# Characteristic Model-Based Adaptive Control for a Class of MIMO Uncertain Nonaffine Nonlinear Systems Governed by Differential Equations

Authors: Duo-Qing Sun; Xiao-Ying Ma
Journal: Complexity
 (2020)

---

## Abstract

This paper addresses the difficulty of designing a controller for a class of multi-input multi-output uncertain nonaffine nonlinear systems governed by differential equations. We first derive the first-order characteristic model composed of a linear time-varying uncertain system for such nonaffine systems and then design an adaptive controller based on this first-order characteristic model for position tracking control. The designed controller exhibits a simple structure that can effectively avoid the controller singularity problem. The stability of the closed-loop system is analyzed using the Lyapunov method. The effectiveness of our proposed method is validated with a numerical example.


---

## Body

## 1. Introduction

The study of complex nonlinear systems governed by differential equations has attracted considerable attention [1–4]. It is worth noting that in the real world, there are many nonaffine nonlinear systems, such as certain flight control systems [5, 6], tank reactor system [7, 8], biochemical processes [9, 10], and carrier landing control systems for unmanned aerial vehicles [11]. The study of nonaffine systems is often much more difficult and complicated than that of affine ones because control inputs always appear implicitly in the nonlinear functions [7], and finding the explicit inversion of nonaffine functions is quite difficult, even when the existence of such nonlinear function inversion can be proved using the implicit function theorem [5, 9]. On the other hand, in practical applications, uncertainties often arise owing to the presence of unknown parameter variations, modeling simplifications, unmodeled dynamics, and external disturbances, among others [12–14]. In certain practical situations, establishing accurate and detailed models of nonlinear systems may be difficult, costly, and time consuming. Therefore, the control design for uncertain nonaffine systems is an interesting yet challenging problem. Over the past decades, nonaffine nonlinear system control has received considerable attention, and many important results have been presented on the design of such controllers [5–12, 14, 15]. Owing to the universal approximation theorem, some adaptive control schemes based on fuzzy logic systems [16–22] and neural networks [8, 10, 23–25] have been developed for uncertain nonaffine nonlinear systems. Nevertheless, these approaches typically incur heavy computational costs. Apart from these approaches, other adaptive control methods have also been considered for uncertain nonaffine-in-control nonlinear systems, such as adaptive dynamic surface control [7] and generalized PI control [26].Recently, a majority of representative adaptive control schemes focuses on the continuous-time domain for nonaffine nonlinear systems described by differential equations. However, in engineering implementations, computers are usually utilized to produce digital control signals, which require controllers to be designed in discrete time [27].The above discussion motivates us to use a novel method, called characteristic modeling proposed by Wu [28–35], to solve the control design problem for uncertain nonaffine systems. The key idea of the characteristic modeling is to use a lower-order discrete time-varying linear system to express an original continuous system equivalently based on both the dynamic characteristics of controlled plants and performance specifications. This discrete system is called the “characteristic model of the original system.” It is worth mentioning that the characteristic modeling differs from the conventional model reduction methods in that all information of the original dynamic model, including uncertainties and nonlinearities, is compressed/integrated into the time-varying coefficients of the characteristic model instead of truncating parts of the plant model [28, 32, 33]. The time-varying coefficients of the characteristic model are referred to as the characteristic parameters [33, 35, 36], and the characteristic parameters are identified online adaptively [31, 33, 35, 37]. The characteristic model-based adaptive control might be a fresh perspective in the field of adaptive control: creating simplicity out of complexity [31], and it has already been applied successfully to more than 400 engineering systems including spacecraft, e.g., the reentry control of a manned spaceship [28, 29, 32], the control of servo systems with backlash and friction [30], and the control for the swing arm in a Fourier transform spectrometer [38].However, in the existing literature, the main focus is on the second-order characteristic model; the first-order model has received scarce attention for related results on controller design based on the characteristic model, and few studies have reported on the control method for nonaffine nonlinear systems based on the characteristic model. In the controller design method based on the second-order characteristic model, almost all the existing control methods belong to the indirect method in adaptive control. In other words, the model parameters are identified by the input and output of the system, and then the control law is designed. For the second-order characteristic model, the most successful application in practical engineering is the characteristic model-based golden section adaptive control method, and it must also be combined with the integral controller and the differential controller to improve the control performance [30, 33, 35]. This makes it very difficult to analyze the stability of the control system. Stability analysis, especially for MIMO systems, remains a challenging problem for adaptive control based on the second-order characteristic model despite its practical successes. Compared with the second-order characteristic model, the first-order model has the advantages of simpler form, fewer identified parameters, and more convenient applicability for practical systems.In this paper, inspired by [28, 39], we derive the first-order characteristic model for multi-input multi-output (MIMO) uncertain nonaffine nonlinear systems. Then, we develop a characteristic model-based adaptive control law for the position tracking problem. By constructing a suitable Lyapunov functional, we investigate the stability of the closed-loop system involving the first-order characteristic model-based control law.Compared with the existing results, the primary contributions and novelties of our study can be highlighted as follows:(1)
The first-order characteristic model composed of a linear time-varying uncertain system is derived for a class of MIMO uncertain nonaffine nonlinear systems governed by differential equations. To the best of the authors’ knowledge, first-order characteristic models have only been presented by Xu and Hu [39] and Sun et al. [40]. We note that SISO systems are only considered in [39, 40] and linear systems are only considered in [40].(2)
An adaptive controller based on this first-order characteristic model for position tracking control (see formula (22)) is designed by adaptively estimating the time-varying coefficients of the characteristic model (see formula (23)). In the existing literature, no results have been reported on the adaptive control design based on the first-order characteristic model for our considered class of nonaffine multivariable systems.(3)
The ranges of controller parameters, obtained (see Lemma1) by utilizing some mathematical technique, guarantee the stability of the closed-loop system (see Theorem 2). In [39], no intervals of controller parameters are presented.(4)
The designed controller exhibits a simple structure and can effectively avoid possible controller singularity problems.The paper is organized as follows. The problem formulation and the characteristic model are given in Section2. Then, we present the design of the adaptive controller based on the characteristic model and the stability analysis by using the Lyapunov theory in Section 3. The effectiveness of the proposed method is illustrated by a numerical example in Section 4. Section 5 concludes this paper.


