Optimal and sub-optimal relay control by state-variable transformation†

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Abstract
A control method is proposed which permits the application of the relatively simple time-optimal feedback control laws for the double and triple integrator plants to a wide range of more general second and third-order plants having real non-positive eigenvalues. The method, which is based on a state-variable transformation technique, gives time-optimal control for certain plants with eigenvalues in a simple ratio. In other cases the control is sub-optimal and yields settling times typically within 2% and 4% of the corresponding optimal value3 for second and third-order plants, respectively. Thus one may avoid the difficulty of synthesizing- the logarithmic, exponential or other complicated functions of state variables which are generally present in the exact time-optimal control laws (where known) for such plants.