Optimal control of grinding mill circuit using model predictive static programming: A new nonlinear MPC paradigm

The recently developed reference-command tracking version of model predictive static programming (MPSP) is successfully applied to a single-stage closed grinding mill circuit. MPSP is an innovative optimal control technique that combines the philosophies of model predictive control (MPC) and approximate dynamic programming. The performance of the proposed MPSP control technique, which can be viewed as a `new paradigm' under the nonlinear MPC philosophy, is compared to the performance of a standard nonlinear MPC technique applied to the same plant for the same conditions. Results show that the MPSP control technique is more than capable of tracking the desired set-point in the presence of model-plant mismatch, disturbances and measurement noise. The performance of MPSP and nonlinear MPC compare very well, with definite advantages offered by MPSP. The computational speed of MPSP is increased through a sequence of innovations such as the conversion of the dynamic optimization problem to a low-dimensional static optimization problem, the recursive computation of sensitivity matrices and using a closed form expression to update the control. To alleviate the burden on the optimization procedure in standard MPC, the control horizon is normally restricted. However, in the MPSP technique the control horizon is extended to the prediction horizon with a minor increase in the computational time. Furthermore, the MPSP technique generally takes only a couple of iterations to converge, even when input constraints are applied. Therefore, MPSP can be regarded as a potential candidate for online applications of the nonlinear MPC philosophy to real-world industrial process plants. (C) 2014 Elsevier Ltd. All rights reserved.

Inclusion of nonlinear demand-supply relationships within large-scale partial equilibrium linear programming models

This paper presents a new method for the inclusion of nonlinear demand and supply relationships within a linear programming model. An existing method for this purpose is described first and its shortcomings are pointed out before showing how the new approach overcomes those difficulties and how it provides a more accurate and 'smooth' (rather than a kinked) approximation of the nonlinear functions as well as dealing with equilibrium under perfect competition instead of handling just the monopolistic situation. The workings of the proposed method are illustrated by extending a previously available sectoral model for the UK agriculture.

Development of new scenario decomposition techniques for linear and nonlinear stochastic programming

Une approche classique pour traiter les problèmes d’optimisation avec incertitude à deux- et multi-étapes est d’utiliser l’analyse par scénario. Pour ce faire, l’incertitude de certaines données du problème est modélisée par vecteurs aléatoires avec des supports finis spécifiques aux étapes. Chacune de ces réalisations représente un scénario. En utilisant des scénarios, il est possible d’étudier des versions plus simples (sous-problèmes) du problème original. Comme technique de décomposition par scénario, l’algorithme de recouvrement progressif est une des méthodes les plus populaires pour résoudre les problèmes de programmation stochastique multi-étapes. Malgré la décomposition complète par scénario, l’efficacité de la méthode du recouvrement progressif est très sensible à certains aspects pratiques, tels que le choix du paramètre de pénalisation et la manipulation du terme quadratique dans la fonction objectif du lagrangien augmenté. Pour le choix du paramètre de pénalisation, nous examinons quelques-unes des méthodes populaires, et nous proposons une nouvelle stratégie adaptive qui vise à mieux suivre le processus de l’algorithme. Des expériences numériques sur des exemples de problèmes stochastiques linéaires multi-étapes suggèrent que la plupart des techniques existantes peuvent présenter une convergence prématurée à une solution sous-optimale ou converger vers la solution optimale, mais avec un taux très lent. En revanche, la nouvelle stratégie paraît robuste et efficace. Elle a convergé vers l’optimalité dans toutes nos expériences et a été la plus rapide dans la plupart des cas. Pour la question de la manipulation du terme quadratique, nous faisons une revue des techniques existantes et nous proposons l’idée de remplacer le terme quadratique par un terme linéaire. Bien que qu’il nous reste encore à tester notre méthode, nous avons l’intuition qu’elle réduira certaines difficultés numériques et théoriques de la méthode de recouvrement progressif.

