Observer-based event-triggered control for uncertain fractional-order systems

Some remarks on the Gronwall integral inequality

“…[3][4][5][6] The integer-order Gronwall integral inequality (IOGII) has been widely applied to study the stability of fractional-order systems. [7][8][9] However, it should be pointed out that the multiplier function in the fractional-order inequality is k(t, 𝜚) = (t−𝜚) 𝜔−1 Γ(𝜔) rather than k(𝜚). In other words, multiplier function k(t, 𝜚) does depend on the variable 𝜚 of the integration besides the variable t, which was ignored in many existing literatures.…”

“…Stability analysis is one of most crucial issues in fractional‐order systems 3–6 . The integer‐order Gronwall integral inequality (IOGII) has been widely applied to study the stability of fractional‐order systems 7–9 . However, it should be pointed out that the multiplier function in the fractional‐order inequality is $$$ k\left(t,\varrho \right)=\frac{{\left(t-\varrho \right)}^{\omega -1}}{\Gamma \left(\omega \right)} $$$ rather than $$$ k\left(\varrho \right) $$$.…”

“…In other words, multiplier function k(t, 𝜚) does depend on the variable 𝜚 of the integration besides the variable t, which was ignored in many existing literatures. [7][8][9] Therefore, the solution of the fractional-order integral inequality cannot be directly evaluated by the IOGII. 10 Although this key point has been mentioned in Cong et al, 11 a persuasive counterexample is missing and the effect of fractional-order 𝜔 on the application of Gronwall integral inequality has not been discussed in detail.…”

Event-triggered robust tracking control for fractional-order uncertain systems

“…On the one hand, output TC (also called model reference control) is of tremendous importance in many control applications (Elmokadem et al, 2016;Matraji et al, 2018;Va et al, 2016), and the main objective of TC is to make the output of a plant, via a controller, track the output of a given reference model as close as possible. On the other hand, as a generalization of classical integral calculus, fractional calculus can describe a wider range of practical systems, and fit the complex dynamic processes in the real world more accurately and efficiently (Feng et al, 2020;Ge and Chen, 2020).…”

“…Meanwhile, considering that in some practical situations, it is not easy to obtain full-state information directly for feedback, different from our preliminary work which utilized the observer's state to design ET controllers (Feng et al, 2020), the issue of ET control based on the tracking error is investigated in this paper, to achieve the asymptotic robust tracking for a class of fractional-order (FO) uncertain systems. Accordingly, the main contributions we try to make lie in two aspects: (a) an error-based triggered inequality is proposed to design the ET controller, then a sufficient condition of the asymptotic stability for the corresponding FO augmented closed-loop (CL) system is presented; and (b) based on the obtained result, a feasible condition in the form of linear matrix inequality (LMI) is further derived, as well as the error feedback controller gain matrix.…”