# Partial-State-Constrained Adaptive Intelligent Tracking Control of Nonlinear Nonstrict-Feedback Systems with Unmodeled Dynamics and Its Application

Authors: Yuzhuo Zhao; Ben Niu; Xiaoli Jiang; Ping Zhao; Huanqing Wang; Dong Yang
Journal: Complexity
 (2020)

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## Abstract

In this paper, an adaptive intelligent control scheme is presented to investigate the problem of adaptive tracking control for a class of nonstrict-feedback nonlinear systems with constrained states and unmodeled dynamics. By approximating the unknown nonlinear uncertainties, utilizing Barrier Lyapunov functions (BLFs), and designing a dynamic signal to deal with the constrained states and the unmodeled dynamics, respectively, an adaptive neural network (NN) controller is developed in the frame of the backstepping design. In order to simplify the design process, the nonstrict-feedback form is treated by using the special properties of Gaussian functions. The proposed adaptive control scheme ensures that all variables involved in the closed-loop system are bounded, the corresponding state constraints are not violated. Meanwhile, the tracking error converges to a small neighborhood of the origin. In the end, the proposed intelligent design algorithm is applied to one-link manipulator to demonstrate the effectiveness of the obtained method.


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## Body

## 1. Introduction

Over the past few decades, nonlinear control systems, which can be employed to model numerous applications such as biological systems, chemical processes, and aerospace vehicles, have aroused a wide range of concerns among researchers. In this area, adaptive control of nonlinear systems with uncertainties is a very active research subject. The backstepping method, as we all know, has been proposed in [1] as an effective method to solve the adaptive control problem of nonlinear strict-feedback systems with mismatched uncertainties. With the rapid development of adaptive control theory, the backstepping method has been widely used in the control design of different complex nonlinear systems such as interconnected large-scale systems, MIMO systems, and unmodeled dynamic systems [2–4].On the other hand, unmodeled dynamics are common phenomenon in practical applications, which are mainly caused by modeling errors and external disturbances. The existence of unmodeled dynamics usually degrades control performance or leads to the instability for a control system, and thus dealing with unmodeled dynamics has drawn considerable attention from many scholars in the control field. In recent years, adaptive control of uncertain nonlinear systems with unmodeled dynamics has become a research hot spot, and many related achievements have been reported [5–9]. To just name a few, a robust controller is designed in [5] for a class of nonlinear systems with unmodeled dynamics by using the method of adaptive backstepping and introducing a dynamic signal; and an adaptive output feedback controller is designed in [6] for a class of stochastic nonlinear systems with output unmodeled dynamics by using a stochastic small-gain theorem. Obviously, all the above-mentioned references are nonlinear strict-feedback systems with unmodeled dynamics, and there are few results on nonlinear nonstrict-feedback systems with unmodeled dynamics at present. Therefore, how to deal with the nonstrict-feedback form is one of the hardest issues in the research field of nonlinear systems.It should be pointed out that none of the above related adaptive results can be used to deal with completely unknown nonlinearities of control systems. To solve this problem, some elegant intelligent adaptive control algorithms are proposed by using NN or fuzzy logic systems [10–32]. Up to now, great progress has been made in the area of adaptive intelligent control for uncertain nonlinear systems with unmodeled dynamics, and a large number of valuable results are presented in [33–44]. For example, several adaptive intelligent control schemes are proposed in [33, 44] for several classes of nonlinear systems made up of unmodeled dynamics and strict-feedback form by using fuzzy logic and NN compensators, respectively. A suitable learning controller is proposed in [39] to overcome the disadvantages caused by parameter uncertainties and unmodeled dynamics for a class of multi-input and multi-output nonlinear systems. Meanwhile, the corresponding results have been obtained for interconnected nonlinear systems with unmodeled dynamics in [40]. However, the problem of constraints inevitably appears in various systems. The research in the field of handling complicated constraints has been paid more and more attention by researchers, and by utilizing the BLFs or the nonlinear mappings (NMs) to deal with state constraints or output constraints, a series of significant results have been obtained [45–48]. However, there exist a few intelligent control algorithms for nonstrict-feedback nonlinear systems with unmodeled dynamics to deal with the state constraints until now.Motivated by the above research situation, this paper proposes an adaptive tracking control strategy for a class of nonstrict-feedback nonlinear systems with unmodeled dynamics and state constraints. The unknown functions are estimated by NN, then a dynamic signal is designed to handle the dynamic uncertainties to ensure that the considered system can be controlled effectively. Meanwhile, by using the BLFs to handle the state constraints, the proposed adaptive control approach can guarantee the boundedness of all the signals in the closed-loop system, and the tracking error converges to a small neighborhood of the origin. The main contributions of the proposed method are summarized as follows: (1) Compared with the variable partition technique in [36], this paper uses the essential property of Gaussian functions to deal with the nonstrict-feedback form, so that the controller design process is relatively simpler. (2) Barrier functions are applied in the design process to constrain state variables into the specified regions, despite the presence of unmodeled dynamics at the front end of the studied system. (3) This paper adopts a dynamic signal to handle the dynamic uncertainties to ensure the considered system can be controlled effectively so that the conservative assumption about unmodeled dynamics in [5] is not used.