## 2. Problem Formulation and Characteristic Modeling

The following notation is used in the paper.x denotes the Euclidean norm for a vector x. A denotes the Frobenius norm for a matrix A. Rn denotes the n-dimensional Euclidean space. I is the n-dimensional identity matrix. Let Ts > 0 be the sampling period.Consider a class of MIMO uncertain nonaffine nonlinear systems:(1)x˙=fx,u,where x=x1,…,xnT∈Rn and u=u1,…,unT∈Rn represent the state (output) and the input vector of the controlled plant, respectively.fx,u=f1x,u,…,fnx,uT is an unknown smooth nonlinear function vector.Letri∈0,1 for i=1,2,…,n. For Jacobian matrix ∂fx,u/∂u, u in the ith row is replaced by riu, and the resulting matrix is denoted as fux,⋅. We suppose that system (1) satisfies the following assumptions.Assumption 1.
f
u
x
,
⋅ is nonsingular.Assumption 2.
Eachfix,0 and fux,⋅−1 are bounded.
The control objective is to design the sampled data controlleruit=uikTs, kTs≤t≤k+1Ts, such that the output xit of the controlled system (1) tracks the preset value xir,i=1,2,…,n.
We first define the following set:(2)Ωi=xi∈Rxi≤ηi,i=1,2,…,n,where ηi is a small positive constant.
Denotexi,k=xikTs and ui,k=uikTs for i=1,2,…,n. Let(3)yi,k=xi.k+Didi,i=1,2,…,n,where di>ηi is a constant; Di=1 if xi,k∈Ωi and Di=0 if xi,k∉Ωi; we refer to yi,k as the unified variable. From (3), we can see that unified variables yi,k are not close to zero.Theorem 1.
Given an MIMO nonaffine nonlinear system (1) satisfying Assumption 1, if position keeping or position tracking control is desired, then its characteristic model can be described by the following system of first-order difference equations:(4)yi,k+1=θi,kyi,k+Tsui,k,i=1,2,…,n,where θi,k is the so-called characteristic parameter as follows:(5)θi,k=1+εi,k,εi,k=gi,kyi,k,where gi,k is the ith component of the vector ∫kTsk+1TsI−fux,⋅−1x˙+fux,⋅−1fx,0ds.Proof.
For a fixedii=1,…,n, the following equation can be obtained based on the differential mean value theorem:(6)x˙i=fix,0+f¯ix,u¯iu,i=1,2,…,n,where u¯i=riu, 0<ri<1,(7)f¯ix,u¯i=f¯i1x,u¯i,f¯i2x,u¯i,…,f¯inx,u¯i=∂fix,u∂uu=u¯i,f¯ilx,u¯i=∂fix,u∂ulu=u¯i.
Therefore, we have(8)x˙=fx,0+f¯u,where f¯=fux,⋅=f¯1x,u¯1T,f¯2x,u¯2T,…,f¯nx,u¯nTT.
