# Dynamic Feedback Backstepping Control for a Class of MIMO Nonaffine Block Nonlinear Systems

Authors: Hai-Yan Li; Yun-An Hu; Jian-Cun Ren; Min Zhu
Journal: Mathematical Problems in Engineering
 (2012)

---

## Abstract

For a class of MIMO nonaffine block nonlinear systems, a neural network- (NN-) based dynamic feedback backstepping control design method is proposed to solve the tracking problem. This problem is difficult to be dealt with in the control literature, mainly because the inverse controls of block nonaffine systems are not easy to resolve. To overcome this difficulty, dynamic feedback, backstepping design, sliding mode-like technique, NN, and feedback linearization techniques are incorporated to deal with this problem, in which the NNs are used to approximate and adaptively cancel the uncertainties. It is proved that the whole closed-loop system is stable in the sense of Lyapunov. Finally, simulations verify the effectiveness of the proposed scheme.


---

## Body

## 1. Introduction

In the last two decades, a number of efforts have been made on developing systematic design tools for control of uncertain nonlinear systems. Among the obtained results, feedback linearization techniques [1], adaptive backstepping design [2], and NN control [3, 4] are the representative theoretical achievements. The common assumptions made in most of the researches are that the systems to be controlled are affine and the nonlinearities are linearly parameterized by unknown parameters [2, 5]. NN-based adaptive control has relaxed the assumption on linear parameterized nonlinearities mostly in affine systems [4, 6, 7], which can deal with nonlinear parameterized nonlinearities. But some systems, such as chemical reactions [8] and flight control systems [9], cannot be expressed in an affine form.There are three kinds of methods to deal with the controller design for nonaffine systems.The idea of the first method is to transform a nonaffine system into an affine system with respect to a new control input by introducing an integrator [9–12]. In these attempts, an augmented system affine in u̇ is derived for control design by differentiation [13].The second method directly controls a nonaffine system without transformation to an affine system [14–19]. Under the assumption that a control Lyapunov function (CLF) was available, Moulay and Perruquetti [16] obtained a sufficient condition to guarantee the existence of a continuous stabilizing control for nonaffine systems. Lin [14, 15] presented how nonaffine passive systems theory, together with the techniques of feedback equivalence and bounded control, could be used to explicitly construct a smooth state feedback control law that solved the problem of global stabilization for nonaffine nonlinear systems. New state feedback stabilizing controllers and sufficient conditions of asymptotic stability were proposed by Shiriaev and Fradkov [18] under assumptions similar to those in [14]. But it is difficult to find a CLF and to deal with controller design for systems with uncertainties.The third one employs NNs, PI, or fuzzy-neural models to approximate the inverse system or the uncertainties in controller design for nonaffine nonlinear systems [20–28]. For a class of general nonaffine nonlinear systems, virtual-linearized-system- (VLS-) based design methods were proposed, in which the T-S fuzzy-neural model was employed to approximate a VLS of a real system with modelling errors and external disturbances [20, 26, 27]. Teo et al. [25] constructed a proportional-integral (PI) controller for the minimum-phase nonaffine system, which was an equivalent realization of an approximate dynamic inversion controller. Ge and Zhang [21] suggested using NNs as emulators of inverse systems for controller design of general nonlinear systems. Using the implicit function theory and the mean value theorem, an NN was employed to approximate an ideal control signal which solved the tracking problem in [22]. In [29–32], instead of seeking a direct solution to the inverse problem, a solution was obtained by introducing an analytically invertible model  and then employing an NN to compensate inversion error. By using implicit function theorem  and Taylor series expansion, an observer-based adaptive fuzzy-neural control scheme was presented for the nonaffine nonlinear system in the presence of unknown structure of nonlinearities [33]. A neural synthesis method was considered for a class of multivariable nonaffine uncertain systems [28]. The method extended the previous approach developed in a single-input single-output system to a multi-input multi output system without resorting to a fixed-point assumption or boundedness assumption on the time derivative of a control effectiveness term. The difficulty associated with these methods for nonaffine control systems is that an explicit inverting control design is, in general, not possible even if the inverse exists by the implicit function theorem [28]. Moreover, this kind of method relies on the approximation ability of NN.Backstepping method is one of the breakthroughs in design of nonlinear control systems. Therefore, it has become one of the important and popular approaches for nonlinear systems [2]. This approach is based on a systematic procedure for the design of feedback control strategies suitable for the design of a large class of nonlinear systems with unmatched uncertainties, and it guarantees global regulation and tracking for the class of nonlinear systems transformable into the strict-feedback form. Developing a systematic synthesis method for general nonaffine systems still remains a challenging problem.In this paper, we discussed the NN-based backstepping design for a class of uncertain nonaffine systems in block control form. The main contributions of this paper can be summarized as follows: (1) the proposed method avoids the difficulties to solve the inverse control in most literatures; (2) it does not rely on implicit function theorem and Taylor series expansion which makes the output tracking difficult; (3) it can deal with the systems with unmatched uncertainties; (4) introducing the sliding mode surface-like variables into backstepping procedure makes the design and stability analysis clear and simple; (5) a systematic procedure is proposed for tracking control design for a class of block nonlinear systems that are nonaffine in the control inputs.The rest of the work is organized as follows. The problem formulation is introduced in Section2. The controller design and stability analysis are given in Section 3. Simulation example is given in Section 4 and followed by Section 5 which concludes the work.


