Research Article
Adaptive fuzzy dynamic surface tracking control for a class of nonlinear systems with unknown distributed time delays and backlash-like hysteresis
International Journal of Advanced Robotic Systems September-October 2017: 1–17 ª The Author(s) 2017 DOI: 10.1177/1729881417733886 journals.sagepub.com/home/arx
Hongyun Yue, Zongtian Wei, Qingjiang Chen and Xiaoyan Zhang
Abstract In this article, an adaptive fuzzy backstepping dynamic surface control approach is developed for a class of nonlinear systems with unknown backlash-like hysteresis and unknown state discrete and distributed time-varying delays. Fuzzy logic systems are used to approximate the unknown nonlinear functions and a fuzzy state observer is designed for estimating the immeasurable states. Then, by combining the backstepping technique and the appropriate Lyapunov–Krasovskii functionals with the dynamic surface control approach, the output-feedback adaptive fuzzy tracking controller is designed. The main advantages of this article are (i) the existence of the state discrete and distributed time-varying delays such that the investigated systems are more general than that of the existing results, (ii) the proposed control scheme can eliminate the problem of “explosion of complexity” inherent in the backstepping design method, and (iii) for the nth nonlinear system, only one fuzzy logic system is used to approximate the unknown continuous time-varying delay functions since all of them are lumped into one unknown nonlinear function, which makes our design scheme easier to be implemented in practical applications. It is proven that the proposed design method is able to guarantee that all the signals in the closed-loop system are bounded and the tracking error can converge to a small neighborhood of origin with an appropriate choice of design parameters. Finally, the simulation results demonstrate the effectiveness of the proposed approach. Keywords Adaptive fuzzy backstepping control, observer, Lyapunov–Krasovskii functionals, dynamic surface control, time-varying delays, backlash-like hysteresis Date received: 14 March 2016; accepted: 3 September 2017 Topic: Robot Manipulation and Control Topic Editor: Andrey V Savkin
Introduction In the past decades, many new control theories have been gradually established because of the controlled plants and the control objectives becoming more and more complex, such as the nonlinear sliding mode control,1 neural network control,2,3 fuzzy control,4 and so on. It should be emphasized that fuzzy control with heuristic knowledge or linguistic information has been successfully applied to many nonlinear systems since it does not need an accurate mathematical model of the system.5–9 In a study by Qin et al.,5 a fuzzy adaptive robust control strategy was put forward for the trajectory tracking control of space robot,
which can control the space manipulator to achieve a good trajectory tracking effect in different gravity environments without changing the structure or parameters. In a study by Bakdi et al.,6 the off-line path planning problem for mobile robots was dealt with by combining with the
School of Science, Xi’an University of Architecture and Technology, Xi’an, China Corresponding author: Hongyun Yue, School of Science, Xi’an University of Architecture and Technology, Xi’an 710055, China. Email:
[email protected]
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) 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 fuzzy adaptive control. And in a study by Cazarez-Castro et al.,7–9 the output regulation problem for the servomechanisms with nonlinear backlash was solved using type-1 or type-2 fuzzy logic controllers. Moreover, in recent years, the control design of nonlinear systems with hysteresis has become a challenging yet rewarding problem. One of the main reasons is that the hysteresis phenomenon can be often encountered in a wide range of physical systems and devices.10 On the other hand, since the hysteresis nonlinearity is non-differentiable, the system performance is often severely deteriorated and usually exhibits undesirable inaccuracies or oscillations and even instability.11,12 And in the literature,13–18 in order to control uncertain nonlinear systems with unknown backlashlike hysteresis, fuzzy logic systems (FLSs) of Mamdani type were used to approximate the uncertain smooth nonlinear functions, and then an adaptive backstepping technique was applied to design controllers. In a study by Boulkroune et al.,13 the authors investigated the fuzzy adaptive control design for uncertain multivariable systems with unknown backlash-like hysteresis and unknown control direction that possibly exhibited time delay. In a study by Wang et al.,14 in order to handle the nonlinear properties of hysteretic systems, an indirect adaptive fuzzy controller was proposed. Furthermore, in a study by Li et al.,15–17 the proposed adaptive fuzzy control schemes with a state observer can be used to deal with those uncertain single-input single-output or multiple-input multiple-output (MIMO) nonlinear systems with unknown backlash-like hysteresis. And in a study by Cui et al.,18 an adaptive tracking control problem was studied for a class of switched stochastic nonlinear pure-feedback systems with unknown backlash-like hysteresis under arbitrary switching. Moreover, it should be pointed out that a drawback with the backstepping technique is the problem of “explosion of complexity,” that is, the complexity of the controller grows drastically as the order n of the system increases. This “explosion of complexity” is caused by repeated differentiations of certain nonlinear functions. To overcome the “explosion of complexity,” dynamic surface control (DSC) was proposed in many existing results, for example,19–25 In a study by Swaroop et al.,19 a DSC technique was proposed to eliminate this problem by introducing a first-order filtering of the synthetic input at each step of the traditional backstepping approach. In a study by Swaroop et al.,20 DSC was considered for the tracking problem of non-Lipschitz systems. In a study by Zhang and Ge,21 the control scheme was developed using the DSC approach for pure-feedback nonlinear systems with an unknown dead zone. In a study by Wang et al.,22 the authors constructed an adaptive neural controller for a class of pure-feedback nonlinear time-delay systems through the DSC technique. Moreover, in the lierature,23–25 the adaptive fuzzy backstepping control approaches were proposed for nonlinear systems with unknown time delays and immeasurable
International Journal of Advanced Robotic Systems states, MIMO nonlinear systems with immeasurable states, and uncertain stochastic nonlinear strict-feedback systems without the measurements of the states, respectively. And in a study by Zhang and Lin,26 for a class of nonlinear systems with unknown backlash-like hysteresis at the input, based on a high-gain observer, an adaptive DSC scheme was proposed which can be able to mitigate the effect of hysteresis, to eliminate the explosion of terms inherent in backstepping control. However, by combining the state observer with the FLS and the DSC technique, how to design an adaptive fuzzy output-feedback controller for a class of uncertain nonlinear systems with unknown backlash-like hysteresis is a challenging subject. Since time-delay phenomena are often encountered in various engineering systems such as biological reactors, rolling mills, and so on, the existence of time delay is a significant cause of instability and deteriorative performance, so the control design of nonlinear time-delay systems has been receiving much attention. Recently, several fuzzy adaptive control schemes have been reported which combine the Lyapunov–Krasovskii functional with the adaptive backstepping fuzzy control for nonlinear systems with time delays.2,13,17,27–30 However, it is worth noting that the results in them were obtained in the context of continuous fuzzy systems with constant 27,28 or timevarying delays.2,13,17,29,30 When the number of summands in a system equation is increased and the differences between the neighboring argument values are decreased, systems with distributed delays will arise. Thus, the topic of distributed delay systems has been an attractive research topic in the past years.31–34 However, for a nonlinear system with unknown backlash-like hysteresis and unknown distributed time-varying delays, how to design an effective controller is worth studying. Motivated by the above observations, an adaptive fuzzy DSC output-feedback control approach is presented in this article for a class of uncertain nonlinear systems, under the conditions of unknown backlash-like hysteresis, unmeasured states, unknown state discrete time-varying delays, and unknown state distributed time-varying delays. A fuzzy state observer is constructed to estimate the unmeasured states, the appropriate Lyapunov–Krasovskii functionals are used to deal with the unknown discrete and distributed timevarying delay terms, the FLS is employed to approximate the nonlinear functions, and finally, the adaptive backstepping approach is utilized to construct the fuzzy controller. The main contributions of this article are listed as follows: (i)
(ii)
Considering15,17,23 the distributed time-varying delays, the systems investigated in this article are more general. And the existence of the distributed time-varying delays such that the investigated systems are much more complex and difficult. By combining the DSC approach, the proposed control scheme can eliminate the problem of
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(iii)
3
“explosion of complexity” inherent in the backstepping design method. All the unknown time-delay terms are lumped into one unknown nonlinear function which can be approximated by only one FLS such that the suggested adaptive fuzzy controller contains less adaptive parameters and this makes our design scheme easier to be implemented in practical applications. Moreover, in the studies by Li et al. and Tong et al.,17,23 it was assumed that the time-delay nonlinear terms were bounded by known functions, but in this article, the assumption is relaxed.