According to Assumption1, the inverse function f¯−1 must exist. Thus, multiplying both sides of (8) with f¯−1 yields(9)f¯−1x˙=f¯−1fx,0+u.
Therefore, we have(10)x˙=I−f¯−1x˙+f¯−1fx,0+u.
Integrating both sides of the above equation yields(11)xt=x0+∫0tI−f¯−1x˙+f¯−1fx,0ds+∫0tusds.
Now sett=k+1Ts; then, (11) can be rewritten as(12)xk+1Ts=x0+∫0k+1TsI−f¯−1x˙+f¯−1fx,0ds+∫0k+1Tsusds.
Settingt=kTs, (11) can also be rewritten as(13)xkTs=x0+∫0kTsI−f¯−1x˙+f¯−1fx,0ds+∫0kTsusds.
Subtracting (13) from (12) yields(14)xk+1Ts=xkTs+∫kTsk+1TsI−f¯−1x˙+f¯−1fx,0ds+∫kTsk+1Tsusds=xkTs+∫kTsk+1TsI−f¯−1x˙+f¯−1fx,0ds+TsukTs.
Denote(15)gk=g1,k,g2,k,…,gn,kT=∫kTsk+1TsI−f¯−1x˙+f¯−1fx,0ds,(16)uk=u1,k,u2,k,…,un,kT=ukTs.
Then, we havexk+1Ts=xkTs+gk+uk, or equivalently,(17)xik+1Ts=xikTs+gi,k+Tsui,k,i=1,2,…,n.
Next, we analyze two cases individually.Case 1.
Ifxik∉Ωi, then the ith output from the controlled object is far away from zero. Therefore, (17) can be rewritten as(18)xik+1Ts=1+gi,kxikTsxikTs+Tsui,k.Case 2.
Ifxik∈Ωi, then the ith output from the controlled object is close to zero. In this case, (17) can be rewritten as(19)xik+1Ts+di=xikTs+di+gi,k+Tsui,k=1+gi,kxikTs+dixikTs+di+Tsui,k.
Combining Cases1 and 2 from (3), we can obtain the characteristic model described by (4).Remark 1.
From Assumption2, we can derive the range of the characteristic parameter θi,k. In fact, from Assumption 2, there exist positive constants Mf and Mu such that fx,0≤Mf and f¯−1=fux,⋅−1≤Mu. We note that x˙ is continuous. Thus, x˙ is bounded on each closed interval kTs,k+1Ts. That is, there exist positive constants Mxk such that x˙≤Mxk on kTs,k+1Ts. Further, x˙≤Mx=SupremumkMxk. This is not restrictive, and the supremum of Mxk exists in practical engineering systems [41, 42]. Therefore, I−f−1x˙+f−1fx,0≤I−f¯−1x˙+f¯−1fx,0≤I−f¯−1⋅x˙+f¯−1fx,0≤I⋅x˙+f¯−1⋅x˙+f¯−1⋅fx,0≤I⋅Mx+Mu⋅Mx+Mu⋅Mf.
We denoteG=I⋅Mx+Mu⋅Mx+Mu⋅Mf. Thus, we have g¯i,k≤G, where g¯i,k is the ith component of the vector I−f¯−1x˙+f¯−1fx,0; then, gi,k≤GTs, and we obtain εi,k<εi, where εi=GTs/ηi if xikTs∉Ωi and εi=GTs/di−ηi if xikTs∈Ωi. From (5), we get(20)1−εi=δi1<θi,k<δi2=1+εi.


