In this paper, we classify nonsymmetric Dunkl-classical linear functionals. Firstly, we reduce the characterization of a Dunkl-classical linear functional given by the first author in a previous work ...to a
-distributional equation of Pearson's type. Secondly, after rescaling the parameters, we prove that the unique nonsymmetric Dunkl-classical linear functional is the perturbed generalized Gegenbauer form.
The author examines legal regulation of holographic and written will made before witnesses which represent private and regular forms of wills in the Serbian Law and other modern European Laws. A ...special attention has been paid to issues related to the requirements that must be fulfilled so that a will may be valid, and the procedure of drafting wills in domestic legislation in order to recognize good normative solutions and certain weaknesses in the legislation, particularly when it comes to the fact that the will has a protective function. The comparative analysis of legal solutions dedicated to the above forms of wills focuses on the differences, common solutions and basic principles that modern European legislation uses when regulating the above mentioned forms of wills. In the conclusion of this work, the author highlights advantages and disadvantages of holographic and written will made before witnesses, examines the extent to which their normative regulation in the domestic law is similar to and different from the existing normative regulation of these forms of wills in other European legal systems, takes into account the advantages of certain legal solutions noticed in the comparative law, and suggests possible directions legal regulation of these forms of wills may take in the future.
In this paper, we present three characterizations of Dunkl-classical orthogonal polynomials. The first one is a
-distributional equation of Pearson type fulfilled by its associated form, where
is the ...Dunkl operator. The second is a first order linear
-difference equation with polynomial coefficients satisfied by the corresponding Stieltjes function. The third is the so-called structure relation fulfilled by Dunkl-classical polynomials.
This paper investigates sliding mode control and observation problems for descriptor systems via a linear switching function approach. A generalized regular form and a generalized observer regular ...form, which are counterparts of the regular form and the observer regular form for normal systems respectively, are first introduced for descriptor systems. Then systematic ways to design the sliding mode controller and the descriptor sliding mode observer for descriptor systems are presented by virtue of linear switching functions. Necessary and sufficient conditions are established to determine the existence of the proposed sliding mode controller and descriptor sliding mode observer. In terms of the proposed sliding mode control and observation method, a descriptor sliding mode observer-based sliding mode controller is also developed for descriptor systems. It shows that under mild assumptions, the associated sliding motion for descriptor systems is of reduced order and the separation principle holds if the amplitude of high-order sliding mode controllers in the control input can be selected appropriately.
In this paper, a robust global fast terminal attractor based full flight trajectory tracking control law has been developed for the available regular form which is operated under matched ...uncertainties. Based on the hierarchical control principle, the aforesaid model is first subdivided into two subsystems, i.e., a fully-actuated subsystem and an under-actuated subsystem. In other words, the under-actuated subsystem is further transformed into a regular form whereby the under-actuated characteristics are decoupled in terms of control inputs. In the proposed design, the nonlinear drift terms, which certainly varies in full flight, are estimated via functional link neural networks to improve the performance of the controller in full flight. Besides, a variable gain robust exact differentiator (VG-RED) is designed to provide us with estimated flight velocities. It has consequently reduced the noise in system’s velocities and has mapped this controller as a practical one. The finite-time sliding mode enforcement and the states’ convergence are shown, for all flight loops, i.e., forward flight and backward flight, via the Lyapunov approach. All these claims are verified via numerical simulations and experimental implementation of the quadcopter system in a Matlab environment. For a more impressive presentation, the developed simulation results are compared with standard literature.
•Firstly, we are going to highlight that we have developed for the first time a regular form for the quadcopter system which is, quite confidently, a significant contribution. This regular form facilitates all the control strategies to be implemented on the under study system.•Secondly, we have defined a novel integro-differential sliding surface which provide us a novel non-singular control input for each sub-system of quadcopter. This strategy, on one hand, provides fast finite time convergence. On the other hand, this methodology also provides robustness against matched uncertainties.•Thirdly, a variable gain robust exact differentiator (VG-RED) has been utilized for the velocities estimation of each DOF. These estimations are very robust precise and almost noise free. These features certainly enhance overall robustness and practicability of the proposed strategy.•Fourthly, we have used feed forward neural network (FFNN) for the estimation of nonlinear unknown drift terms. The use of these FFNNs is one more step to the utilization of the proposed strategy in practical scenarios. So, our proposed control design along VG-RED and FFNN is far appealing to be used in practical scenarios.•Finally, the feasibility and benefits of the proposed control algorithm are validated via numerical simulation and experimental implementation.
In this paper, a continuous control strategy for robust stabilization of a class of uncertain multivariable linear systems with delays in both the state and control variables is proposed. A predictor ...is designed to compensate the delay effect in the control input, and then an integral sliding mode control technique along with super-twisting algorithm is applied to compensate partially the effect of the perturbation term. Finally, a nominal delay-free part of the control input is designed to stabilize the sliding mode dynamics. The proposed control scheme is extended to the class of systems modeled in Regular form. For this class of perturbed systems with delay in the state, a transformation to the systems with the delay-free state is proposed. The stability conditions of the closed-loop uncertain system are derived, and the results obtained in this work are compared against previous works. To show the effectiveness of the proposed method, simulation results are presented.
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