convex lifting theory and control applications this paper presents the concept of convex lifting which will be proven to enable significant implementation benefits for the class of piecewise affine controllers accordingly two different algorithms to construct a convex lifting for a given polyhedral polytopic partition will be presented these two algorithms rely on either the vertex or the halfspace representation of the related polyhedra also we introduce an algorithm to refine a polyhedral partition which does not admit a convex lifting into a convexly liftable one furthermore two different schemes will be put forward to considerably reduce both the memory footprint and the online evaluation effort which play a key role in implementation of piecewise affine controllers finally these results will be illustrated via numerical examples and a complexity analysis

constrained linear quadratic regulation the paper is a contribution to the theory of the infinite horizon linear quadratic regulator lqr problem subject to inequality constraints on the inputs and states extending an approach first proposed by sznaier and damborg 1987 a solution algorithm is presented which requires solving a finite number of finite dimensional positive definite quadratic programs the constrained lqr outlined does not feature the undesirable mismatch between open loop and closed loop nominal system trajectories which is present in the other popular forms of model predictive control mpc that can be implemented with a finite quadratic programming algorithm the constrained lqr is shown to be both optimal and stabilizing the solution algorithm is guaranteed to terminate in finite time with a computational cost that has a reasonable upper bound compared to the minimal cost for computing the optimal solution inherent to the approach is the removal of a tuning parameter the control horizon which is present in other mpc approaches and for which no reliable tuning guidelines are available two examples are presented that compare constrained lqr and two other popular forms of mpc the examples demonstrate that constrained lqr achieves significantly better performance than the other forms of mpc on some plants and the computational cost is not prohibitive for online implementation

3 d traveltimes and amplitudes by gridded rays seismic imaging in three dimensions requires the calculation of traveltimes and amplitudes of a wave propagating through an elastic medium they can be computed efficiently and accurately by integrating the eikonal equation on an elemental grid using finite difference methods unfortunately this approach to solving the eikonal equation is potentially unstable unless the grid sampling steps satisfy stability conditions or wavefront tracking algorithms are used we propose a new method for computing traveltimes and amplitudes in 3 d media that is simple fast unconditionally stable and robust defining the slowness vector as p px py pz and assuming an isotropic medium the ray velocity v is related to the slowness vector by the relation v p p p rays emerging from gridpoints on a horizontal plane are propagated downward a single vertical grid step to a new horizontal plane the components of the slowness vector are then interpolated to gridpoints on this next horizontal plane this is termed regriddi

upwind finite difference calculation of traveltimes seismic traveltimes can be computed efficiently on a regular grid by an upwind finite difference method the method solves a conservation law that describes changes in the gradient components of the traveltime field the traveltime field itself is easily obtained from the solution of the conservation law by numerical integration the conservation law derives from the eikonal equation and its solution depicts the first arrival time field the upwind finite difference scheme can be implemented in fully vectorized form in contrast to a similar scheme proposed recently by vidale the resulting traveltime field is useful both in kirchhoff migration and modeling and in seismic tomography many reliable methods exist for the numerical solution of conservation laws which appear in fluid mechanics as statements of the conservation of mass momentum etc a first order upwind finite difference scheme proves accurate enough for seismic applications upwind schemes are stable because they mimic the behavior of flu

corrigendum corrigendum to the explicit linear quadratic regulator for constrained systems automatica 38 1 2002 3 20 we apologize that example 7 1 as publishedin bemporad morari dua andpistikopoulos 2002 is incorrect due to a miscalculation of the weight matrix p on the terminal state

explicit hybrid model predictive control this article presents an algorithm for the exact solution of explicit hybrid model predictive control problems of time invariant discrete time mixed logical dynamical systems using multiparametric programming this control problem is formulated as a multiparametric mixed integer quadratic programming problem where the initial states of the system are treated as parameters in conjunction with decomposition type or branch and bound type approaches the proposed solution strategy first creates affine relaxations of nonconvex critical regions that result from the comparison of two quadratic objective functions these relaxations are then used to generate an affine outer approximation for the critical regions upon termination the bounded space of the initial states is partitioned into possibly nonconvex critical regions and corresponding optimal control laws

