# Adaptive Sliding Mode Control for a Class of Manipulator Systems with Output Constraint

Authors: Guangshi Li
Journal: Complexity
 (2021)

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## Abstract

In this paper, an adaptive sliding mode control method based on neural networks is presented for a class of manipulator systems. The main characteristic of the discussed system is that the output variable is required to keep within a constraint set. In order to ensure that the system output meets the time-varying constraint condition, the asymmetric barrier Lyapunov function is selected in the design process. According to Lyapunov stability theory, the stability of the closed-loop system is analyzed. It is demonstrated that all signals in the resulted system are bounded, the tracking error converges to a small compact set, and the system output limits in its constrained set. Finally, the simulation example is used to show the effectiveness of the presented control strategy.


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## Body

## 1. Introduction

The application of manipulator is of great significance to the development of human society. So far, many effective control strategies have been produced for manipulator control, such as the PID control [1, 2], the robust control [3–5], and the sliding mode control [6–8]. Among them, the PID control is more suitable for the control theory of other technologies difficult to adopt, or the mathematical models are not more accurate. In [9], a new adaptive neural network control method, which is based on a simple structure-like PID control, is constructed to deal with the robot system with incomplete model and disturbance. The feature of the robust control is that once the controller is designed, its parameters are fixed; that is, it is not affected by external interference, and it can guarantee the control performance. In [10, 11], the robust control approaches are proposed to solve the position tracking problem of the manipulator systems. However, if the function of the considered system needs to be improved or made major changes, then the controller requires to be redesigned, which cannot be realized for the robust control method. The sliding mode control has the good stability against external disturbance and system unmodeled dynamics. For this, the sliding mode control will be considered in this paper.In recent years, with the rapid development of science and technique, the adaptive control technology has become particularly important. Especially, the fuzzy logic system-based and neural network-based adaptive control is one of the current research hot topics. These two methods are more suitable for nonlinear systems with unknown dynamics. In this aspect, a large number of research results have been obtained [12–15]. Concrete results such as that in [16], a neural network-based adaptive FTC controller was proposed for the nonstrict-feedback interconnected large-scale systems, in which the neural networks were used to approximate the unknown terms of the considered systems. In [17], the fuzzy logic systems were presented to approximate the unknown terms, and then an adaptive fuzzy controller was designed by using backstepping technique, which ensured the finite-time convergence.In addition, under the influence of environment and other factors, variables in the controlled system are often limited [18–20]; that is, variables need to be within a certain range and cannot be arbitrarily large. Therefore, constraint control has been widely concerned by researchers, and abundant achievements have been obtained [21–24]. In [25], the nonstrict feedback system with full-state constraints was considered, as well as input saturation and unmodeled dynamics. By introducing a logarithm nonlinear mapping, the considered system with full-state constraints was transformed into an unconstrained one.According to the above analysis, an adaptive sliding mode controller with time-varying output constraint is designed. In order to ensure that the system output is with the specified time-varying constraint range, the asymmetric logarithmic Lyapunov function is selected. The unknown nonlinear function existed in the considered system is approximated by the radial basis function neural networks (RNFNNs). Then, an adaptive sliding mode controller is constructed by combining the backstepping technique and the sliding mode control approach.Compared with the existing results, the innovations of this paper mainly include the following two aspects:(1)
Compared with the existing results in [21–25], where the constrained problems were considered and the symmetric barrier Lyapunov functions were employed, in this paper, the asymmetric barrier Lyapunov functions were used to solve the output constraint problem. Compared with symmetric barrier Lyapunov functions, asymmetric barrier Lyapunov functions are more widely used and less conservative.(2)
A new adaptive sliding mode control approach for manipulator systems is proposed by combining the barrier Lyapunov function method and the backstepping technique. A sufficient condition for the adaptive constrained control problem is derived. Moreover, the uncertainties are considered in this paper, which increase the design complexity.The rest of this paper is organized as follows: Section2 presents the problem description which includes the manipulator system description, the required assumption, and the control objective. Section 3 shows the design process of the adaptive NN controller, and the main result is also proposed. Section 4 contains the simulation example, and Section 5 is conclusion.


