Adaptive neural dynamic surface control for a class of uncertain nonlinear systems with disturbances

Adaptive neural dynamic surface control for a class of uncertain nonlinear systems with disturbances

Author's Accepted Manuscript Adaptive Neural Dynamic Surface Control for a Class of Uncertain Nonlinear Systems with Disturbances Yang Cui, Huaguang ...

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Author's Accepted Manuscript

Adaptive Neural Dynamic Surface Control for a Class of Uncertain Nonlinear Systems with Disturbances Yang Cui, Huaguang Zhang, Yingchun Wang, Zhao Zhang

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S0925-2312(15)00274-X http://dx.doi.org/10.1016/j.neucom.2015.03.004 NEUCOM15215

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Neurocomputing

Received date: 22 August 2014 Revised date: 5 February 2015 Accepted date: 2 March 2015 Cite this article as: Yang Cui, Huaguang Zhang, Yingchun Wang, Zhao Zhang, Adaptive Neural Dynamic Surface Control for a Class of Uncertain Nonlinear Systems with Disturbances, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.03.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Adaptive Neural Dynamic Surface Control for a Class of Uncertain Nonlinear Systems with Disturbances Yang Cui, Huaguang Zhang∗, Yingchun Wang, Zhao Zhang College of Information Science and Engineering, Northeastern University, Shenyang, 110819, China

Abstract In this paper, adaptive dynamic surface control is investigated for a class of uncertain nonlinear systems with unknown bounded disturbances in strictfeedback form. Dynamic surface control technique is connected with radial basis function neural networks (RBFNNs) based control framework to avoid the explosion problem of complexity. The composite laws are constructed by prediction error and compensated tracking error between system state and serial-parallel estimation model for NN weights updating. Using Lyapunov techniques, the uniformly ultimate boundedness stability of all the signals in the closed-loop systems is graranteed. Simulation results illustrate the superiority of the proposed scheme and verify the theoretical analysis. Keywords: Dynamic surface control, RBF neural networks, tracking control, strict-feedback, disturbances 1. Introduction Neural networks (NNs) can approximate continuous unknown nonlinear functions to any desired accuracy by learning and parallel processing [1]. Owing to such a property, a lot of effort has been invested on adaptive NN control for nonlinear systems in recent years [2-9]. In [3], it presented a robust adaptive neural control scheme for a class of uncertain chaotic systems ∗

Corresponding author Email addresses: [email protected] (Yang Cui), [email protected] (Huaguang Zhang), [email protected] (Yingchun Wang), [email protected] (Zhao Zhang) Preprint submitted to Neurocomputing

March 18, 2015

in the disturbed strict-feedback form, together with both unknown nonlinearities and uncertain disturbances. For a class of affine nonlinear systems, it provided two different backstepping neural network control approaches in [4]. The proposed controller was feasible by using the neural network approximation performance. For uncertain multiple-input-multiple-output (MIMO) nonlinear systems with unknown control coefficient matrices and input nonlinearities in [10], the variable structure control associated with back-stepping was proposed to design adaptive NN control, and then the systems were stable. A fuzzy adaptive control of stochastic nonlinear systems with unknown virtual control gain function has been proposed in [11]. In [12], an adaptive neural output feedback tracking control of uncertain nonlinear multi-inputmulti-output (MIMO) systems in the discrete-time form was studied. An adaptive neural network tracking control was studied for a class of MIMO nonlinear systems in [13]. In [14], a robust adaptive NN output feedback control was proposed to control a class of uncertain discrete-time nonlinear MIMO systems. For high-order strict-feedback systems, the “explosion of complexity” was caused by the repeated differentiations of some nonlinear functions. In the conventional back-stepping technique, it designed the virtual controllers at each process. Thus, the controller become more and more complexity as the order of the system increases. In the recent years, the dynamic surface control (DSC) technique was proposed to eliminate the “explosion of complexity” problem, the first-order low-pass filter was added in the conventional backstepping design procedure [15]-[21]. In [15], a new robust adaptive control method was presented for a class of nonlinear systems subject to uncertainties. The approach was based on the adaptive dynamic surface control techuque, where the system uncertainties are approximately modeled by interval type-2 fuzzy neural networks. In [16], the tracking control problem has been considered for a class of strict-feedback uncertain nonlinear systems. Combining the DSC and MLP techniques, an RBF NN-based robust adaptive tracking control algorithm has been developed. An adaptive neural network controller for a class of uncertain nonlinear systems in strict-feedback form has been developed in [17]. A precise positioning robust hybrid intelligent control scheme was based on the effective compensation of nonsmooth nonlinearities, such as friction, deadzone, and uncertainty in a dynamic system in [18]. In [19], the problem of tracking control for a class of uncertain time-delay non-linear system with state constraint was addressed. To prevent constraint violation, the tangent barrier Lyapunov function (TBLF) is 2

