Control of Linear Systems with Large Time Delays in the Control |
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admissible controls analogous delay-free system arg min assumed augmented performance index bang-bang control calculus of variations Chapter COMMAND LINE commands occur conditions for optimality control delay control function control history control problem control signal control vector controllability matrix Definitions 3.1 Delay-Free and Time-Delay discussion double commands Equation 3.1 Example Trajectories feedback control law feedback solution finite identical initial KEY td=0 large time delay LIMIT CYCLE Linear Command Curve linear systems logarithmic spiral magnitude necessary conditions non-zero obtained optimal control law order system ordinary differential equations origin phase plane radians satisfied scalar Second-Order Example Second-Order System set of admissible shown in Figure Single Commands specified switching function system behavior system of Equation systems possessing tâ‚‚ ta,t ta,tp taste te[t telt terminal constraints Theorem 4.1 time-delay system Trajectories for Second-Order transition matrix unbounded control vector function x(tp Zero Damping zero-command curve