## 2. Problem Formulation and Preliminaries

### 2.1. Problem Formulation

In this paper, we study a class of nonstrict-feedback nonlinear systems with unmodeled dynamics as follows:(1)χ˙=pχ,ξ,ξ˙i=fiξ¯iξi+1+giξ+Οiξ,χ,t,i=1,…,n−1,ξ˙n=fnξ¯nu+gnξ+Οnξ,χ,t,η=ξ1,where ξ=ξ¯n=ξ1,ξ2,…,ξnT represents the state vector, and u and η depict the system input and output, severally; partial states are constrained in the compact sets; i.e., ξi is required to remain in the sets ξi<kci with kci being positive constants, i=1,2,…,n; χ∈Rn0 are the unmodeled dynamics, Οi⋅, i=1,2…,n represent nonlinear disturbances, and gi⋅ and fi⋅, i=1,2…,n are smooth uncertain functions with gi0=0. It is assumed that Οi⋅ and p⋅ are indeterminate continuous Lipschitz functions.Remark 1.
Plant (1) has a nonstrict-feedback structure, where the diffusion terms gi⋅, i=1,2…,n are the functions of ξ=ξ1,…,ξiT, which is different from the strict-feedback structure in [49] and the semistrict-feedback structure in [50], because the functions gi⋅ are relevant to all stats of ξ.
We will establish an adaptive intelligent controller for system [1] so that the output η can track a given trajectory ηd, the corresponding state constraints are not violated, and all the reference signals of the closed-loop system are bounded. Therefore, we give the following assumptions.Assumption 1.
(see [51]). For system (1), there is an unknown constant bm>0 satisfying(2)0<bm≤fiξ¯i<∞.Assumption 2.
(see [51]). ηdt is a reference signal, its up to nth-order are bounded and smooth. There exists a positive constant d such that ηdt≤d<kc1.Assumption 3.
(see [5]). For i=1,…,n the function Οi⋅ in (1) satisfies the following inequalities:(3)Οiξ,χ,t≤φi1ξ¯i+φi2χ,where φi1⋅ and φi2⋅ are unknown nonnegative increasing smooth functions with φi20=0.Assumption 4.
(see [5]). For χ˙=pχ,ξ in (1), there is a Lyapunov function Vχ such that(4)ω1χ≤Vχ≤ω2χ,(5)∂Vχ∂χpχ,ξ≤−k0Vχ+γξ1+d0,where ω1,ω2 and γ represent class K∞-functions, and k0>0 and d0>0 are known scalars.Remark 2.
Assumption1 implies that the unknown functions fikχi are strictly positive or negative. Further, let us assume that 0<bm≤fiξ¯i<∞ for generality. Assumption 2 is required in many literatures on the tracking control problem such as [36, 40], since we need to figure out its time derivatives up to n th in the design process. There exists the similar assumption in [5], where φi1⋅ and φi2⋅ are assumed to be available. However, Assumption 3 does not need this restriction and is thus more relaxed. Assumption 4 is the key condition to ensure the stability of the unmodeled dynamics in (1).


### 2.2. Preliminaries

To facilitate the design and analysis, the following Lemmas are given.Lemma 1.
(see [51]). For χ˙=pχ,ξ, there is a Lyapunov function V that satisfies (4) and (5), then for any values k¯ in 0,k0, functions γ¯ξ1≥γ¯ξ1 and initial value χ0=χ00, there is a limited time T0=T0k¯,v0,χ0, Bt≥0 for all t>0 and a signal is represented by(6)v˙=−k¯v+γ¯ξ1t+d0,v0=v0,such that Bt=0 for all t≥T0(7)Vχt≤vt+Bt.The solutions are specified for∀t>0. We can select γ¯s=s2γ0s2, where γ¯⋅>0 is a smooth function. Now, (7) makes(8)v˙=−k¯v+ξ12γ0ξ12+d0,v0=v0,where γ0 is a nonnegative smooth function.Lemma 2.
(see [52]). For any ζ∈R and ωn>0, the following relation holds: 0≤ζ−ζtanhζ/ε≤δω0,δ=0.2785.Lemma 3.
(see [53]). For ∀x,y∈Rn, it can be obtained that xy≤εp/pxp+1/qεqyq where ε>0, p>1, q>1, and p−1q−1=1.Lemma 4.
(see [54]). Assume that Ωx1 is defined as Ωx1:=x1|x1<0.8814v, and the inequality 1−tanh2x1/r≤0 is satisfied for any x1∉Ωx1.In this paper, the smooth functiongX:Rn⟶R is estimated by radial basis functions (RBF) NN gnnX. The RBF NN can be written as(9)gnnX=ETHX,where E=e1,…,elT∈Rl with l>1 is weight vector, X∈Rq is input vector, and HX=h1X,…,hlXT is the basis function vector of the Gaussian function. hiX can be expressed as(10)hiX=exp−X−γiTX−γiy2,i=1,…,l,where γi=γi1,…,γiqT is the center of the receptive field, and y is the width of the Gaussian function. The RBF NN (9) with sufficiently large node number l can approximate any continuous function gX over a compact set ΩX∈Rq to arbitrarily accuracy ε>0 as(11)gX=E∗THX+δX,∀X∈ΩX∈Rq,where E∗ is the desired weight vector and chosen as E∗=argminE∈R¯lsupX∈ΩXgX−ETHX and δX denotes the approximation error for δX<ε.Lemma 5.
(see [54]). Let ξ¯q=ξ1,…,ξqT and Hξ¯q=H1ξ¯q,…,Hlξ¯qT be the basis function vector of the RBF NN. Now, for ∀k,q∈N+ and k≤q, we have Hξ¯q2≤Hξ¯k2.


