# Incremental Adaptive Control of a Class of Nonlinear Nonaffine Systems

Authors: Yizhao Zhan; Shengxiang Zou; Xiongxiong He; Mingxuan Sun
Journal: Complexity
 (2022)

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## Abstract

As a class of familiar nonlinear systems, nonaffine systems are frequently encountered in practical applications. Currently, in the context of learning control, there is a lack of research results about such general class of nonlinear systems, especially for the case of performing infinite interval tasks. This article focuses on the incremental adaptive control for nonlinear systems in nonaffine form, without requiring periodicity or repeatability. Instead of using the integral adaptation, incremental adaptive mechanisms are developed and the corresponding control schemes are presented, by which the numerical integration for implementation can be avoided. With the proposed incremental adaptation, the implicit function theorem is introduced to solve the intractability problem of the nonaffine structure. The robustness (robust convergence) of the tracking error is characterized, with the aid of a proposed key lemma, while the boundedness of all the variables is examined. Numerical results are presented to verify the effectiveness of the proposed control design.


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## Body

## 1. Introduction

In recent years, with the acceleration of energy transition, complex energy systems have become a fascinating research area. In particular, more and more attention has been paid to its complex nonlinear dynamic analysis and advanced control method design. Many complex systems in the real world can be described by nonaffine systems, such as wind turbine systems, chemical reactions, and flight control systems. Due to its important theoretical and practical significance, plenty of research effort has been paid on the topic over the years. Research works devoted in this topic can be found in many publications (see [1–6] and references therein).In comparison with affine systems, the challenge for nonaffine systems is that the control input couples with the system state and acts on the system dynamics in a nonlinear implicit way. Therefore, there is usually no concept of control gain or control direction, which in fact increases the difficulty of controller design. By utilizing Taylor series expansion [2, 5], median value theorem [3], and Hadamard theorem [4], the system model subjected to nonaffine characteristics can be simplified before analysis and design. With these mathematical approximations, nonaffine systems can be expressed as an affine-like system or an exact affine form within a neighborhood of the equilibrium. In [2], by introducing the implicit function theorem, the existence of an ideal controller for nonaffine system stabilization was demonstrated. To deal with nonaffine problems, the dynamic inversion technology via time-scale separation was proposed for the system dynamics that are exactly or partially known [6]. Using the singular perturbation theory, this dynamic approach has been shown to be effective in obtaining equivalent controllers that satisfy the control requirements. In order to further reduce the dependence on the knowledge of system dynamics, the approximation technology [7], represented by the neural network (NN) and the fuzzy logic system (FLS), is widely used in nonaffine systems [8–14], by applying the well approximation ability on compact sets. In existing work, many efficient adaptive neural/fuzzy control schemes have been developed. In [8], the proposed neural control synthesis method avoids the fixed-point problem brought by nonaffine structures. Exploiting input-to-state stability and the small gain theorem, the adaptive neural control design was proposed in [14] for the nonaffine systems with a mild assumption, which not only can overcome the problem of intractability when controlling nonaffine nonlinear systems, but also has the characteristics of dynamic learning. Regarding the NN approximation errors, [12] merged it with the external disturbance as a compounded disturbance and designed the disturbance observer for disturbance compensation. With the stochastic generalized Takagi–Sugeno fuzzy approximation models, an approach to a stabilization controller design was developed [13]. These results were generalized to MIMO nonaffine nonlinear systems [9, 10]. More recently, interesting results have been ever-increasing and reported in [15–23], wherein various adaptive NN/FLS-based control schemes have been shown to efficiently tackle challenges such as input saturation [15], nonlinear faults [17, 19], state/output constraints [23], and the consensus tracking control problem for multiagent systems [20].In the above methods, a variety of effective adaptive mechanisms were proposed, in order to ensure the stability and convergence of the closed-loop system. Taking from the perspective of construction, these adaptive laws are integral adaptation, which are conducted in the time domain. As an alternative and supplement, the parameter adaptive learning mechanisms in the iteration domain are provided, in the context of learning control. As a control method that can significantly improve the control performance of systems that perform repeating tasks, learning control has been well established and comprehended theoretically and technically during the past several decades [24–35]. In the context of adaptive control, the theoretical frameworks of adaptive iterative learning control [28, 29, 32, 33] and adaptive repetitive control [25, 30, 31] are gradually developed and formed. Both are important research topics in the field of adaptive control and learning control, and different from traditional learning control with input learning features [24, 26], they emphasize parameter learning. Systematic methodologies were proposed in [25, 28], where [25] explored the Lyapunov-like approach for a repetitive controller design, while [28] developed the composite-energy-function approach for iterative learning control schemes. The effectiveness of the Lyapunov-like synthesis method in learning control performance analysis was further systematically demonstrated in [29]. In addition to the parameter adaptation in the iteration domain, adaptive learning control also provides the time-domain parameter adaptation [27], which can still perform well as the conventional adaptive control. In [32, 33], the composite learning algorithm was proposed, by combining the time- and iteration-domain parameter adaptations. Moreover, in [34], the projection modification was suggested, through which the parameter estimates can be guaranteed vary within a prespecified region. Similar correction ideas were also reflected in [30, 31], in which the properties of saturated functions [25] have been mainly applied, and the partially [30] and fully saturated [31] learning algorithms were developed. In [35], the Barbalat-like lemma was proposed to further address the problem of uniform convergence of learning control systems. The superiority and efficiency of learning control depend largely on the repeatability of system operation. However, there are often potential implementation problems for such strict requirements, which limit the practical application of learning control. To relax some fundamental constraints encountered, a great deal of efforts have been put into related research topics in recent years, and the main results are concentrated on the initial condition issue [36–38], the nonequal task problem [39–42], and so on. For the arbitrary initial condition, various effective techniques have been proposed to solve this problem, including the boundary layer method [36], the trajectory rectifying strategy [37], and the error tracking approach [38]. In addition, the identical trial length assumption was removed in the framework of adaptive learning control, and for the parameter learning, the reference signals are permitted to vary with iteration, and the controller design can be easily conducted [39–42]. With the rapid development of adaptive learning control theory, numerous results of the above control methods have been successfully applied in various practical systems [43–46].It should be pointed out that most of the abovementioned studies on learning control focus on the tracking problem on a finite time interval or that of tracking a periodic reference signal. In addition, the existing learning control theories mainly concern with the convergence analysis. At present, there is still a lack of relevant research for the learning systems performing the task over an infinite interval, without involving repositioning and repeatability [47–49]. Furthermore, few results on adaptive control have been available to tackle such nonlinear systems with nonaffine form based on learning control.In this article, novel incremental adaptive control design methods are presented for nonaffine systems performing infinite-duration tasks, where the incremental adaptive mechanisms are exploited, instead of using the conventional integral adaptation. Drawing on existing learning control strategies, numerous incremental adaptations are developed and the corresponding performances are characterized, including the so-called closed-loop, open-loop, and fully saturated learning mechanisms. With the proposed incremental adaptation, the numerical integration required by the implementation for an integral adaptive algorithm is avoided. What’s more, systems allow arbitrary bounded reference signals and there is no periodicity or repeatability requirements. Moreover, the implicit function theorem is applied to deal with the nonaffine-in-control problem, based on the incremental adaptation. Furthermore, a key lemma is provided as a technical tool, which can be used for the robust convergence analysis of incremental adaptive control (IAC) systems.This article is organized as follows. The problem formulation is provided in Section2, and preliminaries are given, which are helpful for the performance analysis of the IAC algorithms. In Section 3, the IAC problem for a simple class of uncertain nonaffine systems is addressed, in which the basic forms of the incremental adaptation with the corresponding system synthesis are described. In Section 4, the learning control design methods are introduced to the high-order nonlinear nonaffine systems, and both the IAC control design and performance analysis are presented. In Section 5, the control performance is verified by the numerical examples. The conclusion is finally drawn in Section 6.


