Fixed-time Active Disturbance Rejection Consensus Tracking Control of A Multi-Agent System via Event-Triggered Mechanism

A fixed-time active disturbance rejection control (FTADRC) consensus tracking strategy is proposed for a class of non-affine nonlinear multi-agent systems with an event-trigger-based communication. Non-affine followers are transformed into affine ones by combining the implicit function theorem with the mean value theorem. A distributed event-triggered estimator is introduced based on its neighbor output information. It is for estimation of a leader’s signal for parts of followers, who are not able to access the leader signal in a direct manner. A distributed FTADRC control strategy is then developed via an event-triggered communication in the framework of backstepping technology. With the help of the fixed-time control, the settling time of an MAS is assignable and independent on initial conditions. Extended state observers and tracking differentiators are employed to compensate unknown dynamics of each follower in real time and estimate derivatives of virtual control laws, respectively. It is proven theoretically that the MAS achieves input-to-state practically stability and the consensus tracking error converges to a neighborhood around the origin in a fixed time. Also, Zeno behavior is excluded. Finally, two examples are performed to illustrate the effectiveness of the proposed strategy.

The Transmission Dynamics of Hepatitis B Virus via the Fractional-Order Epidemiological Model

We investigate and analyze the dynamics of hepatitis B with various infection phases and multiple routes of transmission. We formulate the model and then fractionalize it using the concept of fractional calculus. For the purpose of fractionalizing, we use the Caputo–Fabrizio operator. Once we develop the model under consideration, existence and uniqueness analysis will be discussed. We use fixed point theory for the existence and uniqueness analysis. We also prove that the model under consideration possesses a bounded and positive solution. We then find the basic reproductive number to perform the steady-state analysis and to show that the fractional-order epidemiological model is locally and globally asymptotically stable under certain conditions. For the local and global analysis, we use linearization, mean value theorem, and fractional Barbalat’s lemma, respectively. Finally, we perform some numerical findings to support the analytical work with the help of graphical representations.

An Error Compensation Method for Improving the Properties of a Digital Henon Map Based on the Generalized Mean Value Theorem of Differentiation

Continuous chaos may collapse in the digital world. This study proposes a method of error compensation for a two-dimensional digital system based on the generalized mean value theorem of differentiation that can restore the fundamental performance of chaotic systems. Different from other methods, the compensation sequence of our method comes from the chaotic system itself and can be applied to higher-dimensional digital chaotic systems. The experimental results show that the improved system is highly consistent with the real chaotic system, and it has excellent chaotic characteristics such as high complexity, randomness, and ergodicity.

Numerical approach of riemann-liouville fractional derivative operator

<p>This article introduces some new straightforward and yet powerful formulas in the form of series solutions together with their residual errors for approximating the Riemann-Liouville fractional derivative operator. These formulas are derived by utilizing some of forthright computations, and by utilizing the so-called weighted mean value theorem (WMVT). Undoubtedly, such formulas will be extremely useful in establishing new approaches for several solutions of both linear and nonlinear fractionalorder differential equations. This assertion is confirmed by addressing several linear and nonlinear problems that illustrate the effectiveness and the practicability of the gained findings.</p>

A New Conformable Fractional Derivative and Applications

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

Model-Free Tracking Control with Prescribed Performance for a Shape Memory Alloy-Based Robotic Hand

The shape memory alloy (SMA)-based robotic hand has been a new emerging technology with potential applications ranging from life service to surgical treatment, because of the characteristics of SMA, such as high power-to-weight ratio, small volume and low driving voltage. However, due to the complex dynamic model and nonlinear aspects of SMA, it is complicated to control an SMA-based robotic hand. This paper presents a novel model free adaptive control for the SMA-based robotic hand system. By applying the Taylor series expansion method and the differential mean value theorem, the SMA based robotic hand system can be transformed into an equivalent linearization model, which merely depends on measurement data without any information on the system. Combined with prescribed performance control, the novel control method can constrain the tracking error in a preassigned domain. Experiments are conducted on the SMA-based robotic hand system to verify the performance of the presented control method.

On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis

The discrete delta Caputo-Fabrizio fractional differences and sums are proposed to distinguish their monotonicity analysis from the sense of Riemann and Caputo operators on the time scale Z. Moreover, the action of Q− operator and discrete delta Laplace transform method are also reported. Furthermore, a relationship between the discrete delta Caputo-Fabrizio-Caputo and Caputo-Fabrizio-Riemann fractional differences is also studied in detail. To better understand the dynamic behavior of the obtained monotonicity results, the fractional difference mean value theorem is derived. The idea used in this article is readily applicable to obtain monotonicity analysis of other discrete fractional operators in discrete fractional calculus.

On a basic mean value theorem with explicit exponents

We follow a paper by Sedunova regarding Vaughan’s basic mean value Theorem to improve and complete a more general demonstration for a suitable class of arithmetic functions as started by Cojocaru and Murty. As an application we derive a basic mean value theorem for the von Mangoldt generalized functions.