, = up xd two yd two (24) (23)Assumption three. The best heading angle d given by
, = up xd two yd two (24) (23)Assumption three. The ideal heading angle d given by the guidance program is usually accurately tracked by the dynamics controller, namely – d = 0. In accordance with Assumption 3 and Formula (22), sin arctan – ye – tan cos arctan=- =^ ye tan ^ 2 (ye ) ^ 2 two (ye )- ye- tan(25)Substituting Equations (23) and (25) into Equation (17), we can acquire ^ xe = -k s xe F ye – u sin( – F )(tan – tan ) ^ ye = -Cye – F xe C1 (tan – tan ) where C1 = u ^ 2 two (ye tan ) . (26)^ In line with Lemma 4, we know (tan – tan ) 0. Style Lyapunov function for guidance technique, V1 = 1 2 ( x y2 g2 ) e 2 e (27)Derivation on the above formula and substituting Formulas (21) and (26) to acquire,2 V1 = -k s xe – C1 y2 – k g2 g g e(28)Sensors 2021, 21,8 of3.2. Path Following Controller Design In this portion, initially, a finite-time disturbance observer is designed to accurately estimate the external disturbance as well as the perturbation parameter. Then, so that you can track the yaw angle d and forward velocity ud , the attitude GLPG-3221 Purity & Documentation tracking controller plus the velocity tracking controller are designed based on the quick non-singular terminal sliding mode. The introduction of the auxiliary energy technique solves the issue of saturation in the actuator during the actual heading. The block diagram of the proposed controller is shown in Figure two.Figure 2. The Block Diagram of the Path Following Controller.3.2.1. Design from the Finite-Time Lumped Disturbance Observer Look at the under-driven unmanned ship model with lumped disturbances as follows, m11 u = Fu (u, v, r ) u (29) m22 v = Fv (u, v, r ) m33 r = Fr (u, v, r ) r exactly where Fu = m11 f u du , Fv = m22 f v dv , Fr = m33 f r dr . The finite-time lumped disturbance observer is developed as follows, M = = – 1 L 2 sig two ( M – M) F F = -2 Lsign( F – ) m11 where M = 0 0 0, two 0. 0 m221(30)0 0 , = [u, v, r]T , = [u , v , r ] T , L = diag(l1 , l2 ) 0, 1 mTheorem 1. Based on the created finite-time disturbance observer, the unknown external distur^ bance d can be accurately estimated inside a finite time. Proof. The definition error is as follows, M = -1 L 2 sig 2 ( M) F – Mv1=1 1 -1 L 2 sig 2 ( M) F(31)F = -2 Lsign( F – ) – F-2 Lsign( M) [- D, D ](32)Sensors 2021, 21,9 ofwhere = – , F = F – F . Applying Lemma 1, it can be concluded that the error on the finite-time disturbance observer can converge to zero, i.e., there is a finite time T0 to ensure that, (t) (t), F F , t T0 (33)three.two.two. Attitude Tracking Controller Design Define the heading angle tracking error e as, e = – d Then derivation in the e might be obtained, e = r – d (35) (34)Style of rapid non-singular terminal sliding surface s for heading angle error as, s = e e (e ) (36)where 0, 0. The specific style with the piecewise function (e ) is as follows, (e ) = sig a (e ), s = 0 or (s = 0 and |e | ) 2 , s = 0 and | | r1 e r2 sig e e (37)where s = e e sig a (e ), 0 a , r1 = (two – a) a-1 , r2 = ( a – 1) a-2 , is usually a smaller good constant. Continue to derive the s , s = e e (e ) where (e ) expressed as, (e ) = a|e | a-1 e , s = 0 or (s = 0, |e | ) e 2r2 |e |e , s = 0 and |e | r1 (39) (38)Based on the above analysis, the adaptive synovial heading tracking control law r is designed as follows, r = -m33 ( Fr – d e (e )) – m33 (r kr (t))s m33 (40)Among them, the AAPK-25 site introduced adaptive term updates the switching term achieve kr (t) in real time, and its adaptive law is updated in the following type, kr (t) = -r (t)sgn(r (t)) rr (t) = r |r (t)| r0,r r sgn(er (t)) where r , r0,r.