Een that the sliding surface is the very same as that of the conventional SMC in Equation (25). As a result, the point where qe,i and i develop into zero is definitely the equilibrium point. Now, let us investigate the stability of the closed-loop attitude handle system using CSMC to make sure that the motion with the sliding surfaces function properly. Stability evaluation is expected for every single sliding surface. For the stability of the closed-loop technique, the representative Lyapunov candidate by the very first sliding surface in Equation (49), is defined as VL = 1 T s s two (52)Inserting Equation (45) in to the time derivative from the Lyapunov candidate results in VL = s T s 1 = s T aD (q qe,four I3) 2 e (53)Then, let us substitute Equation (five) in to the above equation, and replace the control input with Equation (48). Then, the time derivative of the Lyapunov candidate is rewritten as VL = s T J -1 (-J f u)= s T -k1 s – k2 |s| sgn(s)(54)Note that D is zero within this case. Moreover, the second term on the right-hand side on the above equation is always positive. Which is, k2 s T |s| sgn(s) = ki =|si ||si |(55)Thus, the time derivative in the Lyapunov candidate is provided by VL = -k1 s- k2 |si ||si | i =(56)where s R denotes the two-norm of s. Because the time derivative from the Lyapunov candidate is generally negative, the closed-loop technique is asymptotically stable. This implies that to get a given initial condition of and qe , the sliding surface, si , in Equation (49) will converge towards the initial equilibrium point, i = -m sign(qe,i). When once again, for the closed-loop system stability by the second equilibrium point, the identical Lyapunov candidate by the sliding surface in Equation (50) is also defined as VL = 1 T s s 2 (57)Electronics 2021, 10,ten ofBy proceeding identically together with the prior case, the time derivative in the Lyapunov candidate is also written as 1 VL = s T J -1 (-J f u) aD (q qe,4 I3) two e= s T -k1 s – k2 |s| sgn(s)(58)Note that the variable D will not disappear within this case. However, applying the control input in Equation (48), the remaining procedure is identical with that of the previous case. Because the closed-loop technique is asymptotically stable for the offered situation of – L qe,i L, the sliding surface, si , in Equation (50) will converge for the second equilibrium point, that’s, i = qe,i = 0, which is proven by Lemma 1. three.4. Summary For the attitude handle of fixed-wing UAVs which can be in a position to be operated inside limited angular rates, the sliding mode handle investigated in this section, comparable to variable Barnidipine Antagonist structure control technologies, is summarized as follows. This technique consists of two handle laws separated by the volume of the attitude errors induced by the attitude commands and also the allowable maximum angular price with the UAV. If the attitude errors are bigger than the limiter, for instance, |qe,i | L, then the connected sliding surface and control law are offered respectively by s = m sgn(qe) u = –(59) (60)- J f J k1 s k2 |s| sgn(s)otherwise, the relevant sliding surface and the control law are expressed respectively as s = aqe 1 u = –1 -J f aJ (q qe,four I3) J k1 s k2 |s| sgn(s) 2 e 4. 3D Path-Following Approach In this section, a three-dimensional guidance algorithm for the path following of waypoints is moreover employed to make sure that the manage law in Equation (48) performs successfully. To provide the recommendations on the angular price for a provided UAV to be operated safely inside the allowable forces and moment, the idea on the Dubins curve is intr.