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In this paper, an optimal finite discrete-time linear quadratic tracking (LQT) control method is proposed to control the altitude and attitude of a quadrotor. First, the dynamic model of the quadrotor is derived using Newton-Euler equations. Next, non-linear equations of the quadrotor are written in the state space form and linearized around an equilibrium point. Then, continuous-time linear state-space equations are converted into discrete-time equations considering a specific sampling time. Moreover, the controller design process is completed by determining the performance index and the weighting matrices, and the optimal control input is acquired for the closed-loop system. In the end, the simulation results are shown to demonstrate the robustness of the controller against parameter uncertainties and show its performance in attenuating the external disturbance effect.

The unique plasmonic character of silver and gold nanoparticles has a wide range of applications, and tailoring this property by changing electronic and geometric structures has received a great deal of attention. Herein, we study the role of the quantum properties in controlling the plasmonic excitations of gold and silver atomic chains and rods. The influence of relativistic effects, scalar as well as spin–orbit, on the intensity and energy of plasmonic excitations is investigated. The intensity quenching and the red shift of energy in the presence of relativistic effects are introduced via the appearance of d orbitals directly in optical excitations in addition to the screening of s-electrons by mixing with the occupied orbitals. For the linear gold system, it will be demonstrated that by increasing the length the relativistic behavior declines and the contribution of d orbitals to the plasmonic excitations evidently decreases. Furthermore, silver atoms are doped in gold chains and rods (with two different arrangements) to realize how gold–silver interactions decrease the relativistic effects and enhance the intensity of collective excitations. Finally, to strengthen the plasmonic behavior of gold, the elongation of chain and doping with suitable atoms such as silver (with the classical plasmonic behavior) can be introduced as the manipulating ways to control the influence of scalar-relativistic and spin–orbit effects and, consequently, reinforce the plasmonic properties.