Discretization of homogeneous systems using Euler method with a state-dependent step

Abstract

Numeric approximations to the solutions of asymptotically stable homogeneous systems by Euler method, with a step of discretization scaled by the state norm, are investigated (for the explicit and implicit integration schemes). It is proven that for a sufficiently small discretization step the convergence of the approximating solutions to zero can be guaranteed globally in a finite or a fixed time depending on the degree of homogeneity of the system, but in an infinite number of discretization iterations. The maximal admissible step can be estimated by analyzing the system properties on the sphere. It is shown that the absolute and relative errors of the discretizations are globally bounded functions, thus the approximations approaching the solutions with the step converging to zero. In addition, it is established that the proposed discretization approach preserves robustness with respect to exogenous perturbations. Efficiency of the designed discretization algorithms is demonstrated in simulations.

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

“…We shall see an example of such solver in section 3.3. More details are also provided in section D. Onceσ k+1 has been computed from (37), then u s k can be obtained using (35).…”

“…From (35) it follows that the fundamental operator associated with the implicit algorithm is given by:…”

Finite-time and fixed-time input-to-state stability: Explicit and implicit approaches

“…Then, from [9, Lemma 1] we know that the inequality (7) holds with β(r, t) = 0 ∀t ≥ ∼ T (r) and that sup r∈R ≥0 ∼ T (r) < +∞. Since the estimates (24) and (25) also hold in this case, we conclude FXISS of the origin of (1).…”

“…Nonasymptotic convergence rates are a major feature in Sliding Mode Control [11,12,13,14] and some further developments in non-asymptotic convergence include a bound on the convergence time that is not only fixed but also arbitrarily selected [15,16,17], a better control performance when initial conditions are far away from the origin by separating low and high growing terms [18,19] and finite-time stable controllers with an enhancement of the domain of attraction for state-constrained systems [20]. Although systems with nonasymptotic rates of convergence may exhibit numerical inconsistencies [21] or lose some of its properties under discretization algorithms [22], recent advances in consistent discretization provide algorithms that overcome these issues [23,24].…”

Robust stability analysis and implementation of Persidskii systems

“…To this end a special structure of Lyapunov function is proposed, and it is shown that its properties, as well as for its derivative, can be investigated by solving linear matrix inequalities (LMIs). Next, it is demonstrated that for discretization of this class of nonlinear dynamical systems, the Euler method has an advantage to preserve the stability of trajectories under mild restrictions (for nonlinear systems it is not always the case [17], [18] Lotka-Volterra system is considered as an example of application of the proposed theory demonstrating its efficiency.…”