firstly used for time-delay nonlinear system. An adaptive fuzzy backstepping DSC approach has been developed for a class of nonlinear MIMO systems with immeasurable states in [20]. In [21], the DSC method was extended to adaptive regulation control for systems in parametric strict-feedback form. All these control methods have been used the first-order low-pass filter to solved the problem of “explosion of complexity”. Due to simplicity, the DSC method is widely applied on flexible-joint robot control [22], ship control [23], power system control [24], marine vessels control [25], and flight control [26], [27]. Actual systems are almost nonlinear systems, and external disturbances exist everywhere, so how to solve the nonlinear systems with perturbation is very meaningful. Now in this paper, we use DSC to solve the control problem of SISO strict-feedback nonlinear systems with perturbation. It is different from [28], the composite laws are constructed by prediction error and compensated tracking error between system state and serial-parallel estimation model for NN weights updating. The composite DSC approach is employed to handle both compensated tracking error and prediction error. Using the new composite DSC approach, tracking performance is very ideal. The main contributions of this paper are as follows. 1. This paper considers the tracking problem of a class of non-affine nonlinear systems with uncertain disturbance and system functions unknown by neural networks estimation technique. 2. By employing DSC technology, differential explosion problem is avoided effectively. 3. By introducing compensating signal errors and prediction signal errors in updating law of neural networks, the performance of tracking is effectively improved. We organize the paper as follows. In Section 2, a class of SISO nonlinear systems with perturbation is characterized. In Section 3, we give a simple presentation of radial-basis-function neural networks (RBF NNs). In Section 4, the composite DSC control is designed. The stability analysis is given in Section 5. The simulation results by the proposed approach are verified in Section 6. In Section 7, the conclusion of this paper is drawn.

3

2. Problem statement and preliminaries 2.1. Problem statement The class of SISO nonlinear system can be expressed in the following state-space representation: ⎧ ⎪ xi , xi+1 , di (t)), i = 1, 2, · · · , n − 1 ⎨x˙ i = hi (¯ (1) xn , u, dn(t)), x˙ n = hn (¯ ⎪ ⎩ y = x1 , where xi (t) ∈ R is the state, x¯i = [x1 , x2 , . . . , xi ]T ∈ Ri , u ∈ R is the input, y ∈ R is the output, hi (·), i = 1, 2, . . . , n are smooth functions of their arguments, with satisfying the Lipschitz condition. di , i = 1, 2, . . . , n are the external disturbance, and bounded by unknown positive constant d∗i , that is, |di (t)| ≤ d∗i . For a given reference yr (t), t ≥ 0, which is a smooth function of t with its time derivative y˙ r bounded for t ≥ 0. The control objective is to find an adaptive neural control law such that all signals and states defined in the closed-loop system are bounded and the output tracks the bounded reference trajectory yr . 2.2. Transformation of the system representation For the system (1), define gi (·) :=