## 2. Problem Formulation

The uncertain block nonaffine system considered in this paper is given by(2.1)ẋ1=f1(x1,x2),ẋ2=f2(x̅2,x3),⋮ẋn=fn(x̅n,u),y=x1,
where xi=[xi1⋯xim]T∈Rm, x̅i=[x1T⋯xiT]T, i=1,2,…,n, u∈Rm, and y∈Rm, are state variables, input and output, respectively.Remark 2.1.
Although we assumexi∈Rm, the proposed method is easy to extend to the other cases by using pseudoinversion.Assumption 2.2.
fi(x̅i+1)=[fi1(x̅i+1),fi2(x̅i+1),…,fim(x̅i+1)]T, i=1,2,…,n, is an unknown smooth function vector.
The control objective is to design an adaptive NN controller for the system (2.1) such that the output tracks the desired signal yd and all signals in the closed-loop system remain bounded. Let ∥·∥ denote the 2-norm, and let ∥·∥F denote the Frobenius norm.


## 3. Controller Design and Stability Analysis

In the following, introducing sliding mode-like technique, a systematic design method is proposed for a class of the uncertain block nonaffine systems.Consider the following NN:(3.1)h(x)=WTS(VTx),
where W=[w1w2⋯wl]T∈Rp×l and V=[v1v2⋯vi]T∈RN×l are the first-to-second layer and the second-to-third layer weights, respectively, h(x)∈Rp, p≥1, x∈RN is the input vector, and the node number is l>1:
(3.2)S(VTx)=[sv(v1Tx)sv(v2Tx)⋯sv(vl-1Tx)1]T,where sv(xa)=1/(1+e-γxa) with the constant γ>0.Before designing the controller, we make the following assumptions.Assumption 3.1.
One has a function vectorh(x):Ω→Rp; for any σ>0, there always exist a Gauss function array S:RN→Rl and an optimal weight matrix W* such that ∥h(x)-W*TS(V*Tx)∥≤σ, for all x∈Ω, where Ω is a compact subset of RN, W*TS(V*Tx) is the optimal approximation of h(x) using NN, and h(x)-W*TS(V*Tx)≝Δh(x) is called reconstruction error. Define W̃=Ŵ-W* and  Ṽ=V̂-V*, where Ŵ and V̂are the estimated values of W*  and V*.Assumption 3.2.
