Search Results (195)

Robust disturbance rejection in voice coil motor (VCM) motion stages is often limited by model uncertainties and the difficulty of obtaining accurate plant inverses. To address this issue, this paper develops a data-based Youla parameterization method for designing a robust disturbance observer (DOB) without relying on an analytical plant model. Frequency response data from the VCM stage are measured directly under multiple operating conditions. The Youla parameter Q is expanded using a Laguerre orthogonal basis, and its coefficients are optimized by solving a convex problem that enforces H∞ robust stability and H₂ average tracking error constraints on a finite frequency grid. Experiments on a VCM motion stage demonstrate that the optimized Q filter effectively estimates and rejects electromagnetic noise and other disturbances. A total of 30 groups of data covering the full range of operating conditions were used for optimization, and 10 randomly designed experiments were conducted to validate the controller, with the maximum average error below 0.05%. Repetitive tests were carried out to verify the tracking performance for 1 Hz sinusoidal and triangular signals. The results show that the average RMSEs of the proposed method is 0.87% and 0.59%, respectively, which are lower than those of the ITAE-PID, ADRC and K₀ controllers. Finally, the robustness of the proposed method is further verified by analyzing the sensitivity function of the closed-loop system. Full article

We classify the continuous reciprocally symmetric cost functions $J:(0,\infty )\to \mathbb{R}$ with $J\left(1\right)=0$ and strictly convex log-substitution $G\left(t\right):=J\left(e^t\right)$ (admissible costs) for which the symmetric compound $J\left(xy\right)+J(x/y)$ depends on $(x,y)$ only through $\left(J\right(x),J(y\left)\right)$. We first prove that this dependence is automatic: for every admissible J, there exists a unique continuous combiner (the auxiliary function P that encodes the compound) $P:{[0,\infty )}^2\to \mathbb{R}$ with $J\left(xy\right)+J(x/y)=P\left(J\right(x),J(y\left)\right)$ for all $x,y>0$ (Theorem 1); P is symmetric, non-negative, satisfies $P(u,0)=2u$, and inherits monotonicity and coercivity from admissibility. When P is required to be a polynomial, a growth rate comparison between two recursions for G forces $degP\le 2$ (Theorem 4), so $P(u,v)=cuv+2u+2v$ with $c\ge 0$, and the corresponding admissible costs are exhausted by two explicit families (Theorem 8)—the hyperbolic family $J\left(x\right)=c^{-1}(x^{\lambda }+x^{-\lambda })-2c^{-1}$ ($c,\lambda >0$) and the degenerate quadratic family $J\left(x\right)=a{(lnx)}^2$ ($a>0$)—with the latter arising as the Inönü–Wigner contraction $\lambda \to 0^{+}$, ${\lambda }^2/c\to a$ of the former (Theorem 9). Two regularity extensions are obtained: a Lebesgue-measurable cost satisfying explicit regularity hypotheses admits a continuous representative (Theorem 5), and in the entire finite-order regime, the diagonal combiner $Q\left(u\right):=P(u,u)$, when polynomial of degree $d^{\prime }$, obeys the sharp bound $d^{\prime }\ge 2^{\rho }$ (Theorem 6), attained with equality in both classified families. The normalisations $P(1,1)=6$ and $G^{″}\left(0\right)=1$ single out the canonical representative $J_{\mathrm{cost}}\left(x\right)=\frac{1}{2}(x+x^{-1})-1$. Full article

This paper is devoted to defining and analyzing a new subclass $M(\nu ,\tau ,q)$ of q-valent harmonic mappings in the unit disk D, as well as investigating its connection with close-to-convex analytic functions. First, we prove that this newly defined class is non-empty and discuss its relationship with several known classes of harmonic mappings. Using arguments similar to those employed in the study of Mocanu-type harmonic mappings, we establish the close-to-convexity of functions belonging to this class. Necessary coefficient estimates for the analytic part are obtained, and auxiliary lemmas which play an essential role in the investigation of geometric properties of the class are derived. In particular, we establish distortion estimates for the derivative of the analytic part, which lead to a growth and distortion theorem for functions in the newly defined class $M(\nu ,\tau ,q)$. Furthermore, a covering theorem is obtained for these harmonic mappings. In addition, we also derive sharp bounds for the Fekete–Szegö-type functionals and several graphical examples are presented to analyze the geometric structure of the mappings in the class $M(\nu ,\tau ,q)$ that demonstrate how the parameters $q,\nu ,$ and $\tau ,$ affect the deformation of the unit disk. Full article

