duality relations and equilibrium crystal shapes of potts models on triangular and honeycomb lattices abstract we consider the q state potts models on triangular and honeycomb lattices for each lattice it is shown that the anisotropic interfacial tension is related to the anisotropic correlation length on the dual lattice using the star triangle transformation we also find that the anisotropic correlation length on the triangular lattice is the same as that of the honeycomb lattice for q 4 at the first order transition point the anisotropic correlation length is exactly calculated at the first order transition point by a method which introduces auxiliary edges into the models from the calculated anisotropic correlation length the equilibrium droplet shape of one ordered phase embedded inside another is derived via the wulff construction the equilibrium shape is written as a simple algebraic curve which corresponds to the dispersion relation of bound states in the terminology of spin chains

the relation between amplitudes and critical exponents in finite size scaling the relation between critical exponents and the amplitude of the correlation length divergence at the critical point of two dimensional systems as a function of finite size is investigated and generalised in two ways correlations more general than those of order order type are included a form appropriate for anisotropic systems is proposed the authors present a exact results for the ising and gaussian models and b numerical results for the symmetric eight vertex baxter continuous q state potts and continuous n component cubic models

technical communique who needs qp for linear mpc anyway conventional mpc uses quadratic programming qp to minimise on line a cost over n linearly constrained control moves however stability constraints often require the use of large n thereby increasing the on line computation rendering the approach impracticable in the case of fast sampling here we explore an alternative that requires a fraction of the computational cost which increases only linearly with n and propose an extension which in all but a small class of models matches to within a fraction of a percent point the performance of the optimal solution obtained through qp the provocative title of the paper is intended to point out that the proposed approach offers a very attractive alternative to qp based mpc

implicit lyapunov functions and isochrones of linear systems lyapunov s direct method is based on a lyapunov function that explicitly depends on the system s state vector this article describes implicit lyapunov functions and corresponding stability theorems that are counterparts of the explicit case we also define isochrones of linear systems and describe their relationship to implicit lyapunov functions determining settling times is considered as an application

on analytical approximation of the maximal invariant ellipsoids for linear systems with bounded controls analytical approximation of the maximal invariant ellipsoids for linear systems with actuator saturation is established simultaneously a necessary and sufficient condition for such approximation equalling to the exact maximal invariant ellipsoid is presented in addition a geometric property of the maximal invariant ellipsoid in ldquominimal energy control with guaranteed convergence raterdquo problem is proposed all results are given in explicit forms and easy to implement in system analysis and design two numerical examples are used to show the effectiveness of the proposed approach

design techniques for the control of discrete time systems subject to state and control constraints the regulation problem of linear discrete time systems under state and control constraints is investigated in the first part of the paper necessary and sufficient conditions for the existence of a solution to the constrained regulation problem are established the constructive form of the proof of this result also provides a method for the derivation of a control law transferring to the origin any state belonging to a given set of initial states while respecting the state and control constraints in the second part of the paper design techniques are proposed in which the determination of the control law is reduced to simple linear programming problems

robust hybrid control incorporating over saturation a switching controller aimed at dealing with a class of uncertain systems subject to input saturation is presented the switching strategy is such that a pre specified level of over saturation is allowed forcing the input deliberately into saturation the goal of this method is to use the full authority of the available control the proposed scheme allows for up to 100 over saturation under these conditions robust stability of the hybrid scheme is established

composite quadratic lyapunov functions for constrained control systems a lyapunov function based on a set of quadratic functions is introduced in this paper we call this lyapunov function a composite quadratic function some important properties of this lyapunov function are revealed we show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids these results are used to study the set invariance properties of continuous time linear systems with input and state constraints we show that for a system under a given saturated linear feedback the convex hull of a set of invariant ellipsoids is also invariant if each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators then their convex hull can also be made invariant by the same actuators for a set of ellipsoids each invariant under a separate saturated linear feedback we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant

the bethe ansatz for the six vertex model with rotated boundary conditions a method is suggested for derivation of the bethe ansatz equations for the six vertex model on a square lattice rotated at an arbitrary angle with respect to the coordinate axes the method is based on the random walk representation for configurations of the model the equations for the ice model on the rotated lattice are derived and some numerical results are obtained

technical communique a fast nonlinear control method for linear systems with input saturation we present a novel fast saturating nonlinear feedback law for single input systems with linear dynamics and input saturation it is fast in the sense that it yields a better performance than a saturating linear control law the control law is based on implicit soft variable structure control a convex optimization procedure for the controller synthesis based on linear matrix inequalities lmis is derived at the price of some conservatism as an example we consider the control of a submarine

