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In this paper, a fuzzy adaptive tracking control for uncertain strict-feedback nonlinear systems with unknown bounded disturbances is proposed. The generalized fuzzy hyperbolic model (GFHM) with better approximation performance is used to approximate the unknown nonlinear function in the system. The dynamic surface control (DSC) is used to design the controller, which not only avoids the “explosion of complexity” problem in the process of repeated derivation, but also makes the control system simpler in structure and lower in computational cost because only one adaptive law is designed in the controller design process. Through the Lyapunov stability analysis, all signals in the closed loop system designed in this paper are semi-globally uniformly ultimately bounded (SGUUB). Finally, the effectiveness of the method is verified by a simulation example.

As an effective tool to solve the uncertainty of nonlinear systems, fuzzy logic systems are widely used in adaptive control design [

Recently, the research on adaptive fuzzy backstepping control for nonlinear system tracking control problem has been widely reported [

In addition, it is well known that the dynamic disturbance signal is an important factor that leads to the instability of the system, and even leads to the serious degradation of the control system performance. Therefore, it is very meaningful to study nonlinear systems with dynamic disturbance and uncertainty in the field of control. In [

Inspired by the previous studies, an improved fuzzy adaptive tracking control technique combining DSC method and GFHM approximator is proposed for a class of strict feedback nonlinear systems with dynamic disturbance signals. It not only avoids the problem of calculation expansion, but also obtains higher tracking accuracy. In addition, only one adaptive law is designed in the controller design, which greatly reduces the calculation cost. It is proved that all signals of the closed-loop system are semi globally asymptotically stable by Lyapunov method.

This paper considers the following SISO strictly feedback uncertain nonlinear systems with unknown disturbances:

( x ˙ j = g j ( x _ j ) + x j + 1 + d j ( t ) ,1 ≤ j ≤ n − 1, x ˙ n = g n ( x _ n ) + u + d n ( t ) , y = x 1 , (1)

where x _ j = [ x 1 , x 2 , ⋯ , x j ] T ∈ R j ( j = 1,2, ⋯ , n ) is the state variable of the system, u is the control input and y is the output variable of the system. g j ( ⋅ ) ( j = 1,2, ⋯ , n ) is an unknown smooth nonlinear function. d j ( t ) is an unknown bounded disturbance signal satisfying | d j | ≤ d j * and d j * is a constant.

Because the generalized fuzzy hyperbolic model makes multiple linear transformations on the input variables, the model has universal approximation property, that is, it can approach the actual model with arbitrary precision.

The membership functions λ P x and λ N x of fuzzy sets P x and N x are defined as follows:

λ P x ( x s ) = e − 1 2 ( x s − η s ) 2 , λ N x ( x s ) = e − 1 2 ( x s + η s ) 2 , (2)

where the constant η s > 0 .

Definition 1 [

1) IF ( x 1 − β 11 ) is F x 11 and ⋯ and ( x 1 − β 1 σ 1 ) is F x 1 σ 1 and ( x 2 − β 21 ) is F x 21 and ⋯ and ( x n − β n 1 ) is F x n 1 and ⋯ and ( x n − β n σ n ) is F x n σ n THEN

y = c F 11 + ⋯ + c F 1 σ 1 + c F 21 + ⋯ + c F n 1 + ⋯ + c F n σ n , (3)

where F x s i is the fuzzy subset corresponding to x s − β s i , including two linguistic values of P x and N x . c F s i is the output constant corresponding to F x s i .

2) The output constant c F s i ( s = 1 , ⋯ , n , i = 1 , ⋯ , σ s ) corresponds to F x s i one by one, that is, if F x s i is included in IF part, then c � s i should be included in THEN part. Instead, c F s i is not included in the THEN part. c P j is used to replace c F s i + and c N j to replace c F s i − .

3) There are 2 m fuzzy rules in this rule base, that is, the fuzzy input variables in the IF part include all possible positive ( P x ) and negative ( N x ) combinations, and the constant parameters in the THEN part include all the output constant combinations.

Lemma 1 [

y = ∑ j = 1 m c P j e η x j x _ j + c N j e − η x j x _ j e η x j x _ j + e − η x j x _ j = ∑ j = 1 m a j + ∑ j = 1 m b j e η x j x _ j − e − η x j x _ j e η x j x _ j + e − η x j x _ j = ξ + B T t a n h ( Φ x _ ) = J ( x ) , (4)

where a j = c P j + c N j 2 , b j = c P j − c N j 2 , ξ = ∑ j = 1 m a j , B = [ b 1 , ⋯ , b m ] T ,

Φ = d i a g [ η x 1 , ⋯ , η x m ] where t a n h ( Φ x _ ) is given by t a n h ( Φ x _ ) = [ t a n h ( η x 1 x _ 1 ) , ⋯ , t a n h ( η x m x _ m ) ] T . We call model (4) the generalized fuzzy hyperbolic model.