It can be proven that all the signals in the closed-loop system are bounded and the tracking errors can converge to a small neighborhood around zero. The effectiveness of the developed scheme is illustrated by a simulation example.
Problem formulation and preliminaries Problem formulation Consider the system described by 8 x_i ¼ xiþ1 þ fi ð xi Þ þ hi ð yðt d1i ðtÞÞÞ > > > ðt > > > > mi ðyðsÞÞds þ > > > td 2i ðtÞ < x_n ¼ ððtÞÞ þ fn ð xn Þ þ hn ð yðt d1n ðtÞÞÞ > ðt > > > > > mn ð yðsÞÞds þ > > > td 2n ðtÞ > : y ¼ x1
where DFi ¼ fi ð xi Þ fi ðx^i Þ, x^i is the estimate of xi and the observer will be given later. (1)
Control objective
where x ¼ ½x 1 ; . . . ; xn T 2 Rn is the state vector of the system with xi ¼ ½x 1 ; . . . ; xi T 2 Ri and y 2 R is the output signal. fi ðÞ, hi ðÞ, and mi ðÞ are all unknown smooth nonlinear functions. ðtÞ 2 R is the control input and ðÞ denotes the hysteresis type of nonlinearity. This article assumes that the states of the system (1) are unknown and only the output yðtÞ is available for measurement. Assumption 1. The discrete and distributed time-varying delays d1i ðtÞ and d2i ðtÞ satisfy 0 d1i ðtÞ d 1 and 0 d2i ðtÞ d2 with d_ 1i ðtÞ d1 < 1 and d_ 2i ðtÞ d2 < 1, respectively. In particular, the constants d1 , d2 , d1 , and d2 may be unknown. Assumption 2. There exists a known constant Li such that j fi ð xi Þ fi ðx^i Þj Li ðjx 1 x^1 j þ þ jxi x^i jÞ
Remark 1. Practically speaking, assumption 1 is common and relaxed, we can see it in many results, for example2,17,29,34,35 and the references therein. Moreover, it should be emphasized that the constants d1 , d2 , d1 , and d2 are introduced only for the stability analysis and they are not used in the controller design, thus they can be unknown in the controller design procedure. Moreover, assumptions 3 and 4 are both common, which can be found in many existing literature. 25,15,17,23,36 Due to the assumption of jðtÞj M (M may be a large constant), the stability of the closed-loop control system of this article is built in the sense of semi-globally uniformly ultimately bounded (SUUB).15,17 Moreover, since the states of the system (1) are unmeasurable, (1) can be rewritten as follows 8 x_i ¼ xiþ1 þ fi ðx^i Þ þ DFi þ hi ðyðt d1i ðtÞÞÞ > > > ðt > > > > mi ðyðsÞÞds þ > > > td 2i ðtÞ < (3) x_n ¼ ððtÞÞ þ fn ðx^n Þ þ DFn > > ð > t > > > þ hn ðyðt d1n ðtÞÞÞ þ mn ðyðsÞÞds > > > > td 2n ðtÞ : y ¼ x1
(2)
where x^i is the estimate of xi . Assumption 3. The desired trajectory yr ðtÞ is continuous and known. Moreover, yr ðtÞ, y_r ðtÞ, and y€r ðtÞ are all assumed to be bounded. Assumption 4. There exists a large positive constant M such that jðtÞj M.
Our control objective is to design the output-feedback controller ðtÞ and parameter adaptive laws such that all the signals involved in the resulting closed-loop system are SUUB, and the output yðtÞ tracks the given reference signal yr ðtÞ as desired.
Fuzzy logic systems In this article, the following rules are used to develop the adaptive fuzzy controller. Rl : if x 1 is F 1l , x 2 is F2l and xn is Fnl , then y is Gl ; l ¼ 1; 2:::; Q where x ¼ ½x 1 ; . . . ; xn T and y are the FLS input and output, respectively. Fuzzy sets Fil and Gl are associated with the membership functions Fil ðxi Þ and Gl ðyÞ, respectively. Q is the rule number. Through singleton function, center average defuzzification, the FLS can be expressed as follows XQ yl Pni¼1 Fil ðxi Þ yðxÞ ¼ Xl¼1 (4) Q n P l ðxi Þ F i¼1 l¼1 i where x ¼ ½x 1 ; x 2 ; . . . ; xn T 2 Rn ; Fil ðxi Þ is the membership function of Fil , and yl ¼ arg supy2R Gl ðyÞ. Define y ¼ ½y 1 ; . . . ; yQ T and ’ðxÞ ¼ ½’ 1 ðxÞ; . . . ; ’Q ðxÞT with the fuzzy basis function l given by
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International Journal of Advanced Robotic Systems
Pni¼1 Fil ðxi Þ ’l ðxÞ ¼ XQ Pn l ðx Þ l¼1 i¼1 Fi i
(5)
Then, the FLS (4) can be rewritten as yðxÞ ¼ yT ’ðxÞ
(6)
Our first choice for the membership function is the 1 Gaussian function Fil ðxi Þ ¼ exp 2 ððxi ali Þ=sli Þ 2 , where sli and ali are fixed parameters. It has been proven that when the membership functions are chosen as Gaussian functions, the above FLS is capable of uniformly approximating any continuous nonlinear function over a compact set with any degree of accuracy. This property is shown by the following lemma Lemma 1. Let f ðxÞ be a continuous function defined on compact set O.4 Then, for any constant > 0, there exists an FLS (6) such that sup j f ðxÞ yT ’ðxÞj
(7)
x2O
Lemma 2. Given a positive definite matrix W 2 Rnn and two scalars g > d 0 for any vector oðtÞ 2 Rn , we have ð td ð td oT ðsÞds W oðsÞds tg tg (8) ð td oT ðsÞW oðsÞds
ðd gÞ
tg
By lemma 1, the FLS (6) is a universal approximator, that is, it can approximate any unknown continuous function on a compact set.31 Therefore, it can be assumed that the unknown function fi ðx^i Þ in system (3) can be approximated by the following FLS. Define the optimal parameter vector yi as ( ) T y ’ ðx^i Þj (9) y ¼ arg min sup jfi ðx^i Þ ^ i