lagrange stability and performance analysis of discrete time piecewise affine systems with logic states in this paper we consider discrete time piecewise affine systems with boolean inputs outputs and states and show that they can be represented in a logic canonical form where the logic variables influence the switching between different submodels but not the continuous valued dynamics we exploit this representation for studying lagrange stability and developing performance analysis procedures based on linear matrix inequalities moreover by using arguments from dissipativity theory for non linear systems we generalize our approach to solve the h analysis problem

on the structure of the set of active sets in constrained linear quadratic regulation the constrained linear quadratic regulation problem is solved by a continuous piecewise affine function on a set of state space polytopes it is an obvious question whether this solution can be built up iteratively by increasing the horizon i e by extending the classical backward dynamic programming solution for the unconstrained case to the constrained case unfortunately however the piecewise affine solution for horizon n is in general not contained in the piecewise affine law for horizon n 1 we show that backward dynamic programming does in contrast result in a useful structure for the set of the active sets that defines the solution essentially every active set for the problem with horizon n 1 results from extending an active set for horizon n if the constraints are ordered stage by stage consequently the set for horizon n 1 can be found by only considering the constraints of the additional stage furthermore it is easy to detect which polytopes and affine pieces are invariant to increasing the horizon and therefore persist in the limit n several other aspects of the structure of the set of active sets become evident if the active sets are represented by bit tuples there exists for example a subset of special active sets that generates a positive invariant and persistent i e horizon invariant set around the origin it is very simple to identify these special active sets and the positive invariant and persistent region can be found without solving optimal control or auxiliary optimization problems the paper briefly discusses the use of these results in model predictive control some opportunities for uses in computational methods are also briefly summarized

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

cardinality constrained linear quadratic optimal control as control implementation often incurs not only a variable cost associated with the magnitude or energy of the control but also a setup cost we consider a discrete time linear quadratic lq optimal control problem with a limited number of control implementations termed in this technical note the cardinality constrained linear quadratic optimal control cclq we first derive a semi analytical feedback policy for cclq problems using dynamic programming dp due to the exponential growth of the complexity in calculating the action regions however dp procedure is only efficient for cclq problems with a scalar state space recognizing this fact we develop then two lower bounding schemes and integrate them into a branch and bound bnb solution framework to offer an efficient algorithm in solving general cclq problems adopting the devised solution algorithm for cclq problems we can solve efficiently the linear quadratic optimal control problem with setup costs

technical communique on the facet to facet property of solutions to convex parametric quadratic programs in some of the recently developed algorithms for convex parametric quadratic programs it is implicitly assumed that the intersection of the closures of two adjacent critical regions is a facet of both closures this will be referred to as the facet to facet property it is shown by an example whose solution is unique that the facet to facet property does not hold in general consequently some existing algorithms cannot guarantee that the entire parameter space will be explored a simple modification applicable to several existing algorithms is presented for the purpose of overcoming this problem numerical results indicate that compared to the original algorithms for parametric quadratic programs the proposed method has lower computational complexity for problems whose solutions consist of a large number of critical regions

enumeration based approach to solving parametric linear complementarity problems this paper presents a solution method for parametric linear complementarity problems plcp that relies on an enumeration technique to discover all feasible bases the enumeration procedure is based on evaluating all possible combinations of active constraints and testing for feasibility although the enumeration approach is known to grow exponentially in the number of constraints the formulation of the plcp allows incorporation of cheap rank tests to quickly prune the infeasible directions in the exploration the motivation for the development of the enumeration based plcp solver is that it represents a direct method to solve parametric linear and quadratic optimization problems as well as their mixed integer counterparts these types of problems often arise in the field of model predictive control for linear and hybrid systems the enumeration based plcp solver offers another alternative to compute explicit solutions in the field of hybrid model predictive control that can be extremely effective in some important cases

