adaptive neural dynamic surface sliding mode control for uncertain nonlinear systems with unknown input saturation

Research Article Adaptive neural dynamic surface sliding mode control for uncertain nonlinear systems with unknown input saturation International Journal of Advanced Robotic Systems September-October 2016: 1–14 ª The Author(s) 2016 DOI: 10.1177/1729881416657750 arx.sagepub.com Qiang Chen1, Linlin Shi1, Yurong Nan1, and Xuemei Ren2 Abstract In this article, an adaptive neural dynamic surface sliding mode control scheme is proposed for uncertain nonlinear systems with unknown input saturation The non-smooth input saturation nonlinearity is firstly approximated by a smooth non-affine function, which can be further transformed into an affine form according to the mean value theorem Then, one simple sigmoid neural network is employed to approximate the uncertain nonlinearity including the input saturation, and the approximation error is estimated using an adaptive learning law Virtual controls are designed in each step by combing the dynamic surface control and integral sliding mode technique, and thus the problem of complexity explosion inherent in the conventional backstepping method is avoided With the proposed control scheme, no prior knowledge is required on the bound of input saturation, and comparative simulations are given to illustrate the effectiveness and superior performance Keywords Dynamic surface control, integral sliding mode control, neural network, input saturation, nonlinear system Date received: October 2015; accepted: 21 May 2016 Topic: Robot Manipulation and Control Topic Editor: Andrey V Savkin Associate Editor: Jayantha Katupitiya Introduction In many practical dynamic systems, lots of nonlinear and uncertain characteristics are encountered, such as saturation, hysteresis, dead zone, and so on.1–3 Input saturation is well known as one of the most common non-smooth input nonlinearities The magnitude of control signal is always limited due to physical constraints or safety consideration of actual actuators If the physical input saturation is ignored in the control process, unfortunately, the designed controller may severely degrade the system performance or even lead to its instability So far, much attention has been paid to controllers design for nonlinear systems with input saturation.4–8 In the study by Gao and Selmic,4 Chen et al.5 and Chen et al.,6 significant results have been obtained for controlling saturated nonlinear systems with the bounds of input saturation being known or estimated in prior Recently, some research work has been investigated without the prior knowledge of saturation parameter bounds Wen et al.7 uses a smooth non-affine function of the control input signal to approximate the non-smooth saturation function, and a Nussbaum function is introduced to compensate for the nonlinear term arising from the input saturation Due to good approximation abilities of nonlinear functions, neural networks (NNs) are employed in controllers design to College of Information Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang, China School of Automation, Beijing Institute of Technology, Beijing, China Corresponding author: Yurong Nan, College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, Zhejiang, China Email: nyr@zjut.edu.cn Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage) 2 International Journal of Advanced Robotic Systems approximate the saturated nonlinear systems.8,9 However, in most aforementioned works, multiple NNs are used for nonlinearity approximation in each step, which may lead to increasing complexities of the controller design Sliding mode control (SMC) is regarded as one of the robust control techniques against matched uncertainties and bounded disturbances In the study by Zhu et al.,10 two adaptive SMC laws are designed to force the state variables of the closed-loop system to achieve the attitude stabilization The backstepping method relaxes the matching condition at the expense of a high-gain feedback required for robustness, making it prone to chattering.11 In the study by Taheri et al.,12 backstepping technique is combined with the SMC to relax the matching condition in SMC design However, a