Abstract
In this paper, some estimators are proposed for nonlinear dynamical systems with the general conformable derivative. In order to analyze the stability of these estimators, some Lyapunov-like theorems are presented, taking into account finite-time stability. Thus, to prove these theorems, a stability function is defined based on the general conformable operator, which implies exponential stability. The performance of the estimators is assessed by means of numerical simulations. Furthermore, a comparison is made between the results obtained with the integer, fractional, and general conformable derivatives.
Highlights
Fractional calculus, the generalization of calculus to noninteger orders, besides looking to extend the classical mathematical results, has had many applications to physical systems since the 1970s of the twentieth century
Given that the Riemann–Liouville and Caputo derivatives may deal with singularity issues in the kernel, some operators with nonsingular kernel have been proposed, such as Caputo–Fabrizio [11] and Atangana–Baleanu [12] derivatives; these operators are currently being studied extensively, and they have been used for both theoretical results and applications [13,14,15,16,17]
Another reason for proposing other noninteger operators is that the Riemann–Liouville and Caputo derivatives do not satisfy the main results of classical calculus, such as the Leibniz product rule, the chain rule, the semigroup property, and the fundamental theorem, which would be expected to occur naturally for their use in applications
Summary
Fractional calculus, the generalization of calculus to noninteger orders, besides looking to extend the classical mathematical results, has had many applications to physical systems since the 1970s of the twentieth century. Definition 11 The general conformable exponential function is defined as follows: Eαψ (γ , t, t0) = exp γ t dτ t0 ψ(τ , α). 3. There exist a positive definite continuous function α : R+ → R+ such that the settling time with respect to the initial conditions of system (4) satisfies. V (t, x) ≤ –r V (t, x) with a positive definite continuous function r : R+ → R+, r(0) = 0, such that for some >0 dz < +∞, 0 r(z) the origin of system (1) is finite-time stable. Proof From i and ii, the origin of system (1) is GCES (from Theorem 3), and there exists a continuous and differentiable Lyapunov function V (t, x). Remark 12 An estimator is said to be finite-time general conformable exponentially stable if the estimation error e obtained with it is FGCES.
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