eigenvectors of the multi component generalization of the six vertex model abstract the most general multi component solution of the star triangle equations under an ice type restriction is considered the transfer matrix of this model is diagonalized and the eigenvectors are written down explicitly using the quantum inverse scattering formalism the eigenvectors are shown to have simple symmetry properties under an exchange of arguments

brief predictive pi control of linear plants under positional and incremental input saturations a predictive control strategy is developed for set point tracking of lti plants in the presence of joint positional and incremental rate input saturation constraints the resulting control algorithm is built so as to provide an integral action capable to asymptotically reject arbitrary constant disturbances this task is achieved by possibly modifying on line the set point in the case this becomes incompatible with constraints and disturbances when applied to asymptotically null controllable with bounded inputs ancbi plants viz all stabilizable linear systems with eigenvalues in the closed unit circle the control law is shown to globally provide set point tracking in the presence of arbitrary constant disturbances and initial state for non ancbi plants viz systems with exponentially unstable modes the control law features the mentioned properties locally viz in bounded regions of initial states and disturbances

conformal spectrum of the six vertex model by way of a generalization of analytical methods recently introduced the conformal spectrum of the six vertex model with twisted boundary conditions is analysed nonlinear integral equations are derived from which all scaling dimensions and the spins can be extracted analytically in terms of rogers dilogarithms the scaling dimensions of low lying excitations are given by the so called gaussian dimensions

hard hexagon model calculation of anisotropic interfacial tension from asymptotic degeneracy of largest eigenvalues of row row transfer matrix to find the directional dependence of the interfacial tension of the hard hexagon model an inhomogeneous system is studied this system is defined on a square lattice with 1 v m columns so that the lhs of the m 1 th column is the hard hexagon model and the rhs of the m 1 th column works as the operator which shifts the particle configuration of a column downward a triplet of the largest eigenvalues of the row row transfer matrix are asymptotically degenerate asm under the conditions that 1 v m 0 mod 3 withv being fixed to be constant the interfacial tension of a tilted interface is calculated from the finite correction terms in this limit

finite size corrections and scaling dimensions of solvable lattice models an analytic method a general analytic method for calculating finite size corrections central charges and scaling dimensions of solvable lattice models is presented the approach is to solve the special functional equations or inversion identities satisfied by the commuting row transfer matrices of these lattice models at criticality for purposes of illustration the method is applied to calculate the central charge c 4 5 and leading magnetic scaling dimension x 2 15 of hard hexagons these numbers are rational due to special values of rogers dilogarithms

a nonlinear controller for input amplitude and rate constrained linear systems this technical note presents a nonlinear state feedback controller for systems with linear dynamics subject to an amplitude and rate limited actuator the nonlinear state feedback facilitates faster convergence compared to linear state feedback while guaranteeing domains of attraction of similar size the parametrization of the controller is done solving a convex optimization problem based on linear matrix inequalities lmis finally an example illustrates that the controller achieves large stability regions and short settling times

hard squares with diagonal attractions the authors complete the solution of the square lattice gas with nearest neighbour exclusion and attractive next nearest neighbour diagonal interactions on the special surface corresponding to regimes iii and iv of the generalised hard hexagon model the interfacial tension correlation length and sublattice density difference are calculated throughout these regimes by obtaining the eigenvalues of the row to row and corner transfer matrices the associated critical exponents are found to be mu v 5 4 beta 3 32 in regime iii and mu v 5 2 beta 1 4 in regime iv in particular their results confirm the recent proposal by huse that regime iii is the first order coexistence surface separating the disordered fluid phase from the square ordered solid phases and that the regime iii regime iv boundary is a line of tricritical points

on two types of obstructed random walks abstract two examples of random walks on a lattice which are subject to obstructing conditions are discussed i a random walk on a d dimensional lattice containing a periodic array of traps ii a random walk in a set of colliding random walks in d 1 expressions are derived for the probability distribution of the lattice point where the walk comes to an end and the average total number of distinct lattice points visited in example i and for the probability distribution of a walker s displacement in particular the mean square displacement for long times in example ii

on maximizing the convergence rate for linear systems with input saturation in this note we consider a few important issues related to the maximization of the convergence rate inside a given ellipsoid for linear systems with input saturation for continuous time systems the control that maximizes the convergence rate is simply a bang bang control through studying the system under the maximal convergence control we reveal several fundamental results on set invariance an important consequence of maximizing the convergence rate is that the maximal invariant ellipsoid is produced we provide a simple method for finding the maximal invariant ellipsoid and we also study the dependence of the maximal convergence rate on the lyapunov function