possible issue in conventional backstepping method is the problem of complexity explosion caused by the differentiation operation of virtual controls in each step To remedy this issue, dynamic surface control (DSC) has been investigated by introducing a first-order filter in each recursive design step In the study by Xu et al.,13 Wang14 and Li et al.,15 a NN-based dynamic surface technique has been proposed for nonlinear pure-feedback and strictfeedback systems, respectively However, the effect of input saturation is not considered in the aforementioned works Thus, it is a challenge work to develop an effective robust control scheme for uncertain nonlinear systems with unknown input saturation Motivated by the aforementioned discussion, this article develops a new neural dynamic surface SMC scheme for a class of uncertain nonlinear systems with unknown input saturation The main contributions are summarized as follows We transform the nonlinear pure-feedback system into the canonical form using the first-order Taylor expansion and coordinate transformation Besides, to deal with the non-smooth input saturation nonlinearity, a smooth non-affine function is used to approximate the input saturation function Integral sliding mode surface is combined with DSC to design the controller, in which only one simple NN is employed for approximating uncertain nonlinearities, and thus the complexity of controller design has been reduced With the proposed control scheme, no prior knowledge is required on the bound of input saturation, and the explosion of complexity in backstepping method is avoided The rest of this article is organized as follows Problem formulation and preliminaries are provided in the section ‘‘Problem formulation and preliminaries.’’ Controller design and stability analysis are given in sections ‘‘Controller design’’ and ‘‘Stability analysis,’’ respectively The section ‘‘Simulations’’ provides comparative simulation results to validate the proposed scheme Some conclusions are given in the section ‘‘Conclusion.’’ Problem formulation and preliminaries System description Consider a class of nonlinear system in the following purefeedback form xi ; xiỵ1 ị ; i n À > < x_i ¼ fi ð (1) x_n ¼ fn ð xn ; vðuÞÞ > : y ¼ x1 where xi ẳ ẵx ; ; xi ŠT Ri is the vector of states of the ith differential equations, and xn ẳ ẵx ; ; xn ŠT Rn ; fi , i ¼ 1; ; n À 1, are unknown smooth functions of ðx ; xiỵ1 ị satisfying fi 0; ; 0ị ẳ 0; y R is the output; vðuÞ R is the control input subject to saturation nonlinearity described as & v max ðuÞ; sgnjuj ! v max vuị ẳ satuị ẳ (2) u; ; juj v max where vmax is a positive but unknown parameter Assumption The state variables xi of equation (1) are measurable, and the nonlinear functions fi , i ¼ 1; ; n, are continuously differentiable to n-order with respect to the state variables xi and the input v(u) Since the unknown functions fi , i ¼ 1; ; n, are continuously differentiable with respect to xi , we apply the first-order Taylor expansion for fi , i ¼ 1; ; n as  @fi  xi ; xiỵ1 ị  fi  xi ; xiỵ1 ị ẳ fi  xi ; xiỵ1 ị ỵ @xiỵ1 xiỵ1 ẳxiỵ1 xiỵ1 xiỵ1 ị ;1 fn  xn ; vị ẳ fn  xn ; v ị ỵ i n1 (3) @fn  xn ; vị jvẳvan v v ị @u i where xaiỵ1 ẳ xiỵ1 ỵ ịxiỵ1 , with < < 1, an i n À 1, and v ẳ an v ỵ an ịv , with < ¼ and v ¼ 0, equation (3) can an < By choosing xiỵ1 be rewritten as  @fi  xi ; xiỵ1 ị  xi ; xiỵ1 ị ẳ fi  xi ; 0ị ỵ fi  @xiỵ1 xiỵ1 ẳxiỵ1 xiỵ1 ; n1  @fn  xn ; vị  fn  xn ; vị ẳ fn  xn ; 0ị ỵ vẳvan v @v i For the analysis convenience, it is defined that  @fi ð xi ; xiỵ1 ị  i ; xi ; xaiỵ1 ịẳ i n1 gi  @xiỵ1 xiỵ1 ẳxiỵ1  @fn ð xn ; vÞ  an gn ð xn ; v ị ẳ vẳvan @v (4) (5) Chen et al which are unknown nonlinear functions From equations (4) and (5), equation (1) can be re-expressed as i xi ; 0ị ỵ gi  xi ; xaiỵ1 ị xiỵ1 ; i n x_ ẳ fi ð > > < i x_n ¼ fn  xn ; 0ị ỵ gn  xn ; v Þ Â v > > : y ¼ x1 (6) System coordinate transformation In the following, it will be shown that the original system (1) can be transformed into the canonical form with respect to the newly defined state variables.16 Let z1 ¼ y ¼ x1 z ¼ z_1 ẳ f x ị ỵ g x ; xa21 Þx (7) The time derivative of z is derived as ỵ g1 x ; xa21 Þx_ 1 @f @g @g 1 ẳ@ ỵ x Af ỵ g1 x ị ỵ @ x ỵ g1 Af2 ỵ g2 x ị @x @x @x D ẳ a2  x ị ỵ b2 ð x ; xa32 Þx (8)    @f @g 1 where a2 ð x ị ẳ @x ỵ @g @x x f ỵ g x ịỵ @x x ỵ g f 1  and b2  x ; xa32 ị ẳ @g @x x ỵ g g Again, let z ẳ a ỵ  b2 x , and its time derivative is X @a2 jẳ1 @xj x_j ỵ X @b2 jẳ1 @xj x_j x ỵ b2 x_ X @a @b 2 @ Afj ỵ gj xjỵ1 ị ẳ ỵ @xj @xj jẳ1 @b x ỵb2 Af ỵ g3 x ị ỵ@ @x (9) D x ị ỵ b3  x ; xa4 ịx ẳ a     @b 2 ỵ @b @xj fj ỵ gj xjỵ1 ịỵ @x x ỵb f jẳ1   a3 and b3  x ; x ị ẳ @b x ỵb g3 When defining ai1 and @x x3 ị ẳ where a3   P @a @xj biÀ1 , i ¼ 2; ; n, we can obtain D zi ẳ ai1  xi1 ị ỵ bi1  xi1 ; xai i1 ịxi z_i ẳ  xi ị ỵ bi  xi ịxiỵ1 where i1 X @a @b @ i1 ỵ i1 xi Afj ỵ gj xjỵ1 Þ xi Þ ¼ ð @xj @xj j¼1 @b i1 xi ỵbi1 Afi ỵ@ @xi D @bi1 bi  xi ; xiỵ1 ị ẳ@ xi ỵbi1 Agi @xi D a1 a1 @f1 ðx Þ @g ðx ; x Þ @g ðx ; x Þ 1 1 2 x_ þ @ x_ þ x_ Ax z_2 ¼ @x 1 @x @x z_3 ¼ Figure Saturation satðuÞ (solid line) and smooth function gðuÞ (dot line) (10) (11) Thus, the pure feedback system (6) can be rewritten in the canonical form with respect to the newly defined state variables as > < z_i ẳ ziỵ1 ; i ẳ 1; ; n z_n ẳ an  (12) xn ị þ bn ð xn ; van Þ v > : y ¼ z1 To proceed the design procedure, the control function bn ð xn ; van Þ in equation (12) is assumed to be positive and bounded satisfying < b1 < bn ð xn ; van Þ < b2 , where b and b2 are positive constants It is pointed out that this condition has been widely used in the literature17–20 as a necessary condition for the controllability of equation (1) The control objective of this article is to design a dynamic surface sliding-mode controller vðtÞ for the system (12), such that the system output y can track the desired reference signal yd and all signals in the closed-loop system are bounded Nonlinear saturation model As shown in Figure 1, the control input vðtÞ R is the output of the following nonlinear input saturation and uðtÞ R is the input of the saturation (practical control signal) The saturation is approximated by a smooth nonaffine function defined as International Journal of Advanced Robotic Systems  gðuÞ ¼ v max  u  v max e eu=v max eu=v max ỵ eu=v max u=v max ¼ v max  (13) Then, vðuÞ ¼ satðuÞ in equation (2) can be expressed as vuị ẳ satuị ẳ guị ỵ duị (14) where duị ẳ satuị guị is a bounded function and its bound is À Á jduịj ẳ j satuị guịj v max tanh1ị ẳ D (15) where D is the upper bound of jdðuÞj According to the mean value theorem,8 there exists a constant x with < x < 1, such that guị ẳ gu0 ị ỵ gux u u0 Þ (16)   where gux ¼ @gðuÞ @u u¼ux > 0; ux ẳ xu ỵ xịu and u0 ẵ0 ;u By choosing u0 ẳ 0, equation (16) can be rewritten in the following affine form guị ẳ gux u methods and can be replaced by any other approximation approaches such as spline functions, RBF functions, or fuzzy systems However, the structure of the employed NN in this article is simpler than the other NNs that are commonly used in other works There is no hidden layer in the employed NN, in which five inputs and one output are included and the corresponding weight matrix is  (17) Substituting equations (17) and (14) into equation (12), we can obtain z_ ẳ ziỵ1 ; i ¼ 1; ; n À > < i (18) xn ị ỵ b xn ; van ị u z_n ẳ a > : y ẳ z1 where a xn ị ẳ an  xn ị ỵ d, b xn ; van ị ẳ bn  xn ; van Þgux Controller design In this section, we will incorporate the DSC and integral sliding mode techniques into a NN-based adaptive control design scheme for the nth-order system described by equation (12) Similar to the traditional backstepping design, the recursive design procedure contains n steps From step to step n 1, virtual control ziỵ1 , i ¼ 1; n À 1, is designed at each step, and the integral sliding mode surface is proposed in the first step Finally, the control law u is obtained at step n Step In this step, we consider the first equation of equation (12), that is, z_1 ¼ z Define the tracking error and its sliding surface as < e ¼ y yd (21) : s ẳ e ỵ l e dt where yd is the desired reference signal and l is a positive constant The derivatives of e and s1 are & e_ ¼ y_ À y_d ¼ z y_d (22) s_1 ẳ e_ ỵ le ẳ z y_d ỵ le_ Choose a virtual control z as NN approximation z2 ¼ Àk s ỵ y_d le Due to good capabilities in function approximation, NNs are usually used for the approximation of nonlinear functions.8,9 The following NN will be used to approximate