Ross' π lemma
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Ross' π lemma, named after I. Michael Ross,[1][2][3] is a result in computational optimal control. Based on generating Carathéodory-π solutions for feedback control, Ross' π-lemma states that there is fundamental time constant within which a control solution must be computed for controllability and stability. This time constant, known as Ross' time constant,[4][5] is proportional to the inverse of the Lipschitz constant of the vector field that governs the dynamics of a nonlinear control system.[6][7]
The proportionality factor in the definition of Ross' time constant is dependent upon the magnitude of the disturbance on the plant and the specifications for feedback control. When there are no disturbances, Ross' π-lemma shows that the open-loop optimal solution is the same as the closed-loop one. In the presence of disturbances, the proportionality factor can be written in terms of the Lambert W-function.