THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Aeronautical and Astronautical Engineering QUARTERLY PROGRESS REPORT for January, February, March 1962 by; #"/Y Elmer GJ Gilbert ORA Project 04487 Prepared for George C. Marshall Space Flight Center National Aeronautics and Space Administration Huntsville, Alab ma under Contract No. NAS8-1569.*. ~... i..'.:.''''..'' administered by The University of Michigan Office of Research Administration Ann Arbor, Michigan

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1. Introduction This report describes work accomplished in the period of January through March 1962 in the contractor'sresearch program on various aspects of space vehicle booster control systems. The study of recovery regions of linear systems with limited control effort described herein is by doctoral student Joseph L. LeMay under the supervision of Professor Elmer G. Gilbert. 2. Regions of Recoverability of Linear Systems with Input Constraints Concepts of a more general nature have been applied to what is essentially the same problem as that described in the previous progress report. We study a system described by a general linear differential equation with an input constrained to be in a specified set 2. The set D is assumed to be compact and convex. The set of states which can be driven to the origin is called the maximum region of recoverability, RRM (t, to). This set is a function of two variables —starting time t and stopping time t. In the constant coefficient case, RRM(T) depends only on elapsed time T. The results for RRM(T) of the previousreport assumed implicitly that T was arbitrarily large. It has been shown that RRM(t, t ) is compact and convex and does.0 not diminish in size as t increases. Another result is a necessary and sufficient condition that RRM(oo, t ) be the whole state space. For the constant case this condition requires the eigenvalue real parts to be non-positive. Some computational procedures for determining RRM(t, t ) and 0 bounds thereof have been developed. Work on determining RRM(T) for the yaw (pitch) control of a space vehicle booster (in actual coordinates) is in progress. ~Some of these results exist elsewhere. - 1 -

3. Simplified Stability Analysis of Booster Control System Professional degree student Guy H. Risley, Jr. began work under Professor Edward 0. Gilbert on a simplified method of stability analysis for the booster control system. Based on the method of Krylov and Bogoliubov, it should be useful in determining the damping of both structural and slosh modes. -2

Statement of Man-Hours Expended and Summary of Expenditures for January, February, March 1962 Three Month January 1962 February 1962 March 1962 Period Man-Hours Expended Faculty Participants 46 46 66 158 Graduate Students 83 88 80 251 Total 129 134 146 409 Salaries and Wages Faculty Participants $403.61 $447. 73 $601. 90 $1, 453.24 Graduate Students 269. 75 330. 00 300. 00 899. 75 Total $673.36 $777.73 $901.90go $2,352.99 Overhead $336.68 $396.37 $453.00 $1,186.05 Materials and Supplies 3.65 1. 00 8.14 12.79 Reports 0.00 15.00 4.10 19.10 Travel 0,00 0.00 0.00 0,00 Totals $1,013.69 $1,190.10 $1, 367,14 $3,570.93

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