Control of linear systems with large time delays in the control, Volume 1553
National Aeronautics and Space Administration, 1969 - Astronautics - 64 pages
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admissible controls analogous delay-free system assumed assumption attitude control attitude-control problem augmented performance index bang-bang control bounded control calculus of variations Chapter conditions for optimality control delay control function control history CONTROL OF LINEAR control problem control signal control vector controllability matrix Definition Delay-Free and Time-Delay discussion double commands Equation 3.1 Equation l.l Example Trajectories feedback control law finite identical initial large time delay limit cycle Linear Command Curve linear systems logarithmic spiral magnitude Minimum-Time Switching Curve necessary conditions non-zero o o d obtained optimal control law ordinary differential equations phase plane possessing time delays radians revised optimization problem satisfied scalar Second-Order Example Second-Order System set of admissible shown in Figure Single Commands sinvT system behavior system of Equation systems possessing td,t td,tf te[t te[tQ terminal constraints Theorem time-delay system Trajectories for Second-Order transition matrix vector function x(tf x(tQ Zero Damping zero-command curve