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

For a class of MIMO nonaffine block nonlinear systems, a neural networkNNbased 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.


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 u 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 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.

Problem Formulation
The uncertain block nonaffine system considered in this paper is given by The control objective is to design an adaptive NN controller for the system 2.1 such that the output tracks the desired signal y d and all signals in the closed-loop system remain bounded.Let • denote the 2-norm, and let • F denote the Frobenius norm.

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: and the second-to-third layer weights, respectively, h x ∈ R p , p ≥ 1, x ∈ R N is the input vector, and the node number is l > 1: where s v x a 1/ 1 e −γx a with the constant γ > 0.

Mathematical Problems in Engineering
Before designing the controller, we make the following assumptions.
Assumption 3.1.One has a function vector h x : Ω → R p ; for any σ > 0, there always exist a Gauss function array S : R N → R l and an optimal weight matrix W * such that is the optimal approximation of h x using NN, and , where W and V are the estimated values of W * and V * .
∂f n x n , u /∂u denote the Jacobians with respect to x i and u, respectively.Let x i , and J u0 x n , u are the nominal parts of the functions f i−1 x i−1 , x i , f n x n , u , J x i x i−1 , x i , and J u x n , u , respectively, and Δf i−1 0 x i−1 , x i , Δf n0 x n , u , ΔJ x i 0 x i−1 , x i , and ΔJ u0 x n , u are the unknown parts.
Remark 3.3.Assumption 3.2 is not a strong condition imposed on the system.In fact, because J x i 0 x i−1 , x i and J u0 x n , u are the nominal parts of the functions J x i x i−1 , x i and J u x n , u , respectively, we can modify the values of the elements of J x i 0 x i−1 , x i and J u0 x n , u such that J −1 x i 0 , and J −1 u0 , i 2, . . ., n, exist.
Lemma 3.4 see 31, 34 .For the NN approximator, the approximation error can be described as where S S V T X , S diag{s v1 , s v2 , . . ., s vl } s vi s v v T i X d s v x a /dx a | x a v T i X , and the residual term d u satisfies the following inequality: Step 1.Consider the first subsystem of 2.1 ẋ1 f 1 x 1 , x 2 .Taking its derivative gives where /∂x 1 denotes the Jacobian with respect to x 1 .Equation 3.5 can be rewritten as where Let z 1 x 1 − x 1d and s 1 z 1 c 1 ż1 , where c 1 > 0 is a constant, s 1 is a sliding mode surface-like vector, and x 1d is the reference signal of x 1 .Taking the time derivative of s 1 , we can obtain

3.8
Let z 2 x 2 − x 2d and s 2 z 2 c 1 ż2 , where x 2d is the desired signal of x 2 and s 2 is a sliding mode surface-like vector.
Choose the virtual control as where . ., n, v 1r will be defined in 3.20 , k i is a diagonal matrix with its elements positive, and żi is the output of a tracking differentiator 35 with z i as its input.The error between żi and żi can be approximated by a neural network.v NNi 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 where W * T i S i V * T i X i ε i is the optimal approximation of Δf i x i , x i 1 Δ i , Δ i is the uncertainty induced by the error between the output of the tracking differentiator żi and żi , namely, Δ i żi − żi , and

3.11
Substituting 3.9 into 3.8 leads to

3.12
Substituting the expressions of v 1 and v NN1 into 3.12 gives

Mathematical Problems in Engineering
According to Lemma 3.4, 3.13 can be transformed into

3.14
We choose Lyapunov function as where

3.16
Choose the following adaptive tuning laws as

3.18
With the property tr{yx T } x T y, 3.18 can be simplified as

3.19
Design the robust term v ir as where η i > 0 is a small constant.After applying Lemma 3.4 and substituting 3.20 into 3.19 , V1 is upper bounded by

3.21
With the following properties 31 :

3.22
Equation 3.21 can be simplified as is a bounded constant.
Step 2. Let us consider the subsystem ẋ2 f 2 x 2 , x 3 .Taking its derivative leads to ẍ2 where J x 2 x 2 , x 3 ∂f 2 x 2 , x 3 /∂x 2 denotes the Jacobian with respect to x 2 .Equation 3.24 can be rewritten as where Taking its time derivative of s 2 , we can obtain

