Stereographic projection of probabilistic frequency-domain uncertainty - Nyström, Anton; Renganathan, Venkatraman; Cantoni, Michael (IEEE, 2024)
This paper investigates the stereographic projection of points along the Nyquist plots of single input single output (SISO) linear time invariant (LTI) systems subject to probabilistic uncertainty. At each frequency, there corresponds a complex-valued random variable with given probability distribution in the complex plane. The chordal distance between the stereographic projections of this complex value and the corresponding value for a nominal model, as per the well-known ν-Gap metric of Vinnicombe, is also a random quantity. The main result provides the cumulative density function (CDF) of the chordal distance at a given frequency. Such a stochastic distance framework opens up a fresh and a fertile research direction on probabilistic robust control theory.
Keywords: 4901 Applied Mathematics; 49 Mathematical Sciences; 4007 Control Engineering, Mechatronics and Robotics; 40 Engineering; Robust control; Measurement; Linear systems; Uncertainty; SISO; Frequency-domain analysis; Stochastic processes; Probabilistic logic; Probability distribution; Random variables