Endpoint Constraints Research Articles

Pointwise Second-Order Necessary Optimality Conditions for the Mayer Problem with Control Constraints

This paper is devoted to second-order necessary optimality conditions for the Mayer optimal control problem when the control set $U$ is a closed subset of $\mathbb{R}^m$. We show that, in the absence of endpoint constraints, if an optimal control $\bar u(\cdot)$ is singular and integrable, then for almost every $t$ such that $\bar u(t)$ is in the interior of $U$, both the Goh and a generalized Legendre--Clebsch condition hold true. Moreover, when the control set is a convex polytope, similar conditions are verified on the tangent subspace to $U$ at $ \bar u(t)$ for almost all $t$'s such that $ \bar u(t)$ lies on the boundary $\partial U$ of $U$. The same conditions are valid also for $U$ having a smooth boundary at every $t$ where $\bar u(\cdot)$ is singular and locally Lipschitz and $\bar u(t) \in \partial U.$ In the presence of a smooth endpoint constraint, these second-order necessary optimality conditions are satisfied whenever the Mayer problem is calm and the maximum principle is abnormal. If it is ...

A maximum principle for optimal control system with endpoint constraints

Pontryagin’s maximum principle for an optimal control system governed by an ordinary differential equation with endpoint constraints is proved under the assumption that the control domain has no linear structure. We also obtain the variational equation, adjoint equation and Hamilton system for our problem.

Concept for the integration of geometric and servo dynamic errors for predicting volumetric errors in five-axis high-speed machine tools: an application on a XYC three-axis motion trajectory using programmed end point constraint measurements

A modelling approach for volumetric error prediction taking into account geometric and servo dynamic errors in a five-axis high-speed machine tool is proposed in this paper. Polynomial functions are used to represent and then predict the geometric errors. A simple second-order transfer function model is used to model and predict the servo dynamic error. The servo dynamic errors are added to the axis position geometric errors and propagated to the tool and workpiece using matrix transformations. The validity of the error integration concept is tested for a XYC three-axis motion trajectory. Two experimental setups are used. The first experimental test used the KGM grid encoder instrument to estimate the parameters of the servo dynamic error models of the X- and Y-axes. The second experimental test used a programmed end point constraint procedure with measurement of the 3D volumetric positioning errors between a point on the tool holder and another fixed to the machine table. The tests involve maintaining the nominal coincidence of these two points whilst exercising the three axes. These last tests are used to estimate the geometric error parameters and also to validate the prediction performance of the integrated geometric and dynamic model. The result shows the effectiveness of the error integration concept.

Practically Efficient and Robust Free Energy Calculations: Double-Integration Orthogonal Space Tempering.

The orthogonal space random walk (OSRW) method, which enables synchronous acceleration of the motions of a focused region and its coupled environment, was recently introduced to enhance sampling for free energy simulations. In the present work, the OSRW algorithm is generalized to be the orthogonal space tempering (OST) method via the introduction of the orthogonal space sampling temperature. Moreover, a double-integration recursion method is developed to enable practically efficient and robust OST free energy calculations, and the algorithm is augmented by a novel θ-dynamics approach to realize both the uniform sampling of order parameter spaces and rigorous end point constraints. In the present work, the double-integration OST method is employed to perform alchemical free energy simulations, specifically to calculate the free energy difference between benzyl phosphonate and difluorobenzyl phosphonate in aqueous solution, to estimate the solvation free energy of the octanol molecule, and to predict the nontrivial Barnase-Barstar binding affinity change induced by the Barnase N58A mutation. As demonstrated in these model studies, the DI-OST method can robustly enable practically efficient free energy predictions, particularly when strongly coupled slow environmental transitions are involved.

On the Use of Temporal Reachout Technique for Characteristics Method with Time-Line Cubic Spline Interpolation

ABSTRACTThe study had indicated that the computational performances of the characteristics method with the time-line cubic spline interpolation are related to the endpoint constraint, especially for large Courant number in which the foot of the characteristic trajectory is located near the endpoint. The first derivative endpoint constraint with higher-order central difference approximation provides better simulation results among various endpoint constraints, but it still induces some degree of numerical error. In this study, by locating the foot of the characteristic trajectory away from the endpoint, the temporal reachout technique is proposed to avoid the effect of endpoint constraint on the time-line cubic spline interpolation. Modeling the transport of a Gaussian concentration distribution in a uniform flow with constant diffusion coefficient and the viscous Burgers equation is used to examine the temporal reachout technique. The outcomes show that the temporal reachout technique yields much better simulation results than the first derivative endpoint constraint with higher-order central difference approximation. The effect of endpoint constraint on the time-line cubic spline interpolation can be greatly diminished by the use of the temporal reachout technique.

