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Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2012, Article ID 493715, 18 pages doi:10.1155/2012/493715 Research Article Dynamic Feedback Backstepping Control for a Class of MIMO Nonaffine Block Nonlinear Systems Hai-Yan Li, Yun-An Hu, Jian-Cun Ren, and Min Zhu Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai, Shandong 264001, China Correspondence should be addressed to Yun-An Hu, [email protected] Received 23 November 2011; Revised 26 January 2012; Accepted 4 April 2012 Academic Editor: Christos H. Skiadas Copyright q 2012 Hai-Yan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. 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. 2 Mathematical Problems in Engineering 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 Mathematical Problems in Engineering 3 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 Section 2. 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 ẋ1 f1 x1 , x2 , ẋ2 f2 x2 , x3 , .. . 2.1 ẋn fn xn , u, y x1 , T where xi xi1 · · · xim T ∈ Rm , xi x1T · · · xiT , i 1, 2, . . . , n, u ∈ Rm , and y ∈ Rm , are state variables, input and output, respectively. Remark 2.1. Although we assume xi ∈ Rm , the proposed method is easy to extend to the other cases by using pseudoinversion. Assumption 2.2. fi xi1 fi1 xi1 , fi2 xi1 , . . . , fim xi1 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: hx W T S V T x , 3.1 where W w1 w2 · · · wl T ∈ Rp×l and V v1 v2 · · · vi T ∈ RN×l are the first-to-second layer and the second-to-third layer weights, respectively, hx ∈ Rp , p ≥ 1, x ∈ RN is the input vector, and the node number is l > 1: T T x 1 , S V T x sv v1T x sv v2T x · · · sv vl−1 where sv xa 1/1 e−γxa with the constant γ > 0. 3.2 4 Mathematical Problems in Engineering Before designing the controller, we make the following assumptions. Assumption 3.1. One has a function vector hx : Ω → Rp ; for any σ > 0, there always exist a Gauss function array S : RN → Rl and an optimal weight matrix W ∗ such that hx − W ∗T SV ∗T x ≤ σ, for all x ∈ Ω, where Ω is a compact subset of RN , W ∗T SV ∗T x def is the optimal approximation of hx using NN, and hx − W ∗T SV ∗T x Δhx is called W − W ∗ and V V − V ∗ , where W and V are the estimated reconstruction error. Define W ∗ ∗ values of W and V . −1 Assumption 3.2. Jx−1i 0 and Ju0 , i 2, . . . , n, exist. Jxi xi−1 , xi ∂fi−1 xi−1 , xi /∂xi and Ju xn , u ∂fn xn , u/∂u denote the Jacobians with respect to xi and u, respectively. Let fi−1 xi−1 , xi fi−10 xi−1 , xi Δfi−10 xi−1 , xi , fn xn , u fn0 xn , u Δfn0 xn , u, Jxi xi−1 , xi Jxi 0 xi−1 , xi ΔJxi 0 xi−1 , xi , and Ju xn , u Ju0 xn , u ΔJu0 xn , u, where fi−10 xi−1 , xi , fn0 xn , u, Jxi 0 xi−1 , xi , and Ju0 xn , u are the nominal parts of the functions fi−1 xi−1 , xi , fn xn , u, Jxi xi−1 , xi , and Ju xn , u, respectively, and Δfi−10 xi−1 , xi , Δfn0 xn , u, ΔJxi 0 xi−1 , xi , and ΔJu0 xn , u are the unknown parts. Remark 3.3. Assumption 3.2 is not a strong condition