## 3. Controller Design and Stability Analysis

We define the tracking errors asei,k=xi,k−xir for i=1,2,…,n. The reference tracking signal of the unified variable yi,k can be expressed by yir=xir+Didi. Note that the tracking errors of the unified variables equal the ones of the original object, which can be seen from(21)yi,k−yir=xi,k+Didi−xir+Didi=xi,k−xir=ei,k.We present the basic approach to obtain an adaptive controller that achieves our control objective. If the functionf is known, then each θi,k is also known. The control laws ui,k∗=−θi,kyi,k+ai0ei,k+yir/Ts i=1,2,…,n applied to (4) result in yi,k+1=ai0ei,k+yir, that is, ei,k+1−ai0ei,k=0. Notice that if one chooses ai0 such that ai0<1, where i=1,2,…,n, then the roots of the polynomials Hqi=qi−ai0 related to the characteristic equation of ei,k+1−ai0ei,k=0 are inside unit circles, which implies that limk⟶+∞ei,k=0. This is main objective of control. Because f is unknown, the optimal control ui,k∗ cannot be implemented. Thus, our purpose is to design an adaptive controller to approximate this optimal control.Based on the above ideas, the adaptive controller is designed as(22)ui,k=1Ts−θ^i,kyi,k+ai0ei,k+yir,i=1,2,…,n,where ai0∈0,1 and θ^i,k denotes the estimation of the parameter θi,k.Note thatai0∈−1,0 is not considered for the convenience of theoretical analysis.The adaptive law of parameterθi,k is designed as(23)θ^i,k=miθ^i,k−1+λiai0yi,kei,k1+λiμiyi,k2,where λi>0,μi>0, and 0<mi<1 represent design constants.The adaptive controller (22) includes estimated parameters θ^i,k, and the performance of asymptotic tracking with zero error or near zero can be obtained by adjusting the design parameters λi>0, μi>0, and 0<mi<1.Letφi,k=θi,k−θ^i,k be the parameter estimation error. The error equation of the closed-loop system can be expressed by substituting (22) into (4) as follows:(24)ei,k+1=θi,k−θ^i,kyi,k+ai0ei,k=φi,kyi,k+ai0ei,k.From (23), we have(25)θ^i,k−miθ^i,k−1=λiai0yi,kei,k−μiyi,k2θ^i,k.Thus, as the basis for preparing the stability analysis of the closed-loop system, the relation between the parameter estimation errors at two adjacent instants can be obtained as follows:(26)φi,k=θi,k−θ^i,k+miφi,k−1−miφi,k−1=miφi,k−1+θi,k−θ^i,k−miθi,k−1−θ^i,k−1=miφi,k−1+θi,k−miθi,k−1−θ^i,k−miθ^i,k−1=miφi,k−1+θi,k−miθi,k−1−λiai0yi,kei,k−μiyi,k2θ^i,k=miφi,k−1+θi,k−miθi,k−1−λiai0yi,kei,k+λiμiyi,k2θi,k−φi,k=miφi,k−1+wi,k,where(27)wi,k=θi,k−miθi,k−1−λiai0yi,kei,k+λiμiθi,kyi,k2−λiμiφi,kyi,k2.For subsequent use, we denote(28)λ`i1=mi2Δi−mi2Δi2−ci2ci2,(29)λ`i2=mi2Δi+mi2Δi2−ci2ci2,where Δi=−ai0ci+1−ai02μi, ci=μiδi2−ai0; i=1,2,…,n. Then, we have the following lemma.Lemma 1.
