# Fuzzy Adaptive Switching Control for an Uncertain Robot Manipulators with Time-Varying Output Constraint

Authors: Yongqing Fan; Wenqing Wang; Ying Liu; Minjuan Wang
Journal: Complexity
 (2018)

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## Abstract

An adaptive control strategy based on a fuzzy logic system by introducing a nonzero time-varying parameter is studied for ann-link manipulator system with the condition of a complex environment. At the beginning, a universal approximator with a one time-varying parameter is proposed based on the analysis of the fuzzy logic system, which is utilized to equalize uncertainties in robot manipulators with time-varying output constraints. The novel design method is used to reduce greatly the online learning computation burden compared with traditional fuzzy adaptive control. The output and the position of robotic manipulators are constrained with time-varying, a good tracking performance can be guaranteed with the condition of unknown dynamics of robot manipulators, and the violation of constraints can be conquered by the analysis based on the barrier Lyapunov function. A switching adaptive control is proposed to extend the semiglobal stability to global stability. Effectiveness of the approach is demonstrated by simulation results.


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## Body

## 1. Introduction

In recent decades, more challenges in an intelligent controller design or learning algorithms are urgent to be developed for the growing different applications in engineering environments. Intelligent technologies for manifold aims as a hot topic research and development emerged in many different regions or fields, like robotics and automation [1, 2], manufacturing [3], process industries [4], and so on. Accordingly, the intelligent control has become a formidable problem owing to the different environment in complex systems. Especially in various sophisticated scenarios, intelligent control is a challenging issue because of the various changes of the target object or other complicated phenomenon. In [5], an intelligent shared control system combined with human-computer interaction and a brain computer interface was developed. Based on a spatiotemporal context learning algorithm, good tracking performance can be achieved for various challenging environments in [6]. Under the condition of a sophisticated environment, a coupling interface proposed to learn human adaptive skills in [7]. Extreme learning machine (ELM) was employed to compensate for the uncertain nonlinearities of manipulator systems or be used to the robot teaching systems such that satisfactory performance can be achieved in [8, 9].Adaptive technologies play a pivotal role in dealing with controlled systems in various complex environments and received more favor by scholars. Uncertain nonlinearities or unknown models commonly existed in systems with many complex environments; thus, fuzzy logic systems (FLSs) or neural networks (NNs) are employed to tackle with the performance indicators of the systems. Especially, FLSs and NNs are universal approximators [10–14], which can be used to approximate unknown terms by combining with an adaptive algorithm technique. In [15], a novel teleoperation control combining with NNs and wave variable is proposed to guarantee the stability of the closed-loop systems. For the unknown parameters of robotic systems, adaptive estimation design methods are developed in [16, 17]. For uncertain multiagent systems, adaptive NN synchronization controls by using the information of the neighboring agents are established in [18]. A new idea of finite-time quantized feedback control with NNs is addressed for quantized nonlinear systems with unknown states in [19]. Tracking controllers with prescribed performance are constructed for dynamical nonlinear systems in [20–23]. A variety of different forms of control technologies are designed for robot manipulators with unknown dynamics based on NNs or FLSs, like output feedback control [24–27], robust position control [28], decentralized control [29, 30], and so on. Nevertheless, it should be noted that the problem of constraint control was not considered in these literatures.A barrier function as a useful tool was firstly proposed to protect the boundary on the feasible region in [31, 32], and this pioneer work induced many excellent research results in preventing constraint violation. For example, constant output constraint can be ensured by the analysis of the barrier Lyapunov function (BLF) in [33–35]. An adaptive neural tracking controller is designed by the stability of the integral barrier Lyapunov functions for robot manipulators in [36]. Adaptive fuzzy output controllers united in wedlock backstepping are also be proposed in [37–41]. Adaptive controls by employing the BLF are developed for a class of nonlinear systems with full-state constraints in [40, 41]. However, it should be pointed out that the output constraint is assumed to be a constant value in these references [33–41], and thus, these methods will be invalided for time-varying output constraints.In many special complex work environments, varying constraints are necessary to be taken into account for meeting specific predefined targets, so how to exploit some novelty controls is essential. In [42], an asymmetric bimanual coordinate control based on a novel BLF with the position and velocity constraints in tasks is developed, which can be used to make the robot behavior adaptive and flexible to detect human operator motion. In [11], time-varying constraints are considered, and an adaptive neural network control of robot manipulators is designed to satisfy the tracking performance. It should be needed to point out that these above-mentioned NN or FLS control methods only have been guaranteed for semiglobal uniformly ultimately bounded (SGUUB) because the approximation holds only on a certain compact domain; wherefore, the range of the domain must be ensured such that approximation accuracies can be obtained. However, it is impossible that the accurate domain can be chosen precisely in advance, especially in some complicated systems such as multiple input multiple output (MIMO) systems; hence, it is worth to exploit NN or FLS control that can be achieved by global uniformly ultimately bounded (GUUB). In [43], the GUUB stability can be ensured, but output constraint was not considered in the systems. A global adaptive NN control with presetting tracking performance is designed for a class hypersonic flight vehicle in [44]. A switching adaptive NN controller was projected for the GUUB stability of bimanual robot systems in [45], and tailored transient performances can be achieved at the same time. Nevertheless, these global controllers are the main focus on the approximation of NNs to equalize the unknown nonlinearities or dynamic model in systems. From the viewpoint of engineering, FLSs have advantages of high interpretability of rich expert experiences over NNs, so it is important to design a fuzzy adaptive control of the GUUB stability in practical engineering.Inspired by the above descriptions, we attempt to solve these problems by using FLSs with abundant experiences or languages of experts in this paper. A fuzzy adaptive control based on the BLF technique is proposed for an unknown robot manipulator with time-varying constraints. The stability of GUUB can be guaranteed for the closed-loop system by introducing switching mechanism. The main advantages of this design method are as follows:(1)
A nonzero time-varying parameter is introduced in FLSs, and a fuzzy adaptive controller is designed to get the good tracking performance with time-varying output constraint.
(2)
All the signals can be guaranteed GUUB in the closed-loop robot manipulator.