imposed on the system. In fact, because Jxi 0 xi−1 , xi and Ju0 xn , u are the nominal parts of the functions Jxi xi−1 , xi and Ju xn , u, respectively, we can modify the values of the elements of Jxi 0 xi−1 , xi and Ju0 xn , u such that −1 , i 2, . . . , n, exist. Jx−1i 0 , and Ju0 Lemma 3.4 see 31, 34. For the NN approximator, the approximation error can be described as T S − S V T X W T S V T X du , T S V T X − W ∗T S V ∗T X W W 3.3 where S SV T X, S diag{sv1 , sv2 , . . . , svl } svi sv viT X dsv xa /dxa |x vT X , and the a i residual term du satisfies the following inequality: T S X W ∗ F S V T X W ∗ F l. du ≤ V ∗ F W F F 3.4 Step 1. Consider the first subsystem of 2.1 ẋ1 f1 x1 , x2 . Taking its derivative gives ẍ1 Jx1 x1 , x2 f1 x1 , x2 Jx2 x1 , x2 ẋ2 , 3.5 where Jx1 x1 , x2 ∂f1 x1 , x2 /∂x1 denotes the Jacobian with respect to x1 . Equation 3.5 can be rewritten as ẍ1 Jx1 0 x1 , x2 f10 x1 , x2 Jx2 0 x1 , x2 ẋ2 Δf1 x1 , x2 , 3.6 where Δf1 x1 , x2 ΔJx1 0 x1 , x2 f10 x1 , x2 Jx1 0 x1 , x2 Δf10 x1 , x2 ΔJx1 0 x1 , x2 Δf10 x1 , x2 ΔJx2 0 x1 , x2 ẋ2 . 3.7 Mathematical Problems in Engineering 5 Let z1 x1 − x1d and s1 z1 c1 ż1 , 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 ṡ1 ż1 c1 z̈1 ż1 c1 Jx1 0 x1 , x2 f10 x1 , x2 c1 Jx2 0 x1 , x2 ẋ2 c1 Δf1 x1 , x2 − c1 ẍ1d ż1 c1 Jx1 0 x1 , x2 f10 x1 , x2 c1 Jx2 0 x1 , x2 ẋ2d 3.8 − Jx2 0 x1 , x2 z2 c1 Δf1 x1 , x2 − c1 ẍ1d Jx2 0 x1 , x2 s2 . Let z2 x2 − x2d and s2 z2 c1 ż2 , where x2d is the desired signal of x2 and s2 is a sliding mode surface-like vector. Choose the virtual control as 1 − Jx2 0 x1 , x2 z2 v1 , ẋ2d −c1 Jx2 0 x1 , x2 −1 c1 Jx1 0 x1 , x2 f10 x1 , x2 ż 3.9 where vi ki si − ci ẍid vNNi − vir , i 1, 2, . . . , n, v1r will be defined in 3.20, ki is a diagonal i is the output of a tracking diﬀerentiator 35 with zi matrix with its elements positive, and ż as its input. The error between żi 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 ci Δfi xi , xi1 Δi Wi∗T Si Vi∗T Xi εi , 3.10 where Wi∗T Si Vi∗T Xi εi is the optimal approximation of Δfi xi , xi1 Δi , Δi is the i and uncertainty induced by the error between the output of the tracking diﬀerentiator ż T i , and Xi xTi , xT , ẋTi , 1 is the input of NN. εi ≤ εiu is the żi , namely, Δi żi − ż i1 approximation error with constant εiu > 0. Let T Si V T Xi , vNNi W i i i 1, 2, . . . , n. 3.11 Substituting 3.9 into 3.8 leads to ṡ1 c1 Δf1 x1 , x2 Δ1 − c1 ẍ1d Jx2 0 x1 , x2 s2 − v1 −c1 ẍ1d Jx2 0 x1 , x2 s2 − v1 W1∗T S1 V1∗T X1 ε1 . 3.12 Substituting the expressions of v1 and vNN1 into 3.12 gives T S1 V T X1 W ∗T S1 V ∗T X1 ε1 Jx2 0 x1 , x2 s2 v1r . ṡ1 −k1 s1 − W 1 1 1 1 3.13 6 Mathematical Problems in Engineering According to Lemma 3.4, 3.13 can be transformed into ṡ1 −k1 s1 Jx2 0 x1 , x2 s2 T S V T X1 − du1 ε1 v1r . T S1 − S V T X1 − W −W 1 1 1 1 1 1 3.14 We choose Lyapunov function as V1 1 T 1 T −1 1 T −1 s1 s1 tr W tr V1 ΓV 1 V1 , 1 ΓW1 W1 2 2 2 3.15 where ΓW1 ΓTW1 > 0 and ΓV 1 ΓTV 1 > 0 are constant design parameters. Taking the derivative of