long time versus steady state optimal control this paper analyzes the convergence of optimal control problems for an evolution equation in a finite time horizon 0 t toward the limit steady state ones as t to infty we focus on linear problems we first consider linear time independent finite dimensional systems and show that the optimal controls and states exponentially converge in the transient time as t tends to infinity to the ones of the corresponding steady state model for this to occur suitable observability assumptions need to be imposed we then extend the results to infinite dimensional systems including the linear heat and wave equations with distributed controls

on convergence and performance certification of a continuous time economic model predictive control scheme with time varying performance index this paper addresses the design of convergence and performance certified sampled data model predictive control mpc laws with a time dependent economic performance index more precisely using a dissipativity property of the system we provide a set of sufficient conditions that guarantee convergence of the closed loop state trajectory to a possibly time varying average economically optimal state trajectory moreover the average performance of the closed loop system is shown to be no worse than the one obtained by operating the system at the average economically optimal state trajectory constructive methods to design an appropriate terminal set and terminal cost that satisfy the proposed sufficient conditions are presented and illustrated with numerical examples

on necessity and robustness of dissipativity in economic model predictive control in this paper we study a dissipativity property which was recently used in several results on economic model predictive control to ensure optimal operation of a system at steady state as well as stability in particular we first investigate whether this dissipativity property is not only sufficient but also necessary for optimal steady state operation in the most general case this is not true nevertheless under an additional controllability assumption we show that dissipativity is in fact necessary second we provide a robustness analysis of the dissipativity property with respect to changes in the constraint set which can result in a change in the considered supply rate

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

a linear framework on the distributed model predictive control of positive systems abstract this paper investigates the distributed model predictive control dmpc for positive systems with interval and polytopic uncertainties respectively different from the traditional quadratic dmpc a new linear dmpc framework is established for positive systems first a linear cooperative cost function performance index function is introduced for the dmpc of positive systems accordingly some linear constraint conditions are imposed on the systems then a linear co positive lyapunov function is constructed using the linear lyapunov function the dmpc controllers are designed for interval positive systems and polytopic positive systems respectively the controller of each subsystem contains not only the state information of the subsystem but also the state information of other correlated subsystems linear programming based optimization algorithm is addressed to compute the controller parameters to guarantee the feasibility and stability of the systems a cone invariant set is proposed finally two case studies are provided to illustrate the effectiveness of the theoretical findings

survey paper well posed systems the lti case and beyond this survey is an introduction to well posed linear time invariant lti systems for non specialists we recall the more general concept of a system node classical and generalized solutions of system equations criteria for well posedness the subclass of regular linear systems some of the available linear feedback theory motivated by physical examples we recall the concepts of impedance passive and scattering passive systems conservative systems and systems with a special structure that belong to these classes we illustrate this theory by examples of systems governed by heat and wave equations we develop local and global well posedness results for lti systems with nonlinear in particular bilinear feedback by extracting the abstract idea behind various proofs in the literature we apply these abstract results to derive well posedness results for the burgers and navier stokes equations

robust economic model predictive control using stochastic information in this paper we develop a new tube based robust economic mpc scheme for linear time invariant systems subject to bounded disturbances with given distributions by using the error distribution in the predictions of the finite horizon optimal control problem we can incorporate stochastic information in order to improve the expected performance while being able to guarantee strict feasibility for this new framework we can provide bounds on the asymptotic average performance of the closed loop system moreover a constructive approach is presented in order to find an appropriate terminal cost leading to a slight degradation of the bound on the guaranteed average performance

correlation and independence in the neural code the decoding scheme of a stimulus can be different from the stochastic encoding scheme in the neural population coding the stochastic fluctuations are not independent in general but an independent version could be used for the ease of decoding how much information is lost by using this unfaithful model for decoding there are discussions concerning loss of information nirenberg latham 2003 schneidman bialek berry 2003 we elucidate the nirenberg latham loss from the point of view of information geometry