Jxi0-1 and  Ju0-1, i=2,…,n, exist. Jxi(x̅i-1,xi)=∂fi-1(x̅i-1,xi)/∂xi and Ju(x̅n,u)=∂fn(x̅n,u)/∂u denote the Jacobians with respect to xi and u, respectively. Let fi-1(x̅i-1,xi)=f(i-1)0(x̅i-1,xi)+Δf(i-1)0(x̅i-1,xi), fn(x̅n,u)=fn0(x̅n,u)+Δfn0(x̅n,u), Jxi(x̅i-1,xi)=Jxi0(x̅i-1,xi)+ΔJxi0(x̅i-1,xi), and Ju(x̅n,u)=Ju0(x̅n,u)+ΔJu0(x̅n,u), where f(i-1)0(x̅i-1,xi), fn0(x̅n,u), Jxi0(x̅i-1,xi),  and Ju0(x̅n,u) are the nominal parts of the functions fi-1(x̅i-1,xi), fn(x̅n,u), Jxi(x̅i-1,xi), and  Ju(x̅n,u), respectively, and Δf(i-1)0(x̅i-1,xi), Δfn0(x̅n,u), ΔJxi0(x̅i-1,xi), and ΔJu0(x̅n,u) are the unknown parts.Remark 3.3.
Assumption3.2 is not a strong condition imposed on the system. In fact, because Jxi0(x̅i-1,xi) and Ju0(x̅n,u) are the nominal parts of the functions Jxi(x̅i-1,xi) and  Ju(x̅n,u), respectively, we can modify the values of the elements of  Jxi0(x̅i-1,xi) and  Ju0(x̅n,u) such that Jxi0-1, and  Ju0-1, i=2,…,n, exist.Lemma 3.4 (see [31, 34]).
For the NN approximator, the approximation error can be described as(3.3)ŴTS(V̂TX)-W*TS(V*TX)=W̃T(Ŝ-Ŝ′V̂TX)+ŴTŜ′ṼTX+du,where Ŝ=S(V̂TX), Ŝ′=diag{sv1′,sv2′,…,svl′}ŝvi′=sv′(v̂iTX)=d[sv(xa)]/dxa|xa=v̂iTX, and the residual term du satisfies the following inequality:
(3.4)‖du‖≤‖V*‖F‖ŴTŜ′‖F‖X‖+‖W*‖F‖Ŝ′V̂TX‖F+‖W*‖Fl.Step 1.
Consider the first subsystem of (2.1)  ẋ1=f1(x1,x2). Taking its derivative gives
(3.5)ẍ1=Jx1(x1,x2)f1(x1,x2)+Jx2(x1,x2)ẋ2,whereJx1(x1,x2)=∂f1(x1,x2)/∂x1 denotes the Jacobian with respect to x1. Equation (3.5) can be rewritten as(3.6)ẍ1=Jx10(x1,x2)f10(x1,x2)+Jx20(x1,x2)ẋ2+Δf1(x1,x2),
where(3.7)Δf1(x1,x2)=ΔJx10(x1,x2)f10(x1,x2)+Jx10(x1,x2)Δf10(x1,x2)+ΔJx10(x1,x2)Δf10(x1,x2)+ΔJx20(x1,x2)ẋ2.Letz1=x1-x1d and s1=z1+c1ż1, where c1>0 is a constant, s1 is a sliding mode surface-like vector, and x1d is the reference signal of x1. Taking the time derivative of s1, we can obtain(3.8)ṡ1=ż1+c1z̈1=ż1+c1Jx10(x1,x2)f10(x1,x2)+c1Jx20(x1,x2)ẋ2+c1Δf1(x1,x2)-c1ẍ1d=ż1+c1Jx10(x1,x2)f10(x1,x2)+c1Jx20(x1,x2)ẋ2d-Jx20(x1,x2)z2+c1Δf1(x1,x2)-c1ẍ1d+Jx20(x1,x2)s2.
Let z2=x2-x2d and s2=z2+c1ż2, where x2d is the desired signal of x2 and s2 is a sliding mode surface-like vector.Choose the virtual control as(3.9)ẋ2d=-[c1Jx20(x1,x2)]-1[c1Jx10(x1,x2)f10(x1,x2)+ż̂1-Jx20(x1,x2)z2+v1],
