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A modified tracking differentiator is proposed. Firstly, a nonlinear odd exponent continuous function is adopted which is only stable at one equilibrium point and proved the global asymptotic stability of the modified tracking differentiator by select a Lyapunov function. Through combining of the nonlinear and linear function properly, it can be sure that the state converges to the equilibrium point with high speed automatically no matter that the state was far away from the equilibrium point or near to it, and it can prevent the chattering. Simulation results show that the modified tracking differentiator tracking results are superior to the classical nonlinear tracking differentiator, and the respons e of state variables tracking differentiator estimated is almost coincide with the real state of the variables of the given system.

In actual engineering, it is often necessary to extract the differential signal from a given signal, and it is important to obtain accurate differential signal for the controller design, especially for the design of modern control theory based on the exact mathematical model. Usually a first order inertial component or difference method is used to obtain differential signals. But when the signal contains noise, the noise can be amplified when the differential signal is obtained with the above two methods (general noise change rate is better than the useful signal changes rate). In particular, a noise differential signal is obtained, and even the differential signal of the desired signal is drowned. In the literature 1, a second-order fastest switching system is used to construct a mechanism to extract the approximate differential signal, and the concept of the nonlinear tracking differentiator is proposed. The general form of the nonlinear tracking differentiator is given. The linear tracking differentiator changes the differential equation before the first-order inertia or differential method, and can better track the differential signal of a given signal [

In order to improve the performance of the tracking differentiator, it is necessary to remove the complex solution of the tracer, in particular to remove the solution on the imaginary axis, and to satisfy the requirement that the tracking differentiator converge quickly [

Assuming that for system S 1 [

{ x ˙ 1 ( t ) = x 2 ( t ) x ˙ 2 ( t ) = f [ x 1 ( t ) , x 2 ( t ) ] (1)

where X ( t ) = [ x 1 ( t ) , x 2 ( t ) ] has the following theorem:

Theorem 1: For system S 1 , if its solution x 1 ( t ) , x 2 ( t ) satisfies: lim t → ∞ x 1 ( t ) = 0 , lim t → ∞ x 2 ( t ) = 0 , then there will be system S 2

{ x ˙ 1 ( t ) = x 2 ( t ) x ˙ 2 ( t ) = R 2 ⋅ f [ ( x 1 ( t ) − v ( t ) ) , x 2 ( t ) R ] (2)

For any given bounded integrable function v ( t ) and constant T , the solution of Equation (2) satisfies: lim t → ∞ ∫ 0 T | x 1 ( t ) − v ( t ) | d t = 0 (or lim t → ∞ ∫ 0 + t 0 T + t 0 | x 1 ( t ) − v ( t ) | d t = 0 ), lim t → ∞ x 2 ( t ) = 0 . Proof process see the literature 1.

The key to selecting the differential tracker is to select f [ x 1 ( t ) , x 2 ( t ) ] , which is set here as follows：

x ˙ 2 ( t ) = f [ x 1 ( t ) , x 2 ( t ) ] (3)

Equal

x ˙ n ( t ) = − a 1 x 1 ( t ) − b 1 x 2 ( t ) − b 2 x 2 ( t ) n (4)

The following theorems are obtained:

Theorem 2: The system S 3

{ x ˙ 1 ( t ) = x 2 ( t ) x ˙ 2 ( t ) = − a 1 x 1 ( t ) − b 1 x 2 ( t ) − b 2 x 2 ( t ) n (5)

When a 1 > 0 , b 1 > 0 , b 2 > 0 , n > 0 , and n is an odd number, the solution of the Formula (5) satisfies the conclusion of the system S 1 in theorem 1, and the system S 3 is stabilized and stabilized at ( x 1 , x 2 ) T = ( 0 , 0 ) T , that is lim t → ∞ x 1 ( t ) = 0 , lim t → ∞ x 2 ( t ) = 0 .