∂hi (·) , i = 1, 2, . . . , n − 1, ∂xi+1

∂hn (·) , i = n. ∂u Using the mean value theorem, it is as follows gn (·) :=

(2) (3)

xi , xi+1 , di (t)) = fi (¯ xi ) + gi (¯ xi , xqi+1 )xi+1 + Δi (t), hi (¯

(4)

xn , u, dn(t)) = fn (¯ xn ) + gn (¯ xn , uq )u + Δn (t), hn (¯

(5)

where fi (¯ xi ) = hi (¯ xi , o), fn (¯ xn , u) = hn (¯ xn , o), xqi+1 ∈ [min{xi+1 , 0}, max{xi+1 , 0}], 4

uq ∈ [min{u, 0}, max{u, 0}], Δi (t) = hi (¯ xi , xi+1 , di (t)) − hi (¯ xi , xi+1 , 0), xn , u, dn (t)) − hn (¯ xn , u, 0). Δn (t) = hn (¯ Because hi (·) , i = 1, 2, . . . , n , are unknown smooth nonlinear functions. Therefore, fi (·) and gi (·) ,i = 1, 2, . . . , n, are unknown smooth functions also. Moreover, since hi (·) satisfies the Lipschitz condition, and there exists a constant d∗i > 0 such that | Δi |≤ d∗i . From (4) and (5), system (1) can be rewritten as follows ⎧ ⎪ xi ) + gi (¯ xi , xqi+1 )xi+1 + Δi (t), i = 1, 2, · · · , n − 1 ⎨x˙ i = fi (¯ (6) x˙ n = fn (¯ xn ) + gn (¯ xn , uq )u + Δn (t), ⎪ ⎩ y = x1 , where gi (·),i = 1, 2, . . . , n, are smooth functions, they are bounded in some compact set. Conveniently for later work, the following assumption is made for (6). Assumption 1: The signs of gi (·) are known. 3. RBF neural networks In control engineering, RBF NNs[16] are usually used as a tool for modeling nonlinear functions because of their good capabilities in function approximation. They belong to a class of linearly parameterized networks. In this paper, we use the following RBF NNs to approximate a smooth function f (x): RQ → R ˆ T S(Xin ), fˆ(Xin ) = W (7) where Xin ∈ RM is the input vector of the RBF NNs, fˆ ∈ R is the NNs output, W ∈ RL is the weight vector, L > 1 is the NN nodes number, and S(·) : RM → RL is the basis function vector with si being the commonly used Gaussian functions, that is si (Xin ) = √

1  Xin − ξi 2 exp(− ), 2σi2 2πσi

(8)

where ξi is the center of the receptive field, and σi is the width of the Gaussian functions. 5

It has been shown that any continuous function f over a compact set ΩXin ∈ Rm can be approximated to any arbitrary accuracy by using (7), and an arbitrary εM > 0, there exist RBF NNs in the form of (9) such that f (Xin ) = W ∗T S(Xin ) + ε,

(9)

where W ∗ denotes ideal constant weights, and εM > 0 denotes the supremum value of the error ε that is inevitably generated. 4. Composite DSC design with prediction error In this paper, the prediction error derived from the difference between system state and serial-parallel estimation model, and it will be incorporated into the DSC technique for the nth order system described by (6). Similar to the traditional backstepping method, the recursive design procedure contains n steps. At each step, the virtual controller xdi+1 , i = 1, 2, . . . , n − 1 is developed. The control law u is constructed at the final step n. step 1: Considering the first equation in (6), and using NN to approximate an unknown function f1 (x1 ), we have x˙ 1 = f1 (¯ x1 ) + g1 x2 + Δ1 (t) = W1∗T S1 (¯ x1 ) + ε1 + g1 x2 + Δ1 (t),

(10)

where W1∗ is the ideal NN weights vector, and ε1 is the NN approximation error. The tracking error is defined as e1 = x1 − yr ,

