Search Results (14)

There are many techniques for the extension and generalization of fractional theories, one of which improves fractional operators by means of their kernels. This paper is devoted to the most general concept of interval-valued functions, studying fractional integral operators for interval-valued functions, along with the multi-variate extension of the Bessel–Maitland function, which acts as kernel. We discuss the behavior of Hermite–Hadamard Fejér (HHF)-type inequalities by using the convex fuzzy interval-valued function (C-FIVF) with generalized fuzzy fractional operators. Also, we obtain some refinements of Hermite–Hadamard(H-H)-type inequalities via convex fuzzy interval-valued functions (C-FIVFs). Our results extend and generalize existing findings from the literature. Full article

We apply known special functions from the literature (and these include the Fox $\mathbb{H}–$function, the exponential function, the Mittag-Leffler function, the Gauss Hypergeometric function, the Wright function, the $\mathbb{G}–$function, the Fox–Wright function and the Meijer $\mathbb{G}–$function) and fuzzy sets and distributions to construct a new class of control functions to consider a novel notion of stability to a fractional-order system and the qualified approximation of its solution. This new concept of stability facilitates the obtention of diverse approximations based on the various special functions that are initially chosen and also allows us to investigate maximal stability, so, as a result, enables us to obtain an optimal solution. In particular, in this paper, we use different tools and methods like the Gronwall inequality, the Laplace transform, the approximations of the Mittag-Leffler functions, delayed trigonometric matrices, the alternative fixed point method, and the variation of constants method to establish our results and theory. Full article

Due to its significant influence on numerous areas of mathematics and practical sciences, the theory of integral inequality has attracted a lot of interest. Convexity has undergone several improvements, generalizations, and extensions over time in an effort to produce more accurate variations of known findings. This article’s main goal is to introduce a new class of convexity as well as to prove several Hermite–Hadamard type interval-valued integral inequalities in the fractional domain. First, we put forth the new notion of generalized convexity mappings, which is defined as $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convexity on coordinates with regard to fuzzy-number-valued mappings and the up and down ($\mathsf{U}\mathsf{D}$) fuzzy relation. The generic qualities of this class make it novel. By taking into account different values for $\mathrm{Ԓ}$, we produce several known classes of convexity. Additionally, we create some new fractional variations of the Hermite–Hadamard ($\mathsf{H}\mathsf{H}$) and Pachpatte types of inequalities using the concepts of coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convexity and double Riemann–Liouville fractional operators. The results attained here are the most cohesive versions of previous findings. To demonstrate the importance of the key findings, we offer a number of concrete examples. Full article

This paper proposes a new observer approach used to simultaneously estimate both vehicle lateral and longitudinal nonlinear dynamics, as well as their unknown inputs. Based on cascade observers, this robust virtual sensor is able to more precisely estimate not only the vehicle state but also human driver external inputs and road attributes, including acceleration and brake pedal forces, steering torque, and road curvature. To overcome the observability and the interconnection issues related to the vehicle dynamics coupling characteristics, tire effort nonlinearities, and the tire–ground contact behavior during braking and acceleration, the linear-parameter-varying (LPV) interconnected unknown inputs observer (UIO) framework was used. This interconnection scheme of the proposed observer allows us to reduce the level of numerical complexity and conservatism. To deal with the nonlinearities related to the unmeasurable real-time variation in the vehicle longitudinal speed and tire slip velocities in front and rear wheels, the Takagi–Sugeno (T-S) fuzzy form was undertaken for the observer design. The input-to-state stability (ISS) of the estimation errors was exploited using Lyapunov stability arguments to allow for more relaxation and an additional robustness guarantee with respect to the disturbance term of unmeasurable nonlinearities. For the design of the LPV interconnected UIO, sufficient conditions of the ISS property were formulated as an optimization problem in terms of linear matrix inequalities (LMIs), which can be effectively solved with numerical solvers. Extensive experiments were carried out under various driving test scenarios, both in interactive simulations performed with the well-known Sherpa dynamic driving simulator, and then using the LAMIH Twingo vehicle prototype, in order to highlight the effectiveness and the validity of the proposed observer design. Full article

