^{1}

^{2}

^{1}

^{1,3}

^{1}

^{1}

^{2}

^{3}

This paper is concerned with the problem of designing disturbance observer for fractional order systems, of which the disturbance is in time series expansion. The stability of a special observer with the selected nonlinear weighted function and transient dynamics function is rigorously analyzed for slowly varying disturbance. In addition, the result is also extended to estimate slope forms disturbance and higher order disturbance of fractional order systems. The efficacy of the proposed method is validated through numerical examples.

In recent years, fractional order systems (FOSs) have attracted considerable attention from control community, since many engineering plants and processes cannot be described concisely and precisely without the introduction of fractional order calculus [

Like the integer order system, the disturbance always exists in the FOSs and usually it is not possible or practical to obtain the exact model of the FOSs, so disturbance or uncertainty observation has been one of the major issues in the control field. A rich body of results about disturbance observation have been reported in the literature with different methods. For example, the so-called Q-filter method estimates the disturbance depending on the inversion of the transfer function. Many applications based on Q-filter have been reported in process control fields [

A special kind of disturbance in time series expansion is considered to solve the friction compensation problem in [

The following section is devoted to some basic background materials and the main problems. Discussing the more general disturbance using a special nonlinear function, the main results are presented in Section

Consider the following fractional order system:

The reduced-order system can be expressed as

Consider the disturbance has the following form:

slowly varying disturbance

slope forms disturbance

higher order disturbance

The following Caputo definition [

The so-called fractional order integral is just the dual operation of the fractional order differential. If

Consider polynomials

Given a constant matrix

Defining the observation error

Substituting (

From the Caputo definition of fractional derivatives, one has

The Laplace transform of (

According to the initial condition, one has that

Regarding stability, any of the matrices whose eigenvalues are all in the region

The functions

Considering

Consider the slope forms disturbance; Theorem

Given two constant matrices

Consider the same observation error

Set

When

When

Overall, when

When

If

When

If

To sum up the above arguments, if

When

When

When

Also considering that

According to the understanding of the aforementioned approaches, one generalizes Theorem

Given constant matrices

The theorem can be generalized by using the similar aforementioned approach; wherefore the proof is omitted here.

Set

Under certain conditions, all the above mentioned theorems (Theorems

The proposed method has a great design freedom. With the appropriate parameters, the FDOBp is a fast nonovershooting disturbance observer for the FOSs with the disturbance (

Consider the system [

Simulations are performed for

Chaotic attractor of the system with

Set the numerical simulation parameters

Results of Example

Results of Example

Results of Example

Results of Example

Figure

Considering the system in (

Results of Example

Considering the same system in (

Results of Example

Considering the same system in (

This simulation is implemented with

Results of Example

In this paper, the methods of observer for fractional order systems in time series expansion disturbance have been investigated. According to the maximum degree of disturbance polynomials, the disturbances are divided into three categories. And then, different observers have been designed for three disturbances, respectively. Compared to existing integer order results, the new proposed approaches have greater design freedom and the designed observers have a faster convergent speed. The numerical examples have shown the advantages and the efficiency of the proposed design methods. It is believed that the approaches provide a new avenue to solve such problems. The interesting future topics involve the following cases:

to study the problem of noise effect reduction in case the measured state is mixed with the measurement noise;

to discuss the problem of output-based method when only partial states are measurable;

to investigate the problem considering the missing measurement data.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the associate editor and the anonymous reviewers for their keen and insightful comments which greatly improved the contents and the presentation. This work is supported by the National Nature Science Foundation under Grant no. 61004017.