Letμi satisfy(30)ai0δi2<μi<ai0δi2−1+ai0,i=1,2,…,n.Then,μi, λi1, λi2>0, and λi1<λi2.Proof.
For the sake of simplicity, the subscripti for all variables is omitted in this proof. For example, a0 written in this proof actually represents ai0.
We first need to prove that (30) is well defined. It should be noted that δ2>δ2−1+a0=1+ε−1+a0>0. Therefore, (30) is well defined and clearly μ>0.
To prove thatλ1 and λ2 are also well defined, we need to first prove that Δ2−c2>0. To do this, let us consider the following equality:(31)Δ−c=−a0c+1−a02μ−c=−1+a0c+1−a02μ=−1+a0δ2−1+a0μ−a0.
Sinceμ<a0/δ2−1+a0, we have Δ>c. Considering that μ>a0/δ2, we obtain c=μδ2−a0>0. Thus, we further have Δ>0, and hence Δ2−c2>0. Therefore, both λ1 and λ2 are well defined. It is clear that λ1,λ2>0 and λ1<λ2.We setTi=minηi,di−ηi, i=1,2,…,n; Tmin=min1≤i≤nTi/Gi. We require the following lemma to summarize the performance of our proposed control strategy.Lemma 2 (see [43], Lemma 6).
Letx,y∈R. Then,(32)2xy≤εx2+1εy2,for any ε>0.Theorem 2.
Consider an MIMO nonaffine nonlinear system (1) satisfying Assumptions 1 and 2. If the adaptive tracking controller is designed as (22) with update law (23), the sampling period Ts<Tmin, and controller parameters μi∈ai0/δi1,ai0/δi2−1+ai0, λi∈λi1,λi2 i=1,2,…,n, then the following properties are guaranteed:(1)
The tracking errorsei,k i=1,2,…,n and each parameter estimation error φi,k are both bounded.(2)
The tracking errorsei,k i=1,2,…,n converge to the neighborhoods of the origin.Proof.
First, we note that from (20), εi<1 when the sampling period Ts<Tmin. Thus, δi1=1−εi>0. Since μi>ai0/δi1, it is easy to see that μi>ai0/δi2. From Lemma 1, there exist real numbers λi such that λi∈λi1,λi2 for i=1,2,…,n. Now, consider the Lyapunov function given by(33)Vi,k=ei,k+12+φi,k2.Part 1.
Calculating the difference of the Lyapunov function (33).
By utilizing (24), we can derive the first difference of the Lyapunov function (33) as(34)ΔVi,k=Vi,k−Vi,k−1=ei,k+12−ei,k2+φi,k2−φi,k−12=φi,kyi,k+ai0ei,k2−ei,k2+φi,k2−φi,k−12=ai02ei,k2+φi,k2yi,k2+2ai0φi,kyi,kei,k−ei,k2+φi,k2−φi,k−12.
Using (26), the last term −φi,k−12 in (34) can be written as(35)−φi,k−12=−φi,k−wi,kmi2=−1mi2φi,k2−wi,k2mi2+2φi,kwi,kmi2.
Now, substituting (27) into φi,kwi,k in the last term of (35) and employing yi,k=ei,k+yir, we then have(36)φi,kwi,k=φi,kθi,k−miθi,k−1−λiai0yi,kei,k+λiμiθi,kyi,k2−λiμiφi,kyi,k2=θi,k−miθi,k−1φi,k−λiai0φi,kyi,kei,k+λiμiθi,kyi,k2φi,k−λiμiφi,k2yi,k2=θi,k−miθi,k−1φi,k−λiai0φi,kyi,kei,k+λiμiθi,kφi,kyi,kei,k+yir−λiμiφi,k2yi,k2=−λiμiφi,k2yi,k2+θi,k−miθi,k−1φi,k+λiφi,kyi,kei,k−ai0+μiθi,k+λiμiθi,kφi,kyi,kyir.