## 2. Problem Formulation and Preliminaries

Let us begin with the following systems with unknown parameters:(1)x˙=θTφx+u,where x and u are the scalar system state and the control input, respectively, θ∈Rn denotes a vector of unknown parameters, and φx∈Rn represents a known state-dependent vector-valued function, satisfying that φ is bounded as the state is bounded.The control goal is to design an adaptive controller for system (1) such that the system state x converges to 0, that is, limt⟶∞xt=0, while the boundedness of all signals in the closed-loop system are guaranteed.For the adaptive system consisting of system (1), the controller can be designed as follows:(2)u=−θ^Tφx−βx,and the adaptation law(3)θ^˙=γφxx,where β,γ>0 are adjustable parameters, and θ^ is the estimate of θ. Let us denote by θ˜=θ−θ^ the estimation error. With control law (2), (1) can be expressed as(4)x˙=θ˜Tφx−βx.To ensure the convergence of the adaptive system, the Lyapunov functionV=x2/2+2γ−1θ˜Tθ˜. is chosen. By applying (3), the derivative of V along (4) can be rewritten as V˙=−βx2, which implies that V˙ is negative semidefinite; that is, Vt is bounded, as V0 is bounded. Because of the boundedness of V, the boundedness of x and θ^ is obvious, and then, we can say that x˙ and u are bounded, as well as ∫0tx2sds<+∞, as t⟶∞. Invoking Barbalat’s lemma shows that limt⟶∞xt=0.Now let us look into (3), which is usually referred to as the integral adaptive law, because it can be rewritten as(5)θ^t=θ^0+γ∫0tφxsxsds,which implies that, for t>T, where T is a positive constant,(6)θ^t−T=θ^0+γ∫0t−Tφxsxsds.From the above two equations, we obtain(7)θ^t=θ^t−T+γ∫t−Ttφxsxsds.Furthermore, by the integral mean value theorem, an incremental form of (3) can be given as(8)θ^t=θ^t−T+Tγφxτxτ,with τ being a certain moment located between t−T and t.Looking into the second item of the right-hand side of (8), Tγφxτxτ, both the values of φxτ and xτ depend on τ, which should be given a priori. However, as time varies, τ also changes that the values of these items cannot be determined in advance at different moments. Due to the application of digital control technology, the integral adaptive law (3) can only be approximated in the implementation; that is, φxτxτ can be replaced by φxtxt or φxt−Txt−T. Obviously, such these approximations have a deviation from the integral adaptive law (3) itself. In fact, no matter what kind of interpolation is used, the long-term drifting phenomenon may occur, since the small but unavoidable error at each numerical integration step accumulates.Instead of using the integral adaptation, we intend to explore the incremental adaptive control method and present the corresponding performance analysis of the closed-loop system in detail, aiming at avoiding the numerical integration problems encountered when implementing the integral adaptation.The following lemmas are useful for the control design and the performance analysis to be presented:Lemma 1 (mean Value Theorem).
Functionf:a,b⟶Ris continuous on the closed intervala,band differentiable on the open intervala,b. Then, there existsc∈a,bsuch thatf′c=fb−fa/b−a.Lemma 2 (implicit Function theorem [2]).
Assume that functionf:Rn×R⟶Ris continuously differentiable, and there is a positive constantdsuch that for allx,u∈Rn×R,∂fx,u/∂u>d>0holds. Then, there exists only a continuous functiong:Rn⟶Rthat makesfx,gx=0.Lemma 3.
Letdkbe a sequence of real numbers (need not be positive). Assume that both sequences of positive numbersfkandgksatisfy(9)fk≤fk−1−gk+dk,wheref0is given and bounded. Then,(i)
if for allk,dk≤d¯, then limk⟶∞supgk≤d¯;(ii)
Asdk=0,limk⟶∞gk=0.Lemma 4.
(see [31]) Fora,b∈Rm, ifasatisfiesbimin≤a≤bimax, then(10)ζ+1a−ζb+satbTΛb−satb≤0,whereζ≥0is a constant andΛ>0is a diagonal matrix of appropriate dimensions. In particular, whenζ=0andΛis the identity matrix,a−satbTb−satb≤0holds.Throughout this article, radial basis function neural network (RBFNN) will be used to model nonlinear functions. As an efficient approximation tool, RBFNN is widely used in the adaptive controller design. It has been shown that, by selecting enough neurons, RBFNN can approximate any smooth function on a given compact set with the expected approximation accuracy.For smooth nonlinear functionfZ:ΩZ⊂Rq⟶R, using RBF network θ∗TφZ, fZ can be approximated as(11)fZ=θ∗TφZ+εZ,∀Z∈ΩZ,where θ∗ is an N-dimensional ideal constant weight vector, defined as(12)θ∗=argminθ∈RNsupZ∈ΩZfZ−θTφZ,where θ=θ1,θ2,…θNT is an N-dimensional weight vector, N is the number of neural network nodes, φZ=φ1Z,φ2Z,…φNZT∈RN is the basis vector consisting of a series of basis functions φiZ,i=1,…,N, and εZ is the approximation error that satisfies ε¯≥εZ>0,∀Z∈ΩZ.A commonly used basis function is the Gauss function, which has the following form:(13)φiZ=e−Z−ci2/2bi2,i=1,2,…,N,where Z is the network input, ci,i=1,2,…,N are the receptive field centers, and bi is the width of the Gaussian function. For Gauss RBF networks, the following lemma provides an upper bound on φZ, which is useful for the convergence analysis to be presented.Lemma 5.
[7] For Gauss RBF network, letρ=1/2mini≠jμi−μj, the upper bound ofφZcan be presented as(14)φZ≤∑i=0∞3qi+1q−1e−2ρ2i2/ζ2=χ.


## 3. Incremental Adaptive Control Algorithm

For simplify the presentation, we address the problem of trajectory tracking of a simple class of uncertain nonaffine systems as an illustrative example. The proposed IAC strategies are shown to be able to deal with the difficulty in controlling nonaffine nonlinear systems, and the control design will be extended to the higher-order systems.Let us consider the first-order nonaffine systems:(15)x˙=fx,u,where x∈R is the system state, u∈R is the control input, and fx,u:R×R⟶R is a smooth unknown nonlinear function.Definefu=∂fx,u/∂u, for which the following assumptions are made:Assumption 1.
The sign offuis known, andfu>0,∀x,u∈R×R.Assumption 2.
There exists a continuous functionβx>0,∀x∈R, such thatf˙u/fu≤βx.Remark 1.
Assumption1means that∂fx,u/∂uis strictly either positive or negative. Without loss of generality, suppose that there is a positive constantbfsuch thatfu≥bf>0. It should be noted that in the controller design to be presented, only Assumptions 1 and 2are required, and there is no need to know the true value ofbfand the exact expression ofβx.Here, the control goal is to design the controlleru for system (15) to guarantee that the system state x tracks the given desired trajectory xd with acceptable accuracy, while all closed-loop system signals are ensured to be bounded. For the given desired trajectory xdt, both itself and the first derivative with respect to time are bounded. To achieve this control objective, this article proposes IAC approaches for the adaptive control design, with incremental adaptive mechanisms, instead of using the conventional integral adaptation.


### 3.1. Controller Design

Let us define the tracking error of the system undertaken ase=x−xd. Its derivative can be written as follows, by adding and subtracting βxe:(16)e˙=fx,u−x˙d−βxe+βxe,=−βxe+fx,u+υx,xd,x˙d,where υx,xd,x˙d=−x˙d+βxx−xd. According to Lemma 2, for the given desired trajectory xd and the system state x, there always exists a smooth function αx,xd,x˙d such that(17)fx,u|u=αx,xd,x˙d+υx,xd,x˙d=0.Therefore, if the controller is taken as u=αx,xd,x˙d, the error system (16) can be rewritten as(18)e˙=−βxe+fx,αx,xd,x˙d+υx,xd,x˙d,=−βxe.Sinceβx>0, according to assumption 2, the error system e˙=−βxe is asymptotically stable, and e converges to zero asymptotically, that is, limt⟶∞et=0.To proceed, we apply RBF network to approximate the unknown smooth nonlinear functionαx,xd,x˙d, that is,(19)αx,xd,x˙d=θTφz+εz,z=x,xd,x˙dT,where θ represents the weight vector, φ indicates the vector of basis functions, and εz is the approximation error bounded by εz≤ε¯, with ε¯ being a positive constant.Now, the following controller can be designed as(20)u=−κe+θ^Tφz,where κ>0 is the adjustable gain, and θ^ is the estimated value of θ, for which the adaptation law is to be presented.Moreover, by applying Lemma1, there is a constant 0<λ<1 that satisfies(21)fx,u−fx,αx,xd,x˙d=fuλu−αx,xd,x˙d,where fuλ=∂fx,u¯/∂fx,u¯|u¯=uλ, and uλ=λu+1−λαx,xd,x˙d. Apparently, fuλ≥bf>0 according to Assumption 1.According to (19) and (21), and using controller (20), then the error dynamics (16) can be rewritten as follows:(22)e˙=−βxe+fx,αx,xd,x˙d+fuλu−αx,xd,x˙d+υx,xd,x˙d,=−βxe+fuλ−κe−θ˜Tφz−εz,where θ˜=θ−θ^ is the weight estimation error.