## 2. Problem Description

Consider the manipulator system in [26](1)x˙1=x2,x˙2=1Mx3−NMsinx1−BMx2,x˙3=1Lu−KBLx2−RLx3,y=x1,where x1,x2,x3 are the system states, u∈R and y∈R are the control input and output, respectively, and the definitions of related notations can be found in [26]. In this paper, the system output is required to satisfy a certain constraint condition. That is to say the system output yt satisfies the constraint condition that k¯t≤yt≤k¯t, where k¯t and k¯t are continuous functions.Assumption 1.
(see [27]). The desired reference signal ydt is bounded, which means that there exists a constant Y¯ such that ydt≤Y.
The control target of this paper is to design an adaptive NN controller such that the closed-loop system is stable, and the system output can track the desired reference signalydt as well as satisfy its constraint condition.


## 3. The Adaptive NN Controller Design

In this paper, the backstepping method is used to design the adaptive NN controller, and the following coordinate transformation is defined:(2)z1=x1−yd,z2=x2−α1,z3=x3−α2,where z1, z2, and z3 are the error variables and α1 and α2 are the virtual controllers, which will be designed in the later procedure.Step 1.
From (1) and (2), one gets(3)z˙1=x˙1−y˙d=z2+α1−y˙d.
Consider the following asymmetric obstacle Lyapunov function:(4)V1=qz12logkz12tkz12t−z12+1−qz12logkz22tkz22t−z12,where qz1 is a piecewise function, which means that if z1>0, then qz1=1, otherwise qz1=0. kz1t and kz2t are positive definite smooth bounded function, which satisfy kz1t=k¯t−Y and kz2t=−k¯t−Y, respectively.
For convenience of expression, define the following error coordinate transformation:(5)e1=z1kz1,e2=z1kz2,e=qz1e1+1−qz1e2,where e1, e2, and e are transfer variables. Based on the above transformation, we know that if z1>0, then qz1=1 and e=e1. Furthermore,(6)V1=12log11−e12=12log11−e2.
Similarly, ifz1≤0, then qz1=0 and e=e2. Thus we have(7)V1=12log11−e22=12log11−e2.
Combining (6) and (7), one has(8)V1=12log11−e2.
Then, the time derivative ofV1 is obtained as(9)V˙1=qz1e1kz11−e12z2+α1−y˙d−z1k˙z1kz1+1−qz1e2kz21−e22z2+α1−y˙d−z1k˙z2kz2.
Design the following virtual controller:(10)α1=−γ1+γ¯1tz1+y˙d,where γ1>0 is a constant and γ¯1t=k˙z1/kz12+k˙z2/kz22+γ0 is a time-varying gain with γ0 being a positive constant.
Substituting (10) into (8), the following holds:(11)V˙1≤−γ1e21−e2+qz1kz12−z12+1−qz1kz22−z12z1z2≤−γ1e21−e2+μz1z2,where μ=qz1/kz12−z12+1−qz1/kz22−z12.Step 2.
From (1) and (2), z˙2 can be described as(12)z˙2=1Mx3−Nsinx1−Bx2−Mα˙1.
In this paper, we consider the case that the load mass is uncertain, and then according to the definitions of the system parameter in [26], it is easy to know that M and N are unknown, which implies that the function −Nsinx1−Bx2−Mα˙1 is also unknown. Here, the RBFNNs are used to approximate it, i.e.,(13)−Nsinx1−Bx2−Mα˙1=ω2∗Tφ2X2+ε2X2,where ω2∗T stands for the ideal weight vector of the NNs, φ2X2 is a Gaussian function with X2=x1,x2,x3,yd,y˙dT being the NN input, and εX2 is the approximation error. In addition, ε2X2≤ε¯2.Assumption 2.
(see [28]). The ideal weight vectors of RBFNNs are bounded.
Substituting (13) into (12), one gets(14)z˙2=1Mx3+ω2∗Tφ2X2+ε2X2.
Select the following Lyapunov function:(15)V2=V1+M2z22+12δ2W˜22,where δ2>0 is a design parameter. W˜2=W2∗−W^2 is the estimation error with W^2 being the estimation of W2∗, and W2∗=ω2∗2.
Then, combining (11) and (14), the time derivative of V2 is(16)V˙2=V˙1+Mz2z˙2−1δ2ω˜2Tω^˙2≤−γ1e21−e2+μz1z2+z2x3+ω2∗Tφ2X2+ε2X2−1δ2W˜2W^˙2.
Based on Young’s inequality, it has(17)z2ω2∗Tφ2X2≤z22W2∗φ2TX2φ2X24ξ2+ξ2z2ε2X2≤12z22+12ε¯22,where ξ2>0 is a parameter. Then, the derivative of V2 becomes(18)V˙2≤−γ1e21−e2+z2z3+α2+μz1+12z2+z2W^2φ2TX2φ2X24ξ2+W˜2z22φ2TX2φ2X24ξ2−1δ2W^˙2+12ε¯22+ξ2.
Select the following virtual controller and adaptive law(19)α2=−γ2z2−μz1−12z2−z2W^2φ2TX2φ2X24ξ2,(20)W^˙2=δ2z22φ2TX2φ2X24ξ2−σ2W^2,where γ2 and σ2 are the positive constants.