rogers ramanujan identities in the hard hexagon model the hard hexagon model in statistical mechanics is a special case of a solvable class of hard square type models in which certain special diagonal interactions are added the sublattice densities and order parameters of this class are obtained and it is shown that many rogers ramanujan type identities naturally enter the working

properties of the composite quadratic lyapunov functions a composite quadratic lyapunov function introduced recently was shown to be very useful in the study of set invariance properties for linear systems with input and state constraints and for systems with a class of convex concave nonlinearities in this note more properties about this function are revealed in particular we study the continuity of the optimal parameter involved in this function this continuity is crucial in the construction of a continuous feedback law which makes the convex hull of a group of ellipsoids invariant

nonlinear regulation in constrained input discrete time linear systems the use of composite linear and non linear feedback laws for the control of constrained input discrete time linear systems is re examined by making use of the delta operator formulation of a discrete time system an apparent restriction on the magnitude of the non linear control law is removed and the similarities between the continuous and discrete time solutions to the problem are elucidated in order to develop the results unconstrained systems are treated initially but it is shown that locally the sufficient conditions for the stabilization of such systems are actually equivalent to those for the stabilization of the corresponding constrained systems

piecewise linear constrained control for continuous time systems in the control of linear systems with nonsymmetrical input constraints larger domains of initialization can be obtained by allowing slow initial dynamics of the system in closed loop and making it faster during its evolution the positive invariance concept application leads to a piecewise linear constrained control using nonsymmetrical nested polyhedral sets

controller design for a class of nonlinear systems with input saturation using convex optimization abstract in this paper we propose a new design strategy for nonlinear systems with input saturation the resulting nonlinear controllers are locally asymptotically stabilizing the origin the proposed methodology is based on exact feedback linearization which is used to reformulate the nonlinear system as a linear system having state dependent input saturation linear saturating state feedback controllers and soft variable structure controllers are developed based on this system formulation the resulting convex optimization problems can be written in terms of linear matrix inequalities and sum of squares conditions for which efficient solvers exist polynomial approximation based on legendre polynomials is used to extend the methodology to a more general class of nonlinear systems to demonstrate the benefit of this design method a stabilizing controller for a single link manipulator with flexible joint is developed

efficient robust constrained model predictive control with a time varying terminal constraint set abstract an efficient robust constrained model predictive control algorithm with a time varying terminal constraint set is developed for systems with model uncertainty and input constraints the approach is novel in that it off line constructs a continuum of terminal constraint sets and on line achieves robust stability by using a relatively short control horizon even n 0 with a time varying terminal constraint set this algorithm not only dramatically reduces the on line computation but also significantly enlarges the size of the allowable set of initial conditions moreover this control scheme retains the unconstrained optimal performance in the neighborhood of the equilibrium the controller design is illustrated through a benchmark problem

on stabilizing linear systems with input saturation via soft variable structure control laws abstract a large number of control methods deal with stabilizing single input linear systems with input saturation the linear full state feedback is one of the most used yielding either nonsaturating or high gain controls whereby the saturating effects of the latter are reduced by means of anti windup structures among the nonlinear controls the time optimal control in this case a bang bang type control yields the fastest time response but its switching surface is generally not characterizable related to the speed of the time response is the convergence rate which can be determined using invariant sets the most used ones are the ellipsoidal sets since they can be analyzed using powerful tools such as the lyapunov equation and designed via convex optimization for this reason they are also used for designing soft variable structure controls the paper presents a nonconservative design method for a stabilizing control of this type employing implicit lyapunov functions ilf a nonsaturating control law is given including some infinitely densely nested and contractive invariant sets of the equilibrium state the control law is then optimized by maximizing the ilf based lower bound of the convergence rate the maximal convergence control is shown to be of bang bang type with a parameter dependent switching scheme to overcome possible difficulties of a switching controller a saturating high gain control is also presented

