Control of Linear Systems with Large Time Delays in the Control |
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admissible controls analogous delay-free system arg min assumed assumption attitude control attitude-control problem augmented performance index bang-bang control bounded control calculus of variations Chapter Command Curve k₁ conditions for optimality control delay control function control history control problem control signal control vector controllability matrix Definition 3.3 Delay-Free and Time-Delay discussion Equation 3.1 Example Trajectories feedback control law finite identical initial large time delay LIMIT CYCLE Linear Command Curve linear systems logarithmic spiral magnitude necessary conditions non-zero obtained optimal control law ordinary differential equations phase plane radians revised optimization problem satisfied scalar Second-Order Example Second-Order System set of admissible shown in Figure Single Commands specified switch point switching function system behavior system of Equation t₁ ta,t taste tastpl te[t te[to telt terminal constraints Theorem 4.2 time-delay system Trajectories for Second-Order transition matrix vector function x(to x(tp Zero Damping zero-command curve