Esign controller into the expression of Vp2 , we are able to obtain: Vp
Esign controller into the expression of Vp2 , we can obtain: Vp2 = Vp1 + sp s2 = -cp e2 + e1 e2 – hp s2 – hp p sp + Fsp – Fsp p p 1 -cp e2 + e1 e2 – hp s2 – hp p sp p 1 Taking Qp = on account of cp + hp k2 hp kp – 1 T p 2 e1 e2 1 hp kp – two hp = cp e2 – e1 e2 + hp k2 e2 + 2hp kp e1 e2 + hp e2 = cp e2 – e1 e2 + hp k2 p 1 p 2 1 1 eT Qp e = e1 e2 cp + hp k2 p hp kp – 1 2 hp kp – hp1 2 . .(49)(50)(51)Aerospace 2021, eight,10 ofwhere eT =ee2 . If Qp is guaranteed to be a optimistic definite matrix, there is Vp2 -eT Qp e – hp p sp.(52) 1due to Qp = hp cp hp + hp k2 – hp kp – p 1= hp kp + cp -(53)By taking the values of hp , cp and kp , we can make Qp 0 to ensure that Qp is often a good definite matrix, in order that Vp2 0. As outlined by the principle of Lasalle invariance, when Vp2 0 is taken, then e 0, sp 0, sp 0 , therefore, e1 0 , e2 0 , then . p pdes , v p des . four. Simulation Analysis Within this study, the performance on the proposed control algorithm is illustrated via a numerical simulation. Considering the mathematical model offered in (17)20), the basic parameters of a coaxial rotor aircraft are listed in Table 1, plus the initial conditions of all states are zero, p = v = = = 0. The attitude robust backstepping sliding mode controller defined by (33) along with the position robust backstepping sliding mode controller defined by Equation (48) was made use of. Taking L1 = 1, L2 = 1, the handle parameters are presented in Table 2. The desired trajectory was selected as follows: (t + 0.five) sin(0.5t) = (t + two) cos(0.5t) t + 0.. .pdes(54)Aerodynamic force and moment F, D are selected as: sin(0.1t) F = sin(0.1t) sin(0.1t) 0.2 sin(0.1t) D = 0.2 sin(0.1t) 0.two sin(0.1t)(55)Table 1. Model parameters of coaxial rotor aircraft. Parameter g m d Ixx Iyy Izz k TU k TL k MU k ML Table two. Control parameters. cp ten hp 20 kp 15 c 5 h ten k ten Value 9.81 two 80 eight.21 10-3 eight.21 10-3 eight.21 10-3 5.12 10-4 4.63 10-4 6.34 10-6 eight.36 10-6 Unit m/s2 kg m kg m2 kg m2 kg m2 N/rad2 s2 N/rad2 s2 Nm/rad2 s2 Nm/rad2 sThe desired attitude angle and desired position had been set to zero. To discover the effectiveness on the proposed manage algorithm, the following two situations were considered, and every single simulation lasted for 30 s.Aerospace 2021, 8,11 of4.1. Numerical Simulation below Aerodynamic Interference Inside the case of external aerodynamic interference, the position and attitude-tracking manage of a coaxial rotor aircraft are numerically simulated. Figure 4a shows the threedimensional trajectory tracking of a coaxial rotor aircraft. In position manage, backstepping sliding mode handle makes use of a symbolic Etiocholanolone medchemexpress function to deal with the uncertainty dilemma and shows excellent robustness, exhibiting good tracking efficiency with little uncertainty and almost no chattering. Figure 4b shows the tracking on the desired position as well as the actual position from the coaxial rotor aircraft. Figure 4c shows the tracking in the preferred attitude angle and also the actual attitude angle on the coaxial rotor aircraft. In attitude manage, the backstepping sliding mode exhibits a steady C2 Ceramide medchemexpress response that completely tracks the control command because the vehicle attitude is adjusted inside the initial phase to make a sharp change, and it shows a Aerospace 2021, 8, x FOR PEER Overview 11 of 17 good impact under a sharp modify within the handle command. Figure 4d shows the output control on the coaxial rotor aircraft, and its control is continuous, which is suitable for application to an actual model. As shown in the figure, when the external aerodynamic.
