# A New Adaptive Controller for Nonlinear Systems with Uncertain Virtual Control Gains

Authors: Wei Xiao; Zhixiang Yin; Tianyue Zhou; Zi Ye; Xiaoqi Yang
Journal: Complexity
 (2022)

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## Abstract

This paper addresses the adaptive asymptotic tracking control problem for nonlinear systems whose virtual control gains are unknown nonlinear functions of system states. Only in the first step, the Nussbaum gain technique is utilized to handle the uncertain virtual control gain. In the remaining steps, virtual control gains are dealt with by constructing novel control laws without the approximation of the uncertain nonlinear functions and external disturbances by neural networks or fuzzy logic. New adaptive laws are defined to compensate for unknown virtual control gains, uncertain parameters, and external disturbances. Finally, an adaptive tracking controller is designed and applied to the control of a 3-order robot system, which guarantees the boundedness of all the signals in the closed-loop system and asymptotic stability of the tracking error.


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## Body

## 1. Introduction

The tracking control has received considerable attention for the purpose of ensuring the output of the system is tracking a desired trajectory. To deal with the uncertainties in nonlinear systems, one of the most popular methods, adaptive control, has been introduced for controller design. There have been many related results in this area [1–11]. An adaptive data-driven controller was designed for nonlinear systems using goal representation heuristic and dynamic programming [4]. In view of unknown nonlinear fractional-order systems, an adaptive control scheme was proposed [5]. For nonlinear systems with dead-zone and actuator failure, two novel finite-time adaptive tracking controllers were developed [7–9]. By introducing a new performance function, [10, 11] they constructed two adaptive tracking controllers via prescribed performance control and funnel control, respectively. It needs to be emphasized that the VCGs (virtual control gains) of systems in the above papers are assumed to be known.However, the VCGs are uncertain for many actual systems. In response to this challenge, the Nussbaum gain technique was first proposed [12]. There have been many achievements for nonlinear systems, whose VCGs were unknown constants [13–16]. Two adaptive control strategies were given for unknown nonlinear SISO (single input, single output) systems and stochastic nonlinear systems based on state observers [13, 14]. In consideration of unmodeled dynamics and an unknown dead-zone, an improved control strategy was addressed [15]. In view of nonlinear systems with actuator faults and state/input constraints, a controller was designed based on dynamic surface and Nussbaum gain [16]. Taking into account uncertain time-varying VCGs, an improved adaptive control method was given [17], which was further improved [18] such that the Nussbaum gain technique was applied to nonlinear systems whose virtual control gains were unknown nonlinear functions of system states. To handle time-varying uncertain control gains, new Nussbaum functions were defined [19]. Two adaptive robust control schemes were proposed [20] for nonlinear systems with certain and uncertain signs of VCGs. Aiming at nonlinear systems with uncertain dead-zone output, an adaptive fuzzy control scheme was designed using a state observer [21]. To achieve full state constraints, adaptive tracking control was studied by the Barrier Lyapunov function [22]. Considering MIMO (multiple input, multiple output) nonlinear systems with input saturations and uncertain control gains, two adaptive NN (neural network) controllers were constructed [23, 24]. A dynamic surface control strategy was given for nonlinear systems with uncertain VCGs using fuzzy logic [25].Fuzzy logic and neural networks were also usually applied to handle the unknown VCGs. The VCGs were supposed to have known upper and lower bounds and were approximated by fuzzy logic, and then the adaptive fuzzy tracking controllers were constructed for strict feedback [26] and switched nonlinear systems [27]. Two adaptive fuzzy control schemes were proposed for nonaffine [28] and nonstrict-feedback [29] nonlinear systems based on funnel control and dynamic surface control, respectively. Several adaptive NN tracking controllers were constructed for different-type uncertain nonlinear systems [30–32]. Besides, by utilizing z−z2/z2+δ2<δ with δ>0 and z being any real number, two controllers were designed by invoking the lower bounds of unknown VCGs [33, 34]. By describing UVCCs in terms of their known and unknown parts, a new adaptive tracking controller was constructed and applied to the attitude control of quadrotors [35]. More research results on nonlinear systems with unknown VCGs can be seen in [36, 37].In summary, only the boundedness of tracking error can be obtained in the aforementioned achievements. The better tracking performance of asymptotic stability was not researched for nonlinear systems with uncertain VCGs being functions of system states. Recently, an adaptive asymptotic tracking control method was presented in [38]. There were also some others that dealt with unknown nonlinear functions and external disturbances by neural networks. However, according to the literature review, there has been no work reported on adaptive asymptotic tracking controllers for nonlinear systems with VCGs being uncertain state functions without using fuzzy logic or neural networks.Inspired by the mentioned research achievements, the paper is concerned with adaptive tracking control for nonlinear systems with external disturbances, whose VCGs are uncertain functions of system states. An adaptive tracking controller is designed in a stepwise strategy and applied to the control of a robot system, which ensures both the boundedness of all the signals in the closed-loop system and the asymptotic stability of the tracking error. The effectiveness and practicability of the developed control strategy are validated by both theoretical analysis and simulations.The paper possesses the following features:(1)
Different from the control methods based on the Nussbaum gain technique [13–25], a Nussbaum function is only employed in the first step and an improved adaptive law for the Nussbaum variable is defined such that the asymptotic stability of the tracking error is achieved. This is the main improvement on [13–32], which can only guarantee that the tracking error is bounded. Without using fuzzy logic or neural networks [26–32], and the lemma of z−z2/z2+δ2<δ [33, 34], novel control laws are constructed in the remaining steps without the bounds of the unknown VCGs.(2)
To compensate for unknown VCGs, unknown external disturbances, and parameter uncertainties, new adaptive laws are adopted so that the assumption of unknown VCGs can be relaxed as in Assumption2, where unknown VCGs only have unknown lower and upper bounds [22, 25, 26, 29, 31, 34, 38]. Unknown VCGs are assumed to be bound by with known upper and lower bounds [14, 18], known lower bounds [27, 30], and known upper bounds [13, 16]. In [32], unknown VCGs are assumed to be strictly either positive or negative. All the above assumptions are more restrictive. Therefore, the proposed controller can be suitable for more nonlinear systems.(3)
The reference signal is only required to be differentiable and the assumption is written as Assumption1, which is less restrictive than the related assumption that the reference signal and its time derivatives up to the n-th order are continuous and bounded [13, 17, 18, 20–22, 26, 28–31, 33, 38].The outline of this paper is presented below. In Section2, the preliminaries and problem formulation are described. The design of an adaptive control algorithm and stability analysis are given in Section 3. Sections 4 and 5 provide the simulation examples of a second-order nonlinear system and a robotic system, respectively, and the summary of the paper.


