Geometric Homogeneity and Stabilization

Fast Control Systems: Nonlinear Approach

“…Homogeneity widely studied in control theory [23], [41], [32], [15], [37] is a sort of symmetry of an object (e.g. function or vector field) with respect to some group of transformations called dilation.…”

“…The geometric dilation [24], [23], [42] is more general since it allows the map d(s) : R n → R n (s ∈ R) to be non-linear.…”

“…[22], [41], [23], [35]). For monotone dilations we can introduce the canonical homogeneous norm as follows:…”

“…The next theorem gives the most important result about scalability solutions to dhomogeneous evolution equations [49], [23], [41], [7], [28], [35].…”

Finite-time output stabilization of the double integrator

“…This is why the homogeneity plays a central role in the FTS system design. The reader may found additional properties and results on homogeneity in [6], [7], [8], [9], [10]. The homogeneity property was used many times to design FTS state controls [11], [12], [13], [14], [15], [16], FTS observers [17], [18] and FTS output feedback [19], [20].…”