generalized terminal state constraint for model predictive control a terminal state equality constraint for model predictive control mpc laws is investigated where the terminal state input pair is not fixed a priori but it is a free variable in the optimization the approach named generalized terminal state constraint can be used for both tracking mpc i e when the objective is to track a given steady state and economic mpc i e when the objective is to minimize a cost function which does not necessarily attains its minimum at a steady state it is shown that the proposed technique provides in general a larger feasibility set with respect to the existing approaches given the same prediction horizon moreover a new receding horizon strategy is introduced exploiting the generalized terminal state constraint under mild assumptions the new strategy is guaranteed to converge in finite time with arbitrarily good accuracy to an mpc law with an optimally chosen terminal state constraint while still enjoying a larger feasibility set the features of the new technique are illustrated by an inverted pendulum example in both the tracking and the economic contexts

primitive group rings and noetherian rings of quotients let k be a field and let g be a countable nilpotent group with centre z we show that the group algebra kg is primitive if and only if k is countable g is torsion free and there exists an abelian subgroup a of g of infinite rank with a fn z 1 suppose now that g is torsion free then kg has a partial quotient ring q kg kz 1 the above characterisation of the primitivity of kg is intimately connected with the question when is q a noetherian ring we determine this for those groups g as above all of whose finite rank subgroups are finitely generated in this case q is noetherian if and only if g has no abelian subgroup a of infinite rank with a fn z 1

population coding with correlation and an unfaithful model this study investigates a population decoding paradigm in which the maximum likelihood inference is based on an unfaithful decoding model umli this is usually the case for neural population decoding because the encoding process of the brain is not exactly known or because a simplified decoding model is preferred for saving computational cost we consider an unfaithful decoding model that neglects the pair wise correlation between neuronal activities and prove that umli is asymptotically efficient when the neuronal correlation is uniform or of limited range the performance of umli is compared with that of the maximum likelihood inference based on the faithful model and that of the center of mass decoding method it turns out that umli has advantages of decreasing the computational complexity remarkably and maintaining high level decoding accuracy moreover it can be implemented by a biologically feasible recurrent network pouget zhang deneve latham 1998 the effect of correlation on the decoding accuracy is also discussed

turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems we analyze the connection between turnpike behaviors and strict dissipativity properties for discrete time finite dimensional linear quadratic optimal control problems we first use strict dissipativity as a sufficient condition for the turnpike property next we characterize strict dissipativity and the newly introduced property of strict pre dissipativity in terms of the system matrices related to the linear quadratic problem these characterizations then lead to new necessary conditions for the turnpike properties under consideration and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities one of the key novelties which distinguishes the results in the present paper from earlier ones on linear quadratic optimal control problems is the consideration of state and input constraints

on turnpike and dissipativity properties of continuous time optimal control problems this paper investigates the relations between three different properties which are of importance in optimal control problems dissipativity of the underlying dynamics with respect to a specific supply rate optimal operation at steady state and the turnpike property we show in a continuous time setting that if along optimal trajectories a strict dissipation inequality is satisfied then this implies optimal operation at this steady state and the existence of a turnpike at the same steady state finally we establish novel converse turnpike results i e we show that the existence of a turnpike at a steady state implies optimal operation at this steady state and dissipativity with respect to this steady state we draw upon a numerical example to illustrate our findings

exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations abstract we analyze the sensitivity of linear quadratic optimal control problems governed by general evolution equations with bounded or admissible control operator we show that if the problem is stabilizable and detectable the solution of the extremal equation can be bounded by the right hand side including initial data with the bound being independent of the time horizon consequently the influence of perturbations of the extremal equations decays exponentially in time this property can for example be used to construct efficient space and time discretizations for a model predictive control scheme furthermore a turnpike property for unbounded but admissible control of general semigroups can be deduced

homogeneity and rigidity of the brookes groves invariant for the non commutative torus in 4 bieri and strebel defined a geometric invariant for finitely generated modules over the group algebras of finitely generated abelian groups they used this to define a criterion for when metabelian groups are finitely presented this invariant was further developed by bieri strebel and groves and has many interesting applications in 2 bieri and groves showed that when the group algebra is defined over a dedekind domain the complement of must be a closed rational spherical polyhedral cone in 6 7 brookes and groves defined a similar invariant for modules over the crossed product of a division ring by a free finitely generated abelian group such a crossed product is often known as the coordinate ring of the non commutative torus since in the special case where it is commutative it is the coordinate ring of an algebraic torus if in the commutative case we take the complement of and identify points that differ by a positive scalar multiple we obtain brookes and groves were unable to prove that their invariant must be a rational polyhedral cone although using the methods of 2 they do prove a weaker version of the result they show that for any finitely generated module m m must contain a rational polyhedral cone m of dimension equal to the gelfand kirillov dimension of m and moreover that the complement m m must be contained inside a rational polyhedral cone of strictly smaller dimension