## 2. System Descriptions and Preliminaries

### 2.1. Descriptions and Assumptions of Robot Manipulators

Then-link robot manipulator is considered as the following dynamical equation:
(1)Mqq¨+Cq,q̇q̇+Gq=τt−JTqft,where q∈Rn×1 denotes the joint angular positions, q̇∈Rn×1 is the velocity, and q¨∈Rn×1 denotes acceleration vectors; Mq∈Rn×n is a symmetric positive known inertia matrix, Cq,q̇∈Rn×n denotes the centripetal and Coriolis force vector, Gq∈Rn×1 denotes the gravity vector, τt∈Rn×1 represents the input torque, Jq is the unknown reversible Jacobian matrix, and ft denotes the constrained force, which satisfies ft≤f¯ for t>0,f¯>0 is a known constant.In this paper, we assume the position of system (1) to be constrained in a time-varying compact set as
(2)h¯cht≤q≤h¯ct,∀t≥0,where h¯ct=h¯c1,h¯c2,…,h¯cnT and h¯ct=h¯c1t,h¯c2t,…,h¯cntT with hcit>h¯cit,∀t∈R+,i=1,…,n.Ifq=x1=x11,x12,…,x1nT and q̇=x2=x21,x22,…,x2nT, then the dynamical system (1) can be transformed into following dynamical equation:
(3)ẋ1=x2,ẋ2=M−1x1τ−JTx1f−Cx1,x2x2−Gx1,y=x1.For a given reference signal vectoryd=yd1,yd2,…,ydnT, the control aim is to design a fuzzy adaptive controller such that the output y=x1=q1,q2,…,qnT tracks this ideal reference signal vector with small errors, and all the signals in the closed-loop robot manipulators (1) are all be bounded. Meanwhile, the time-varying constrains must be guaranteed without damage.Assumption 1.
Unknown function vectorΔX=Δ1X,…,ΔnXT on the compact set U and Δi satisfies the Lipschitz condition with coefficient li which satisfies ΔiX1−ΔiX2≤liX1−X2.Assumption 2 [11].
There exist constantsH¯cij and H¯cij satisfying that h¯cij≤H¯cij and h¯cij≥H¯cij,j=0,1,…,n,∀t>0.Assumption 3 [11].
There exist two function vectorsX¯0t<h¯c and X¯0>h¯c,∀t>0 and exist positive constants Xi which satisfies ydi≤Xi and X¯0t≤ydt≤X¯0t.Define(4)hait=ydit−h¯cit,hbit=h¯cit−ydit.If Assumptions2 and 3 are hold, then there exist some positive constants h¯bi,h¯bi,h¯ai, and h¯ai, which can make the following inequalities true:
(5)h¯αit≤h¯αit≤h¯ait,h¯bit≤h¯bit≤h¯bit.The tracking error vector is defined asz1=z11,z11,…,z1nT=x11−yd1,x12−yd2,…,x1n−ydnT and z2=z21,z22,…,z2nT=x21−β11,x22−β12,…,x2n−β1nT. We introduce the following error coordinates that will be used in the procedure of the control design:
(6)ζai=z1ihai,ζbi=z1ihbi,ζi=piz1iζbi+1−piz1iζai,where
(7)piz1i=1,z1i>0,0,z1i≤0.The following Lemmas1 and 2 are introduced.Lemma 1 [11].
For a givenζi<1 and any integer r>0, then log1/1−ζi2r<ζi2r/1−ζi2r is hold.Lemma 2 [11].
If−hait<z1it<hbit, then ζit<1 can be obtained.