^ yi 2Xi
x^i 2Oi
i
i
where Xi and Oi are compact regions for ^ y i and x^i , respectively. The fuzzy minimum approximation error and fuzzy approximation error "i ð xi Þ and δ i ð xi Þ are defined as "i ðx^i Þ ¼ fi ðx^i Þ yi T ’i ðx^i Þ T δ i ðx^i Þ ¼ fi ðx^i Þ ^ y i ’i ðx^i Þ
(10)
Assumption 5. j"i j "i and jδ i j δ i , where "i > 0 and δ i > 0 are unknown constants, i ¼ 1; . . . ; n. Denote oi ¼ "i δ i , by assumption 5, we have joi j "i þ δ i ¼ oi with oi being an unknown constant. By (10), system (3) can be rewritten as follows
8 x 1 Þ þ " 1 ð^ x 1 Þ þ DF1 x_ 1 ¼ x 2 þ yi T ’ 1 ð^ > > ðt > > > > > þ h ð yðt d ðtÞÞÞ þ m 1 ð yðsÞÞds 1 11 > > > td 21 ðtÞ > > > > x_i ¼ xiþ1 þ yi T ’i ðx^i Þ þ "i ðx^i Þ þ DFi > > ðt < þ h ð yðt d ðtÞÞÞ þ mi ð yðsÞÞds i 1i > > td 2i ðtÞ > > > > x_n ¼ ðÞ þ yn T ’n ðx^n Þ þ "n ðx^n Þ þ DFn > > > ðt > > > > þ hn ð yðt d1n ðtÞÞÞ þ mn ðyðsÞÞds > > > td 2n ðtÞ : y ¼ x1
(11)
From10,15,17 the control input and the hysteresis type of nonlinearity ðÞ, system (1) can be described by d dððtÞÞ d ¼ ðc Þ þ B 1 (12) dt dt dt where ; c, and B1 are constants and c > 0 is the slope of the lines satisfying c > B1 12. In this article, the parameters of hysteresis in (12), that is, ; c, and B1 are completely unknown.17 Based on the analysis of the literature,10,11,15 (12) can be solved explicitly as ðÞ ¼ cðtÞ þ d1 ðÞ d1 ðÞ ¼ ½ 0 c 0 eð0 Þ sgn_ ð þ e sgn_ ½B 1 ce sgn_ d 0
(13)
where ð0Þ ¼ 0 and ð0Þ ¼ 0 . Remark 2. According to (12) and (13), the backlash-like hysteresis is dealt with successfully and then using the backstepping technique, the input signal ðtÞ is designed finally. Based on the above solution, it is shown that d1 ðÞ is bounded.10,11,15 In this article, we assume that d1 ðÞ d with d being a constant. Thus, using (11) and (13) gives that 8 x_ 1 ¼ x 2 þ y1 T ’ 1 ð^ x 1 Þ þ " 1 ð^ x 1 Þ þ DF1 > > ðt > > > > > þ h1 ð yðt d11 ðtÞÞÞ þ m 1 ð yðsÞÞds > > > td 21 ðtÞ > > > > x_i ¼ xiþ1 þ yi T ’i ðx^i Þ þ "i ðx^i Þ þ DFi > > ðt > > < þ hi ð yðt d1i ðtÞÞÞ þ mi ð yðsÞÞds (14) td 2i ðtÞ > > > T > x_n ¼ cðtÞ þ d1 ðÞ þ yn ’n ðx^n Þ > > > > þ "n ðx^n Þ þ DFn þ hn ð yðt d1n ðtÞÞÞ > > ðt > > > > > mn ðyðsÞÞds þ > > > td 2n ðtÞ : y ¼ x1
Fuzzy adaptive observer design Note that the states xn of system (1) are not available for feedback, thus a state observer must be designed to estimate the unmeasured states. In this article, a fuzzy adaptive observer is designed for (14) as follows
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5
8 T > y 1 ’ 1 ð^ x 1 Þ þ k 1 ðy x^1 Þ x^_ ¼ x^2 þ ^ > < 1 T y i ’i ðx^i Þ þ ki ðy x^1 Þ x^_i ¼ x^iþ1 þ ^ > > : T y n ’n ðx^n Þ þ kn ðy x^1 Þ x^_n ¼ c^ðtÞ þ ^
where E ¼ (15)
Rewriting (15) in the following form x^_ ¼ A^ x þ Ky þ Fð^ xj^ y n Þ þ En c^ðtÞ (16) y^ ¼ ET1 x^1 2 3 k 1 6 . 7 T 7 xj^ ynÞ ¼ where A ¼6 In1 5, K ¼ ½k1 ; . . . ; kn , Fð^ 4 .. kn . . . 0 T T ^ x 1 Þ; ; ^y n ’n ðx^n ÞT , EnT ¼ ½0; ; 1; ET1 ¼ ½1; ; 0, ½y 1 ’ 1 ð^ and c^ is the estimate of c. Given a positive definite matrix Q ¼ QT > 0, appropriate constants and ", choose the vector K in A such that the following matrix inequality is satisfied 0 @
n X i¼1
5 1 1 þ ðL 2 þ 1ÞAI þ AT P þ PA þ PPT Q 8" 4" 1
(17) where PT ¼ P > 0 is a positive definite matrix. Remark 3. To obtain the positive definite matrix P and vector K from (17), we can decompose A into A ¼ A þ KD with 0 In1 and D ¼ ½1 0 . . . 0. Then, (17) can be A¼ 0 0 transformed into a standard linear matrix inequality (LMI) E P 0, we
have jz 1 j"1 z 1 "1 tanh zk1 0:2785k ¼ k0 , then 1 2 z1 2 0 _ ~ V 1 z 1 þ 2"z 1 þ k þ z 1 " 1 tanh 2 k z1 1 þ z 1 ^" 1 tanh þ ðL21 þ 1Þjjejj2 (29) 4" k þ z 1 ð 1 þ y1 T ’1 ð^ x 1 Þ y_r Þ þ z 1 z 2 ðt 2 þ h1 ð yðt d 11 ðtÞÞÞ þ d2 m21 ð yðsÞÞds td 21 ðtÞ
Step 2. The time derivative of z 2 is given by T
z_ 2 ¼ x^_2 _ 1 ¼ x^3 þ ^y 2 ’ 2 ðx^2 Þ þ k 2 ðy x^1 Þ _ 1 ¼ x^3 þ y2 T ’ 2 ðx^2 Þ þ f 2 ðx^2 Þ y2 T ’2 ðx^2 Þ T
þ ^y 2 ’ 2 ðx^2 Þ f2 ðx^2 Þ þ k 2 ðy x^1 Þ _ 1 (30) ¼ x^3 þ y2 T ’ 2 ðx^2 Þ þ " 2 δ 2 þ k 2 ðy x^1 Þ _ 1 T T ¼ x^3 þ ^y 2 ’ 2 ðx^2 Þ þ ~y 2 ’ 2 ðx^2 Þ þ o 2 þ k 2 ðy x^1 Þ _ 1
where ~y 2 ¼ y2 y^ 2 . Consider the Lyapunov function candidate V2 as follows 1 (31) V 2 ¼ z 22 2 then, the time derivative of V2 along the solutions of (30) is given by T T V_ 2 ¼ z 2 z_2 ¼ z 2 x^3 þ ^y 2 ’2 ðx^2 Þ þ ~y 2 ’ 2 ðx^2 Þ þ o 2 (32) þ k 2 ðy x^1 Þ _ 1 by using z 2 o 2 jz 2 jo2 , we have T T V_ 2 ¼ z 2 z_2 z 2 x^3 þ z 1 þ ^y 2 ’ 2 ðx^2 Þ þ ~y 2 ’2 ðx^2 Þ þ k 2 ðy x^1 Þ _ 1 z 2 z 1 þ jz 2 jo2 T T ¼ z 2 z 3 þ 2 þ z 1 þ ^y 2 ’2 ðx^2 Þ þ ~y 2 ’ 2 ðx^2 Þ þ k 2 ðy x^1 Þ _ 1 z 2 z 1 þ jz 2 jo2 z2 z2 z2 ~ 2 tanh ^ 2 tanh z 2 o2 tanh þ z2 o þ z2 o k k k (33)
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7