a rigidity property for the set of all characters induced by valuations if k is a field and g a finitely generated multiplicative subgroup of k then every real valuation on k induces a character g r it is known that the set ls g c r 1 of all characters induced by valuations is polyhedral we prove that a g satisfies a certain rigidity property and apply this to give a new and conceptual proof of the brewster roseblade result 4 on the group of automorphisms of k stabilizing g 1 lntroduction 1 1 let k be a field and k c k a subfield by a real valuation on k over k we mean a homomorphism w kx r of the multiplicative group of k into the additive group of the reals which is trivial on kx and satisfies w a b mint w a w b for all a b e kx with a b o throughout the paper g shall dersote a finitely generated multiplicative subgroup of k then w induces a character x wlc g 1t on g and so following g m bergman 1 we consider the set a g of all characters of g induced by real valuations on k over k thus a g is a

optimal neumann control for the 1d wave equation finite horizon infinite horizon boundary tracking terms and the turnpike property we consider a vibrating string that is fixed at one end with neumann control action at the other end we investigate the optimal control problem of steering this system from given initial data to rest in time t by minimizing an objective functional that is the convex sum of the l 2 norm of the control and of a boundary neumann tracking term we provide an explicit solution of this optimal control problem showing that if the weight of the tracking term is positive then the optimal control action is concentrated at the beginning and at the end of the time interval and in between it decays exponentially we show that the optimal control can actually be written in that case as the sum of an exponentially decaying term and of an exponentially increasing term this implies that if the time t is large the optimal trajectory approximately consists of three arcs where the first and the third short time arcs are transient arcs and in the middle arc the optimal control and the corresponding state are exponentially close to 0 this is an example for a turnpike phenomenon for a problem of optimal boundary control if t infinite horizon time problem then only the exponentially decaying component of the control remains and the norms of the optimal control action and of the optimal state decay exponentially in time in contrast to this situation if the weight of the tracking term is zero and only the control cost is minimized then the optimal control is distributed uniformly along the whole interval 0 t and coincides with the control given by the hilbert uniqueness method

convergence in economic model predictive control with average constraints in this paper we thoroughly investigate various aspects of economic model predictive control with average constraints i e constraints on average values of state and input variables in particular we first show that a certain time varying output constraint has to be included into the mpc problem formulation in order to ensure fulfillment of these average constraints optimizing a general possibly economic performance criterion may result in a non converging behavior of the corresponding closed loop system while such a behavior might be acceptable in some cases it may be undesirable for other types of applications hence as a second contribution we provide a lyapunov like analysis to conclude that indeed asymptotic convergence to the optimal steady state follows if the system satisfies a certain dissipativity condition finally for the case that this dissipativity property is not satisfied but still a convergent behavior of the closed loop is required we examine two different methods how convergence can be enforced within an economic mpc setup by imposing additional average constraints on the system in the first method an additional average constraint is defined which results in the system being dissipative while the second consists of imposing an additional even zero moment average constraint we illustrate our results with various examples

implications of dissipativity on stability of economic model predictive control the indefinite linear quadratic case abstract in contrast to the conventional model predictive control mpc approach to control of a given system where a positive definite objective function is employed economic mpc employs a generic cost which is related to the economics of the process as the objective function in the regulation layer often stability proofs of the closed loop system are based on strict dissipativity of the system with respect to this objective function in this paper we focus on linear systems with indefinite quadratic costs we show that while strict dissipativity guarantees stability of the closed loop system it is not required hence we formulate a necessary and sufficient condition that guarantees asymptotic stability of the closed loop system this condition comes down to the existence of two distinct storage functions for which the system is dissipative