Substituting (36) into (35) and then into (34) yields(37)ΔVi,k=−1+ai02ei,k2+φi,k2yi,k2+2ai0φi,kyi,kei,k+φi,k2−1mi2φi,k2−wi,k2mi2+2mi2−λiμiφi,k2yi,k2+θi,k−miθi,k−1φi,k+λiφi,kyi,kei,k−ai0+μiθi,k+λiμiθi,kφi,kyi,kyir=−1+ai02ei,k2+−1mi2+1φi,k2−1mi2wi,k2+φi,k2yi,k2−2λiμimi2+1+2φi,kyi,kei,kai0+1mi2−λiai0+λiμiθi,k+2mi2λiμiθi,kφi,kyi,kyir+2mi2θi,k−miθi,k−1φi,k.
Denote(38)Si1=ai0+1mi2−λiai0+λiμiθi,k,Si2=1mi2λiμiθi,kyir,Si3=1mi2θi,k−miθi,k−1.
Then, (37) can be rewritten as(39)ΔVi,k=−1+ai02ei,k2+−1mi2+1φi,k2−1mi2wi,k2+φi,k2yi,k2−2λiμimi2+1+2Si1φi,kyi,kei,k+2Si2φi,kyi,k+2Si3φi,k.Part 2.
Enlarging the differenceΔVi,k denoted by (39).
For further analysis, we derive the ranges ofSi1,Si2,andSi3 shown in (38). Let Sij1 and Sij2 denote the minimum and maximum values of Sijj=1,2,3, respectively. Based on the expressions of Si1 and Si3, we have(40)Si11=ai0+1mi2λiμiδi1−λiai0,Si12=ai0+1mi2λiμiδi2−λiai0,Si31=1mi2δi1−miδi2,Si32=1mi2δi2−miδi1.
We denoteSi1m=maxSi11,Si12 and Si3m=maxSi31,Si32. Then, Si1≤Si1m and Si3≤Si3m. From (20), Si2≤Si2m=1/mi2λiμi1+εi⋅yir.
Now, employing Lemma2, we obtain(41)2Si1φi,kyi,kei,k≤2Si1mφi,kyi,k⋅ei,k≤Si1mγi1φi,k2yi,k2+γi1Si1mei,k2,(42)2Si2φi,kyi,k≤Si2mγi2φi,k2yi,k2+γi2Si2m,2Si3φi,k≤Si3mγi3φi,k2+γi3Si3m,provided that γi1,γi2,andγi3>0. Substituting these three inequalities into (39) yields(43)ΔVi,k≤−1+ai02ei,k2+−1mi2+1φi,k2−1mi2wi,k2+φi,k2yi,k2−2λiμimi2+1+Si1mγi1φi.k2yi,k2+γi1Si1mei,k2+Si2mγi2φi.k2yi,k2+γi2Si2m+Si3mγi3φi,k2+γi3Si3m≤−1+ai02+γi1Si1mei,k2+−2λiμimi2+1+Si1mγi1+Si2mγi2φi.k2yi,k2+−1mi2+1+Si3mγi3φi,k2+γi2Si2m+γi3Si3m.
Denote(44)Ni1=1−ai02−γi1Si1m,Ni2=2λiμimi2−1−Si1mγi1−Si2mγi2,Ni3=1mi2−1−Si3mγi3,δim=γi2Si2m+γi3Si3m.
Then, we have(45)ΔVi,k≤−Ni1ei,k2−Ni2φi,k2yi,k2−Ni3φi,k2+δim,this is,(46)ΔVi,k=ei,k+12+φi,k2−ei,k2−φi,k−12≤−Ni1ei,k2−Ni2φi,k2yi,k2−Ni3φi,k2+δim.Part 3.
Giving the properties ofNi1, Ni2, and Ni3 in (46).
Based on the bounds of the characteristic parameters provided by (20), we can prove that there exist γi1,γi2,andγi3>0, such that Ni1∈0,1; Ni2andNi3>0. To prove these, it suffices to show that(47)2λiμi−mi2>0,mi2Si1m2λiμi−mi2<1−ai02Si1m.
Indeed, if (47) holds, then there exist γi1 that satisfy(48)0<mi2Si1m2λiμi−mi2<γi1<1−ai02Si1m.
Further, we haveNi1=1−ai02−γi1Si1m∈0,1, and 2λiμiγi1−mi2γi1−mi2Si1m>0. Thus, there exists a positive number(49)γi2>mi2γi1Si2m2λiμiγi1−mi2γi1−mi2Si1m.
Note that(50)Ni2=2λiμiγi1−mi2γi1−mi2S1mmi2γi1−Si2mγi2.
Consequently, we can obtainNi2>0 for any γi1 and γi2 that satisfy (48) and (49), respectively.