V1 , we have ˙ T Γ−1 V˙ 1 T Γ−1 W tr V̇1 sT1 ṡ1 tr W V 1 1 W1 1 V1 T S V T X1 − du1 ε1 v1r T S1 − S V T X1 − W sT1 −k1 s1 Jx2 0 x1 , x2 s2 − W 1 1 1 1 1 1 ˙ T Γ−1 W T Γ−1 V˙ 1 tr W tr V 1 1 W1 1 V1 3.16 ˙ T T −1 ˙ T Γ−1 W −k1 s1 2 tr W 1 tr V1 ΓV 1 V 1 s1 Jx2 0 x1 , x2 s2 1 W1 T S V T X1 sT −du1 ε1 v1r . T S1 − S V T X1 W − sT1 W 1 1 1 1 1 1 1 Choose the following adaptive tuning laws as ˙ Γ W 1 W1 1 , S1 − S1 V1T X1 sT1 − σW1 W T S − σV 1 V1 , V˙ 1 ΓV 1 X1 sT1 W 1 1 3.17 where σW1 > 0 and σV 1 > 0 are small design parameters. Substituting 3.17 into 3.16 results in 1 T S1 − S V T X1 sT − σW1 W V̇1 −k1 s1 2 tr W 1 1 1 1 T S − σV 1 V1 − sT du1 − ε1 − v1r tr V 1T X1 sT1 W 1 1 1 3.18 T S V T X1 sT Jx2 0 x1 , x2 s2 . T S1 − S V T X1 W − sT1 W 1 1 1 1 1 1 1 With the property tr{yxT } xT y, 3.18 can be simplified as 1 − σV 1 tr V T V1 − sT du1 − ε1 − v1r sT Jx2 0 x1 , x2 s2 . T W V̇1 −k1 s1 2 − σW1 tr W 1 1 1 1 3.19 Mathematical Problems in Engineering 7 Design the robust term vir as vir − T 2 T 2 2 si Wi Si Xi Si Vi Xi 2 F F ηi , i 1, 2, . . . , n, 3.20 where ηi > 0 is a small constant. After applying Lemma 3.4 and substituting 3.20 into 3.19, V̇1 is upper bounded by 1 − σV 1 tr V T V1 T W V̇1 ≤ −k1 s1 2 − σW1 tr W 1 1 T 2 T 2 2 S X 2 S V s1 2 W X 1 1 1 1 1 1 F F sT1 Jx2 0 x1 , x2 s2 η1 ∗ T W ∗ ε1u . W V T X1 S sT1 V1∗ F W l S X 1 1 F 1 1 1 F 1 1 − F 3.21 F With the following properties 31: 2 1 1 ≤ σW1 W ∗ 2 − σW1 T W −σW1 tr W W , 1 F 1 F 2 2 σ σV 1 2 V 1 ∗ 2 −σV 1 tr V 1T V1 ≤ V1 F − V1 , F 2 2 T 2 s η1 2 T 2 T ∗ T S1 X1 2 V1∗ F , s1 V1 F W1 S1 X1 ≤ 1 W 1 F F η1 4 T 2 2 η s 1 2 T ∗ T s1 W1 F S1 V1 X1 ≤ 1 S1 V1T X1 W1∗ F , F F η1 4 2sT 2 lη1 T 1 ∗ W ∗ 2 η1 ε2 , l W1 F ε1u ≤ s1 1 F η1 4 4 1u 3.22 Equation 3.21 can be simplified as V̇1 ≤ −k1 s1 2 − σW1 2 σV 1 2 W1 − V1 b1 sT1 Jx2 0 x1 , x2 s2 , F F 2 2 3.23 2 where b1 η1 1/4W1∗ 2F 1/4V1∗ 2F l/4W1∗ 2F ε1u σW1 /2W1∗ 2F σV 1 /2V1∗ 2F is a bounded constant. Step 2. Let us consider the subsystem ẋ2 f2 x2 , x3 . Taking its derivative leads to ẍ2 Jx2 x2 , x3 f2 x2 , x3 Jx3 x2 , x3 ẋ3 , 3.24 8 Mathematical Problems in Engineering where Jx2 x2 , x3 ∂f2 x2 , x3 /∂x2 denotes the Jacobian with respect to x2 . Equation 3.24 can be rewritten as ẍ2 Jx2 0 x2 , x3 f20 x2 , x3 Jx3 0 x2 , x3 ẋ3 Δf2 x2 , x3 , 3.25 where Δf2 x2 , x3 ΔJx2 0 x2 , x3 f20 x2 , x3 Jx2 0 x2 , x3 Δf20 x2 , x3 ΔJx2 0 x2 , x3 Δf20 x2 , x3 ΔJx2 0 x2 , x3 ẋ3 . 3.26 Taking its time derivative of s2 , we can obtain ṡ2 ż2 c1 Jx2 0 x2 , x3 f20 x2 , x3 c1 Jx3 0 x2 , x3 ẋ3d − Jx3 0 x2 , x3 z3 c1 Δf2 x2 , x3 − c1 ẍ2d Jx3 0 x2 , x3 s3 . 3.27 Let z3 x3 − x3d and s3 z3 c1 ż3 , where x3d is the desired signal of x3 and s3 is a sliding mode surface-like vector. Design a virtual control signal as ẋ3d −c1 Jx3 0 x2 , x3 −1 2 − Jx3 0 x2 , x3 z3 JxT 0 x1 , x2 s1 v2 . · c1 Jx2 0 x2 , x3 f20 x2 , x3 ż 2 3.28 Choose Lyapunov function as 1 1 T −1 1 T −1 V2 V1 sT2 s2 tr W tr V2 ΓV 2 V2 , 2 ΓW2 W2 2 2 2 3.29 where ΓW2 ΓTW2 > 0 and ΓV 2 ΓTV 2 > 0 are constant design parameters. Choose the following adaptive tuning laws: ˙ Γ W 2 W2 2 , S2 − S2 V2T X2 sT2 − σW2 W T S − σV 2 V2 , V˙ 2 ΓV 2 X2 sT2 W 2 2 where σW2 > 0 and σV 2 > 0 are small constants. 