(11)

yr is given tracking signal. Design the virtual control as xd2 =

ˆ T S1 (¯ −W x1 ) − k1 e1 + y˙r 1 , g1

(12)

ˆ 1 is the estimation of W ∗ , and k1 is a positive constant. where W 1 Let xd2 be the smoothed counterpart of xc2 through a low-pass filter α2 x˙ c2 + xc2 = xd2 ,

xc2 (0) = xd2 (0),

(13)

with a properly chosen α2 > 0. The smoothed xc2 can be equivalently considered the required xd2 . 6

Define e2 = x2 − xc2 . The derivative of e1 can be obtained as e˙ 1 =x˙ 1 − y˙ r =W1∗T S1 (¯ x1 ) + ε1 + g1 x2 + Δ1 (t) − y˙r ˜ T S1 (¯ =W x1 ) − k1 e1 + ε1 + g1 e2 + g1 (xc2 − xd2 ) + Δ1 (t), 1

(14)

˜1 = W∗ − W ˆ 1. where W 1 In order to eliminate the effect of the known error xc2 − xd2 , the compensating signal z1 is designed as z˙1 = −k1 z1 + g1 z2 + g1 (xc2 − xd2 ), z1 (0) = 0,

(15)

where zi will be defined in the next step. Let the compensated tracking error signals obtained as v1 = e1 − z1 ,

v2 = e2 − z2 .

(16)

Defined the prediction error as z1nn = x1 − xˆ1

(17)

with the serial-parallel estimation model, and the derivative of NN modeling information is defined as ˆ 1T S1 (¯ x1 ) + g1 x2 + β1 z1nn , xˆ˙ 1 = W

(18)

where β1 > 0 is the user-defined positive constant. The signal z1nn is employed to construct the learning design for the NN updating law, we design ˆ 1 ], ˆ˙ 1 = γ1 [(v1 + γz1 z1nn )S1 (¯ W x1 ) − δ1 W

(19)

where γ1 , γz1 , and δ1 are positive design constants. step i: i = 1, 2, . . . , n − 1. Considering the ith equation in (6), and using NN to approximate an unknown function fi (¯ xi ), we have xi ) + gi xi+1 + Δi (t) = Wi∗T Si (¯ xi ) + εi + gi xi+1 + Δi (t), x˙ i = fi (¯

(20)

where Wi∗ is the ideal NN weights vector, and εi is the NN approximation error. 7

Define the ith error surface ei to be ei = xi − xci .

(21)

Design the virtual control as xdi+1

ˆ T Si (¯ xi ) − ki ei − gi−1 ei−1 + x˙ ci −W i = , gi

(22)

ˆ i is the estimation of W ∗ , and ki is a positive constant. where W i Let xdi+1 be the smoothed counterpart of xci+1 through a low-pass filter αi+1 x˙ ci+1 + xci+1 = xdi+1 ,

xci+1 (0) = xdi+1 (0)

(23)

with a properly chosen αi+1 > 0, the smoothed xci+1 can be equivalently considered the required xdi+1 . Define ei+1 = xi+1 − xci+1 . The derivative of ei can be obtained as e˙ i =x˙ i − x˙ ci =Wi∗T Si (¯ xi ) + εi + gi xi+1 + Δi (t) − x˙ ci ˜ iT Si (¯ =W xi ) − ki ei + εi − gi−1 ei−1 + gi ei+1 + gi (xci+1 − xdi+1 ) + Δi (t),

(24)

˜i = W∗ − W ˆ i. where W i In order to eliminate the effect of the known error xci+1 − xdi+1 , the compensating signal zi is designed as z˙i = −ki zi − gi−1 zi−1 + gi zi+1 + gi (xci+1 − xdi+1 ), zi (0) = 0,

(25)

where zi+1 will be defined in the next step. Let the compensated tracking error signals obtained as vi = ei − zi .