### 2.2. Descriptions of Fuzzy Logic Systems

The following Mamdani type FLS withq fuzzy rules and the kth rule is to be considered:Ifx1 is A1k, x2 is A2k, ⋯, and xn is Ank, then
(8)y=Bk,k=1,2,…,q,where Ahk and Bk denote fuzzy sets. Ahkxhh=1,2,…,n is the membership function of xh. Now, a nonzero time-varying parameter ρ=ρt is introduced in the fuzzy rule (8). According to the defuzzification in [12], the output of the fuzzy logic system (8) can be written as
(9)y=∑k=1qyk∏h=1nAhkxh/ρ∑k=1q∏h=1nAhkxh/ρ,where yk=maxyk∈RμBkyk. On a given compact set, we know that the FLS (8) can be utilized to approximate any given uncertain continuous function with any degree of accuracy. Therefore, based on this property of universal approximation, Lemma 3 can be easily obtained.Lemma 3.
If the uncertain continuous nonlinear function∆iX satisfies the Lipschitz condition with coefficient li and existing FLSs FiX make supX∈UΔiX−FiX≤εi, where U=X∣X≤αρ and εi is the approximate error, then the following inequality is hold:
(10)supX∈UΔiX−FiXρ≤liαρ−1+εi.Proof.
In terms ofΔiX, it satisfies the Lipschitz condition with coefficient li, such that ΔiX−ΔiX/ρ≤liX−X/ρ. If X∈X∣X≤αρ is true, then the following result is also true:
(11)ΔiX−FiXρ≤ΔiX−ΔiXρ+ΔiXρ−FiXρ≤liαρ−1+εi.
This completed the proof for Lemma1.LetFX=F1X,…,FnXT; supX∈UΔX−FX≤∑i=1nεi2≜ε is easily obtain. From Assumption 1, we define supX∈Ul1,…,ln≤∑i=1nli2≜L. The values of ε and L are all unknown in this paper, so we introduce ε^=ε^t and L^=L^t to denote the estimation of ε and L, respectively. ε~=ε^−ε and L~=L^−L denote the estimate error.


