Rivative. . . . .=-(26)1 = – Then, the attitude angle tracking des = derivative error
Rivative. . . . .=-(26)1 = – Then, the attitude angle tracking des = derivative error – des Define the Lyapunov function [30]:=-1=-(27)V1 Define the Lyapunov function [30]:= 2.Define = two + des – c 1 , exactly where c is usually a normal = and 2 is really a virtual control V numberDefine = +. . .(28) – c two =wheredes + c 1 standard quantity and is actually a virtual , – c is a.1Then 1 = – des = 2 – c 1 , and.=-.+c(29)= two c , – Then = – V1 = 2 -= 2 ( anddes ) = -c 2 + 1 2Define switching functions: V= = -.= -c +.Define switching k 1 + 2 = k 1 + 1 + c e1 = (k + c )1 + 1 s = functions:(30)s = k + = k + + c e = (k + c ) + Since k + c 0, it really is clear that if s = 0, then = 0, = 0 and V fore, the following design and style is needed to define the Lyapunov function:Aerospace 2021, eight,eight ofSince k + c 0, it’s clear that if s = 0, then 1 = 0, 2 = 0 and V1 0. Thus, the following design is necessary to define the Lyapunov function: 1 V2 = V1 + s2 2 Then V2 = V1 + s s2 = -c 2 + 1 2 + s k 1 + two. . . . . . ..(31)= -c 2 + 1 two + s k (2 – c 1 ) + – des + c 1 1 = -c two 1 + 1 two + s k (two – c 1 ) +. 1 J M + D – des(32).+ u1 + cThe design and style ML-SA1 custom synthesis controller is:. .. 1 u1 = -s (2 – c 1 ) – M – L2 sgn(s ) + des – c 1 – h [s + sgn(s )] J(33)where, h and are optimistic continuous. . D-Fructose-6-phosphate disodium salt Metabolic Enzyme/Protease Substituting the design and style controller into the expression of V2 , we can get: V2 = V1 + s s2 = -c two + 1 2 – h s2 – h sp + Ds – L2 s 1 -c 2 + 1 two – h s2 – h |s | 1 Taking Q = due to c + h k2 h k – 1 T two 1 2 h k – 1 h 2 = c 2 – 1 2 + h k2 2 + 2h k 1 2 + h 2 = c 2 – 1 two + h k2 1 2 1 1 T Q = 1 2 exactly where T = 1 2 . If Q is guaranteed to be a optimistic definite matrix, there is: V2 -T Q – h |s | 0 as a consequence of: 12 . . .(34)c + h k2 h k – 1h k – h1(35)(36)(37) 1|Q | = h c h + h k2 – h k -= h (k + c ) -(38)By taking the values of h , c and k , we are able to make |Q | 0 to ensure that Q is really a . optimistic definite matrix, to ensure that V2 0.Based on the principle of Lasalle invariance, . when V2 0 is .taken, then 0, s 0, 0, s 0 , hence, 1 0, two 0 , then des , des . 3.2. Position Control Method Similarly, a robust backstepping sliding mode handle algorithm for position handle is designed. Within this control algorithm, position p and velocity v track the anticipated position T T pdes = xdes ydes zdes and the expected velocity vdes = vxdes vydes vzdes below the action of external disturbance F. Position manage tracking error e1 is as follows: e1 = p – pdes (39)Aerospace 2021, 8,9 ofthen the attitude angle tracking error derivative e1 e1 = p – pdes = v – pdes Define the Lyapunov function: Vp1 =. . . . ..(40)1 2 e two(41)Define v = e2 + pdes – cp e1 , exactly where cp is usually a optimistic continual and e2 is usually a virtual handle, e2 = v – pdes + cp e1 then e1 = v – pdes = e2 – cp e1 and Vp1 = e1 e1 = e1 v – pdes = -cp e2 + e1 e2 1 Define switching functions: sp = kp e1 + e2 where kp 0, given that e1 = e2 – cp e1 , then sp = kp e1 + e2 = kp e1 + e1 + cp e1 = kp + cp e1 + e. . . . . . . . . .(42)(43)(44)(45)Since kp + cp 0, it is actually clear that if sp = 0, then e1 = 0, e2 = 0, and Vp1 0. Therefore, the following design is essential. 1 Vp2 = Vp1 + s2 2 p Then Vp2 = Vp1 + sp s2 = -cp e2 + e1 e2 + sp kp e1 + e2 p 1 .. . . = -cp e2 + e1 e2 + sp kp (e2 – cp e1 + v – pdes + cp e1 ] 1 .. . 1 = -cp e2 + e1 e2 + sp kp (e2 – cp e1 + m T + g + F – pdes + up1 + cp e1 ] 1 the design and style controller is: up1 = -kp (e2 – cp e1 ) -. .. . 1 T – g – L1 sgn sp + pdes – cp e1 – hp sp + p sgn sp m . . . .(46)(47)(48)exactly where, hp and p are positive continuous. Substituting the d.