where vi=kisi-ciẍid+vNNi-vir, i=1,2,…,n,v1r will be defined in (3.20), ki is a diagonal matrix with its elements positive, and ż̂i is the output of a tracking differentiator [35] with zi as its input. The error between ż̂i and żi can be approximated by a neural network. vNNi is the NN compensator, which is used to overcome the influence of the uncertainties in the system. According the approximation ability, we can assume that(3.10)ciΔfi(x̅i,xi+1)+Δi=Wi*TSi(Vi*TXi)+εi,
where Wi*TSi(Vi*TXi)+εi is the optimal approximation of Δfi(x̅i,xi+1)+Δi, Δi is the uncertainty induced by the error between the output of the tracking differentiator ż̂i and żi, namely, Δi=żi-ż̂i, and Xi=[x̅iT,xi+1T,x̅̇iT,1]T is the input of NN. ∥εi∥≤εiu is the approximation error with constant εiu>0.Let(3.11)vNNi=ŴiTSi(V̂iTXi),i=1,2,…,n.
Substituting (3.9) into (3.8) leads to(3.12)ṡ1=c1Δf1(x1,x2)+Δ1-c1ẍ1d+Jx20(x1,x2)s2-v1=-c1ẍ1d+Jx20(x1,x2)s2-v1+W1*TS1(V1*TX1)+ε1.
Substituting the expressions of v1 and vNN1 into (3.12) gives(3.13)ṡ1=-k1s1-Ŵ1TS1(V̂1TX1)+W1*TS1(V1*TX1)+ε1+Jx20(x1,x2)s2+v1r.According to Lemma3.4, (3.13) can be transformed into (3.14)ṡ1=-k1s1+Jx20(x1,x2)s2-W̃1T(Ŝ1-Ŝ1′V̂1TX1)-Ŵ1TŜ1′Ṽ1TX1-du1+ε1+v1r.
We choose Lyapunov function as(3.15)V1=12s1Ts1+12tr⁡{W̃1TΓW1-1W̃1}+12tr⁡{Ṽ1TΓV1-1Ṽ1},
where ΓW1=ΓW1T>0 and ΓV1=ΓV1T>0 are constant design parameters. Taking the derivative of V1, we have
(3.16)V̇1=s1Tṡ1+tr⁡{W̃1TΓW1-1Ŵ̇1}+tr⁡{Ṽ1TΓV1-1V̂̇1}=s1T[-k1s1+Jx20(x1,x2)s2-W̃1T(Ŝ1-Ŝ1′V̂1TX1)-Ŵ1TŜ1′Ṽ1TX1-du1+ε1+v1r]+tr⁡{W̃1TΓW1-1Ŵ̇1}+tr⁡{Ṽ1TΓV1-1V̂̇1}=-k1‖s1‖2+tr⁡{W̃1TΓW1-1Ŵ̇1}+tr⁡{Ṽ1TΓV1-1V̂̇1}+s1TJx20(x1,x2)s2-s1T[W̃1T(Ŝ1-Ŝ1′V̂1TX1)+Ŵ1TŜ1′Ṽ1TX1]+s1T(-du1+ε1+v1r).
Choose the following adaptive tuning laws as(3.17)Ŵ̇1=ΓW1[(Ŝ1-Ŝ1′V̂1TX1)s1T-σW1Ŵ1],V̂̇1=ΓV1(X1s1TŴ1TŜ1′-σV1V̂1),
where σW1>0 and σV1>0 are small design parameters. Substituting (3.17) into (3.16) results in(3.18)V̇1=-k1‖s1‖2+tr⁡{W̃1T[(Ŝ1-Ŝ1′V̂1TX1)s1T-σW1Ŵ1]}+tr⁡{Ṽ1T(X1s1TŴ1TŜ1′-σV1V̂1)}-s1T(du1-ε1-v1r)-s1T[W̃1T(Ŝ1-Ŝ1′V̂1TX1)+Ŵ1TŜ1′Ṽ1TX1]+s1TJx20(x1,x2)s2.
