High-order integral-chain differentiator and application to acceleration feedback

Differentiation of signals is an old and well-known problem [3]- [5] and has attracted more attention in recent years [6]- [9].In [6,7], a differentiator via second-order (or high-order) sliding modes algorithm has been proposed. The information one needs to know on the signal is the upper bounds for Lipschitz constant of the derivatives of the signal. It constrains the types of input signals. And for this differentiator, the chattering phenomenon is inevitable. The popular high-gain differentiators in [8,9] provide for an exact derivative when their gains tend to infinity. Noises exist in each layer of differential equations, therefore, all the derivatives estimations of input signal are all affected by noises directly.

In [10], a finite-time-convergent differentiator based on singular perturbation technique has been presented. However, the differentiators are complicated and difficult to be implemented in practice, due to the long computation time. In [11], a nonlinear tracking-differentiator with high speed is designed, and it succeeds in application to velocity estimation for low-speed regions only based on position measurements [12], in which only the convergence of signal tracking is described for this differentiator, but the convergence of derivative tracking is not given.

In this paper, we give the equivalence between integral-chain differentiator and use high-gain differentiator under suitable coordinate transformation. Because the structure of high-order integral chains exists in differentiator, and noises only exist in the last differential equation, each layer integrator can restrain noises. Therefore, integral-chain differentiator can restrain noises more thoroughly than usual high-gain linear differentiator. A nonlinear integral-chain differentiator is designed which is the expansion of linear integral-chain differentiator to increase convergent velocity.

Moreover, a 3-order differentiator is applied to the estimation of acceleration for a second-order uncertain system.

First of all, the concepts related to finite-time stability are given [13]- [17].

Definition 1 [13]: Consider a time-invariant system in the form of ( ) ( )

where

, ,

Theorem [17]: The origin is a finite-time stable equilibrium of f if and only if the origin is an asymptotically stable equilibrium of f and k < 0.

( ) ( )

We know that linear high-gain differentiator is [8,9]:

( )

where ( )

An integral-chain differentiator is designed as:

( ) ( )

where the parameters are the same with those in high-gain differentiator (5). We can also obtain the results of ( ) ( )

In the following, we will give the equivalence between ( 5) and (7).

Theorem 1: High-gain differentiator (5) and integral-chain differentiator are equivalent if the following relations are satisfied.

Proof:

then from (7) and (10) we can obtain that ( )

Therefore, we have ( ) ( )

From ( 10) and ( 7), we have

From ( 9), we can get

Therefore, ( 13) can be written as

From ( 10) and ( 7), we have

, then we get ( ) ( )

Because of

From ( 9), we can get

Therefore, we have ( ) ( )

Finally, from ( 10) and ( 7), we have

a a a = in (9), we have ( ) ( )

Then we get high-gain differentiator (5). This concludes the proof.

From Theorem 1, integral-chain linear differentiator has the same tracking results with that of usual high-gain differentiator in the condition that ε is sufficiently small.

Usually there are noises in the input signal. From (7), noises only exist in the last layer of differential equation.

Therefore, noises are retrained by layers of integral chains thoroughly. However, noises exist in every layers of usual high-gain differentiator. The noise cannot be restrained thoroughly. Therefore, integral-chain differentiator has better ability of restraining noises than usual high-gain differentiator.

We first give nonlinear integral-chain differentiator as follow:

A theorem about nonlinear integral-chain differentiator is given in the following.

Theorem 2: For nonlinear integral-chain differentiator, we have that: there exist 0 γ > (where 0 ργ > ) and 0 Γ > such that ( ) ( ) ( )

Where 0 ε > is the perturbation parameter and ( )

order [18] between i x and ( ) ( )

In order to prove Theorem 2, we will give a lemma in the following.

The equilibrium z = 0 of system ( )

is finite-time stable. Where

Proof: From [19], we know that system (24) is asymptotically stable. In the following, we will prove that the equilibrium z = 0 of system (24) is finite-time stable. From definition 2 and (24), we have ( )

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