on feasible sets for mpc and their approximations in this paper we are interested in the computation of feasible sets for linear model predictive control techniques based on set relations and not on the conventional orthogonal projection further the problem of computing suitable inner approximations of the feasible sets is considered such approximations are characterized by simpler polytopic representations and preserve essential properties as convexity positive invariance inclusion of the set of expected initial states

reduced explicit constrained linear quadratic regulators in this note it is studied how structural properties of certain linear systems can be exploited to derive reduced dimension multiparametric quadratic programs that lead to explicit piecewise linear feedback solutions to the state and input constrained linear quadratic regulation problem the reduced dimensionality typically results in suboptimal controllers of lower complexity with associated computational advantages in the online implementation at the heart of the methods are state space projections based on the singular value decomposition

global attractivity of positive periodic solution to periodic lotka volterra competition systems with pure delay abstract we consider a periodic lotka volterra competition system without instantaneous negative feedbacks i e pure delay systems x i t x i t r i t j 1 n a i j t x j t i j t i 1 2 n we establish some 3 2 type criteria for global attractivity of a positive periodic solution of the system which generalize the well known wright s 3 2 criteria for the autonomous delay logistic equation and thereby address the open problem proposed by both kuang y kuang global stability in delayed nonautonomous lotka volterra type systems without saturated equilibria differential integral equations 9 1996 557 567 and teng z teng nonautonomous lotka volterra systems with delays j differential equations 179 2002 538 561

set membership approximation theory for fast implementation of model predictive control laws in this paper the use of set membership sm methodologies is investigated in the approximation of model predictive control mpc laws for linear systems such approximated mpc laws are derived from a finite number n of exact control moves computed off line properties in terms of guaranteed approximation error closed loop stability and performance are derived assuming only the continuity of the exact predictive control law these results are achieved by means of two main contributions at first it will be shown that if the approximating function enjoys two key properties i e fulfillment of input constraints and explicit evaluation of a bound on the approximation error which converges to zero as n increases then it is possible to guarantee the boundedness of the controlled state trajectories inside a compact set their convergency to an arbitrary small neighborhood of the origin and satisfaction of state constraints moreover the guaranteed performance degradation in terms of maximum state trajectory distance can be explicitly computed and reduced to an arbitrary small value by increasing n then two sm approximations are investigated both enjoying the above key properties the first one minimizes the guaranteed approximation error but its on line computational time increases with n the second one has higher approximation error but lower on line computational time which is constant with n when the off line computed moves are suitably chosen the presented approximation techniques can be systematically employed to obtain an efficient mpc implementation for fast processes the effectiveness of the proposed techniques is tested on two numerical examples

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

using interpolation to improve efficiency of multiparametric predictive control multi parameteric quadratic programming mpqp gives a full offline solution to a time varying quadratic programming qp problem arising during constrained predictive control however coding and implementation of this solution may be more burdensome than simply solving the original qp this paper shows how interpolation can be used in conjunction with mpqp to achieve a large decrease in both the online computation and data storage requirements with negligible deterioration of performance extensive simulation results are given to back this claim and contrast the two proposed interpolation schemes

harmless delays for uniform persistence abstract this paper studies a two dimensional predator prey system with a finite number of discrete delays our purpose is to demonstrate that the time delays are harmless for uniform persistence of the solutions of the system the results are obtained by constructing suitable persistence functionals

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

efficient model predictive control for nonlinear systems via function approximation techniques this note deals with the problem of fast implementation of nonlinear model predictive control using approximated control laws at first accuracy properties of a generic approximated controller are introduced together with their influence on closed loop stability and performance then exploiting such results it is shown how set membership sm function approximation theory can be systematically employed to improve the accuracy performance of the resulting controller given by the sum of with a sm approximating function satisfies the above mentioned properties even if they are not met by alone a nonlinear oscillator example shows the effectiveness of the proposed methodology

stochastic lotka volterra population dynamics with infinite delay we discover in this paper that when environmental noise is strongly dependent on the population size this noise may suppress the population explosion in a finite time and guarantee the global positive solution when the environmental noise is weakly dependent on the population size the conditions that guarantee the global positive solution are independent of this environmental noise we also discuss the pth moment boundedness stochastic ultimate boundedness and moment average boundedness in time under two classes of conditions these properties are natural requirements from the biological point of view