data stability in clustering a closer look we consider the model introduced by bilu and linial 2010 13 who study problems for which the optimal clustering does not change when distances are perturbed they show that even when a problem is np hard it is sometimes possible to obtain efficient algorithms for instances resilient to certain multiplicative perturbations e g on the order of o n for max cut clustering awasthi et al 2012 6 consider center based objectives and balcan and liang 2012 9 analyze the k median and min sum objectives giving efficient algorithms for instances resilient to certain constant multiplicative perturbations here we are motivated by the question of to what extent these assumptions can be relaxed while allowing for efficient algorithms we show there is little room to improve these results by giving np hardness lower bounds for both the k median and min sum objectives on the other hand we show that constant multiplicative resilience parameters can be so strong as to make the clustering problem trivial leaving only a narrow range of resilience parameters for which clustering is interesting we also consider a model of additive perturbations and give a correspondence between additive and multiplicative notions of stability our results provide a close examination of the consequences of assuming stability in data

explicit sub optimal linear quadratic regulation with state and input constraints optimal feedback solutions to the infinite horizon lqr problem with state and input constraints based on receding horizon real time quadratic programming are well known in this paper we develop an explicit solution to the same problem eliminating the need for real time optimization it is shown that the resulting feedback controller is piecewise linear this explicit functional structure is exploited for efficient real time implementation a suboptimal strategy based on a suboptimal choice of a finite horizon and imposing additional limitations on the allowed switching between active constraint sets on the horizon is suggested in order to address the computer memory and processing capacity requirements of the explicit solution

diagonalization of the transfer matrix of a nonintersecting string model abstract in this paper we obtain the eigenvectors of the transfer matrix of a nonintersecting string model using the bethe ansatz the result is in agreement with baxter s conjecture of weak equivalence of the present model with the six vertex model

on improving the convergence speed of linear continuous time systems under constrained control sylvester equation approach a controller design methodology is proposed for linear continuous time systems with asymmetrical constrained control because of the trade off between the size of the domain of attraction and convergence speed one can improve the convergence speed during the system evolution the closed loop system starts with bad performances and a large domain of attraction and ends with good performances near the origin the objective of this work is to overcome some discontinuities resulting from a piecewise linear control law previously proposed to improve the convergence speed of a linear system during its evolution the proposed method is based on the use of the sylvester equation together with a quadratic lyapunov function it guaranties the improvement of the speed of convergence without any discontinuities on the control vector components and respecting always the imposed asymmetric constraints without saturations

reply to comment on equilibrium crystal shape of the potts model at the first order transition point the eight vertex model is defined on the square lattice rotated through an arbitrary angle with respect to the coordinate axes we re examine the analysis of the anisotropic correlation length in a previous paper fujimoto m 1996 physica a 233 485 502 we point out that the asymptotic form of the correlation function is expressed by the use of differential forms on a riemann surface of genus 1 combined with the symmetry of the square lattice this fact explains that the anisotropic correlation length is represented in terms of simple algebraic curves the argument is applicable to a wide class of lattice models including unsolvable ones

anisotropic interfacial tension and the equilibrium crystal shape of kagome lattice eight vertex model we exactly calculate the anisotropic interfacial tension of the kagome lattice eight vertex model with the help of two types of square lattice models each square lattice model is inhomogeneous but still possesses a one parameter family of commuting transfer matrices from the anisotropic interfacial tension the equilibrium shape of an inclusion of one ordered phase in an environment of another is obtained by applying the wulff construction calculations are expressed by the use of differential forms on a riemann surface of genus 1 and the equilibrium shape is written as an elliptic curve we suggest the possibility of interpreting critical phenomena in terms of the moduli space of riemann surfaces of genus 1

feedback control of linear discrete time systems under state and control constraints in this paper the problem of stabilizing linear discrete time systems under state and control linear constraints is studied based on the concept of positive invariance existence conditions of linear state feedback control laws respecting both the constraints are established these conditions are then translated into an algorithm of linear programming

perimeter bethe ansatz the partition function of a finite z invariant six vertex lattice model with a prescribed arrow configuration at the boundary is given the expression is of the same type that occurs in the bethe ansatz but there are no wavenumber equations to solve

generalized linear quadratic control we consider the problem of stochastic finite and infinite horizon linear quadratic control under power constraints the calculations of the optimal control law can be done off line as in the classical linear quadratic gaussian control theory using dynamic programming which turns out to be a special case of the new theory developed in this technical note a numerical example is solved using the new methods