With the property tr{yxT}=xTy, (3.18) can be simplified as(3.19)V̇1=-k1‖s1‖2-σW1tr⁡{W̃1TŴ1}-σV1tr⁡{Ṽ1TV̂1}-s1T(du1-ε1-v1r)+s1TJx20(x1,x2)s2.
Design the robust term vir as(3.20)vir=-si(‖ŴiTŜi′‖F2‖Xi‖2+‖Ŝi′V̂iTXi‖F2+2)ηi,i=1,2,…,n,
where ηi>0 is a small constant. After applying Lemma 3.4 and substituting (3.20) into (3.19), V̇1 is upper bounded by(3.21)V̇1≤-k1‖s1‖2-σW1tr⁡{W̃1TŴ1}-σV1tr⁡{Ṽ1TV̂1}-‖s1‖2(‖Ŵ1TŜ1′‖F2‖X1‖2+‖Ŝ1′V̂1TX1‖F2+2)η1+s1TJx20(x1,x2)s2+‖s1T‖(‖V1*‖F‖Ŵ1TŜ1′‖F‖X1‖+‖W1*‖F‖Ŝ1′V̂1TX1‖F+l‖W1*‖F+ε1u).With the following properties [31]:(3.22)-σW1tr⁡{W̃1TŴ1}≤σW12‖W1*‖F2-σW12‖W̃1‖F2,-σV1tr⁡{Ṽ1TV̂1}≤σV12‖V1*‖F2-σV12‖Ṽ1‖F2,‖s1T‖‖V1*‖F‖Ŵ1TŜ1′‖F‖X1‖≤‖s1T‖2η1‖Ŵ1TŜ1′‖F2‖X1‖2+η14‖V1*‖F2,‖s1T‖‖W1*‖F‖Ŝ1′V̂1TX1‖F≤‖s1T‖2η1‖Ŝ1′V̂1TX1‖F2+η14‖W1*‖F2,‖s1T‖(l‖W1*‖F+ε1u)≤2‖s1T‖2η1+lη14‖W1*‖F2+η14ε1u2,Equation (3.21) can be simplified as(3.23)V̇1≤-k1‖s1‖2-σW12‖W̃1‖F2-σV12‖Ṽ1‖F2+b1+s1TJx20(x1,x2)s2,
where b1=[η1((1/4)∥W1*∥F2+(1/4)∥V1*∥F2+(l/4)∥W1*∥F2+ε1u2)+(σW1/2)∥W1*∥F2+(σV1/2)∥V1*∥F2] is a bounded constant.Step 2.
Let us consider the subsystemẋ2=f2(x̅2,x3). Taking its derivative leads to
(3.24)ẍ2=Jx̅2(x̅2,x3)f2(x̅2,x3)+Jx3(x̅2,x3)ẋ3,whereJx̅2(x̅2,x3)=∂f2(x̅2,x3)/∂x̅2 denotes the Jacobian with respect to x̅2. Equation (3.24) can be rewritten as
(3.25)ẍ2=Jx̅20(x̅2,x3)f20(x̅2,x3)+Jx30(x̅2,x3)ẋ3+Δf2(x̅2,x3),
where(3.26)Δf2(x̅2,x3)=ΔJx̅20(x̅2,x3)f20(x̅2,x3)+Jx̅20(x̅2,x3)Δf20(x̅2,x3)+ΔJx̅20(x̅2,x3)Δf20(x̅2,x3)+ΔJx̅20(x̅2,x3)ẋ3.Taking its time derivative ofs2, we can obtain(3.27)ṡ2=ż2+c1Jx̅20(x̅2,x3)f20(x̅2,x3)+c1Jx30(x̅2,x3)ẋ3d-Jx30(x̅2,x3)z3+c1Δf2(x̅2,x3)-c1ẍ2d+Jx30(x̅2,x3)s3.
Let z3=x3-x3d and  s3=z3+c1ż3, where x3d is the desired signal of x3 and s3 is a sliding mode surface-like vector.Design a virtual control signal as(3.28)ẋ3d=-[c1Jx30(x̅2,x3)]-1⋅{c1Jx̅20(x̅2,x3)f20(x̅2,x3)+ż̂2-Jx30(x̅2,x3)z3+Jx20T(x1,x2)s1+v2}.
Choose Lyapunov function as(3.29)V2=V1+12s2Ts2+12tr⁡{W̃2TΓW2-1W̃2}+12tr⁡{Ṽ2TΓV2-1Ṽ2},