periodic optimal control dissipativity and mpc recent research has established the importance of strict dissipativity for proving stability of economic mpc in the case of an optimal steady state in many cases though steady state operation is not economically optimal and periodic operation of the system yields a better performance in this technical note we propose ways of extending the notion of strict dissipativity for periodic systems we prove that optimal p periodic operation and mpc stability directly follow similarly to the steady state case which can be seen as a special case of the proposed framework finally we illustrate the theoretical results with several simple examples

cost to travel functions a new perspective on optimal and model predictive control abstract this paper concerns a class of functions named cost to travel functions which find applications in model based control for a given potentially nonlinear control system the cost to travel function associates with any given start and end point in the state space and any given travel duration the minimum economic cost of the associated point to point motion cost to travel functions are a generalization of cost to go functions which are often used in the context of dynamic programming as well as model predictive control we discuss the properties of cost to travel functions their relations to existing concepts in control such as dissipativity but also a variety of control theoretic applications of this function class in particular we discuss how cost to travel functions can be used to analyze the properties of economic model predictive control with return constraints

analysis and design of unconstrained nonlinear mpc schemes for finite and infinite dimensional systems we present a technique for computing stability and performance bounds for unconstrained nonlinear model predictive control mpc schemes the technique relies on controllability properties of the system under consideration and the computation can be formulated as an optimization problem whose complexity is independent of the state space dimension based on the insight obtained from the numerical solution of this problem we derive design guidelines for nonlinear mpc schemes which guarantee stability of the closed loop for small optimization horizons these guidelines are illustrated by a finite and an infinite dimensional example

dynamics and computation of continuous attractors continuous attractor is a promising model for describing the encoding of continuous stimuli in neural systems in a continuous attractor the stationary states of the neural system form a continuous parameter space on which the system is neutrally stable this property enables the neutral system to track time varying stimuli smoothly but it also degrades the accuracy of information retrieval since these stationary states are easily disturbed by external noise in this work based on a simple model we systematically investigate the dynamics and the computational properties of continuous attractors in order to analyze the dynamics of a large size network which is otherwise extremely complicated we develop a strategy to reduce its dimensionality by utilizing the fact that a continuous attractor can eliminate the noise components perpendicular to the attractor space very quickly we therefore project the network dynamics onto the tangent of the attractor space and simplify it successfully as a one dimensional ornstein uhlenbeck process based on this simplified model we investigate 1 the decoding error of a continuous attractor under the driving of external noisy inputs 2 the tracking speed of a continuous attractor when external stimulus experiences abrupt changes 3 the neural correlation structure associated with the specific dynamics of a continuous attractor and 4 the consequence of asymmetric neural correlation on statistical population decoding the potential implications of these results on our understanding of neural information processing are also discussed

a causal perspective on the analysis of signal and noise correlations and their role in population coding the role of correlations between neuronal responses is crucial to understanding the neural code a framework used to study this role comprises a breakdown of the mutual information between stimuli and responses into terms that aim to account for different coding modalities and the distinction between different notions of independence here we complete the list of types of independence and distinguish activity independence related to total correlations conditional independence related to noise correlations signal independence related to signal correlations coding independence related to information transmission and information independence related to redundancy for each type we identify the probabilistic criterion that defines it indicate the information theoretic measure used as statistic to test for it and provide a graphical criterion to recognize the causal configurations of stimuli and responses that lead to its existence using this causal analysis we first provide sufficiency conditions relating these types second we differentiate the use of the measures as statistics to test for the existence of independence from their use for quantification we indicate that signal and noise correlation cannot be quantified separately third we explicitly define alternative system configurations used to construct the measures in which noise correlations or noise and signal correlations are eliminated accordingly we examine which measures are meaningful only as a comparison across configurations and which ones provide a characterization of the actually observed responses without resorting to other configurations fourth we compare the commonly used nonparametric approach to eliminate noise correlations with a functional model based approach showing that the former approach does not remove those effects of noise correlations captured by the tuning properties of the individual neurons and implies nonlocal causal structure manipulations these results improve the interpretation of the measures on the framework and help in understanding how to apply it to analyze the role of correlations

