properties of symmetric matrices abstract we find the extreme points and the smooth points of the unit ball of the banach space n of all real symmetric n n matrices

the lambrechts stanley model of configuration spaces we prove the validity over mathbb r of a commutative differential graded algebra model of configuration spaces for simply connected closed smooth manifolds answering a conjecture of lambrechts stanley we get as a result that the real homotopy type of such configuration spaces only depends on the real homotopy type of the manifold we moreover prove if the dimension of the manifold is at least 4 that our model is compatible with the action of the fulton macpherson operad weakly equivalent to the little disks operad when the manifold is framed we use this more precise result to get a complex computing factorization homology of framed manifolds our proofs use the same ideas as kontsevich s proof of the formality of the little disks operads

the cohomology of certain function spaces we study a spectral sequence converging to the cohomology of the configuration space of n ordered points in a manifold a chain complex is constructed with homology equal to the e2 term if the field is the rationals and the manifold is formal then the spectral sequence is shown to collapse the results are applied to compute the anderson spectral sequence converging to the cohomology of a function space 0 introduction from the point of view of homotopy theory a very natural problem is to describe the topology of function spaces in 1956 r thom 13 wrote a seminal paper in 1972 d anderson 1 using the notion of a cosimplicial space introduced by bousfield and kan 4 obtained a spectral sequence for the homology of function spaces in 1969 1970 gelfand and fuks 6 wrote a series of papers on the continuous cohomology of the lie algebra of vector fields on a manifold m they introduced a pair of spectral sequences for such a study based on the results of gelfand fuks bott and fuks conjectured that the continuous cohomology of the lie algebra of vector fields was the usual cohomology of a certain function space namelys that of the space of sections of a certain associated bundle to the tangent bundle of m haefliger 7 and bott and segal 3 in 1978 using entirely different methods proved the bott fuks conjecture the method followed by bott and segal was to utilize the ideas of anderson the cohomology of configuration spaces appears in the computation of the el term of the above spectral sequences in 1978 f cohen and l taylor 5 were able to compute the cohomology of the configuration spaces for a certain class of manifolds and applied these results to describe the gelfand fuks cohomology of these manifolds also there has been interest in the configuration spaces as they appear as approximations to certain function spaces see 2 9 in this paper we study the configuration spaces and the function spaces as outlined by bott segal in both instances the cohomology can be obtained from a received by the editors october 3 1987 and in revise l form july 1 1989 1980 mathematics subject classification 1985 revision primary 54c35

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

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

properties of generalized predictive control control design using long range prediction based on a dynamic model of the plant has become an important contender for high performance applications the model can be derived analytically but more typically it is provided by simple experiments or in the adaptive context by recursive estimation many methods have been proposed in the literature depending on the assumed model structure and the choice of cost function here the approach called generalized predictive control or gpc is reviewed of special interest is the derivation of prediction equations using an observer polynomial and a demonstration that the selection of particular horizons the costing horizons and the control horizon leads to well understood basic techniques such as dead beat pole placement and lq simulated examples show that gpc is suitable for controlling a complex plant such as unstable inverse unstable systems and how the observer polynomial independently tailors the response to the disturbances preprogrammed set points as with robot trajectory control and actuator nonlinearities can also be catered for a discussion of adaptive applications of gpc to several industrial process concludes that the method is easy to use and effective

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

constrained robust model predictive control for time delay systems with polytopic description this paper considers an infinite horizon min max model predictive control mpc problem for time delay discrete systems with a polytopic uncertainty description which has rarely been addressed in the literature several new solutions to this problem are proposed which are inter complementary if there is an input delay then the single state feedback solution requires all the system matrices to be stable this deficiency can be removed by adding free control moves before a modified state feedback law if there is no input delay then a computationally efficient off line state feedback solution can be applied in order to obtain desirable region of attraction while at the same time preserve reasonably low on line computational effort one can choose the standard solution simulation results are given to illustrate these new solutions

geometry of 2 homogeneous polynomials on lp spaces 1 p abstract we characterise extreme 2 homogeneous polynomials of diagonal form on l p with p 2 and give a class of examples of smooth points we describe the unit ball of p 2 l p 2 for both the cases p 2 and p we apply the results thus found to study the geometry of the predual of p 2 l p 2 and to show that for p 2 an extreme 2 homogeneous polynomial on l p 2 remains extreme when regarded as a polynomial on l p