## 2.1. Descriptions and Assumptions of Robot Manipulators

Then-link robot manipulator is considered as the following dynamical equation:
(1)Mqq¨+Cq,q̇q̇+Gq=τt−JTqft,where q∈Rn×1 denotes the joint angular positions, q̇∈Rn×1 is the velocity, and q¨∈Rn×1 denotes acceleration vectors; Mq∈Rn×n is a symmetric positive known inertia matrix, Cq,q̇∈Rn×n denotes the centripetal and Coriolis force vector, Gq∈Rn×1 denotes the gravity vector, τt∈Rn×1 represents the input torque, Jq is the unknown reversible Jacobian matrix, and ft denotes the constrained force, which satisfies ft≤f¯ for t>0,f¯>0 is a known constant.In this paper, we assume the position of system (1) to be constrained in a time-varying compact set as
(2)h¯cht≤q≤h¯ct,∀t≥0,where h¯ct=h¯c1,h¯c2,…,h¯cnT and h¯ct=h¯c1t,h¯c2t,…,h¯cntT with hcit>h¯cit,∀t∈R+,i=1,…,n.Ifq=x1=x11,x12,…,x1nT and q̇=x2=x21,x22,…,x2nT, then the dynamical system (1) can be transformed into following dynamical equation:
(3)ẋ1=x2,ẋ2=M−1x1τ−JTx1f−Cx1,x2x2−Gx1,y=x1.For a given reference signal vectoryd=yd1,yd2,…,ydnT, the control aim is to design a fuzzy adaptive controller such that the output y=x1=q1,q2,…,qnT tracks this ideal reference signal vector with small errors, and all the signals in the closed-loop robot manipulators (1) are all be bounded. Meanwhile, the time-varying constrains must be guaranteed without damage.Assumption 1.
Unknown function vectorΔX=Δ1X,…,ΔnXT on the compact set U and Δi satisfies the Lipschitz condition with coefficient li which satisfies ΔiX1−ΔiX2≤liX1−X2.Assumption 2 [11].
There exist constantsH¯cij and H¯cij satisfying that h¯cij≤H¯cij and h¯cij≥H¯cij,j=0,1,…,n,∀t>0.Assumption 3 [11].
There exist two function vectorsX¯0t<h¯c and X¯0>h¯c,∀t>0 and exist positive constants Xi which satisfies ydi≤Xi and X¯0t≤ydt≤X¯0t.Define(4)hait=ydit−h¯cit,hbit=h¯cit−ydit.If Assumptions2 and 3 are hold, then there exist some positive constants h¯bi,h¯bi,h¯ai, and h¯ai, which can make the following inequalities true:
(5)h¯αit≤h¯αit≤h¯ait,h¯bit≤h¯bit≤h¯bit.The tracking error vector is defined asz1=z11,z11,…,z1nT=x11−yd1,x12−yd2,…,x1n−ydnT and z2=z21,z22,…,z2nT=x21−β11,x22−β12,…,x2n−β1nT. We introduce the following error coordinates that will be used in the procedure of the control design:
(6)ζai=z1ihai,ζbi=z1ihbi,ζi=piz1iζbi+1−piz1iζai,where
(7)piz1i=1,z1i>0,0,z1i≤0.The following Lemmas1 and 2 are introduced.Lemma 1 [11].
For a givenζi<1 and any integer r>0, then log1/1−ζi2r<ζi2r/1−ζi2r is hold.Lemma 2 [11].
If−hait<z1it<hbit, then ζit<1 can be obtained.


## 2.2. Descriptions of Fuzzy Logic Systems

The following Mamdani type FLS withq fuzzy rules and the kth rule is to be considered:Ifx1 is A1k, x2 is A2k, ⋯, and xn is Ank, then
(8)y=Bk,k=1,2,…,q,where Ahk and Bk denote fuzzy sets. Ahkxhh=1,2,…,n is the membership function of xh. Now, a nonzero time-varying parameter ρ=ρt is introduced in the fuzzy rule (8). According to the defuzzification in [12], the output of the fuzzy logic system (8) can be written as
(9)y=∑k=1qyk∏h=1nAhkxh/ρ∑k=1q∏h=1nAhkxh/ρ,where yk=maxyk∈RμBkyk. On a given compact set, we know that the FLS (8) can be utilized to approximate any given uncertain continuous function with any degree of accuracy. Therefore, based on this property of universal approximation, Lemma 3 can be easily obtained.Lemma 3.
If the uncertain continuous nonlinear function∆iX satisfies the Lipschitz condition with coefficient li and existing FLSs FiX make supX∈UΔiX−FiX≤εi, where U=X∣X≤αρ and εi is the approximate error, then the following inequality is hold:
(10)supX∈UΔiX−FiXρ≤liαρ−1+εi.Proof.
In terms ofΔiX, it satisfies the Lipschitz condition with coefficient li, such that ΔiX−ΔiX/ρ≤liX−X/ρ. If X∈X∣X≤αρ is true, then the following result is also true:
(11)ΔiX−FiXρ≤ΔiX−ΔiXρ+ΔiXρ−FiXρ≤liαρ−1+εi.
This completed the proof for Lemma1.LetFX=F1X,…,FnXT; supX∈UΔX−FX≤∑i=1nεi2≜ε is easily obtain. From Assumption 1, we define supX∈Ul1,…,ln≤∑i=1nli2≜L. The values of ε and L are all unknown in this paper, so we introduce ε^=ε^t and L^=L^t to denote the estimation of ε and L, respectively. ε~=ε^−ε and L~=L^−L denote the estimate error.