optimal complexity reduction of polyhedral piecewise affine systems this paper focuses on the np hard problem of reducing the complexity of piecewise polyhedral systems e g polyhedral piecewise affine pwa systems the results are fourfold firstly the paper presents two computationally attractive algorithms for optimal complexity reduction that under the assumption that the system is defined over the cells of a hyperplane arrangement derive an equivalent polyhedral piecewise system that is minimal in the number of polyhedra the algorithms are based on the cells and the markings of the hyperplane arrangement in particular the first algorithm yields a set of disjoint non overlapping merged polyhedra by executing a branch and bound search on the markings of the cells the second approach leads to non disjoint overlapping polyhedra by formulating and solving an equivalent and well studied logic minimization problem secondly the results are extended to systems defined on general polyhedral partitions and not on cells of hyperplane arrangements thirdly the paper proposes a technique to further reduce the complexity of piecewise polyhedral systems if the introduction of an adjustable degree of error is acceptable fourthly the paper shows that based on the notion of the hyperplane arrangement pwa state feedback control laws can be implemented efficiently three examples including a challenging industrial problem illustrate the algorithms and show their computational effectiveness in reducing the complexity by up to one order of magnitude

finite difference solution of the eikonal equation along expanding wavefronts we show that a scheme to solve the 2 d eikonal equation by a finite difference method can violate causality for moderate to large velocity contrasts v2 v1 2 as an alternative we present a finite difference scheme in which the solution region progresses outward from an expanding wavefront rather than an expanding square and therefore honors causality our method appears to be stable and reasonably accurate for a variety of velocity models with moderate to large velocity contrasts the penalty is a large increase in computational cost and programming effort

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

fast algorithm for a constrained infinite horizon lq problem this paper presents an efficient algorithmic solution to the infinite horizon linear quadratic optimal control problem for a discrete time siso plant subject to bound constraints on a scalar variable the solution to the corresponding quadratic programming problem is based on the active set method and on dynamic programming it is shown that the optimal solution can be updated after inclusion or removal of an active constraint by a simple procedure requiring in the order of kn operations n being the system order and k the time at which the constraint is included or removed

on constrained infinite time linear quadratic optimal control abstract this work presents a solution to the infinite time linear quadratic optimal control itlqoc problem with state and control constraints it is shown that a single finite dimensional convex program of known size can yield this solution properties of the resulting value function with respect to initial conditions are also established and are shoen to be useful in determining the aforementioned problem size an example illustrating the method is finally presented

complexity reduction of explicit model predictive control via separation the problem of reducing complexity of explicit mpc feedback laws for linear systems is considered we propose to simplify controllers defined by continuous piecewise affine pwa functions by employing separating functions if a state resides in a region where the optimal control action attains a saturated value the optimal control move is determined from the sign of the separator thus instead of storing all regions only the unconstrained regions and the separator are needed we propose several approaches to construct separators with different efficacy and properties

inverse minimax optimality of model predictive control policies we present novel results linking model predictive control mpc and minimax optimal control theory specifically we show that for linear discrete time systems with constraints and disturbance inputs the closed loop optimal solutions of a particular class of minimax optimal control problems are a class of typical mpc policies we also present conditions which ensure that the inverse optimal mpc policies achieve a prescribed regional l2 gain from the disturbance input

derangements ehrhart theory and local h polynomials abstract the eulerian polynomials and derangement polynomials are two well studied generating functions that frequently arise in combinatorics algebra and geometry when one makes an appearance the other often does so as well and their corresponding generalizations are similarly linked this is this case in the theory of subdivisions of simplicial complexes where the eulerian polynomial is an h polynomial and the derangement polynomial is its local h polynomial separately in ehrhart theory the eulerian polynomials are generalized by the h polynomials of s lecture hall simplices here we show that derangement polynomials are analogously generalized by the box polynomials or local h polynomials of the s lecture hall simplices and that these polynomials are all real rooted we then connect the two theories by showing that the local h polynomials of common subdivisions in algebra and topology are realized as local h polynomials of s lecture hall simplices we use this connection to address some open questions on real rootedness and unimodality of generating polynomials some from each side of the story

simplices for numeral systems the family of lattice simplices in r n formed by the convex hull of the standard basis vectors together with a weakly decreasing vector of negative integers include simplices that play a central ro