the continuous function hX ị ẳ W T jX ị ỵ e jX ị ẳ r1 ỵ r4 r ỵ expX =r ị where k is a positive constant Introduce a new state variable b and let z pass through a first-order filter with time constant t > 0, and we have t2 b_2 ỵ b2 ẳ z ; (19) where W à Rn Ân2 is the ideal weight matrix, jðX Þ Rn Â1 is the basis function of the NN,e à is the NN approximation error satisfying je à j eN , and jðX Þ can be chosen as the commonly used sigmoid function, which is in the following form Remark The employed NN with sigmoid function represents a class of linearly parameterized approximation b2 ð0Þ ¼ z2 ð0Þ (24) Define y ¼ b À z (25) Substituting equation (25) into equation (24), we can obtain (20) where r1, r2, r3, and r4 are appropriate parameters, and exp is an exponential function (23) z2 À b2 y2 b_2 ¼ ¼À t2 t2 (26) z_2 ¼ z (27) Step Consider Let Chen et al s2 ¼ z2 À b2 (28) Given a compact set Ozn Rn , let W à and eà be such that for any ðz ; ; zn Þ Ozn which is called the second error surface Then, we have s_2 ¼ z À b_2 (29) with jeà j Choose a virtual control  z as z3 ¼ Àk s s ỵ b_2 b 0ị ¼  z ð0Þ (31) Define y3 ¼ b3 À  z3 (32) Substituting equation (32) into equation (31), we can obtain z3 À b3 y3 b_3 ¼ ¼À t3 t3 (33) z_i ẳ ziỵ1 (34) s i ẳ z i À bi (35) Step i Consider Let which is called the ith error surface Then, we have s_i ẳ ziỵ1 b_ i (36) biỵ1 0ị ẳ ziỵ1 0ị (38) Define yiỵ1 ẳ biỵ1 ziỵ1 (39) Substituting equation (39) into equation (38), we can obtain ziỵ1 biỵ1 yiỵ1 b_ iỵ1 ẳ ẳ tiỵ! tiỵ1 (40) Step n The final control law will be derived in this step Consider z_n ẳ a xn ị ỵ b xn ; van ị u eN Let sn ẳ zn À bn (43) which is called the nth error surface From equations (41) and (43), we have s_n ¼ að xn ị ỵ b xn ; van ịu b_ n (44) Finally, design the final law u as s  T n zn ị ^eN u ẳ Àkn sn À snÀ1 À W^ fð d (45) where W^ is the estimation of W à and ^eN is the estimation of the upper limit for jeà j The adaptive learning laws of W^ and ^eN are given by _ _ > > zn Þsn À sW^ Š < W^ ẳ W~ ẳ Gẵj  i h (46) _ N ¼ ~e_ N ¼ veN sn sn > ^ e > : d where s and d are positive small constants and G ¼ GT > is a constant matrix In this section, a theorem is provided to show the boundedness of all signals in system (12) and convergence of tracking error e as well as sliding surface s (37) where ki is a positive constant Introduce a new state variable biỵ1 and let ziỵ1 pass through a first-order filter with time constant tiỵ1 > 0, and we have tiỵ1b_ iỵ1 ỵ biỵ1 ẳ ziỵ1 ; (42) Stability analysis Choose a virtual control  ziỵ1 as ziỵ1 ẳ ki si si1 ỵ b_ i a xn ị b_n ẳ W T f zn ị þ eà bð xn ; van Þ (30) where k is a positive constant Again, introducing a new state variable b3 and let z pass through a first-order filter with time constant t > 0, we have t3 b_3 ỵ b3 ẳ  z3 ; Hẳ (41) Theorem Consider the nonlinear system (12) with unknown input saturation (14), the integral sliding mode surface (21), control law (45), and adaptive learning laws (46) Given any positive number, for all initial conditions ! satisfying nÀ1 P T si ỵ yiỵ1 ịỵ b1 sn2 ỵ W~ G1 W~ ỵ v ~eN2 2p, all the eN jẳ1 closed-loop signals are semi-global uniformly ultimately bounded, and the tracking error can be made arbitrarily small by properly choosing the design parameters Proof Firstly, define the estimation error as ( W~ ¼ W^ À W à ~e ¼ ^e À eà (47) Then, the closed-loop system in the new coordinates, si , bi , and W~ i , can be expressed as follows International Journal of Advanced Robotic Systems s_1 ẳ s ỵ b2 ỵ le y_d s_2 ẳ s ỵ b3 b_2 s_i ẳ siỵ1 ỵ biỵ1 b_ i ; i ẳ 3; n À (48) s  s_n n ¼ Àkn sn À snÀ1 À W~ f zn ị ỵ eN ^eN b d From equation (48), we have biỵ1 ẳ yiỵ1 ỵ ziỵ1 ; i ¼ 1; n À Then, we can obtain s_1 ẳ s2 ỵ y k s1 s_2 ẳ s3 ỵ y À k s2 À s1 À b_2 y_ ¼ b_2 À z_ ¼ À ¼À y2 _ k s_1 ỵ yă d leị t2 y2 ỵ B2 s ; s ; y ; yd ; y_ d ; yă d Þ t2 (50) _ where B2 ðs ; s ; y ; yd ; y_ d ; yă d ị ẳ k s_1 ỵ yă d À leÞ, which is a continuous function Similarly, for i ¼ 2; n À 1, we have y_iỵ1 ẳ yiỵ1 y ki s_i s_i1 ẳ iỵ1 tiỵ1 tiỵ1 (51) ỵ Biỵ1 s ; ; siỵ1 ; y ; yi ; yd ; y_ d ; yă d ị: Consider the Lyapunov function candidate s_i ẳ siỵ1 ỵ yiỵ1 ki si si1 b_ i ; i ¼ 3; n À s_n ¼ Àkn sn À snÀ1 À W~ f zn ị ỵ eN ^eN tanhsn =dị: b (49) V_ ¼ Besides, we