### 3.2. Performance Analysis

For the parameter adaptation, we first consider the following incremental adaptive law:(23)θ^t=θ^t−T−γφztet,t≥0,θ^0,t∈−T,0,where the constant θ^0>0 is the initial value of θ^t, T>0 is the learning cycle, and γ>0 is the adjustable parameter. By applying the incremental adaptation (23), we summarize the theoretical results including the boundedness of variables in the closed-loop system and the convergence of the tracking error in the following theorem.Theorem 1.
Considering the learning control system consisting of system (15) and controller (20), and the learning law (23), under assumptions 1-2. For any bounded initial condition,e,∫t−Ttθ^s2dsand∫t−Ttu2sdsare bounded, andeconverges to a neighborhood to the origin whose radius is proportional toε¯.Proof. Let us choose the Lyapunov-like function: L=2fuλ−1e2+2γ−1∫t−Ttθ˜Tsθ˜sds. Its derivative along (12) can be calculated as(24)L˙=1fuλe−βxe+fuλ−κe−θ˜Tφz−εz−f˙uλ2fuλ2e2+12γθ˜Ttθ˜t−θ˜Tt−Tθ˜t−T,≤−κe2−θ˜Ttφze−βxfuλe2−f˙uλ2fuλ2e2−eεz−1γθ˜Ttθ^t−θ^t−T,where(25)θ˜Ttθ˜t−θ˜Tt−Tθ˜t−T=−2θ^t−θ^t−TTθ˜t−θ^t−θ^t−TTθ^t−θ^t−T≤−2θ^t−θ^t−TTθ˜t,Employing the learning law (23) gives rise to(26)L˙≤−κe2−βxfuλe2−f˙uλ2fuλ2e2−eεz.According to Assumptions1 and 2, we have the following relationship:(27)−f˙uλ2fuλ2≤f˙uλ2fuλ2≤βx2fuλ<βxfuλ.In addition, using Young’s inequality yields(28)−eεz≤14ηe2+ηε2z≤14ηe2+ηε¯2,where η>0 is a constant. Substituting (27) and (28) into (26) gives rise to(29)L˙≤−κ−14ηe2+ηε¯2.In this way, whenκ>4η−1 and e>2ηε¯/4ηκ−1, L˙<0, ensuring that e is uniformly ultimately bounded, which means that there exists T1>0 such that et≤2ηε¯/4ηκ−1, for t≥T1. Then, integrating both sides of (29) over 0,t, we have(30)Lt≤L0−κ−14η∫0te2−4η2ε¯24ηκ−1ds.For any bounded initial condition,L0 is bounded. Due to the positive definiteness of Lt, for any finite time t, ∫0te2sds is bounded; otherwise, the right side of (30) tends to negative infinity, which contradicts the positive definiteness of Lt. Therefore, for any finite time, Lt is bounded, and according to the definition of Lt, the boundedness of ∫t−Ttθ^2ds can be guaranteed.By the inequalitya+b2≤2a2+2b2, it follows from (20) that(31)u2≤2κ2e2+2θ^Tφ2≤d1+d2θ^2,where d1=2κ2supt∈0,+∞e2 and d2=2supt∈0,+∞φ2. Furthermore,(32)∫t−Ttu2sds≤∫t−Ttd1+d2θ^s2ds≤d1T+d2∫t−Ttθ^s2ds<+∞,which implies that∫t−Ttu2sds is bounded..Remark 2.
As for the incremental adaptation law (23), the main feature is that the correction item,γφztet, depends entirely on the information at the current moment. We may call it the closed-loop incremental adaptive learning law. As an alternative, an open-loop learning law,(33)θ^t+T=θ^t−γφztet,t≥T,θ^0,t∈0,T,
is also valid for system (15).The following theorem illustrates the flexibility of choice of incremental adaptive learning mechanisms.Theorem 2.
Let the control laws(34)u=−κe+θ^Tφ+u1,(35)u1=−12γφTφe.Together with the learning law (33) be applied to system (15), then the same results as in Theorem 1 hold.Proof. We choose the Lyapunov–Krasovskii function candidate, Lt=Vt+2γ−1∫tt+Tθ˜Tsθ˜sds, where V=1/2fuλe2. By applying the control law (34), the error dynamics (16) can be rewritten as(36)e˙=−βxe+fuλ−κe−θ˜Tφ−εz+u1.Further using the control law (35), the derivative of V can be expressed as(37)V˙=−κe2−θ˜Tφze−βxfuλe2−f˙uλ2fuλ2e2−eεz−12γφTφe2.According to (36) and (37), the derivative of L is(38)L˙=−κe2−θ˜Tφze−βxfuλe2−f˙uλ2fuλ2e2−εe−12γφTφe2+12γθ˜Tt+Tθ˜t+T−θ˜Ttθ˜t.For the itemθ˜Tt+Tθ˜t+T in the right side of (38), we have(39)θ˜Tt+Tθ˜t+T=−2θ^t+T−θ^tTθ˜t+θ˜Ttθ˜t+θ^t+T−θ^tTθ^t+T−θ^t.Substituting (33) into (38), L˙ can be rewritten as(40)L˙=−κe2−eεz−βxfuλe2−f˙uλ2fuλ2e2−θ˜Tφe−12γφTzφze2+θ˜Tφe+12γφTzφze2≤−κe2−eεz,where −βx/fuλe2−f˙uλ/2fuλ2e2≤0, according to (27). Then, the proof can be carried out with similar lines to those of the proof for Theorem 1.Remark 3.
In (34), an additional componentu1is added into (20), for cancelling the extra term appeared when applying (33).Remark 4.
By Theorems 1 and 2, both the proposed incremental adaptive mechanisms (23) and (33) are applicable to the aforementioned system (15). In comparison with (8), noτappears in the correction item of the adaptation laws (23) and (33). Furthermore, taking the learning cycleTas the sampling period or an integer multiple of the sampling period, both the incremental adaptive learning laws (23) and (33) can be applied to the actual digital computer control system directly.Remark 5.
With incremental adaptation (23) or (33), the tracking errore,∫t−Ttθ^2dsand∫t−Ttu2sdsare assured to be bounded, whileeconverges to a neighborhood to the origin with the radius being proportional toε¯. It is worth noting that the boundedness of the parameter estimates and control input themselves are not guaranteed. In the following, we will provide more incremental adaptive learning mechanisms to achieve better control performance.


## 3.1. Controller Design

Let us define the tracking error of the system undertaken ase=x−xd. Its derivative can be written as follows, by adding and subtracting βxe:(16)e˙=fx,u−x˙d−βxe+βxe,=−βxe+fx,u+υx,xd,x˙d,where υx,xd,x˙d=−x˙d+βxx−xd. According to Lemma 2, for the given desired trajectory xd and the system state x, there always exists a smooth function αx,xd,x˙d such that(17)fx,u|u=αx,xd,x˙d+υx,xd,x˙d=0.Therefore, if the controller is taken as u=αx,xd,x˙d, the error system (16) can be rewritten as(18)e˙=−βxe+fx,αx,xd,x˙d+υx,xd,x˙d,=−βxe.Sinceβx>0, according to assumption 2, the error system e˙=−βxe is asymptotically stable, and e converges to zero asymptotically, that is, limt⟶∞et=0.To proceed, we apply RBF network to approximate the unknown smooth nonlinear functionαx,xd,x˙d, that is,(19)αx,xd,x˙d=θTφz+εz,z=x,xd,x˙dT,where θ represents the weight vector, φ indicates the vector of basis functions, and εz is the approximation error bounded by εz≤ε¯, with ε¯ being a positive constant.Now, the following controller can be designed as(20)u=−κe+θ^Tφz,where κ>0 is the adjustable gain, and θ^ is the estimated value of θ, for which the adaptation law is to be presented.Moreover, by applying Lemma1, there is a constant 0<λ<1 that satisfies(21)fx,u−fx,αx,xd,x˙d=fuλu−αx,xd,x˙d,where fuλ=∂fx,u¯/∂fx,u¯|u¯=uλ, and uλ=λu+1−λαx,xd,x˙d. Apparently, fuλ≥bf>0 according to Assumption 1.According to (19) and (21), and using controller (20), then the error dynamics (16) can be rewritten as follows:(22)e˙=−βxe+fx,αx,xd,x˙d+fuλu−αx,xd,x˙d+υx,xd,x˙d,=−βxe+fuλ−κe−θ˜Tφz−εz,where θ˜=θ−θ^ is the weight estimation error.