The proof of theorem 2 is as follows:

Select the Lyapuvov energy function V , so V = a 1 x 1 ( t ) 2 / 2 + x 2 ( t ) 2 / 2 + 1 , because a 1 is greater than 0, then the function V coefficient is greater than 0, and because x i ( t ) 2 ≥ 0 , ( i = 1 , 2 ) , so for any state ( x 1 , x 2 ) T have V = a 1 x 1 ( t ) 2 / 2 + x 2 ( t ) 2 / 2 + 1 > 0 .

The derivative of V on both sides is obtained:

V ˙ = ∂ V ∂ x 1 x ˙ 1 ( t ) + ∂ V ∂ x 2 x ˙ 2 ( t ) (6)

V ˙ = ∂ V ∂ x 1 x 2 ( t ) + ∂ V ∂ x 2 [ − a 1 x 1 ( t ) − b 1 x 2 ( t ) − b 2 x 2 ( t ) n ] (7)

Also because:

∂ V ∂ x 1 = a 1 x 1 ( t ) (8)

∂ V ∂ x 2 = x 2 ( t ) (9)

The Formula (8) and the Formula (9) are given to the Formula (7):

V ˙ = a 1 x 1 ( t ) x 2 ( t ) + x 2 ( t ) [ − a 1 x 1 ( t ) − b 1 x 2 ( t ) − b 2 x 2 ( t ) n ] (10)

Finished:

V ˙ = − b 1 x 2 ( t ) 2 − b 2 x 2 ( t ) n + 1 (11)

Because b 1 , b 2 , n are greater than 0, the function V ˙ of the coefficient are less than 0, and because v is greater than 0 odd, it is x, so for any state x have v. That is, the system is gradual and stable, that can eventually think that x tends to a certain stability does not change, then this time there must be x. Because xx, there xxx.

Since b 1 , b 2 , n are greater than 0, the coefficients of function V ˙ are less than 0, and because n is odd number greater than 0, there is x 2 ( t ) n + 1 ≥ 0 , so ( x 1 , x 2 ) T for any state has V ˙ = − b 1 x 2 ( t ) 2 − b 2 x 2 ( t ) n + 1 ≤ 0 . That is, the system is asymptotically stable that can eventually think that x 1 ( t ) tends to a certain stability does, then this time there must be x ˙ 1 ≡ 0 ， x 2 = x ˙ 1 ≡ 0 . Because x 2 ≡ 0 , there x ˙ 2 ≡ 0 .

x 2 ≡ 0 ， x ˙ 2 ≡ 0 are given to the Formula (5):

− a 1 x 1 ( t ) = 0 (12)

Answer is:

x 1 ( t ) = 0 (13)

That is, the system stability value is:

( x 1 , x 2 ) T = ( 0 , 0 ) T (14)

So far, theorem 2 is proved.

According to theorem 1 and theorem 2, the form of an improved tracking differentiator can be obtained, as shown in Equation (15):

{ x ˙ 1 ( t ) = x 2 ( t ) x ˙ 2 ( t ) = − R 2 { a 1 [ x 1 ( t ) − v ( t ) ] − b 1 x 2 ( t ) R − b 2 x 2 ( t ) n R n } (15)

In Formula (15), a 1 > 0 , b 1 > 0 , b 2 > 0 , n > 0 , n is odd.

For any given bounded integrable function v ( t ) and constant T , the solution of Equation (15) satisfies: lim t → ∞ ∫ 0 T | x 1 ( t ) − v ( t ) | d t = 0 (or lim t → ∞ ∫ 0 + t 0 T + t 0 | x 1 ( t ) − v ( t ) | d t = 0 ), lim t → ∞ x 2 ( t ) = 0 .

