partial b where 0 for positive p we assume 5 n 12 and p frac n 4 n 4 or n 13 and frac n 4 n 4 0 such that if s there exists a radial weakly singular solution for s there exist infinitely many regular radial solutions and the number of radial regular solutions goes to infinity as s

aspects of variational arguments in the theory of elasticity fact and folklore abstract an account of the use of the calculus of variations in elastostatics is given variations of both the independent and dependent variables are used to obtain the equations of balance and natural boundary conditions the meaning and use of the necessary transversality conditions for movable internal or external boundaries are examined satisfaction of the transversality conditions is shown to provide immediate access to the forces that act on such boundaries through a virtual work type argument misconceptions concerning the general form of lagrangian densities for which the field equations are identically satisfied are corrected the complete characterization of such lagrangian densities is shown to lead to methods for realization of a wide class of traction boundary conditions as natural boundary conditions and to a significantly richer class of invariance transformations and associated conservation laws

sampled data control of continuous time systems with an h optimality criterion abstract the synthesis of sampled data controllers for continuous time systems is studied in the context of h control theory the sampled data h optimal control problem consists of finding a sampled data controller which minimizes the l2 induced norm from the continuous time disturbance to the continuous time output the problem is here solved via a discrete system representation this is achieved by describing the closed loop system as a discrete system with finite state dimension and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite dimensional spaces via finite rank approximations of a compact integral operator associated with the system the sampled data h control problem can be solved to any degree of accuracy from the structure of the problem it follows that irrespective of the order of the approximation a feasible sampled data controller can be found which has the same order as the system

on the sensitivity minimization problem for linear time varying periodic systems an optimal control problem is formulated in the context of linear discrete time periodic systems the cost is the supremum over all exogenous inputs in a weighted ball of plant inputs the controller is required to be causal periodic of the fixed order of the system and to achieve internal stability existence of an optimal controller is proved and a formula for the minimum cost is derived

state space solution to the periodic multirate h sub spl infin control problem a lifting approach a two riccati equation solution to the h sub spl infin control problem for periodic multirate systems is derived via the lifting method the solution is expressed in terms of two algebraic riccati equations the causality constraints are represented by a set of positive definiteness conditions and coupling criteria as a by product the study shows that there is a close connection between the solution of the causally constrained lifted problem and the solution obtained by solving the problem directly as a periodic h sub spl infin problem defined for the multirate system

application of material forces to hyperelastostatic fracture mechanics ii computational setting the concern of this work is a novel algorithmic treatment of hyperelastostatic fracture mechanics problems consistent to the notion of material forces within the geometrically nonlinear setting of continuum mechanics to this end we consider the continuum mechanics of material forces as outlined in part i of this work p steinmann int j solid struct 37 7371 7391 which act contrary to the common physical forces on the material manifold or rather in the material space in the sequel it is proposed to discretize the corresponding quasi static balance of pseudo momentum by a standard galerkin finite element procedure as a result we obtain global discrete node point quantities the material node point forces which prove to be of the same qualitative and quantitative importance for the assessment of fracture mechanics problems as the classical j integral

optimal and robust controllers for periodic and multirate systems the problem of optimal rejection of bounded persistent disturbances is solved in the case of linear discrete time periodic systems the solution consists of solving an equivalent time invariant standard l sup 1 optimization problem subject to an additional constraint this constraint assures the causality of the resulting periodic controller by the duality theory the problem is shown to be equivalent to a linear programming problem which is no harder than the standard l sup 1 problem also it is shown that the method of solution presented applies exactly to the problem of disturbance rejection in the case of multirate sampled data systems finally the results are applied to the problem of robust stabilization of periodic and multirate systems

on existence and completeness of conservation laws associated with elementary beam theory abstract from a variational principle with varying boundaries conservation laws in the context of elementary beam theory are derived necessary and sufficient conditions are given for transformations leading to conservation laws the formulas obtained are applied to several examples

positive periodic solutions of discrete three level food chain model of holling type ii with the help of differential equations with piecewise constant arguments we first derive a discrete analogy of continuous three level food chain model of holling type ii which is governed by difference equations with periodic coefficients a set of sufficient conditions is derived for the existence of positive periodic solutions with strictly positive components by using the continuation theorem in coincidence degree theory particularly the upper and lower bounds of the periodic solutions are also established

controller parametrization for time varying multirate plants the authors obtain a parametrization of all stabilizing controllers for a given discrete time time varying linear finite dimensional multirate system the parametrization is a natural extension of the youla parametrization of all stabilizing controllers to the setting of multirate systems the set of all stabilizing controllers is shown to be the set of all closed loop systems obtained by terminating a fixed two port multirate system in a stable multirate system state space formulas for the parametrization are also given

a new optimal multirate control of linear periodic and time invariant systems the optimal multirate design of linear continuous time periodic and time invariant systems is considered it is based on solving the continuous linear quadratic regulation lqr problem with the control being constrained to a certain piecewise constant feedback necessary and sufficient conditions for the asymptotic stability of the resulting closed loop system are given an explicit multirate feedback law that requires the solution of an algebraic discrete riccati equation is presented such control is simple and can be easily implemented by digital computers when applied to linear time invariant systems multirate optimal feedback optimal control provides a satisfactory response even if the state is sampled relatively slowly compared to the classical single rate sampled data feedback in which the state is always sampled at the same rate the multirate system can provide a better response with a considerable reduction in the optimal cost in general the multirate scheme offers more flexibility in choosing the sampling rates

a parametrization of stabilizing controllers for multirate sampled data systems all linear multirate controllers for a given multirate sampled data system plant are parameterized in terms of a single parameter q z that is allowed to be any stable transfer matrix provided that certain causality conditions are met the result parallels a well known parameterization result for single rate sampled data plants the parameterization suggests architectures for multirate controllers