(26)

Define the prediction error as zinn = xi − xˆi

(27)

with the serial-parallel estimation model, and the derivative of NN modeling information is defined as ˆ T Si (¯ xˆ˙ i = W xi ) + gi xi+1 + βi zinn , x ˆi (0) = xi (0), i 8

(28)

where βi > 0 is the user-defined positive constant. The signal zinn is employed to construct the learning design for the NN updating law, we design ˆ˙ i = γi [(vi + γzi zinn )Si (¯ ˆ i ], W xi ) − δi W

(29)

where γi , γzi , and δi are positive design constants. step n: Considering the nth equation in (6), and using NN to approximate an unknown function fn (¯ xi ), we have xn ) + gn u + Δn (t) = Wn∗T Sn (¯ xn ) + εn + gn u + Δn (t), x˙ n = fn (¯

(30)

where Wn∗ is the ideal NN weights vector, and εn is the NN approximation error. Define the nth error surface en to be en = xn − xcn .

(31)

Design the final control u as u=

ˆ T Sn (¯ −W xn ) − kn en − gn−1en−1 + x˙ cn n , gn

(32)

ˆ n is the estimation of W ∗ , kn is a positive constant. where W n The derivative of en can be obtained as e˙ n =x˙ n − x˙ cn =Wn∗T Sn (¯ xn ) + εn + gn u + Δn (t) − x˙ cn ˜ T Sn (¯ =W xn ) − kn en + εn − gn−1 en−1 + Δn (t),

(33)

n

˜n = W∗ − W ˆ n. where W n The compensating signal zn is design as z˙n = −kn zn − gn−1 zn−1 ,

zn (0) = 0.

(34)

Let the compensated tracking error signals obtained as vn = en − zn .

(35)

Define the prediction error as znnn = xn − xˆn 9

(36)

with the serial-parallel estimation model, the derivative of NN modeling information is defined as ˆ T Sn (¯ xn ) + gn u + βn znnn , (37) xˆ˙ n = W n where βn > 0 is the user-defined positive constant. The signal znnn is employed to construct the learning design for the NN updating law, we design ˆ˙ n = γn [(vn + γzn znnn )Sn (¯ ˆ n ], W xn ) − δn W (38) where γn , γzn , and δn are positive design constants. Remark 1: Compared with back-stepping design (28), only the information of yr (t) and y˙ r (t) is required in this paper while back-stepping design (i) needs the knowledge of yr (t). Remark 2: The prediction error of NN modeling together with the compensated tracking error is included in the composite NN updating law to provide fast convergence speed. The convergence speed is related to the design parameters of the prediction errors and compensating errors. But there is no analytical result in the NN literature to quantify the relationship of the design parameters, and an explicit expression of the stability condition is not available at present. We can only rely on a large number of simulation experiments. 5. Stability analysis Theorem 1: For any continuous and bounded reference signal yr and y˙ r , if exist the DSC laws (12),(22),and (32), the NN updating law (19),(29),and (38),and the compensated error signals vi in (26), then for system (6), all the signals vi ,w˜i and zinn are uniformly ultimately bounded. Proof: Choose a Lyapunov function candidate as n n 1 1 2 T −1 ˜ 2 ˜ (vi + Wi γi Wi ) + γzi zinn . (39) V = 2 i=1 2 i=1 The derivative of the compensated tracking error can be obtained as v˙ 1 =e˙ 1 − z˙1 ˜ T S1 (¯ =W x1 ) − k1 e1 + ε1 + g1 e2 + g1 (xc2 − xd2 ) + Δ1 (t) 1 + k1 z1 − g1 z2 − g1 (xc2 − xd2 ) ˜ 1T S1 (¯ =W x1 ) + ε1 − k1 v1 + g1 v2 + Δ1 (t), 10

(40)

v˙ i =e˙ i − z˙i ˜ T Si (¯ =W xi ) − ki ei + εi − gi−1 ei−1 + gi ei+1 + gi (xc i

i+1

− xdi+1 ) + Δi (t)