Mathematical programming and optimization problems related to fluid dynamics are heavily influenced by stochastic processes associated with integral and variational inequalities. Furthermore, symmetry and convexity are intrinsically related. Over the last few years, both have become increasingly interconnected so that we can learn from one and apply it to the other. The objective of this note is to convert ordinary stochastic processes into interval stochastic processes due to the wide range of applications in various disciplines. We have developed Hermite–Hadamard ($\mathbb{H}.\mathbb{H}$), Ostrowski-, and Jensen-type inequalities using interval h-convex stochastic processes. Our main results can be applied to a variety of new and well-known outcomes as specific situations. The results of this study are expected to stimulate future research on inequalities using fractional and fuzzy integral operators. Furthermore, we validate our main findings by providing some non-trivial examples. To demonstrate their general properties, we illustrate the connections between the examined results and those that have already been published. The results discussed in this article can be seen as improvements and refinements to results that have already been published. This is a fascinating subject that can be investigated in the future to identify equivalent inequalities for various convexity types. Full article

This paper investigates the observer-based non-fragile output feedback tracking control problem for nonlinear networked systems with randomly occurring gain variations. The considered nonlinear networked systems are represented by a Takagi–Sugeno (T–S) fuzzy model. The dynamical quantization methodology is employed to achieve the reasonable and efficacious utilization of the limited communication resources. The objective is to design the observer-based non-fragile output feedback tracking controller, such that the resulting system is mean-square asymptotically stable with the given ${\mathcal{H}}_{\infty }$ tracking performance. Based on the descriptor representation strategy combined with the S-procedure, sufficient conditions for the existence of the desired dynamic quantizers and observer-based non-fragile tracking controller are proposed in the form of linear matrix inequalities. Finally, simulation results are provided to show the effectiveness of the proposed design method Full article

Convex and non-convex fuzzy mappings are well known to be important in the research of fuzzy optimization. Symmetry and the idea of convexity are closely related. Therefore, the concept of symmetry and convexity is important in the discussion of inequalities because of how its definition behaves. This study aims to consider new class of generalized fuzzy variational-like inequality for fuzzy mapping which is known as perturbed fuzzy mixed variational-like inequality. We also introduce strongly fuzzy mixed variational inequality, as a particular case of perturbed fuzzy mixed variational-like inequality which is also a new one. Furthermore, by using the generalized auxiliary principle technique and some new analytic techniques, some existence results and efficient numerical techniques of perturbed fuzzy mixed variational-like inequality are established. As exceptional cases, some known and new results are obtained. Results obtained in this paper can be viewed as refinement and improvement of previously known results. Full article

This study aims to consider new kinds of generalized convex fuzzy mappings (convex-FM), which are called strongly α-preinvex fuzzy mappings. We investigated the characterization of preinvex-FMs using α-preinvex-FMs, which can be viewed as a novel and innovative application. Some different types of strongly α-preinvex-FMs are introduced, and their properties are investigated. Under appropriate conditions, we establish the relationship between strongly α-invex-FMs and strongly αj-monotone fuzzy operators. Then, the minimum of strongly α-preinvex-FMs are characterized by strongly fuzzy α-variational-like inequalities. Results obtained in this paper can be viewed as a refinement and improvement of previously known results. Full article

This study inspects the issue of robust reliable sampled data control (SDC) for a class of Takagi-Sugeno (TS) fuzzy CE151 Helicopter systems with time-varying delays and linear fractional uncertainties. Specifically, both the variation range and the distribution probability of the time delay are considered in the control input. The essential aspect of the suggested results in this study is that the time variable delay in the control input is dependent not only on the bound but also on the distribution probability of the time delay. The prime intent of this study is to enhance a state feedback reliable sampled-data controller. By constructing an appropriate Lyapunov-Krasovskii functional (LKF) and employing a linear matrix inequalities (LMIs) approach, a new set of delay-dependent necessary conditions is obtained to ensure the asymptotic stabilisation of a TS fuzzy CE151 Helicopter system with a prescribed mixed H_∞ and passivity (MH_∞P) performance index. The acquired results are expressed as LMIs, which are easily addressed using standard optimization algorithms. In addition, an exemplary scenario based on the CE151 helicopter model is presented to demonstrate the less conservative nature of the obtained results as well as the application of the recommended unique design approaches. Full article

