1. Introduction

Recently, fractional-order control systems have attracted increasing interest [15, 9, 11]. On the one hand, this is mainly due to the fact that many real-world physical systems are well characterized by fractional-order state equations [15], i.e., equations involving the so-called fractional derivatives and integrals. On the other hand, with the success in the synthesis of real noninteger differentiators and the emergence of a new electrical circuit element called “fractance” [10, 21], fractional-order controllers [16, 18, 12] have been designed and applied to control a variety of dynamical processes, including integer-order and fractional-order systems, so as to enhance the robustness and performance of the control systems. Singular fractional systems (known as generalized, descriptor of Fractional systems) describe a large class of systems, which are not only theoretical interest but also have a great importance in practice.

Stability is fundamental to all control systems, certainly including fractional-order control systems [20, 19]. Recently, stability and stabilization problems of fractional-order linear time-invariant interval systems have been investigated in [1], [2, 4]. For example, for fractional-order linear time-invariant interval systems described in the transfer function form, the stability issue was discussed first in [13] and then further in [14]. In this paper we consider the problem of the robust asymptotical Stabilization for uncertain descriptor fractional-order systems.

The descriptor multi-fractional-order systems by applying a derivative multi-controller and a state feedback.

Controller is given to achieve the robust asymptotical stabilization of the obtained two sub system, first is fractional-order systems and the second is zero state.

We using canonical form for the descriptor fractional-order systems and by applying a derivative controller and a state feedback controller is given to achieve the robust asymptotical stabilization of the fractional-order systems.

In section II, the paper are organized as follow in section II, we introduce the definition of fractional derivative in brief; we present also some mathematical results. In section III, we propose robust linear uncertainty descriptor multi-fractional controller for the stabilization of system.

2. Preliminaries

A. Some definition

Now we review some important and definition:

The Caputo derivative on the other, defined [17],

For and is the well-known Euler’s gamma function.

Definition (2.1), [22]:

Let , . Their Kronecker product (i.e., the direct product or tensor product), denoted as is defined by

Now we consider the fractional –order linear system.

(1)

Where is the state vector.

The system (1) is stable if the condition is satisfied

[5], with the condition(2),

(2)

Where spec(A) represents the eigenvalues of matrix A.

B.Some mathematical inequalities.

Lemma (2.1), [8]:

Let and . The fractional- order system is asymptotically stable that means

if and only if there exist two real symmetric matrices and two skew-symmetric matrices where

such that

where

Lemma (2.2), [7]:

For any matrices X and Y with appropriate dimensions, we have

For any

Remark (2.1), [5]:

Consider the singular fractional linear system if the following conditions hold:

i. the matrix pair (E, A) is regular .

ii. the matrix pair (E, A) is regular an impulse free.

Then there exist there two invertible matrices. Satisfying:

where is an invertible, are the identity matrices of dimensions r, n-r respectively.

Definition (2.2), [5, 6]:

i. A matrix pair (E, A) is called regular if E and A are square and for some value it is called singular otherwise. Where is the set of all Finite Spectrum Eigenvalues.

ii. The matrix pair (E, A) is said to be impulse free if

3. The Main Result

The singular linear fractional order uncertainty control system

(3)

where is the singular matrix, , , is the semi-state vector, is the input vector, is invertible and are time-invariant matrix representing norm-bounded parameter uncertainty, with the following conditions:

i. The singular matrix E has the form, where nonsingular matrix

defined in (iii) and equation (7) later on.

ii. where , , , such that

iii. , is invertible, where ,,

iv.

where and are known real constant matrices of appropriate dimensions, and the uncertain matrices satisfies

(5)

v. The pencil matrix is regular and impulse free. Consider the feedback control for system (3) in the following form

(6)

where has the form

(7)

and are gain matrices such that,

(8)

where

and is invertible matrix. We substituting (6) into system (3), we obtain

We have

We gets

We obtain

(9)

We have

By condition (i), we have

(10)

Such that

The right side of equation (9) has the form

(11)

is invertible sub-matrix of the system (9) and by using the remark (2.1), then there exist two invertible matrices satisfying:

(12)

where is an invertible which defined later on. One can get:

Then

Since and , hence the finite eigenvalue of the matrix pair

is also an eigenvalue of matrix

with

We assume that M and N has the forms:

where

and

Where

We obtain

By using M and N, we have

(13)

We have

We have

(14)

0ne can get

Then we have

We obtain

We have

We have

(15)

We get

(16)

Where

also

Then, we have

One can get

We have

We obtain the formula as follows:

where The design of the gain matrix K which robustly stabilization the descriptor fractional-order system (3) for the fractional order belonging to (18.a), are derived.

Theorem (3.1)

Assume that (3) is regular and impulse free, then there exists again matrix such that descriptor fractional order (3) with fractional-order controlled by the control (6) is asymptotically stable, if there exist matricesand two real scalars such that

(19)

Where

Satisfy Lemma (2.1).

Proof:-

Under the assumption regular and impulse free that system(3), then there exists a gain matrix L such that system (3)can be written in the form (18), in this case the matrix K can be determined from the stability of system (18).It follows from Lemma (2.1) that is equivalent to

(20)

Where and Satisfy Lemma (2.1). By assume in (20) one can conclude that

(21)

Suppose that there exists matrices and , Such that

(22)

Substituting in (22) with we obtain

(23)

By equation (5) , then we obtain

(24)

Also

Then by (24) and Lemma (2.2) that for any real scalar

(25)

By using equation (24), we obtain

(26)

By substituting (26) into (23), we have

(27)

Inequality (27) is equivalent to (18) by the well-known Schur Complement by [3].

4. Conclusions

The necessary conditions of robust asymptotically stabilization for special Uncertain singuler fractional-order systems with feedback fractional control for the fractional order α belonging to with parameter uncertainties in the state matrix have been given in details . The problem of canonical of descriptor fractional-order systems by derivative fractional controller has been proposed with implosive free condition.