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35
Direct Trajectory Optimization and Costate Estimation of FiniteHorizon and InfiniteHorizon Optimal Control Problems Using a Radau pseudospectral Method
 Computational Optimization and Applications
, 2011
"... A method is presented for direct trajectory optimization and costate estimation using global collocation at LegendreGaussRadau (LGR) points. The method is formulated first by casting the dynamics in integral form and computing the integral from the initial point to the interior LGR points and the ..."
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Cited by 30 (24 self)
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A method is presented for direct trajectory optimization and costate estimation using global collocation at LegendreGaussRadau (LGR) points. The method is formulated first by casting the dynamics in integral form and computing the integral from the initial point to the interior LGR points and the terminal point. The resulting integration matrix is nonsingular and thus can be inverted so as to express the dynamics in inverse integral form. Then, by appropriate choice of the approximation for the state, a pseudospectral (i.e., differential) form that is equivalent to the inverse integral form is derived. As a result, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Moreover, the formulation derived in this paper enables solving general finitehorizon problems using global collocation at the LGR points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem (NLP) to the costates of the optimal control problem. Finally,
Theory and implementation of numerical methods based on RungeKutta integration for solving optimal control problems
, 1996
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Retrospective on Optimization
 25 TH YEAR ISSUE ON COMPUTERS AND CHEMICAL ENGINEERING
"... In this paper we provide a general classification of mathematical optimization problems, followed by a matrix of applications that shows the areas in which these problems have been typically applied in process systems engineering. We then provide a review of solution methods of the major types of op ..."
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Cited by 27 (1 self)
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In this paper we provide a general classification of mathematical optimization problems, followed by a matrix of applications that shows the areas in which these problems have been typically applied in process systems engineering. We then provide a review of solution methods of the major types of optimization problems for continuous and discrete variable optimization, particularly nonlinear and mixedinteger nonlinear programming. We also review their extensions to dynamic optimization and optimization under uncertainty. While these areas are still subject to significant research efforts, the emphasis in this paper is on major developments that have taken place over the last twenty five years.
Bonvin D. Dynamic optimization of batch processes: I. Characterization of the nominal solution. Comp Chem Eng
"... The optimization of batch processes has attracted attention in recent years because, in the face of growing competition, it is a natural choice for reducing production costs, improving product quality, meeting safety requirements and environmental regulations. The main bottleneck in using optimizat ..."
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Cited by 18 (9 self)
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The optimization of batch processes has attracted attention in recent years because, in the face of growing competition, it is a natural choice for reducing production costs, improving product quality, meeting safety requirements and environmental regulations. The main bottleneck in using optimization in industry is the presence of uncertainty. The most natural way to compensate for uncertainty, and thus to improve process operations, is through the use of measurements. This forms the subject of this series of two papers. In this first part, the optimal input profiles are expressed in terms of arcs and switching times, of which some push the system to the constraints of the problem while the others exploit the intrinsic compromise present in the system for the purpose of optimality. Such a characterization improves considerably the interpretability of the solution, enhances the numerical efficiency, and acts as a necessary first step towards a measurementbased optimization framework.
Direct Trajectory Optimization Using a Variable LowOrder Adaptive Pseudospectral Method
 AIAA Journal of Spacecraft and Rockets
"... A variableorder adaptive pseudospectral method is presented for solving optimal control problems. The method developed in this paper adjusts both the mesh spacing and the degree of the polynomial on each mesh interval until a specified error tolerance is satisfied. In regions of relatively high cur ..."
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Cited by 14 (12 self)
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A variableorder adaptive pseudospectral method is presented for solving optimal control problems. The method developed in this paper adjusts both the mesh spacing and the degree of the polynomial on each mesh interval until a specified error tolerance is satisfied. In regions of relatively high curvature, convergence is achieved by refining the mesh, while in regions of relatively low curvature, convergence is achieved by increasing the degree of the polynomial. An efficient iterative method is then described for accurately solving a general nonlinear optimal control problem. Using four examples, the adaptive pseudospectral method described in this paper is shown to be more efficient than either a global pseudospectral method or a fixedorder method. Nomenclature C = path constraint function D = N N 1 Radau pseudospectral differentiation matrix E = maximum absolute solution error Fd = magnitude of drag force, N Fg = magnitude of gravity force, N Fl = magnitude of lift force, N
Optimal Configuration of Tetrahedral Spacecraft Formations,” The
 Journal of the Astronautical Sciences
"... The problem of determining minimumfuel maneuver sequences for a fourspacecraft formation is considered. The objective of this paper is to find fueloptimal spacecraft trajectories that transfer four spacecraft from an initial parking orbit to a desired terminal reference orbit while satisfying a s ..."