turnpike theorem for an infinite horizon optimal control problem with time delay an optimal control problem for systems described by a special class of nonlinear differential equations with time delay is considered the cost functional adopted could be considered as an analogue of the terminal functional defined over an infinite time horizon the existence of optimal solutions as well as the asymptotic stability of optimal trajectories that is the turnpike property are established under some quite mild restrictions on the nonlinearities of the functions involved in the description of the problem such mild restrictions on the nonlinearities allowed us to apply these results to a blood cell production model

a lyapunov function for economic optimizing model predictive control standard model predictive control mpc yields an asymptotically stable steady state solution using the following procedure given a dynamic model a steady state of interest is selected a stage cost is defined that measures deviation from this selected steady state the controller cost function is a summation of this stage cost over a time horizon and the optimal cost is shown to be a lyapunov function for the closed loop system in this technical note the stage cost is an arbitrary economic objective which may not depend on a steady state and the optimal cost is not a lyapunov function for the closed loop system for a class of nonlinear systems and economic stage costs this technical note constructs a suitable lyapunov function and the optimal steady state solution of the economic stage cost is an asymptotically stable solution of the closed loop system under economic mpc both finite and infinite horizons are treated the class of nonlinear systems is defined by satisfaction of a strong duality property of the steady state problem this class includes linear systems with convex stage costs generalizing previous stability results and providing a lyapunov function for economic mpc or mpc with an unreachable setpoint and a linear model a nonlinear chemical reactor example is provided illustrating these points

optimal control problems over large time intervals abstract the paper focuses on problems of optimal control over a large time interval with end states prescribed for a wide class of systems and performance indices an approximate solution is characterized by piecing together two solutions for two infinite time problems these solutions exhibit similar transient behaviour

the turnpike property in finite dimensional nonlinear optimal control turnpike properties have been established long time ago in finite dimensional optimal control problems arising in econometry they refer to the fact that under quite general assumptions the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces the first and the last of which being transient short time arcs and the middle piece being a long time arc staying exponentially close to the optimal steady state solution of an associated static optimal control problem we provide in this paper a general version of a turnpike theorem valuable for nonlinear dynamics without any specific assumption and for very general terminal conditions not only the optimal trajectory is shown to remain exponentially close to a steady state but also the corresponding adjoint vector of the pontryagin maximum principle the exponential closedness is quantified with the use of appropriate normal forms of riccati equations we show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method or a shooting method in particular we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector not at the initial time but at the middle of the trajectory

losslessness feedback equivalence and the global stabilization of discrete time nonlinear systems in this paper a necessary and sufficient condition for a nonlinear system of the form spl sigma given by x k 1 f x k g x k u k y k h x k j x k u k to be lossless is given and it is shown that a lossless system can be globally asymptotically stabilized by output feedback if and only if the system is zero state observable then we investigate conditions under which spl sigma can be rendered lossless via smooth state feedback in particular we show that this is possible if and only if the system in question has relative degree spl lcub 0 0 spl rcub and has lossless zero dynamics under suitable controllability like rank conditions we prove that nonlinear systems having relative degree spl lcub 0 0 spl rcub and lossless zero dynamics can be globally stabilized by smooth state feedback as a consequence we obtain sufficient conditions for a class of cascaded systems to be globally stabilizable the global stabilization problem of the nonlinear system spl sigma without output is also investigated in this paper by means of feedback equivalence some of the results are parallel to analogous ones in continuous time but in many respects the theory is substantially different and many new phenomena appear

an exponential turnpike theorem for dissipative discrete time optimal control problems we investigate the exponential turnpike property for finite horizon undiscounted discrete time optimal control problems without any terminal constraints considering a class of strictly dissipative systems we derive a boundedness condition for an auxiliary optimal value function which implies the exponential turnpike property two theorems illustrate how this boundedness condition can be concluded from structural properties like controllability and stabilizability of the control system under consideration