+ ki zi + gi−1 zi−1 − gi zi+1 − gi (xci+1 − xdi+1 ) ˜ iT Si (¯ =W xi ) + εi − ki vi − gi−1 vi−1 + gi vi+1 + Δi (t), v˙ n =e˙ n − z˙n ˜ T Sn (¯ =W xn ) − kn en + εn − gn−1 en−1 + Δn (t) + kn zn + gn−1 zn−1 n ˜ =WnT Sn (¯ xn )

(41)

(42)

+ εn − kn vn − gn−1 vn−1 + Δn (t).

Using (12), (22), and (32) with (18), (28), and (37), the derivative of the NN prediction error can be obtained as z˙inn =x˙ i − xˆ˙ i ˆ T Si (¯ =Wi∗T Si (¯ xi ) + εi + gi xi+1 + Δi (t) − W xi ) − gi xi+1 − βi zinn (43) i T ˜ i Si (¯ =W xi ) + εi − βi zinn + Δi (t). Then the following equation can be obtained as 2 , z˙inn zinn = zinn (mi + εi + Δi (t)) − βi zinn

(44)

˜ T Si (¯ where mi = W xi ). i The derivative of V is derived as n n   ˜ T γ −1 W ˆ˙ i ) + (vi v˙ i − W γzi zinn z˙inn V˙ = i i i=1

=

n 

i=1

(−ki vi2

+ vi mi + vi εi ) +

i=1

+

n 

˜ iT W ˆi + δi W

i=1

=

n 

vi Δi (t) −

i=1

γzi zinn (mi + εi ) −

i=1

n 

i=1 n 

mi (vi + γzi zinn )

2 γzi βi zinn

i=1

+

n 

γzi zinn Δi (t)

i=1

2 ˜ TW ˜ i + δi W ˜ T W ∗) (−ki vi2 + vi εi + γzi zinn εi − γzi βi zinn − δi W i i i

i=1

+

n 

n 

n  i=1

vi Δi (t) +

n 

γzi zinn Δi (t).

i=1

(45) 11

It is true that vi εi − ki vi2 = −ki (vi −

εi 2 1 2 ) + ε, 2ki 4ki i

2 zinn εi − βi zinn = −βi (zinn −

εi 2 1 2 ) + ε, 2βi 4βi i

∗ ˜ TW ˜ i − Wi 2 + 1 W ∗ 2 , ˜i = −  W ˜ TW∗ − W W i i i i 2 4 ci 1 vi Δi (t) ≤ vi2 + Δ2i (t), 4 ci bi 2 1 γzi zinn Δi (t) ≤ zinn + (γzi Δi (t))2 . 4 bi Then we obtained

V˙ ≤ −

n 

εi 2 εi 2 Wi∗ 2 ˜  ) (ki (vi − ) + γz βi (zinn − ) + δi Wi − 2ki 2βi 2 i=1

n n   ci 2 bi 2 1 γzi 2 δi 1 1 2 2 ( vi + zinn ) + ( ε2i + εi + Wi∗ 2 + Δ2i + γzi Δi ) + 4 4 4k 4β 4 c b i i i i i=1 i=1

≤−

n  i=1

∗  cmax 2 ˜ i − Wi 2 − )(vi − δmin W ((kmin − ε i )2 2 4 4k − c i i i=1

+ (γzmin βmin −

n

bmax 2γzi )(zinn − εi )2 ) + P, 4 4γzi βi − bi

(46)

where kmin = min[ki ], βmin = min[βi ], γzmin = min[γzi ], γzmax = max[γzi ], Wmax = max[Wi∗ ], cmin = min[ci ], cmax = max[ci ] , bmin = min[bi ], bmax = max[bi ], d∗ = max[d∗i ], δmin = min[δi ], δmax = max[δi ], and P =