## 3.2. Performance Analysis

For the parameter adaptation, we first consider the following incremental adaptive law:(23)θ^t=θ^t−T−γφztet,t≥0,θ^0,t∈−T,0,where the constant θ^0>0 is the initial value of θ^t, T>0 is the learning cycle, and γ>0 is the adjustable parameter. By applying the incremental adaptation (23), we summarize the theoretical results including the boundedness of variables in the closed-loop system and the convergence of the tracking error in the following theorem.Theorem 1.
Considering the learning control system consisting of system (15) and controller (20), and the learning law (23), under assumptions 1-2. For any bounded initial condition,e,∫t−Ttθ^s2dsand∫t−Ttu2sdsare bounded, andeconverges to a neighborhood to the origin whose radius is proportional toε¯.Proof. Let us choose the Lyapunov-like function: L=2fuλ−1e2+2γ−1∫t−Ttθ˜Tsθ˜sds. Its derivative along (12) can be calculated as(24)L˙=1fuλe−βxe+fuλ−κe−θ˜Tφz−εz−f˙uλ2fuλ2e2+12γθ˜Ttθ˜t−θ˜Tt−Tθ˜t−T,≤−κe2−θ˜Ttφze−βxfuλe2−f˙uλ2fuλ2e2−eεz−1γθ˜Ttθ^t−θ^t−T,where(25)θ˜Ttθ˜t−θ˜Tt−Tθ˜t−T=−2θ^t−θ^t−TTθ˜t−θ^t−θ^t−TTθ^t−θ^t−T≤−2θ^t−θ^t−TTθ˜t,Employing the learning law (23) gives rise to(26)L˙≤−κe2−βxfuλe2−f˙uλ2fuλ2e2−eεz.According to Assumptions1 and 2, we have the following relationship:(27)−f˙uλ2fuλ2≤f˙uλ2fuλ2≤βx2fuλ<βxfuλ.In addition, using Young’s inequality yields(28)−eεz≤14ηe2+ηε2z≤14ηe2+ηε¯2,where η>0 is a constant. Substituting (27) and (28) into (26) gives rise to(29)L˙≤−κ−14ηe2+ηε¯2.In this way, whenκ>4η−1 and e>2ηε¯/4ηκ−1, L˙<0, ensuring that e is uniformly ultimately bounded, which means that there exists T1>0 such that et≤2ηε¯/4ηκ−1, for t≥T1. Then, integrating both sides of (29) over 0,t, we have(30)Lt≤L0−κ−14η∫0te2−4η2ε¯24ηκ−1ds.For any bounded initial condition,L0 is bounded. Due to the positive definiteness of Lt, for any finite time t, ∫0te2sds is bounded; otherwise, the right side of (30) tends to negative infinity, which contradicts the positive definiteness of Lt. Therefore, for any finite time, Lt is bounded, and according to the definition of Lt, the boundedness of ∫t−Ttθ^2ds can be guaranteed.By the inequalitya+b2≤2a2+2b2, it follows from (20) that(31)u2≤2κ2e2+2θ^Tφ2≤d1+d2θ^2,where d1=2κ2supt∈0,+∞e2 and d2=2supt∈0,+∞φ2. Furthermore,(32)∫t−Ttu2sds≤∫t−Ttd1+d2θ^s2ds≤d1T+d2∫t−Ttθ^s2ds<+∞,which implies that∫t−Ttu2sds is bounded..Remark 2.
As for the incremental adaptation law (23), the main feature is that the correction item,γφztet, depends entirely on the information at the current moment. We may call it the closed-loop incremental adaptive learning law. As an alternative, an open-loop learning law,(33)θ^t+T=θ^t−γφztet,t≥T,θ^0,t∈0,T,
is also valid for system (15).The following theorem illustrates the flexibility of choice of incremental adaptive learning mechanisms.Theorem 2.
Let the control laws(34)u=−κe+θ^Tφ+u1,(35)u1=−12γφTφe.Together with the learning law (33) be applied to system (15), then the same results as in Theorem 1 hold.Proof. We choose the Lyapunov–Krasovskii function candidate, Lt=Vt+2γ−1∫tt+Tθ˜Tsθ˜sds, where V=1/2fuλe2. By applying the control law (34), the error dynamics (16) can be rewritten as(36)e˙=−βxe+fuλ−κe−θ˜Tφ−εz+u1.Further using the control law (35), the derivative of V can be expressed as(37)V˙=−κe2−θ˜Tφze−βxfuλe2−f˙uλ2fuλ2e2−eεz−12γφTφe2.According to (36) and (37), the derivative of L is(38)L˙=−κe2−θ˜Tφze−βxfuλe2−f˙uλ2fuλ2e2−εe−12γφTφe2+12γθ˜Tt+Tθ˜t+T−θ˜Ttθ˜t.For the itemθ˜Tt+Tθ˜t+T in the right side of (38), we have(39)θ˜Tt+Tθ˜t+T=−2θ^t+T−θ^tTθ˜t+θ˜Ttθ˜t+θ^t+T−θ^tTθ^t+T−θ^t.Substituting (33) into (38), L˙ can be rewritten as(40)L˙=−κe2−eεz−βxfuλe2−f˙uλ2fuλ2e2−θ˜Tφe−12γφTzφze2+θ˜Tφe+12γφTzφze2≤−κe2−eεz,where −βx/fuλe2−f˙uλ/2fuλ2e2≤0, according to (27). Then, the proof can be carried out with similar lines to those of the proof for Theorem 1.Remark 3.
In (34), an additional componentu1is added into (20), for cancelling the extra term appeared when applying (33).Remark 4.
By Theorems 1 and 2, both the proposed incremental adaptive mechanisms (23) and (33) are applicable to the aforementioned system (15). In comparison with (8), noτappears in the correction item of the adaptation laws (23) and (33). Furthermore, taking the learning cycleTas the sampling period or an integer multiple of the sampling period, both the incremental adaptive learning laws (23) and (33) can be applied to the actual digital computer control system directly.Remark 5.
With incremental adaptation (23) or (33), the tracking errore,∫t−Ttθ^2dsand∫t−Ttu2sdsare assured to be bounded, whileeconverges to a neighborhood to the origin with the radius being proportional toε¯. It is worth noting that the boundedness of the parameter estimates and control input themselves are not guaranteed. In the following, we will provide more incremental adaptive learning mechanisms to achieve better control performance.


## 4. Extension to High-Order Nonaffine Nonlinear Systems

In this section, the proposed design methods are extended to the following class ofn th-order nonaffine nonlinear systems:(41)yn=fy,y1,…,yn−1,u,where y is the measurement output, u is the control input, yi,i=1,…,n is the i th time derivative of y, and the unknown smooth nonlinearity f⋅:Rn+1⟶R is an implicit function with respect to u.Defining the state vectorx=x1,x2,…,xnT=y,y1,…,yn−1T, system (41) can be rewritten in a state space model:(42)x˙1=x2,x˙2=x3,⋮x˙n=fx,u,y=x1.Letfu=∂fx,u/∂u, and the following assumption is used for system (42).Assumption 3.
Without loss of generality, we shall assume thatfu>0, and there are positive constantsf¯uandf¯uthat satisfies0<f¯u≤fu≤f¯u<∞,∀x,u∈Rn+1.The control goal of this section is to design an incremental-adaptation-based IAC controller for system (42) such that the system output follows the given smooth desired trajectory yd with acceptable accuracy, while all the signals in the closed-loop system are bounded. The given desired trajectory yd is continuous and differentiable, and ydt,yd1t,…,ydnt are all bounded for all t≥0.