## 2.1. Problem Formulation

In this paper, we study a class of nonstrict-feedback nonlinear systems with unmodeled dynamics as follows:(1)χ˙=pχ,ξ,ξ˙i=fiξ¯iξi+1+giξ+Οiξ,χ,t,i=1,…,n−1,ξ˙n=fnξ¯nu+gnξ+Οnξ,χ,t,η=ξ1,where ξ=ξ¯n=ξ1,ξ2,…,ξnT represents the state vector, and u and η depict the system input and output, severally; partial states are constrained in the compact sets; i.e., ξi is required to remain in the sets ξi<kci with kci being positive constants, i=1,2,…,n; χ∈Rn0 are the unmodeled dynamics, Οi⋅, i=1,2…,n represent nonlinear disturbances, and gi⋅ and fi⋅, i=1,2…,n are smooth uncertain functions with gi0=0. It is assumed that Οi⋅ and p⋅ are indeterminate continuous Lipschitz functions.Remark 1.
Plant (1) has a nonstrict-feedback structure, where the diffusion terms gi⋅, i=1,2…,n are the functions of ξ=ξ1,…,ξiT, which is different from the strict-feedback structure in [49] and the semistrict-feedback structure in [50], because the functions gi⋅ are relevant to all stats of ξ.
We will establish an adaptive intelligent controller for system [1] so that the output η can track a given trajectory ηd, the corresponding state constraints are not violated, and all the reference signals of the closed-loop system are bounded. Therefore, we give the following assumptions.Assumption 1.
(see [51]). For system (1), there is an unknown constant bm>0 satisfying(2)0<bm≤fiξ¯i<∞.Assumption 2.
(see [51]). ηdt is a reference signal, its up to nth-order are bounded and smooth. There exists a positive constant d such that ηdt≤d<kc1.Assumption 3.
(see [5]). For i=1,…,n the function Οi⋅ in (1) satisfies the following inequalities:(3)Οiξ,χ,t≤φi1ξ¯i+φi2χ,where φi1⋅ and φi2⋅ are unknown nonnegative increasing smooth functions with φi20=0.Assumption 4.
(see [5]). For χ˙=pχ,ξ in (1), there is a Lyapunov function Vχ such that(4)ω1χ≤Vχ≤ω2χ,(5)∂Vχ∂χpχ,ξ≤−k0Vχ+γξ1+d0,where ω1,ω2 and γ represent class K∞-functions, and k0>0 and d0>0 are known scalars.Remark 2.
Assumption1 implies that the unknown functions fikχi are strictly positive or negative. Further, let us assume that 0<bm≤fiξ¯i<∞ for generality. Assumption 2 is required in many literatures on the tracking control problem such as [36, 40], since we need to figure out its time derivatives up to n th in the design process. There exists the similar assumption in [5], where φi1⋅ and φi2⋅ are assumed to be available. However, Assumption 3 does not need this restriction and is thus more relaxed. Assumption 4 is the key condition to ensure the stability of the unmodeled dynamics in (1).


## 2.2. Preliminaries

To facilitate the design and analysis, the following Lemmas are given.Lemma 1.
(see [51]). For χ˙=pχ,ξ, there is a Lyapunov function V that satisfies (4) and (5), then for any values k¯ in 0,k0, functions γ¯ξ1≥γ¯ξ1 and initial value χ0=χ00, there is a limited time T0=T0k¯,v0,χ0, Bt≥0 for all t>0 and a signal is represented by(6)v˙=−k¯v+γ¯ξ1t+d0,v0=v0,such that Bt=0 for all t≥T0(7)Vχt≤vt+Bt.The solutions are specified for∀t>0. We can select γ¯s=s2γ0s2, where γ¯⋅>0 is a smooth function. Now, (7) makes(8)v˙=−k¯v+ξ12γ0ξ12+d0,v0=v0,where γ0 is a nonnegative smooth function.Lemma 2.
(see [52]). For any ζ∈R and ωn>0, the following relation holds: 0≤ζ−ζtanhζ/ε≤δω0,δ=0.2785.Lemma 3.
(see [53]). For ∀x,y∈Rn, it can be obtained that xy≤εp/pxp+1/qεqyq where ε>0, p>1, q>1, and p−1q−1=1.Lemma 4.
(see [54]). Assume that Ωx1 is defined as Ωx1:=x1|x1<0.8814v, and the inequality 1−tanh2x1/r≤0 is satisfied for any x1∉Ωx1.In this paper, the smooth functiongX:Rn⟶R is estimated by radial basis functions (RBF) NN gnnX. The RBF NN can be written as(9)gnnX=ETHX,where E=e1,…,elT∈Rl with l>1 is weight vector, X∈Rq is input vector, and HX=h1X,…,hlXT is the basis function vector of the Gaussian function. hiX can be expressed as(10)hiX=exp−X−γiTX−γiy2,i=1,…,l,where γi=γi1,…,γiqT is the center of the receptive field, and y is the width of the Gaussian function. The RBF NN (9) with sufficiently large node number l can approximate any continuous function gX over a compact set ΩX∈Rq to arbitrarily accuracy ε>0 as(11)gX=E∗THX+δX,∀X∈ΩX∈Rq,where E∗ is the desired weight vector and chosen as E∗=argminE∈R¯lsupX∈ΩXgX−ETHX and δX denotes the approximation error for δX<ε.Lemma 5.
(see [54]). Let ξ¯q=ξ1,…,ξqT and Hξ¯q=H1ξ¯q,…,Hlξ¯qT be the basis function vector of the RBF NN. Now, for ∀k,q∈N+ and k≤q, we have Hξ¯q2≤Hξ¯k2.


## 3. Main Result

For system (1), this part gives the concrete design process of the controller through the backstepping algorithm. The adaptive neural backstepping design requires n steps. The virtual control input in step i is designed as αii=1,…,n−1, and the real controller u is added in step n to form a stabilized closed-loop system. They are represented separately as(12)αi=−cixi−12ai2xikbi2−xi2θ^iHiTXiHiXi,1≤i≤n−1,(13)u=−cnxn−12an2xnkbn2−xn2θ^nHnTXnHnXn,where the design parameters are ci>0 and ai>0, Xi=ξ¯iT,θ^¯iT,η¯diT,vT with ξ¯i=ξi,ξ2,…,ξiT, θ^¯i=θ^1,θ^2,…,θ^iT, η¯di=ηd,η˙d,…,ηdiT with ηdi=diηd/dti, and the uncertain parameter θi is estimated to be θ^i. The transformations of coordinates are selected as follows:(14)xi=ξi−αi−1,i=1,2,…,n,where α0=ηdt.The adaption laws are expressed as(15)θ^˙i=κi2ai2xikbi2−xi22HiTXiHXi−μiθ^i,i=1,2,…,n,where the design parameters are κi>0 and μi>0.For clarity, let us abbreviate the functionsfiξ¯i to fi and set Οiξ,χ,t=Οi and HiXi=Hi.Now, let us start the design process.Step 1.
Based onx1=ξ1−ηd, one has(16)x˙1=f1ξ2+g1ξ+Ο1−η˙d.
Next, choose the following Lyapunov function:(17)V1=12logkb12kb12−x12+1κ0v+bm2κ1θ˜12,where logϑ stands for the natural logarithm of ϑ, θ˜1=θ1−θ^1 denotes the parameter error, and κ0 and κ1 are positive constants. In the set Ωx1, V1 is continuous.
Thus, according to Assumption2, the derivative of V1 along with (8) leads to(18)V˙1≤x1kb12−x12f1ξ2+g1ξ−η˙d+x1kb12−x12φ11ξ1+x1kb12−x12φ12χ+1κ0ξ12γ0ξ12+d0−k¯κ0v−bmκ1θ˜1θ^˙1.
Now, we conductx1/kb12−x12φ11ξ1 and x1/kb12−x12φ12χ in (18). According to Lemma 2, the following inequality holds:(19)x1kb12−x12φ11ξ1≤λ1′+x1kb12−x12φ11ξ1tanhx1/kb12−x12φ11ξ1λ1=x1kb12−x12φ^11ξ1+λ1′,where λ1′=0.2785λ1 and φ^11ξ1=φ11ξ1tanhx1φ11ξ1/λ1kb12−x12 is a smooth function.
The same method in [32] is repeated(20)x1kb12−x12φ12χ≤x1kb12−x12φ¯12v+14x1kb12−x122+d1t≤x1kb12−x12φ¯12vtanhx1/kb12−x12φ¯12vℓ1+ℓ1′+14x1kb12−x122+d1t=x1kb12−x12φ^12ξ1,v+ℓ1′+14x1kb12−x122+d1t,where ℓ1′=0.2785ℓ1,φ¯12v=φ12∘ω1−12v,d1t=φ12∘ω1−12Bt2, and φ^12ξ1,v=φ¯12tanhx1φ¯12v/ℓ1kb12−x12.
Combing (18), (19), and (20), it yields(21)V˙≤x1kb12−x12f1ξ2+g1ξ−η˙d+φ^11ξ1+φ^12ξ1,v+14x1kb12−x12+d0κ0−k¯κ0v+λ1′+ℓ1′+d1−bmκ1θ˜θ^˙+x1ξ12γξ12x1κ0,where d1t≤d1.
Forx1=0, 1/κ0x1ξ12γξ12 in (21) is discontinuous, and the NN cannot be directly modeled, so we introduce a hyperbolic tangent function tanh2x1/r and (21) becomes(22)V˙≤x1kb12−x12f1ξ2+g^1χ1−12x1kb12−x122−bmκ1θ˜θ^˙+1−2tanh2x1rξ12γ0ξ12κ0−k¯κ0v+d0κ0+λ1′+ℓ1′+d1,where r is the positive constant, and the unknown nonlinear function g^1X1 is expressed as(23)g^1X1=12x1kb12−x12−ηd+14x1kb12−x12+g1ξ+φ^11ξ1+φ^12ξ1,v+2x1kb12−x12tanh2x1rξ12γ0ξ12κ0.
For∀ε1>0, g^1X1 is approximated to NN E1TH1X1, such that(24)g^1X1=E1∗TH1X1+δ1X1,δ1X1≤ε1,where δ1X1 is the error of this model and X1=ξT,ηd,η˙d,vT.
Based on Lemma3 and Lemma 5, the following inequality holds:(25)x1kb12−x12g^1X1=x1kb12−x12E1∗TH1X1+δ1X1,≤x1kb12−x12E1∗H1X1+ε1≤x1kb12−x12E1∗H1P1+ε1≤12a12x1kb12−x122bmθ1H1TP1H1P1+a122+12x1kb12−x122+ε122,where θ1=E1∗2/bm and P1=ξ1,ηd,η˙d,vT.