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Cited by 11 (11 self)
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The problem of determining minimumfuel maneuver sequences for a fourspacecraft formation is considered. The objective of this paper is to find fueloptimal spacecraft trajectories that transfer four spacecraft from an initial parking orbit to a desired terminal reference orbit while satisfying a set of constraints on the formation at the terminal time. Trajectories involving both one and two allowable maneuvers per spacecraft are considered. The resulting nonlinear optimal control problem is solved numerically using a recently developed direct transcription method called the Gauss pseudospectral method. The results presented in this paper highlight interesting features of the fueloptimal formation and control. Furthermore, by showing that the discretized firstorder optimality conditions from an indirect formulation are satisfied, a postoptimality analysis of the results demonstrates the accuracy and usefulness of the Gauss pseudospectral method.
Optimization Strategies for Dynamic Systems
 In C. Floudas, P. Pardalos (Eds), Encyclopedia of Optimization
, 1999
"... Introduction and Problem Statement Interest in dynamic simulation and optimization of chemical processes has increased significantly during the last two decades. Common problems include control and scheduling of batch processes; startup, upset, shutdown and transient analysis; safety studies and th ..."
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Cited by 9 (2 self)
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Introduction and Problem Statement Interest in dynamic simulation and optimization of chemical processes has increased significantly during the last two decades. Common problems include control and scheduling of batch processes; startup, upset, shutdown and transient analysis; safety studies and the evaluation of control schemes. Chemical processes are modeled dynamically using differentialalgebraic equations (DAEs). The DAE formulation consists of differential equations that describe the dynamic behavior of the system, such as mass and energy balances, and algebraic equations that ensure physical and thermodynamic relations. The general dynamic optimization problem can be stated as follows: min z(t);y(t);u(t);t f ;p '(z(t f ); y(t f ); u(t<F8
A Boundary Value Problem Approach to the Optimization of Chemical Processes Described by DAE Models
, 1997
"... An efficient and robust technique for the optimization of dynamic chemical processes is presented. In particular, we address the solution of large, multistage optimal control and design optimization problems for processes described by DAE models of index one. Our boundary value problem approach (a ..."
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Cited by 6 (0 self)
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An efficient and robust technique for the optimization of dynamic chemical processes is presented. In particular, we address the solution of large, multistage optimal control and design optimization problems for processes described by DAE models of index one. Our boundary value problem approach (a simultaneous solution strategy) is based on a piecewise parametrization of the control functions and a multiple shooting discretization of the DAEs, combined with a specifically tailored SQP technique. The inherent problem structure is exploited on various levels in order to obtain an efficient overall method. In addition, the formulation lends itself well to parallel computation. Unlike other simultaneous strategies based on collocation, direct use is made of existing advanced, fully adaptive DAE solvers. An implementation of this strategy is provided by the recently developed modular optimal control package MUSCODII. Apart from a difficult DAE test problem with control and path constrain...
Analyse und Restrukturierung eines Verfahrens zur direkten Lösung von OptimalSteuerungsproblemen (The Theory of MUSCOD in a Nutshell)
, 1995
"... MUSCOD (MU ltiple Shooting COde for Direct Optimal Control) is the implementation of an algorithm for the direct solution of optimal control problems. The method is based on multiple shooting combined with a sequential quadratic programming (SQP) technique; its original version was developed in the ..."
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Cited by 5 (0 self)
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MUSCOD (MU ltiple Shooting COde for Direct Optimal Control) is the implementation of an algorithm for the direct solution of optimal control problems. The method is based on multiple shooting combined with a sequential quadratic programming (SQP) technique; its original version was developed in the early 1980s by Plitt under the supervision of Bock [Plitt81, Bock84]. The following report is intended to describe the basic aspects of the underlying theory in a concise but readable form. Such a description is not yet available: the paper by Bock and Plitt [Bock84] gives a good overview of the method, but it leaves out too many important details to be a complete reference, while the diploma thesis by Plitt [Plitt81], on the other hand, presents a fairly complete description, but is rather difficult to read. Throughout the present document, emphasis is given to a clear presentation of the concepts upon which MUSCOD is based. An effort has been made to properly reflect the structure of the a...
Optimization Framework for the Synthesis of Chemical Reactor Networks
, 1998
"... The reactor network synthesis problem involves determining the type, size, and interconnections of the reactor units, optimal concentration and temperature profiles, and the heat load requirements of the process. A general framework is presented for the synthesis of optimal chemical reactor networks ..."
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Cited by 3 (1 self)
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The reactor network synthesis problem involves determining the type, size, and interconnections of the reactor units, optimal concentration and temperature profiles, and the heat load requirements of the process. A general framework is presented for the synthesis of optimal chemical reactor networks via an optimization approach. The possible design alternatives are represented via a process superstructure which includes continuous stirred tank reactors and cross flow reactors along with mixers and splitters that connect the units. The superstructure is mathematically modeled using differential and algebraic constraints and the resulting problem is formulated as an optimal control problem. The solution methodology for addressing the optimal control formulation involves the application of a control parameterization approach where the selected control variables are discretized in terms of time invariant parameters. The dynamic system is decoupled from the optimization and solved as a func...