## 2. Problem Formulation and Preliminaries

The uncertain nonlinear system is considered as follows:(1)x˙i=gix¯ixi+1+θTφix¯i+dit,i=1,…,n−1,x˙n=gnx¯nu+θTφnx¯n+dnt,y=x1,where x¯i=x1,…,xiT∈Ri,u∈R,y∈R are the state vector, input, and output of the system, respectively, i=1,…,n. gix¯i∈R denotes the VCG, which is an unknown smooth non-zero nonlinear function. θ∈Rs1 is an unknown parameter vector, and φix¯i∈Rs1. denotes a nonlinear function, which is known and smooth, s1 is a positive integer; dit∈R is an unknown bounded external disturbance.Assumption 1 (see [24]).
The ideal output vectoryd1t=ydt,y˙dt.T is absolutely continuous and bounded.Assumption 2 (see [22]).
Let us assume that the sign ofgix¯i. does not change for all xi and gix¯i. satisfies(2)0<g¯im≤gix¯i≤g¯iM<+∞,with g¯im and g¯iM being unknown positive constants.
It is reasonable thatgix¯i. is bigger than a positive constant g¯im due to the system controllable condition of gix¯i being away from 0, which could eliminate the controller singularity problem and has been given in many control strategies [22, 25, 26, 29, 31, 34, 38] and the references therein. It is worth noting that the VCGs in the above references possess the following synthesis: the lower and upper bounds of the VCGs are only used in the procedures of the controller design and are not required in the controller, which means that g¯im and g¯iM can be unknown. Without loss of generality, we assume that gix¯i>0.Assumption 3.
There exists an unknown positive constantd¯i satisfying(3)dit≤d¯i.Definition 1 (see [12]).
A continuous functionNς is named as Nussbaum function when the following conditions hold:(4)limε⟶Esup1ε∫0εNςdς=+∞,limε⟶∞inf1ε∫0εNςdς=−∞.
There are many Nussbaum-type functions, such aseς2sinπ/2ς, eς2cosπ/2ς, ς2sinς, and ς2cosς.Lemma 1 (see [18]).
Letgx∈gl,gu with gl, gu being non-zero real numbers and 0∉gl,gu, Nς is an even Nussbaum function, zt is an absolutely continuous function, and xt is another function. If there exist l>0 and a real constant γ such that a function Vt should be subject to(5)Vt≥e−lt∫0tgxτNζτ+1ζ˙τeltdτ+γ,∀t∈0,tf,then ζt, ∫0tgxτNζτ+1ζ˙τeltdτ and Vt are all bounded on 0,tf.
According to Proposition 2 in [39], tf can be extended to +∞, when the solution of the closed-loop system is bounded.