The article derives sufficient conditions under which the normalized Rabotnov function becomes q-close-to-convex relative to specific starlike functions on the open unit disk. To enhance the impact of our results, we include some consequences derived from the main theorems, along with graphical illustrations. The starlikeness of the Rabotnov function with respect to different aspects also falls within the scope of this study. Full article

In this paper, certain subclasses of meromorphic harmonic functions which are formulated using a q-differential operator are meticulously analyzed. Initially, two new subclasses $W_H^q(k;E,F)$ and $W_{\eta }^q(k;E,F)$ associated with the Janowski function with relevance to the idea of weak subordination are defined. These classes are further studied through their various analytical and geometric properties. Some of these explored properties include the necessary and sufficient coefficient condition, the radii of starlikeness, characterizations of extreme points, distortion estimation, closeness under convolution, and convex combination features. Additionally, the asymptotic behavior of the coefficients is also examined, and to express the findings, the Big-O, little-o, and asymptotic equivalency notations are used. These findings significantly represent the interaction between the growth, dominant terms, and limiting behavior of functions within these subclasses. Full article

Some equivalent statements and basic properties for the generalized convexity assumption $p\left(x\right)f\left(x\right)+q\left(x\right)f^{\prime }\left(x\right)+f^{″}\left(x\right)\ge 0$ are proved. Then based on these conclusions, various Jensen type inequalities under the generalized convexity are established. The idea is to transform such $p\left(x\right),q\left(x\right)$-convex functions to some simpler $p\left(x\right)$, 0-convex or 0, $q\left(x\right)$-convex functions, or even convex functions. Ky Fan and Wang-Wang type inequalities are also generalized as applications. Full article

We introduce and study a new subclass of Janowski-type harmonic close-to-convex functions in the open unit disk defined via the Jackson q-derivative operator. The structure of the operator naturally reflects certain symmetric properties in the analytic representation of the considered harmonic mappings. By applying subordination techniques, we establish sufficient conditions for sense-preserving close-to-convexity and distortion estimates. The extreme points of the class are determined, and its topological properties are examined, showing that the class is convex and compact. We further obtain the radius of starlikeness and prove that the class is closed under convolution. Moreover, as $q\to 1^{-}$, the operator reduces to the classical derivative, and our results recover several known results in the existing literature, demonstrating that the present work extends and generalizes earlier findings. Full article

In this paper, we investigate a class of analytic functions associated with the generalized Marcum Q-function and its Alexander transform. We establish sufficient conditions under which these functions exhibit important geometric properties in the open unit disk, including strong starlikeness, strong convexity, and pre-starlikeness. The results presented are believed to be new and are supported by illustrative examples and consequences. Full article

The concept of fuzzy differential subordination was introduced in 2011 as a natural generalization of classical differential subordination, reflecting the contemporary trend of incorporating fuzzy set theory into well-established mathematical frameworks. This work aims to explore multiple fuzzy differential subordinations (FDS) and fuzzy differential superordinations (FDSs) associated with the linear multiplier fractional q-differintegral operator. Utilizing the linear multiplier fractional q-differintegral operator, we introduce a novel fuzzy subclass of analytic functions, denoted by $\mathcal{S}{\mathfrak{D}}_{\mathfrak{F}}^{\sigma ,\mathfrak{m}}(q,\lambda ,\gamma )$. Using the concept of FDS and FDSs, we identify important characteristics and analytical aspects of the class $\mathcal{S}{\mathfrak{D}}_{\mathfrak{F}}^{\sigma ,\mathfrak{m}}(q,\lambda ,\gamma )$. Furthermore, we derive a collection of FDS and FDSs results specifically related to the linear multiplier fractional q-differintegral operator. Full article

