Generalized Homogeneity in Systems and Control

Digital implementation of sliding‐mode control via the implicit method: A tutorial

“…His notion of homogeneity is still well known in the context of the so-called homogeneous polynomials. Homogeneity of nonlinear systems is studied, for example, in [135,44,108,17,94,73], see [99] for a general presentation. Obviously, the sign function is standard homogeneous, indeed: sgn(e s x) = e 0s sgn(x)= sgn(x) (at x = 0 this is understood as equality of sets).…”

“…Obviously, the generator of the standard dilation is the identity matrix I n , and the generator of the weighted dilation is a diagonal matrix with r i on the main diagonal. Strictly monotone dilations satisfy a coercivity condition [99,Proposition 6.5]. The monotonicity of a dilation may depend on a norm • in IR n .…”

“…The monotonicity of a dilation may depend on a norm • in IR n . A dilation d in IR n is strictly monotone (see [98,99] for more details) with respect to the weighted Euclidean norm x P ∆ = √…”

“…) defines a strictly monotone (increasing) function of x 1,k+1 , hence one infers that the nonlinear equation 99 shows that it is hopeless to guarantee finite-time convergence of the scheme in (99). Let us now proceed with the derivation of a consistent method.…”

“…• Infinite dimensional homogeneous systems [99] (would maximal monotone operators as used for the first time in sliding-mode control in [83,86,85,87], be useful in this context? ).…”