Consistent Discretization of Finite-time Stable Homogeneous Systems

Abstract

An algorithm of implicit discretization for generalized homogeneous system having discontinuity only at the origin is developed. It is based on transformation of the original system to an equivalent standard homogeneous system which admits implicit discretization preserving finite-time convergence property. The scheme is demonstrated for a version of the socalled "quasi-continuous" high-order sliding mode algorithm.

Consistent Discretization of Locally Homogeneous Finite-time Stable Control Systems

“…The homogeneity allows local properties (e.g., smoothness) of vector fields (functions) to be extended globally [7], [8]. For instance [21], the vector field f ∈ F d (R n ) is Lipschitz continuous on R n \{0} if and only if it satisfies Lipschitz condition on the unit sphere S. Similarly, since the map s → d(s)x is locally uniformly continuous, then uniform continuity of f ∈ F d (R n ) on the unit sphere implies its local uniform continuity on R n \{0} (see [24]).…”

“…is asymptotically stable then the system (1) is locally finitetime stable. A consistent discretization scheme for finite-time stable homogeneous systems is developed [24] based on Theorem 2. Below we use similar ideas in order to design a consistent discretization scheme locally (close to 0), where the dhomogeneous approximation f 0 is a dominating nonlinearity.…”

“…Let us denote q i+1 = y i+1 , z i+1 = yi+1 yi+1 andà = A 0 + BK + G d . Similarly to [24] the discretized control can be defined as u i = u(x i+1 ) = u Φ −1 (y) = Kz i+1 . If y i (I n −Ã) − P (I n −Ã) −1 y i ≤ h 2 then y i ∈ hF (0), q i+1 = 0 and z i+1 = h −1 (I n −Ã) −1 y i ;…”

“…Otherwise, according the scheme proposed above the value of y i+1 (as well as q i+1 = 0 and z i+1 ) are assumed to be derived by means of a Newton method. Notice that the matrix I −Ã is invertible [24], so the control u i is well defined. The original state is given by x i+1 = d(ln y i+1 )y i+1 / y i+1 .…”

“…It is shown [24] that an implementation of finite-time controllers using the developed scheme rejects numerical chattering and essentially improves the control precision. This paper continues this research direction and proposes a consistent discretization scheme for non-homogeneous finitetime stable systems, which admit homogeneous approximation at the origin.…”