This work introduces new bi-univalent function classes defined using the fractional q-Ruscheweyh operator and characterized by subordination to q-Hermite polynomials. We derive coefficient bounds and Fekete–Szegö inequalities for these classes and show that our results generalize several earlier findings in both the classical and q-analytic settings. The approach highlights the effectiveness of q-Hermite structures in analyzing operator-defined subclasses of bi-univalent functions. Full article

In this paper, we investigate new geometric properties of normalized analytic functions associated with the generalized Marcum Q-function. In particular, we focus on two analytic forms derived from a normalized derivative of a representation involving the Marcum Q-function, and its Alexander transform. For these functions, we establish sufficient conditions ensuring membership in the classes of lemniscate starlike and lemniscate convex functions. Special attention is given to the case $\nu =1$, where explicit admissible parameter ranges for b are derived. We further examine inclusion relations between these normalized analytic forms and lemniscate subclasses, complemented by several corollaries, illustrative examples, and graphical visualizations. These results extend and enrich the geometric function theory of special functions related to the generalized Marcum Q-function. Full article

The purpose of this research is to develop a set of Hermite–Hadamard–Mercer-type inequalities that involve different types of fractional integral operators such as classical Riemann–Liouville fractional integral operators. Furthermore, some fractional integral inequalities are obtained for three-times differentiable convex functions with respect to the right-hand side of the Hermite–Hadamard–Mercer-type inequality. Moreover, several new results regarding Young’s inequality, bounded function and L-Lipschitzian function are deduced. The paper presents additional remarks and comments on the results to make sense of them. To illustrate the key findings, graphical representations are provided, and applications involving special means, midpoint formula, q-digamma function and modified Bessel function are presented to demonstrate the practical utility of the derived inequalities. Full article

The goal of this paper is to use a Boole-type inequality framework to provide better estimates for differentiable functions. Using majorization theory, fractional integral operators are incorporated into a new auxiliary identity. The method establishes sharp bounds by combining the properties of convex functions with classical inequalities like the Power mean and Hölder inequalities, as well as the Niezgoda–Jensen–Mercer (NJM) inequality for majorized tuples. Additionally, the study presents real-world examples involving special functions and examines pertinent quadrature rules. This work’s primary contribution is the extension and generalization of a number of results that are already known in the current body of mathematical literature. Full article

Quantum calculus is a powerful extension of classical calculus, providing novel tools for deriving sharper and more efficient analytical results without relying on limits. This study investigates error estimations for Milne formula-type inequalities within the framework of quantum calculus, offering a fresh perspective on numerical integration theory. New variants of Milne’s formula-type inequalities are established for q-differentiable convex functions by first deriving a key quantum integral identity. The primary aim of this work is to obtain sharper and more accurate bounds for Milne’s formula compared to existing results in the literature. The validity of the proposed results is demonstrated through illustrative examples and graphical analysis. Furthermore, applications to special means of real numbers, the Mittag–Leffler function, and numerical integration formulas are presented to emphasize the practical significance of the findings. This study contributes to advancing the theoretical foundations of both classical and quantum calculus and enhances the understanding of integral inequality theory. Full article

We introduce and analyze a subclass of analytic functions with negative coefficients, denoted by ${\mathcal{P}}_{q,\sigma }^{m,\ell ,p}(\alpha ,\eta )$, constructed through a generalized q-calculus operator in combination with a multiplier-type transformation. For this class, we obtain sharp coefficient bounds, growth and distortion estimates, and closure results. The radii of close-to-convexity, starlikeness, and convexity are determined, and further consequences, such as integral means inequalities and neighborhood characterizations, are derived. The results presented provide a broad framework that incorporates and extends several earlier families of analytic and geometric function classes. Full article