then, similar to (29) yields that T y 2 ’2 ðx^2 Þ V_ 2 z 2 z 3 þ z 2 2 þ z 1 þ ^
1 Vn ¼ zn2 2
T þ~ y 2 ’2 ðx^2 Þ þ k 2 ðy x^1 Þ _ 1 z 2 z 1 z2 z2 0 ~ 2 tanh ^ 2 tanh þ k þ z2 o þ z2 o k k
(34)
Step i. Similar to step 2, the time derivative of zi is given by T T z_i ¼ x^iþ1 þ ^ y i ’i ðx^i Þ þ ~ y i ’i ðx^i Þ þ oi þ ki ðy x^1 Þ _ i1
(35)
Consider the Lyapunov function candidate Vi as follows 1 Vi ¼ zi2 2
(36)
the derivative of Vi is given by T V_ i ¼ zi z_i zi ziþ1 þ zi i þ zi1 þ ^ y i ’i ðx^i Þ T þ~ y i ’i ðx^i Þ þ ki ðy x^1 Þ _ i1 zi zi1 þ k0 (37) zi zi ~ i tanh ^ i tanh þ zi o þ zi o k k
the derivative of Vi is given by T T y n ’n ðx^n Þ þ ~ y n ’n ðx^n Þ V_ n ¼ zn z_n zn c^ðtÞ þ zn1 þ ^ þ kn ðy x^1 Þ _ n1 zn zn1 (40) zn zn 0 ~ n tanh ^ n tanh þ zn o þ k þ zn o k k Choosing the following Lyapunov function candidate n X 1 1 2 V ¼ eT Pe þ z þL (41) 2 2 i i¼1 where egðtd 1 Þ L¼ 1 d1
ðt
d2 þ 1 d2
td 11 ðtÞ
ðt
egs h21 ð yðsÞÞds ðt
td 21 ðtÞ r
egðtd 2 sÞ m 21 ð yðsÞÞdsdr
ðt n X e 2 þ 8"jjPjj egs hj2 ðyðsÞÞds 1 d1 j¼1 td 1j ðtÞ ðt n ðt 8d2 "jjPjj2 X þ egðtd 2 sÞ mj2 ð yðsÞÞdsdr 1 d 2 j¼1 td 2j ðtÞ r gðtd 1 Þ
Step n: The time derivative of zn is given by T z_n ¼ c^ðtÞ þ ^ y n ’n ðx^n Þ þ kn ðy x^1 Þ _ n1
(39)
(38)
Consider the Lyapunov function candidate Vi as follows
(42) then, we can get that
n n X X 1 1 5 V_ eT AT P þ PA þ PPT þ Li2 I þ ðL21 þ 1ÞI þ I e þ zi ki ðy x^1 Þ 4" 8" i¼1 i¼2 n n T X X zi " T 2 2 2 0 ^ ~ ^ ~i þ o ^ i Þ tanh þ zi y i þ y i ’i ðxi Þ þ 32"c jjPjj þ nk þ zi ðo þ c~2 jjPjj2 2 2 k i¼2 i¼2 z1 1 2 2 T 8"~ cc^jjPjj þ z 1 ð~" 1 þ ^" 1 Þ tanh x 1 Þ y_r þ z 1 þ 2"z 1 þ z 1 1 þ y1 ’ 1 ð^ 2 k þ
n1 X
zi ði þ zi1 _ i1 Þ þ zn ð^ cðtÞ þ zn1 _ n1 Þ þ h 21 ð yðt d11 ðtÞÞÞ þ 8"jjPjj 2
i¼2
hj2 ð yðt d1j ðtÞÞÞ
j¼1
ðt
þ d2
n X
m21 ð yðsÞÞds þ 8"jjPjj 2 d2
ðt n X j¼1
td 21 ðtÞ
mj2 ðyÞds þ
td 2j ðtÞ
egd 1 h21 ðyÞ egðd1 d 11 ðtÞÞ ð1 d_ 11 ðtÞÞ 2 h1 ðyðt d11 ðtÞÞÞ 1 d1 1 d1
ð n d2 ð1 d_ 21 ðtÞÞ t d2 d21 ðtÞ gd 2 2 egd1 X egðtd 2 sÞ m21 ðyðsÞÞds þ e m 1 ðyÞ þ 8"jjPjj2 hj2 ðyÞ 1 d2 1 d ð1 d Þ td 21 ðtÞ 2 1 j¼1
n X 8"jjPjj 2 egðd 1 d 1j ðtÞÞ ð1 d_ 1j ðtÞÞ j¼1
þ
1 d1
n X d2 d2j ðtÞ8"jjPjj2 egd 2 j¼1
1 d2
hj2 ðyðt
d1j ðtÞÞÞ
n X 8d2 "jjPjj2 ð1 d_ 2j ðtÞÞegd2 j¼1
mj2 ðyÞ þ D gL
1 d2
ðt td 2j ðtÞ
egðtsÞ mj2 ðyÞds
(43)
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International Journal of Advanced Robotic Systems 2
where D ¼ 8"jjPjj 2 jjδ jj2 þ 8"jjPjj2 d . From assumption 1 and (17) yields that V_ eT Qe þ
n X
zi ki ðy x^1 Þ þ
n X T T zi ð^ yi þ ~ y i Þ’i ðx^i Þ
i¼2
i¼2 X n zi zi ~ i tanh ^ i tanh þ zi o zi o þ þ nk0 k k i¼2 i¼2 " 2 2 2 2 þ c~ jjPjj þ 32"c jjPjj2 2 8"~ cc^jjPjj 2 2 2 z1 z1 þ z 1 ~" 1 tanh þ z 1 ^" 1 tanh k k 1 2 T þ z 1 1 þ y1 ’1 ð^ x 1 Þ y_r þ z 1 þ 2"z 1 2 n X
þ þ
n1 X
zi ði i¼2 gd 1
where Z 1 ¼ ½z 1 ; yT 2 OZ1 R 2 and OZ 1 is some known
compact set in R 2 . Notice that in (46), the term HðyÞ z 1 is discontinuous at z 1 ¼ 0. Therefore, it cannot be approximated by the FLS. Similar to the study by Wang and Chen, we introduce hyperbolic tangent function tanh
j¼1
where # is a positive design parameter. Note that
2 z1 limz 1 !0 16 z 1 tanh # HðyÞ exists, thus the nonlinear function J1 ðZ 1 Þ can be approximated by an FLS T ’ 1 ðZ1 Þ such that 1
J1 ðZ1 Þ ¼
þ zi1 _ i1 Þ þ zn ð^ cðtÞ _ n1 þ zn1 Þ
T1 ’ 1 ðZ1 Þ
then, (44) can be rewritten as
i¼2
n X T y i ’i ðx^i Þ zi ^ i¼2
n n X X zi T ~ ~ i tanh þ zi y i ’i ðx^i Þ þ zi o þ nk0 k i¼2 i¼2 n X zi " ^ i tanh þ zi o þ c~2 jjPjj 2 2 þ 32"c 2 jjPjj 2 2 2 k i¼2 z1 z1 8"~ cc^jjPjj2 2 þ z 1 ~" 1 tanh þ z 1 ^" 1 tanh k k þ z 1 ð1 þ y1 T ’ 1 ð^ x 1 Þ þ J 1 ðZ1 Þ y_r Þ n1 X zi ði þ zi1 _ i1 Þ þ
(45) where
d22 egd 2 2 m ðyÞ þ 1 d2 1
j¼1
n X
T
zi ~y i ’i ðx^i Þ
i¼2
n1 X zi i þ zi1 _ i1 þ ki ðy x^1 Þ
2 l 2 þ zn c^ðtÞ _ n1 þ zn1 þ kn ðy x^1 Þ þ þ 1 2 2 z1 þ 1 16 tanh 2 H þ D gL #
(46) in order to obtain the control laws i;d , now, define the boundary layer errors as
n egd 1 egd1 X h21 ðyÞ þ 8"jjPjj2 hj2 ðyÞ ð1 d1 Þ ð1 d1 Þ j¼1
þ
T
zi ^y i ’i ðx^i Þ þ
(49)
HðyÞ 1 þ z 1 þ 2"z 1 z1 2
n X
n X
i¼2
þ zn ð^ cðtÞ _ n1 þ zn1 Þ þ D gL
HðyÞ ¼
(48)
X n n X zi zi ~ i tanh ^ i tanh zi o zi o þ þ k k i¼2 i¼2 " cc^jjPjj2 2 þ nk0 þ c~2 jjPjj 2 2 þ 32"c 2 jjPjj 2 2 8"~ 2 z1 z1 þ z 1~" 1 tanh þ z 1^" 1 tanh k k 1 T z1 T x 1 Þ þ 2 ’1 ’1 z 1 þ y_r þ z 1 1 þ y1 ’ 1 ð^ 2 2l þ
i¼2
J 1 ðZ1 Þ ¼
V_ eT Qe þ
i¼2
zi ki ðy x^1 Þ þ
z 21 2 l21 þ þ 2l 2 2
where 1 ¼ T1 1 is an unknown parameter and is the upper bound of 1 ðZ 1 Þ. We can have (44)
n X
þ 1 ðZ 1 Þ
then, by using z 1 T1 ’ 1 ðZ1 Þ þ 1 ðZ1 Þ 12 ’T1 ðZ1 Þ’ 1 ðZ1 Þz 21 2 þ
d22 8"jjPjj 2 egd 2 mj2 ðyÞ þ D gL 1 d2
V_ eT Qe þ
to deal
z HðyÞ 16 1 HðyÞ þ tanh 2 J1 ð Z1 Þ ¼ J 1 ð Z 1 Þ z1 z1 #
e d22 h21 ðyÞ þ egd2 m21 ðyÞ 1 d1 1 d2
n X
#
with the term. Define
n egd1 X þ 8"jjPjj 2 hj2 ðyÞ 1 d1 j¼1
þ
z1
yi ¼ i i;d ;
d22 8"jjPjj2 egd 2 mj2 ðyÞ 1 d2 (47)
i ¼ 1; . . . ; n 1
(50)
where ti _ i þ i ¼ i;d ; i ð0Þ ¼ i;d ð0Þ;
i ¼ 1; . . . ; n 1 (51)
Yue et al.