Lemma 2 [

s u p x ∈ U | h ( x ) − J ( x ) | < ε . (5)

Remark 1 [

y = ϑ T ψ ( x ) , (6)

where ϑ = [ ξ , B T ] T , ψ ( x ) = [ 1, t a n h ( η x 1 x _ 1 ) , ⋯ , t a n h ( η x m x _ m ) ] T . The output function y ( t ) is linear with respect to the adjustment parameter ϑ . The optimization parameters ϑ * are defined as follows

ϑ * = arg m i n ϑ ∈ R n { s u p x ∈ U | ϑ * T ψ ( x ) − y | } .

In this section, a fuzzy adaptive DSC control method based on GFHM is designed by using the preparatory knowledge in the previous section for system (1). In addition, let the normal number W be W = max { ‖ ϑ j * ‖ 2 : j = 1 , 2 , ⋯ , n } . According to backstepping, the design process of the controller includes n steps.

First of all, in the DSC design, there are the following transformations:

e 1 = x 1 − y r (7)

e ¯ j = e j − s j (8)

e j + 1 = x j + 1 − ν ¯ j + 1 (9)

s ˙ j = − k j s j + ν ¯ j + 1 − ν j + 1 (10)

where j = 1 , ⋯ , n − 1 , y r is the output reference signal, e j is the tracking error, e ¯ j is the tracking error of the transformation, s j is the design parameter, k j is the positive constant, ν j is the virtual control law and ν ¯ j is the first-order filter signal with time constant ι j > 0 .

ι j ν ¯ ˙ j + ν ¯ j = ν j , ν ¯ j ( 0 ) = ν j ( 0 ) . (11)

where j = 2 , ⋯ , n .

Then, the transformed error system is as follows

e ¯ ˙ j = − k j e ¯ j + g j ( x j ) + d j + ε j − 1 2 e ¯ j W ^ ψ j T ( x j ) ψ j ( x j ) − e ¯ j − 1 + e ¯ j + 1 e ¯ ˙ n = − k n e ¯ n + g n ( x n ) + d n + ε n − 1 2 e ¯ n W ^ ψ n T ( x n ) ψ n ( x n ) − e ¯ n − 1 (12)

where j = 1 , ⋯ , ( n − 1 ) , e ¯ 0 = 0 , W ˜ = W − W ^ and W ^ is the estimated value of W. The equivalent function ϑ * ψ ( x ) of GFHM is used to approximate the nonlinear function g j ( ⋅ ) , and the design virtual control laws ν j + 1 are

ν 2 = − k 1 e 1 − s 2 + y ˙ r − 1 2 e ¯ 1 W ^ ψ 1 T ( x 1 ) ψ 1 ( x 1 ) (13)

ν j + 1 = − k j e j − s j + 1 + ν ¯ ˙ j − 1 2 e ¯ j W ^ ψ j T ( x j ) ψ j ( x j ) − e ¯ j − 1 (14)

where j = 2 , ⋯ , ( n − 1 ) .

Finally, according to the design program, the parameter s n is redesigned to meet the following conditions

s ˙ n = − k n s n (15)

Therefore, we can design controller u and adaptive law according to the following formula

u = − k n e n + ν ¯ ˙ n − 1 2 e ¯ n W ^ ψ n T ( x n ) ψ n ( x n ) − e ¯ n − 1 (16)

W ^ ˙ = ∑ j = 1 n ( r 2 e ¯ j 2 ψ j T ( x j ) ψ j ( x j ) ) − q W ^ (17)

where r and q are the positive constants of the design.

Remark 2: Compared with [

Remark 3: In this method, n adaptive laws need not be designed. Only one adaptive parameter needs to be adjusted, which greatly reduces the computational burden.

The stability of the designed control method is proved in this section.

Theorem 1: If there are virtual control variables such as Equation (13) and Equation (14), such as the actual control variables of Equation (16), and the adaptive law of Equation (17), then the nonlinear system (1) is semi globally uniformly ultimately bounded, and the tracking error is kept in a small range.

Proof: Define Lyapunov functional as

V = 1 2 ∑ j = 1 n e ¯ j 2 + 1 2 r W ˜ 2 . (18)

The derivative of V is obtained

V ˙ = ∑ j = 1 n e ¯ j e ¯ ˙ j − 1 r W ˜ W ^ ˙ = − ∑ j = 1 n k j e ¯ j 2 + ∑ j = 1 n e ¯ j ϑ j * ψ j ( x _ j ) + ∑ j = 1 n e ¯ j ε j + ∑ j = 1 n e ¯ j d j − ∑ j = 1 n 1 2 e ¯ j 2 W ^ ψ j T ( x _ j ) ψ j ( x _ j ) − 1 r W ˜ W ^ ˙ (19)

Next, we know

e ¯ j ϑ j * ψ j ( x _ j ) ≤ 1 2 e ¯ j 2 ϑ j * T ϑ j * ψ j T ( x _ j ) ψ j ( x _ j ) + 1 2 ≤ 1 2 e ¯ j 2 W ψ j T ( x _ j ) ψ j ( x _ j ) + 1 2 (20)

e ¯ j ε j ≤ 1 2 e ¯ j 2 + 1 2 ε j * 2 (21)

e ¯ j d j ≤ 1 2 e ¯ j 2 + 1 2 d j * 2 (22)