3.30 Mathematical Problems in Engineering 9 7 6 5 x11 4 3 2 1 0 0 5 10 15 20 Time (s) Figure 1: Curve of x11 in case of the system without uncertainties. Taking 3.25–3.30 into account, we have V̇2 ≤ −k1 s1 2 − k2 s2 2 − − σV 2 tr V 2T V2 σW1 2 σV 1 2 2 T W W1 − V1 − σW2 tr W 2 F F 2 2 T 2 T 2 2 S X 2 S V s2 2 W X 2 2 2 2 2 2 F 3.31 F sT2 Jx3 0 x2 , x3 s3 η2 ∗ T W ∗ ε2u b1 . W V T X2 S s2 V2∗ F W l S X 2 2 F 2 2 2 F 2 2 − F F Similar to Step 1, 3.31 can be simplified as σW1 2 σV 1 2 W1 − V1 F F 2 2 2 2 σW2 σV 1 − W2 − V2 b1 b2 sT2 Jx3 0 x2 , x3 s3 , F F 2 2 V̇2 ≤ −k1 s1 2 − k2 s2 2 − 3.32 2 where b2 η2 1/4W2∗ 2F 1/4V2∗ 2F l/4W2∗ 2F ε2u σW2 /2W2∗ 2F σV 2 /2V2∗ 2F is a bounded constant. Steps 3 to n − 1 are similar to Step 2, which are omitted here. Step n. Let us consider the system ẋn fn xn , u. Taking its derivative leads to ẍn Jxn xn , ufn xn , u Ju xn , uu̇, 3.33 10 Mathematical Problems in Engineering 0.35 0.3 0.25 x12 0.2 0.15 0.1 0.05 0 −0.05 0 5 10 15 20 Time (s) Figure 2: Curve of x12 in case of the system without uncertainties. 0.1 0 −0.1 x21 −0.2 −0.3 −0.4 −0.5 −0.6 0 5 10 15 20 Time (s) Figure 3: Curve of x21 in case of the system without uncertainties. where Jxn xn , u ∂fn xn , u/∂xn denotes the Jacobian with respect to xn . Equation 3.33 can be rewritten as ẍn Jxn 0 xn , ufn0 xn , u Ju0 xn , uu̇ Δfn xn , u, 3.34 where Δfn xn , u ΔJxn 0 xn , ufn0 xn , u Jxn 0 xn , uΔfn0 xn , u ΔJxn 0 xn , uΔfn0 xn , u ΔJxn 0 xn , uu̇. 3.35 Mathematical Problems in Engineering 11 20 15 x22 10 5 0 −5 0 5 10 15 20 Time (s) Figure 4: Curve of x22 in case of the system without uncertainties. Let zn xn − xnd and sn zn c1 żn , where xnd is the desired signal of xn and sn is a sliding mode surface-like vector. Taking its time derivative, we can obtain ṡn żn c1 z̈n żn c1 Jxn 0 xn , uf20 xn , u 3.36 c1 Ju xn , uu̇ c1 Δfn xn , u − c1 ẍnd . We choose the control law as n JxT 0 xn−1 , xn sn−1 vn . u̇ −c1 Ju0 xn , u−1 c1 Jxn 0 xn , ufn0 xn , u ż n−1 3.37 Let nT Sn VnT Xn . vNNn W 3.38 Let us consider the following Lyapunov function: Vn n−1 1 1 T −1 1 T −1 tr Vn ΓV n Vn , Vi sTn sn tr W n ΓWn Wn 2 2 2 i1 where ΓWn ΓTWn > 0 and ΓV n ΓTV n > 0 are constant design parameters. 3.39 12 Mathematical Problems in Engineering 7 6 5 x11 4 3 2 1 0 0 5 10 15 20 Time (s) Figure 5: Curve of x11 in case of the system with uncertainties. 0.35 0.3 0.25 x12 0.2 0.15 0.1 0.05 0 −0.05 0 5 10 15 20 Time (s) Figure 6: Curve of x12 in case of the system with uncertainties. Choose the adaptive tuning law as ˙ Γ W n Wn n , Sn − Sn VnT Xn sTn − σWn W nT Sn − σV n Vn , V˙ n ΓV n Xn sTn W where σWn > 0 and σV n > 0 are small parameters. 