Hamilton-Jacobi inequalities for optimal impulsive control problems

AbstractSufficient and necessary global optimality conditions for nonlinear impulsive dynamic optimization problems with endpoint constraints are obtained. Proofs of these results are based on Hamilton-Jacobi canonical optimality theory. As consequence, a Maximum Principle reverse into sufficient optimality conditions is proposed.

Reducing the Prediction Horizon in NMPC: An Algorithm Based Approach⋆

AbstractIn order to guarantee stability, known results for MPC without additional terminal costs or endpoint constraints often require rather large prediction horizons. Still, stable behavior of closed loop solutions can often be observed even for shorter horizons. Here, we make use of the recent observation that stability can be guaranteed for smaller prediction horizons via Lyapunov arguments if more than only the first control is implemented. Since such a procedure may be harmful in terms of robustness, we derive conditions which allow to increase the rate at which state measurements are used for feedback while maintaining stability and desired performance specifications. Our main contribution consists in developing two algorithms based on the deduced conditions and a corresponding stability theorem which ensures asymptotic stability for the MPC closed loop for significantly shorter prediction horizons.

Stability of Constrained Adaptive Model Predictive Control Algorithms

AbstractRecently, suboptimality estimates for model predictive controllers (MPC) have been derived for the case without additional stabilizing endpoint constraints or a Lyapunov function type endpoint weight. The proposed methods yield a posteriori and a priori estimates of the degree of suboptimality with respect to the infinite horizon optimal control and can be evaluated at runtime of the MPC algorithm. Our aim is to design automatic adaptation strategies of the optimization horizon in order to guarantee stability and a predefined degree of suboptimality for the closed loop solution. Here, we present a stability proof for an arbitrary adaptation scheme and state a simple shortening and prolongation strategy which can be used for adapting the optimization horizon.

Intensity-Modulated Radiotherapy for Locally Advanced Non–Small-Cell Lung Cancer: A Dose-Escalation Planning Study

To evaluate the potential for dose escalation with intensity-modulated radiotherapy (IMRT) in positron emission tomography-based radiotherapy planning for locally advanced non-small-cell lung cancer (LA-NSCLC). For 35 LA-NSCLC patients, three-dimensional conformal radiotherapy and IMRT plans were made to a prescription dose (PD) of 66 Gy in 2-Gy fractions. Dose escalation was performed toward the maximal PD using secondary endpoint constraints for the lung, spinal cord, and heart, with de-escalation according to defined esophageal tolerance. Dose calculation was performed using the Eclipse pencil beam algorithm, and all plans were recalculated using a collapsed cone algorithm. The normal tissue complication probabilities were calculated for the lung (Grade 2 pneumonitis) and esophagus (acute toxicity, grade 2 or greater, and late toxicity). IMRT resulted in statistically significant decreases in the mean lung (p <.0001) and maximal spinal cord (p = .002 and 0005) doses, allowing an average increase in the PD of 8.6-14.2 Gy (p ≤.0001). This advantage was lost after de-escalation within the defined esophageal dose limits. The lung normal tissue complication probabilities were significantly lower for IMRT (p <.0001), even after dose escalation. For esophageal toxicity, IMRT significantly decreased the acute NTCP values at the low dose levels (p = .0009 and p <.0001). After maximal dose escalation, late esophageal tolerance became critical (p <.0001), especially when using IMRT, owing to the parallel increases in the esophageal dose and PD. In LA-NSCLC, IMRT offers the potential to significantly escalate the PD, dependent on the lung and spinal cord tolerance. However, parallel increases in the esophageal dose abolished the advantage, even when using collapsed cone algorithms. This is important to consider in the context of concomitant chemoradiotherapy schedules using IMRT.

On optimal service capacity allocation policy in an advance selling environment in continuous time

Based on some general reasonable assumptions, this paper employs the techniques of calculus of variations and optimal control theory to derive 10 main propositions associated with a service provider who aims at maximizing the present value of revenues/profits over a planning horizon in continuous time. Employing an aggregate pricing-response function that is dependent on excess capacity and a convex service cost formulation, the results show that (i) for a zero discount rate and unanticipated competitive entry, units of service capacity are allocated evenly over time, (ii) for a zero discount rate and anticipated competitive entry, units of allocated service capacity are increasing over time, (iii) for a positive discount rate and unanticipated competitive entry, units of allocated service capacity are decreasing over time, and (iv) for a positive discount rate and anticipated competitive entry, units of allocated service capacity are increasing over time, or are decreasing at first then increasing later. In addition, the questions of how deep a service firm should advance sell, whether advance selling is optimal in continuous time, and whether excess capacity could be an optimal strategy have been addressed. Furthermore, related sensitivity analyses are performed and endpoint constraints together with alternative shapes of the service-cost function are discussed. The managerial implications of the study’s results have been demonstrated.