γzmax 2 δmax 2 n ∗2 n 2 n 2 Wmax + εM + εM + d + γ d∗2 4kmin 4βmin 4 cmin bmin zmax n n + ε2M + γ 2 ε2 . 4kmin − cmax 4γzmin βmin − bmax zmax M

(47)

If for any δmin > 0, kmin − cmac > 0, and γzmin βmin − bmax > 0, if 4 4   ∗ P ˜ i − Wi  ≥ W or |vi − 4ki2−ci εi | ≥ kmin −P cmax or |zinn − 4γzi2γβzii −bi εi | ≥ 2 δmin 4

12



P γzmin βmin − bmax 4

˜ i , vi , and zinn are , then V˙ ≤ 0. Then we know that W

invariant to the sets defined as follows:  ˜ i | W ˜ i ≤ ΩW˜ i = (W Ωvi = (vi | |vi | ≤ Ωzinn = (zinn | |zinn | ≤

P kmin −

P

+

δmin cmax 4

P γzmin βmin −

bmax 4

+

Wmax ), 2

2 εM ), 4kmin − cmax

+

(48)

(49)

2γzmax εM ). (50) 4γzmin βmin − bmax

Therefore, all the signals are uniformly ultimately bounded, and this theorem is completely proved. 6. Simulation In this section, we will present a practical example and a numerical simulation comparison to demonstrate the effectiveness and merits of the proposed scheme in this paper. The objective is to design a control law u such that the output of the closed-loop system can approximately track a reference input yr asymptotically. To simulate the overall control system, the differential equations of the system and the adaptive NN updating law are integrated with the MATLAB function ODE45. The adaptive neural control law and NN updating law are chosen as follows: u=

ˆ T Sn (¯ −W xn ) − kn en − gn−1en−1 + x˙ cn n , gn (¯ xn )

ˆ˙ i = γi [(vi + γzi zinn )Si (¯ ˆ i ]. W xi ) − δi W 6.1. Example one: One-link manipulator with a BDC motor At first, we will use a popular benchmark of application example, the trajectory tracking control of a one-link manipulator actuated by a brush dc (BDC) motor, to validate the effectiveness of the proposed scheme in this

13

paper. The dynamics of a one-link manipulator actuated by a BDC motor can be expressed as follows [29]:

D q¨ + B q˙ + Nsin(q) = I + ΔI (51) M I˙ = −HI − Km q˙ + V where q, q, ˙ and q¨ denote the link angular position, velocity, and acceleration, respectively. I is the motor current. ΔI denotes the current disturbance. V is the input control voltage. The parameter values with appropriate units are given in [30] by D = 1, B = 1, M = 0.05, H = 0.5, N = 10, and Km = 10. The torque disturbance ΔI = 4sin(t). Note that the first subsystem in (51) represents the dynamics of the mechanical subsystem, the one-link robot, and the second subsystem denotes the dynamics of the BDC electrical subsystem. Setting x1 = q, x2 = q, ˙ x3 = I, and u = V , then (51) can be expressed in the form of (6) as ⎧ x˙ 1 = x2 ⎪ ⎪ ⎪ ⎨x˙ = −10sin(x ) − x + x + 4sin(t) 2 1 2 3 (52) ⎪ x ˙ 3 = −200x2 − 10x3 + 20u ⎪ ⎪ ⎩ y = x1 . 2

The reference signal is yr = (π/2)sin(t)(1 − e−0.1t ). The initial state of the system is 1. The parameters chosen for simulation are: k1 = 200, k2 = k3 = 20, γ1 = γ2 = γ3 = 10, δ1 = δ2 = δ3 = 1, α2 = α3 = 0.005, β1 = β2 = β3 = 5, γz1 = γz2 = γz3 = 1. For the NN design, the centers for x1 , x2 and x3 are evenly spaced in [−1, 1][−3, 3][−5, 5]. Fig.1 is the tracking performance. The tracking error is presented in Fig.2. 6.2. Example two In this section, we will consider the following second-order uncertain nonlinear system in the general form: ⎧ 2 2 ⎪ ⎨x˙ 1 = 0.5x1 + (1 + 0.1x1 )x2 + 0.1(1 − cos (x1 x2 )) (53) x˙ 2 = x1 x2 + (2 + cos(x1 ))u + 0.2x2 sin(x1 x2 ) ⎪ ⎩ y = x1 . The reference signal is yr = sin(t). The initial state of the system is 1. The parameters chosen for simulation are: k1 = 200, k2 = 20, γ1 = γ2 = 10, 14