In what follows, we shall prove that (47) holds indeed.
For simplicity, the subscripti of variables is omitted until the end of the proof of (47), except for Si11 and Si12. For example, we use the notation μ instead of μi.(i)
First, we prove the first formula in (47).Since(51)2μΔ−c22−4μ2Δ2−c2=4μ2Δ2+c4−4μΔc2−4μ2Δ2+4μ2c2=c4−4μΔc2+4μ2c2=c4−4μ−a0c+1−a02μc2+4μ2c2=c4+4μa0c3−4μ2c2+4μ2a02c2+4μ2c2>0.We have2μΔ−c2>2μΔ2−c2. Consequently, we obtain 2μΔ−Δ2−c2/c2>1. Multiplying both sides of the above inequality by m2 yields 2μm2Δ−m2Δ2−c2/c2>m2. That is, 2λ1μ−m2>0. Thus, 2λμ−m2>0.(ii)
Next, we prove the second formula in (47).Note thatδ1=1−ε>0 when the sampling period Ts<Tmin. Since μ>a0/δ1, we have μδ1>a0 and μδ1−a0>0. Therefore, Si11>0. Thus, we obtain S1m=Si12=a0+1/m2λμδ2−λa0. Then, from c=μδ2−a0, we have(52)m2S1m2−1−a022λμ−m2=1m2cλ+m2a02−1−a022λμ−m2=1m2c2λ2−2−m2a0c+m21−a02μλ+m4.Fromλ∈λ1,λ2, we have c2λ2−2−m2a0c+m21−a02μλ+m4<0, i.e.,(53)m2S1m2<1−a022λμ−m2.
Therefore, we havem2S1m/2λμ−m2<1−a02/S1m. Thus, the second formula in (47) is proved.
Now, we notice thatNi3 > 0 can be guaranteed by taking positive numbers γi3>mi2Si3m/1−mi2.Part 4.
Proving the boundedness for the tracking errorsei,k and parameter estimation error φi,k and convergence for tracking errors ei,k.
From the above results and (46), we can obtain the following result:(54)ei,k+12+1+Ni3φi,k2≤1−Ni1ei,k2+φi,k−12−Ni2φi,k2yi,k2+δim≤1−Ni1ei,k2+φi,k−12+δim.
DenoteNi=max1/1+Ni3,1−Ni1; then, Ni∈0,1. Thus, we have(55)ei,k+12+1+Ni3φi,k2≤1−Ni1ei,k2+11+Ni31+Ni3φi,k−12+δim≤Niei,k2+1+Ni3φi,k−12+δim.
In other words, we obtained a recursion relation that can yield(56)ei,k+12+1+Ni3φi,k2≤Ni2ei,k−12+1+Ni3φi,k−22+Niδim+δim≤⋯≤Nikei,12+1+Ni3φi,02+Nik−1δim+Nik−2δim+⋯+Ni2δim+Niδim+δim=Nikei,12+1+Ni3φi,02+1−Nik1−Niδim<ei,12+1+Ni3φi,02+11−Niδim.
From this inequality, bothei,k and φi,k are bounded, and we can obtain(57)ei,k+12≤Nikei,12+1+Ni3φi,02+1−Nik1−Niδim,which implies limsupk⟶+∞ei,k+12≤δim/1−Ni. Thus, limsupk⟶+∞ei,k≤δim/1−Ni. By the definition of the supremum limit, there exist integers Mi>0 for all sufficiently small ε¯i>0 such that(58)ei,k≤ρi=maxei,1,…,ei,Mi,ε¯i+δim1−Ni.
Hence, Theorem2 is proved.Remark 2.
Iffix,u is bounded, that is, there exists a constant Li>0 such that fix,u≤Li, then the tracking error eit=xit−xir of system (1) converges to the neighborhood of the origin under the conditions in Theorem 2. In fact, for any t∈kTs,k+1Ts, integrating (1) from kTs to t and noting e˙it=x˙it, one can easily obtain eit≤eikTs+TsLi≤ρi+TsLi, where ρi is defined in (58).