robust receding horizon control of constrained nonlinear systems we present a method for the construction of a robust dual mode receding horizon controller which can be employed for a wide class of nonlinear systems with state and control constraints and model error the controller is dual mode in a neighborhood of the origin the control action is generated by a linear feedback controller designed for the linearized system outside this neighborhood receding horizon control is employed existing receding horizon controllers for nonlinear continuous time systems which are guaranteed to stabilize the nonlinear system to which they are applied require the exact solution at every instant of an optimal control problem with terminal equality constraints these requirements are considerably relaxed in the dual mode receding horizon controller presented in this paper stability is achieved by imposing a terminal inequality rather than an equality constraint only approximate minimization is required a variable time horizon is permitted robustness is achieved by employing conservative state and stability constraint sets thereby permitting a margin of error the resultant dual mode controller requires considerably less online computation than existing receding horizon controllers for nonlinear constrained systems

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

worst case formulations of model predictive control for systems with bounded parameters two different predictive control formulations are developed based on minimization of the worst case quadratic cost for systems with bounded parameters the two formulations differ on the assumptions made about the future inputs in optimizing the current input one assumes open loop control while the other considers closed loop control their closed loop properties such as asymptotic stability are examined we then focus on a moving average model with an integrator and derive computationally simpler suboptimal algorithms

multilinear forms subsymmetric polynomials and spreading models on banach spaces abstract some applications of ramsey theory to the study of multilinear forms and polynomials on banach spaces are given related to the existence of lower l q estimates of sequences we give an explicit representation of subsymmetric polynomials in banach spaces with subsymmetric basis finally we apply our results to get approximation of polynomials by subsymmetric polynomials and to the problem of stabilization of a certain chain of algebras generated by polynomials

general interpolation in mpc and its advantages linear terminal control laws in model predictive control imply the need to compromise between performance and size of terminal region a remedy allows the control law to vary with time but requires very large online computation here use is made of nonlinear terminal laws comprising an interpolation between predefined laws each with its own precomputed terminal set this reduces the online computation significantly yet results in large terminal sets defined by the convex hull of invariant ellipsoids the use of a different prediction class also produces significant benefits with respect to performance

cyclomonotonicity and stabilizability properties of solutions of the difference periodic riccati equation the kalman filter associated with a discrete time linear t periodic system is tested the problem considered is that of selecting an initial covariance matrix such that the periodic filter based on the first t values of the kalman filter gain is stabilizing sufficient conditions are given that hinge on the cyclomonotonicity of the solution of the periodic riccati equation potential applications are found in filter design quasi linearization techniques for the periodic riccati equation and the design of receding horizon control strategies for periodic and multirate systems when specialized to time invariant systems the results give rise to new sufficient conditions for the cyclomonotonicity of the solutions of the time invariant riccati equation and the existence of periodic stabilizing feedback

moving horizon control of nonlinear systems with input saturation disturbances and plant uncertainty we present a moving horizon feedback system based on constrained optimal control algorithms for nonlinear plants with input saturation disturbances and plant uncertainty the system is a non conventional sampled data system its sampling periods vary from sampling instant to sampling instant and the control during the sampling time is not constant but determined by the solution of an open loop optimal control problem we show that the proposed moving horizon control system is robustly stable and is capable of attenuating l bounded disturbances

flat connections on configuration spaces and braid groups of surfaces abstract we construct an explicit bundle with flat connection on the configuration space of n points on a complex curve this enables one to recover the 1 formality isomorphism between the lie algebra of the prounipotent completion of the pure braid group of n points on a surface and an explicitly presented lie algebra and to extend it to a morphism from the full braid group of the surface to the semidirect product of the associated group with the symmetric group s n

extreme polynomials and multilinear forms onl1 abstract given a 2 homogeneous polynomialp x y ax2 by2 cxywith real coefficients let p rand p cdenote the norms ofpon the real and complex banach spacel21 respectively we show p r p c and obtain a sufficient and necessary condition on the coefficientsa b andcforpto have norm 1 applying these results we characterize extreme points of the unit ball of p 2l1 for the real banach spacel21and examine them for the complex banach spacel21 we apply them to find extreme points and strongly extreme points of the unit ball of p 2l1 and get an extremal 2 homogeneous polynomial onl1that is not an extreme point we also characterize extreme points and strongly extreme points of the unit ball of l ml1 or l ml1 0 1

a linear programming approach to constrained robust predictive control a receding horizon predictive control algorithm for systems with model uncertainty and input constraints is developed the proposed algorithm adopts the receding horizon dual mode i e free control moves and invariant set paradigm the approach is novel in that it provides a convenient way of combining predictions of control moves which are optimal in the sense of worst case performance with large target invariant sets thus the proposed algorithm has a large stabilizable set of states corresponding to a cautious state feedback law while enjoying the good performance of a tightly tuned but robust control law unlike earlier approaches which are based on qp or semidefinite programming here computational complexity is reduced through the use of lp

