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1.0 a variable adaptive control ANALOG SIMULATION average value band-pass filter bang-bang chapter chatter motion coefficients complex contactor control system control problem control processes control signal u(t control switching surface differential equation Doctor of Philosophy eigenvalues error state space greater than zero high-pass filter Hopkin and Wang initial conditions linear filter magnitude minimum-time control minimum-time problem optimal order plant parametric equations Peak to Peak performance feedback performance measure performance surface plant output plant parameters plant state variables plant zeros plants with zeros plotted power-series approximation prediction proper sign quasi-optimum control switching quasi-optimum switching surfaces reference input regulator switching surface reversal surface Second Order second-order plant self-optimizing control system shown in Fig solution Square Wave Variable state-space origin state-space trajectories surface equations switching line switching-surface parameters third-order control THIRD-ORDER REGULATOR PROCESSES tion tractable transition matrix value of f(e variable Square Wave various derivatives vector yields