## 3. Design and Analysis of the Controller

The procedure of the adaptive controller design and stability analysis are presented.Firstly, the common coordinate transformation is introduced as(6)z1=x1−yd,zi=xi−αi−1,i=2,…,nwith αi−1 being the intermediate control law\designed later.The system (1) is changed into(7)z˙1=g1x1x2+θTφ1x1+d1t−y˙d,z˙i=gix¯ixi+1+θTφix¯i+d2t−α˙i−1,i=2,…,n−1,z˙n=gnx¯nu+θTφnx¯n+dnt−α˙n−1,y=x1.In order to estimate the unknown VCGsgix¯i, unknown parameter vector θ, and uncertain external disturbances dit, define(8)ϖi∗=Γi,i=1,2,…,n,(9)ϖ˜i=ϖi∗−ϖi,where ⋅ is the 2-norm of vectors, Γi is an unknown bounded vector given later; ϖ˜i is the approximation error with ϖi being the estimation of ϖi∗.In what follows, the controller will be constructed in a stepwise strategy via backstepping. For convenience, letgi=gix¯i, φi=φix¯i, and di=dit.Step 1.
We choose the first Lyapunov function candidate as(10)V1=12g¯1mz12+12q1ϖ˜12,where q1 is a positive constant.
DifferentiatingV1 and invoking (9) produce(11)V˙1=1g¯1mz1z˙1−1q1ϖ˜1ϖ˙1∗.
Substitutingz˙1 in (7) into (11) and considering (6) yield(12)V˙1=z1g¯1mg1z2+g1α1+θTφ1+d1−y˙d−1q1ϖ˜1ϖ˙1.
Based on Young inequality, we have(13)g1g¯1mz1z2≤12g¯1Mg¯1mz12+12g¯1Mg¯1mz22,1g¯1mθTz1φ1≤12a1θTθg¯1mφ1Tφ1z12+a12g¯1m,1g¯1mz1d1≤12b11g¯1md12z12+b12g¯1m≤12b11g¯1md¯12z12+b12g¯1m,−1g¯1mz1y˙d≤12c11g¯1my˙d2z12+c12g¯1m≤12c11g¯1my˙dM2z12+c12g¯1m,where a1, b1, and c1 are positive constants and y˙dM is the maximum of y˙d and unknown.
By invoking (13), V˙1 is described as(14)V˙1≤z1g¯1mg1α1+12g¯1Mg¯1mz12+12g¯1Mg¯1mz22+12a1θTθg¯1mφ1Tφ1z12+12b11g¯1md¯12z12+a12g¯1m+b12g¯1m+12c11g¯1my˙dM2z12+c12g¯1m−1q1ϖ˜1ϖ˙1.
Let us define(15)Γ1=g¯1Mg¯1m,θTθg¯1m,1g¯1md¯12,y˙dM2g¯1m,(16)Υ1=12,12a1φ1Tφ1,12b1,12c1.
Substituting (15) and (16) into (14) results in (17)V˙1≤g1g¯1mz1α1+12g¯1Mg¯1mz22+Γ1ϒ1Tz12+a12g¯1m+b12g¯1m+c12g¯1m−1q1ϖ˜1ϖ˙1=g1g¯1mz1α1+12g¯1Mg¯1mz22+ϖ1∗ϒ1z12+a12g¯1m+b12g¯1m+c12g¯1m−1q1ϖ˜1ϖ˙1.
In the following, the virtual control signalα1 is defined as(18)α1=Nςα¯1,(19)α¯1=k1z1+ϖ1ϒ1z1,where Nς is a Nussbaum-type even function and k1 is a positive design parameter.
ς is adjusted according to the following law:(20)ς˙=λz1α¯1,where λ>0 is a design parameter.Remark 1.
From (19) and the definition of α¯1, ϖ1, it can be known that all the terms of ς˙ are nonnegative, which is the key to proving the asymptotic stability of the tracking error later.
By invoking (18) and (20), V˙1 is shown as(21)V˙1≤g1g¯1mz1Nςα¯1+z1α¯1−z1α¯1+ϖ1∗ϒ1z12+a12g¯1m+b12g¯1m+c12g¯1m+12g¯1Mg¯1mz22−1q1ϖ˜1ϖ˙1=−k1z12+g1λg¯1mNςς˙+ς˙λ−ϖ˜1q1ϖ˙1−q1ϒ1z12+12g¯1Mg¯1mz22+a12g¯1m+b12g¯1m+c12g¯1m.
The adaptive lawϖ˙1 is defined as(22)ϖ˙1=q1ϒ1z12−σ1ϖ1.
Withσ1>0 being a design parameter.
Replacingϖ˙1 in (21) by (22), V˙1 can be given as(23)V˙1≤−k1z12+g1λg¯1mNςς˙+1λς˙+σ1q1ϖ˜1ϖ1+12g¯1Mg¯1mz22+a12g¯1m+b12g¯1m+c12g¯1m.Step 2.
A Lyapunov function candidate is given as(24)V2=V1+12g¯2mz22+12q2ϖ˜22,where q2>0 is a constant.
Based on (23) and z˙2 in (7), we have (25)V˙2=V˙1+1g¯2mz2z˙2−1q2ϖ˜2ϖ˙2≤−k1z12+g1λg¯1mNςς˙+1λς˙+σ1q1ϖ˜1ϖ1+12g¯1Mg¯1mz22+a12g¯1m+b12g¯1m+1g¯2mz2g2x3+θTφ2+d2−α˙1+c12g¯1m−1q2ϖ˜2ϖ˙2.
From (18), α˙1 is expressed as(26)α˙1=∂α1∂x1x˙1+∂α1∂ydy˙d+∂α1∂ςς˙+∂α1∂ϖ1ϖ˙1.
Substituting (26) into (25) leads to (27)V˙2≤−k1z12+g1λg¯1mNςς˙+1λς˙+σ1q1ϖ˜1ϖ1+12g¯1Mg¯1mz22+a12g¯1m+b12g¯1m+c12g¯1m+1g¯2mz2g2z3+α2+θTφ2+d2−1q2ϖ˜2ϖ˙2−1g¯2mz2∂α1∂x1g1x2+θTφ1−1g¯2mz2∂α1∂ydy˙d+∂α1∂ςς˙+∂α1∂ϖ1ϖ˙1.
According to Young inequality, the following inequalities hold(28)g2g¯2mz2z3≤12g¯2Mg¯2mz22+12g¯2Mg¯2mz32,1g¯2mz2d2≤12b21g¯2md¯22z22+b22g¯2m,1g¯2mθTz2φ2−∂α1∂x1φ1≤12a2θTθg¯2mφ2−∂α1∂x1φ12z22+a22g¯2m,−1g¯2mz2∂α1∂ydy˙d≤12c21g¯2my˙dM2∂α1∂yd2z22+c22g¯2m,−1g¯2mz2∂α1∂x1g1x2≤12l21g¯2mg¯1M2x22∂α1∂x12z22+l22g¯2m,−z2g¯2m∂α1∂ςς˙+∂α1∂ϖ1ϖ˙1≤12m21g¯2m∂α1∂ςς˙+∂α1∂ϖ1ϖ˙12z22+m22g¯2m,where a2, b2, c2, l2, and m2 are positive constants.
Substituting (28) into (27) leads to(29)V˙2≤−k1z12+g1λg¯1mNςς˙+1λς˙+σ1q1ϖ˜1ϖ1+12g¯1Mg¯1mz22+∑i=12ai+bi+ci2g¯im+g2g¯2mz2α2+12g¯2Mg¯2mz22+12g¯2Mg¯2mz32−1q2ϖ˜2ϖ˙2+12a2θTθg¯2mφ2−∂α1∂x1φ12z22+12b21g¯2md¯22z22+12c21g¯2my˙dM2∂α1∂yd2z22+12l2g¯1M2g¯2mx22∂α1∂x12z22+l22g¯2m+12m2z22g¯2m∂α1∂ςς˙+∂α1∂ϖ1ϖ˙12+m22g¯2m.
