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In this work, the susceptible-infectious-removed (SIR) dynamics are considered in relation to the effects on the health system. With the help of the Caputo derivative fractional-order method, the SIR epidemic model for childhood diseases is designed. Subsequenly, a set of sufficient conditions ensuring the existence and uniqueness of the addressed model by choosing proper fuzzy approximation methods. In particular, the fuzzy Laplace method along with the Adomian decomposition transform were employed to better understand the dynamical structures of childhood diseases. This leads to the development of an efficient methodology for solving fuzzy fractional differential equations using Laplace transforms and their inverses, specifically with the Caputo sense derivative. This innovative approach facilitates the numerical resolution of the problem and numerical simulations are executed for considering parameter values. [ABSTRACT FROM AUTHOR]

In this study, we delve into the intricacies of addressing the challenge posed by simultaneous external disturbances and ever-changing actuator and sensor faults in the context of linear parameter-varying (LPV) systems. Our focus is on fault estimation (FE) and the pursuit of fault-tolerant tracking control (FTTC). LPV systems are described through a polytopic LPV representation with measurable gain scheduling functions. An adaptive LPV sliding mode observer (ASMO) is developed for the purpose of simultaneously estimating the system states and faults despite external disturbances. Compared with other conventional ASMO designs, the proposed observer has the capability to reconstruct the actuator faults by exploiting the equivalent output error injection signal required to maintain sliding motion and to directly estimate sensor faults using an adaptive algorithm. Based on online FE information, an FTTC is synthesized to compensate for the fault effect and to force closed-loop system states to track their desired reference signals. Sufficient conditions to ensure the stability of the state estimation errors and closed-loop system are established using Lyapunov stability theory together with H ∞ techniques. These requirements are articulated using linear matrix inequalities (LMIs), which can be effortlessly addressed through optimization problem-solving methods. To illustrate the potency of the proposed approaches, an illustrative example is provided. To illustrate the potency of the proposed approaches and to validate their practical effectiveness, we offer an illustrative example featuring a vertical takeoff and landing aircraft. This real-world case study serves as a practical application of our theoretical contributions, demonstrating the adaptability and robustness of our approach in the face of complex, real-world challenges. [ABSTRACT FROM AUTHOR]