### 4.1. IAC Design and Analysis

To design an IAC controller for the high-order nonaffine system (42), let us define the state error e=x−xd=e1,e2,…,enT, with xd=ydt,yd1t,…ydn−1tT, and the filtered error is defined as(43)ef=ΛT1e,with Λ=λ1,λ2,…λn−1T that is chosen to guarantee the polynomial sn−1+λn−1sn−2+⋯+λ1 is a Hurwitz. Obviously, e will converge to zero, as ef approaches zero. At the same time, the output error e1=y−yd converges to zero too. Then, the ef-dynamics can be given as(44)e˙f=0ΛTe+fx,u−ydn=Fx,υ,u,where Fx,υ,u=fx,u+υ, and υ=0ΛTe−ydn with ∂υ/∂u=0.By applying Lemma2, there is a continuous smooth function αx,υ that assures(45)Fx,υ,u|u=αx,υ=fx,αx,υ+υ=0.RBFNN is used for approximating the unknown smooth nonlinear functionαx,υ, in the manner of(46)αx,υ=θTφz+εz,z∈x,υT,where εz is the approximation error that satisfies εz<ε¯, and ε¯ is a positive constant.To proceed, applying Lemma1 again, we obtain that(47)Fx,υ,u=Fx,υ,αx,υ+Fuλu−αx,υ,where Fuλ=∂Fx,υ,u¯/∂u¯|u¯=uλ, and uλ=λu+1−λαx,υ, with λ∈0,1. According to Assumption 3 and the fact that ∂υ/∂u=0, it is obvious that(48)f¯u≤Fuλ=∂Fx,υ,uλ∂uλ,=∂fx,uλ+υ∂uλ≤f¯u.Then, the adaptive neural controller can be designed as(49)u=−κef+θ^Tφz,where κ>0 is the adjustable control gain, and θ^ is the estimate of the ideal weight θ.According to (45)–(49), the error system (44) can be rewritten as(50)e˙f=Fuλ−κef−θ˜Tφz−εz,where θ˜=θ−θ^ is the weight estimation error.Givenε¯, let θ∈RN be such that (46) hold. The results about the closed-loop stability and the error convergence of the proposed IAC scheme are summarized in the following theorem.Theorem 3.
Consider the IAC system described by system (42), adaptive controller (49), and the following incremental adaptive learning law:(51)1+γσθ^t=θ^t−T+γφzteft,and θ^t=0fort≤0, whereγandσare the adjustable parameters, under the assumption 3. Then, for bounded initial conditions, by selecting the parameters appropriately, all the signals in the closed-loop system are bounded, and the statexremains inΩx=xt‖eft|≤σ1/2κf−1/2ε¯2+θ2/2,t≥T1, where the positive constantκfsatisfiesκf≤κf¯u−2f¯u2χ2+f¯u2/4+2χ2/σ. Moreover, the filtered error satisfieslimt⟶∞sup∫t−Ttef2sds≤κf−1σTε¯2+θ2/2.Proof. Let us consider the Lyapunov function candidate, Lt=1/2ef2+1/2γ∫t−Ttθ˜Tsθ˜sds.Then, the derivative of Lt along (50) is(52)L˙t=Fuλ−κef2t−θ˜Ttφzeft−eftεz+12γθ˜Ttθ˜t−θ˜Tt−Tθ˜t−T≤Fuλ−κef2t−θ˜Ttφzeft−eftεz−1γθ˜Ttθ^t−θ^t−T,where the algebraic equation a−bTa−b−a−cTa−c=c−bT2a−b+b−c is used.Using the fact thatεz≤ε¯, and applying Young’s inequality yields(53)−Fuλθ˜Ttφzeft≤2Fuλ2σef2tφTzφz+σ8θ˜Ttθ˜t≤2Fuλ2χ2σef2t+σ8θ˜Ttθ˜t,(54)−Fuλefεz≤Fuλ24σef2+σε2z≤Fuλ24σef2+σε¯2,where φz≤χ, according to Lemma 5.Substituting (53) and (54) and the learning law (51) into (52) implies(55)L˙t≤−κFuλef2t+2Fuλ2χ2σef2t+σ8θ˜Ttθ˜t+Fuλ24σef2t+σε¯2−1γθ˜Ttθ^t−θ^t−T=−κFuλ−2Fuλ2χ2σ−Fuλ24σef2t+σ8θ˜Ttθ˜t+σε¯2−θ˜Ttφzeft+σθ˜Ttθ^t≤−κFuλ−2Fuλ2χ2σ−Fuλ24σ−2χ2σef2t+σ4θ˜Ttθ˜t+σε¯2+σθ˜Ttθ^t,where −θ˜Ttφzeft≤2/σef2tφz2+σ/8θ˜Ttθ˜t≤2χ2/σef2t+σ/8θ˜Ttθ˜t with φz≤χ, by using Young’s inequality. Subsequently, the following inequality can be derived by completing the square:(56)σθ˜Ttθ^t=σθ˜Ttθ−σθ˜Ttθ˜t≤σ2θ˜Ttθ˜t+σ2θ2−σθ˜Ttθ˜t=−σ2θ˜Ttθ˜t+σ2θ2,implying that(57)L˙t≤−κFuλ−2Fuλ2χ2σ−Fuλ24σ−2χ2σef2t−σ4θ˜Ttθ˜t+σε¯2+θ22≤−κf¯u−2f¯u2χ2σ−f¯u24σ−2χ2σef2t−σ4θ˜Ttθ˜t+σε¯2+θ22.The control parameter is selected appropriately so as to satisfy(58)κ>2f¯u2χ2+f¯u2/4+2χ2σf¯u.Then, it follows that(59)L˙t≤−κfef2t−σ4θ˜Ttθ˜t+σε¯2+θ22,which is ensured to be negative on condition that(60)ef2t≥σκfε¯2+θ22.As such, ifef>σ1/2κf−1/2ε¯2+θ2/2, L˙t will be negative and L will decrease. Therefore, outside this region, ef decreases and finally converges within the bound. This ensures the uniformly ultimate boundedness of ef, implying that there exists T1 such that the system state xt remains in Ωx, for t≥T1. Similarly, from (59), the UUB property of θ^t can also be guaranteed. Due to boundedness of x, θ^, and yd, it is easy from (49) to establish the boundedness of u.Now, integrating (59) over t−T,t results in(61)Lt≤Lt−T−κf∫t−Ttef2sds−σ4∫t−Ttθ˜s2ds+σ∫t−Ttε¯2+θ22ds≤Lt−T−κf∫t−Ttef2sds+σε¯2+θ22T.Considering (59) with t=t0∈0,T, it follows that L˙t0≤−κfef2t0+σε¯2+θ2/2≤σε¯2+θ2/2, which illustrates that Lt0 is finite, due to the boundedness of θ. Furthermore, with the aid of Lemma 3, the error variable ∫t−Ttef2sds could enter the specified bound σκf−1Tε¯2+θ2/2, that is,(62)limt⟶∞sup∫t−Ttef2sds≤σTκfε¯2+θ22.This completes the proof.Remark 6.
As for the learning algorithm (51), the modification is suggested by introducing the parameterσ, by which the finiteness of estimates is assured.