mutual information expansion for studying the role of correlations in population codes how important are autocorrelations the role of correlations in the activity of neural populations responding to a set of stimuli can be studied within an information theory framework regardless of whether one approaches the problem from an encoding or decoding perspective the main measures used to study the role of correlations can be derived from a common source the expansion of the mutual information two main formalisms of mutual information expansion have been proposed the series expansion and the exact breakdown here we clarify that these two formalisms have a different representation of autocorrelations so that even when the total information estimated differs by less than 1 individual terms can diverge more precisely the series expansion explicitly evaluates the informational contribution of autocorrelations in the count of spikes that is count autocorrelations whereas the exact breakdown does not we propose a new formalism of mutual information expansion the poisson exact breakdown which introduces poisson equivalents in order to explicitly evaluate the informational contribution of count autocorrelations with no approximation involved because several widely employed manipulations of spike trains most notably binning and pooling alter the structure of count autocorrelations the new formalism can provide a useful general framework for studying the role of correlations in population codes

the existence of catching up optimal solutions for a class of infinite horizon control problems with time delay the optimal control of a system whose states are governed by a nonlinear autonomous volterra integrodifferential equation with unbounded time interval is considered specifically it is assumed that the delay occurs only in the state variable the results obtained extend those of brock and haurie math oper res 1 1976 pp 337 346 and leizarowitz math oper res 10 1985 pp 450 461 in particular it is shown that under appropriate hypotheses catching up optimal solutions asymptotically approach a unique optimal steady state and thus enjoy the so called turnpike property found in the economics literature by combining this result with an associated optimal control problem the desired existence result is obtained furthermore it is remarked that in addition to extending these earlier works to the time delay case the results presented below utilize convexity seminormality and growth hypotheses that in some cases are weaker than those encountered in the above mentioned papers

a homogeneous and self dual interior point linear programming algorithm for economic model predictive control we develop an efficient homogeneous and self dual interior point method ipm for the linear programs arising in economic model predictive control of constrained linear systems with linear objective functions the algorithm is based on a riccati iteration procedure which is adapted to the linear system of equations solved in homogeneous and self dual ipms fast convergence is further achieved using a warm start strategy we implement the algorithm in matlab and c its performance is tested using a conceptual power management case study closed loop simulations show that 1 the proposed algorithm is significantly faster than several state of the art ipms based on sparse linear algebra and 2 warm start reduces the average number of iterations by 35 40

polyhedrality of the brookes groves invariant for the non commutative torus in 3 bieri and strebel defined a geometric invariant for finitely generated modules over the group algebras of finitely generated abelian groups they used this to define a criterion for when metabelian groups are finitely presented this invariant was further developed by bieri strebel and groves and has many interesting applications in 2 bieri and groves showed that when the group algebra is defined over a dedekind domain the complement of must be a closed rational spherical polyhedral cone in 4 and 5 brookes and groves defined a similar invariant for modules over the crossed product of a division ring by a free finitely generated abelian group such a crossed product is often known as the coordinate ring of the non commutative torus since in the special case where it is commutative it is the coordinate ring of an algebraic torus if in the commutative case we take the complement of and identify points that differ by a positive scalar multiple we obtain brookes and groves were unable to prove that their invariant must be a rational polyhedral cone although using the methods of 2 they do prove a weaker version of the result they show that for any finitely generated module m m must contain a rational polyhedral cone m of dimension equal to the gelfand kirillov dimension of m and moreover that the complement m m must be contained inside a rational polyhedral cone of strictly smaller dimension in this paper we use grobner basis methods to prove the following theorem

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

stability of multiobjective predictive control we propose a utopia tracking strategy to handle multiple conflicting objectives in model predictive control the controller minimizes the distance of its vector of objectives to that of the compromise solution the point along the steady state pareto front closest to the utopia point where all the objectives are independently minimized we establish conditions for asymptotic stability and propose numerical implementation variants one of the key advantages of the approach is that it avoids the computation of pareto fronts in real time environments in addition the approach can handle general objectives of different nature such as economic and regularization