an lqr lqg theory for systems with saturating actuators an extension of the lqr lqg methodology to systems with saturating actuators referred to as slqr slqg where s stands for saturating is obtained the development is based on the method of stochastic linearization whereby the saturation is replaced by a gain calculated as a function of the variance of the signal at its input using the stochastically linearized system and the lagrange multiplier technique solutions of the slqr slqg problems are derived these solutions are given by standard riccati and lyapunov equations coupled with two transcendental equations which characterize both the variance of the signal at the saturation input and the lagrange multiplier associated with the constrained minimization problem it is shown that under standard stabilizability and detectability conditions these equations have a unique solution which can be found by a simple bisection algorithm when the level of saturation tends to infinity these equations reduce to their standard lqr lqg counterparts in addition the paper investigates the properties of closed loop systems with slqr slqg controllers and saturating actuators in this regard it is shown that slqr slqg controllers ensure semi global stability by appropriate choice of a parameter in the performance criterion finally the paper illustrates the techniques developed by a ship roll damping problem

finite size corrections for ground states of the xxz heisenberg chain the method of de vega and woynarovitch 1985 is used to calculate finite size corrections to the ground state energy in different sectors for the xxz heisenberg chain finite size scaling amplitudes and correction to scaling exponents in the critical region are derived using conformal invariance a scaling dimension x pi gamma 2 pi is extracted corresponding to the electric field operator in the 8 vertex model this confirms a conjecture of baxter and kelland 1974 finite size scaling properties near the kosterlitz thouless critical point delta 1 are discussed

regulator problem for linear discrete time systems with non symmetrical constrained control the regulator problem is studied for linear discrete time systems with non symmetrical constrained control i e systems described by the state equation x k 1 ax k bu k where u k and u k fx k necessary and sufficient conditions allowing us to obtain the largest non symmetrical polyhedral domain of positive invariance and contractivity with respect to motions of the system in the closed loop are established the case of symmetrically constrained control is obtained as a particular case

on translational motion planning of a convex polyhedron in 3 space let b be a convex polyhedron translating in 3 space amidst k convex polyhedral obstacles a1 ak with pairwise disjoint interiors the free configuration space space of all collision free placements of b can be represented as the complement of the union of the minkowski sums p i a i oplus b for i 1 k we show that the combinatorial complexity of the free configuration space of b is o nk log k and that it can be omega nk alpha k in the worst case where n is the total complexity of the individual minkowski sums p1 pk we also derive an efficient randomized algorithm that constructs this configuration space in expected time o nk log k log n

on the tightness of a recent set invariance condition under actuator saturation a sufficient condition for an ellipsoid to be invariant was obtained recently and an lmi approach was developed to find the largest ellipsoid satisfying the condition this condition was later shown to be necessary for the single input case this paper is dedicated to the multi input case we will examine the conservatism of the condition for multi input systems our investigation is conducted by studying the optimal solution to a related lmi problem a criterion is presented to determine when the condition is not conservative and when the largest invariant ellipsoid has been obtained by using the lmi method

six vertex model with rotated boundary conditions we investigate the six vertex model on a square lattice rotated through an arbitrary angle with respect to the coordinate axes a model recently introduced by litvin and priezzhev auxiliary vertices are used to define an inhomogeneous system which leads to a one parameter family of commuting transfer matrices a product of commuting transfer matrices can be interpreted as a transfer matrix acting on zigzag walls in the rotated system using an equation for commuting transfer matrices we calculate their eigenvalues finite size properties of the model are discussed from the viewpoint of the conformal field theory

positively invariant polyhedral sets of discrete time linear systems the problem of the existence of positively invariant polyhedral sets for linear discrete time dynamical systems is studied in the first part of the paper necessary and sufficient conditions for a

computationally efficient algorithms for constraint handling with guaranteed stability and near optimality predictive control strategies allow for the systematic handling of constraint performance and stability however the associated algorithms can be computational burdensome and or difficult to unravel the aim of this paper is to discuss and compare algorithms based on invariant sets which meet the additional requirement for computational simplicity there may of course be a concomitant loss of optimality but as illustrated this can be minimal and often is a small price to pay when one considers the significant improvements in efficiency

control sharing and merging control lyapunov functions given two control lyapunov functions clfs a merging is a new clf whose gradient is a positive combination of the gradients of the two parents clfs the merging function is an important trade off since this new function may for instance approximate one of the two parents functions close to the origin while being close to the other far away for nonlinear control affine systems some equivalence properties are shown between the control sharing property i e the existence of a single control law which makes simultaneously negative the lyapunov derivatives of the two given clfs and the existence of merging clfs it is shown that even for linear time invariant systems the control sharing property does not always hold with the remarkable exception of planar systems the class of linear differential inclusions is also discussed and similar equivalence results are presented for this class of systems linear matrix inequalities conditions are provided to guarantee the control sharing property finally a constructive procedure based on the recently considered r functions is defined to merge two smooth positively homogeneous clfs