Γ2 and ϒ2 are written as(30)Γ2=g¯1Mg¯1m+g¯2Mg¯2m,θTθg¯2m,d¯22g¯2m,y˙dM2g¯2m,g¯1M2g¯2m,1g¯2m,(31)ϒ2=12,φ2−∂α1/∂x1φ122a2,12b2,∂α1/∂yd22c2,ϒ2∗,ϒ2∗=∂α1/∂x1x222l2,∂α1/∂ςς˙+∂α1/∂ϖ1ϖ˙122m2.
Invoking (30) and (31) produces(32)V˙2≤−k1z12+g1λg¯1mNςς˙+1λς˙+σ1q1ϖ˜1ϖ1+∑i=12ai+bi+ci2g¯im,+l22g¯2m+m22g¯2m+g2g¯2mz2α2+ϖ2∗ϒ2z22+12g¯2Mg¯2mz32−1q2ϖ˜2ϖ˙2.
Designing the following virtual control signal as(33)α2=−k2z2−ϖ2ϒ2z2.and substituting it into (32) give(34)V˙2≤−k1z12+g1λg¯1mNςς˙+1λς˙+σ1q1ϖ˜1ϖ1−g2g¯2mk2z22+ϖ2ϒ2z22+ϖ2∗ϒ2z22+∑i=12ai+bi+ci2g¯im+l22g¯2m+m22g¯2m+12g¯2Mg¯2mz32−1q2ϖ˜2ϖ˙2,where k2>0 is a design parameter.Remark 2.
. Based on (32), Assumption 2, and the definition of ϖ2, it is easy to check that all the items of −g2/g¯2mk2z22+ϖ2ϒ2z22 are nonpositive, and g2/g¯2m≥1, which means that g2/g¯2m can be replaced by 1 in the controller design.
V˙2 is changed to(35)V˙2≤−∑i=12kizi2+g1λg¯1mNςς˙+1λς˙+σ1q1ϖ˜1ϖ1+∑i=12ai+bi+ci2g¯im+l22g¯2m+m22g¯2m−1q2ϖ˜2ϖ˙2+q2ϒ2z22+12g¯2Mg¯2mz32.
We construct an adaptive law as(36)ϖ˙2=q2ϒ2z22−σ2ϖ2,where σ2 is a positive constant.
Substituting (36) into (35) produces(37)V˙2≤−∑i=12kizi2+g1λg¯1mNςς˙+1λς˙+∑i=12σiqiϖ˜iϖi+∑i=12ai+bi+ci2g¯im+l22g¯2m+m22g¯2m+12g¯2Mg¯2mz32.
Stepj3≤j≤n−1: For the j th subsystem of (6), a Lyapunov function candidate is iteratively chosen from the previous step.(38)Vj=Vj−1+12g¯jmzj2+12qjϖ˜j2,where qj>0 is a design parameter.
DifferentiatingVj produces(39)V˙j=V˙j−1+1g¯jmzjz˙j−1qjϖ˜jϖ˙j.
By an induction argument,V˙j−1 is deduced as(40)V˙j−1≤−∑i=1j−1kizi2+g1λg¯1mNςς˙+1λς˙+∑i=1j−1ai+bi+ci2g¯im+∑i=1j−1σiqiϖ˜iϖi+∑i=2j−1li+mi2g¯im+12g¯j−1,Mg¯j−1,mzj2,where ai, bi, ci, li, mi, σi, qi are all positive constants.
Substituting (40) into (39) gives(41)V˙j≤−∑i=1j−1kizi2+g1λg¯1mNςς˙+1λς˙+∑i=1j−1σiqiϖ˜iϖi+∑i=1j−1ai+bi+ci2g¯im+∑i=2j−1li+mi2g¯im+12g¯j−1,Mg¯j−1,mzj2+zjg¯jmgjzj+1+gjαj+θTφj+dj−α˙j−1−1qjϖ˜jϖ˙j,where(42)α˙j−1=∑i=1j−1∂αj−1∂xix˙i+∂αj−1∂ydy˙d+∂αj−1∂ςς˙+∂αj−1∂ϖj−1ϖ˙j−1,ϖ˙j−1=qj−1ϒj−1zj−12−σj−1ϖj−1.
Furthermore,V˙j is expressed as(43)V˙j≤−∑i=1j−1kizi2+g1λg¯1mNςς˙+1λς˙+∑i=1j−1σiqiϖ˜iϖi+∑i=1j−1ai+bi+ci2g¯im+∑i=2j−1li+mi2g¯im+12g¯j−1,Mg¯j−1,mzj2+gjg¯jmzjzj+1+gjg¯jmzjαj+zjg¯jmdj−zjg¯jm∑i=1j−1∂αj−1∂xigixi+1−zjg¯jm∂αj−1∂ydy˙d−zjg¯jm∂αj−1∂ςς˙+∂αj−1∂ϖj−1ϖ˙j−1+zjg¯jmθTφj−∑i=1j−1∂αj−1∂xiφi−1qjϖ˜jϖ˙j.
Based on the completion of squares, one has(44)gjg¯jmzjzj+1≤12g¯jMg¯jmzj2+12g¯jMg¯jmzj+12,1g¯jmzjdj≤12bj1g¯jmd¯j2zj2+bj2g¯jm,θTg¯jmzjφj−∑i=1j−1∂αj−1∂xiφi≤12ajθTθg¯jmφj−∑i=1j−1∂αj−1∂xiφi2zj2+aj2g¯jm,−1g¯jmzj∂αj−1∂ydy˙d≤12cj1g¯jmy˙dM2∂αj−1∂yd2zj2+cj2g¯jm−1g¯jmzj∑i=1j−1∂αj−1∂xigixi+1,≤12lj1g¯jmg¯jM∗∑i=1j−1∂αj−1∂xixi+12zj2+lj2g¯jm−zjg¯jm∂αj−1∂ςς˙+∂αj−1∂ϖj−1ϖ˙j−1,≤12mjzj2g¯jm∂αj−1∂ςς˙+∂αj−1∂ϖj−1ϖ˙j−12+mj2g¯jm,where aj, bj, cj, lj and mj are positive design parameters and g¯jM∗=maxg¯1M2,…,g¯j−1,M2.
Invoking (44), V˙j is rewritten as(45)V˙j≤−∑i=1j−1kizi2+g1λg¯1mNςς˙+1λς˙+∑i=1j−1σiqiϖ˜iϖi+∑i=1jai+bi+ci2g¯im+∑i=2jli+mi2g¯im+12g¯j−1,Mg¯j−1,mzj2+12g¯jMg¯jmzj2+12g¯jMg¯jmzj+12+gjg¯jmzjαj+12bj1g¯jmd¯j2zj2+12ajθTθg¯jmφj−∑i=1j−1∂αj−1∂xiφi2zj2+12cj1g¯jmy˙dM2∂αj−1∂yd2zj2+12lj1g¯jmg¯jM∗∑i=1j−1∂αj−1∂xixi+12zj2−1qjϖ˜jϖ˙j+12mj1g¯jm∂αj−1∂ςς˙+∂αj−1∂ϖ1ϖ˙12zj2.
Similarly, we define(46)Γj=g¯j−1,Mg¯j−1,m+g¯jMg¯jm,θTθg¯jm,d¯j2g¯jm,y˙dM2g¯jm,g¯jM∗g¯jm,1g¯jm,(47)ϒj=12,φj−∑i=1j−1∂αj−1/∂xiφi22aj,12bj,ϒj1∗,ϒj2∗,ϒj1∗=∂αj−1/∂yd22cj,∑i=1j−1∂αj−1/∂xixi+122lj,ϒj2∗=∂αj−1/∂ςς˙+∂αj−1/∂ϖj−1ϖ˙j−122mj.
By invoking (46) and (47), V˙j is shown as(48)V˙j≤−∑i=1j−1kizi2+g1λg¯1mNςς˙+1λς˙+∑i=1j−1σiqiϖ˜iϖi+∑i=1jai+bi+ci2g¯im+∑i=2jli+mi2g¯im+12g¯jMg¯jmzj+12+gjg¯jmzjαj+ϖi∗ϒjzj2−1qjϖ˜jϖ˙j.
Defining thej th virtual control law as(49)αj=−kjzj−ϖjϒjzj.and substituting it into (48) to yield(50)V˙j≤−∑i=1j−1kizi2+g1λg¯1mNςς˙+1λς˙+∑i=1j−1σiqiϖ˜iϖi+∑i=1jai+bi+ci2g¯im+∑i=2jli+mi2g¯im+12g¯jMg¯jmzj+12+ϖ˜jϒjzj2−1qjϖ˜jϖ˙j,where kj>0 is a constant.
Similar to (36), ϖ˙j can be specified as(51)ϖ˙j=qjϒjzj2−σjϖj.with σj>0 being a design parameter.
Then,V˙j is given as(52)V˙j≤−∑i=1j−1kizi2+g1λg¯1mNςς˙+ς˙λ+∑i=1jσiqiϖ˜iϖi+∑i=1jai+bi+ci2g¯im+∑i=2jli+mi2g¯im+g¯jM2g¯jmzj+12.
Stepn: following the similar procedure for (38) in Step j, the Lyapunov function candidate for the last step is designed as(53)Vn=Vn−1+12g¯nmzn2+12qnϖ˜n2.and V˙n is presented as(54)V˙n=1g¯nmznz˙n+V˙n−1−1qnϖ˜nϖ˙n,where qn>0 is a constant.