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

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

polytopic approximation of explicit model predictive controllers a model predictive control law mpc is given by the solution to a parametric optimization problem that can be precomputed offline which provides an explicit map from state to input that can be rapidly evaluated online however the primary limitations of these optimal explicit solutions are that they are applicable to only a restricted set of systems and that the complexity can grow quickly with problem size in this paper we compute approximate explicit control laws that trade off complexity against approximation error for mpc controllers that give rise to convex parametric optimization problems the algorithm is based on the classic double description method and returns a polyhedral approximation to the optimal cost function the proposed method has three main advantages from a control point of view it is an incremental approach meaning that an approximation of any specified complexity can be produced it operates on implicitly defined convex sets meaning that the prohibitively complex optimal explicit solution is not required and finally it can be applied to any convex parametric optimization problem a sub optimal controller based on barycentric interpolation is then generated from this approximate polyhedral cost function that is feasible and stabilizing the resulting control law is continuous although non linear and defined over a non simplical polytopic partition of the state space the non simplical nature of the partition generates significantly simpler approximate control laws which is demonstrated on several examples

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

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

the lyapunov functionals for delay lotka volterra type models by means of lyapunov functionals we have succeeded in establishing the global attractivity of the positive equilibrium of n species lotka volterra systems modeled by equations of the pure delay type with both finite and infinite delays involved as a corollary we show that if the delay system is dissipative and the coefficient matrix is vl stable then the global attractivity of the unique positive equilibrium is maintained provided the delays are small estimates on the allowable sizes of delays are indicated applications to 2 species prey predator models and n species food chains and competition models are included

stabilizing polynomial approximation of explicit mpc a given explicit piecewise affine representation of an mpc feedback law is approximated by a single polynomial computed using linear programming this polynomial state feedback control law guarantees closed loop stability and constraint satisfaction the polynomial feedback can be implemented in real time even on very simple devices with severe limitations on memory storage

global stability for large systems of volterra lotka type integrodifferential population delay equations on considere l equation integrodifferentielle non lineaire x i t x i t e i j 1 n p ij x j t j 1 n o r s ij k ij x j t d i 1 n ou 1 i j n e i p ij s ij sont des constantes on etudie la stabilite globale de la solution q 1 q n par rapport aux solutions positives

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

neural approximations for multistage optimal control of nonlinear stochastic systems two main approximations are used to solve a nonlinear quadratic gaussian lqg optimal control problem the control law is assigned a given structure in which a finite number of parameters have to be determined to minimize the cost function the chosen structure is that of a multilayer feedforward neural network and the control law is given a limited memory the errors resulting front both assumptions are discussed simulation results show that the proposed method may constitute a simple and effective tool for solving to a sufficient degree of accuracy optimal control problems traditionally regarded as difficult ones

approximate explicit receding horizon control of constrained nonlinear systems an algorithm for the construction of an explicit piecewise linear state feedback approximation to nonlinear constrained receding horizon control is given it allows such controllers to be implemented via an efficient binary tree search avoiding real time optimization this is of significant benefit in applications that requires low real time computational complexity or low software complexity the method has a priori guarantee of asymptotic stability with region of attraction being a close inner approximation to the stabilizable set this is achieved by ensuring that the approximation error does not exceed the stability margin

a patchy approximation of explicit model predictive control the explicit solution of multi parametric optimisation problems mpop has been used to construct an off line solution to relatively small and medium sized constrained control problems the control design principles are based on receding horizon optimisation and generally use linear prediction models for the system dynamics in this context it can be shown that the optimal control law is a piecewise linear pwl state feedback defined over polytopic cells of the state space however as the complexity of the related optimisation problems increases the memory footprint and implementation of such explicit optimal solution may be burdensome for the available hardware principally due to the high number of polytopic cells in the state space partition in this article we provide a solution to this problem by proposing a patchy pwl feedback control law which intend to approximate the optimal control law the construction is based on the linear interpolation of the exact solution at the vertices of a feasibl

explicit robustness and fragility margins for linear discrete systems with piecewise affine control law in this paper we focus on the robustness and fragility problem for piecewise affine pwa control laws for discrete time linear system dynamics in the presence of parametric uncertainty of the state space model a generic geometrical approach will be used to obtain robustness fragility margins with respect to the positive invariance properties for pwa control laws defined over a bounded region in the state space it is shown that these margins can be described in terms of polyhedral sets in parameter space the methodology is further extended to the fragility problem with respect to the partition defining the controller finally several computational aspects are presented regarding the transformation from the theoretical formulations to explicit representations vertex halfspace representation of polytopes of these sets