where ΓW2=ΓW2T>0 and  ΓV2=ΓV2T>0 are constant design parameters.Choose the following adaptive tuning laws:(3.30)Ŵ̇2=ΓW2[(Ŝ2-Ŝ2′V̂2TX2)s2T-σW2Ŵ2],V̂̇2=ΓV2(X2s2TŴ2TŜ2′-σV2V̂2),
where σW2>0 and σV2>0 are small constants.Taking (3.25)–(3.30) into account, we have
(3.31)V̇2≤-k1‖s1‖2-k2‖s2‖2-σW12‖W̃1‖F2-σV12‖Ṽ1‖F2-σW2tr⁡{W̃2TŴ2}-σV2tr⁡{Ṽ2TV̂2}-‖s2‖2(‖Ŵ2TŜ2′‖F2‖X2‖2+‖Ŝ2′V̂2TX2‖F2+2)η2+s2TJx30(x̅2,x3)s3+‖s2‖(‖V2*‖F‖Ŵ2TŜ2′‖F‖X2‖+‖W2*‖F‖Ŝ2′V̂2TX2‖F+l‖W2*‖F+ε2u)+b1.
Similar to Step 1, (3.31) can be simplified as(3.32)V̇2≤-k1‖s1‖2-k2‖s2‖2-σW12‖W̃1‖F2-σV12‖Ṽ1‖F2-σW22‖W̃2‖F2-σV12‖Ṽ2‖F2+b1+b2+s2TJx30(x̅2,x3)s3,
where b2=[η2((1/4)∥W2*∥F2+(1/4)∥V2*∥F2+(l/4)∥W2*∥F2+ε2u2)+(σW2/2)∥W2*∥F2+(σV2/2)∥V2*∥F2] is a bounded constant.Steps 3 ton-1 are similar to Step 2, which are omitted here.Step n.
Let us consider the systemẋn=fn(x̅n,u). Taking its derivative leads to
(3.33)ẍn=Jx̅n(x̅n,u)fn(x̅n,u)+Ju(x̅n,u)u̇,whereJx̅n(x̅n,u)=∂fn(x̅n,u)/∂x̅n denotes the Jacobian with respect to x̅n. Equation (3.33) can be rewritten as(3.34)ẍn=Jx̅n0(x̅n,u)fn0(x̅n,u)+Ju0(x̅n,u)u̇+Δfn(x̅n,u),
where(3.35)Δfn(x̅n,u)=ΔJx̅n0(x̅n,u)fn0(x̅n,u)+Jx̅n0(x̅n,u)Δfn0(x̅n,u)+ΔJx̅n0(x̅n,u)Δfn0(x̅n,u)+ΔJx̅n0(x̅n,u)u̇.Letzn=xn-xnd and sn=zn+c1żn, where xnd is the desired signal of xn and sn is a sliding mode surface-like vector. Taking its time derivative, we can obtain(3.36)ṡn=żn+c1z̈n=żn+c1Jx̅n0(x̅n,u)f20(x̅n,u)+c1Ju(x̅n,u)u̇+c1Δfn(x̅n,u)-c1ẍnd.
We choose the control law as(3.37)u̇=-[c1Ju0(x̅n,u)]-1[c1Jx̅n0(x̅n,u)fn0(x̅n,u)+ż̂n+Jxn-10T(x̅n-1,xn)sn-1+vn].
Let(3.38)vNNn=ŴnTSn(V̂nTXn).
Let us consider the following Lyapunov function:(3.39)Vn=∑i=1n-1Vi+12snTsn+12tr⁡{W̃nTΓWn-1W̃n}+12tr⁡{ṼnTΓVn-1Ṽn},
where ΓWn=ΓWnT>0andΓVn=ΓVnT>0 are constant design parameters.Choose the adaptive tuning law as(3.40)Ŵ̇n=ΓWn[(Ŝn-Ŝn′V̂nTXn)snT-σWnŴn],V̂̇n=ΓVn(XnsnTŴnTŜn′-σVnV̂n),
where σWn>0 and σVn>0 are small parameters.Similar to the derivation process in Step1, we have(3.41)V̇n≤-∑i=1n[ki‖si‖2+σWi2‖W̃i‖F2+σVi2‖Ṽi‖F2-bi]≤-kVn+b,
