Ts d ^ ^T dT = DT T ^ ^T du = Du u ^ ^T dr = Dr r (74) (75) (76)^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ exactly where D T = DT1 , DT2 , . . . , DTn , Du = Du1 , Du2 , . . . , Dun , and Dr = Dr1 , Dr2 , . . . , Drn , ^ Ti, Dui, and Dri to each rule i; define ^ ^ respectively, are vectors containing the attributed values D T = [T1, T2, . . . , Tn ], u = [u1, u2, . . . , un ], and r = [r1, r2, . . . , rn ], respectively, are vectors with 24(RS)-Hydroxycholesterol-d7 Biological Activity components Ti = Ti / Ti, ui = ui / ui, and ri = ri / ri; ui, ui, and ui will be the firing strengths of each rule in (73). We propose that the vector of adjustable parameters is often automatically updated by the NPPM 6748-481 manufacturer following adaptation laws to make sure the best achievable estimation. ^ D T = 1 ST T ^ Du = 2 Su u ^ D =Sr three r r i=1 i=1 i=1 n n n(77) (78) (79)exactly where 1 , two , and three are strictly constructive constants related for the adaptation price. Theorem four. Think about the single-span roll-to-roll nonlinear system described in detail in Equations (21)23) and bounded unknown disturbance pointed out in Assumption 1. Then, the system obtains stability in accordance with the Lyapunov theorem by using the control signals (70)72) and adaptive laws in (77)79). Proof of Theorem 4. Let a positive-definite Lyapunov function candidate V3 be defined as V3 = 1 two 1 2 1 T 1 T 1 T 1 S T S u Sr T T u u r two 2 2 21 22 23 r (80)^ ^ ^ ^u ^ ^ ^ ^u ^ exactly where T = DT – D , u = Du – D , r = Dr – Dr and D , D , Dr will be the optimal T T ^ , d , d , respectively. Taking ^ ^ parameter vectors, related using the optimal estimates d T u r the derivative with respect to time, 1 1 T 1 T V3 = ST ST Su Su Sr Sr T T u u r r 1 T 2 3 = ST f T gT u d T – Td Su f u gu Mu du – Wud 1 1 T 1 T Sr f r gr Mr dr – Wrd T T u u r r 1 T two(81)Inventions 2021, six,14 ofSubstituting the manage signals rewritten in (70)72) into (81), we obtain 1 T 1 1 T ^ V3 = T T u u r r ST d T – d T – k T1 sgn(ST) – k T2 .ST T 1 2 3 ^ ^ Su du – du – k u1 sgn(Su) – k u2 .Su Sr dr – dr – kr1 sgn(Sr) – kr2 .Sr(82)^ ^ Defining the minimum approximation errors as T = d – d T , u = d – du , r = u T , = D , = D , Equation (82) becomes ^ – dr and noting that T = DT u ^ ^u r ^r dr 1 ^ ^ ^ V3 = T D T – ST T d T – d k T1 sgn(ST) k T2 ST T 1 T 1 T ^ ^ ^ u Du – Su u du – d k u1 sgn(Su) k u2 Su u 2 1 T ^ ^ ^ r Dr – Sr r dr – dr kr1 sgn(Sr) kr2 Sr 3 1 ^ = T D T – 1 ST T – ST ( T k T1 sgn(ST) k T2 ST) 1 T 1 T ^ u Du – two Su u – Su (u k u1 sgn(Su) k u2 Su) two 1 T ^ r Dr – 3 Sr r – Sr (r kr1 sgn(Sr) kr2 Sr)(83)^ ^ ^ By applying the adaptation laws in (77)79) for D T , Du and Dr , we rewrite V3 as follows:two two V3 = -k T2 S2 – k u2 Su – kr2 Sr – ST T – k T1 |ST | – Su u – k u1 |Su | – Sr r – kr1 |Sr | T(84)Additionally, it can be seen that ^ ^ ^ | T | = d – d T d T – d T d T 1 T ^ ^ ^ |u | = d – du du – du du 2 u ^ ^ ^ |r | = dr – dr dr – dr dr 3 ^ ^ ^ The manage parameters are selected as k T1 d T 1 , k u1 du two , kr1 dr 3 , and k T2 , k u2 , kr2 are strictly positive constants; thus, it may be concluded that V3 0. Remark 3. To deal with the imprecise single-span roll-to-roll nonlinear method, adaptive fuzzy sliding mode manage is an efficient option since the fuzzy disturbance observer doesn’t need to have model facts. The control law in (70)72) in fact ensures not merely the finite-time convergence to a sliding surface but also the asymptotic stability on the closed-loop system, whilst the control law in (60)62) employing a high-gain disturbance observer only drives the system converge to an arbitra.