Therefore, substitute (25) into (22) to get(26)V˙1≤x1kb12−x12f1x2+x1kb12−x12f1α1+12a12x1kb12−x122bmθ1H1TH1+d0κ0+λ1′+1−2tanh2x1rξ12γξ12κ0−kκ0v+ℓ1′+d1+a122+ε122−bmκ1θ˜1θ^˙1,where x2=ξ2−α1.
Next, using Assumption1 and designing a virtual control signal α1 in (12) when i=1,
(26) becomes(27)V˙1≤x1kb12−x12f1x2−c1bmx12kb12−x12+bmκ1θ˜1κ12a12x1kb12−x122H1TH1−θ^˙1+1−2tanh2x1rξ12γ0ξ12κ0−kκ0v+d0κ0+λ1′+ℓ1′+d1+a122+ε122.
Then, we choose an adaptive lawθ^˙1 from (15), when i=1. θ^˙1 can be expressed as(28)θ^˙1=κ12a12x1kb12−x122H1TH1−μ1θ^1,where κ1>0 and μ1>0 are constants.
Substituting (28) into (27), it yields(29)V˙1≤x1kb12−x12f1x2−c1bmx12kb12−x12+1−2tanh2x1rξ12γ0ξ12κ0−kκ0r+d0κ0+λ1′+ℓ1′+d1+a122+ε122+bmμ1κ1θ˜1θ^1.
Under the action of(30)μ1bmκ1θ˜1θ^1=μ1bmκ1θ˜1θ1−θ˜1≤−μ1bm2κ1θ˜12+μ1bm2κ1θ12,we get(31)V˙1≤x1kb12−x12f1x2+1−2tanh2x1rξ12γ0ξ12κ0−kκ0v+d0κ0+B1−k1x12kb12−x12−μ1bm2κ1θ˜12,where k1=c1bm>0 and B1=μ1bm/2κ1θ12+λ1′+ℓ1′+d1+a12/2+ε12/2. Step i2≤i≤n−1: Let xi=ξi−αi−1, then one has(32)x˙i=fiξi+1+giξ+Οi−α˙i−1,where(33)α˙i−1=∑j=1i−1∂αi−1∂ξjgjξ+∑j=1i−1∂αi−1∂ξjΟj+Ξi−1,with(34)Ξi−1=∑j=1i−1∂αi−1∂ξjfjξj+1+∑j=1i−1∂αi−1∂θ^jθ^˙j+∑j=0i−1∂αi−1∂ηdjηdj+1+∂αi−1∂vv˙.
Then, construct a Lyapunov functionVi as(35)Vi=Vi−1+12logkbi2kbi2−xi2+bm2κiθ˜i2,where κi>0 is a design parameter and θ˜i=θi−θ^i is the error. In the set Ωxi, kbi2/kbi2−xi2 is continuous.
Thus, the derivative ofVi is(36)V˙i=V˙i−1+xikbi2−xi2fiξi+1+gix−∑j=1i−1∂αi−1∂ξjgjξ+Ο¯i−Ξi−1−bmκiθ˜iθ^˙i,where Ο¯i=Οi−∑j=1i−1∂αi−1/∂ξjΟj and similar to the method in the Step 1, we can obtain(37)V˙i−1≤xi−1kbi−12−xi−12fi−1xi−∑j=1i−1kjxj2kbj−12−xj−12−∑j=1i−1μjbm2κjθ˜j2+d0κ0+1−2tanh2x1rξ12γ0ξ12κ0−kκ0v+∑j=1i−1Bj,where kj=cjbm>0, djt≤dj and Bj=1/2aj2+1/2εj2+μjbm/2κjθj2+λj′+ℓj′+dj. j=1,2,…,i−1..
By using Assumption3 and the absolute value inequality, it yields(38)xikbi2−xi2Ο¯i≤xikbi2−xi2φi1ξ¯i+∑j=1i−1∂αi−1∂ξjφj1ξ¯j+xikbi2−xi2φi2χ+∑j=1i−1∂αi−1∂ξjφj2χ.
Using the same method as (19) and (20), we can get(39)xikbi2−xi2φi1ξ¯i+∑j=1i−1∂αi−1∂ξjφj1ξ¯j≤xikbi2−xi2φ^i1ξ¯i,θ^¯i−1,v+λi′,(40)xikbi2−xi2φi2χ+∑j=1i−1∂αi−1∂ξjφj2χ≤14xikbi2−xi221+∑j=1i−1∂αi−1∂ξj2+xikbi2−xi2φ^i2ξ¯i,θ^¯i−1,v+ℓi′+di,where(41)φ^i1ξ¯i,θ^¯i−1,v=φi1+∑j=1i−1∂αi−1∂ξjφj1×tanhxiφi1+∑j=1i−1∂αi−1/∂ξjφj1λikbi2−xi2,λi′=0.2785λi,φ^i2ξ¯i,θ^¯i−1,v=φ¯i2ξ¯i,θ^¯i−1,vtanhxiφ¯i2ξ¯i,θ^¯i−1,vℓikbi2−xi2,φ¯i2ξ¯i,θ^¯i−1,v=φi2∘ω1−12v+∑j=1i−1∂αi−1∂ξjφj2∘ω1−12v,ℓi′=0.2785ℓi,dit=∑j=1iφj2∘ω1−12Dt2 and dit≥0 for ∀t≥0.
Then, substitute (36)–(39) into (35) to get(42)V˙i≤−∑j=1i−1kjxj2kbj2−xj2−∑j=1i−1μjbm2κjθ˜j2−k¯κ0v+xikbi2−xi2fiξi+1+g^iχi−12xikbi2−xi22+1−2tanh2x1rξ12γ0ξ12κ0−bmκiθ˜iθ^˙i+∑j=1i−1Bj+d0κ0+λi′+ℓi′+di,where(43)g^iXi=12xikbi2−xi2+xi−1kbi−12−xi−12fi−1+14xikbi2−xi21+∑j=1i−1∂αj−1∂ξj2−Ξi−1+φ^i1ξ¯i,θ^¯i−1,v+φ^i2ξ¯i,θ^¯i−1,v+xikbi2−xi2gix−xikbi2−xi2∑j=1i−1∂αi−1∂ξjglξ.
For∀εi>0, the unknown smooth function g^iXi is estimated by the RBF NN EiTHiXi and we have(44)g^iXi=Ei∗THiX+δiXi,δXi≤εi,where Xi=ξT,η¯dT,θ^¯iT,vT, η¯d=ηd,ηd′,…,ηdiT and θ^¯i=θ˜i,…,θ˜iT.