Substituting Equations (20)-(22) into Equation (19), it can be deduced that

V ˙ ≤ − ∑ j = 1 n ( k j − 1 ) e ¯ j 2 + 1 2 e ¯ n 2 + ∑ j = 1 n 1 2 e ¯ j 2 W ˜ ψ j T ( x _ j ) ψ j ( x _ j ) + ω 1 − 1 r W ˜ W ^ ˙ (23)

where ω 1 = n 2 + ∑ j = 1 n 1 2 ε j * 2 + ∑ j = 1 n 1 2 d j * 2

Then substituting Equation (17) into Equation (23), we can get

V ˙ ≤ − ∑ j = 1 n ( k j − 1 ) e ¯ j 2 + 1 2 e ¯ n 2 + ∑ j = 1 n 1 2 e ¯ j 2 W ˜ ψ j T ( x _ j ) ψ j ( x _ j ) + ω 1 − 1 r W ˜ ( ∑ j = 1 n ( r 2 e ¯ j 2 ψ j T ( x _ j ) ψ j ( x _ j ) ) − k 0 W ^ ) ≤ − ∑ j = 1 n ( k j − 1 ) e ¯ j 2 + 1 2 e ¯ n 2 + ω 1 + k 0 r W ˜ W ^ (24)

Notice the following equation

k 0 r W ˜ W ^ = k 0 r W ˜ ( W − W ˜ ) ≤ − k 0 2 r W ˜ 2 + k 0 2 r W 2 (25)

Then, by rearranging Equation (24), we can get

V ˙ ≤ − ∑ j = 1 n ( k j − 1 ) e ¯ j 2 + 1 2 e ¯ n 2 − k 0 2 r W ˜ 2 + k 0 2 r W 2 + ω 1 ≤ − ∑ j = 1 n ( k j − 3 2 ) e ¯ j 2 − k 0 2 r W ˜ 2 + Λ ≤ − Φ V + Λ , (26)

where Λ = k 0 2 r W 2 + ω 1 , Φ = min { 2 k j − 3 , k 0 , j = 1 , ⋯ , n } .

From Equation (26), we can conclude that

V ( t ) ≤ ( V ( t 0 ) − Λ Φ ) e − Φ ( t − t 0 ) + Λ Φ . (27)

Inequality (27) shows that V ( t ) is ultimately bounded and the boundary

value is Λ Φ . Therefore, we can consider that all the signals of the closed-loop

system (i.e., e ¯ j , j = 1 , ⋯ , n and W ˜ ) are semi global uniformly ultimately bounded.

References [

( C p ¨ + A p ˙ + L sin ( p ) = I + d I Q I ˙ + R I = − K m p ˙ + U , (28)

where p, p ˙ and p ¨ are the angular displacement, velocity and acceleration of the joint, respectively. I is the current of the motor. d I is a random disturbance signal given by d I = 4 sin ( t ) . U is the motor input voltage. Other parameters are set as C = 1 , A = 1 , Q = 1 , R = 0.5 , L = 2.2 and K m = 5 .

Set x 1 = p , x 2 = p ˙ , x 3 = I , so Equation (28) can be expressed as the form of system (1), as follows

( x ˙ 1 = x 2 x ˙ 2 = − 2.2 sin ( x 1 ) − x 2 + x 3 + 4 sin ( t ) x ˙ 3 = − 5 x 2 − 0.5 x 3 + u y = x 1 (29)

The initial state value is set as [ x 1 ( 0 ) , x 2 ( 0 ) , x 3 ( 0 ) ] T = [ 0.5,0.5,0.5 ] . Set parameters k 1 = 2.5 , k 2 = 18 , k 3 = 0.1 , ι 2 = 0.08 , ι 3 = 0.08 , r = 30 , k 0 = 0.1 . y r = ( π / 2 ) s i n ( t ) ( 1 − e − 0.1 t 2 ) is the given reference signal.

The trajectory curve of output signal y tracking reference signal y r are shown in

In this paper, the problem of fuzzy adaptive tracking control for a class of uncertain SISO strict feedback nonlinear systems with disturbance is studied. In this control method, GFHM nonlinear function approximator is introduced to improve the approximation performance, and DSC technology is used to obtain better tracking performance. It not only avoids the problem of calculation expansion, but also obtains higher tracking accuracy. In addition, only one adaptive law is designed in the controller design, which greatly reduces the calculation cost. The SGUUB of the system is proved by Lyapunov stability theory. The effectiveness of the control method is further proved by simulation examples.

The authors declare no conflicts of interest regarding the publication of this paper.

Shi, J.X. and Yang, Z.J. (2020) Fuzzy Adaptive Tracking Control of Uncertain Strict-Feedback Nonlinear Systems with Disturbances Based on Generalized Fuzzy Hyperbolic Model. Journal of Computer and Communications, 8, 50-59. https://doi.org/10.4236/jcc.2020.810006