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

analysis of unconstrained nonlinear mpc schemes with time varying control horizon for discrete time nonlinear systems satisfying an exponential or finite time controllability assumption we present an analytical formula for a suboptimality estimate for model predictive control schemes without stabilizing terminal constraints based on our formula we perform a detailed analysis of the impact of the optimization horizon and the possibly time varying control horizon on stability and performance of the closed loop

shape turnpike for linear parabolic pde models abstract we introduce and study the turnpike property for time varying shapes within the viewpoint of optimal control we focus here on second order linear parabolic equations where the shape acts as a source term and we seek the optimal time varying shape that minimizes a quadratic criterion we first establish existence of optimal solutions under some appropriate sufficient conditions we then provide necessary conditions for optimality in terms of adjoint equations and using the concept of strict dissipativity we prove that state and adjoint satisfy the measure turnpike property meaning that the extremal time varying solution remains essentially close to the optimal solution of an associated static problem we show that the optimal shape enjoys the exponential turnpike property in terms of hausdorff distance for a mayer quadratic cost we illustrate the turnpike phenomenon in optimal shape design with several numerical simulations

enforcing asymptotic orbital stability of economic model predictive control this paper is about economic model predictive control empc a modern receding horizon based control technique that minimizes at each sampling instance an economic objective on a moving horizon subject to a continuous time model dynamics and optionally control and state constraints the contribution of this paper is the introduction of a new type of terminal constraint and terminal cost which allow us to enforce the asymptotic orbital stability of empc under reasonable assumptions on the controllability and well posedness of the problem we provide an associated orbital lyapunov function that is employed for proving stability

partial stability for nonlinear model predictive control while the theory of stability for nonlinear model predictive control nmpc has been under extensive study partial stability is apparently overlooked in this paper we apply the concept of partial stability in order to extend several results for the stability analysis of nmpc and investigate the behavior of nmpc for dynamic systems with uncertain parameters partial stability for nmpc is established without using terminal costs and terminal constraints numerical examples are provided to illustrate the obtained results

robust moving horizon estimation based output feedback economic model predictive control abstract in this work we develop an economic model predictive control scheme for a class of nonlinear systems with bounded process and measurement noise in order to achieve fast convergence of the state estimates to the actual system state as well as the robustness of the observer to measurement and process noise a deterministic high gain observer is first applied for a small time period with continuous output measurements to drive the estimation error to a small value after this initial small time period a robust moving horizon estimation scheme is used on line to provide more accurate and smoother state estimates in the design of the robust moving horizon estimation scheme the deterministic observer is used to calculate reference estimates and confidence regions that contain the actual system state within the confidence regions the moving horizon estimation scheme is allowed to optimize its estimates the output feedback economic model predictive controller is designed via lyapunov techniques based on state estimates provided by the deterministic observer and the moving horizon estimation scheme the stability of the closed loop system is analyzed rigorously and conditions that ensure the closed loop stability are derived extensive simulations based on a chemical process example illustrate the effectiveness of the proposed approach

sensitivity analysis of optimal control for a class of parabolic pdes motivated by model predictive control we analyze the sensitivity of the extremal equations that arise from the first order optimality conditions for time dependent optimization problems more specifically we consider parabolic pdes wi

partitioned dynamic programming for optimal control parallel algorithms for the solution of linear quadratic optimal control problems are described the algorithms are based on a straightforward decomposition of the domain of the problem and are related to multiple shooting methods for two point boundary value problems their arithmetic cost is approximately twice that of the serial dynamic programming approach however they have the advantage that they can be efficiently implemented on a wide variety of parallel architectures extension to the case in which there are box constraints on the controls is simple the algorithms can be used to solve linear quadratic subproblems arising from the application of newton s method or two metric gradient projection methods to nonlinear problems

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

on the relation between strict dissipativity and turnpike properties abstract for discrete time nonlinear systems we study the relation between strict dissipativity and so called turnpike like behavior in optimal control under appropriate controllability assumptions we provide several equivalence statements involving these two properties the relation of strict dissipativity to an exponential variant of the turnpike property is also studied

state estimation based economic model predictive control of nonlinear systems abstract in this work we focus on a class of nonlinear systems and design an estimator based economic model predictive control mpc system which is capable of optimizing closed loop performance with respect to general economic considerations taken into account in the construction of the cost function working with the class of full state feedback linearizable nonlinear systems we use a high gain observer to estimate the nonlinear system state using output measurements and a lyapunov based approach to design an economic mpc system that uses the observer state estimates we prove using singular perturbation arguments that the closed loop system is practically stable provided the observer gain is sufficiently large we use a chemical process example to demonstrate the ability of the state estimation based economic mpc to achieve process time varying operation that leads to a superior cost performance metric compared to steady state operation using the same amount of reactant material