Thus, it yields(45)xikbi2−xi2g^iXi=xikbi2−xi2Ei∗THiXi+δiXi,≤xikbi2−xi2Ei∗HiXi+εi≤xikbi2−xi2Ei∗HiPi+εi≤bm2ai2xikbi2−xi22θiHiTPiHiPi+12ai2+12xikbi2−xi22+12εi2,where ai is the design parameter, Pi=ξ¯iT,η¯dT,θ^¯iT,vT and θi=Ei∗2/bm.
Therefore, substitute (45) into (42) to get(46)V˙i≤−∑j=1i−1kjxj2kbi2−xj2−∑j=1i−1μjbm2κjθ˜j2−k¯κ0v+xikbi2−xi2fixi+1+xikbi2−xi2fiαi+bm2ai2xikbi2−xi22θiHiTHi+d0κ0+λi′+ℓi′+di+12ai2+12εi2+1−2tanh2x1rξ12γ0ξ12κ0−bmκiθ˜iθ^˙i+∑j=1i−1Bj,where xi+1=ξi+1−αi.
Next, designing a virtual control signalαi in (12) and an adaptive law θ^i from (15), then, using the same method as (27)–(31), we can get(47)V˙i≤−∑j=1i−1kjxj2kbi2−xj2−∑j=1i−1μjbm2κjθ˜j2−k¯κ0v+xikbi2−xi2fixi+1+1−2tanh2x1rξ12γ0ξ12κ0+∑j=1i−1Bj+d0κ0,where kj=cjbm>0 and Bj=1/2aj2+1/2εj2+μjbm/2κjθj2+λj′+ℓj′+dj. Step n: In this step, we design the real controller u from (13), so the derivative of x˙n is(48)x˙n=fnu+gnξ+Οn−α˙n−1,where α˙n−1 is specified in (33) when i=n.
Then, choose the Lyapunov function as(49)Vn=Vn−1+12logkbn2kbn2−xn2+bm2κnθ˜n2.
Then, the dynamic equation ofVn is(50)V˙n≤−∑j=1n−1kjxj2kbj2−xj2−∑j=1n−1μjbm2κjθ˜j2−k¯κ0v+d0κ0+∑j=1n−1Bj+xnkbj2−xn2fnu+gnξ−∑j=1n−1∂αn−1∂ξjgjξ−Ξn−1+xn−1kbn−12−xn−12fn−1+xnkbn2−xn2Ο¯n−bmκnθ˜nθ^˙n+1−2tanh2x1rξ12γ0ξ12κ0,where Ο¯n=Οn−∑j=1n−1∂αn−1/∂ξjΟj and Ξn−1 has been defined in (38) with i=n.
Using the same method as (38) to (40), we can get(51)xnkbn2−xn2Ο¯n≤xnkbn2−xn2φ^n1ξ¯n,θ^¯n−1,v+xnkbn2−xn2φ^n2ξ¯n,θ^¯n−1,v+14xnkbn2−xn221+∑j=1n−1∂αn−1∂ξj2+λn′+ℓn′+dnt,where φ^n1ξ¯n,θ^¯n−1,v and φ^n2ξ¯n,θ^¯n−1,v are defined in (47) and (48), respectively.
In view of (51), (50) is written as(52)V˙n≤−∑j=1n−1kjxj2kbj2−xj2−∑j=1n−1μjbm2κjθ˜j2−k¯κ0v+d0κ0+∑j=1n−1Bj+1−2tanh2x1rkb12−x12ξ12γ0ξ12κ0−12xnkbn2−xn22+xnkbn2−xn2fnu+g^nXn−bmκnθ˜nθ^˙n+λn′+ℓn′+dn,where dnt≤dn and(53)g^nXn=12xnkbn2−xn2+gnξ−∑j=1n−1∂αn−1∂ξjgjξ−Ξn−1+φ^n1ξ¯n,θ^¯n−1,v+14xnkbn2−xn221+∑j=1n−1∂αn−1∂ξj2+xn−1kbn−12−xn−12fn−1+φ^n2ξ¯n,θ^¯n−1,v.
For∀εn>0, the unknown smooth function g^nXn is estimated of the RBF NN EnTHnXn and we have(54)g^nXn=En∗THnXn+δnXn,δXn≤εn,where δiXn is the estimated error and εn>0 denotes a given constant.
Just like (45), one has(55)xnkbn2−xn2g^nXn≤bm2an2xnkbn2−xn22θnHnTHn+12an2+12xnkbn2−xn22+12εn2,where the unknown constant θn=En∗2/bm.
By combining (52) with (55), it yields(56)V˙n≤−∑j=1n−1kjxj2kbj2−xj2+xnkbn2−xn2fnu+bm2an2xnkbn2−xn2θnHnTHn+12an2+12εn2+1−2tanh2x1rξ12γ0ξ12κ0−bmκnθ˜nθ^˙n+∑j=1n−1Bj+d0κ0+λn′+ℓn′+dn−∑j=1n−1μjbm2κjθ˜j2−k¯κ0v.
Now, we are ready to design the actual controlleru and adaptive law θ^˙n, which are given as(57)u=−cnxn−12an2xnkbn2−xn2θ^nHnTXnHnXn,θ^˙n=κn2an2xnkbn2−xn22HnTHn−μnθ^n,where the positive parameters are cn,an,κn,μn.
Then, following the same derivations as those used in Step1, one has(58)V˙n≤−∑j=1nkjxj2kbj2−xj2−∑j=1nμjbm2κjθ˜j2−k¯κ0v+1−2tanh2x1rξ12γ0ξ12κ0+∑j=1nBj+d0κ0,where kj=cjbm>0 and Bj=1/2aj2+1/2εj2+μjbm/2κjθj2+λj′+ℓj′+dj,j=1,2,…,n..
So this backstepping control design process is complete. The main result is summarized in the next section.