clipping based complexity reduction in explicit mpc the idea of explicit model predictive control mpc is to characterize optimal control inputs as an explicit piecewise affine pwa function of the initial conditions the function however is often too complex and either requires too much processing power to evaluate on line or consumes a prohibitive amount of memory the paper focuses on the memory issue and proposes a novel method of replacing a generic continuous pwa function by a different function of significantly lower complexity in such a way that the simple function guarantees the same properties as the original the idea is based on eliminating regions of the pwa function over which the function attains a saturated value an extensive case study is presented which confirms that a significant reduction of complexity is achieved in general

fast phase space computation of multiple arrivals we present a fast general computational technique for computing the phase space solution of static hamilton jacobi equations starting with the liouville formulation of the characteristic equations we derive escape equations which are static time independent eulerian pdes they represent all arrivals to the given boundary from all possible starting configurations the solution is numerically constructed through a one pass formulation building on ideas from semi lagrangian methods dijkstra like methods for the eikonal equation and ordered upwind methods to compute all possible trajectories corresponding to all possible boundary conditions the technique is of computational order o n log n where n is the total number of points in the computational phase space domain any particular set of boundary conditions then is extracted through rapid post processing suggestions are made for speeding up the algorithm in the case when the particular distribution of sources is provided in advance as an application we apply the technique to the problem of computing first multiple and most energetic arrivals to the eikonal equation

an algebraic approach to bounded controllability of linear systems in this paper we present an algebraic approach to the proof that a linear system with matrices a b is null controllable using bounded inputs if and only if it is null controllable with unbounde

a unified framework for the study of anti windup designs we present a unified framework for the study of linear time invariant lti systems subject to control input nonlinearities the framework is based on the following two step design paradigm design the linear controller ignoring control input nonlinearities and then add anti windup bumpless transfer awbt compensation to minimize the adverse eflects of any control input nonlinearities on closed loop performance the resulting awbt compensation is applicable to multivariable controllers of arbitrary structure and order all known lti anti windup and or bumpless transfer compensation schemes are shown to be special cases of this framework it is shown how this framework can handle standard issues such as the analysis of stability and performance with or without uncertainties in the plant model the actual analysis of stability and performance and robustness issues are problems in their own right and hence not detailed here the main result is the unification of existing schemes for awbt compensation under a general framework

ultra fast stabilizing model predictive control via canonical piecewise affine approximations this paper investigates the use of canonical piecewise affine pwa functions for approximation and fast implementation of linear mpc controllers the control law is approximated in an optimal way over a regular simplicial partition of a given set of states of interest the stability properties of the resulting closed loop system are analyzed by constructing a suitable pwa lyapunov function the main advantage of the proposed approach to the implementation of mpc controllers is that the resulting stabilizing approximate mpc controller can be implemented on chip with sampling times in the order of tens of nanoseconds

ultimate boundedness control for uncertain discrete time systems via set induced lyapunov functions in this note linear discrete time systems affected by both parameter and input uncertainties are considered the problem of the synthesis of a feedback control assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned rate of convergence is investigated it is shown that the problem has a solution if and only if there exists a certain lyapunov function which does not belong to a preassigned class of functions e g the quadratic ones but it is determined by the target set in which ultimate boundedness is desired one of the advantages of this approach is that we may handle systems with control constraints no matching assumptions are made for systems with linearly constrained uncertainties it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets the resulting compensator may be implemented as a linear variable structure control

brief multiplier theory for stability analysis of anti windup control systems we apply the passivity theorem with appropriate choice of multipliers to develop a general framework for analyzing stability of anti windup bumpless transfer awbt schemes for appropriate choices of the multipliers we reduce the resulting sufficient conditions for stability to equivalent linear matrix inequalities lmis we show that a number of previously reported attempts to analyze awbt stability using seemingly diverse techniques such as the popov circle and off axis circle criteria the optimally scaled small gain theorem generalized m upper bound and describing functions can all be generalized within the framework of this paper

linear machine solution to point location problem linear machine has been recently proposed as an elegant solution for solving the point location problem arising in multi parametric programming mp p based online optimization linear machine associates a linear discriminant function with each polytopic region in the parametric space the solution to the point location problem is then obtained by simply evaluating these discriminant functions and finding their maximum value in this technical note we rigorously establish the correctness of the linear machine generation procedure and identify a necessary condition for existence of linear machine a modified procedure involving systematic subdivision of the parametric space is proposed when this condition is not satisfied analysis of complexity and storage requirements along with computational experiments on a large sized example indicate that linear machine can be an efficient tool for solving the point location problem

