A fractional super-twisting control of electrically driven mechanical systems

The Super Twisting Control Algorithm (STA) constitutes a powerful and robust technique for control and observation problems. The structure of the STA allows inducing second-order sliding modes, such that the sliding variable and its derivative remain at zero after some finite time. However, the STA requires the strong differentiability of the sliding variable and the weak differentiability of disturbances. Thus, the sliding variable should become from an adequate design, ensuring its strong differentiability. Nonetheless, in the more general case of not necessarily integer-order differentiable disturbances, a typical case in electromechanical systems due to non-smooth effects, alternative control methods need to be considered. For that reason, this paper proposes a structural modification of the STA, allowing the integral of the discontinuous function to assume a fractional order to compensate not necessarily integer-order differentiable disturbances. An experimental assessment is conducted, and comparisons to other sliding mode based controllers are presented to demonstrate the reliability of the proposed method.

To the Problem of A Strong Differentiability of Integrals Along Different Directions

It is proved that for any given sequence (σ n , n ∈ ℕ) = Γ0 ⊂ Γ, where Γ is the set of all directions in (i.e., pairs of orthogonal straight lines) there exists a locally integrable function f on such that: (1) for almost all directions σ ∈ Γ\Γ0 the integral ∫ f is differentiable with respect to the family B 2σ of open rectangles with sides parallel to the straight lines from σ; (2) for every direction σ n ∈ Γ0 the upper derivative of ∫ f with respect to B 2σn equals +∞; (3) for every direction σ ∈ Γ the upper derivative of ∫ |f | with respect to B 2σ equals +∞.

A REMARK ON DIFFERENTIABILITY OF THE PRESSURE FUNCTIONAL

We give a short review of results on equilibrium description and description by stochastic dynamics for spin systems on a lattice. We remark also that some coercive inequalities for the generators of stochastic dynamics, as e.g. the Logarithmic Sobolev inequality, can be used in a direct and natural way to prove strong differentiability properties of the pressure functional for lattice spin systems with multiparticle interactions at high temperatures. Motivated by this, we exhibit also a class of examples of multiparticle interactions which do not belong to the space [Formula: see text] of spin interactions, but for which the Gibbs measures exist and are unique at high temperatures.

Strong differentiability of the norm and rotundity of the dual

For normed linear spaces two similar characterizations of strong differentiability of the norm and rotundity of the dual space are established, but it is shown that in general there is no causal relation between these two concepts.