9
using ti _ i þ i ¼ i;d gives that yi y_i ¼ _ i;d ; i ¼ 1; . . . ; n 1 ti
(52)
substituting (50) into (49) yields that V_ eT Qe þ
n n X X T T y i ’i ðx^i Þ þ y i ’i ðx^i Þ zi ^ zi ~
i¼2 X n zi zi ~ i tanh ^ i tanh þ zi o zi o þ k k i¼2 i¼2 " þ nk0 þ c~2 jjPjj 2 2 þ 32"c 2 jjPjj2 2 8"~ cc^jjPjj 2 2 2 z1 þ z 1 ð~" 1 þ ^" 1 Þ tanh x1 Þ þ z 1 y1 þ 1;d þ y1 T ’ 1 ð^ k z1 þ 12 ’T1 ’1 z 1 þ y_r 2 2l n X
þ
n1 X
i¼2
Remark 4. By (47) and (48), we can see that all the unknown time-delay terms are lumped into one unknown nonlinear function HðyÞ, thus only one FLS is used to approximate it. Furthermore, by using the technique 1 ¼ T1 1 such that only one unknown parameter 1 needs to be adjusted, which reduces the online computation burden greatly and makes our design scheme easier to be implemented in practical applications. Then, consider the following Lyapunov function VðtÞ ¼ V ðtÞ þ
n n1 X 1 ~T ~ 1 ~2 X 1 2 y yi yi þ 2 þ 1 2gi 4 2 i i¼1 i¼1
n X 1 1 T 1 ~i o ~ i þ ~"T1 ~" 1 þ c~2 þ o 4& 2 2 i¼2
(58)
the time derivative of V is given by zi ðyi þ i;d þ zi1 _ i1 þ ki ðy x^1 ÞÞ
i¼2
þ zn ð^ cðtÞ _ n1 þ zn1 þ kn ðy x^1 ÞÞ 2 l21 2 z1 þ þ þ 1 16 tanh H þ D gL 2 2 #
V_ eT Qe
i;d ¼ li zi zi1 þ _ i1 ki ðy x^1 Þ T zi ^ oi tanh ^ y i ’i ðx^i Þ k ðtÞ ¼ 1^ cðln zn zn1 þ _ n1 kn ðy x^1 Þ z n ^ n T ’ ðx^n Þ ^ n tanh o y n k
i¼1
þ
n X i¼1
(54)
(55)
(56)
i¼1
n X zi " ~ i tanh þ zi o þ c~2 jjPjj2 2 þ 32"c 2 jjPjj 2 2 2 k i¼2 z1 8"~ cc^jjPjj2 2 þ z 1 ~" 1 tanh k ~1 2 l21 T 2 2 z1 þ 1 16 tanh H þ 2 ’1 ’1 z1 þ þ 2 2 2 # þ D gL þ nk0 (57)
n1 n1 X X yi y_i þ zi yi i¼1
i¼1
" 1 ~T _ y i gi zi ’i ðx^i Þ ^y i þ c~2 jjPjj2 2 gi 2
þ 32"c 2 jjPjj2 2 þ
where li is a positive design parameter. Moreover, combining (53) to (56) gives that n n1 n X X X T V_ eT Qe y i ’i ðx^i Þ li zi2 þ zi yi þ zi ~ i¼1
li zi2 þ
i¼1
(53) then, choose the control laws defined as follows ^ T 1;d ¼ l 1 z 1 ^ y 1 ’ 1 ð^ x 1 Þ 12 ’T1 ’ 1 z 1 2 z1 z1 ^" 1 tanh þ y_r 2l k
n X
~1 ^_ 1 ’T1 ðZ1 Þ’ 1 ðZ1 Þz 21 2 2
c~ _ ðc^ þ 16&"^ cjjPjj2 2 Þ 2&
z1 þ ~"T1 z 1 tanh ^"_i k n X zi 2 l2 T _ ~ ^ þ oi zi tanh oi þ þ 1 k 2 2 i¼2
z1 H þ D gL þ nk0 þ 1 16 tanh # 2
(59) choose the adaptive laws as follows ^_ 1 ¼ ’T ðZ1 Þ’ ðZ1 Þz 2 s1 ^1 1 1 1 z1 ^"_1 ¼ z 1 tanh s3 ^" 1 k
(60)
c c^0 Þ c^_ ¼ 16&"^ cjjPjj 2 2 s2 ð^ ^y_ i ¼ g zi ’ ðx^i Þ s i ^y i i i zi ^i ^_ i ¼ zi tanh i o s o k
(61)
10
International Journal of Advanced Robotic Systems combining (62) and (63) yields that
i , and s i are all positive design parawhere s1 , s 2 , s3 , s meters and c^0 ¼ c^ð0Þ.
V_ eT Qe
n X i¼1
Remark 5. From (60), it can be seen that the function SðtÞ ¼ 16&"jjPjj2 2 þ s2 is nonnegative. This implies that if c^ðtÞ s2 c^0 =SðtÞ, then c^_ 0. Consequently, c^ decreases until c^ðtÞ ¼ s2 c^0 =SðtÞ. So, for any given initial condition c^0 > 0, c^ > 0 holds for all t > t 0 . Then, using zi yi 4 1 i yi2 þ i zi2 yields that n1 X 1 2 yi2 _ T 2 V e Qe li zi þ y 4 i i ti i¼1 i¼1
s2 " s1 ^ ~ c~c^ þ D þ c~2 jjPjj 2 2 þ 2 2 2& 2 1 1
(62)
n X T 2 z1 ~i o ^ i þ 1 16 tanh i o H þ s # i¼2 þ s 3~"T1 ^" 1 þ
i¼1
gi
T ^ yi~ yi
n X
~ Ti o ^i i o s
2 l 21 þ gL þ nk0 2 2
i¼2
i¼2
s3~"T1 ^" 1
2
þ
n i T s1 2 X 2 l 2 s 1 þ yi yi þ þ 1 2 4 2gi 2 2 i¼1
n n1 n X X X i ~ T ~ s yi yi li zi2 yi _ id 2g i i¼1 i¼1 i¼1
0 1 n1 2 X s3 T 1 y @ y2 i A ~" 1 ~" 1 þ 4 i i 2 ti i¼1
2
~i ~ Ti o o
(65)
0
0 11 z1 þ@1 16 tanh 2 @ AAH þ C gL #
s3 T s3 " " 1 ~"T1 ~" 1 2 1 2
where C ¼ 32"c 2 jjPjj 2 M 2 þ D þ
s2 s2 s2 c~ð^ c c^0 Þ ¼ ðc c~Þ~ c c~c^0 2& 2& 2&
(64)
n i T s " s2 s1 ~ 2 X ~i ~ o þ c~2 jjPjj 2 2 c~2 2 o 1 2 4& 4 2 i i¼2
n X i s i¼2
1 2 1 y 2 i @ y A þ 32"c 2 jjPjj2 2 þ 4 i i ti i¼1
V_ eT Qe
s1 2 s1 ~ 2 4 2 1 4 2 1 oTi oi
0
According to assumption 4 gives that 32"c 2 jjPjj2 2 32"c 2 jjPjj2 M 2 , thus we can obtain
n n X X i T i ~ T ~ s s yi yi yi yi 2g 2g i i i¼1 i¼1
n X i s
n i T s s2 2 s1 ~ 2 X s3 ~ i ~"T1 ~"1 ~i o c~ 2 o 1 4& 4 2 2 i¼2
0 11 z1 þ@1 16 tanh2 @ AAH þ D gL þ nk0 #
s1 ^ ~ s1 ~ 1 Þ ~1 ð ¼ 2 2 1 1 2 2 1
0
moreover, we have n X i s
i¼1
n X i T s s3 s2 s2 oi oi þ "T1 "1 þ c 2 þ c^20 þ 2 2 4& 4& i¼2
n1 n X X i ^ ~ T s y i y i þ 32"c 2 jjPjj 2 2 yi _ id þ g i¼1 i¼1 i
þ
n1 X yi _ id
n X i ~ T ~ s " y i y i þ c~2 jjPjj 2 2 2 2g i i¼1
n1 X
n X
li zi2
n P s i i¼2
s2 2 s2 2 s2 2 s2 2 s2 2 c þ c~ c~ þ c^0 þ c~ 2& 8& 2& 2& 8&
s2 s2 s2 ¼ c~2 þ c^0 2 þ c 2 4& 2& 2& (63)
2
oTi oi þ s23 "T1 " 1 þ
n s2 c 2 þ s2 c^2 þ s 1 2 þ P s i yT y þ 2 þ l 21 þ nk0 . 4& 4& 0 2gi i i 2 2 4 2 1 i¼1
Assumption 6. 23 For all initial conditions, there exists positive constant p satisfying Vð0Þ p. According to 12 l min ðPÞeT e 12 eT Pe and assumption 6 gives that
Yue et al.