It follows from (52) that(55)V˙n−1≤−∑i=1n−1kizi2+g1λg¯1mNςς˙+1λς˙+∑i=1n−1σiqiϖ˜iϖi+∑i=1n−1ai+bi+ci2g¯im+∑i=2n−1li+mi2g¯im+12g¯n−1,Mg¯n−1,mzn2,where ai, bi, ci, li, mi, σi, and qi are all positive constants.
V˙n is given as(56)V˙n≤−∑i=1n−1kizi2+g1λg¯1mNςς˙+1λς˙+∑i=1n−1σiqiϖ˜iϖi+∑i=1n−1ai+bi+ci2g¯im+∑i=2n−1li+mi2g¯im+1g¯nmzngnu+θTφn+dn−α˙n−1+12g¯n−1,Mg¯n−1,mzn2−1qnϖ˜nϖ˙n,where(57)α˙n−1=∑i=1n−1∂αn−1∂xix˙i+∂αn−1∂ydy˙d+∂αn−1∂ςς˙+∂αn−1∂ϖn−1ϖ˙n−1,ϖ˙n−1=qn−1Υn−1zn−12−σn−1ϖn−1.
Furthermore,V˙n is reformulated as(58)V˙n≤−∑i=1n−1kizi2+g1λg¯1mNςς˙+1λς˙+∑i=1n−1σiqiϖ˜iϖi+∑i=1n−1ai+bi+ci2g¯im+∑i=2n−1li+mi2g¯im+12g¯n−1,Mg¯n−1,mzn2+gng¯nmznu+zndn−zng¯nm∑i=1n−1∂αn−1∂xigixi+1−zng¯nm∂αn−1∂ςς˙+∂αn−1∂ϖn−1ϖ˙n−1−zng¯nm∂αn−1∂ydy˙d+zng¯nmθTφn−∑i=1n−1∂αn−1∂xiφi−1qnϖ˜nϖ˙n.
Repeating the similar procedures inStep j, we have (59)θTg¯nmznφn−∑i=1n−1∂αn−1∂xiφi≤12anθTθg¯nmφn−∑i=1n−1∂αn−1∂xiφi2zn2+an2g¯nm,1g¯nmzndn≤12bn1g¯nmd¯n2zn2+bn2g¯nm,−1g¯nmzn∂αn−1∂ydy˙d≤12cn1g¯nmy˙dM2∂αn−1∂yd2zn2+cn2g¯nm,−1g¯nmzn∑i=1n−1∂αn−1∂xigixi+1≤12ln1g¯nmg¯nM∗∑i=1n−1∂αn−1∂xixi+12zn2+ln2g¯nm,−zng¯nm∂αn−1∂ςς˙+∂αn−1∂ϖn−1ϖ˙n−1≤zn22g¯nmmn∂αn−1∂ςς˙+∂αn−1∂ϖn−1ϖ˙n−12+mn2g¯nm,(60)Γn=g¯n−1,Mg¯n−1,m,θTθg¯nm,d¯n2g¯nm,y˙dM2g¯nm,g¯nM∗g¯nm,1g¯nm,(61)ϒn=12,φn−∑i=1n−1∂αn−1/∂xiφi22an,12bn,Υn1∗,Υn2∗,ϒn1∗=∂αn−1/∂yd22cn,∑i=1n−1∂αn−1/∂xixi+122ln,ϒn2∗=∂αn−1/∂ςς˙+∂αn−1/∂ϖn−1ϖ˙n−122mn,where an, bn, cn, ln, and mn are positive design parameters and g¯nM∗=maxg¯1M2,…,g¯n−1,M2.
Substituting (60) and (61) into (58) results in (62)V˙n≤−∑i=1n−1kizi2+g1λg¯1mNςς˙+1λς˙+∑i=1n−1σiqiϖ˜iϖi+∑i=1nai+bi+ci2g¯im+∑i=2nli+mi2g¯im+gng¯nmznu+ϖn∗Υnzn2−1qnϖ˜nϖ˙n.
Similar to (49) and (51), the real control law and n th adaptive law can be defined, respectively, as follows: (63)u=−knzn−ϖnϒnzn,and(64)ϖ˙n=qnϒnzn2−σnϖn,where kn and σn are positive design parameters.
By some direct calculations,V˙n can be described as(65)V˙n≤−∑i=1nkizi2+g1λg¯1mNςς˙+1λς˙+∑i=1nσiqiϖ˜iϖi+∑i=1nai+bi+ci2g¯im+∑i=2nli+mi2g¯im.
According to the above controller design, the main conclusion is summarized as
Theorem 1. For system (1) with Assumptions1–3, there exists an adaptive controller with the adaptive laws in (21), (35), (51), (65), and real control signal u in (64), which can guarantee the boundedness of all the signals in the closed-loop system for t∈0,+∞ and asymptotic convergence of the tracking error.Proof: .
sinceϖ˜iϖi=ϖ˜iϖi∗−ϖ˜i≤1/2ϖi∗2−1/2ϖ˜i2, it is easy to check that(66)∑i=1nσiqiϖ˜iϖi=∑i=1nσiqiϖ˜iϖi∗−ϖ˜i≤∑i=1nσi2qiϖi∗2−∑i=1nσi2qiϖ˜i2.
Substituting (66) into (65) leads to(67)V˙n≤−∑i=1nkizi2+g1λg¯1mNςς˙+1λς˙+∑i=1nσi2qiϖi∗2−∑i=1nσi2qiϖ˜i2+∑i=1nai+bi+ci2g¯im+∑i=2nli+mi2g¯im≤−κVn+ι+g1λg¯1mNςς˙+1λς˙.with(68)κ=min2k1,…,2kn,σ1,…,σn,ι=∑i=1nai+bi+ci2g¯im+∑i=2nli+mi2g¯im+∑i=1nσi2qiϖi∗2.
Moving−κVn to the left-hand side and then multiplying both sides of (67) by eκt produce(69)dVneκtdt≤ιeκt+g1λg¯1mNςς˙eκt+1λς˙eκt.
Integrating (69) gives(70)Vnteκt−Vn0≤∫0tg1x1λg¯1mNςτς˙τeκτdτ+∫0tς˙τλeκτdτ+ικeκt−1.
By some direct calculations, (70) is described as(71)Vnt≤e−κtλ∫0tg1x1g¯1mNςτ+1ς˙τeκτdτ+γ.where γ=Vn0+ι/κ is a positive constant.
By virtue of Lemma1, it is shown that Vnt., Nςτ., ςτ are bound for t∈0,+∞. Therefore, the boundedness of all the signals in the closed-loop system is certified for t∈0,+∞.
Furthermore, using the definition ofς˙, it can be seen that(72)ς˙=λk1z12+λϖ1ϒ1z12≥λk1z12.
Based on the boundedness ofςt, the following inequality holds.(73)limt⟶+∞∫0tλk1z12dτ≤limt⟶+∞ςt−ς0<+∞.
From the definition ofz˙1 in (6) and the boundedness of x1, x2, and y˙d, it is easily shown that z˙1 is bounded.
Thus, by utilizing Barbalat’s Lemma, one has(74)limt⟶∞z1t=0.which means that the tracking error is asymptotically stable.
Control parameters selection. The selection ranges of σi and ki are wide. The bigger the parameters σi and ki, the faster the asymptotic convergence and response. Besides, λ is the other key parameter, which affects the asymptotic convergence of the tracking error. According to the simulation tests (see Figure 1), the bigger λ the faster the transient process and the smaller the steady-state errors. But it cannot be too big. In addition, the choices of ai, bi, ci, and qi are flexible.Figure 1
The tracking errors with differentλ.