know the fact that V¼ nÀ1 1X 1 T 2 ~e ðs ỵ yiỵ1 ịỵ sn2 ỵ W~ G1 W~ ỵ i¼1 i 2b 2veN N (52) The derivative of the Lyapunov function is  nÀ1 n nÀ1  X X X yiỵ1 1 T _ _ ~ ^ ~eN ^eN ẳ si s_i ỵ yiỵ1 y_iỵ1 ị ỵ sn s_n ỵ W G W þ ðÀki si Þ þ si yiþ1 À þ Biþ1 yiỵ1 b veN tiỵ1 iẳ1 iẳ1 iẳ1  n n1   s  X X yiỵ1 T À1 _ n à _ ^ ~ ~ ~eN ^eN ẳ zn ị ỵ eN ^eN ki si ị ỵ si yiỵ1 ỵ Biỵ1 yiỵ1 ỵ sn W f ỵW G W ỵ veN d tiỵ1 i¼1 i¼1  n nÀ1   s  X X y2 T n ~eN ^e_ N ðÀki si2 ị ỵ si yiỵ1 iỵ1 ỵ Biỵ1 yiỵ1 ỵ sn eN ^eN sW~ W^ ỵ v eN d tiỵ1 iẳ1 iẳ1 n  s  s  s  X T n n n ~eN ^e_ N ỵ eN jsn j sn ki si2 ị ỵ eN sn sn^eN sW~ W^ ỵ veN d d d iẳ1  n1   s  X yiỵ1 T n ỵ si yiỵ1 ỵ Biỵ1 yiỵ1 ỵ eN jsn j sn sW~ W^ t d iỵ1 iẳ1 (53) Using the following property with regard to function tanh(.), we have x jxj À x 0:2785 d (54) d Using the fact T sW~ W^ T sW~ W~ ỵ W ị skW~ k ỵ skW~ k kW k s s skW~ k ỵ kW~ k ỵ kW k 2 s s kW~ k ỵ kW k2 2 s kW à k 2 and substituting equations (54) and (55) into equation (53), we can obtain  n n1  X X y2 V_ ki si2 ị ỵ si yiỵ1 iỵ1 ỵ Biỵ1 yiỵ1 tiỵ1 iẳ1 iẳ1 s (56) ỵ 0:2785eN d kW k2 Using the fact si2 ỵ yiỵ1 ! si yiỵ1 (55) (57) we have  n n1  X X yiỵ1 2 _ V ki si ị ỵ si ỵ yiỵ1 ỵ Biỵ1 yiỵ1 tiỵ1 iẳ1 iẳ1 s ỵ 0:2785eN d kW à k2 (58) Chen et al Choose R and R ki ẳ ỵ a0 ; i ¼ 1; ; n À kn ẳ a 1 M2 ẳ ỵ iỵ1 þ a tiþ1 2Z where a0 and Z are positive constants and jBiỵ1 j (59) V_ ~2 veN eN ị jẳ1 iỵ2 Pi , respectively Q Q i is also com- iỵ5 pact in RQ jẳ1 Therefore, jBiỵ1 j has a maximum Q Miỵ1 on i Noting that, for any positive number Z Miỵ1 Remark As pointed out in the study by Li et al.15 for any Q :ẳ fyd ; y_d ; yă d ị yă d2 þ B0 > and p > 0, the sets  Q P i1 2 y_2d ỵ yă d2 B0 g and i :ẳ jẳ1 si ỵ yiỵ1 ịỵ b sn ỵ T W~ G1 W~ ỵ Pi 2 yiỵ1 Biỵ1 Z ỵ ! jBiỵ1 yiỵ1 j 2Z (60) we can obtain 2p, i ¼ 2; ; n, are compact in    n nÀ1  X X M y2 B2 Miỵ1 Z s yiỵ1 ỵ ỵ a0 yiỵ1 a si2 ị þ þ iþ1 iþ1 2iþ1 þ þ 0:2785eÃN d À kW à k 4 2 2Z 2ZM iỵ1 iẳ1 iẳ1   2  n n1  X X B2 Miỵ1 yiỵ1 s a si2 ị ỵ iỵ1 a0 yiỵ1 þ 0:2785eÃN d À kW à k2 2 2Z M iỵ1 iẳ1 iẳ1 (61) n X s a si2 ị ỵ 0:2785eN d kW k2 i¼1 Hence, we can conclude V_ if sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:2785eÃN d ỵ skW~ k kW k j si j ! a0 ð 1) (62) Then, the ultimate boundedness of si is guaranteed, and si will converge to the following positively invariant set O ¼ fjsi j gs g (63) q 0:2785eN dỵskW~ k kW k with g s ! a0 When s reaches the positively invariant set g s , it remains inside thereafter From equation (12), the error dynamics inside g s is e_ ỵ le ¼ s_1 (64) Due to the boundedness of s and s_1 , the ultimate boundedness of the tracking error e can be directly concluded using the Lyapunov function W ẳ 1=2ị e 21 IAE ẳ jetịjdt, which is the integrated absolute error to measure the system tracking performance 2) ISDE ẳ etị e Þ dt, where e0 is the average error of whole process ISDE is the integrated square error and used to demonstrate the smoothness of the ð profile 3) IAV ¼ jvðtÞj dt, which is the integrated absolute control and taken as a measurement of the overall amount ofð control effort 4) ISDV ẳ vtị v ị dt, where v is the average control input of whole process ISDV is the integrated square control and used as a measurement of the fluctuations of control signal around its mean value In the following, two simulation examples are adopted for the fair comparison of different control schemes Simulations In order to show the superior tracking performance of the proposed scheme, we consider three different control schemes for comparison: (S1) neural dynamic SMC with saturation compensation; (S2) neural dynamic SMC without saturation compensation; and (S3) neural DSC without saturation compensation.15 The following four indices are adopted to compare the tracking performance of each control algorithm Spring mass and damper system The considered spring mass and damper system represents a class of widely used second-order electromechanical servo systems, such as hydraulic systems, rigid robots, or turntable systems.22–24 In those systems, the number of freedom degrees is always equal to the number of control inputs, and thus the controller design is relatively easier 8 International Journal of Advanced Robotic Systems Figure Spring mass and damper system As shown