Dynamic and geometric error assessment of an XYC axis subset on five-axis high-speed machine tools using programmed end point constraint measurements

This paper presents a technique for assessing the volumetric errors on a five-axis machine tool for motion involving two linear axes and one rotary axis at selected feed rates using data from two sources. The first source of data is obtained through a programmed end point constraint procedure with measurement of the 3D volumetric positioning errors between a point on the tool holder and another fixed to the machine table reference frame. The tests involve maintaining the nominal coincidence of these two points whilst exercising the three axes. The second source of data is the position feedback signal from the encoder provided by the machine controller. Tests were carried out at low and high feed rates to evaluate the effect of geometric and dynamic errors. Polynomial functions are used to represent and then predict the geometric errors. The predicted geometric errors are then added to the dynamic errors provided by the servo errors from position feedback signals and propagated to the tool centre point and are compared with the measured volumetric errors. It shows that the influence of the geometric errors are dominant at low feed, whereas the effects of the servo errors of the linear axes become dominant as the feed increases, reaching 80% of the total error at a feed of 10,000 mm/min.

Exponential Stability Based Design of Constrained Fuzzy Predictive Control

Predictive control of nonlinear systems subject to output and input constraints is considered. A fuzzy model is used to predict the future behavior. Three new ideas are proposed here. First, an added constraint on the applied control action is used to ensure the decrease of a quadratic Lyapunov function, and so guarantee Lyapunov exponential stability of the closed-loop system. Second, the feasibility of the finite-horizon optimization problem with the added constraints is ensured based on off-line solution of a set of linear matrix inequalities (LMIs). Third, the proposed method is extended to observer-based feedback case. The novel stability method, which we call First Point Constraint Method, is compared to existing methods, such as the techniques based on the end-point constraints (Terminal Constraint Set), and the robust stability techniques based on the small gain theory. The proposed method ensures Lyapunov exponential stability, can be used for open loop unstable plants, and doesn’t need an auxiliary controller. Illustrative examples including the predictive control of a highly nonlinear continuous stirred tank reactor (CSTR) and the observer-based predictive control of a steam generator are discussed.

LMI based design of constrained fuzzy predictive control

Predictive control of nonlinear systems subject to output and input constraints is considered. A fuzzy model is used to predict the future behavior. Two new ideas are proposed here. First, an added constraint on the applied control action is used to ensure the decrease of a quadratic Lyapunov function, and so guarantee Lyapunov exponential stability of the closed-loop system. Second, the feasibility of the finite-horizon optimization problem with the added constraints is ensured based on an off-line solution of a set of LMIs. The novel stability method is compared to the existing methods, such as the techniques based on the end-point constraints (terminal constraint set), and the robust stability techniques based on the small gain theory. The proposed method ensures Lyapunov exponential stability, does not need an auxiliary controller and can be used with any feasible controller parameters. Illustrative examples including the predictive control of a highly nonlinear chemical reactor (CSTR) are discussed.

Optimal control of delay-differential inclusions with functional endpoint constraints in infinite dimensions

This paper is devoted to the study of a general class of optimal control problems described by delay-differential inclusions with equality and inequality endpoint constraints and multivalued initial conditions. We use the method of discrete approximations and advanced tools of variational analysis and generalized differentiation in infinite dimensions to derive necessary optimality conditions in the extended Euler–Lagrange form. This method is fully realized for the delay-differential systems under consideration.

Investigation of Effect of Endpoint Constraint on Time-Line Cubic Spline Interpolation

AbstractIn this study, the effects of various endpoint constraints, including the first derivative, natural, quadratic, and not-a-knot endpoint constraints, on the time-line cubic spline interpolation are examined by solving the advection-diffusion equation with constant and variable velocities, and the viscous Burgers equation. The natural endpoint constraint could produce large numerical diffusion. The quadratic endpoint constraint can decrease the numerical diffusion from the natural endpoint constraint, but it could trigger large numerical oscillation. The first derivative endpoint constraint with higher-order central difference approximation has better simulated results than the natural and quadratic endpoint constraints. It can significantly reduce not only the numerical diffusion produced by the natural endpoint constraint, but also the numerical oscillation caused by the quadratic endpoint constraint. The applicability of the time-line cubic spline interpolation together with the not-a-knot endpoint constraint is limited, since the computational instability is caused while the Courant number is greater than unity. Therefore, as far as accuracy and applicability are concerned, the first derivative endpoint constraint with higher-order central difference approximation rather than the commonly used natural endpoint constraint should be a better choice for the time-line cubic spline interpolation.