## 3. Fuzzy Adaptive Control Design

In this section, the fuzzy adaptive controller is designed based on backstepping technology and the asymmetric BLF.From, an asymmetric TVBLF in [11], the following is considered:
(12)V1=12r∑i=1npiz1iloghbi2rthbi2rt−z1i2r+12r∑i=1n1−piz1iloghai2rthai2rt−z1i2r,where r≥1 is an integer of differentiability of β1ii=1,2,…,n. We know that V1 is continuously differentiable on the set ζi<1; then
(13)V̇1=∑i=1npiζbi2r−1hbi1−ζbi2rż1i−z1iḣbihbi+∑i=1n1−piζai2r−1hai1−ζai2rż1i−z1iḣaihai.Sinceż1i=ẋ1i−ẏdi=z2i+β1i−ẏdi, (13) can be rewritten as
(14)V̇=∑i=1npiζbi2r−1hbi1−ζbi2rz2i+β1i−ẏdi−z1iḣaihai+∑i=1n1−piζai2r−1hai1−ζai2rz2i+β1i−ẏdi−z1iḣaihai.Chooseβ1=−K1+K~1tz1+ẏd, and
(15)K1=k110⋯00k12⋯0⋮⋮⋱⋮00⋯k1n,K~1=k~110⋯00k~12⋯0⋮⋮⋱⋮00⋯k~1n,with k~1it=ḣai/hai2+ḣai/hai2+oi; positive parameters k1i and oi are to be given suitable positive constants by the designer.Based on (6), (14) becomes
(16)V̇1=∑i=1nσiz1i2r−1z2i−∑i=1nζi2r1−ζi2rk1i+k~1i+p1iḣbihbi+1−p1iḣaihai,where σi=p1i/hbi2r−z1i2r+1−p1i/hai2r−z2i2r, because k~1i+p1iḣbi/hbi+1−p1iḣai/hai≥0; then we have
(17)V̇1=−∑i=1nk1iζi2r1−ζi2r+∑i=1nσiz1i2r−1z2i.Theorem 1.
Consider the robot system described by (1) under Assumptions 1–3, with the condition of z2>αρ. The following control scheme design as shown in (18) can make sure that the state vector X¯=x1T,x2T,βT,β̇T,L~,ε~T reaches the sliding model surface s as shown in (19) in limited time period.
(18)τ=0,0,…,01×nT,ρ̇=1α2ρλ+∑i=1nσiz1i2r−1z2i+Δ¯M−1z2,L^̇=0,ε^̇=0,where the parameters λ and α are known positive constants.Proof.
Sincez2>αρ and X≥z2≥αρ is satisfied, open control is adopted with this condition; choose the sliding model surface as
(19)s=sx1T,x2T,βT,β̇T,L~,ε~T=V1+12z2Tz2−12α2ρ2+12μ1L~2+12θ1ε~2.
Define the positive functionV¯=1/2s2, and according to (17) and (18), the derivative of V¯ can be obtained as
(20)V¯̇=s−∑i=1nk1iζi2r1−ζi2r+∑i=1nσiz1r2r−1z2i+z2TM−1x1τ−JTx1f−Cx1,x2x2−Gx1−β̇−α2ρρ̇+μ1−1L~L^̇+θ1−1ε~ε^̇.
We denote the function vectorΔX=JTx1f+Cx1,x2x2+Gx1+Mx1β̇ and satisfy Assumption 1; we have
(21)V¯̇≤s−∑i=1nk1iζi2rζi2r+∑i=1nσnz1n2r−1z2n+Δ¯M−1⋅z2−β̇−α2ρρ̇+μ1−1L~L^̇+θ1−1ε~ε^̇≤−s∑i=1nk1iζi2r1−ζi2r+λ<0.