in Figure 2, a second-order system is described as7 x_ ¼ x x_ ¼ À k c x x ỵ vuị ỵ Etị m m m (65) Figure Tracking performance of yd ¼ À0:2cosð3p tị ỵ 0:2 where y ẳ x , x and x are the position and velocity, respectively; m is the mass of the object; k is the stiffness constant of the spring; and c is the damping According to equation (8), equation (65) can be transformed into x_ ẳ x x ị ỵ b x ; vị vuị x_ ẳ a where x2 ẳ ẵx ; x T a x ; vị ẳ (66) k c x x ỵ Etị, m m b x ; vị ẳ 1=m In the simulation, two different signal waves are adopted as the desired reference signals, and the system parameters are fixed for various reference signals The initial states of the system are ½x ; x T ẳ ẵ0; 0T The constants of adaptive leaning laws d ¼ 0:5, s ¼ 0:01, ^eN ¼ 0:01, veN ¼ The time constant is t ¼ 0:01 The NN parameters are r ¼ 1, r ¼ 5, r ¼ 5, r ¼ À0:1, and G ¼ diagf5g The system parameters are set as m ¼ kg, c ¼ N Á s= m, and k ¼ N= m, which are not needed to be known in our controller design The external disturbance is Etị ẳ 0:1 sin2p tị The control parameters are k ¼ 10, k ¼ 8, and l ¼ Case yd ¼ À0:2cosð3p tị ỵ 0:2 is employed as the reference signal The input saturation bound is v max ¼ 14 N Comparative tracking performance, tracking errors, and control input are shown in Figures 3, 4, and 5, respectively From Figures and 4, we can see that compared with the proposed S1 method, S2 has the larger overshoot, and S2 and S3 have larger tracking errors From Figure 5(a), compared with S2 and S3, the control input of S1 is more smooth, and the compensation effect of input saturation is shown in Figure 5(b) From the figures, we can clearly observe the significantly improved performances with the S1 Figure Tracking errors of yd ẳ 0:2cos3p tị ỵ 0:2 (a) Control inputs of three methods (b) The saturated control vðtÞ and the practical control uðtÞ in S2 In order to compare the control performance, four indices are given in Table From Table 1, we can obtain that S3 controller gives the largest IAV and ISDV, while S2 controller gives the largest IAE and ISDE The proposed S1 has the smallest IAE, ISDE, IAV, and ISDV, which means it performs best among three controllers The comparative result from Table is consistent with the Figures to Case yd ¼ 0:5 sintị ỵ sin0:5tịị is employed as the reference signal The first reference trajectory is sinusoidal signal, and all parameters are tuned based on this signal In order to show the high robustness of the proposed method, we give the second reference trajectory (sinusoidal signal with harmonics) The control parameters are set the same as case 1, and the input saturation bound becomes v max ¼ N, which is more stringent than that of case Comparative tracking Chen et al   Figure Tracking performance of yd ẳ 0:5 sintị ỵ sin0:5tị   Figure Tracking errors of yd ẳ 0:5 sintị ỵ sin0:5tị (a) Control inputs of three methods (b) The saturated control vðtÞ and the practical control uðtÞ in S2 Figure Control inputs of yd ẳ 0:2cos3p tị ỵ 0:2 Table Comparison for yd ẳ 0:2cos3p tị ỵ 0:2 IAE ISDE IAV ISDV S1 S2 S3 0.0311 0.0002 43.3104 452.2407 0.1190 0.0037 55.4190 688.8545 0.1436 0.0013 51.0590 609.4313 IAE: integrated absolute error; ISDE: integrated square error; ISDV: integrated square control; IAV: integrated absolute control performance, tracking errors, and control inputs are shown in Figures 6, 7, and 8, respectively From Figures and 7, we can see that S2 and S3 have larger tracking errors, while the proposed S1 method achieves the smallest tracking errors and fastest convergence speed From Figure 8(a), compared with S2 and S3, the control input of S1 is more smooth, and the compensation effect of input saturation is shown in Figure 8(b) In conclusion, S1 has the best performance when tracking the sinusoidal signal with harmonics In addition, the comparative results of the IAE are shown in Table From Table 2, we can see that the proposed S1 method has the smallest IAE among all the three control schemes Besides, other three indices (i.e ISDE, IAV, and ISDV) of the proposed S1 method are also smallest, which means S1 has the smoothness of tracking error and control signal Therefore, S1 has the best tracking preference, which is consistent with the results