where bn=[ηn((1/4)∥Wn*∥F2+(1/4)∥Vn*∥F2+(l/4)∥Wn*∥F2+εnu2)+(σWn/2)∥Wn*∥F2+(σVn/2)∥Vn*∥F2] is a bounded constant, k=minj=1,…,n{2ki,σWj/λmax(ΓWj-1),σVj/λmax(ΓVj-1)}, and b=∑j=1nbj.Integrating (3.41) over [0,t], it can be shown that(3.42)Vn(t)≤Vn(0)e-kt+1kb≤Vn(0)+bk,∀t≥0.Definingλmin(ΓW-1)=minj=1,2,⋯n{λmin(ΓWj-1)}, and λmin(ΓV-1)=minj=1,2,⋯n{λmin(ΓVj-1)}, and from (3.39), it can be shown that(3.43)∑j=1n‖W̃j‖F2≤2Vn(t)λmin⁡(ΓW-1),∑j=1n‖Ṽj‖F2≤2Vn(t)λmin⁡(ΓV-1).
It is clear that(3.44)Vn(t)≥12∑j=1n‖sj‖2.
It can be seen from (3.41)–(3.44) that all the closed-loop signals are uniformly ultimately bounded. Inequality (3.41) implies that V̇n(t)≤(-k/2)∑j=1n∥sj∥2+b holds. Integrating it yields(3.45)∫0t‖sj(τ)‖2dτ≤2[Vn(0)+tb]k,j=1,…,n.Summarizing the previous discussion, we have the following results.Theorem 3.5.
Considering the system (2.1), if Assumptions 2.2–3.2 hold, the NN  weights are updated according to (3.17), (3.30), (3.40), and the control u̇ is given in (3.37), and then the following results hold.(1)
The sliding surfacessj and the estimated parameter errors of  NN are bounded and converge to the neighbourhoods of the origins exponentially:
(3.46)Ωj={sj,W̃j,Ṽj|∑j=1n‖sj‖2≤2[Vn(0)+(b/k)],∑j=1n‖W̃j‖2≤2[Vn(0)+(b/k)]λmin⁡(ΓW-1),∑j=1n‖Ṽj‖2≤2[Vn(0)+(b/k)]λmin⁡(ΓV-1),j=1,…,n}.(2)
The following inequality holds:(3.47)lim⁡t→∞1t∫0t‖sj(τ)‖2dτ≤2bk.Remark 3.6.
It is obvious that the boundedsj, j=1,…,n, implies the bounded zj and xj. Furthermore, if sj→0 as t→∞, we also can conclude that zj→0 and xj→xjd as t→∞.Remark 3.7.
The result (1) of  Theorem3.5 indicates that adjusting the values of ki, ΓWi, ΓVi, σWi, and σVi can control the convergence rate and the size of the convergence region. It is shown from the expression (3.46) that larger gains ki, ΓWi, ΓVi, σWi, and σVi may result in smaller convergence region. However, in practice, we do not suggest the use of high adaptation gains because such a choice may cause large oscillations in the control outputs [36].


