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SELECTION OF CONTROLLERS
AN ILLUSTRATIVE APPLICATION
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achieves stability approaches zero assumed Automatic Control bang-bang behavior CART DAMPING cart position chatter motion closed form coefficient considered constant constructed control system CONTROLLABILITY BOUNDARY controllable region controller which achieves decay rate defined differential equations dynamic system entire region equilibrium point f v v max finite region flow of probability Fokker-Planck equation Gaussian given initial conditions inverted pendulums ISOMETRIC PROJECTION Jordan canonical form k-dimensional limited linear switching function matrix maximum minimum time switching minimum-time motor negative nonlinear obtained optimal control origin parameters periodic solution physical coordinates physical system piecewise linear switching plane positive eigenvalues positive real eigenvalues probability flux probability of stable problem quasi-steady random disturbance reduced system region of controllability region of stability section III-C shown in Fig signum function sketch space stability boundary stable operation stable region switching line system of Eq trajectories unstable systems variables vector w(y_ white noise