11 y2 M 2 Mi on P 1 Pi . Using jyi χ i j t2 þ i2ti with t being a positive design parameter gives that 0 1 n X s " 2 2 li zi2 @ jjPjj 2 Ac~2 V_ eT Qe 4& 2 i¼1
n n n X X 1 1 2 X 1 ~T ~ 1 ~2 l min ðPÞ ei2 þ zi þ yi yi þ 2 1 2 2 2g 4 i i¼1 i¼1 i¼1
þ
n1 X 1 i¼1
2
yi2 þ L þ
n X 1 i¼2
2
~ Ti o ~i þ o
1 2 1 T c~ þ ~" 1 ~" 1 4& 2
n n X 1 1 2 X 1 ~T ~ 1 ~2 zi þ yi yi þ 2 eT Pe þ 1 2 2 2g 4 i j¼1 i¼1
þ
n1 X 1 i¼1
2
yi2 þ L þ
n X 1 i¼2
2
~ Ti o ~i þ o
1 2 1 T c~ þ ~" 1 ~" 1 p 4& 2 (66)
then, from (54) and (55) yields that _ 1d
T _T ¼ l1 z_1 ^ y 1 ’_ 1 ð^ x1 Þ ^ y 1 ’ 1 ð^ x1 Þ ^_ k 1 ðy_ x^_1 Þ 12 ’T1 ’ 1 z 1 2
0 1 0 0 110 z_1 z z1 1 ^"_1 tanh@ A ^" 1 @ tanh@ AA þ y€r 2l k k ^ 1 ;^" 1 ;yr ;y_ ;€ ¼ χ 1 ðe 1 ;e 2 ;z 1 ;z 2 ;y 1 ;^ y 1 ; r yr Þ y_ _ id ¼ li z_i z_i1 i1 ki ðy_ x^_1 Þ ti1
n n X i T i ~ T ~ s s1 ~ 2 X s ~i ~ o yi yi 2 o 1 2gi 4 2 i i¼1 i¼2 0 1 n1 2 X s3 @ 1 1 þ Mi Ay 2 þ C ~"T1 ~" 1 þ i 4 i ti 2t 2 i¼1 0 0 11 n1 X z1 t gL þ@1 16 tanh 2 @ AAH þ 2 # i¼1
choosing ¼ s4&2 "2 jjPjj 2 M 2 > 0 yields that V_ eT Qe
0 1 0 0 110 z zi i ^ i @ tanh@ AA ^_ i tanh@ A o o k k
n X
li zi2 ~ c2
i¼1
n X i ~ T ~ s yi yi 2g i i¼1
n i T s s1 ~ 2 X s3 ~ i ~"T1 ~" 1 ~i o o 1 4 2 2 2 i¼2 0 1 n1 n1 2 X X t @ 1 1 þ Mi Ay 2 þ þ i 4 i ti 2t 2 i¼1 i¼1 0 0 11 z1 þ C gL þ @1 16 tanh 2 @ AAHðyÞ #
T _T ^ y i ’i ðx^i Þ ^ y i ’_ i ðx^i Þ
¼ χ i ðe 1 ; . . . ;eiþ1 ;z 1 ; . . . ;ziþ1 ;y 1 ; . . . ;yi ; ^ 1 ; . . . ;y^ i ;^" 1 ;o ^ 2 ; . . . ;o ^ i ;yr ;y_ ;€ yÞ y r
(67) where χ i is a continuous function. For any B0 > 0 and p > 0, the sets P :¼ fðyr ; y_r ;€ yr Þ : yr2 þ y_r2 þ y€r2 B0 g and 2 2 P P T ~ 2 þ 1 ~"T ~"1 þ y1 ~ y 1 þ 41 2 P1 :¼ l min2ðPÞ ej2 þ 12 zj2 þ 2g1 ~ 1 2 1 1 1 j¼1
(68)
^ ^_ 1 ’_ T1 ’ 1 z 1 12 ’T1 ’1 z_ 1 2 2
r
n n X i T i ~ T ~ s s1 ~ 2 X s ~i ~ o yi yi 2 o 1 2gi 4 2 i i¼1 i¼2 0 1 n1 2 X s3 T 1 1 M @ ~" 1 ~" 1 þ þ i Ayi2 4 i ti 2t 2 i¼1 0 0 11 n1 X z1 t 2 þ C gL þ@1 16 tanh @ AAH þ 2 # i¼1 0 1 n X s2 " eT Qe li zi2 @ jjPjj2 M 2 Ac~2 4& 2 i¼1
j¼1
p are compact in R 3 and R 8 , respectively; therefore, P P 1 is also compact in R 11 , and χ 1 has a maximum M 1 on P P 1 . Similarly, the sets P :¼ fðyr ; y_r ; y€r Þ : iP þ1 iP þ1 1 2 yr2 þ y_r2 þ y€r2 B0 g and Pi :¼ f12 l min ðPÞ ej2 þ 2 zj þ 1 2 2 y1
(69)
Stability analysis Before proposing the main theorem, we first introduce the following lemma
þ 211 ~"T1 ~"1 pg
Lemma 3. Consider the set O# defined by O# :¼ fzjjzj < 0:2554#g. Then, for any z 2 = O# , the inequality 28 2 z 1 16 tanh 0 is satisfied. # The main results are summarized as follows.
are compact in R 3 and R 5iþ3 , respectively; therefore, P 1 Pi is also compact in R 5iþ6 and χ i has a maximum
Theorem 1. Consider the closed-loop system consisting of system (1) under assumptions 1 to 6, the control laws (54)
j¼1
i P j¼1
1 ~T ~ 2gj y j y j
~2 þ 41 2 1
þ
i P 1 j¼1
2 2 yj
þ
i P j¼2
1 ~T ~ 2j oj oj
j¼1
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International Journal of Advanced Robotic Systems
2.5
1.5
, y r and z
1
2
the signals x
1
1 0.5 0 −0.5 −1 −1.5
0
5
10
15
20
25 Time (s)
30
35
40
45
50
35
40
45
50
Figure 1. The trajectories of x1 , yr (solid line) and z1 (dot line).
8
the estimates of x1 and x2
6
4
2
0
−2
−4
0
5
10
15
20
25 Time (s)
30
Figure 2. The signals ^x 1 (dash–dot line) and ^x 2 (solid line).
to (56), and the parameter adaptive laws (60) to (61). Suppose that the packaged uncertain function J1 ðZ 1 Þ can be approximated by the FLSs in the sense that approximation i , errors are bounded. Then, there exists li ; ti , s 1 , s2 , s 3 , s i such that the solution of the closed-loop system is and s SUUB, and the steady-state tracking error is smaller than a prescribed error bound. In the following, similar to the above discussion, the proof process is also divided into two cases. Case 1. z 1 2 O#1 . In this case, jz 1 j 0:2554# 1 : Since z 1 ¼ x 1 yr , the boundedness of yr and z 1 , we can obtain that x1 is bounded.Then, according to lemma 3 yields that 2 z1 0 < 1 16 tanh 1. Moreover, since HðyÞ is a # continuous function of the variable x 1 , we obtain that HðyÞ is bounded. Let jHðyÞj D 1 , thus
z 1 H D1 0 < 1 16 tanh 2 #
(70)
combining (69) with (70) gives that V_ eT Qe
n X
li zi2 ~ c2
i¼1 n X
n X i ~ T ~ s yi yi 2g i i¼1
i T s s1 ~ 2 s3 ~ i ~"T1 ~"1 ~i o 1 o 2 4 2 2 i¼2 0 1 n1 X 1 1 M2 @ i Ayi2 ti 4 i 2t i¼1
þC þ D1 þ
n1 X t i¼1
2
gL
(71)
Yue et al.