## 4. Simulation Results

To illustrate the feasibility of the developed adaptive control approach, an uncertain nonlinear system and a single-link robot are considered.


### 4.1. Simulation of a Second-Order Nonlinear System

Let us consider the strict-feedback nonlinear system as(75)x˙1=g1x1x2+θTφ1x1+d1t,x˙2=g2x¯2u+θTφ2x¯2+d2t,y=x1,where g1=1+0.1sinx1 with g1m=0.9 and g¯1M=1.1, g2=1+0.1cosx1sinx2 with g2m=0.9, and g¯2M=1.1, θ=1,1Tφ1=cosx1, φ2=sinx1cosx2, d1=e−tsint, d2=e−tcost.The desired trajectories are selected asyd1t=4sin0.2πt. and yd2t=2.5sin0.5t+sint. In order to see the controller performance under different initial conditions for the system states, the initial value of x1 is given as 1.5 and 3, respectively; ς0=1; and other initial conditions are chosen to be 0.The intermediate control signalα1, Nussbaum variable ς˙, and actual control law u of the developed control strategy are defined as (17), (19), and (64), where n=2, k1=5, and k2=10, a1=a2=b1=b2=c1=c2=l2=m2=10, λ=0.1, 0.5, 1, and 3. It is worth noting that the upper limit of λ is 3.5. Otherwise, the controller is easy to be ill-defined such that the states in the system are divergent.ϖ˙1 and ϖ˙2 are designed as (21) and (65), with n=2, q1=q2=1, and σ1=σ2=0.1.The proposed controller (PC) is compared with the adaptive fuzzy (AFC) [28] and adaptive controllers (AC) [34], respectively.The AFC is presented as(76)α1′=−k1z1+0.5z1+12a12z1θ1R1TR1,u′=−k2z2+0.5z2+12a22z2θ2R2TR2,θ.1=q12a12z12R1TR1−σ1θ1,θ.2=q22a22z22R2TR2−σ2θ2.where θi and Ri are defined in [28], ki, ai, qi, and σi are taken the same values as the PC, i=1 , 2.The AC is given as(77)α1=−z1α¯12z12α¯12+δ12,α¯1=k1z1+ϖ1ϒ1z1,u=−z2α¯22z22α¯22+δ22,α¯2=k2z2+ϖ2ϒ2z2,ϖ˙1=q1ϒ1z12−σ1ϖ1,ϖ˙2=q2ϒ2z22−σ2ϖ2,where ki, ai, bi, ci, l2, m2, qi, and σi are as same as the PC, i=1, 2. δ1, δ2 are positive constants, δ1=0.1, δ2=1.Figures.1–11 display the simulation results. In Figure 1, the tracking errors are drawn under different λ. It can be seen that the transient processes of the tracking errors are faster and the steady-state errors decrease as λ increase. Under different initial values x10=1.5 and 3, the comparisons of tracking performance for the desired trajectory yd1 between the AFC, AC, and PC are displayed in Figures 2–4. It is shown that the tracking performance with the PC is better and the tracking error is much smaller. Furthermore, the AFC can only ensure the boundedness of tracking errors, and the PC guarantees the asymptotic convergence of tracking errors, which is obvious in Figure 2. In addition, the absolute mean and root mean square values of tracking errors with the PC are smaller than the values with AFC and AC. Figures 5–7 draw the comparisons of tracking errors and tracking performance for the ideal trajectory yd2, and the similar conclusion can be obtained as Figures 2–4. The inputs u are given in Figure 8, which are large at the initial stage and converge rapidly to a range defined by ±10. The states x2 and adaptive parameters ϖ1∗, ϖ2∗ are presented in Figures 9 and 10, which are all bounded. Finally, Figure 11 shows the Nussbaum variables ς, which tends to positive constants from the initial values 1 as time goes on.Figure 2
The comparison of tracking errors between PC and AFC withyd1.Figure 3
The comparison of tracking errors PC and AC withyd1.Figure 4
The trajectories withx10=1.5.Figure 5
The comparison of tracking errors between PC and AFC withyd2.Figure 6
The comparison of tracking errors between PC and AC withyd2.Figure 7
The trajectories withx10=1.5.Figure 8
The control inputsu with x10=1.5.Figure 9
The state variablesx2 with x10=1.5.Figure 10
The adaptive parametersϖ1,ϖ2 with x10=1.5.Figure 11
The Nussbaum variablesς with x10=1.5.


