High-gain fractional disturbance observer control of uncertain dynamical systems

Robust stabilisation of distributed‐order systems

“…[Lemma 1 of Muñoz-Vázquez et al 35 ] Let (V•x)(t) be a real-valued function that is Lipschitz continuous on x, with convex domain Ω ⊆ R n , and let x ∈ Ω be a real-valued-vector function such that D 𝛼 x(t) exists for some 𝛼 ∈ (0, 1). If V(x(t)) is convex, then the following holds:…”

“…Lemma [Lemma 1 of Muñoz‐Vázquez et al 35 ] Let $$$ \left(V\circ \boldsymbol{x}\right)(t) $$$ be a real‐valued function that is Lipschitz continuous on $$$ \boldsymbol{x} $$$, with convex domain $$$ \Omega \subseteq {\mathbb{R}}^n $$$, and let $$$ \boldsymbol{x}\in \Omega $$$ be a real‐valued‐vector function such that $$$ {D}^{\alpha}\boldsymbol{x}(t) $$$ exists for some $$$ \alpha \in \left(0,1\right) $$$. If $$$ V\left(\boldsymbol{x}(t)\right) $$$ is convex, then the following holds: $$$$ {D}^{\alpha }V\left(\boldsymbol{x}(t)\right)\le \underset{\boldsymbol{\zeta} \left(\boldsymbol{x}\right)\in \partial V\left(\boldsymbol{x}\right)}{\operatorname{inf}}{\boldsymbol{\zeta}}^T\left(\boldsymbol{x}(t)\right)\kern0.1em {D}^{\alpha}\boldsymbol{x}(t).$$…”

Synchronization of time-varying fractional-order chaotic systems in the presence of unknown disturbance using a fast disturbance identification

“…Therefore, significant efforts have been accomplished to investigate in this regard by researchers. For example, in Munoz-Vazquez et al (2021), a high-gain fractional smooth proportional-integral (PI)-like disturbance observer based on a state-feedback controller is proposed to observe continuous but not necessarily differentiable disturbances. In Chen et al (2012), disturbance observer-based robust synchronization control of uncertain chaotic systems has been studied.…”

Lyapunov functions for fractional‐order nonlinear systems with Atangana–Baleanu derivative of Riemann–Liouville type

“…In recent years, fractional differential equations have been employed in nonlinear control theory, signal processing, physical systems modeling, and even in social and biological models. [1][2][3][4][5][6][7][8][9] Considering that, explicit solutions of nonlinear fractional-order systems are difficult to obtain, many stability analysis methods have been developed. In the case of linear systems, some advances have been obtained for commensurate-and incommensurate-order systems.…”