## 4. Simulation Study

In order to check the effectiveness of the algorithm, consider the following system:(4.1)ẋ1=f1(x1,x2),ẋ2=f2(x̅2,u),y=x1,
where x1=[x11x12]T, x2=[x21x22]T,
(4.2)f1(x1,x2)=[x12+x22x11x122-x11x12+(2+0.3sin⁡⁡x12)x21+cos⁡⁡(0.1x21)]T,f2(x̅2,u)=[x22+0.8cos⁡⁡(u1)-x21+x22+u2+(x212+x222)(1-e-u2)(1+e-u2)-x212x22]T.Let the desired output of the system beyd=[x11dx12d]T=[50]T, and let the initial conditions be [x11(0)x12(0)x21(0)x22(0)]=[0.50.250.010], W1(0)=W2(0)=[0]11×2, and V1(0)=V2(0)=[0]5×11.According to Remark3.7, we choose the parameters of the controller as follows: k111=6.6, k122=6.6, k211=12.4, k222=12.4, ΓW1=0.01, ΓW2=0.28, σW1=σW2=0.05, ΓV1=0.11, ΓV2=0.07, σV1=σV2=0.052, η1=η2=0.001, and c1=0.69.The following two cases will be considered.Case 1.
All the parameters in (4.1) are known.Case 2.
One has the system with uncertaintiesf10(x1,x2)=0.8f1(x1,x2), Δf10(x1,x2)=0.2f1(x1,x2),f20(x̅2,u)=0.8f2(x̅2,u), and Δf20(x̅2,u)=0.2f2(x̅2,u).The simulation results are shown in Figures1–8. Figures 1–4 show the state responses in case of the system without uncertainties (Case 1), where Figure 1 shows the tracking response curve of the state x11, Figure 2 shows the response curve of the state x12, Figure 3 shows the response curve of the state x21, and Figure 4 shows the response curve of the state x22. Figures 5–8 show the state responses in case of the system with uncertainties (Case 2), where Figure 5 shows the tracking response curve of the state x11, Figure 6 shows the response curve of the state x12, Figure 7 shows the response curve of the state  x21, and Figure 8 shows the response curve of the state x22. The control signals are shown in Figures 9 and 10.Figure 1
Curve ofx11 in case of the system without uncertainties.Figure 2
Curve ofx12 in case of the system without uncertainties.Figure 3
Curve ofx21 in case of the system without uncertainties.Figure 4
Curve ofx22 in case of the system without uncertainties.Figure 5
Curve ofx11 in case of the system with uncertainties.Figure 6
Curve ofx12 in case of the system with uncertainties.Figure 7
Curve ofx21 in case of the system with uncertainties.Figure 8
Curve ofx22 in case of the system with uncertainties.Figure 9
Control signalu1 in case of the system with uncertainties.Figure 10
Control signalu2 in case of the system with uncertainties.Although no exact model of the plant is available and the initial NN weights are set to zero, through the NN learning phase and the action of the robust term, it can be seen that the output tracking performance shown in Figure5 is quite well and the output tracking error converges to a quite small set after 4 s in Case 2.From the figures, one can conclude that the proposed control method presents a good quality control in both cases.


## 5. Conclusions

In this paper, an NN-based sliding mode-like controller is presented for a class of uncertain block nonaffine systems. The controller is designed using NN control, dynamic feedback, backstepping design, sliding mode-like technique, and feedback linearization techniques, which makes the stability analysis simple for block nonaffine systems and guarantees the stability of the closed-loop system. The sliding mode-like technique can be applied to other classes of nonlinear systems in strict feedback form. The simulation results show the effectiveness of the proposed scheme.



---

*Source: 493715-2012-06-03.xml*