### 4.2. Saturated Learning

In order to assure the finiteness of estimates within a prespecified bound, the fully/partially saturate learning algorithms are applicable. Taking advantage of the fully saturated learning, the weight estimateθ^ defined in (49) can be updated as follows:(63)θ^t=satθ^∗t,θ^∗t=satθ^∗t−T+γφzteft.Then, the results of boundedness and convergence are summarized as Theorem4.Theorem 4.
If the control law given by (49) with the fully saturation incremental adaptive learning law (63) is applied to system (42), then, by selecting the parameters appropriately, all the signals in the closed-loop system are bounded, and the statexremains inΩx=xt‖eft|≤κf−1/2η1/22Bθ2+ε¯21/2,t≥T1, where the positive constantκfsatisfiesκf≤κf¯u−f¯u2χ2+f¯u2+χ2/4η, andBθis to be specified. Moreover, the filtered error satisfieslimt⟶∞sup∫t−Ttef2sds≤κf−1ηT2Bθ2+ε¯2.Proof. Let us consider the same Lyapunov-like function candidate as given in the proof for Theorem 3. Similarly, by following the same line used in (52), and applying (63), the time derivative of L can be obtained:(64)L˙t≤Fuλ−κef2t−θ˜Ttφzeft−eftεz−1γθ˜Ttθ^t−θ^t−T=Fuλ−κef2t−θ˜Ttφzeft−eftεz−1γθ˜Ttθ^t−θ^∗t−θ˜Ttφzeft.By invoking Lemma4, and we have the following inequality:(65)−1γθ˜Ttθ^t−θ^∗t=1γθ−satθ^∗tTθ^∗t−satθ^∗t≤0,implying that(66)L˙t≤−κFuλef2t−Fuλθ˜Ttφzeft−Fuλeftεz−θ˜Ttφzeft.By Young’s inequality, we obtain the following relationships:(67)−Fuλθ˜Ttφzeft≤Fuλ24ηφz2ef2t+ηθ˜Ttθ˜t≤Fuλ2χ24ηef2t+ηθ˜Ttθ˜t,(68)−εFuλeft≤Fuλ24ηef2t+ηε2≤Fuλ24ηef2t+ηε¯2,(69)−θ˜Ttφzeft≤14η‖φz‖2ef2t+ηθ˜Ttθ˜t≤χ24ηef2t+ηθ˜Ttθ˜t,where φz≤χ, by using Lemma 5, and the constant η>0, which can be arbitrarily small. It follows that(70)L˙t≤−κFuλ−Fuλ2χ24η−Fuλ24η−χ24ηef2t+2ηθ˜Ttθ˜t+ηε¯2≤−κf¯u−f¯u2χ24η−f¯u24η−χ24ηef2t+η2θ˜t2+ε¯2.By setting(71)κ>f¯u2χ2+f¯u2+χ24ηf¯u,we have(72)L˙t≤−κfef2t+η2θ˜t2+ε¯2.There exists a positive constantBθ such that θ˜t≤Bθ, for t≥0, due to the properties of fully saturated learning algorithm (63). Thus, as long as ef>κf−1/2η1/22Bθ2+ε¯21/2, L will decrease, and we conclude the UUB of ef. Then, xt will remain in Ωx, for t∈T1,+∞. The conclusions follow with similar lines to those in the proof for Theorem 3.Remark 7.
By Theorems 3 and 4, the robustness of the proposed IAC schemes is established, with the aid of Lemma 3. This lemma plays a key role to finalize the robust convergence analysis.Remark 8.
Both the incremental adaptive control schemes proposed in Theorems 3 and 4 are applicable for performing infinite-duration tasks. The presented theoretical results hold, as long asydtand its derivatives are bounded, whereas the periodicity thatydt=ydt−Tis not required, whereT>0is the period.Remark 9.
Similarly, taking advantage of the partially saturated learning algorithms developed in [30], we can also propose a partially saturated incremental adaptive learning mechanism:(73)θ^t=satθ^t−T+γφzteft.
However, in the case of using incremental adaptation (73), a dilemma arises in the corresponding stability and convergence analysis. Below, we briefly explain this difficulty. Consider the IAC system described by the system (42), the adaptive controller (33), and the learning law (73), under Assumption 3. In the case, a Lyapunov–Krasovskii function candidate,Lt=ef2t/2+2γ−1∫t−Ttθ−satθ^sTθ−satθ^sds, is suggested. Differentiating the functionLt,(74)ΔLt=Lt−Lt−T=12ef2t−12ef2t−T+12γ∫t−Ttθ−satθ^sTθ−satθ^s−θ−satθ^s−TTθ−satθ^s−Tds.whereef2t−ef2t−T/2=∫t−TtFuλ−κef2s−θ˜Tsφzefs−εefsds, according to (50). Using Young’s inequality for−Fuλθ˜Tsφzefsand−εFuλefssame as (67) and (68), we have−Fuλθ˜Tsφzefs≤Fuλ2χ2/4ηef2s+ηθ˜s2, and−εFuλefs≤Fuλ2/4ηef2s+ηε2. Furthermore, the following inequality can be derived by utilizing the incremental adaptation law (73):(75)θ−satθ^s−TTθ−satθ^s−T=θ−θ^s+γφzefsTθ−θ^s+γφzefs,=θ˜sTθ˜s+2γθ˜Tsφzefs+γ2φz2ef2s.
It follows that(76)ΔLt=∫t−Tt−κFuλef2s−Fuλθ˜Tsφzefs−εFuλefsds+12γ∫t−Ttθ−satθ^sTθ−satθ^s−θ˜sTθ˜s−2γθ˜Tsφzefs−γ2φs2ef2sds≤∫t−Tt−κFuλ−Fuλ2χ24η−Fuλ24ηef2s+ηθ˜s2+ε¯2ds+12γ∫t−Ttθ−satθ^sTθ−satθ^s−θ˜sTθ˜s−2γθ˜Tsφzefs−γ2φz2ef2sds≤∫t−Tt−κFuλ−Fuλ2χ24η−Fuλ24η−γ2χ2ηef2s+η2θ˜s2+ε¯2ds+12γ∫t−Ttθ−satθ^sTθ−satθ^s−θ˜sTθ˜sds.where−2γθ˜Tsφzefs≤γ2χ2/ηef2s+ηθ˜s2, upon using Young’s inequality. Besides, according to [30], θ−satθ^sTθ−satθ^s−θ˜sTθ˜s≤0, then(77)ΔLt≤∫t−Tt−κFuλ−Fuλ2χ24η−Fuλ24η−4γ2χ24ηef2s+η2θ˜s2+ε¯2ds.
Obviously, for the above inequality, the weight estimation errorθ˜is not guaranteed to be bounded, because the learning law (73) cannot ensure that the parameter estimateθ^is bounded in advance. Therefore, it is difficult to continue the performance analysis. Compared with partially saturated incremental adaptation (73), by the property of the saturation function, the entire right-hand side of the learning algorithm (63) is saturated, and the estimate for each parameter is guaranteed to be constrained in a prespecified region.


