integration of quantum many body problems by affine knizhnik zamolodchikov equations abstract an equivalence of the affine analogues of knizhnik zamolodchikov and dunkl equations is established for arbitrary root systems as a corollary an isomorphism is obtained from the solutions of the above kz equations in the induced representations of hecke algebras to the eigenfunctions for the corresponding matrix quantum calogero sutherland operators which generalizes the matsuo map

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

periodic solutions of a class of dynamical systems of second order abstract we give some sufficient conditions for the existence of infinitely many periodic solutions with prescribed period for a class of dynamical systems of second order variational methods and finite dimensional approximations are used

periodic solutions for a class of first order super quadratic hamiltonian system abstract based on a developed local linking theorem by replacing the palais smale condition with a cerami one we consider the following unbounded hamiltonian system 0 1 t u x u v x u h v t x u v t v x v v x v h u t x u v for t x r where r n n 1 is a bounded domain with smooth boundary the nonlinearities are assumed to be super quadratic at the infinite which are different from those investigated previously

on the sutherland spin model of b n type and its associated spin chain the b n hyperbolic sutherland spin model is expressed in terms of a suitable set of commuting dunkl operators this fact is exploited to derive a complete family of commuting integrals of motion of the model thus establishing its integrability the dunkl operators are shown to possess a common flag of invariant finite dimensional linear spaces of smooth scalar functions this implies that the hamiltonian of the model preserves a corresponding flag of smooth spin functions the discrete spectrum of the restriction of the hamiltonian to this spin flag is explicitly computed by triangularization the integrability of the hyperbolic sutherland spin chain of b n type associated with the dynamical model is proved using polychronakos s freezing trick

approximate explicit constrained linear model predictive control via orthogonal search tree solutions to constrained linear model predictive control problems can be precomputed off line in an explicit form as a piecewise linear state feedback on a polyhedral partition of the state space avoiding real time optimization we suggest an algorithm that can determine an approximate explicit piecewise linear state feedback by imposing an orthogonal search tree structure on the partition this leads to a real time computational complexity that is logarithmic in the number of regions in the partition and the algorithm yields guarantees on the suboptimality asymptotic stability and constraint fulfillment

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

integrable spin chain with reflecting end a new integrable spin chain of the haldane shastry type is introduced it is interpreted as the inverse square interacting spin chain with a reflecting end the lattice sites of this model consist of the square roots of the zeros of the laguerre polynomial using the exchange operator formalism the integrals of motion for the model are explicitly constructed

computational complexity certification for real time mpc with input constraints based on the fast gradient method this paper proposes to use nesterov s fast gradient method for the solution of linear quadratic model predictive control mpc problems with input constraints the main focus is on the method s a priori computational complexity certification which consists of deriving lower iteration bounds such that a solution of pre specified suboptimality is obtained for any possible state of the system we investigate cold and warm starting strategies and provide an easily computable lower iteration bound for cold starting and an asymptotic characterization of the bounds for warm starting moreover we characterize the set of mpc problems for which small iteration bounds and thus short solution times are expected the theoretical findings and the practical relevance of the obtained lower iteration bounds are underpinned by various numerical examples and compared to certification results for a primal dual interior point method

exact complexity certification of active set methods for quadratic programming active set methods are recognized to often outperform other methods in terms of speed and solution accuracy when solving small size quadratic programming qp problems making them very attractive in embedded linear model predictive control mpc applications a drawback of active set methods is the lack of tight bounds on the worst case number of iterations a fundamental requirement for their implementation in a real time system extensive simulation campaigns provide an indication of the expected worst case computation load but not a complete guarantee this paper solves such a certification problem by proposing an algorithm to compute the exact bound on the maximum number of iterations and floating point operations required by a state of the art dual active set qp solver the algorithm is applicable to a given qp problem whose linear term of the cost function and right hand side of the constraints depend linearly on a vector of parameters as in the case of linear mpc in addition a new solver is presented that combines explicit and implicit mpc ideas guaranteeing improvements of the worst case computation time the ability of the approach to exactly quantify memory and worst case computation requirements is tested on a few mpc examples also highlighting when online optimization should be preferred to explicit mpc