13
25 20 15
the trajecory of υ
10 5 0 −5 −10 −15 −20 −25
0
5
10
15
20
25 Time (s)
30
35
40
45
50
5
10
15
20
25 Time (s)
30
35
40
45
50
Figure 3. The signal ðtÞ.
0.7
2
L norms of the estimates of θ1 and θ2
0.6 0.5 0.4 0.3 0.2 0.1 0
0
Figure 4. The signals jj^y 1 jj (dashed line) and jj^y 2 jj (solid line).
Now, choose l 1 ¼ l 2 ¼ . . . ¼ ln ¼ l and t1i ¼ 4 1 i þ 2t þ ti with l and ti being positive constants, then we have n n X X i ~ T ~ s V_ eT Qe y yi l zi ~ c2 2gi i i¼1 i¼1
Mi2
n i T s1 ~ 2 X s3 s ~ i ~"T1 ~"1 ~ o o 4 2 1 i¼2 2 i 2 n1 X
(72)
ti yi2 þ C gL 2 0 V þ C
i¼1
where C ¼ C þ D 1 þ
n t and ¼ min l min ðQÞ ; l ; 0 2 l max ðPÞ i¼1
n1 P
2& ; s2i ; s21 ; s2i ; s23 ; ti g: Let 0 >
C 2p ;,
then V 0 on
Vð0Þ ¼ p. Thus, V p is an invariant set, that is, if Vð0Þ p, then VðtÞ p for all t 0. Therefore, (72) holds for all VðtÞ p and all t 0. Solving (72) gives that C C þ Vð0Þ 2 which e20 t ; 8t 0; 0 VðtÞ 2 0 0 C means that VðtÞ eventually is bounded by 2 0 . Furthermore, ~ 1 , zi ðtÞ, ei ðtÞ, c~, ~" 1 ; o ~ i , and yi are the boundedness of y~ i , obtained. In addition, according to the boundedness of the ^ 1 , c, ^" 1 , and o ^ i are all signals yi , 1 , c, " 1 , oi gives that ^y i , bounded, and from (54) to (56) yields that 1;d ; . . . ; n1;d and are all bounded. Finally, from (51), the boundedness of i is ensured for i ¼ 1; . . . ; n 1: Case 2. z 1 2 = O# 1 . In this case, jz 1 j > 0:2554# 1 : Accord ing to lemma 3 gives that 1 16 tanh 2 z1 H 0: Sim# ilar to (72), we obtain
14
International Journal of Advanced Robotic Systems
2 1.8
the estimate of β 1
1.6 1.4 1.2 1 0.8 0.6 0.4
0
5
10
15
20
25
30
35
40
45
50
30
35
40
45
50
Time (s)
^ 1. Figure 5. The trajectory of
5 4.5
the estimate of c
4 3.5 3 2.5 2 1.5 1 0.5
0
5
10
15
20
25 Time (s)
Figure 6. The trajectory of ^c.
V_ eT Qe
n X i¼1
l zi ~ c2
n X i ~ T ~ s y yi 2gi i i¼1
n i T s s1 ~ 2 X s3 ~ i ~"T1 ~"1 ~i o o 1 2 4 2 2 i¼2 n1 X
Simulation
(73)
ti yi2 þ C1 gL 2 0 V þ C 1
i¼1
n o where 0 ¼ min l min ðQÞ ; l ; 2& ; s2i ; s21 ; s2i ; s23 ; ti and l max ðPÞ n1 Pt C1 ¼ C þ 2 . Then, similar to case 1, the boundedness i¼1
of all the signals in the closed-loop can be proven. This concludes the proof.
In this section, to illustrate the validity of the proposed scheme, consider the following nonlinear time-varying delay system 8 x 21 > _ x ¼ x e 2 sin x ðtÞ t d > 1 2 1 11 > > > ðt > > > > > x 1 ðsÞds > > > td 21 ðtÞ > < 1 (74) _ x x ¼ ðtÞ cosðx x Þ ðtÞ t d 2 1 2 2 12 > > 2 > > > > ðt > > > > sin x ðsÞ ds > 1 > > > td 22 ðtÞ : y ¼ x1
Yue et al.
15
the trajectory of φ(υ (t))
150 100 50 0 −50 −100 −150
0
5
10
15
20
25
30
35
40
45
50
Time (s)
Figure 7. The trajectory of ððtÞÞ.
In this simulation, we choose d 11 ðtÞ ¼ 1 þ 0:4 sinðtÞ, d21 ðtÞ ¼ 1 0:4 sinðtÞ, d12 ðtÞ ¼ 0:2 0:5 sinðtÞ, and d22 ðtÞ ¼ 0:2 sinðtÞ þ 0:5. The upper bounds of them are d1 ¼ 1:4 and d2 ¼ 1:4. The simulation objective is to apply the developed adaptive fuzzy controller such that (1) boundedness of all the signals in the closed-loop system is guaranteed and (2) the system output y follows the reference signal yr to a small neighborhood of zero with yr ¼ 0:5 sinð0:5tÞ þ 0:5 sinðtÞ. Choose the first intermediate control function a1;d as (54) and the input of the backlash-like ðtÞ as (56) and the parameter update laws as (60) and (61), respectively. Given the symmetric positive matrix Q ¼ 0:02I 22 , " ¼ 75, ¼ 12:5 and L 1 ¼ L2 ¼ 0:02, by solving LMI (18), the symmetric positive matrix P and observer gain matrix K are obtained 0: 1046 0: 0692 as P ¼ and K ¼ ½8:2356; 8:4658. 0: 0692 0: 0704 The initial states are chosen as x 1 ð0Þ ¼ x 2 ð0Þ ¼ ^ 1 ¼ 0:001, " 1 ð0Þ ¼ x^1 ð0Þ ¼ x^2 ð0Þ ¼ 0, y 1 ð0Þ ¼ 0, y 2 ð0Þ ¼ 0. Then, select the 0:001, c^ð0Þ ¼ 5, ^ y 1 ð0Þ ¼ 0, and ^ design parameters as ¼ 0:5; k ¼ 0:1, l ¼ 0:5; l 1 ¼ 299; 1 ¼ 0:1, s 2 ¼ 0:1, l 2 ¼ 300; & ¼ 0:0028, g1 ¼ g2 ¼ 1; s 2 ¼ 0:1, s 1 ¼ 0:2, and s3 ¼ 0:2. Furthermore, when s t 2 ½di ; 0, for i ¼ 1; 2; j ¼ 1; 2, choose x 1 ðtÞ ¼ x 2 ðtÞ ¼ 0. Membership functions are specified as 2 Fi;jl ðxi;j Þ ¼ exp 0:5 xi;j þ 1:5 0:5ðl 1Þ ;1 l 7; i ¼ 1; 2; j ¼ 1; 2. ððtÞÞ represents an output of the following backlashlike hysteresis d ðtÞ d d ¼ ðc Þ þ B1 (75) dt dt dt where ¼ 6; c ¼ 3:1635, and B 1 ¼ 0:345. The simulation results are shown in Figures 1 to 7, where Figure 1 illustrates the trajectories of the tracking error, the output, and tracking signals; Figures 2 and 3
exhibit the boundedness of the trajectories of x^1 , x^2 , and ðtÞ; and Figure 4 shows that jj^y 1 jj and jj^y 2 jj are all bounded. Finally, in Figures 5 to 7, the boundedness of ^ 1 , c^, and ððtÞÞ are illustrated, respectively.