given by Figures to Furthermore, the comparative results of IAE for different input saturation values of case and case are shown in Tables and 4, respectively From Table 3, we can see that the proposed S1 method has the smallest IAE in case compared with S2 and S3, although the input saturation 10 International Journal of Advanced Robotic Systems   Table Comparison for yd ẳ 0:5 sintị ỵ sin0:5tị IAE ISDE IAV ISDV S1 S2 S3 0.0821 0.0005 50.9398 192.0874 0.4360 0.0303 58.3204 263.9002 0.5413 0.0246 53.8104 223.3383 IAE: integrated absolute error; ISDE: integrated square error; ISDV: integrated square control; IAV: integrated absolute control Table Comparison of IAE for case vmax ¼ 10 vmax ¼ 14 vmax ¼ 18 S1 S2 S3 0.0248 0.0311 0.0257 0.6036 0.1190 0.0289 0.3756 0.1436 0.0853 IAE: integrated absolute error Table Comparison of IAE for case vmax ¼ vmax ¼ vmax ¼ S1 S2 S3 0.0915 0.0821 0.0850 3.8849 0.0430 0.0924 1.7634 0.5413 0.5194 IAE: integrated absolute error twice that of the rigid robots, and the number of freedom degrees is larger than the number of control inputs, which may lead the control task more difficult As shown in Figure (figure in the study by Talole25), the mechanical dynamics of the robotic manipulator system can be described as ( I qă ỵ Kq yị ỵ MgL sinqị ẳ (67) J yă Kq yị ẳ vuị   Figure Control inputs of yd ẳ 0:5 sintị ỵ sin0:5tị values are changed from 10 to 18 Table shows that for case 2, the proposed S1 scheme with the fixed parameters can still achieve the best tracking performance in the case of various input saturation values It should be noted that with the input saturation becoming more stringent, the IAE of S2 and S3 will be changed much larger than that of S1 In conclusion, the proposed S1 scheme can achieve a satisfactory tracking performance for different input saturation values A single-link flexible-joint robotic manipulator system In the following, we give the second example, that is, a single-link flexible-joint robotic manipulator system.25 Due to the introduction of joint flexibility in the robot model, the motion equations become more complicated In particular, the order of the related dynamics becomes where q and y are the position of the link and motor angles, respectively, I is the link inertia, J is the inertia of the motor, K is the spring stiffness, M and L are the mass and length of link, respectively, and vðuÞ R is the plant input subject to saturation nonlinearity For convenience of the controller design, defining x ¼ q, x ¼ q_ ¼ x_ , x ¼ y, and x ¼ y_ ¼ x_ , equation (67) is transformed into x_ ¼ x > > > > MgL K > > > > x_ ¼ À I sinx À I ðx À x Þ < (68) > x_ ¼ x > > > > K > > > : x_ ẳ J vuị ỵ J ðx À x Þ K Let z ¼ x , z ¼ x , z ¼ À MgL I sinx À I ðx À x Þ K and z ¼ Àx MgL I cosx À I ðx À x Þ, and equation (68) can be rewritten in terms of the new coordinates as Chen et al 11 Figure Schematic of flexible-joint manipulator Figure 11 Tracking errors of step signal yd ¼ (a) Control inputs of three methods (b) The saturated control vðtÞ and the practical control uðtÞ in S1 The NN parameters are r ¼ 10, r ¼ 10, r ¼ 1, r ¼ À1, G ¼ diagf5g, and veN ¼ 0:01 The system parameters are Mgl ¼ 5, I ¼ 1, J ¼ 1, and K ¼ 40, which are not known in prior for the proposed controller design The control parameters are k ¼ 0:5, k ¼ 8, k ¼ 8, k ¼ 2, and l ¼ Figure 10 Tracking performance of step signal yd ¼ z_1 ¼ z > > > > < z_2 ¼ z z_3 ¼ z > > z_ ¼ a1 ð zị ỵ b1  z; vị vuị > > : y ¼ z1 (69) À KÁ where z ẳ ẵz ; z ; z T , a1  zị ẳ MgL I sinz z À J À ÀMgL Á K K z; vị ẳ IJK I cosz ị ỵ J þ I z , and b ð According to equation (8), equation (69) can be rewritten as z_1 ¼ z > > > > z < _2 ¼ z3 (70) z_3 ¼ z > > z _ ẳ a  z ị ỵ b  z ; vÞ Â u > 2 > : y ¼ z1 À KÁ where z ¼ ½z ; z ; z ŠT , a2  zị ẳ MgL I sinz z J MgL K K z; vị ẳ IJK gu I cosz ỵ J ỵ I z ỵ d, and b  x In the simulation, two different signal waves are adopted as the desired reference signals, and the system parameters are fixed for various reference signals The initial states of the system are ½z ; z ; z ; z T ẳ ẵ0; 0; 0; 0T The constants of adaptive leaning laws are d ¼ 0:1, s ¼ 0:01, and ^eN ¼ 0:01 The time constants are t2 ¼ t3 ¼ t4 ¼ 0:01 Case Step signal yd ¼ is employed as the reference signal