### 4.2. Simulation of a 3-Order Robot

In order to demonstrate the practicability, the developed controller is applied to the following 3-order robot system [40].(78)x˙1=x2,x˙2=KmM¯x3+−Nsinx1−Bx2M¯+τeM¯,x˙3=1LV+−Rx3−Kex2L,where x1, x2, and x3 denote the angle, angular velocity, and motor current of the manipulator, respectively. M¯ is the system inertia, Km=KNc represents the coefficient between the current and moment with K being the constant torque, and Nc expressing the joint reduction ratio. Furthermore, τe represents the uncertain disturbance of torque by the external environment, L is the armature inductance and unknown, Ke denotes the back electromotive force constant, R displays the circuitous resistance, and V is the input voltage. The formulae of B, N, and M are introduced as follows:(79)B=B0KτN=m1dg2Kτ+m2dgKτM=JKτ+m1d23Kτ+m2d2Kτ+2m2δ25Kτwhere Kτ expresses the coefficient of the electromechanical conversion of armature current to torque and B0 shows the coefficient of viscous friction at the joint. g is the gravity coefficient, J denotes the rotor inertia, d represents the link length, δ is the radius of the load, and m1 and m2 display the mass of the link and load. Moreover, the parameter values are as(80)J=1.625×10−3kg⋅m2,d=0.305m,m1=0.506kg,m2=0.434kg,Ke=0.9Vsrad,B0=16.25×10−3N.m.srad,δ=0.023m,Kτ=0.9NmA,L=25×10−3H.The external disturbanceτe is defined as the Gaussian white noise written as wgn1,L,0.2 with L being the time series.The initial condition vector is selected asx10,x20,x30,ϖ10,ϖ20,ϖ30,ς=0.1rad,0,0,0,0,0,0.7. The reference trajectory is chosen to be a step signal. To satisfy the first order differentiability, it is written as yd=0.5tanhpt−5+1 with p=2.The Nussbaum variableς˙, control laws α1, α2u, and adaptive laws ϖ˙1, ϖ˙2, ϖ˙3 are defined as (19), (17), (32), (64), (21), (35), and (65) with k1=10, k2=12, k3=20, q1=0.02, q2=0.01, q3=0.025, σ1=15, σ2=12, σ3=28, and λ=10. The other parameters are equal to 1.Figures12–15 show the simulation results of the 3-order single-link robot. The angle tracking error is displayed in Figure 12, which converges to the initial value 0.1rad rapidly and oscillates at the step time t=5s. In Figure 13, the output can track the reference trajectory well. This is a step signal. The input voltage u is presented in Figure 14. It is easily seen that the oscillation also occurs at the step time. The angular velocity x2 and motor current x3 are given in Figure 15, respectively, both of which are bounded.Figure 12
The angle error.Figure 13
The output and desired trajectories.Figure 14
The input voltage.Figure 15
The angular velocity and motor current.