According to [46], (21) means that Theorem 1 can be guaranteed.Theorem 2.
If Assumptions1–3 are hold and z2≤αρ can be satisfied, then the proposed fuzzy adaptive control (22) can make the tracking errors converge to a small zero field. All the signals in the closed-loop system can be ensured to be uniformly ultimately bounded.
(22)τ=−Mx1K2z2+Γ+FXρ,ρ̇=12γ1ρ−δ1αM−1L^α1−ρ+ε^sign~ρ,L^̇=−γ2L^+μ2α2ρ⋅1−ρ⋅M−1,ε^̇=−γ3ε^+θ2αρ⋅M−1,where
(23)K2=diagk21,k22,…,k2n,sign~ρ=1,ρ>0,−1,ρ≤0,FXρ=F1Xρ,F2Xρ,…,FnXρT,Γ=σ1z112r−1z21,σ2z122r−1z22,…σiz1i2r−1z2iT.The parameters γ1,γ2,γ3,δ1,µ2, and θ2 are positive constants proposed by the designer.Proof.
Because ofz2≤αρ and X≥z2, so if X>αρ is satisfied, the control scheme will go back to Theorem 1. But if X>αρ, we consider the following candidate Lyapunov function:
(24)V2=V1+12z2Tz2.
In differentiationV2, then
(25)V̇2=V̇1+z2Tż2.We know ż2=ẋ2−β̇1=M−1x1τ−Δ, with β̇1=β̇11,β̇12,…,β̇1nT and β̇1i=∂β1i/∂x1ix2i+∑k=01∂β1i/∂ξik+1ξi=ydi,hai,hbiT, so the following inequality is true:
(26)V̇2=−∑i=1nk1iζi2r1−ζi2r+∑i=1nσiz1i2r−1z2i+z2TM−1x1τ−Δ.
Due to the function vectorΔ=Δ1,Δ2,…ΔnT which is unknown, so the approximation of the fuzzy logic systems (9) can be used to deal with the unknown function; then
(27)V̇2≤−∑i=1nk1iζi2r1−ζi2r−z2TK2z2+z2M−1x1FXρ−Δ≤−∑i=1nk1iζi2r1−ζi2r−∑i=1nz2ik2iz2i+z2M−1⋅FXρ−Δ≤−∑i=1nk1iζi2r1−ζi2r−∑i=1nz2ik2iz2i+αρ·M−1αL1−ρ+ε.
Define the candidate Lyapunov function.(28)V3=V2+12δ1ρ2+12μ2L~2+12θ2ε~2.Then the derivative of V3 yields
(29)V̇3=V̇2+δ1−1ρρ̇+μ2−1L~L^̇+θ1−1ε~ε^̇≤−∑i=1nk1iζi2r1−ζi2r−∑i=1nz2ik2iz2i+δ1−1ρρ̇+αρ⋅M−1αL^1−ρ+ε^+μ2−1L~L^̇−α2L~ρ⋅1−ρ⋅M−1+θ2−1ε~ε^̇−αρ⋅M−1ε~=−∑i=1nk1iζi2r1−ζi2r−∑i=1nz2ik2iz2i−γ12δ1ρ2−γ2μ2L~L^−γ3θ2ε~ε^.
The following inequality is true:(30)−γ2μ2L~L^=−γ2μ2L~2−γ2μ2L~L≤−γ22μ2L~2+γ22μ2L2.
Similarly,(31)−γ3θ2ε~ε^=−γ3θ2ε~2−γ3θ2ε~ε≤−γ32θ2ε~2+γ32θ2ε2.
Substituting (30) and (31) into (29), it follows that
(32)V̇3=−∑i=1nk1ilog11−ζi2r−∑i=1nz2ik2iz2i−γ12δ1ρ2−γ22μ2L~2−γ32θ2ε~2+γ22μ2L2+γ32θ2ε2
Denoteϖ=min2rk1i,2k2i,γ1/δ1,γ2/μ2,γ3/θ2 and υ=γ2/2μ2L2+γ3/2θ2ε2; inequality (32) becomes
(33)V̇3≤−ϖV3+υ.