Substituting (19) and (20) into V˙2, one has(21)V˙2≤−γ1e21−e2−γ2z22+z2z3+σ2W˜2W^2+12ε¯22≤−γ1e21−e2−γ2z22−12σ2W˜22+z2z3+12σ2W2∗2+12ε¯22+ξ2.Step 3.
In order to ensure that the tracking error of the system tends to zero in finite time, improve the convergence accuracy, and avoid singular problems; an integral end face sliding surface is designed as(22)s=z3+η∫0tz3l/ndw,where η>0, l>0, n>0, and 1<l/n<2.
According toz3=x3−α2 and x˙3=1/Lu−KB/Lx2−R/Lx3, s˙ can be obtained as(23)s˙=1Lu−KBx2−Rx3−Lα˙2+Lηz3l/n.
The function−KBx2−Rx3−Lα˙2+Lηz3l/n is unknown due to the unknown parameter KB, and thus the RBFNNs are used to approximate it(24)−KBx2−Rx3−Lα˙2+Lηz3l/n=ω3∗Tφ3X3+ε3X3.
Therefore,s˙ becomes(25)s˙=1Lu+ω3∗Tφ3X3+ε3X3.
Consider the Lyapunov function(26)V3=V2+L2s2+12δ3W˜32,where δ3 is a positive design parameter. W˜3=W3∗−W^3 is the estimation error with W^3 being the estimation of W3∗, and W3∗=ω3∗2.
Taking the derivative ofV3 along time, one has(27)V˙3=V˙2+Lss˙−1δ3W˜3W^˙3.
Substituting (21) and (25) into (27), it has(28)V˙3≤−γ1e21−e2−γ2z22−12σ2W˜22+z2z3+12σ2W2∗2+12ε¯22+ξ2−1δ3W˜3W^˙3+su+ω3∗Tφ3X3+ε3X3.
Using Young’s inequality, we have(29)sω3∗Tφ3X3≤s2W3φ3TX3φ3X34ξ3+ξ3sε3X3≤12s2+12ε¯32,and then, we can deduce that(30)V˙3≤−γ1e21−e2−γ2z22−12σ2W˜22+12σ2W2∗2+12ε¯22+12ε¯32+su+sW^3φ3TX3φ3X34ξ3+12s+z2z3s+W˜3s2φ3TX3φ3X34ξ3−1δ3W^˙3+ξ2+ξ3.
Design the following actual controller and adaptive law:(31)u=−γ3s−12s−z2z3s−sW^3φ3TX3φ3X34ξ3,(32)W^˙3=δ3s2φ3TX3φ3X34ξ3−σ3W^3,where γ3 and σ3 are the positive parameters.
On the basis of (31) and (32), we get(33)V˙3≤−γ1e21−e2−γ2z22−γ3s2−12∑j=23σjW˜j2+12∑j=23σjWj∗2+12∑j=23ε¯j2+ξj.
LetC=minγ1,γ2/M,γ3/L,σ2δ2,σ3δ3, and D=1/2∑j=22σjωj∗Tωj∗+ε¯j2+ξj, from which C and D are the constants.
Furthermore, consider the inequality−γ1e2/1−e2≤−γ1log1/1−e2. Then, V˙3 can be described as(34)V˙3≤−CV3+D.
Based on the above analysis, the following theorem is obtained.Theorem 1.
Consider the manipulator system (1); based on Assumptions 1-2, the controllers (10), (19), (31) and adaptive laws (20), (32) are designed such that all signals in the resulted system are bounded, the system output can track the reference signal to a compact set, and it does not violate its constraint.Proof.
. Select the Lyapunov function as follows:(35)V=12log11−e2+M2z22+L2s2+12∑j=231δjW˜j2.
According to the above analysis, it is easily known that(36)V˙≤−CV+D.
The above inequality is integrated on both sides over0,t, and then the following holds:(37)Vt≤V0e−Ct+DC.
According to the definition ofV, we can know that e, z2, z3, ω˜2, and ω˜3 are all bounded for the bounded initial condition. When z1>0, e=e1, so e1 is bounded. Since e1=z1/kz1 and kz1 is a bounded function, z1 is also bounded. Based on z1=x1−yd and the boundness of yd, x1 is bounded. According to the definition of α1, it is easy to know α1 is bounded. Furthermore, on the basis of z2=x2−α1, x2 is also bounded. Similarly, α2, x3, and u are bounded. Thus, all signals in the resulted system are bounded.
In addition, from (35) and (36), one has(38)12log11−e2≤V0e−Ct+DC.
Furthermore, whenz1>0, e=e1=z1/kz1, and then(39)z1≤kz11−e−V0e−Ct−D/C.
Similarly, whenz1≤0, we have(40)z1≤kz21−e−V0e−Ct−D/C.
Therefore, whent⟶∞, the tracking error z1 converges to the compact set that(41)z1≤kz1−e−D/C,where kz=minkz1,kz2. That is to say the system output can track the reference signal to a compact set.
Furthermore, fromz1=x1−yd, one has −z1−yd≤x1≤z1+yd. As stated at the beginning of this paper, we can deduce that when z1>0,x1≤k¯t and when z1≤0, x1≥k¯t. Thus, the system output satisfies the constraint condition.This completes the proof of Theorem1.Remark 1.
It is worth noting that equation (41) actually provides a direction on how to select some key design parameters: on the one hand, increasing γj,j=1,2,3, σj,j=2,3, and δj,j=2,3 may help increase C, which further reduces D/C. On the other hand, decreasing ξj,j=2,3, may help reduce D and reduces D/C. The smaller D/C leads to larger e−D/C, which implies that the tracking error z1 can be smaller, and then the tracking performance is better.