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

an improved algorithm for combinatorial multi parametric quadratic programming the goal of multi parametric quadratic programming mpqp is to compute analytic solutions to parameter dependent constrained optimization problems e g in the context of explicit linear mpc we propose an improved combinatorial mpqp algorithm that is based on implicit enumeration of all possible optimal active sets and a simple saturation matrix pruning criterion which uses geometric properties of the constraint polyhedron for excluding infeasible candidate active sets in addition techniques are presented that allow to reduce the complexity of the discussed algorithm in the presence of symmetric problem constraints performance improvements are discussed for two example problems from the area of explicit linear mpc

explicit model predictive control a connected graph approach abstract the ability to solve model predictive control mpc problems of linear time invariant systems explicitly and offline via multi parametric quadratic programming mp qp has become a widely used methodology the most efficient approaches used to solve the underlying mp qp problem are either based on combinatorial considerations which scale unfavorably with the number of constraints or geometrical considerations which require heuristic tuning of the step size and correct identification of the active set in this paper we describe a novel algorithm which unifies these two types of approaches by showing that the solution of a mp qp problem is given by a connected graph where the nodes correspond to the different optimal active sets over the parameter space using an extensive computational study as well as the explicit mpc solution of a combined heat and power system the merits of the proposed algorithm are clearly highlighted

brief paper a predictive controller with artificial lyapunov function for linear systems with input state constraints this paper copes with the problem of satisfying input and or state hard constraints in set point tracking problems stability is guaranteed by synthesizing a lyapunov quadratic function for the system and by imposing that the terminal state lies within a level set of the function procedures to maximize the volume of such an ellipsoidal set are provided and interiorpoint methods to solve on line optimization are considered

multi resolution explicit model predictive control delta model formulation and approximation this paper deals with the explicit solution and approximation of the constrained linear finite time optimal control problem for systems with fast sampling rates to this aim the recently developed explicit model predictive control empc is reformulated and characterized using the operator to enjoy its promising advantages compared to the time shift operator using the proposed model empc formulation a systematic method is proposed for first designing a low complexity approximate empc solution and then improving its closed loop action without first determining an exact optimal solution that might be of prohibitive complexity it is shown that the stability and feasibility of the proposed sub optimal solution is guaranteed

analytical expression of explicit mpc solution via lattice piecewise affine function an analytical expression of the explicit solution to linear model predictive control mpc is proposed by the introduction of a lattice piecewise affine pwa function a systematic procedure is developed for building a lattice pwa representation from a continuous explicit mpc solution obtained by a multi parametric program a simple method is presented to remove the redundant parameters in the lattice expression of mpc control laws the effectiveness of this approach is supported by the study of three benchmark mpc problems the proposed analytical expression provides a very efficient and practically viable method for implementing the explicit mpc solutions regarding its online calculation and memory space requirements

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

a novel approach to multiparametric quadratic programming multiparametric mp programming pre computes optimal solutions offline which are functions of parameters whose values become apparent online this makes it particularly well suited for applications that need a rapid solution of online optimization problems in this work we propose a novel approach to multiparametric programming problems based on an enumeration of active sets and use it to obtain a parametric solution for a convex quadratic program qp to avoid the combinatorial explosion of the enumeration procedure an active set pruning criterion is presented that makes the enumeration implicit the method guarantees that all regions of the partition are critical regions without any artificial cuts and further that no region of the parameter space is left unexplored

a dynamic programming approach to solving constrained linear quadratic optimal control problems abstract the solution of a constrained linear quadratic regulator problem is determined by the set of its optimal active sets we propose an algorithm that constructs this set of active sets for a desired horizon n from that for horizon n 1 while it is not obvious how to extend the optimal feedback law itself for horizon n 1 to horizon n a simple relation between the optimal active sets for two successive horizon lengths has recently been established essentially we show how to use this recent result to improve the efficiency of existing active set enumeration algorithms