quantum versus classical integrability in calogero moser systems calogero moser systems are classical and quantum integrable multiparticle dynamics defined for any root system the quantum calogero systems having 1 q2 potential and a confining q2 potential and the sutherland systems with 1 sin2q potentials have integer energy spectra characterized by the root system various quantities of the corresponding classical systems e g minimum energy frequencies of small oscillations the eigenvalues of the classical lax pair matrices etc at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems to our surprise most of these classical data are also integers or they appear to be quantized to be more precise these quantities are polynomials of the coupling constant s with integer coefficients the close relationship between quantum and classical integrability in calogero moser systems deserves fuller analytical treatment which would lead to better understanding of these systems and of integrable systems in general

cooperative distributed model predictive control abstract in this paper we propose a cooperative distributed linear model predictive control strategy applicable to any finite number of subsystems satisfying a stabilizability condition the control strategy has the following features hard input constraints are satisfied terminating the iteration of the distributed controllers prior to convergence retains closed loop stability in the limit of iterating to convergence the control feedback is plantwide pareto optimal and equivalent to the centralized control solution no coordination layer is employed we provide guidance in how to partition the subsystems within the plant we first establish exponential stability of suboptimal model predictive control and show that the proposed cooperative control strategy is in this class we also establish that under perturbation from a stable state estimator the origin remains exponentially stable for plants with sparsely coupled input constraints we provide an extension in which the decision variable space of each suboptimization is augmented to achieve pareto optimality we conclude with a simple example showing the performance advantage of cooperative control compared to noncooperative and decentralized control strategies

conditions under which suboptimal nonlinear mpc is inherently robust abstract we address the inherent robustness properties of nonlinear systems controlled by suboptimal model predictive control mpc i e when a suboptimal solution of the generally nonconvex optimization problem rather than an element of the optimal solution set is used for the control the suboptimal control law is then a set valued map and consequently the closed loop system is described by a difference inclusion under mild assumptions on the system and cost functions we establish nominal exponential stability of the equilibrium and with a continuity assumption on the feasible input set we prove robust exponential stability with respect to small but otherwise arbitrary additive process disturbances and state measurement estimation errors these results are obtained by showing that the suboptimal cost is a continuous exponential lyapunov function for an appropriately augmented closed loop system written as a difference inclusion and that recursive feasibility is implied by such nominal exponential cost decay these novel robustness properties for suboptimal mpc are inherited also by optimal nonlinear mpc we conclude the paper by showing that in the absence of state constraints we can replace the terminal constraint with an appropriate terminal cost and the robustness properties are established on a set that approaches the nominal feasibility set for small disturbances the somewhat surprising and satisfying conclusion of this study is that suboptimal mpc has the same inherent robustness properties as optimal mpc

new proof and generalizations of the demidowitsch schneider criterion a new proof of the demidowitsch schneider criterion on the absence of closed trajectories of r3 is given the new proof is generalized and applied in several directions rn vector fields n 3

efficient active set optimization in triple mode mpc an active set algorithm tailored to quadratically constrained quadratic programming in model predictive control mpc is presented it enables efficient use of augmented ellipsoidal invariant sets in conjunction with polytopic constraints in triple mode mpg the algorithm gives improved optimality and larger stabilizable initial condition sets than conventional quadratic programming mpc algorithms of comparable online computational burden

a free energy bound for the hopfield model the author gives a simple upper and lower bound on the free energy density of the hopfield model of size n with p stored patterns in the limit where n and p tend to infinity with p n to alpha 1 the two bounds coincide for alpha 0

subharmonics near an equilibrium for some second order hamiltonian systems this work is devoted to the study of subharmonic solutions of a second order hamiltonian system near an equilibrium point say q 0 the problem of existence of periodic solutions from the global point of view is also considered this problem has been studied for the case where the potential is positive and superquadratic in this work a potential v that has change in sign is considered the potential is decomposed as where p is homogeneous superquadratic and nondegenerate and is of higher order near 0 in this paper the existence is shown of a sequence of subharmonic solutions of the equation above that converges to the equilibrium such that their minimal periods converge to infinity this problem is approached from a variational point of view existence of subharmonic and periodic solutions is obtained via minimax techniques