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

on the rationality homotopy type of configuration spaces in this article we prove that the rational homotopy type of the configuration space of n distinct points on a smooth projective variety x over c is determined by the rational cohomology ring of x together with its canonical orientation class this strengthens a previous result of fulton and macpherson fm see previous issue in this volume a few words are due on the history of differential graded algebras by a dga we shall mean a graded commutative differential graded algebra fulton and macpherson fm gave a model f n for the space f x n by resolving the singularities of the inclusion f x n c xn and using a result of morgan m on the rational homotopy type of smooth quasiprojective varieties over c surprisingly the dga f n depended not only on h x and its orientation class but also on the chern classes of x fulton suggested that one should be able to find another explicit model of f x n independent of chern classes the dga used here e n was found independently by the author and by totaro t totaro s idea was to study the leray spectral sequence of the inclusion f x n c xn by a weight filtration argument he showed that the only nontrivial differential in this spectral sequence is d2m the e2 term with the differential d2m is isomorphic to e n thus by a result of deligne d2 h e n h f x n as rings finally the relations 1 3 of theorem 1 2 are much older than any of the results mentioned above they were discovered by f cohen c see also clma in the study of configuration spaces of ir what happens could be described by saying that h f x n for a smooth projective variety x is related to h f lr2m n in the simplest possible way

admissible sets for slowly varying discrete time systems abstract this note describes a method to calculate admissible sets for dynamical systems with slowly varying parameters a set of initial states is said to be admissible if all state trajectories starting within this set respect safety critical constraints for all future time instances for many classes of systems the maximal admissible set mas equals the maximal robust positive invariant mrpi set therefore much of the literature focus on algorithms that produce approximations of the mrpi set however for slowly varying systems it turns out that the mas can be larger than the mrpi set a property referred to as the m proximal contraction property is defined for such systems to state sufficient conditions for sets to be admissible it is then shown that algorithm 2 produces an admissible set that is a superset of the mrpi set

holomorphic mappings on we describe the holomorphic mappings of bounded type and the arbitrary holomorphic mappings from the complex banach space 11 into a complex banach space x it is shown that these mappings have monomial expansions and the growth of the norms of the coefficients is characterized in each case this characterization is used to give new descriptions of the compact open topology and the nachbin ported topology on the space a 11 x of holomorphic mappings and to prove a lifting property for holomorphic mappings on 11 we also show that the monomials form an equicontinuous unconditional schauder basis for the space 11 so of holomorphic functions on 11 with the topology of uniform convergence on compact sets

the unit ball of the complex mathcal p 3h let h be a two dimensional complex hilbert space and mathcal p 3h the space of 3 homogeneous polynomials on h we give a characterization of the extreme points of its unit ball mathsf b mathcal p 3h from which we deduce that the unit sphere of mathcal p 3h is the disjoint union of the sets of its extreme and smooth points we also show that an extreme point of mathsf b mathcal p 3h remains extreme as considered as an element of mathsf b mathcal l 3h finally we make a few remarks about the geometry of the unit ball of the predual of mathcal p 3h and give a characterization of its smooth points

stabilizing tube based model predictive control terminal set and cost construction for lpv systems abstract this paper presents a stabilizing tube based mpc synthesis for lpv systems we employ terminal constraint sets which are required to be controlled periodically contractive periodically or finite step contractive sets are easier to compute and can be of lower complexity than true contractive ones lowering the required computational effort both off line and on line under certain assumptions on the tube parameterization recursive feasibility of the scheme is proven subsequently asymptotic stability of the origin is guaranteed through the construction of a suitable terminal cost based on a novel lyapunov like metric for compact convex sets containing the origin a periodic variant on the well known homothetic tube parameterization that satisfies the necessary assumptions and yields a tractable lpv mpc algorithm is derived the resulting mpc algorithm requires the on line solution of a single linear program with linear complexity in the prediction horizon the properties of the approach are demonstrated by a numerical example

technical communique extended invariance and its use in model predictive control the use of time invariant linear state feedback control laws for the definition of terminal invariant regions can be conservative thereby reducing the efficacy of predictive control in terms of size of stabilisable sets and closed loop performance this difficulty which is particularly pronounced in the case of nonlinear and or uncertain dynamics can be remedied through the use of time varying control laws and terminal invariant sets in existing mpc schemes however these have to be computed online thereby rendering implementation impracticable for anything other than low dimensional systems here the definition of invariance is extended to apply over n predicted control moves thereby enabling the use of pre determined offline time varying state feedback gains more importantly this extension allows for the use of local uncertainty or linear difference inclusion sets and thus affords significant improvements e g in terms of the size of terminal regions

