Observability of Nonlinear Fractional Dynamical Systems

Abstract

We study the observability of linear and nonlinear fractional differential systems of order 0<𝛼<1 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 0<𝛼<1 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 0<𝛼<1 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 𝛼∈ℂ with 𝑛−1<𝛼≤𝑛, 𝑛∈ℕ, for a suitable function 𝑓, is defined as ^𝐶𝐷^𝛼₀₊𝑓𝑡)=1(Γ(𝑛−𝛼)^𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑓^𝑛(𝑠)d𝑠, (1) where 𝑓^𝑛(𝑠)=d^𝑛𝑓/d𝑠^𝑛. In particular, if 0<𝛼<1 then ^𝐶𝐷^𝛼₀₊𝑓𝑡)=1(Γ(1−𝛼)^𝑡₀(𝑡−𝑠)^−𝛼𝑓^′(𝑠)d𝑠. (2) For brevity, the Caputo fractional derivative ^𝐶𝐷^𝛼₀₊ is taken as ^𝐶𝐷^𝛼.

Definition 2. The Miller-Ross sequential fractional derivative is defined as 𝐷^𝑘𝛼𝑓(𝑥)=𝐷^𝛼𝐷^{(𝑘−1)𝛼}𝑓(𝑥), (3) where 𝑘=1,…,𝑛, (𝐷⁰𝑓)(𝑥)=𝑓(𝑥) and 𝐷^𝛼 is any fractional differential operator, for example, it could be ^𝐶𝐷^𝛼_𝑎+.

Definition 3 (see [25]). The Mittag-Leffler matrix function for an arbitrary square matrix 𝐴 is 𝐸_𝛼,𝛽(𝐴)=^∞_𝑘=0𝐴^𝑘Γ𝐸𝛼𝑘+𝛽,𝛼,𝛽>0,_𝛼,1(𝐴)=𝐸_𝛼(𝐴)with𝛽=1. (4) Consider the fractional differential equation of order 0<𝛼<1^𝐶𝐷^𝛼𝑥(𝑡)(𝑡)(𝑡)[]𝑥(0)=𝐴𝑥+𝑓,𝑡∈0,𝑇∶=𝐽,=𝑥₀, (5) where 𝐴 is an 𝑛×𝑛 matrix and 𝑓(𝑡) is a continuous function on 𝐽. The solution of (5) is given by [26] 𝑥(𝑡)=𝐸_𝛼𝐴𝑡^𝛼𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑓(𝑠)d𝑠. (6) We use this solution representation to study the observability results.

3. Linear Systems

Consider the fractional order linear time invariant system ^𝐶𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡),0<𝛼<1,𝑡∈𝐽, (7) with linear observation 𝑦(𝑡)=𝐻𝑥(𝑡), (8) where 𝑥∈ℝ^𝑛, 𝑦∈ℝ^𝑚,  𝐴 is a 𝑛×𝑛 matrix, and 𝐻 is an 𝑚×𝑛 matrix.

Definition 4. The system (7) and (8) is observable on an interval 𝐽 if 𝑦(𝑡)=𝐻𝑥(𝑡)=0,𝑡∈𝐽, (9) implies that 𝑥(𝑡)=0,𝑡∈𝐽. (10)

Theorem 5. The linear system (7) and (8) is observable on 𝐽 if and only if the observability Gramian matrix 𝑊(0,𝑇)=^𝑇₀𝐸_𝛼𝐴^∗𝑡^𝛼𝐻^∗𝐻𝐸_𝛼𝐴𝑡^𝛼d𝑡 (11) is positive definite.

Proof. The solution 𝑥(𝑡) of (7) corresponding to the initial condition 𝑥(0)=𝑥₀ is given by 𝑥(𝑡)=𝐸_𝛼𝐴𝑡^𝛼𝑥₀, (12) and we have, for 𝑦(𝑡)=𝐻𝑥(𝑡)=𝐻𝐸_𝛼(𝐴𝑡^𝛼)𝑥₀, 𝑦²=^𝑇₀𝑦^∗(𝑡)𝑦(𝑡)d𝑡=𝑥^∗₀^𝑇₀𝐸_𝛼𝐴^∗𝑡^𝛼𝐻^∗𝐻𝐸_𝛼𝐴𝑡^𝛼d𝑡𝑥₀=𝑥^∗₀𝑊(0,𝑇)𝑥₀, (13) a quadratic form in 𝑥₀. Clearly, matrix 𝑊(0,𝑇) is 𝑛×𝑛 symmetric. If 𝑊(0,𝑇) is a positive definite, then 𝑦=0 implies that 𝑥^∗₀𝑊(0,𝑇)𝑥₀=0. Therefore, it yields that 𝑥₀=0. Hence, the system (7) and (8) is observable on 𝐽. If 𝑊(0,𝑇) is not positive definite, then there is some 𝑥₀≠0 such that 𝑥^∗₀𝑊(0,𝑇)𝑥₀=0. Then, 𝑥(𝑡)=𝐸_𝛼(𝐴𝑡^𝛼)𝑥₀≠0, for 𝑡∈𝐽, but ‖𝑦‖²=0, so 𝑦=0, and we conclude that the system (7) and (8) is not observable on 𝐽. Hence, the desired result.