Conclusion In this article, based on an appropriate observer, an adaptive fuzzy DSC control scheme is presented for a class of nonlinear time-varying delay systems with unknown backlash-like hysteresis. By choosing appropriate Lyapunov–Krasovskii functionals, the adaptive output-feedback fuzzy controller is designed. The proposed adaptive fuzzy controller guarantees that the closed-loop system is stable in the sense of SUUB. Moreover, all the unknown timedelay terms are lumped into one unknown nonlinear function which can be approximated by only one FLS such that the suggested adaptive fuzzy controller contains less adaptive parameters and this makes our design scheme easier to be implemented in practical applications. The simulation results have been given to illustrate the effectiveness of the proposed scheme. 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 project is supported by the Special Funds of the National Natural Science Foundation of China (11626183, 11661066), Shaanxi Province Natural Science Fund of China (2017JQ6059, 2016JM1035), Youth Foundation of Xi’an University of Architecture and Technology (QN1436), and Talent Foundation of Xi’an University of Architecture and Technology (RC1425).
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International Journal of Advanced Robotic Systems
References 1. Matveev AS, Hoy M, Katupitiya J, et al. Nonlinear sliding mode control of an unmanned agricultural tractor in the presence of sliding and control saturation. Rob Auton Sys 2013; 61(9): 973–987. 2. Zhang TP and Ge SS. Adaptive neural control of MIMO nonlinear state time-varying delay systems with unknown dead-zones and gain signs. Automatica 2007; 32(6): 1021–1033. 3. Shafiei SE and Soltanpour MR. Robust neural network control of electrically driven robot manipulator using backstepping approach. Int J Adv Rob Sys 2009; 6(4): 285–292. 4. Wang LX. Adaptive Fuzzy Systems and Control: Design and Stability Analysis. Englewood Cliff, New Jersey: PrenticeHall, 1994. 5. Qin L, Liu FC, Liang LH, et al. Fuzzy adaptive robust control for space robot considering the effect of the gravity. Chin J Aeronaut 2014; 27(6): 1562–1570. 6. Bakdi A, Hentout A, Boutami H, et al. Optimal path planning and execution for mobile robots using genetic algorithm and adaptive fuzzy-logic control. Rob Auton Sys 2017; 89: 95–109. 7. Cazarez-Castro NR, Aguilar LT and Castillo O. Designing type-1 and type-2 fuzzy logic controllers via fuzzy Lyapunov synthesis for nonsmooth mechanical systems. Eng Appl Artif Intel 2012; 25(5): 971–979. 8. Cazarez-Castro NR, Aguilar LT, and Castillo O. An observer for the type-1 fuzzy control of a servomechanism with backlash using only motor measurements. Soft Comput Intell Control Mobile Robo 2011; 318: 405–421. 9. Cazarez-Castro NR, Aguilar LT, and Castillo O. Fuzzy logic control with genetic membership function parameters optimization for the output regulation of a servomechanism with nonlinear backlash. Expert Syst Appl 2010; 37(6): 4368–4378. 10. Su CY, Stepanenko Y, Svoboda J, et al. Robust adaptive control of a class of nonlinear systems with unknown backlash-like hysteresis. IEEE Trans Autom Control 2000; 45: 2427–2432. 11. Shahnazi R, Pariz N, and Kamyad AV. Adaptive fuzzy output feedback control for a class of uncertain nonlinear systems with unknown backlash-like hysteresis. Commun Nonlin Sci Numer Simul 2010; 15: 2206–2221. 12. Wen CY and Zhou J. Decentralized adaptive stabilization in the presence of unknown backlash-like hysteresis. Automatica 2007; 43: 426–440. 13. Boulkroune A, MSaadand M, and Chekireb H. Design of a fuzzy adaptive controller for MIMO nonlinear time-delay systems with unknown actuator nonlinearities and unknown control direction. Inform Sci 2010; 180: 5041–5059. 14. Wang CH, Wang JH, and Chen CY. Analysis and design of indirect adaptive fuzzy controller for nonlinear hysteretic systems. Int J Fuzzy Sys 2015; 17(1): 84–93. 15. Li YM, Tong SC, and Li TS. Adaptive fuzzy output feedback control of uncertain nonlinear systems with
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
unknown backlash-like hysteresis Inform Sci 2012; 198: 130–146. Li YM, Tong SC, and Li TS. Adaptive fuzzy output feedback control of nonlinear uncertain systems with unknown backlash-like hysteresis based on modular design. Neural Comput Appl 2013; 23: 261–270. Li YM, Tong SC, and Li TS. Adaptive fuzzy output feedback control of MIMO nonlinear uncertain systems with timevarying delays and unknown backlash-like hysteresis. Neurocomputing 2012; 93: 56–66. Cui GZ, Xu SY, Zhang BY, et al. Adaptive tracking control for uncertain switched stochastic nonlinear pure-feedback systems with unknown backlash-like hysteresis. J Franklin Inst 2017; 354: 1801–1818. Swaroop D, Gerdes JC, Yip PP, et al. Dynamic surface control of nonlinear systems. Proceedings of American Control Conference, Albuquerque, NM, USA, 6 June 1997, 3028–3034, IEEE. Swaroop D, Hedrick JK, Yip PP, et al. Dynamic surface control for a class of nonlinear systems. IEEE Trans Autom Control 2000; 45(10): 1893–1899. Zhang TP and Ge SS. Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form. Automatica 2008; 44(7): 1895–1903. Wang M, Liu XP, and Shi P. Adaptive neural control of purefeedback nonlinear time-delay systems via dynamic surface technique. IEEE Trans Sys Man Cybern B Cybern 2011; 41(6): 1681–1692. Tong S, Li Y, Feng G, et al. Observer-based adaptive fuzzy backstepping dynamic surface control for a class of nonlinear systems with unknown time delays. IET Control Theory Appl 2011; 5(12): 1426–1438. Tong SC, Li YM, Feng G, et al. Observer-based adaptive fuzzy backstepping dynamic surface control for a class MIMO nonlinear systems. IEEE Trans Sys Man Cybern B Cybern 2011; 41(4): 1124–1135. Tong SC, Li Y, Li YM, et al. Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strictfeedback systems. IEEE Trans Sys Man Cybern B Cybern 2011; 41(6): 1693–1704. Zhang X and Lin Y. An adaptive output feedback dynamic surface control for a class of nonlinear systems with unknown backlash-like hysteresis. Asian J Control 2013; 15(2): 489–500. Wang M, Chen B, Liu KF, et al. Adaptive fuzzy tracking control of nonlinear time-delay systems with unknown virtual control coefficients. Inform Sci 2008; 178: 4326–4340. Wang M and Chen B. Adaptive fuzzy tracking control for a class of perturbed strict-feedback nonlinear time-delay systems. Fuzzy Sets Sys 2008; 159: 949–967. Tong SC and Li YM. Observer-based fuzzy adaptive robust control of nonlinear systems with time delays and unmodeled dynamics. Neurocomputing 2010; 74: 369–378. Tong SC, Li YM, Xia YQ, et al. Adaptive fuzzy backstepping output feedback control of nonlinear time-delay systems with unknown high-frequency gain sign. Int J Autom Comput 2011; 8(1): 14–22.
Yue et al. 31. Karimi HR. Adaptive H1 synchronization of master-slave systems with mixed time-varying delays and nonlinear perturbations: an LMI approach. Int J Autom Comput 2011; 8(4): 381–390. 32. Xu S and Chen T. An LMI approach to the H1 filter design for uncertain systems with distributed delays. IEEE Trans Circuits Sys II Exp Briefs 2004; 51(4): 195–201. 33. Xu S, Lam J, Chen T, et al. A delay-dependent approach to robust H1 filtering for uncertain distributed delay systems. IEEE Trans Sig Process 2005; 53(10): 3764–3772.
17 34. Yue HY and Li JM. Adaptive fuzzy dynamic surface control for a class of perturbed nonlinear time-varying delay systems with unknown dead-zone. Int J Autom Comput 2012; 9(5): 545–554. 35. Li YM, Ren CE, and Tong SC. Adaptive fuzzy backstepping output feedback control for a class of MIMO time-delay nonlinear systems based on high-gain observer. Nonlin Dyn 2012; 67(2): 1175–1191. 36. Li Y, Li YM, and Tong SC. Adaptive fuzzy decentralized output feedback control for stochastic nonlinear large-scale systems. Neurocomputing 2012; 83: 38–46.