stability of model predictive control with mixed constraints we derive stability conditions for model predictive control mpc with hard constraints on the inputs and soft constraints on the outputs for an infinitely long output horizon we show that with state feedback mpc is globally asymptotically stabilizing if and only if all the eigenvalues of the open loop system are in the closed unit disk with output feedback we show that the results hold if all the eigenvalues are strictly inside the unit circle the online optimization problem defining mpc can be posed as a finite dimensional quadratic program even though the output constraints are specified over an infinite horizon

robust explicit model predictive control via regular piecewise affine approximation this paper proposes an explicit model predictive control design approach for regulation of linear time invariant systems subject to both state and control constraints in the presence of additive disturbances the proposed control law is implemented as a piecewise affine function defined on a regular simplicial partition and has two main positive features first the regularity of the simplicial partition allows one to efficiently implement the control law on digital circuits thus achieving extremely fast computation times moreover the asymptotic stability or the convergence to a set including the origin of the closed loop system can be enforced a priori rather than checked a posteriori via lyapunov analysis

admissible sets and feedback control for discrete time linear dynamical systems with bounded controls and states for linear discrete time dynamical systems necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set a variable structure linear state feedback controller given in terms of the controls at the vertices of the polyhedral state constraint set is presented

brief paper using hash tables to manage the time storage complexity in a point location problem application to explicit model predictive control the online computational burden of linear model predictive control mpc can be moved offline by using multi parametric programming so called explicit mpc the solution to the explicit mpc problem is a piecewise affine pwa state feedback function defined over a polyhedral subdivision of the set of feasible states the online evaluation of such a control law needs to determine the polyhedral region in which the current state lies this procedure is called point location its computational complexity is challenging and determines the minimum possible sampling time of the system a new flexible algorithm is proposed which enables the designer to trade off between time and storage complexities utilizing the concept of hash tables and the associated hash functions the proposed method solves an aggregated point location problem that overcomes prohibitive complexity growth with the number of polyhedral regions while the storage processing trade off can be optimized via scaling parameters the flexibility and power of this approach is supported by several numerical examples

finite difference solution of the eikonal equation using an efficient first arrival wavefront tracking scheme first break traveltimes can be accurately computed by the finite difference solution of the eikonal equation using a new corner node discretization scheme it offers accuracy advantages over the traditional cell centered node scheme a substantial efficiency improvement is achieved by the incorporation of a wavefront tracking algorithm based on the construction of a minimum traveltime tree for the traditional discretization scheme an accurate average value for the local squared slowness is found to be crucial in stabilizing the numerical scheme for models with large slowness contrasts an improved method based on the traditional discretization scheme can be used to calculate traveltimes in arbitrarily varying velocity models but the method based on the corner node discretization scheme provides a much better solution

on the stabilizing property of siorhc it is shown that the stabilizing property of siorhc stabilizing i o receding horizon control holds for general stabilizable discrete time linear plants irrespective of the condition that the plant has no poles at the origin

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

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

inherent robustness properties of quasi infinite horizon nonlinear model predictive control we consider inherent robustness properties of model predictive control mpc for continuous time nonlinear systems with input constraints and terminal constraints we show that mpc with a nominal prediction model and persistent but bounded disturbances has some degree of inherent robustness when the terminal control law and the terminal penalty matrix are chosen as the linear quadratic control law and the related lyapunov matrix respectively we emphasize that the input constraint sets can be any compact set rather than convex sets and our results do not depend on the continuity of the optimal cost function or of the control law in the interior of the feasible region

efficient on line computation of constrained optimal control we consider constrained finite time optimal control problems for discrete time linear time invariant systems with constraints on inputs and outputs based on linear and quadratic performance indices the solution to such problems is a time varying piecewise affine pwa state feedback law and can be computed by means of multiparametric programming by exploiting the properties of the value function and the piecewise affine optimal control law of the constrained finite time optimal control cftoc we propose two new algorithms that avoid storing the polyhedral regions the new algorithms significantly reduce the on line storage demands and computational complexity during evaluation of the pwa feedback control law resulting from the cftoc