Multiplyingeϖt on both sides in (33), then
(34)ddtV3eϖt≤υeϖt.
Integrating both sides of (34) over 0,t, get
(35)0≤V3≤V3−υϖe−ϖt+υϖ.
From (12), (24), and (28), we have
(36)12rlog11−ζi2r≤V3t≤V¯30,in which V¯30=1/2r∑i=1nlog1/1−ζi2r0+1/2z202+1/2δ2ρ20+1/2μ2L~20+1/2θ2ε~20+υ/ϖ. We know that ζi2r≤1−e−2rV¯30 and ζi≤1−e−2rV¯301/2r.
Based on (6), then
(37)−Q¯it≤z1it≤Q¯it,where Q¯it=hait1−e−2rV¯301/2r and Q¯it=hbit1−e−2rV¯301/2r.
From Assumptions2 and 3, we know that h¯c0<y0<h¯c0, and according to hai and hbi, then −hai0<zi0<hbi0, which is equal to ζi0<1. Based on Lemma 2, ζit<1 can be ensured, and −hait<z1it<hbit is hold. Because x1i=z1i+ydi, so it gets
(38)ydit−hait<x1it<hbit+ydit.
Finally,h¯ct≤yt≤h¯ct∀t≥0, so the output signal can not be violated.
We know that the virtual controlβ1i is bounded, from (35) and (36); z2≤2V¯301/2 obtained. Since z2=x2−β1, then x2 is also bounded. From the updated law in (22), the parameters ρ,L^ and ε^ are all bounded. This is the proof procedure of Theorem 2.Based on the analysis of Theorem1 and Theorem 2, we can draw the following Theorem 3 about global stability.Theorem 3.
For the robot manipulator (1) with Assumptions 1–3 and with the initial output, h¯c0<y0<h¯c0. When the tracking run out of the domain U=X∣X≤αρ, the updated laws in control (18) will pull the states back to the sliding surface s, and then the proposed switching controls (18) and (22) can guarantee that all the signals in the closed-loop systems are to be GUUB, and all the tracking errors z1i will went to a small field Φz1i=z1i∣−Q¯it≤z1it≤Q¯it in a finite time.Remark 1.
According to the analysis in Theorem3, we know that the approximation accuracies only updated the online time-varying parameter in a fuzzy logic system, which broke the limitation of exponential growth with the linguistic variables in traditional fuzzy rules such as [24, 27, 28, 33]. This advantage can make designers focus on his/her energy to convey information from human experience to controller action in a complex environment and neglect the computational burden because of the adaptive laws independent with fuzzy rules.


