Abstract and Applied Analysis

Volume 2013 (2013), Article ID 346041, 7 pages

http://dx.doi.org/10.1155/2013/346041

Review Article

Observability of Nonlinear Fractional Dynamical Systems

K. Balachandran,¹ V. Govindaraj,¹ M. Rivero,² J. A. Tenreiro Machado,³ and J. J. Trujillo⁴

¹Department of Mathematics, Bharathiar University, Coimbatore 641 046, India

²Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, 4200-072 Porto, Portugal

³Departamento de Matemática Fundamental, Universidad de La Laguna, La Laguna, 38271 Tenerife, Spain

⁴Departamento de Análisis Matemático, Universidad de La Laguna, La Laguna, 38271 Tenerife, Spain

Received 19 March 2013; Accepted 7 June 2013

Academic Editor: Hossein Jafari

Copyright © 2013 K. Balachandran et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the observability of linear and nonlinear fractional differential systems of order by using the Mittag-Leffler matrix function and the application of Banach’s contraction mapping theorem. Several examples illustrate the concepts.

1. Introduction

Dynamical systems represented by ordinary differential equations are extensively studied in the literature. Many dynamical systems are modeled by fractional differential equations in various fields of science and technology [1–6]. In fact a fractional differential equation is considered as an alternative model to a nonlinear differential equation [7–9]. Recently, fractional differential equations are attracting a large number of researchers which results an increasing number of publications [10, 11]. Especially, the publications on the theory of fractional differential equations are growing exponentially, but the number of studies on qualitative behaviors of fractional dynamical systems is reduced, and its scope is limited. Therefore, it is important to study the qualitative behavior of fractional dynamical systems.

Observability is one of the structural properties of a fractional dynamical system defined as the possibility to deduce the initial state of the system from observing its input-output behavior. Several authors [12–16] have established the results for observability of linear fractional dynamical system of order using Gramian matrix and rank conditions. Also the recent monograph [17] reported the same results using rank condition. Motivated by the above discussion, in the present paper we study the observability of linear and nonlinear fractional order systems of order with corresponding linear observation using the Mittag-Leffler matrix function and the Banach's contraction mapping theorem. Examples are provided to illustrate the results.

Bearing these ideas in mind, this paper is organized as follows. Section 2 introduces the main fundamental concepts. Sections 3 and 4 analyse the observability of linear and nonlinear systems, respectively. Section 5 presents some examples that illustrate the concepts. Finally, Section 6 outlines the main conclusions.

2. Preliminaries

In this section, we introduce the definitions and preliminary results from fractional calculus which are used throughout this paper [18–23].

Definition 1 (see [24]). The Caputo fractional derivative of order with , , for a suitable function , is defined as

where . In particular, if then

For brevity, the Caputo fractional derivative is taken as .

Definition 2. The Miller-Ross sequential fractional derivative is defined as

where , and is any fractional differential operator, for example, it could be .

Definition 3 (see [25]). The Mittag-Leffler matrix function for an arbitrary square matrix is

Consider the fractional differential equation of order

where is an matrix and is a continuous function on . The solution of (5) is given by [26]

We use this solution representation to study the observability results.

3. Linear Systems

Consider the fractional order linear time invariant system

with linear observation

where , ,  is a matrix, and is an matrix.

Definition 4. The system (7) and (8) is observable on an interval if

implies that

Theorem 5. The linear system (7) and (8) is observable on if and only if the observability Gramian matrix

is positive definite.

Proof. The solution of (7) corresponding to the initial condition is given by

and we have, for ,

a quadratic form in . Clearly, matrix is symmetric. If is a positive definite, then implies that . Therefore, it yields that . Hence, the system (7) and (8) is observable on . If is not positive definite, then there is some such that . Then, , for , but , so , and we conclude that the system (7) and (8) is not observable on . Hence, the desired result.

If the linear system (7) and (8) is observable on an interval , then , and the initial state, for the solution on that interval, is reconstructed directly from the observation .

Definition 6. The matrix function defined on is an reconstruction kernel if and only if

Theorem 7. There exists a reconstruction kernel on if and only if the system (7) and (8) is observable on .

Proof. If a reconstruction kernel exists and satisfying

and , then . So , and we conclude that the system (7) and (8) is observable on . If, on the other hand, the system (7) and (8) is observable on , then from Theorem 5

Let

Then, we have

so that (17) is a reconstruction kernel on .