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

on the log concavity of hilbert series of veronese subrings and ehrhart series for every positive integer n consider the linear operator un on polynomials of degree at most d with integer coefficients defined as follows if we write frac h t 1 t d 1 sum m geq 0 g m t m for some polynomial g m with rational coefficients then frac rm u n h t 1 t d 1 sum m geq 0 g nm t m we show that there exists a positive integer nd depending only on d such that if h t is a polynomial of degree at most d with nonnegative integer coefficients and h 0 1 then for n nd unh t has simple real negative roots and positive strictly log concave and strictly unimodal coefficients applications are given to ehrhart polynomials and unimodular triangulations of dilations of lattice polytopes as well as hilbert series of veronese subrings of cohen macauley graded rings

asymptotically exact stabilisation for constrained discrete takagisugeno systems via set invariance given a takagisugeno ts system this paper proposes a novel methodology to obtain the state feedback controller guaranteeing asymptotically as a polya related complexity parameter grows the largest membership shape independent possible domain of attraction with contraction rate performance based on polyhedral contractive sets from constrained linear systems literature the resulting controller is valid for any realisation of the memberships as usual in most ts literature for a finite complexity parameter an inner estimate of such largest set is obtained the frontier of such approximation can be understood as the level set of a polyhedral control lyapunov function convergence of a proposed iterative algorithm is asymptotically necessary and sufficient for ts system stabilisation for a high enough value of the complexity parameter any conceivable shape independent lyapunov controller design procedure will yield a proven domain of attraction smaller or equal to the algorithm s output

3 d traveltime computation using the fast marching method we present a fast algorithm for solving the eikonal equation in three dimensions based on the fast marching method the algorithm is of the order o n log n where n is the total number of grid points in the computational domain the algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain the method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity in addition the method resolves any overturning propagation wavefronts we begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details we then show examples of traveltime propagation through the seg eage salt model using point source and planewave initial conditions and analyze the error in constant velocity media the algorithm allows for any shape of the initial wavefront while a point source is the most commonly used initial condition initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth surface to another the algorithm presented here is designed for computing first arrival traveltimes nonetheless since it exploits the fast marching method for solving the eikonal equation we believe it is the fastest of all possible consistent schemes to compute first arrivals

robust and efficient upwind finite difference traveltime calculations in three dimensions first arrival traveltimes in complicated 3 d geologic media may be computed robustly and efficiently using an upwind finite difference solution of the 3 d eikonal equation an important application of this technique is computing traveltimes for imaging seismic data with 3 d prestack kirchhoff depth migration the method performs radial extrapolation of the three components of the slowness vector in spherical coordinates traveltimes are computed by numerically integrating the radial component of the slowness vector the original finite difference equations are recast into unitless forms that are more stable to numerical errors a stability condition adaptively determines the radial steps that are used to extrapolate computations are done in a rotated spherical coordinate system that places the small arc length regions of the spherical grid at the earth s surface z 0 plane this improves efficiency by placing large grid cells in the central regions of the grid where wavefields are complicated thereby

a two phase algorithm for the multiparametric linear complementarity problem abstract a new two phase method for solving the multi parametric linear complementarity problem mplcp with sufficient matrices is presented in the first phase an initial feasible solution to mplcp which satisfies certain criteria is determined in the second phase the set of feasible parameters is partitioned into polyhedral regions such that the solution of the mplcp as a function of the parameters is invariant over each region the worst case complexity of the presented algorithms matches that of current methods for nondegenerate problems and is lower than that of current methods for degenerate problems

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

brief paper constrained linear systems with hard constraints and disturbances an extended command governor with large domain of attraction this paper considers a nonlinear feedback control policy that is an extension of those provided by command governors and reference governors as in these control approaches it applies to discrete time linear systems with hard constraints and set bounded disturbances the control policy retains the main properties of traditional governors such as straightforward direct implementation and finite settling time response to arbitrarily specified set points its principal advantage over traditional governors is a significantly larger domain of attraction that may compete in size with those obtained by dynamic programming connections to model predictive control are made numerical examples illustrate advantageous features of the proposed approach