## 4. Numerical Example

To verify the effectiveness of the developed control method, consider a nonaffine double-pendulum system. The dynamic equations of motion are given as [44](59)x˙1x˙2x˙3x˙4=0100000000010000x1x2x3x4+01000001f1x,uf2x,u,yt=10000100xt,with x0=0.2,0.1,0.2,0.1T. x=x1,x2,x3,x4T is the vector of measurable states, u=u1u2T is the vector control inputs, and(60)f1x,u=f11xM1x,u1+f12xM2x,u2+f13x,f2x,u=f21xM1x,u1+f22xM2x,u2+f23x.Here,(61)f11x1,x3=12l124m1+12m2−9m2cos2x3−x1,f12x1,x3=12l2+18l1cosx3−x1l12l29m2cos2x3−x1−4m1−12m2,f21x1,x3=18cosx3−x1l1l29m2cos2x3−x1−4m1−12m2,f22x1,x3=12m1+36m2l22m24m1+12m2−9m2cos2x3−x1+18l2cosx3−x1l1l224m1+12m2−9m2cos2x3−x1,f13x1,x2,x3,x4=9gm2sin2x3−x1l115m2+8m1−9m2cos2x3−2x1−9l1m2x22sin2x3−2x1l115m2+8m1−9m2cos2x3−2x1+12l2m2x42sinx3−x1l115m2+8m1−9m2cos2x3−2x1−15gm2+12gm1sinx1l115m2+8m1−9m2cos2x3−2x1,f23x1,x2,x3,x4=12l12x22sinx3−x1m1+3m2l115m2+8m1−9m2cos2x3−2x1−9l2m2x42sin2x3−2x1l115m2+8m1−9m2cos2x3−2x1−9gsinx3−2x1m1+2m2l115m2+8m1−9m2cos2x3−2x1+3gsinx3m1+6m2l115m2+8m1−9m2cos2x3−2x1;M1u1=tanhu1+0.35u1,M2u2=tanhu2+0.35u2,where m1 and m2 are the mass of pendulums 1 and 2, respectively, and the meaning of other symbols in (61) can be found in [44].Herein, the control objective is to use our adaptive controller such that the statesx1 and x3 are adjusted to 0 from the initial state. In this example, the parameter values of the nonaffine double-pendulum system are m1=m2=1kg, l1=l2=1m, and g=9.81m/s2. The initial values of the parameters θ^i,k are chosen to be θ^i,0=1 for each i=1,2. The parameters of the controller are selected as ai0=0.01Ts, λi=1000, μi=0.001, and mi=0.1Ts for i=1,2. Meanwhile, we set ηi=0.0001, di=0.0002 for i=1,2, and the sampling period Ts=0.001.Figures1–4 depict the simulation results. As shown in the figures, the system can effectively track the expected outputs, thereby confirming the effectiveness of our proposed control approach.Figure 1
Trajectories of the first outputx1 for our controller.Figure 2
Trajectories of the second outputx3 for our controller.Figure 3
Control inputu1 for our controller.Figure 4
Control inputu2 for our controller.To demonstrate the effectiveness of the proposed control strategy, we use the fuzzy adaptive control method in [45] instead of our proposed adaptive controller. The control strategy proposed in [45] can be described as follows.Control laws are designed as follows:(62)ui=udi+ufi+uhi,i=1,2,where udi=KiTe¯i, e¯i=ei,e˙iT, ei is the tracking error of xi, Ki=ki2,ki1T, and all roots of the polynomial hs=s2+ki1s+ki2 are in the open left-half plane; the fuzzy control terms ufi=θiΤξix, in which ξix is fuzzy basis function, and we use the following adaptation laws to adjust the unknown parameter vectors of fuzzy systems:(63)θ˙i=−ηie¯iTPiBξix−σiηie¯iθi,where ηi,σi>0 are adaptive parameter rates;(64)uhi=−1riBTPiei,i=1,2,where ri>0 is a design parameter, B=0,1T, and Pi is a symmetric positive definite matrix satisfying the Riccati equation(65)AiTPi+PiAi+Qi−2ri−1ρi2PiBBTPi=0,where Qi>0 and Ai=01−ki2−ki1.In the simulation, the design parameters areki1=2, ki2=1, Qi=10000100, ri=0.001, and ρi=0.1, for i=1,2. The adaptive parameter rates are chosen as ηi=0.1 and σi=10, for i=1,2. With respect to fuzzy logic systems, Mamdani fuzzy models are employed to obtain the fuzzy control terms uf1 and uf2, and we define five Gaussian membership functions for each variable xi, i=1,2,3,4, distributed uniformly on interval [−0.8, 0.8]. Then, there are 2 × 5 × 5 × 5 × 5 = 1250 rules that are used in the simulation. The initial state of each variable is the same as in the above simulation.The results of this simulation are illustrated in Figures5–8. The figures indicate that the adaptive fuzzy controller can also achieve better performance. However, it should be emphasized that this is the result of more than 2 min of CPU time running on a computer (Lenovo workstation P720 with dual Intel® Xeon® Gold 6136 Processor), whereas the results shown in Figures 1–4 only require 7 s of CPU time on the same computer.Figure 5
Trajectories of the first outputx1 for the fuzzy controller [45].Figure 6
Trajectories of the second outputx3 for the fuzzy controller [45].Figure 7
Control inputu1 for the fuzzy controller [45].Figure 8
Control inputu2 for the fuzzy controller [45].There are 1250 parameters that need to be updated online for the scheme in [45], whereas only 2 parameters are required in our scheme. The fewer parameters used in our scheme help to make the proposed algorithm more efficient.


## 5. Conclusions

In this study, the control method for a class of uncertain nonaffine systems is studied for the first time by using a first-order characteristic model. Specifically, we first transformed the nonaffine uncertain system into a linear first-order characteristic model with bounded time-varying parameters. Based on this model, we designed an adaptive controller with a simple structure and proved that the tracking errors converge to the neighborhoods of the origin. The proposed approach can avoid the controller singularity because there is no inverse operator construction in the controller. As an illustration of the effectiveness of the proposed method, a numerical example is provided.



---

*Source: 3649472-2020-02-26.xml*