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2

0

2

4

6

8

10 12 Time(sec)

14

16

18

20

Figure 1: Trajectories of system output y(solid line) and reference signal yd (dotted line).

1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6

0

0.5

1

1.5

2

2.5 Time(sec)

3

3.5

4

4.5

5

Figure 2: Trajectory of tracking error y − yd .

15

1.5

1

0.5

0

−0.5

−1

−1.5

0

5

10 Time(sec)

15

20

Figure 3: Trajectories of system output y(solid line) and reference signal yd (dotted line).

1

0.5

0

−0.5

0

0.2

0.4

0.6

0.8

1

Time(sec)

Figure 4: Trajectory of tracking error y − yd .

δ1 = δ2 = 1, α2 = 0.005, β1 = β2 = 5, γz1 = γz2 = 1. For the NN design, the centers for x1 and x2 are evenly spaced in [−1, 1][−3, 3]. Fig.3 is the tracking performance. The tracking error is presented in Fig.4. To show the different tracking result, we compare the control method in [31]. The related parameters are selected as c1 = c2 = 3.5, σ1 = σ2 = 2, Γ1 = Γ2 = 0.2. Fig.5 is the tracking performance. The tracking error is presented in Fig.6. By comparing the tracking error of the figure, it is clearly demonstrated that the control method in this paper achieves faster adaption and improved accuracy.

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1.5

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−0.5

−1

−1.5

0

5

10

15

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Figure 5: Trajectories of system output y(solid line) and reference signal yd (dotted line).

1.5

1

0.5

0

−0.5

−1

0

5

10

15

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Figure 6: Trajectory of tracking error y − yd .

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7. Conclusion The composite neural DSC for a class of strict-feedback systems with unknown bounded disturbances has been investigated in this paper. The adaptive neural controller and identification model parameters have been adjusted in the proposed control scheme. The uniformly ultimate boundedness stability of all the signals in the closed-loop systems has been guaranteed by Lyapunov method. The effectiveness of the proposed control algorithm has been shown to make the adaption faster and more reliable. In the future work, the dynamic surface control for a class of stochastic nonlinear systems with unknown control gain function will be researched. Acknowledgement This work was supported by the National Natural Science Foundation of China (61433004), National Basic Research Program of China (2009CB320601), Science and Technology Research Program of the Education Department of Liaoning Province (LT2010040) and the National High Technology Research and Development Program of China (2012AA040104). References [1] S. Ge, C. Hang, T. Lee, T. Zhang, Stable adaptive neural network control, Norwell, MA, USA: Kluwer (2002). [2] S. Ge, C. Wang, Direct adaptive NN control of a class of nonlinear systems, IEEE Trans. Neural Netw. 13 (2002) 214-221. [3] S. Ge, C. Wang, Uncertain chaotic system control via adaptive neural design, Int. j. Bifucat. Chaos 12 (2002) 1097-1109. [4] Y. Li, S. Qiang, X. Zhuang, O. Kaynak, Robust and adaptive backstepping control for nonlinear systems using RBF neural networks, IEEE Trans. Neural Netw. 15 (2004) 693-701. [5] H. Zhang, F. Yang, X. Liu, Q. Zhang, Stability analysis for neural networks with time-varying delay based on quadratic convex combination, IEEE Trans. Neural Netw. Learn. Syst. 24 (2013) 513-521.

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