yang baxter equation in long range interacting systems we consider the su p spin chains with long range interactions and the spin generalization of the calogero sutherland models we show that their properties derive from a transfer matrix obeying the yang baxter equation we obtain the expression of the conserved quantities of the dynamical models and we diagonalise them in the spin chain case we establish the connection between the degeneracies of the spectrum and the representation theory of the yangians we use a correspondence with the dynamical models to diagonalise the hamiltonian finally we extend the previous results to the case of a trigonometric r matrix

brief paper fast large scale model predictive control by partial enumeration partial enumeration pe is presented as a method for treating large linear model predictive control applications that are out of reach with available mpc methods pe uses both a table storage method and online optimization to achieve this goal versions of pe are shown to be closed loop stable pe is applied to an industrial example with more than 250 states 32 inputs and a 25 sample control horizon the performance is less than 0 01 suboptimal with average speedup factors in the range of 80 220 and worst case speedups in the range of 4 9 39 2 compared to an existing mpc method small tables with only 25 200 entries were used to obtain this performance while full enumeration is intractable for this example

a multiparametric quadratic programming algorithm with polyhedral computations based on nonnegative least squares model predictive control mpc is one of the most successful techniques adopted in industry to control multivariable systems under constraints on input and output variables to circumvent the main drawback of mpc i e the need to solve a quadratic program qp on line to compute the control action explicit mpc was proposed in the past to precompute the control law off line using multiparametric qp mpqp the resulting form of the mpc law is piecewise affine which is extremely easy to code can be computed online by simple arithmetic operations and requires a maximum number of iterations that can be exactly determined a priori on the other hand the offline computations to solve the mpqp problem require detecting emptiness full dimensionality and minimal hyperplane representations of polyhedra and other computational geometric operations while most of the existing methods solve such operations via linear programming the approach proposed in this paper relies on a nonnegative least squares nnls solver that is very simple to code and fast to execute in addition the new approach exploits qp duality to identify and construct critical regions and to handle degeneracy issues

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

brief an algorithm for multi parametric quadratic programming and explicit mpc solutions explicit solutions to constrained linear model predictive control problems can be obtained by solving multi parametric quadratic programs mp qp where the parameters are the components of the state vector we study the properties of the polyhedral partition of the state space induced by the multi parametric piecewise affine solution and propose a new mp qp solver compared to existing algorithms our approach adopts a different exploration strategy for subdividing the parameter space avoiding unnecessary partitioning and qp problem solving with a significant improvement of efficiency

the explicit linear quadratic regulator for constrained systems we present a technique to compute the explicit state feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints we show that this closed form solution is piecewise linear and continuous as a practical consequence of the result constrained linear quadratic regulation becomes attractive also for systems with high sampling rates as on line quadratic programming solvers are no more required for the implementation

efficient robust predictive control predictive constrained control of time varying and or uncertain linear systems has been effected through the use of ellipsoidal invariant sets kothare et al 1996 linear matrix inequalities lmis have been used to design a state dependent state feedback law that maintains the state vector inside invariant feasible sets for the purposes of prediction however at each time instant the state feedback law is assumed constant in addition due to the large number of lmis involved online computation becomes intractable for anything other than small dimensional systems here we propose an approach that deploys a fixed state feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state feedback law the problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online over and above the very significant reduction in computational cost the extra degrees of freedom allow for better performance and wider applicability

a global piecewise smooth newton method for fast large scale model predictive control in this paper the strictly convex quadratic program qp arising in model predictive control mpc for constrained linear systems is reformulated as a system of piecewise affine equations a regularized piecewise smooth newton method with exact line search on a convex differentiable piecewise quadratic merit function is proposed for the solution of the reformulated problem the algorithm has considerable merits when applied to mpc over standard active set or interior point algorithms its performance is tested and compared against state of the art qp solvers on a series of benchmark problems the proposed algorithm is orders of magnitudes faster especially for large scale problems and long horizons for example for the challenging crude distillation unit model of pannocchia rawlings and wright 2007 with 252 states 32 inputs and 90 outputs the average running time of the proposed approach is 1 57 ms

online constraint removal we show how to use a lyapunov function to accelerate mpc for linear discrete time systems with linear constraints and quadratic cost our method predicts in the current time step which constraints will be inactive in the next time step these constraints can be removed from the online optimization problem of the next time step the criterion for the detection of inactive constraints is based on the decrease of the lyapunov function along the trajectory of the controlled system the criterion is simple easy to implement in existing mpc algorithms and its computational cost is small