## 4.1. IAC Design and Analysis

To design an IAC controller for the high-order nonaffine system (42), let us define the state error e=x−xd=e1,e2,…,enT, with xd=ydt,yd1t,…ydn−1tT, and the filtered error is defined as(43)ef=ΛT1e,with Λ=λ1,λ2,…λn−1T that is chosen to guarantee the polynomial sn−1+λn−1sn−2+⋯+λ1 is a Hurwitz. Obviously, e will converge to zero, as ef approaches zero. At the same time, the output error e1=y−yd converges to zero too. Then, the ef-dynamics can be given as(44)e˙f=0ΛTe+fx,u−ydn=Fx,υ,u,where Fx,υ,u=fx,u+υ, and υ=0ΛTe−ydn with ∂υ/∂u=0.By applying Lemma2, there is a continuous smooth function αx,υ that assures(45)Fx,υ,u|u=αx,υ=fx,αx,υ+υ=0.RBFNN is used for approximating the unknown smooth nonlinear functionαx,υ, in the manner of(46)αx,υ=θTφz+εz,z∈x,υT,where εz is the approximation error that satisfies εz<ε¯, and ε¯ is a positive constant.To proceed, applying Lemma1 again, we obtain that(47)Fx,υ,u=Fx,υ,αx,υ+Fuλu−αx,υ,where Fuλ=∂Fx,υ,u¯/∂u¯|u¯=uλ, and uλ=λu+1−λαx,υ, with λ∈0,1. According to Assumption 3 and the fact that ∂υ/∂u=0, it is obvious that(48)f¯u≤Fuλ=∂Fx,υ,uλ∂uλ,=∂fx,uλ+υ∂uλ≤f¯u.Then, the adaptive neural controller can be designed as(49)u=−κef+θ^Tφz,where κ>0 is the adjustable control gain, and θ^ is the estimate of the ideal weight θ.According to (45)–(49), the error system (44) can be rewritten as(50)e˙f=Fuλ−κef−θ˜Tφz−εz,where θ˜=θ−θ^ is the weight estimation error.Givenε¯, let θ∈RN be such that (46) hold. The results about the closed-loop stability and the error convergence of the proposed IAC scheme are summarized in the following theorem.Theorem 3.
Consider the IAC system described by system (42), adaptive controller (49), and the following incremental adaptive learning law:(51)1+γσθ^t=θ^t−T+γφzteft,and θ^t=0fort≤0, whereγandσare the adjustable parameters, under the assumption 3. Then, for bounded initial conditions, by selecting the parameters appropriately, all the signals in the closed-loop system are bounded, and the statexremains inΩx=xt‖eft|≤σ1/2κf−1/2ε¯2+θ2/2,t≥T1, where the positive constantκfsatisfiesκf≤κf¯u−2f¯u2χ2+f¯u2/4+2χ2/σ. Moreover, the filtered error satisfieslimt⟶∞sup∫t−Ttef2sds≤κf−1σTε¯2+θ2/2.Proof. Let us consider the Lyapunov function candidate, Lt=1/2ef2+1/2γ∫t−Ttθ˜Tsθ˜sds.Then, the derivative of Lt along (50) is(52)L˙t=Fuλ−κef2t−θ˜Ttφzeft−eftεz+12γθ˜Ttθ˜t−θ˜Tt−Tθ˜t−T≤Fuλ−κef2t−θ˜Ttφzeft−eftεz−1γθ˜Ttθ^t−θ^t−T,where the algebraic equation a−bTa−b−a−cTa−c=c−bT2a−b+b−c is used.Using the fact thatεz≤ε¯, and applying Young’s inequality yields(53)−Fuλθ˜Ttφzeft≤2Fuλ2σef2tφTzφz+σ8θ˜Ttθ˜t≤2Fuλ2χ2σef2t+σ8θ˜Ttθ˜t,(54)−Fuλefεz≤Fuλ24σef2+σε2z≤Fuλ24σef2+σε¯2,where φz≤χ, according to Lemma 5.Substituting (53) and (54) and the learning law (51) into (52) implies(55)L˙t≤−κFuλef2t+2Fuλ2χ2σef2t+σ8θ˜Ttθ˜t+Fuλ24σef2t+σε¯2−1γθ˜Ttθ^t−θ^t−T=−κFuλ−2Fuλ2χ2σ−Fuλ24σef2t+σ8θ˜Ttθ˜t+σε¯2−θ˜Ttφzeft+σθ˜Ttθ^t≤−κFuλ−2Fuλ2χ2σ−Fuλ24σ−2χ2σef2t+σ4θ˜Ttθ˜t+σε¯2+σθ˜Ttθ^t,where −θ˜Ttφzeft≤2/σef2tφz2+σ/8θ˜Ttθ˜t≤2χ2/σef2t+σ/8θ˜Ttθ˜t with φz≤χ, by using Young’s inequality. Subsequently, the following inequality can be derived by completing the square:(56)σθ˜Ttθ^t=σθ˜Ttθ−σθ˜Ttθ˜t≤σ2θ˜Ttθ˜t+σ2θ2−σθ˜Ttθ˜t=−σ2θ˜Ttθ˜t+σ2θ2,implying that(57)L˙t≤−κFuλ−2Fuλ2χ2σ−Fuλ24σ−2χ2σef2t−σ4θ˜Ttθ˜t+σε¯2+θ22≤−κf¯u−2f¯u2χ2σ−f¯u24σ−2χ2σef2t−σ4θ˜Ttθ˜t+σε¯2+θ22.The control parameter is selected appropriately so as to satisfy(58)κ>2f¯u2χ2+f¯u2/4+2χ2σf¯u.Then, it follows that(59)L˙t≤−κfef2t−σ4θ˜Ttθ˜t+σε¯2+θ22,which is ensured to be negative on condition that(60)ef2t≥σκfε¯2+θ22.As such, ifef>σ1/2κf−1/2ε¯2+θ2/2, L˙t will be negative and L will decrease. Therefore, outside this region, ef decreases and finally converges within the bound. This ensures the uniformly ultimate boundedness of ef, implying that there exists T1 such that the system state xt remains in Ωx, for t≥T1. Similarly, from (59), the UUB property of θ^t can also be guaranteed. Due to boundedness of x, θ^, and yd, it is easy from (49) to establish the boundedness of u.Now, integrating (59) over t−T,t results in(61)Lt≤Lt−T−κf∫t−Ttef2sds−σ4∫t−Ttθ˜s2ds+σ∫t−Ttε¯2+θ22ds≤Lt−T−κf∫t−Ttef2sds+σε¯2+θ22T.Considering (59) with t=t0∈0,T, it follows that L˙t0≤−κfef2t0+σε¯2+θ2/2≤σε¯2+θ2/2, which illustrates that Lt0 is finite, due to the boundedness of θ. Furthermore, with the aid of Lemma 3, the error variable ∫t−Ttef2sds could enter the specified bound σκf−1Tε¯2+θ2/2, that is,(62)limt⟶∞sup∫t−Ttef2sds≤σTκfε¯2+θ22.This completes the proof.Remark 6.
As for the learning algorithm (51), the modification is suggested by introducing the parameterσ, by which the finiteness of estimates is assured.