## 4.1. Simulation of a Second-Order Nonlinear System

Let us consider the strict-feedback nonlinear system as(75)x˙1=g1x1x2+θTφ1x1+d1t,x˙2=g2x¯2u+θTφ2x¯2+d2t,y=x1,where g1=1+0.1sinx1 with g1m=0.9 and g¯1M=1.1, g2=1+0.1cosx1sinx2 with g2m=0.9, and g¯2M=1.1, θ=1,1Tφ1=cosx1, φ2=sinx1cosx2, d1=e−tsint, d2=e−tcost.The desired trajectories are selected asyd1t=4sin0.2πt. and yd2t=2.5sin0.5t+sint. In order to see the controller performance under different initial conditions for the system states, the initial value of x1 is given as 1.5 and 3, respectively; ς0=1; and other initial conditions are chosen to be 0.The intermediate control signalα1, Nussbaum variable ς˙, and actual control law u of the developed control strategy are defined as (17), (19), and (64), where n=2, k1=5, and k2=10, a1=a2=b1=b2=c1=c2=l2=m2=10, λ=0.1, 0.5, 1, and 3. It is worth noting that the upper limit of λ is 3.5. Otherwise, the controller is easy to be ill-defined such that the states in the system are divergent.ϖ˙1 and ϖ˙2 are designed as (21) and (65), with n=2, q1=q2=1, and σ1=σ2=0.1.The proposed controller (PC) is compared with the adaptive fuzzy (AFC) [28] and adaptive controllers (AC) [34], respectively.The AFC is presented as(76)α1′=−k1z1+0.5z1+12a12z1θ1R1TR1,u′=−k2z2+0.5z2+12a22z2θ2R2TR2,θ.1=q12a12z12R1TR1−σ1θ1,θ.2=q22a22z22R2TR2−σ2θ2.where θi and Ri are defined in [28], ki, ai, qi, and σi are taken the same values as the PC, i=1 , 2.The AC is given as(77)α1=−z1α¯12z12α¯12+δ12,α¯1=k1z1+ϖ1ϒ1z1,u=−z2α¯22z22α¯22+δ22,α¯2=k2z2+ϖ2ϒ2z2,ϖ˙1=q1ϒ1z12−σ1ϖ1,ϖ˙2=q2ϒ2z22−σ2ϖ2,where ki, ai, bi, ci, l2, m2, qi, and σi are as same as the PC, i=1, 2. δ1, δ2 are positive constants, δ1=0.1, δ2=1.Figures.1–11 display the simulation results. In Figure 1, the tracking errors are drawn under different λ. It can be seen that the transient processes of the tracking errors are faster and the steady-state errors decrease as λ increase. Under different initial values x10=1.5 and 3, the comparisons of tracking performance for the desired trajectory yd1 between the AFC, AC, and PC are displayed in Figures 2–4. It is shown that the tracking performance with the PC is better and the tracking error is much smaller. Furthermore, the AFC can only ensure the boundedness of tracking errors, and the PC guarantees the asymptotic convergence of tracking errors, which is obvious in Figure 2. In addition, the absolute mean and root mean square values of tracking errors with the PC are smaller than the values with AFC and AC. Figures 5–7 draw the comparisons of tracking errors and tracking performance for the ideal trajectory yd2, and the similar conclusion can be obtained as Figures 2–4. The inputs u are given in Figure 8, which are large at the initial stage and converge rapidly to a range defined by ±10. The states x2 and adaptive parameters ϖ1∗, ϖ2∗ are presented in Figures 9 and 10, which are all bounded. Finally, Figure 11 shows the Nussbaum variables ς, which tends to positive constants from the initial values 1 as time goes on.Figure 2
The comparison of tracking errors between PC and AFC withyd1.Figure 3
The comparison of tracking errors PC and AC withyd1.Figure 4
The trajectories withx10=1.5.Figure 5
The comparison of tracking errors between PC and AFC withyd2.Figure 6
The comparison of tracking errors between PC and AC withyd2.Figure 7
The trajectories withx10=1.5.Figure 8
The control inputsu with x10=1.5.Figure 9
The state variablesx2 with x10=1.5.Figure 10
The adaptive parametersϖ1,ϖ2 with x10=1.5.Figure 11
The Nussbaum variablesς with x10=1.5.


## 4.2. Simulation of a 3-Order Robot

In order to demonstrate the practicability, the developed controller is applied to the following 3-order robot system [40].(78)x˙1=x2,x˙2=KmM¯x3+−Nsinx1−Bx2M¯+τeM¯,x˙3=1LV+−Rx3−Kex2L,where x1, x2, and x3 denote the angle, angular velocity, and motor current of the manipulator, respectively. M¯ is the system inertia, Km=KNc represents the coefficient between the current and moment with K being the constant torque, and Nc expressing the joint reduction ratio. Furthermore, τe represents the uncertain disturbance of torque by the external environment, L is the armature inductance and unknown, Ke denotes the back electromotive force constant, R displays the circuitous resistance, and V is the input voltage. The formulae of B, N, and M are introduced as follows:(79)B=B0KτN=m1dg2Kτ+m2dgKτM=JKτ+m1d23Kτ+m2d2Kτ+2m2δ25Kτwhere Kτ expresses the coefficient of the electromechanical conversion of armature current to torque and B0 shows the coefficient of viscous friction at the joint. g is the gravity coefficient, J denotes the rotor inertia, d represents the link length, δ is the radius of the load, and m1 and m2 display the mass of the link and load. Moreover, the parameter values are as(80)J=1.625×10−3kg⋅m2,d=0.305m,m1=0.506kg,m2=0.434kg,Ke=0.9Vsrad,B0=16.25×10−3N.m.srad,δ=0.023m,Kτ=0.9NmA,L=25×10−3H.The external disturbanceτe is defined as the Gaussian white noise written as wgn1,L,0.2 with L being the time series.The initial condition vector is selected asx10,x20,x30,ϖ10,ϖ20,ϖ30,ς=0.1rad,0,0,0,0,0,0.7. The reference trajectory is chosen to be a step signal. To satisfy the first order differentiability, it is written as yd=0.5tanhpt−5+1 with p=2.The Nussbaum variableς˙, control laws α1, α2u, and adaptive laws ϖ˙1, ϖ˙2, ϖ˙3 are defined as (19), (17), (32), (64), (21), (35), and (65) with k1=10, k2=12, k3=20, q1=0.02, q2=0.01, q3=0.025, σ1=15, σ2=12, σ3=28, and λ=10. The other parameters are equal to 1.Figures12–15 show the simulation results of the 3-order single-link robot. The angle tracking error is displayed in Figure 12, which converges to the initial value 0.1rad rapidly and oscillates at the step time t=5s. In Figure 13, the output can track the reference trajectory well. This is a step signal. The input voltage u is presented in Figure 14. It is easily seen that the oscillation also occurs at the step time. The angular velocity x2 and motor current x3 are given in Figure 15, respectively, both of which are bounded.Figure 12
The angle error.Figure 13
The output and desired trajectories.Figure 14
The input voltage.Figure 15
The angular velocity and motor current.


## 5. Conclusion

In the article, an adaptive asymptotic tracking controller has been designed for nonlinear systems with uncertain VCGs. Compared with these existing achievements, the developed control method could not only handle the VCGs in the form of unknown nonlinear functions but also achieve the asymptotic stability of the tracking error, which was carried out without the approximation by fuzzy logic or neural network and repetitive use of Nussbaum-type functions. New adaptive laws were defined to compensate for unknown virtual control gains, uncertain parameters, and external disturbances. Finally, the proposed control scheme was designed and applied to the control of a robot system. Both theoretical analysis and simulation were used to validate the effectiveness and practicability of the developed control strategy. In the future, we plan to apply the proposed controller to MIMO nonlinear systems by combining dead zone, saturation, hysteresis, and so on.



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*Source: 7408077-2022-03-28.xml*