## 4. Stability Analysis

Theorem 1.
Consider the closed-loop system with Assumptions1– 4, which is composed of plant (1), the virtual control inputs αi (12), real controller u (13), and adaptive laws (15), where RBF NN EiTHiXi is employed to estimate the uncertain nonlinear function g^iXi with bounded errors δi1≤i≤n. Suppose that the design parameters ci, ai and μi are appropriately chosen to satisfy kci+1>α¯i+kbi+1 with α¯i=maxαix¯i,θ^j,yd,ydi,j=1,…,i, then the final closed-loop system is SGUUB, and the tracking error converges to a small field around the origin with the partial-state constraints being valid.Proof.
From (17), (35), and (49), we gain(59)Vn=12∑j=1nlogkbj2kbj2−xj2+1k0v+12∑j=1nbmκjθ˜j2.
It is a fact thatlogkbj2/kbj2−xj2<xj2/kbj2−xj2 in the interval xj<kbj. Then, (58) becomes(60)Vn≤12∑j=1nxj2kbj2−xj2+1k0v+12∑j=1nbmκjθ˜j2.
Using (60), inequality (58) can be represented as(61)V˙n≤−r0V+μ0+1−2tanh2x1rξ12γξ12κ0,where r0=min2cibm,μi,k¯:1≤i≤n and μ0=∑j=1nBj+μ0/κ0.
Obviously, we know from (61) that the first item on the right of the inequation must be negative, the second item is a positive constant, and the positive or negative of the last item depends on the size of x1. So we are going to consider the closed-loop system in two different cases, and the results are as follows.Case 1.
x1∈Ωx1=x1|x1<0.8814r, r is a positive constant in (22). Because of the coordinate transformation (14), we can see that ξ1 is bounded because x1 is constructed to be bounded, and the reference signal ηd is also bounded. From (61), it gets(62)V˙n≤−r0V+b0,where b0=μ0+k0.
Besides, (61) satisfies(63)0≤Vn≤V0−b0r0e−r0t+b0r0.Case 2.
x1∉Ωx1. By applying Lemma 4 and ξ12γ0ξ12/κ0≥0, we have(64)1−2tanh2x1rξ12γξ12κ0≤0.
So, (61) is reduced to(65)V˙n≤−r0V+μ0.
Then, using (65) we can get(66)0≤Vnt≤V0−μ0r0e−r0t+μ0r0.
Next, from (63) and (66), we have(67)0≤Vnt≤V0+b0r0,t>0.
Then, it can be concluded from the above inequality and (59) that logkbj2/kbj2−xj2 and θ˜j are bounded. Since θj is bounded and θ^j=θ˜j+θj, θ^j must be bounded.
Fromξ1=x1+ηd and ηd≤d, we can obtain ξ1≤x1+ηd<kb1+d. Let kb1=kc1−d and then, ξ1<kc1. Apparently, α1⋅ is a function of θ^1,ξ1,x1 and η˙d. Because of the boundedness of θ^1,ξ1,x1 and η˙d, α1⋅ is bounded and satisfies α1⋅≤α¯1. Then, ξ2≤α1+x1≤α¯1+kb2.This implies that ξ2<kc2 if kb2=kc2−α¯1. Similarly, it can in turn be proven that ξi+1<kci+1, i=1,…,n−1 as long as kbi+1=kci+1−α¯i. From the definition in (13), we can see that u is a function of θ^n,ξn and ηd,η˙d,…,ηdn. Owing to the boundedness of θ^n,ξn and ηd,η˙d,…,ηdn, the controller u is bounded.
In both cases, we can conclude that all the reference signals of the closed-loop system are bounded. In addition, combining (63) and (66), we can see that the tracking error finally converges to a small region of the origin, and the system partial states are not violated.
From (67), it is easy to obtain(68)logkb12kb12−x12≤2V0−μ0r0e−r0t+2μ0r0.
We take exponentials on both sides of the above inequality; it has(69)kb12kb12−x12≤e2V0−μ0/r0e−r0t+2μ0/r0.
It is straightforward to getx1≤kb11−e−2V0−μ0/r0e−r0t−2μ0/r0=Δ. If Vn0=μ0/r0, then it holds x1≤kb11−e−2μ0/r0=Δ. If Vn0≠μ0/r0, it can be concluded that, given any Δ>kb11−e−2μ0/r0, there exists T such that for any t>T, it has x1≤Δ. As t⟶∞, x1≤kb11−e−2μ0/r0. This implies that x1≤kb11−e−2μ0/r0. We can see that x1 can be made arbitrarily small by selecting the design parameters appropriately.
This completes the proof.