Development of new scenario decomposition techniques for linear and nonlinear stochastic programming

Une approche classique pour traiter les problèmes d’optimisation avec incertitude à deux- et multi-étapes est d’utiliser l’analyse par scénario. Pour ce faire, l’incertitude de certaines données du problème est modélisée par vecteurs aléatoires avec des supports finis spécifiques aux étapes. Chacune de ces réalisations représente un scénario. En utilisant des scénarios, il est possible d’étudier des versions plus simples (sous-problèmes) du problème original. Comme technique de décomposition par scénario, l’algorithme de recouvrement progressif est une des méthodes les plus populaires pour résoudre les problèmes de programmation stochastique multi-étapes. Malgré la décomposition complète par scénario, l’efficacité de la méthode du recouvrement progressif est très sensible à certains aspects pratiques, tels que le choix du paramètre de pénalisation et la manipulation du terme quadratique dans la fonction objectif du lagrangien augmenté. Pour le choix du paramètre de pénalisation, nous examinons quelques-unes des méthodes populaires, et nous proposons une nouvelle stratégie adaptive qui vise à mieux suivre le processus de l’algorithme. Des expériences numériques sur des exemples de problèmes stochastiques linéaires multi-étapes suggèrent que la plupart des techniques existantes peuvent présenter une convergence prématurée à une solution sous-optimale ou converger vers la solution optimale, mais avec un taux très lent. En revanche, la nouvelle stratégie paraît robuste et efficace. Elle a convergé vers l’optimalité dans toutes nos expériences et a été la plus rapide dans la plupart des cas. Pour la question de la manipulation du terme quadratique, nous faisons une revue des techniques existantes et nous proposons l’idée de remplacer le terme quadratique par un terme linéaire. Bien que qu’il nous reste encore à tester notre méthode, nous avons l’intuition qu’elle réduira certaines difficultés numériques et théoriques de la méthode de recouvrement progressif.

Model Predictive Static Programming: A Computationally Efficient Technique For Suboptimal Control Design

Combining the philosophies of nonlinear model predictive control and approximate dynamic programming, a new suboptimal control design technique is presented in this paper, named as model predictive static programming (MPSP), which is applicable for finite-horizon nonlinear problems with terminal constraints. This technique is computationally efficient, and hence, can possibly be implemented online. The effectiveness of the proposed method is demonstrated by designing an ascent phase guidance scheme for a ballistic missile propelled by solid motors. A comparison study with a conventional gradient method shows that the MPSP solution is quite close to the optimal solution.

Application of Mathematical Programming to the Plywood Design and Manufacturing Problem

Plywood manufacture includes two fundamental stages. The first is to peel or separate logs into veneer sheets of different thicknesses. The second is to assemble veneer sheets into finished plywood products. At the first stage a decision must be made as to the number of different veneer thicknesses to be peeled and what these thicknesses should be. At the second stage, choices must be made as to how these veneers will be assembled into final products to meet certain constraints while minimizing wood loss. These decisions present a fundamental management dilemma. Costs of peeling, drying, storage, handling, etc. can be reduced by decreasing the number of veneer thicknesses peeled. However, a reduced set of thickness options may make it infeasible to produce the variety of products demanded by the market or increase wood loss by requiring less efficient selection of thicknesses for assembly. In this paper the joint problem of veneer choice and plywood construction is formulated as a nonlinear integer programming problem. A relatively simple optimal solution procedure is developed that exploits special problem structure. This procedure is examined on data from a British Columbia plywood mill. Restricted to the existing set of veneer thicknesses and plywood designs used by that mill, the procedure generated a solution that reduced wood loss by 79 percent, thereby increasing net revenue by 6.86 percent. Additional experiments were performed that examined the consequences of changing the number of veneer thicknesses used. Extensions are discussed that permit the consideration of more than one wood species.

A single network adaptive critic (SNAC) architecture for optimal control synthesis for a class of nonlinear systems

Even though dynamic programming offers an optimal control solution in a state feedback form, the method is overwhelmed by computational and storage requirements. Approximate dynamic programming implemented with an Adaptive Critic (AC) neural network structure has evolved as a powerful alternative technique that obviates the need for excessive computations and storage requirements in solving optimal control problems. In this paper, an improvement to the AC architecture, called the �Single Network Adaptive Critic (SNAC)� is presented. This approach is applicable to a wide class of nonlinear systems where the optimal control (stationary) equation can be explicitly expressed in terms of the state and costate variables. The selection of this terminology is guided by the fact that it eliminates the use of one neural network (namely the action network) that is part of a typical dual network AC setup. As a consequence, the SNAC architecture offers three potential advantages: a simpler architecture, lesser computational load and elimination of the approximation error associated with the eliminated network. In order to demonstrate these benefits and the control synthesis technique using SNAC, two problems have been solved with the AC and SNAC approaches and their computational performances are compared. One of these problems is a real-life Micro-Electro-Mechanical-system (MEMS) problem, which demonstrates that the SNAC technique is applicable to complex engineering systems.