3.40 Mathematical Problems in Engineering 13 Similar to the derivation process in Step 1, we have V̇n ≤ − n σWi 2 σV i 2 ki si 2 Wi Vi − bi F F 2 2 i1 3.41 ≤ −kVn b, 2 where bn ηn 1/4Wn∗ 2F 1/4Vn∗ 2F l/4Wn∗ 2F εnu σWn /2Wn∗ 2F σV n /2Vn∗ 2F n −1 is a bounded constant, k minj1,...,n {2ki , σWj /λmax ΓWj , σV j /λmax Γ−1 j1 bj . V j }, and b Integrating 3.41 over 0, t, it can be shown that Vn t ≤ Vn 0e−kt 1 b b ≤ Vn 0 , k k ∀ t ≥ 0. 3.42 −1 −1 −1 Defining λmin Γ−1 W minj1,2,···n {λmin ΓWj }, and λmin ΓV minj1,2,···n {λmin ΓV j }, and from 3.39, it can be shown that n 2 Wj ≤ F j1 2Vn t , λmin Γ−1 W n 2 Vj ≤ j1 F 2Vn t . λmin Γ−1 V 3.43 It is clear that Vn t ≥ n 1 sj 2 . 2 j1 3.44 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 nj1 sj 2 b holds. Integrating it yields t sj τ2 dτ ≤ 2Vn 0 tb , k 0 j 1, . . . , n. 3.45 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. 14 Mathematical Problems in Engineering 0.1 0 −0.1 x21 −0.2 −0.3 −0.4 −0.5 −0.6 0 5 10 15 20 Time (s) Figure 7: Curve of x21 in case of the system with uncertainties. 16 14 12 x22 10 8 6 4 2 0 −2 0 5 10 15 20 Time (s) Figure 8: Curve of x22 in case of the system with uncertainties. 1 The sliding surfaces sj and the estimated parameter errors of NN are bounded and converge to the neighbourhoods of the origins exponentially: ⎧ ⎨ n 2 2V 0 b/k n n j j , V j sj 2 ≤ 2Vn 0 b/k , , Ωj sj , W W ≤ −1 ⎩ λ j1 min ΓW j1 ⎫ n 2 ⎬ 2Vn 0 b/k , j 1, . . . , n . Vj ≤ −1 ⎭ λmin Γ j1 V 3.46 Mathematical Problems in Engineering 15 4 2 0 −2 u1 −4 −6 −8 −10 −12 −14 −16 0 5 10 15 20 Time (s) Figure 9: Control signal u1 in case of the system with uncertainties. 300 250 200 u2 150 100 50 0 −50 0 5 10 15 20 Time (s) Figure 10: Control signal u2 in case of the system with uncertainties. 2 The following inequality holds: 1 lim t→∞ t t sj τ2 dτ ≤ 2b . k 0 3.47 Remark 3.6. It is obvious that the bounded sj , 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 Theorem 3.5 indicates that adjusting the values of ki , ΓWi , ΓV i , σWi , and σV i 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 , ΓV i , σWi , and σV i may result 16 Mathematical Problems in Engineering 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: ẋ1 f1 x1 , x2 , ẋ2 f2 x2 , u, 4.1 y x1 , where x1 x11 x12 T , x2 x21 x22 T , T 2 f1 x1 , x2 x12 x22 x11 x12 − x11 x12 2 0.3 sin x12 x21 cos0.1x21 , T −u2 2 2 1 − e 2 . f2 x2 , u x22 0.8 cosu1 −x21 x22 u2 x21 − x x22 x 22 21 1 e−u2 4.2 Let the desired output of the system be yd x11d x12d T 5 0T , and let the initial conditions be x11 0 x12 0 x21 0 x22 0 0.5 0.25 0.01 0, W1 0 W2 0 011 × 2 , and V1 0 V2 0 05 × 11 . According to Remark 3.7, 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 uncertainties f10 x1 , x2 0.8f1 x1 , x2 , Δf10 x1 , x2 0.2f1 x1 , x2 , f20 x2 , u 0.8f2 x2 , u, and Δf20 x2 , u 0.2f2 x2 , u. The simulation results are shown in Figures 1–8. 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. 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 Figure 5 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. Mathematical Problems in Engineering 17 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. 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