Control of linear systems with large time delays in the control, Volume 1553
National Aeronautics and Space Administration, 1969 - Science - 64 pages
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admissible controls analogous delay-free system assumed assumption 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 Cp(tQ deep-space 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 minimized Minimum-Time Switching Curve necessary conditions non-zero obtained optimal control law ordinary differential equations origin 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 switch points system behavior system of Equation systems possessing td,tf td)dT 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