4. Nonlinear Systems

Consider the nonlinear system described by the fractional differential equation

where is an vector and is continuous on , with linear observation

where is an vector with . We assume that the system (19) is observed by the quantity . Then, the problem of observability of (19) is treated as follows: it is required to find the unknown state at the present time , from the quantity over the interval , where is some past time because, since , expression (19) does not allow immediate finding of and .

Definition 8. The system (19) and (20) is said to be observable at time if there exists such that the state of the system at time can be identified from knowledge of the system output over the interval . If the system is observable at every , it is called completely observable.

We will assume that (19) has a unique solution for any initial condition. If we take as , then the solution of (19) is uniquely defined for as the initial condition and is given by

We can rearrange

and is given by

Multiplying the above equation by from the left and integrating from to , we obtain

If the matrix is invertible, that is, if the truncated linear system (19) and (20) is observable, then, from (24), we have

Now let

Then, the following relation is obtained:

This equation represents the relation of the unknown state with the observed output over the interval . Hence, we have the following result.

Theorem 9. The system (19) and (20) is globally observable at and completely observable, if the following conditions hold.

(i) There exists a constant such that

(ii) (27) has a unique solution for any which is continuous on (a) for some , in the case of an observable system at time , and (b) for all and for some , in the case of a completely observable system.

In (27), time may not be necessarily fixed; therefore, can be replaced by . After this change is made, expression (27) is substituted into (22). We obtain that

where

In Theorem 9, if we replace (27) by (29), the same results are also valid with a simple change of variables. Next, we consider the application of Banach's contraction mapping theorem to these nonlinear equations.

Consider a special system of the form

where is a scalar positive constant and there exists a constant such that the nonlinear function satisfies the Lipschitz condition

Theorem 10. The system (31) and (32) is globally observable at time and completely observable, if the following conditions hold.

(i) There exists a constant such that

(ii) A positive constant satisfies

(a) for some , in the case of an observable system at time , and (b) for all and for some , in the case of a completely observable system.

Proof. A general solution for (31) with initial condition is given by

Just as (27) is derived from (22), the next equation is derived from (36):

Substituting (37) into (36), for every , we have

Consequently, for the system (31) and (32) to be observable, it is sufficient that the inverse of exists, and the solution of (38) exists and is unique. If we assume that there exist solutions ,   of (38), for a given such that , then, using (33), we have

where

From this, there exists a such that

where . If satisfies the inequality

it follows that on . This contradiction leads to the sufficient condition for the observability of system (31) and (32), since the condition (42) obviously guarantees the existence of solutions of (38).

Remark 11. The stability of the fractional system (7) and (19) has been discussed in [13, 27]. In the case of integer order system, that is, when , (7) becomes an algebraic equation and so must be greater than zero. When , (7) and (19) become

The stability of these systems are related to the eigenvalues of the matrix and the linear growth condition of the nonlinear function. The solution representation exactly match with the solution of the integer order system, and the stability results are readily follows [28].

5. Examples

In this section, we present two examples that illustrate the previous theoretical concepts.

Example 1. Consider that the sequential linear fractional differential equation is

Let us introduce the auxiliary variables and . Then,

and, therefore, problem (44) can be expressed as , where and . Suppose that observation for the system (44) is . Let us take . We pose the problem of computing and . The Mittag-Leffler matrix function, for the given matrix , is given by

The observability Gramian matrix for this system is

Here, is nonsingular, and then its inverse exists:

The reconstruction formula gives

Then, we conclude that and .

Example 2. Consider that the sequential nonlinear fractional differential equation is

Let us introduce the following auxiliary variables and . Then,

Therefore, problem (50) can be expressed as , where ,  and . Suppose that observation for the system (50) is . Let us take . The Mittag-Leffler matrix function for the given matrix is

where and . The observability Gramian matrix for this system,

is positive definite. Then, exists, and the nonlinear function satisfies the condition (33) with constant . By the application of Banach's contraction mapping theorem, the solution of this system exists and is unique. By Theorem 10, the system (50) is globally observable at time and completely observable.

A final remark noting that is not a state variable in the classical case, but maybe we could consider it as a certain “pseudostate variable,” without any real interpretation till the moment.

6. Conclusions

In this paper, we reviewed the main concepts underlying the observability of linear and nonlinear fractional differential systems of order . We considered the Mittag-Leffler matrix function and the application of Banach's contraction mapping theorem. Two examples were analyzed illustrating the observability concepts.

Acknowledgment

This work is partly supported by Project MTM2010-16499 from the Government of Spain and FEDER from EU.

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