A Note on the Itô and Stratonovich Stochastic Relative Degree and Normal Form

Mellone, Alberto; Scarciotti, Giordano

doi:10.1109/cdc42340.2020.9304504

Cited by 2 publications

(3 citation statements)

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“…This is without loss of generality, as all the results of this section can be obtained also in other formalisms, e.g. Stratonovich's formalism [31].…”

Section: Stochastic Relative Degree and Normal Formmentioning

confidence: 89%

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Stochastic Relative Degree and Path-Wise Control of Nonlinear Stochastic Systems

Mellone

Scarciotti

2023

IEEE Trans. Automat. Contr.

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We address the path-wise control of systems described by a set of nonlinear stochastic differential equations. For this class of systems, we introduce a notion of stochastic relative degree and a change of coordinates which transforms the dynamics to a stochastic normal form. The normal form is instrumental for the design of a state-feedback control which linearises and makes the dynamics deterministic. We observe that this control is idealistic, i.e. it is not practically implementable because it employs a feedback of the Brownian motion (which is never available) to cancel the noise. Using the idealistic control as a starting point, we introduce a hybrid control architecture which achieves practical path-wise control. This hybrid controller uses measurements of the state to perform periodic compensations for the noise contribution to the dynamics. We prove that the hybrid controller retrieves the idealistic performances in the limit as the compensating period approaches zero. We address the problem of asymptotic output tracking, solving it in the idealistic and in the practical framework. We finally validate the theory by means of a numerical example.

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“…This is without loss of generality, as all the results of this section can be obtained also in other formalisms, e.g. Stratonovich's formalism [31].…”

Section: Stochastic Relative Degree and Normal Formmentioning

confidence: 89%

“…The zero dynamics of the system is obtained by equating z 1 = 0 and z 2 = 0, z 3 = η t , which yields x 1 = (3/2)η t , s 2 = −η t . Replacing these in the third equation in (31) we get the zero dynamics as follows…”

Section: Illustrative Examplementioning

confidence: 99%