3.27
Let z 3 x 3 − x 3d and s 3 z 3 c 1 ż3 , where x 3d is the desired signal of x 3 and s 3 is a sliding mode surface-like vector.Design a virtual control signal as

3.28
Choose Lyapunov function as where Γ W2 Γ T W2 > 0 and Γ V 2 Γ T V 2 > 0 are constant design parameters.Choose the following adaptive tuning laws:

3.31
Similar to Step 1, 3.31 can be simplified as 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 f n x n , u .Taking its derivative leads to ẍn J x n x n , u f n x n , u J u x n , u u, 3.33 where J x n x n , u ∂f n x n , u /∂x n denotes the Jacobian with respect to x n .Equation 3.33 can be rewritten as ẍn J x n 0 x n , u f n0 x n , u J u0 x n , u u Δf n x n , u , 3.34 where Let z n x n − x nd and s n z n c 1 żn , where x nd is the desired signal of x n and s n is a sliding mode surface-like vector.Taking its time derivative, we can obtain ṡn żn c 1 zn

3.36
We choose the control law as

3.38
Let us consider the following Lyapunov function: where Γ Wn Γ T Wn > 0 and Γ V n Γ T V n > 0 are constant design parameters.Choose the adaptive tuning law as

Mathematical Problems in Engineering 13
Similar to the derivation process in Step 1, we have and b n j 1 b j .Integrating 3.41 over 0, t , it can be shown that V j }, and from 3.39 , it can be shown that

3.43
It is clear that

3.44
It can be seen from  1 The sliding surfaces s j and the estimated parameter errors of NN are bounded and converge to the neighbourhoods of the origins exponentially: 2 The following inequality holds:

3.47
Remark 3.6.It is obvious that the bounded s j , j 1, . . ., n, implies the bounded z j and x j .Furthermore, if s j → 0 as t → ∞, we also can conclude that z j → 0 and x j → x jd as t → ∞.
Remark 3.7.The result 1 of Theorem 3.5 indicates that adjusting the values of k i , Γ W i , Γ V i , σ Wi , and σ V i can control the convergence rate and the size of the convergence region.It is shown from the expression 3.46 that larger gains k i , Γ W i , Γ V i , σ Wi , and σ V i may result 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 .

Simulation Study
In order to check the effectiveness of the algorithm, consider the following system:
The following two cases will be considered.
Case 1.All the parameters in 4.1 are known.
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 x 11 , Figure 2 shows the response curve of the state x 12 , Figure 3 shows the response curve of the state x 21 , and Figure 4 shows the response curve of the state x 22 .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 x 11 , Figure 6 shows the response curve of the state x 12 , Figure 7 shows the response curve of the state x 21 , and Figure 8 shows the response curve of the state x 22 .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.

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.

11 Figure 1 :
Figure 1: Curve of x 11 in case of the system without uncertainties.

Figure 2 : 6 Figure 3 :
Figure 2: Curve of x 12 in case of the system without uncertainties.

5 Figure 4 :
Figure 4: Curve of x 22 in case of the system without uncertainties.

11 Figure 5 : 12 Figure 6 :
Figure 5: Curve of x 11 in case of the system with uncertainties.

6 Figure 7 : 2 Figure 8 :
Figure 7: Curve of x 21 in case of the system with uncertainties.

1 Figure 9 :Figure 10 :
Figure 9: Control signal u 1 in case of the system with uncertainties.
and y ∈ R m , are state variables, input and output, respectively.
3.41 -3.44 that all the closed-loop signals are uniformly ultimately bounded.Inequality 3.41 implies that Vn t ≤ −k/2 Considering the system 2.1 , if Assumptions 2.2-3.2hold, the NN weights are updated according to 3.17 , 3.30 , 3.40 , and the control u is given in 3.37 , and then the following results hold.