a vertex model with an unbounded number of components a model which is an infinite component generalization of the six vertex model is proposed the method of solution is based on a random walk representation of states the bethe ansatz equations for periodic boundary conditions are obtained the partition function of the finite model with special arrow configurations at the boundary of a square is given in contrast to analogous solutions of korepin 1982 and baxter 1982 it has the form of an evaluable determinant

an algorithm to find maximal state constraint sets for discrete time linear dynamical systems with bounded controls and states an algorithm to find a polyhedral approximant to the maximal state constraint set given state and input bounds is suggested for linear discrete time dynamical systems

scaling dimensions and conformal anomaly in anisotropic lattice spin models the effect of anisotropic interactions on the eigenvalue spectrum of the row to row transfer matrix of critical lattice spin models is investigated it is verified that the predictions of conformal theory apply to anisotropic systems if one allows for spatial rescaling by incorporating an anisotropy factor zeta ay ax sin theta where ay and ax are lattice spacings and theta is an angle describing the anisotropy for exactly solvable models these anisotropy angles can be calculated analytically using corner transfer matrices this is done for the eight vertex model hard hexagons and interacting hard squares and it is found that theta pi u lambda 10u 3 and 5u respectively where u is the spectral parameter and lambda is the crossing parameter for each of these models the amplitude of the finite size corrections to the free energy at criticality is found to be of the form pi zeta c 6n2 where zeta is the anisotropy factor and the central charge or conformal anomaly is given by c 1 4 5 and 7 10 respectively this is an analytic result for the eight vertex model for the hard hexagon and square models the largest eigenvalues are found accurately by numerically solving their inversion identities for various anisotropies and strip widths up to n 48 finally the authors argue that the anisotropy angle for magnetic hard squares and the q state potts models is also given by theta pi u lambda so this result is quite general

exact characterization of invariant ellipsoids for single input linear systems subject to actuator saturation we present a necessary and sufficient condition for an ellipsoid to be an invariant set of a linear system under a saturated linear feedback the condition is given in terms of linear matrix inequalities lmis and can be easily used for optimization based analysis and design

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

scheduled controllers for linear systems with bounded actuators state and output feedback controllers are designed for disturbance attenuation in linear systems with bounded actuators the controller is scheduled according to the proximity to the origin of the state of the plant in the state feedback case and the compensator state in the case of output feedback this procedure yields a linear parameter varying structure for the controller that allows higher gain and hence higher performance controllers as the states move closer to the origin the main results give sufficient conditions for the satisfaction of a parameter dependent performance measure without violating the saturation bounds linear splines are used to obtain solutions that can be obtained by standard lmi software examples including an open loop unstable system highlight the application of the results

fat triangles determine linearly many holes the authors show that for every fixed delta 0 the following holds if f is a union of n triangles all of whose angles are at least delta then the complement of f has o n connected components and the boundary of f consists of o n log log n straight segments where the constants of proportionality depend on delta this latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges

technical communique evaluation of piecewise affine control via binary search tree we present an algorithm for generating a binary search tree that allows efficient computation of piecewise affine pwa functions defined on a polyhedral partition this is useful for pwa control approaches such as explicit model predictive control as it allows the controller to be implemented online with small computational effort the computation time is logarithmic in the number of regions in the pwa partition

surface free energy of the critical six vertex model with free boundaries the bethe ansatz equations are derived for the six vertex model with general boundary weights on a lattice in a diagonal orientation these are solved in the thermodynamic limit finite size corrections to the free energy for a restricted class of boundary weights including those corresponding to a potts model with free boundaries are calculated in the critical region the first order term gives the surface free energy of the model the second order term is found to be pi okt tan pi nu 2 mu c 48n 2 where c 1 6 mu 2 pi 2 pi mu is the conformal anomaly this can be compared to pi kt sin pi nu mu c 6n 2 for a calculation on the lattice in the standard orientation and with periodic boundary conditions this difference can be explained geometrically using conformal invariance

on enlarging the basin of attraction for linear systems under saturated linear feedback we consider the problem of enlarging the basin of attraction for a linear system under saturated linear feedback an lmi based approach to this problem is developed for discrete time system this approach is enhanced by the lifting technique which leads to further enlargement of the basin of attraction the low convergence rate inherent with the large invariant set hence the large basin of attraction is prevented by the construction of a sequence of invariant ellipsoids nested within the large one obtained