It is necessary to make the following 3 points for the improved tracking differentiator shown in Equation (15):

1) For Equation (15), when e = x 1 ( t ) − v ( t ) is large, the acceleration signal x ˙ 1 ( t ) (or x 2 ( t ) ) will be larger, so that the system can be faster to track v ( t ) ,

to meet lim t → ∞ ∫ 0 + t 0 T + t 0 | x 1 ( t ) − v ( t ) | d t = 0 . At this time, b 1 x 2 ( t ) R < b 2 x 2 ( t ) n R n , b 2 x 2 ( t ) n R n

plays a major role in making ( x 1 − v , x 2 ) T → ( 0 , 0 ) T , accelerating the v ( t ) process.

2) When e = x 1 ( t ) − v ( t ) is small, that is, when x 1 ( t ) is close to v ( t ) , the x 1 ( t ) changes slowly, that is, the slope becomes smaller, the acceleration signal x ˙ 1 ( t ) (or x 2 ( t ) ) becomes smaller, and then the acceleration is still needed as much as possible, so that the system can track the v ( t ) at a relatively fast speed, and meet lim t → ∞ ∫ 0 + t 0 T + t 0 | x 1 ( t ) − v ( t ) | d t = 0 .

At this time, just b 1 x 2 ( t ) / R > b 2 x 2 ( t ) n / R n ,

(At this time ( x 1 − v , x 2 ) T → ( 0 , 0 ) T , b 2 x 2 ( t ) n / R n slope close to the x-axis), b 2 x 2 ( t ) / R play a major role.

As a result of the synthesis (1) (2), the improved tracking differential is designed so that the state quantity can accelerate convergence when it is away from the equilibrium point and near the equilibrium point.

3) If: f = − a 1 x 1 ( t ) − a 2 x 1 ( t ) m − b 1 x 2 ( t ) − b 2 x 2 ( t ) n , is it possible to ensure that the formula 15 in e = x 1 ( t ) − v ( t ) larger, in addition to relying on b 2 x 2 ( t ) n / R n but also rely on non-linear link − a 2 x 1 ( t ) m accelerated tracking v ( t ) process, the system can be faster to track v ( t ) , to meet lim t → ∞ ∫ 0 + t 0 T + t 0 | x 1 ( t ) − v ( t ) | d t = 0 ? The answer is negligible. From the theoretical calculation we can see that ( x 1 , x 2 ) T

is convergent to ( 0 , 0 ) T , but also converges to [ ( − a 1 / a 2 ) 1 / ( m − 1 ) , 0 ] T (where

a 1 , a 2 , m is greater than 0 and m is positive odd, then ( − a 1 / a 2 ) 1 / ( m − 1 ) is a complex solution, which is located on the imaginary axis, resulting in critical stability of the tracer system, If we prove that the optional Lyapuvov energy function V , V = a 1 x 1 ( t ) 2 / 2 + a 2 x 1 ( t ) m + 1 / ( m + 1 ) + x 2 ( t ) 2 / 2 , can prove that the two solutions converged by the system are x 1 ( t ) = 0 , x 1 ( t ) = ( − a 1 / a 2 ) 1 / ( m − 1 ) , detailed proof ).

In the simulation, the improved tracking differential is compared with the classical nonlinear differential splitter designed in literature 1. The parameters for the non-linear tracking differentiator are set as follows: The TD setting of ADRC [

The parameters for the improved tracking differentiator are set as follows: R = 5, a = 5, b 1 = b 2 = 1 , n = 3.

The simulation results show that the improved tracking differentiator has stable tracking effect, and the chattering is small in

The improved tracking differentiator designed in this paper is a stable system in the sense of Lyapunov, which has global stability and fast convergence. Through simulation, comparison and verification, it is proved that the improved tracking differentiator has good tracking effect, and it has simple form and no chattering when tracking. It is a better tracking differentiator.

Liu, Z.G. and Jiang, Y.G. (2017) Design of a Modified Tracking Differentiator. World Journal of Engineering and Technology, 5, 668-674. https://doi.org/10.4236/wjet.2017.54055