Single setup estimation of a five-axis machine tool eight link errors by programmed end point constraint and on the fly measurement with Capball sensor

Five-axis machine tools can be programmed to keep a constant nominal tool end point position while exercising all five axes simultaneously. This kinematic capability allows the use of a 3D proximity sensing head mounted at the spindle to track the position changes of a precision steel ball mounted on the machine table effectively measuring the 3D Cartesian volumetric errors of the machine. The new sensing head uses capacitive sensors to gather data on the fly during a synchronized five-axis motion which lasts less than 2 min. Because the measured volumetric errors are strongly affected by the link geometric errors, they can be used to estimate the link errors through an iterative procedure based on an identification Jacobian matrix. The paper presents the new sensor, the identification model and the experimental validation. The approach allows all eight link errors i.e. the three squarenesses of linear axes and the four orientations and center lines offset of the rotary axes to be estimated with the proposed single setup test. The estimation approach is performed on a horizontal five-axis machine tool. Then, using the estimated link errors, the volumetric errors are predicted for axes combinations different from those used for the identification process. The estimated machine model correctly predicts 52–84% of the volumetric errors for the tested trajectories.

A hybrid method for determining particle masses at the Large Hadron Collider with fully identified cascade decays

A new technique for improving the precision of measurements of SUSY particle masses at the LHC is introduced. The technique involves kinematic fitting of events with two fully identified decay chains. We incorporate both event ETmiss constraints and independent constraints provided by kinematic end-points in experiment invariant mass distributions of SUSY decay products. Incorporation of the event specific information maximises the information used in the fit and is shown to reduce the mass measurement uncertainites by ~ 30% compared to conventional fitting of experiment end-point constraints for the SPS1a benchmark model.

An Adaptive Sub-Optimal Energy Management Strategy for Hybrid Drive-Trains

Typically the energy management problem of a hybrid vehicle is formulated as an optimization problem, where the optimal power split between the prime mover and the secondary power converter is calculated off line based on a given driving cycle and solved numerically with dynamic programming techniques. An important constraint is that the energy level of the secondary power source at the end is the same as in the beginning. In real live the future driving cycle is not known a priori, making it difficult to calculate the exact optimal power split beforehand. To arrive at a practical real time control algorithm, a sub-optimal control law can be applied, where the end-point constraint is replaced by a term in the cost function that accounts for the change in energy; in case of a hybrid electric vehicle it represents the fuel equivalence of the stored reversible energy. In this paper it is reasoned that the reversible energy contains also kinetic and potential energy of the vehicle as well as energy stored in the secondary power source. By feedback control of the state of energy of the secondary power source, the amount of stored energy can be kept on a trajectory, such that the total amount of reversible energy remains constant. Kinetic and potential energy is proportional with vehicle mass, therefore this trajectory is adaptive to vehicle loading. In this paper simulations of an on-line strategy are included that show fuel consumption improvements of a distribution truck, close to those obtained with dynamic programming, validating the reasoning.

Semiconcavity results for optimal control problems admitting no singular minimizing controls

Semiconcavity results have generally been obtained for optimal control problems in absence of state constraints. In this paper, we prove the semiconcavity of the value function of an optimal control problem with end-point constraints for which all minimizing controls are supposed to be nonsingular.

Optimal grade transition in industrial polymerization processes via NCO tracking

AbstractIn industrial polymerization processes, several grades of polymer are frequently produced in the same plant by changing the operating conditions. Transitions between the different grades are rather slow and result in the production of a considerable amount of off‐specification polymer. Grade transition improvement is viewed here as a dynamic optimization problem, for which numerous approaches exist. Open‐loop implementation of the input profiles obtained from numerical optimization of a nominal process model is often insufficient, due to the presence of uncertainty in the form of model mismatch and process disturbances. A novel measurement‐based approach that consists of tracking the necessary conditions of optimality (NCO tracking) using a solution model and measurements is considered. The solution model consists of state‐event‐triggered controllers sequenced according to the structure of the nominal optimal solution computed offline. The solution model is generated by expressing the input profiles in terms of arcs and switching times, which are then related to the various parts of the NCO, that is, to the active constraints and sensitivities. These arcs and switching times are then adapted online using appropriate measurements. The application of NCO tracking to an industrial polymerization process for implementing optimal grade transitions is investigated in simulation. The grade transition problem is fairly complex due to a large‐scale process model, many degrees of freedom, as well as path and endpoint constraints. A solution model is generated from the nominal optimal solution, and a control superstructure is considered to handle the possible activation of nominally‐inactive constraints. Simple PI‐type controllers are used to implement the solution model. For different uncertainty scenarios, simulation of the NCO‐tracking approach shows that considerable reduction in transition time is possible, while still guaranteeing feasible operation. © 2007 American Institute of Chemical Engineers AIChE J, 2007