## 5. Simulation Example

Example 1.
In order to test the applicability of the proposed control method, the following one-link manipulator with motor dynamics and disturbances is considered:(70)Dq¨+Bq˙+Nsinq=τ+τd,Mmτ˙+Hmτ=u−Kmq˙,where q, q˙, and q¨ are the link angular position, velocity, and acceleration. τ is the torque, τd=q2cosq˙τ denotes the current disturbance, and u is the control input representing the voltage. Take these parameters as D=1kgm2,B=1Nm,Mm=0.1H,Hm=1.0Ω, and Km=0.2Nm/A. Moreover, the sketch of the one-link manipulator is given in Figure 1.
Letξ1=q,ξ2=q˙, and ξ3=τ. Thus, (70) can be translated into a nonlinear system as follows:(71)χ˙=−1.25χ+0.25ξ12+0.125,ξ˙1=3ξ2+g1ξ+Ο1,ξ˙2=1.2ξ3+g2ξ+Ο2,ξ˙3=8u+g3ξ+Ο3,η=ξ1,where g1⋅=2cos2ξ1−1.75,g2⋅=ξ1ξ2−2sinξ1,g3⋅=−6.25ξ3,Ο1=χsinξ2Ο2=0.2χξ1cosξ1,Ο3=−2χcosξ1ξ2.
The aim is to impel the outputη of the system (70) to follow the reference trajectory ηd=0.5sin1.5t. We can easily check that Assumptions 1– 3 hold. Moreover, to prove that Assumption 4 is correct, Vχχ=2χ2 is chosen, then(72)V˙χχ=4χ−1.25χ+0.25ξ12+0.25,≤−5χ2+14τχ2+τξ14+τ4+χ2τ,When τ=2.5, we have(73)V˙χχ≤−4.5χ2+2.5ξ12+0.625.
Let us selectv1χ=1.5χ2,v2χ=4χ2,k0=1.5,d0=0.625, and γξ1=2.5ξ14. When k¯=1∈0,k0, a dynamical signal is given as(74)v˙=−v+2.5ξ14+0.625.
Then, the virtual control inputα1,α2, and real controller u are expressed as(75)α1=−c1x1−12a12x1kb12−x12θ^1H1TH1,α2=−c2x2−12a22x2kb22−x22θ^2H2TH2,u=−c3x3−12a32x3kb32−x32θ^3H3TH3,and the adaptive laws are expressed as(76)θ^˙i=κi2ai2xikbi2−xi22HiTHi−μiθ^i,i=1,2,3,with x1=ξ1−ηd,x2=ξ2−α1 and x3=ξ3−α2.
The design parameters are chosen asξ10,ξ20,ξ30T=0.0,0.0,0.0T,θ^10,θ^20,θ^30T=0.0,0.0,0.0T, c1=10,c2=15,c3=20,a1=1,a2=1,a3=15,μ1=2,μ2=3,μ3=5,κ1=10,κ2=15,κ3=20, respectively. The states are constrained in. ξ1<0.6,ξ2<0.8,ξ3<2.5.
Figure2 shows the tracking result of the output trajectory η and the reference signal ηd. Figure 3 shows the tracking error of the closed-loop system, which obviously achieves a good tracking performance. The response of the unmodeled dynamics χ and v is shown in Figure 4. Figures 5 and 6 show the state variables ξ2 and ξ3, which show that the system states keep within their bounds. The performance of the adaptive laws is shown in Figure 7. All simulation results show that the proposed control algorithm is effective and applicable.Figure 1
Sketch of one-link manipulator.Figure 2
The outputη and reference signal ηd.Figure 3
The tracking errorη−ηd.Figure 4
The unmodeled dynamicsχ and υ.Figure 5
The state variableξ2.Figure 6
The state variableξ3.Figure 7
The adaptive parametersθ^1, θ^2, and θ^3.


## 6. Conclusions

The paper has investigated the problem of adaptive tracking control for a class of nonstrict-feedback nonlinear systems with partial-state-constraints and unmodeled dynamics. An adaptive neural tracking control method has been presented by using the adaptive backstepping technique. Based on the inherent properties of Gaussian functions, and the universal approximation ability of RBF NN, a new method is proposed to deal with the nonstrict-feedback form of the considered nonlinear system. The proposed control algorithm can guarantee the boundedness of all the resulting closed-loop signals, the tracking error to converge to a small neighborhood of the origin, and the corresponding state constraints are not violated. Finally, the effectiveness and practicability of the obtained result are shown by a practical simulation example.



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*Source: 8835454-2020-10-31.xml*