The hydraulic turbine governing system (HTGS) is a complex nonlinear system that regulates the rotational speed and power of a hydro-generator set. In this work, an incremental form of an HTGS nonlinear model was established and the Takagi–Sugeno (T-S) fuzzy linearization and mixed H₂/H_∞ robust control theory was applied to the design of an HTGS controller. A T-S fuzzy H₂/H_∞ controller for an HTGS based on modified hybrid particle swarm optimization and gravitational search algorithm integrated with chaotic maps (CPSOGSA) is proposed in this paper. The T-S fuzzy model of an HTGS that integrates multiple-state space equations was established by linearizing numerous equilibrium points. The linear matrix inequality (LMI) toolbox in MATLAB was used to solve the mixed H₂/H_∞ feedback coefficients using the CPSOGSA intelligent algorithm to optimize the weighting matrix in the process so that each mixed H₂/H_∞ feedback coefficients in the fuzzy control were optimized under the constraints to improve the performance of the controller. The simulation results show that this method allows the HTGS to perform well in suppressing system frequency deviations. In addition, the robustness of the method to system parameter variations is also verified. Full article

This paper presents a new control technique based on uncertain fuzzy models for handling uncertainties in nonlinear dynamic systems. This approach is applied for the stabilization of a multimachine power system subject to disturbances. In this case, a state-feedback controller based on parallel distributed compensation (PDC) is applied for the stabilization of the fuzzy system, where the design of control laws is based on the Lyapunov function method and the stability conditions are solved using a linear matrix inequalities (LMI)-based framework. Due to the high number of system nonlinearities, two steps are followed to reduce the number of fuzzy rules. Firstly, the power network is subdivided into sub-systems using Thevenin’s theorem. Actually, each sub-system corresponds to a generator which is in series with the Thevenin equivalent as seen from this generator. This means that the number of sub-systems is equal to the number of system generators. Secondly, the significances of the nonlinearities of the sub-systems are ranked based on their limits and range of variation. Then, nonlinearities with non-significant variations are assumed to be uncertainties. The proposed strategy is tested on the Western systems coordinating council (WSCC) integrated with a wind turbine. The disturbances are assumed to be sudden variations in wind power output. The effectiveness of the suggested fuzzy controller is compared with conventional regulators, such as an automatic voltage regulator (AVR) and power system stabilizers (PSS). Full article

It is a familiar fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity plays a significant role in the subject of inequalities. The concepts of convexity and symmetry have a tight connection. We may use whatever we learn from one to the other, thanks to the significant correlation that has developed between both in recent years. Our aim is to consider a new class of fuzzy mappings (FMs) known as strongly preinvex fuzzy mappings (strongly preinvex-FMs) on the invex set. These FMs are more general than convex fuzzy mappings (convex-FMs) and preinvex fuzzy mappings (preinvex-FMs), and when generalized differentiable (briefly, G-differentiable), strongly preinvex-FMs are strongly invex fuzzy mappings (strongly invex-FMs). Some new relationships among various concepts of strongly preinvex-FMs are established and verified with the support of some useful examples. We have also shown that optimality conditions of G-differentiable strongly preinvex-FMs and the fuzzy functional, which is the sum of G-differentiable preinvex-FMs and non G-differentiable strongly preinvex-FMs, can be distinguished by strongly fuzzy variational-like inequalities and strongly fuzzy mixed variational-like inequalities, respectively. In the end, we have established and verified a strong relationship between the Hermite–Hadamard inequality and strongly preinvex-FM. Several exceptional cases are also discussed. These inequalities are a very interesting outcome of our main results and appear to be new ones. The results in this research can be seen as refinements and improvements to previously published findings. Full article

This paper presents a new accurate multiple model of nonlinear pneumatic lateral forces. The bicycle representation is used in order to build up an easy implemented vehicle dynamic model. Moreover, the Takagi–Sugeno fuzzy approach is applied in order to handle the vehicle model nonlinearities. This structure allows for taking into account the small variation of the vehicle longitudinal velocity. Subsequently, a Fault Tolerant Control strategy that is based on a bank of fuzzy Luenberger observers is proposed. The robustness of the control scheme against external noises is guaranteed by applying $H_{\infty }$ performance. Sufficient stability conditions that are based on Lyapunov method are formulated as Linear Matrix Inequality. Thus, allowing the computation of the observers’ and the controllers’ gains by using MATLAB. Finally, the simulation examples are performed to show the effectiveness of our proposal. Full article

In this paper, we consider nonlinear variational inequality problems with fuzzy variables. The fuzzy variables were introduced to deal with the variational inequality containing noise for which historical data is not available. The fuzzy expected residual minimization (FERM) problems were established. We discussed the $S{C}^1$ property of the FERM model. Furthermore, results of convergence analysis were obtained based on an approximation model of the FERM model. The convergence of global optimal solutions and the convergence of stationary points were analysed. Full article