h and lgq control of asynchronous sampled data systems a class of dual rate sampled data systems is considered where the sampling rate is asynchronously related to the rate at which the hold element is operating such asynchronous sampled data systems are aperiodic and hence the lifting technique which is used to represent synchronous sampled data systems as single rate discrete systems cannot be applied to the asynchronous case in this study optimal h and lqg controllers for asynchronous sampled data systems are derived using a two riccati equation approach which does not rely on the lifting method a distribution theorem is used to evaluate the optimal stationary performance levels in the two cases

about the positivity of a class of hybrid dynamic linear systems this paper deals with the characterization of the main properties of positivity excitability output excitability and transparency of a class of single input single output hybrid systems which consist of two coupled continuous time and digital subsystems precise sufficient conditions as well as necessary and sufficient conditions that guarantee the existence of such properties are formulated it is found that on occasions and depending on particular parameterizations the couplings between the digital and continuous time dynamics make possible the achievement or not of such properties an extended discrete system whose state vector is composed of the digital one and of the discretization of the continuous one at sampling instants is used as a key element in the analysis since excitability of such a system implies that of the whole hybrid one while the contrary is not true in general

application of lie transformation group methods to classical linear theories of rods and plates in the present paper a class of partial differential equations governing various rod and plate theories of bernoulli euler and poisson kirchhoff type is studied by lie transformation group methods a system of equations determining the generators of the admitted point lie groups symmetries is derived and the general statement of the associated group classification problem is given a simple relation is deduced allowing to recognize easily the variational symmetries among the ordinary symmetries of a self adjoint equation of the class examined explicit formulae for the conserved currents of the corresponding via bessel hagens extension of noethers theorem conservation laws are suggested solutions of group classification problems are given for subclasses of equations of the foregoing type governing stability and vibration of rods fluid conveying pipes and plates resting on variable elastic foundations the obtained group classification results are used to derive conservation laws and group invariant solutions readily applicable in rod dynamics and plate statics and dynamics new generalized symmetries and conservation laws for the theories of timoshenko beams reissner mindlin plates and three dimensional elastostatics are presented 2002 elsevier science ltd all rights reserved

asymptotic behavior of entire radial solutions of a biharmonic problem with negative exponent abstract we consider some qualitative properties of entire radial solutions of a semilinear biharmonic equation with negative exponent 0 1 2 u u p u 0 x r n where n 3 and p 1 in this paper we study the asymptotic rate of any regular positive radial entire solution u of 0 1 at which the limit x is attained

optimal h and h 2 control of hybrid multirate systems the problem of using a synchronous multirate digital controller for a continuous time plant is considered the performance objectives considered are the h and the h2 norms of the periodically time varying continuous time input output behavior of the closed loop system a continuous time lifting technique is used to solve these hybrid sampled data problems this approach yields equivalent purely discrete time problems while preserving the multirate causality of the systems the later problems can then be solved using known techniques

a new class of shift varying operators their shift invariant equivalents and multirate digital systems a class of linear shift varying operators that generalize the notion of n periodicity is defined it is shown that shift invariant equivalents for these operators exist and that the equivalence is in a strong sense preserving both algebraic and analytic system properties it is shown that multirate sampled data systems although not generally periodic fall into this class kranc vector switch decomposition and block filter implementations for single input single output multirate systems are connected under the unifying framework of shift invariant equivalents and this framework provides a way to extend them both to multi input multi output systems

an lmi solution to the robust synthesis problem for multi rate sampled data systems in this paper we address the asynchronous multi rate sampled data h synthesis problem necessary and sufficient conditions are given for the existence of a controller achieving the desired performance and the problem is shown to be equivalent to a convex optimization problem expressed in the form of linear operator inequalities in the case where the sample and hold rates are synchronous these operator inequalities reduce to linear matrix inequalities for which standard numerical software is available

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

the h 2 problem for sampled data systems m for sampled data systems in this paper we pose and give a complete solution to an analog of the h2 problem for sampled data system we motivate and develop a natural generalization of the h2 cost to periodic systems which is then applied to the continuous time closed loop mapping in a sampled data control system it is shown that the cost criterion developed is actually a norm in an space of hilbert schmidt operator valued functions we give state space solutions to the optimal and suboptimal controllers synthesis problems in this new norm by establishing an equivalent standard h2 problem

on positivity of singular regular linear time delay time invariant systems subject to multiple internal and external incommensurate point delays this paper deals with the positivity properties of singular regular linear time delay time invariant systems subject to multiple internal and external incommensurate constant point delays the main idea behind the investigation is that its main body is performed based on the construction of the whole state space trajectory solution without using as usual equivalence or similarity transformations on the matrix of dynamics in order to split the state trajectory solution into two parts one being typically associated with a nilpotent matrix in that way the whole state trajectory solution contains impulsive terms associated with the initial conditions and inputs some extensions concerning positivity aspects are given for a special canonical form which separates the dynamics associated with the nilpotent matrix obtained from an equivalence transformation on the singular matrix of the dynamic system

state space approach to discrete time h 2 optimal control with a causality constraint abstract controller design with a causality constraint arises in periodic or multirate control systems in this paper complete state space solutions to the optimal and suboptimal h 2 control problems are developed with a causality constraint on controller feedthrough terms explicit formulas for the controllers are given in terms of solutions of two riccati equations the results are more implementable than existing frequency domain solutions

the reachability and observability of hybrid multirate sampling linear systems this paper investigates algebraic sufficient necessary and sufficient conditions for sampling time reachability and observability in hybrid multirate sampling data linear systems for the cases of fast input and fast output sampling rates some implications in the context of closed loop pole placement under both properties under multirate sampling are heuristically discussed