Discrete-time state representations, a new paradigm
Continuing the study of discrete-time dynamical systems, a new state representation of discrete-time dynamics is proposed, whose development has been largely inspired by pioneering works of I. D. Landau. Breaking with the traditional idea of modelling discrete-time processes as difference equations we here propose to represent discrete-time dynamics as coupled difference and differential equations. The difference equation models, through jumps, the free evolution or a nominal one associated with some constant control value while the differential equation models the influence of the control. The differential structure is at the basis of a geometric study which enables us to stress some intriguing analogies with continuous-time, in particular regarding the control dependency of the dynamics compared to the time dependency. The present paper aims at convincing the reader that such a differential modelling might be the proper way to define and represent forced discrete-time behaviours. This would completely renew the understanding and the methodologies used when discussing discrete-time dynamics in modelling or control contexts.
KeywordsVector Field Differential Equation Model Free Evolution Differential Representation Exponential Representation
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