on efficient computation of low complexity controlled invariant sets for uncertain linear systems a method is presented for determining invariant low complexity polytopic sets and associated linear feedback laws for linear systems with polytopic uncertainty conditions based on the relationship between 2 and norms are used to define an initial invariant low complexity polytope as the solution of a semi definite program the problem of computing a maximal controlled invariant low complexity polytopic set is then formulated as a bilinearly constrained problem and a relaxation of this problem is derived as an iterative sequence of convex programs the proposed method scales linearly with the state dimension which allows the possibility of determining low complexity robust controlled invariant sets for high order systems

min max control of constrained uncertain discrete time linear systems for discrete time uncertain linear systems with constraints on inputs and states we develop an approach to determine state feedback controllers based on a min max control formulation robustness is achieved against additive norm bounded input disturbances and or polyhedral parametric uncertainties in the state space matrices we show that the finite horizon robust optimal control law is a continuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric linear programs when the optimal control law is implemented in a receding horizon scheme only a piecewise affine function needs to be evaluated on line at each time step the technique computes the robust optimal feedback controller for a rather general class of systems with modest computational effort without needing to resort to gridding of the state space

a quasi infinite horizon nonlinear model predictive control scheme with guaranteed stability we present in this paper a novel nonlinear model predictive control scheme that guarantees asymptotic closed loop stability the scheme can be applied to both stable and unstable systems with input constraints the objective functional to be minimized consists of an integral square error ise part over a finite time horizon plus a quadratic terminal cost the terminal state penalty matrix of the terminal cost term has to be chosen as the solution of an appropriate lyapunov equation furthermore the setup includes a terminal inequality constraint that forces the states at the end of the finite prediction horizon to lie within a prescribed terminal region if the jacobian linearization of the nonlinear system to be controlled is stabilizable we prove that feasibility of the open loop optimal control problem at time t 0 implies asymptotic stability of the closed loop system the size of the region of attraction is only restricted by the requirement for feasibility of the optimization problem due to the input and terminal inequality constraints and is thus maximal in some sense

on loop spaces of configuration spaces this paper gives a brief introduction to the theory of conguration spaces some recent results about the homology of their loop spaces are also presented we also discuss their relationship to vassiliev invariants for knots

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

invariant sets techniques for youla kucera parameter synthesis this article addresses an invariant sets approach for youla kucera parameter synthesis using linear matrix inequality lmi techniques given a linear discrete time observer based system affected by bounded disturbances and constraints the proposed technique furnishes the best youla parameter in terms of finding an invariant ellipsoidal set satisfying the constraints and having the maximal ellipsoidal projection on the state space compared with the results obtained for an observer based design the synthesis of a youla parameter provides a larger ellipsoidal projection and an improved sensitivity function the price to pay for these achievements in terms of robustness is usually a slow closed loop performance with degraded complementary sensitivity function in order to obtain a compromise between robustness and performance two methods are proposed the first method imposes a new bound on the lyapunov function decreasing speed and the second refers to the pole placement concept the aforementioned appro

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

norm attaining bilinear forms onl1 0 1 abstract we show that the set of norm attaining bilinear forms onl1 0 1 is not dense in the space of all continuous bilinear forms

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

equivalent norms in polynomial spaces and applications abstract in this paper equivalence constants between various polynomial norms are calculated as an application we also obtain sharp values of the hardy littlewood constants for 2 homogeneous polynomials on l p 2 spaces 2 p we also provide lower estimates for the hardy littlewood constants for polynomials of higher degrees

constrained robust model predictive control based on periodic invariance in this paper a receding horizon control method for input state constrained systems with polyhedral uncertainties is proposed the dual mode prediction strategy is adopted to deal with the constraints and periodically invariant sets are used to derive a target invariant set of the dual mode prediction strategy the proposed control method is shown to have novel characteristics earlier approaches do not have i e i the convex hull of all the periodically invariant sets are invariant in the sense that there are feasible feedback gains guaranteeing invariance for any elements of the convex hull and it provides larger target sets than other methods based on ordinary invariant sets ii a particular convex hull of periodically invariant sets that is computable off line can be used as an invariant target set in this case the number of on line variables is only equal to the period of invariance and thus the proposed algorithm is computationally very efficient these on line variables provide interpolation between different feedback gains to yield best performance