curvature properties of the algebraic riccati equation abstract we prove that the solution to the algebraic ricatti equation are is concave with respect to a nonnegative definite symmetric state weighting matrix q when the input weighting matrix r r t 0 we also prove that the solution to the are is concave with respect to a positive definite diagonal input weighting matrix r when q q t 0

on the union of fat tetrahedra in three dimensions we show that the combinatorial complexity of the union of n fat tetrahedra in 3 space i e tetrahedra all of whose solid angles are at least some fixed constant of arbitrary sizes is o n2 e for any e 0 the bound is almost tight in the worst case thus almost settling a conjecture of pach et al 2003 our result extends in a significant way the result of pach et al 2003 for the restricted case of nearly congruent cubes the analysis uses cuttings combined with the dobkin kirkpatrick hierarchical decomposition of convex polytopes in order to partition space into subcells so that on average the overwhelming majority of the tetrahedra intersecting a subcell behave as fat dihedral wedges in as an immediate corollary we obtain that the combinatorial complexity of the union of n cubes in r3 having arbitrary side lengths is o n2 e for any e 0 again significantly extending the result of pach et al 2003 finally our analysis can easily be extended to yield a nearly quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms whose cross sections have arbitrary sizes in r3

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

on improving the performance with bounded continuous feedback laws we present controller design methods to smooth the discontinuity resulting from a piecewise linear control law which was proposed to improve the convergence performance for systems with input constraints the continuous control laws designed in this paper are explicit functions of the state and are easily implementable we also show that the convergence performance can be further improved by using a saturated high gain feedback law the efficiency of the proposed methods is illustrated with the puma 560 robot model

an analytic treatment of finite size corrections in the spin 1 antiferromagnetic xxz chain it is known that the standard method for calculating finite size corrections in bethe ansatz solvable systems is not applicable to the takhtajan babujian model and its anisotropic xxz generalisation the authors develop a new analytic method explicitly avoiding root densities and associated problems nonlinear integral equations are derived whose solutions yield the correct central charges c 1 and c 3 2 for the spin 1 2 and spin 1 xxz chains respectively in the spin 1 case the authors obtain as a by product the finite size deviation of the bethe ansatz roots from the 2 string formation

local stabilization of discrete time linear systems with saturating controls an lmi based approach deals with the problem of local stabilization of linear discrete time systems subject to control saturation a linear matrix inequalities based framework is proposed in order to compute a saturating state feedback that stabilizes the system with respect to a given set of admissible initial states and in addition guarantees some dynamical performances when the system operates in the zone of linear behavior i e when the controls are not saturated

book review actuator saturation control v kapila k m grigoriadis marcel dekker new york isbn 0 8247 0751 6

enlargement of polytopic terminal region in nmpc by interpolation and partial invariance this paper uses the concept of partial invariance to derive a sequence of linear programs in order to maximize offline the volume of an invariant polytopic set with an arbitrary predefined number of vertices subject to a bound on closed loop performance interpolation techniques are used to determine a nonlinear control law which is optimal with respect to a closed loop cost bound through the on line solution of a linear program the invariant polytope is also used to define a receding horizon control law through an appropriate terminal constraint and cost

piecewise linear lq control for systems with input constraints abstract a piecewise linear control plc law based on lq theory is derived ths law raises an lq gain as the controlled error converges towards the origin it can be computed off line and it guarantees that input bounds are never exceeded without causing input saturation a succession of positively invariant sets are defined of diminishing size and associated with each is the corresponding highest lq gain possible in the presence of the input bounds the formulation gives rise to an iteration function whose solution is a fixed point fp the fp defines the optimal gain and the invariant set which comprise ellipsoids fp existence uniqueness and convergence properties are proven we explain the algorithms for the generation and subsequent implementation of the plc law the notion of a switching function is introduced consideration is given to the computational aspects of the fp the plc scheme is demonstrated for the linearized dynamics of the simple pendulum and the puma 560 robot a discrete time plc law can also be derived using the discrete time algebraic riccati equation

the complexity of the union of alpha beta covered objects an alpha beta covered object is a simply connected planar region c with the property that for each point p in partial c there exists a triangle contained in c and having p as a vertex such that all its angles are at least alpha 0 and all its edges are at least beta cdot rm diam c long this notion extends that of fat convex objects we show that the combinatorial complexity of the union of n alpha beta covered objects of constant description complexity is o lambda s 2 n log 2n log log n where s is the maximum number of intersections between the boundaries of any pair of given objects and lambda s n denotes the maximum length of an n s davenport schinzel sequence our result extends and improves previous results concerning convex alpha fat objects