## 4. Simulation Example

In order to illustrate the fuzzy adaptive controller design, a robot manipulator in [11] with 2-DOF is considered, and the output variable is q=q1,q2T=x11,x12T. The dynamic robot system can be written as (1) with
(39)Mq=m¯11m¯12m¯21m¯22,m¯11=m1lr12+m2l12+lr22+2l1lr2cosq2+I1+I2,m¯12=m¯21=m2lr22+l1lr2cosq2+I2,m¯22=m2lr22+I2,Cq,q̇=c¯11c¯12c¯21c¯22,c¯11=−m2l1lr2q̇2sinq2,c¯12=−m2l1lr2q̇1+q̇2sinq2,c¯21=m2l1lr2q̇1sinq2,c¯22=0,Gq=g¯11,g¯21T,g¯11=m1lr2+m2l1gcosq1+m2lr2gcosq1+q2,g¯21=−m2lr2gcosq1+q2.The Jacobian matrix can be chosen as(40)Jq=J¯11J¯12J¯21J¯22,J¯11=−l1sinq1+l2sinq1+q2,J¯12=−l2sinq1+q2,J¯21=l1cosq1+l2cosq1+q2,J¯22=l2cosq1+q2.We know that the order of the system isq=2. The parameters are given as m1=2 kg, m2=0.85 kg, lr1=0.175 m, l1=0.35 m, lr2=0.155 m, l2=0.31 m, l1=0.06125 kgm2, l2=0.02042 kgm2, and g=9.8 m/s2. The initial state vectors are selected as x10=0,2T and x20=0,0T. The reference trajectory was provided as yd=sin0.5t,2cos0.5tT. Initial values of updated laws are given as ρ0=0.5, L^0=0.3, and ε^0=0.9, and the parameters in the controller are defined as α=200, λ=100, δ1=0.0005, o1=0.0002, o1=0.0003, γ1=0.05, γ2=10, γ3=40, µ2=0.0005, and θ2=0.0006.The constrained force is selected asft=1.83+sint,1.26+2costT. We let variable displacement of output constraints as hc1≤q≤hc2, where hc1=hc11,hc12T=0.2sin0.5t−0.9,0.7cos0.5t−1.5T and hc2=hc21,hc22T=0.3+0.8sin0.5t,−0.18cos0.5t+2.25T. Define the parameters as k11=0.1, k12=0.3, k21=2000, and k22=1500.We choose domainU=U1×U2×U3×U4=−200,200×−200,200×−200,200×−200,200, and fuzzy linguistic variables selected as negative big (NB), negative middle (NM), negative small (NS), positive small (PS), positive middle (PM), and positive big (PB) with the fuzzy membership functions are defined as i=1,2;j=1,2:
(41)μNB=11+exji+200,μNM=11+exji+100,μNS=11+exji+0.1,μPS11+exji−0.1,μPM=11+exji−100,μPB=11+exji−200.The fuzzy logic systemF1X is the six rules in Table 1; A1ik×A2ik×A3ik×A4ik→Bikk=1,2,3,4,5,6 can be used to approximate the unknown function Δ1x1x,x2T,β1,β̇1;Table 1
Fuzzy rules of the FLSF1X.
kth rule
x11
x12
x21
x22
∆1
1
NB
PB
PS
NS
PB
2
PM
NS
PB
NM
NM
3
NS
PM
NB
PB
PM
4
PB
PM
NS
NB
PS
5
PS
PB
NB
PM
NS
6
NS
PS
PB
PN
NBSimilarly, theF2X can be constructed in Table 2 by fuzzy rules A¯1ik×A¯2ik×A¯3ik×A¯4ik→B¯ik instead of the unknown function Δ2x1T,x2T,β1,β̇1. The simulation results of tracking performance for the robot system are shown as in Figures 1–7.Table 2
Fuzzy rules of the FLSF2X.
kth rule
x11
x12
x21
x22
∆2
1
NB
NM
PS
PB
NB
2
NM
NS
PS
PM
NM
3
PS
NM
PB
NM
NS
4
PM
PB
NS
NB
PS
5
PB
PS
NB
NM
NM
6
NS
PB
NM
NS
PBFigure 1
The time response of the outputx11 of Link 1 and ideal reference yd1.Figure 2
The time response of the outputx12 of Link 2 and ideal reference yd2.Figure 3
The time response of tracking errorz11 and z12.
(a)
(b)Figure 4
The time response of velocity tracking trajectory of Link 1.Figure 5
The time response of velocity tracking trajectory of Link 2.Figure 6
The time response of tracking errorz21 and z22.
(a)
(b)Figure 7
The time response of parameters in controller (18) and (22).From the simulation results, Figure1 shows the output of the first manipulator and the desired trajectory with time-varying constraint, and Figure 2 indicates the output of the second manipulator and the reference signal with time-varying constraint. The tracking errors are shown in Figure 3. The time response of velocity of the first manipulator and that of the second manipulator are depicted in Figures 4 and 5, respectively, and the speed of tracking errors is shown in Figure 6. All the adaptive parameters of online time response in controls (18) and (22) are shown in Figure 7, which can be achieved GUUB successfully, and a good tracking performance can be obtained with time-varying constraints from these simulation results.


## 5. Conclusions

This paper presented a novel fuzzy adaptive switching control based on a time-varying BLF for a robot manipulator with uncertain models; the FLSs combining with the time-varying parameter are used to compensate for the uncertain model system, which extended the traditional fuzzy control that is only emphasized on approximation accuracies and neglected the interpretability of expert language in the procedure of design controllers. By introducing the time-varying parameter in the FLS, global stability can be guaranteed by the switching fuzzy control scheme with neglecting the outside or the inside of the compact domain of the universal approximator.



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*Source: 8275296-2018-07-22.xml*