Nonlinear Suboptimal Guidance with Impact Angle Constraint for Slow Moving Targets in 1-D Using MPSP

Using a recently developed method named as model predictive static programming (MPSP), a nonlinear suboptimal guidance law for a constant speed missile against a slow moving target with impact angle constraint is proposed. In this paper MPSP technique leads to a closed form solution of the latax history update for the given problem. Guidance command is the latax,which is normal to the missile velocity and the terminal constraints are miss distance and impact angle. The new guidance law is validated by considering the nonlinear kinematics with both lag-free and first order autopilot delay.

Impact-Angle-Constrained Suboptimal Model Predictive Static Programming Guidance of Air-to-Ground Missiles

A nonlinear suboptimal guidance law is presented in this paper for successful interception of ground targets by air-launched missiles and guided munitions. The main feature of this guidance law is that it accurately satisfies terminal impact angle constraints in both azimuth as well as elevation simultaneously. In addition, it is capable of hitting the target with high accuracy as well as minimizing the lateral acceleration demand. The guidance law is synthesized using recently developed model predictive static programming (MPSP). Performance of the proposed MPSP guidance is demonstrated using three-dimensional (3-D) nonlinear engagement dynamics by considering stationary, moving, and maneuvering targets. Effectiveness of the proposed guidance has also been verified by considering first. order autopilot lag as well as assuming inaccurate information about target maneuvers. Multiple munitions engagement results are presented as well. Moreover, comparison studies with respect to an augmented proportional navigation guidance (which does not impose impact angle constraints) as well as an explicit linear optimal guidance (which imposes the same impact angle constraints in 3-D) lead to the conclusion that the proposed MPSP guidance is superior to both. A large number of randomized simulation studies show that it also has a larger capture region.

Generalized model predictive static programming and its application to 3D impact angle constrained guidance of air-to-surface missiles

A new `generalized model predictive static programming (G-MPSP)' technique is presented in this paper in the continuous time framework for rapidly solving a class of finite-horizon nonlinear optimal control problems with hard terminal constraints. A key feature of the technique is backward propagation of a small-dimensional weight matrix dynamics, using which the control history gets updated. This feature, as well as the fact that it leads to a static optimization problem, are the reasons for its high computational efficiency. It has been shown that under Euler integration, it is equivalent to the existing model predictive static programming technique, which operates on a discrete-time approximation of the problem. Performance of the proposed technique is demonstrated by solving a challenging three-dimensional impact angle constrained missile guidance problem. The problem demands that the missile must meet constraints on both azimuth and elevation angles in addition to achieving near zero miss distance, while minimizing the lateral acceleration demand throughout its flight path. Both stationary and maneuvering ground targets are considered in the simulation studies. Effectiveness of the proposed guidance has been verified by considering first order autopilot lag as well as various target maneuvers.

Robust Reentry Guidance of a Reusable Launch Vehicle Using Model Predictive Static Programming

A robust suboptimal reentry guidance scheme is presented for a reusable launch vehicle using the recently developed, computationally efficient model predictive static programming. The formulation uses the nonlinear vehicle dynamics with a spherical and rotating Earth, hard constraints for desired terminal conditions, and an innovative cost function having several components with associated weighting factors that can account for path and control constraints in a soft constraint manner, thereby leading to smooth solutions of the guidance parameters. The proposed guidance essentially shapes the trajectory of the vehicle by computing the necessary angle of attack and bank angle that the vehicle should execute. The path constraints are the structural load constraint, thermal load constraint, bounds on the angle of attack, and bounds on the bank angle. In addition, the terminal constraints include the three-dimensional position and velocity vector components at the end of the reentry. Whereas the angle-of-attack command is generated directly, the bank angle command is generated by first generating the required heading angle history and then using it in a dynamic inversion loop considering the heading angle dynamics. Such a two-loop synthesis of bank angle leads to better management of the vehicle trajectory and avoids mathematical complexity as well. Moreover, all bank angle maneuvers have been confined to the middle of the trajectory and the vehicle ends the reentry segment with near-zero bank angle, which is quite desirable. It has also been demonstrated that the proposed guidance has sufficient robustness for state perturbations as well as parametric uncertainties in the model.