On the Passivity and Positivity Properties in Dynamic Systems: Their Achievement under Control Laws and Their Maintenance under Parameterizations Switching

Abstract

This paper is devoted to discuss certain aspects of passivity results in dynamic systems and the characterization of the regenerative systems counterparts. In particular, the various concepts of passivity as standard passivity, strict input passivity, strict output passivity, and very strict passivity (i.e., joint strict input and output passivity) are given and related to the existence of a storage function and a dissipation function. Later on, the obtained results are related to external positivity of systems and positivity or strict positivity of the transfer matrices and transfer functions in the time-invariant case. On the other hand, how to achieve or how eventually to increase the passivity effects via linear feedback by the synthesis of the appropriate feed-forward or feedback controllers or, simply, by adding a positive parallel direct input-output matrix interconnection gain is discussed.

Discrete-Inverse Optimal Control Applied to the Ball and Beam Dynamical System: A Passivity-Based Control Approach

“…To demonstrate passivity properties in the the dynamical discrete system consider Lemma 1 as presented below [20][21][22]. Theorem 1.…”

“…which confirms control function initially defined in (8) as an optimal control law as it minimizes the functional cost (21). In order to determine the optimal value for the Lagrangian function (21), let us consider that the interval of analysis [0, N], being N a natural number with the following result,…”