## 4.2. Saturated Learning

In order to assure the finiteness of estimates within a prespecified bound, the fully/partially saturate learning algorithms are applicable. Taking advantage of the fully saturated learning, the weight estimateθ^ defined in (49) can be updated as follows:(63)θ^t=satθ^∗t,θ^∗t=satθ^∗t−T+γφzteft.Then, the results of boundedness and convergence are summarized as Theorem4.Theorem 4.
If the control law given by (49) with the fully saturation incremental adaptive learning law (63) is applied to system (42), then, by selecting the parameters appropriately, all the signals in the closed-loop system are bounded, and the statexremains inΩx=xt‖eft|≤κf−1/2η1/22Bθ2+ε¯21/2,t≥T1, where the positive constantκfsatisfiesκf≤κf¯u−f¯u2χ2+f¯u2+χ2/4η, andBθis to be specified. Moreover, the filtered error satisfieslimt⟶∞sup∫t−Ttef2sds≤κf−1ηT2Bθ2+ε¯2.Proof. Let us consider the same Lyapunov-like function candidate as given in the proof for Theorem 3. Similarly, by following the same line used in (52), and applying (63), the time derivative of L can be obtained:(64)L˙t≤Fuλ−κef2t−θ˜Ttφzeft−eftεz−1γθ˜Ttθ^t−θ^t−T=Fuλ−κef2t−θ˜Ttφzeft−eftεz−1γθ˜Ttθ^t−θ^∗t−θ˜Ttφzeft.By invoking Lemma4, and we have the following inequality:(65)−1γθ˜Ttθ^t−θ^∗t=1γθ−satθ^∗tTθ^∗t−satθ^∗t≤0,implying that(66)L˙t≤−κFuλef2t−Fuλθ˜Ttφzeft−Fuλeftεz−θ˜Ttφzeft.By Young’s inequality, we obtain the following relationships:(67)−Fuλθ˜Ttφzeft≤Fuλ24ηφz2ef2t+ηθ˜Ttθ˜t≤Fuλ2χ24ηef2t+ηθ˜Ttθ˜t,(68)−εFuλeft≤Fuλ24ηef2t+ηε2≤Fuλ24ηef2t+ηε¯2,(69)−θ˜Ttφzeft≤14η‖φz‖2ef2t+ηθ˜Ttθ˜t≤χ24ηef2t+ηθ˜Ttθ˜t,where φz≤χ, by using Lemma 5, and the constant η>0, which can be arbitrarily small. It follows that(70)L˙t≤−κFuλ−Fuλ2χ24η−Fuλ24η−χ24ηef2t+2ηθ˜Ttθ˜t+ηε¯2≤−κf¯u−f¯u2χ24η−f¯u24η−χ24ηef2t+η2θ˜t2+ε¯2.By setting(71)κ>f¯u2χ2+f¯u2+χ24ηf¯u,we have(72)L˙t≤−κfef2t+η2θ˜t2+ε¯2.There exists a positive constantBθ such that θ˜t≤Bθ, for t≥0, due to the properties of fully saturated learning algorithm (63). Thus, as long as ef>κf−1/2η1/22Bθ2+ε¯21/2, L will decrease, and we conclude the UUB of ef. Then, xt will remain in Ωx, for t∈T1,+∞. The conclusions follow with similar lines to those in the proof for Theorem 3.Remark 7.
By Theorems 3 and 4, the robustness of the proposed IAC schemes is established, with the aid of Lemma 3. This lemma plays a key role to finalize the robust convergence analysis.Remark 8.
Both the incremental adaptive control schemes proposed in Theorems 3 and 4 are applicable for performing infinite-duration tasks. The presented theoretical results hold, as long asydtand its derivatives are bounded, whereas the periodicity thatydt=ydt−Tis not required, whereT>0is the period.Remark 9.
Similarly, taking advantage of the partially saturated learning algorithms developed in [30], we can also propose a partially saturated incremental adaptive learning mechanism:(73)θ^t=satθ^t−T+γφzteft.
However, in the case of using incremental adaptation (73), a dilemma arises in the corresponding stability and convergence analysis. Below, we briefly explain this difficulty. Consider the IAC system described by the system (42), the adaptive controller (33), and the learning law (73), under Assumption 3. In the case, a Lyapunov–Krasovskii function candidate,Lt=ef2t/2+2γ−1∫t−Ttθ−satθ^sTθ−satθ^sds, is suggested. Differentiating the functionLt,(74)ΔLt=Lt−Lt−T=12ef2t−12ef2t−T+12γ∫t−Ttθ−satθ^sTθ−satθ^s−θ−satθ^s−TTθ−satθ^s−Tds.whereef2t−ef2t−T/2=∫t−TtFuλ−κef2s−θ˜Tsφzefs−εefsds, according to (50). Using Young’s inequality for−Fuλθ˜Tsφzefsand−εFuλefssame as (67) and (68), we have−Fuλθ˜Tsφzefs≤Fuλ2χ2/4ηef2s+ηθ˜s2, and−εFuλefs≤Fuλ2/4ηef2s+ηε2. Furthermore, the following inequality can be derived by utilizing the incremental adaptation law (73):(75)θ−satθ^s−TTθ−satθ^s−T=θ−θ^s+γφzefsTθ−θ^s+γφzefs,=θ˜sTθ˜s+2γθ˜Tsφzefs+γ2φz2ef2s.
It follows that(76)ΔLt=∫t−Tt−κFuλef2s−Fuλθ˜Tsφzefs−εFuλefsds+12γ∫t−Ttθ−satθ^sTθ−satθ^s−θ˜sTθ˜s−2γθ˜Tsφzefs−γ2φs2ef2sds≤∫t−Tt−κFuλ−Fuλ2χ24η−Fuλ24ηef2s+ηθ˜s2+ε¯2ds+12γ∫t−Ttθ−satθ^sTθ−satθ^s−θ˜sTθ˜s−2γθ˜Tsφzefs−γ2φz2ef2sds≤∫t−Tt−κFuλ−Fuλ2χ24η−Fuλ24η−γ2χ2ηef2s+η2θ˜s2+ε¯2ds+12γ∫t−Ttθ−satθ^sTθ−satθ^s−θ˜sTθ˜sds.where−2γθ˜Tsφzefs≤γ2χ2/ηef2s+ηθ˜s2, upon using Young’s inequality. Besides, according to [30], θ−satθ^sTθ−satθ^s−θ˜sTθ˜s≤0, then(77)ΔLt≤∫t−Tt−κFuλ−Fuλ2χ24η−Fuλ24η−4γ2χ24ηef2s+η2θ˜s2+ε¯2ds.
Obviously, for the above inequality, the weight estimation errorθ˜is not guaranteed to be bounded, because the learning law (73) cannot ensure that the parameter estimateθ^is bounded in advance. Therefore, it is difficult to continue the performance analysis. Compared with partially saturated incremental adaptation (73), by the property of the saturation function, the entire right-hand side of the learning algorithm (63) is saturated, and the estimate for each parameter is guaranteed to be constrained in a prespecified region.


## 5. Numerical Simulation

In this section, two examples are used to test the effectiveness of the presented incremental-adaptation-based IAC control schemes. The approach proposed in Theorem4 will be utilized to construct controllers to control the systems under consideration.Example 1.
Consider the following second-order nonaffine system [11]:(78)x˙1=x2,x˙2=fx,u,y=x1,where x=x1,x2T stands for the system state, and the nonlinear function fx,u satisfies(79)fx,u=u+0.5sinu,ifx1+x2<0,sinx1+x2+u+0.5sinu,otherwise.
From the system descriptions above, we can achievefu=∂fx,u/∂u>0 such that Assumption 3 holds. In addition, we assume that there is no a priori of uncertain nonlinearities in the abovementioned nonaffine systems. The reference trajectory is selected as ydt=0.5cos0.5πt.
The controller given by (49) is applied, and the weight adaptation law (63) is adopted in this simulation, with the following parameter settings: κ=30,λ1=12,γ=2,T=0.005,θ^imin=−10,θ^imax=10,i=1,2,…,1089. In addition, the centers of Gaussian RBFNN, ci,i=1,2,…,1089, are evenly spaced on −1,1×−4,4×−10,10, with the width bi=0.06. In the simulation, we set the initial state as x0=0,0T, and the initial weight estimate is θ^0=0.
The simulation results are shown in Figures1–6. Figures 1–4 depict the tracking performance of the IAC system, where Figure 1 shows the output tracking of the system, Figure 2 describes the convergence behavior of the system state, and Figures 3 and 4 give the corresponding state errors e1 and e2, respectively. Adopting the proposed IAC scheme, the output error converges to a neighborhood of the origin and remains there; that is, e1 converges to −2.0,4.0×10−3. The resultant input is shown in Figure 5. In addition, Figure 6 shows the boundedness and convergence of weight estimates θ^. The presented numerical results demonstrate that the boundedness of all variables of the closed-loop system and the convergence of the tracking error are achieved by the proposed IAC scheme.Figure 1
The reference signalyd and output signal y for Example 1.Figure 2
Convergence behaviorx⟶xd for Example 1.Figure 3
State errore1 for Example 1.Figure 4
State errore2 for Example 1.Figure 5
Control inputu for Example 1.Figure 6
Weights estimatesθ^ for Example 1.Example 2.
To further illustrate the effectiveness of the proposed algorithm for higher-order and more complex nonaffine nonlinear systems, we consider the following system:(80)x˙1=x2,x˙2=x3,x˙3=x1−0.35sinx22+x12+2+0.5sinx1x2u+0.34cosu2−cosx3+0.57x12x22,y=x1,where x=x1,x2,x3T is the system state. For the nonlinear function fx,u=x1−0.35sinx22+x12+2+0.5sinx1x2u+0.34cosu/2−cosx3+0.57x12x22, we have fu=∂fx,u/∂u>0 that Assumption 3 is also holds. The reference trajectory is selected as ydt=0.5cos0.5πt.
The same controller, together with the same adaptation laws is applied, and the control parameters are chosen asκ=30,λ1=10,λ2=5,γ=0.5,T=0.005,θ^imin=−10, and θ^imax=10,i=1,2,…,3025. In addition, the centers ci,i=1,2,…,3025 are evenly spaced in −0.6,0.6×−1.0,1.0×−2,3×−15,15, and the width bi=0.1. In this simulation, the initial state is x0=0,0,0T.
The simulation results are shown in Figures7–13. Similarly, the results about the boundedness and the error convergence of the proposed IAC control scheme are obtained. The state tracking performance is shown in Figures 7–11, and it is observed from Figure 9 that the boundary range of the ultimate output error is −3.4,3.6×10−3. Figure 12 shows the corresponding control input. In Figure 13, the convergence of the weight estimates is given.Figure 7
The reference signalyd and output signal y for Example 2.Figure 8
Convergence behaviorx⟶xd for Example 2.Figure 9
State errore1 for Example 2.Figure 10
State errore2 for Example 2.Figure 11
State errore3 for Example 2.Figure 12
Control inputu for Example 2.Figure 13
Weights estimatesθ^ for Example 2.


## 6. Conclusion

The incremental adaptation mechanisms are suggested in this article, including the so-called closed-loop, open-loop, and fully saturated learning mechanisms, and based on them, IAC schemes using neural network for nonaffine systems with infinite interval tracking are developed. With the proposed incremental-adaptation-based IAC schemes, there is no periodicity or repeatability requirements, while the numerical integration for implementation can be effectively avoid. Additionally, the robust convergence of the proposed learning control schemes is established, with the use of a key lemma. Theoretical results of establishing convergence performance of the tracking error and the numerical results have been presented, which demonstrate the effectiveness of the IAC schemes.



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