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

explicit solution of min max mpc with additive uncertainties and quadratic criterion abstract min max model predictive control mmmpc is one of the strategies proposed to control plants subject to bounded uncertainties this technique is very difficult to implement in real time because of the computation time required recently the piecewise affine nature of this control law has been proved for unconstrained linear systems with quadratic performance criterion however no algorithm to compute the explicit form of the control law was given this paper shows how to obtain this explicit form by means of a constructive algorithm an approximation to mmmpc in the presence of constraints is presented based on this algorithm

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

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

low order discontinuous petrov galerkin finite element methods for linear elasticity this paper analyzes lowest order discontinuous petrov galerkin dpg finite element methods fem for the navier lame equations with different norms and side restrictions the focus is on the direct proof of a discrete inf sup condition for a low order test search space the low order finite element spaces in the ultraweak formulation involve the piecewise constant and affine ansatz functions and discontinuous piecewise affine test functions in two and three space dimensions with neumann boundary conditions or the pure dirichlet problem those lowest order discretizations for linear elasticity allow for a direct proof of the discrete inf sup condition and a complete a priori and a posteriori error analysis which is robust in the incompressible limit as lambda rightarrow infty numerical experiments with uniform and adaptive mesh refinements investigate lambda robustness and confirm that one scheme is locking free

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

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

global attractivity of non autonomous lotka volterra competition system without instantaneous negative feedback abstract we consider a non autonomous lotka volterra competition system with distributed delays but without instantaneous negative feedbacks i e pure delay systems we establish some 3 2 type and m matrix type criteria for global attractivity of the positive equilibrium of the system which generalise and improve the existing ones

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

an algorithm for reducing complexity in parametric predictive control multi parametric quadratic programming solutions to predictive control can require large numbers of regions which would make practical implementation difficult this paper demonstrates how an underexplored prediction class can be combined with parametric programming to give far simpler solutions and at little cost to optimality the key proposal is to explore how one can give more flexibility in the shape of the predictions without increasing the required numbers of degrees of freedom this flexibility allows terminal regions to be enlarged

periodicity and stability for a lotka volterra type competition system with feedback controls and deviating arguments abstract this paper deals with the general periodic lotka volterra type competition systems with feedback controls and deviating arguments by employing fixed point index theory on cone an explicit necessary and sufficient condition for the global existence of the positive periodic solution of the systems is proved by constructing a suitable lyapunov functional a set of easily verifiable sufficient conditions for the global asymptotic stability of the positive periodic solution of the systems is given

stability and delays in a predator prey system abstract sufficient conditions for both local and global stability of the positive equilibrium in a predator prey system with time delays are obtained by constructing suitable lyapunov functionals a remark on the mistake by k n murty and m a s srinivas j math anal appl 158 1991 333 341 is also included

semi explicit mpc based on subspace clustering this paper presents a new strategy of simplifying the online computations in linear model predictive control mpc employing a specific type of state dependent parameterization for the optimization variable in mpc advantages of explicit mpc are combined with those of online optimization based mpc into an efficient mpc scheme the parameterization is computed offline applying a tailored subspace clustering algorithm to training data consisting of states and corresponding solutions to the mpc optimization problem it is then refined to guarantee feasibility of the parameterized optimization during the offline design phase complexity of the parameterization can be adjusted and control performance can be traded off against online computational effort and storage requirements numerical examples evaluate the presented methods and illustrate their benefits

a multiresolution approximation method for fast explicit model predictive control a model predictive control law is given by the solution to a parametric optimization problem that can be pre computed offline and provides an explicit map from state to control input in this paper an algorithm is introduced based on wavelet multiresolution analysis that returns a low complexity explicit model predictive control law built on a hierarchy of second order interpolets the resulting interpolation is shown to be everywhere feasible and continuous further tests to confirm stability and to compute a bound on the performance loss are introduced since the controller approximation is built on a grid hierarchy convergence to a stabilizing control law is guaranteed and the evaluation of the control law in real time systems is naturally fast and runs in a bounded logarithmic time two examples are provided a two dimensional example with an evaluation speed of 31 ns and a four dimensional example with an evaluation speed of 119 ns

degenerate lyapunov functionals of a well known prey predator model with discrete delays it is commonly believed that as far as stabilities are concerned small delays are negligible in some modelling processes however to have an affirmative answerfor this common belief is still an open problem for many nonlinear equations in this paper the classical lotka volterra prey predator equation with discrete delays is considered and by using degenerate lyapunov functionals method an affirmative answer to this open problem on both local and global stabilities of the prey predator delay equations is given it is shown that degenerate lyapunov functional method is a powerful tool for studying the stability of such nonlinear delay systems a detailed and explicit procedure of constructing such functionals is provided furthermore some explicit estimates on the allowable sizes of the delays are obtained