The input saturation is v max ¼ 30 ð N Á mÞ, and the results are shown in Figures 10 to 12 From the simulation, we can discover that S1 has a smaller overshoot than S2 S1 and S2 have a faster convergence speed than S3 because of the impact of integral SMC From Figure 12(a), the effect of saturation compensation in the proposed S1 is apparent when the robotic manipulator has the step response characteristic From Figure 12(b), we can conclude that if the input saturation becomes smaller, S2 may not be convergent or even be seriously divergent Consequently, S1 can achieve the best performance among all three control schemes Case Trapezoidal wave is employed as the reference signal This reference signal is expressed by 0; t > > > 5ðt À 2Þ; t > > > t 10; > < À5ðt 8ị; t < 10 (71) yd ẳ 10 t < 12 > À10; > > > 12 t < 16 > 5ðt À 14Þ; > > > > 10; 16 t < 18 > : À5ðt À 20Þ; 18 t 20 12 International Journal of Advanced Robotic Systems Figure 14 Tracking errors of trapezoidal wave (equation (71)) (a) The saturated control vðtÞ and the practical control uðtÞ in S1 (b) Control inputs of three methods Figure 12 Control inputs of step signal yd ¼ Figure 13 Tracking performance of trapezoidal wave (equation (71)) Figure 15 Control input of trapezoidal wave (equation (71)) Chen et al The control parameters are set the same as case Comparative tracking performance, tracking errors, and control inputs are shown in Figures 13, 14 and 15, respectively The input saturation of S2 or S3 becomes v max ẳ 294 Ns_ mị, while the S1 is v max ẳ 75 Ns_ mị From Figures 13 and 14, we can see that S1 has the smallest tracking errors and fastest convergence speed S2 becomes divergent although its tracking error is small at first, and S3 has the largest static error among all three methods It means that control method with integral SMC can decrease the system static errors and improve the convergence speed, which can also lead to system divergence because of its integral term when time goes into infinity From Figures 13 to 15, we have that S1 can achieve the better tracking performance with much less saturated control cost From all the simulation results, we can conclude that compared with S2 and S3, the proposed S1 scheme has the better tracking performance with respect to tracking errors, convergence speed, and control cost Conclusion In this article, an adaptive neural dynamic surface SMC scheme is proposed for uncertain nonlinear systems with unknown input saturation The non-smooth saturation is transformed into an affine form by defining a non-affine function and using the mean value theorem One simple NN is employed for nonlinearity approximation, and the approximation error is estimated by an adaptive learning law By combing the DSC and the integral sliding mode technique, the controller is designed to improve the system robustness, and comparative simulations are given to illustrate the effectiveness of the proposed method Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to thank the support from the National Natural Science Foundation of China under grant numbers 61403343 and 61433003 and the China Postdoctoral Science Foundation funded project under grant number 2015M580521 References Wang H, Chen B, Liu X, et al Robust adaptive fuzzy tracking control for pure-feedback stochastic nonlinear systems with input constraints IEEE Trans Cybernet 2013; 43(6): 2093–2104 Na J, Mahyuddin MN, Herrmann G, et al Robust adaptive finite-time parameter estimation and control for robotic systems Int J Robust Nonlin 2015; 25(16): 3045–3071 13 Yang Y, Ge C, Wang H, et al 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scheme for uncertain nonlinear systems with unknown input saturation Motivated by the aforementioned discussion, this article develops a new neural dynamic surface SMC scheme for. .. speed, and control cost Conclusion In this article, an adaptive neural dynamic surface SMC scheme is proposed for uncertain nonlinear systems with unknown input saturation The non-smooth saturation. .. strict-feedback systems with unknown input saturation Inform Sci 2014; 269(6): 300–315 Chen Q and Tang XQ Nonsingular terminal sliding- mode funnel control for prescribed performance of motor servo systems with