## 4. Simulation Example

In this section, the manipulator system (1) is considered, where(42)M=Jkτ+mL023kτ+M0L02kτ+2M0R025kτ,B=B0kτ,N=mL0G2kτ+M0L0Gkτ,with J=0.001625kg·m2, kτ=0.9N·m/A, m=0.506kg, L0=0.305m, M0=0.434kg, R0=0.022m, B0=0.01625N·m·s/rad, and G=9.8, L=0.225H.In this simulation, the desired signal is chosen asydt=sint, the constraint functions are k¯t=1.3sint+0.5 and k¯t=1.3sint−0.5. The initial values are selected as x10=0.001, x20=0.012, x30=0.001, W20=0.01, and W30=1, and the other related design parameters are chosen as γ1=8, γ2=8, γ3=22, γ0=0.3, δ2=δ3=4, σ2=σ3=3, and ξ2=ξ3=1.Figures1–5 show the simulation results of the adaptive sliding mode control strategy designed in this paper. Figure 1 describes the curves of system state x1 (i.e., yt), the reference signal ydt, and the constraint bounds of x1. It can be seen that the system has a good tracking performance. At the same time, the output of the system is within the constraint range, i.e., it satisfies the constraint condition. Figures 2 and 3 show the trajectories of states x2 and x3, respectively. The adaptive laws for W^2 and W^3 are given in Figure 4. According to Figures 2–4, it is easily known that they are bounded. Figure 5 expresses the trajectory of the control input u, which is also bounded. Thus, the effectiveness of the proposed method is well explained.Figure 1
Trajectories ofx1, yd and the constraint bounds of x1.Figure 2
The trajectory ofx2.Figure 3
The trajectory ofx3.Figure 4
Trajectories of the update laws.Figure 5
The trajectory of control inputu.


## 5. Conclusion

A sliding mode-based adaptive neural network control strategy is proposed for a manipulator system with output constraint. Combining the sliding mode control method with backstepping technique, an adaptive controller is designed, which not only solves the output constraint issue of the considered system, but also makes the error between system output and the reference signal to converge to a small compact set. Moreover, the output satisfies the constraint condition. Finally, the effectiveness of the proposed approach is illustrated by the simulation example.



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