synthesis of controllers for continuous time delay systems with saturating controls via lmis the stabilization of linear continuous time systems with time delay in the state and subject to saturating controls is addressed sufficient conditions obtained via a linear matrix inequality lmi formulation are stated to guarantee both the local stabilization and the satisfaction of some performance requirements the method of synthesis consists in determining simultaneously a state feedback control law and an associated domain of safe admissible states for which the stability of the closed loop system is guaranteed when control saturations effectively occur

auxiliary vertices method for kagome lattice eight vertex model we investigate a class of eight vertex models on a kagome lattice with the help of auxiliary vertices the kagome lattice eight vertex model kevm is related to an inhomogeneous system which leads to a one parameter family of commuting transfer matrices using an equation for commuting transfer matrices we determine their eigenvalues from calculated eigenvalues the correlation length of the kevm is derived with its full anisotropy there are two cases in the first case the anisotropic correlation length acl is the same as that of the triangular honeycomb lattice ising model by the use of an algebraic curve it is shown that the kagome lattice ising model the diced lattice ising model and the hard hexagon model also have essentially the same acl as the kevm in the second case we find that the acl displays 12fold rotational symmetry

stabilization of exponentially unstable linear systems with saturating actuators we study the problem of stabilizing exponentially unstable linear systems with saturating actuators the study begins with planar systems with both poles exponentially unstable for such a system we show that the boundary of the domain of attraction under a saturated stabilizing linear state feedback is the unique stable limit cycle of its time reversed system a saturated linear state feedback is designed that results in a closed loop system having a domain of attraction that is arbitrarily close to the null controllable region this design is then utilized to construct state feedback laws for higher order systems with two exponentially unstable poles

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

eight vertex model anisotropic interfacial tension and equilibrium crystal shape the anisotropic interfacial tension of the eight vertex model is found by a new method which introduces two inhomogeneous systems as the width of the system becomes large a doublet of the largest eigenvalues of the row row transfer matrix is asymptotically degenerate the anisotropic interfacial tension is calculated from their finite size correction terms in this limit by the use of the anisotropic interfacial tension the equilibrium crystal shape of the eight vertex model is derived via wulff s construction the equilibrium crystal shape is represented as a simple algebraic curve we discuss the close relation between the algebraic curve and the form of an elliptic function appearing in the expression of the interfacial tension

h sub 2 control with time domain constraints theory and an application in this paper we study the problem of minimizing the h sub 2 norm of a given transfer function subject to time domain constraints on the time response of a different transfer function to a given test signal the main result of this paper shows that this problem admits a minimizing solution in r h sub 2 moreover rational solutions with performance arbitrarily close to optimal can be found by constructing families of approximating problems each one of these problems entails solving a finite dimensional quadratic programming problem whose dimension can be determined before hand these results are illustrated and experimentally validated by designing a controller for an active vision application

the union of convex polyhedra in three dimensions we show that the number of vertices edges and faces of the union of k convex polyhedra in 3 space having a total of n faces is o k3 kn log k this bound is almost tight in the worst case as there exist collections of polyhedra with omega k 3 kn alpha k union complexity we also describe a rather simple randomized incremental algorithm for computing the boundary of the union in o k3 kn log k log n expected time

spatial anisotropy and rotational invariance of critical hard squares we develop a new method to analyse solvable models defined on a square lattice rotated through an arbitrary angle phi with respect to the coordinate axes in the method we introduce auxiliary faces into a rotated system to relate it with an inhomogeneous one in the natural orientation phi 0 the inhomogeneous system is investigated by commuting a transfer matrix argument as an application of the method we consider interacting hard squares at the critical point we discuss relations between spatial anisotropy and rotational or conformal invariance of the model in calculations of finite size properties

equilibrium crystal shape of hard squares with diagonal attractions the author exactly calculates the anisotropic interfacial tension of a square lattice gas model with nearest neighbour exclusion and next nearest neighbour attractions this is achieved by a method which introduces the shift operator into the standard transfer matrix argument from the calculated anisotropic interfacial tension the equilibrium crystal shape ecs is derived by the use of the wulff construction the ecs is a closed curve in the x y plane represented as cosh lambda x y kbt a3cosh lambda x y kbt a4 2 0 with a scale factor lambda and constants a3 a4 he argues that this shape is a universal one which appears as the ecss of a wide class of models