Authors: G R Hennig A Miele
Publish Date: 2013/08/03
Volume: 12, Issue: 1, Pages: 61-98
Abstract
This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints a state inequality constraint and terminal constraints The problem is to find the statext the controlut and the parameter π so that the functional is minimized while the constraints are satisfied to a predetermined accuracyThe approach taken is a sequence of twophase processes or cycles composed of a gradient phase and a restoration phase The gradient phase involves a single iteration and is designed to decrease the functional while the constraints are satisfied to first order The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy while the norm of the variations of the control and the parameter is minimized The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions the functionsxtut π obtained at the end of each cycle satisfy the constraints to a predetermined accuracy Therefore the functionals of any two elements of the sequence are comparableHere the state inequality constraint is handled in a direct manner A predetermined number and sequence of subarcs is assumed and for the time interval for which the trajectory of the system lies on the state boundary the control is determined so that the state boundary is satisfied The state boundary and the entrance conditions are assumed to be linear inx and π and the sequential gradientrestoration algorithm is constructed in such a way that the state inequality constraint is satisfied at each iteration of the gradient phase and the restoration phase along all of the subarcs composing the trajectoryAt first glance the assumed linearity of the state boundary and the entrance conditions appears to be a limitation to the theory Actually this is not the case The reason is that every constrained minimization problem can be brought to the present form through the introduction of additional state variablesTo facilitate the numerical solution on digital computers the actual time θ is replaced by the normalized timet defined in such a way that each of the subarcs composing the extremal arc has a normalized time length Δt=1 In this way variabletime corner conditions and variabletime terminal conditions are transformed into fixedtime corner conditions and fixedtime terminal conditions The actual times θ1 θ2 τ at which i the state boundary is entered ii the state boundary is exited and iii the terminal boundary is reached are regarded to be components of the parameter π being optimizedThis paper is based in part on a portion of the dissertation which the first author submitted in partial fulfillment of the requirements for the PhD Degree at the Air Force Institute of Technology WrightPatterson AFB Ohio This research was supported in part by the Office of Scientific Research Office of Aerospace Research United States Air Force Grant No AFAFOSR722185 The authors are indebted to Professor H Y Huang Dr R R Iyer Dr J N Damoulakis Mr A Esterle and Mr J R Cloutier for helpful discussions as well as analytical and numerical assistance This paper is a condensation of the investigations reported in Refs 1–2
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