An optimal fractional order Kalman consensus filter (OFOKCF) is developed in this paper to address distributed state estimation challenges within fractional order systems. By leveraging historical state information, the algorithm captures long-term dynamics and model the system more precisely. A consensus mechanism ensures node consistency, while covariance intersection simplifies cross-correlation computations, using trace-based adaptive weights for multi-dimensional states. The simulation validates the effectiveness and accuracy of OFOKCF. The approach offers a scalable solution for distributed estimation in sensor networks and cooperative control, particularly for fractional order systems.

The cooperative-competitive network has broad application prospects due to its advantages in aligning with practical production and life. This work, for the first time, combines the predictive capability of the proportional-integral-derivative (PID) protocol for error and the flexibility of an additional order parameter in fractional calculus to solve the distributed resource allocation problem in cooperative-competitive networks, and provides a discrete time algorithm for the design. The analysis proves that the algorithms can Mittag-Leffler converge to the optimal solution of the distributed resource allocation problem. Finally, numerical simulations are provided, demonstrating the effectiveness of the algorithm, along with a set of comparative simulation that validate the superiority of the proposed algorithm.

This paper proposes a novel optimization algorithm named PIDNAG, which innovatively incorporates the PID control strategy-comprising proportional, and time-dependent integral, and derivative components-into the Nesterov accelerated gradient method for solving convex optimization problems. From a control-theoretic perspective, we conduct rigorous theoretical analysis to demonstrate that the proposed method not only guarantees convergence but also significantly accelerates the optimization process. Extensive experimental results show that PIDNAG achieves remarkable convergence performance in various practical learning tasks, solidly validating its superior capability in convex optimization problems.

Prescribed-time stable systems frequently suffer from infinite gain issues that lead to practical infeasibility. This paper investigates the problem of stabilization for uncertain Euler-Lagrangian systems and proposes a singularity-free prescribed-time stabilization controller. Based on the time space deformation approach, the designed controller ensures singularity avoidance, effectively eliminating the existence of infinite gain. Besides, the considered Euler-Lagrangian systems are subject to unknown nonlinear functions and derivative-bounded external disturbances with unknown bounds. To achieve system stabilization under such uncertainty issue, this paper formulates multiple sliding manifolds with prescribed-time stability. The controller’s effectiveness is validated through simulation studies on a rendezvous formation problem.

This article considers the stability of nabla linear time invariant (LTI) fractional order systems with the order$\alpha \in (0,+\infty)$. First, the stable criterion is developed, by using the nabla Laplace transform. Compared with the existing case of$\alpha \in (0,1)$, our work introduces a wide range of dynamic behaviors for future applications. Second, many essential properties are discussed for the developed criterion, including the changing trend of the stable/unstable region regarding the order, the containment relationship between the imaginary axis, the negative semi-axis and the stable region, the evolution of the modulus with the absolute value of argument for the point lying in the critical stable region. Third, the linear matrix inequality (LMI) condition is tentatively derived to evaluate the stability. Finally, the elaborated results are supported by three illustrative numerical examples.

Exploring the stability of delta fractional order systems is essential for them to be used properly in various applications. Since the existing researches usually focused on the system with$\alpha \in (0,1)$, it is natural to ponder a parallel case with extra order. Motivated by this need, this study addresses the stability of delta delay fractional order systems with$\alpha \in (1,2)$systematically. The main difficulties lie in the approximation of stable region and the derivation of the relevant linear matrix inequalities (LMI) conditions caused by the complicated stable region of the suggested system. Firstly, a novel clustering region is constructed and it is proved that such a region is the subset of the considered stable region. Besides, the scheme on how to construct alternative approximation regions is discussed tentatively. Secondly, the stability conditions are formulated in terms of LMIs, which are sufficient and necessary to evaluate all the eigenvalues of system matrix locating at the approximation region. Thirdly, the complex decision matrices are replaced by the real ones and the formulation is therefore more tractable. Finally, the validity and applicability of the proposed approaches are demonstrated by simulation study.

In this study, a framework for processing gradient algorithms is proposed in accordance with nabla fractional-order system theory. Unlike most of the literature, the gradient algorithm is initially recast into a nabla fractional-order dynamic system. To be specific, the algorithm is designed using control theory and analyzed using the Lyapunov theory, which is a more general and effective way to render high-performance algorithm. Three types of algorithms are built in this study, i.e., the asymptotic convergence case, the finite-time convergence case, and the fixed-time convergence case. Finally, a comprehensive simulation study is conducted verifying the correctness, usefulness, and practicality of the framework.

This article focuses on the conservativeness issue of the existing Lyapunov method for linear time-invariant (LTI) nabla fractional-order systems and proposes a converse Lyapunov theorem to overcome the conservative problem. It is shown that the LTI nabla fractional-order system is asymptotically stable if and only if there exist a positive-definite Lyapunov function whose first-order difference is negative definite. After developing a systematic scheme to construct such Lyapunov candidates, the Lyapunov indirect method is derived for the nonlinear system. Finally, the effectiveness and practicability of the proposed methods are substantiated with four examples.

The well-known generalized Kalman–Yakubovich–Popov lemma is widely used in system analysis and synthesis. However, the corresponding theory for singular systems, especially singular fractional-order systems (SFOSs), is lacking. Therefore, many control problems for this type of systems cannot be optimized in limited frequency ranges. In this article, a universal framework of the finite frequency band generalized Kalman–Yakubovich–Popov lemma for SFOSs is established, the bounded real lemma in the sense of the${L_{\infty }}$-norm is derived for different frequency ranges, and the corresponding controller is designed to improve the${L_{\infty } }$performance index of SFOSs. Three illustrative examples are given to demonstrate the correctness and effectiveness of the theoretical results.

In this paper, a novel approach is presented to approximate the general fractional order nonlinear dynamic systems. Firstly, a generalized T-S fuzzy method is used to approximate the original models. Then a new method to approximate the fractional order T-S models is utilized, and obtain a series of integer order linear model. It is revealed for the first time that a general fractional order nonlinear system (FONS) can be approximated by a series of integer order linear models to any degree of accuracy on any compact set. Finally, numerical simulation results are provided to illustrate the effectiveness of the proposed approach.