If the linear system (7) and (8) is observable on an interval 𝐽, then 𝑥(0)=𝑥₀, and the initial state, for the solution on that interval, is reconstructed directly from the observation 𝑦(𝑡)=𝐻𝐸_𝛼(𝐴𝑡^𝛼)𝑥₀.

Definition 6. The 𝑛×𝑛 matrix function 𝑅(𝑡) defined on 𝐽 is an reconstruction kernel if and only if ^𝑇₀𝑅(𝑡)𝐻𝐸_𝛼𝐴𝑡^𝛼d𝑡=𝐼. (14)

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

Proof. If a reconstruction kernel exists and satisfying ^𝑇₀𝑅(𝑡)𝑦(𝑡)d𝑡=^𝑇₀𝑅(𝑡)𝐻𝐸_𝛼𝐴𝑡^𝛼d𝑡𝑥₀=𝑥₀ (15) and 𝑦(𝑡)=0, then 𝑥₀=0. So 𝑥(𝑡)=0, and we conclude that the system (7) and (8) is observable on 𝐽. If, on the other hand, the system (7) and (8) is observable on 𝐽, then from Theorem 5𝑊(0,𝑇)=^𝑇₀𝐸_𝛼𝐴^∗𝑡^𝛼𝐻^∗𝐻𝐸_𝛼𝐴𝑡^𝛼d𝑡>0. (16) Let 𝑅₀(𝑡)=𝑊⁻¹(0,𝑇)𝐸_𝛼𝐴^∗𝑡^𝛼𝐻^∗,𝑡∈𝐽. (17) Then, we have ^𝑇₀𝑅₀(𝑡)𝐻𝐸_𝛼𝐴𝑡^𝛼d𝑡=𝑊^−1𝑇₀𝐸_𝛼𝐴^∗𝑡^𝛼𝐻^∗𝐻𝐸_𝛼𝐴𝑡^𝛼d𝑡=𝐼, (18) so that (17) is a reconstruction kernel on 𝐽.

4. Nonlinear Systems

Consider the nonlinear system described by the fractional differential equation ^𝐶𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡)),𝑡∈𝐽, (19) where 𝑓 is an 𝑛 vector and is continuous on 𝑡∈𝐽, with linear observation 𝑦(𝑡)=𝐻𝑥(𝑡), (20) where 𝑦 is an 𝑚 vector with 𝑚<𝑛. We assume that the system (19) is observed by the quantity 𝑦. Then, the problem of observability of (19) is treated as follows: it is required to find the unknown state at the present time 𝑡, from the quantity 𝑦 over the interval [𝜃,𝑡], where 𝜃 is some past time because, since 𝑚<𝑛, expression (19) does not allow immediate finding of 𝑥 and 𝑦.

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

We will assume that (19) has a unique solution for any initial condition. If we take 𝜏 as 𝜃<𝜏<𝑡, then the solution of (19) is uniquely defined for 𝑥=𝑥(𝜏) as the initial condition and is given by 𝑥(𝑡)=𝐸_𝛼𝐴(𝑡−𝜏)^𝛼𝑥(𝜏)+^𝑡_𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑓(𝑠,𝑥(𝑠))d𝑠. (21) We can rearrange 𝑥(𝜏)=𝐸_𝛼(𝐴(𝑡−𝜏)^𝛼)⁻¹×𝑥(𝑡)−^𝑡_𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑓(𝑠,𝑥(𝑠))d𝑠, (22) and 𝑦(𝜏) is given by 𝑦(𝜏)=𝐸_𝛼(𝐴(𝑡−𝜏)^𝛼)⁻¹×(𝑡)×𝐻𝑥−𝐻_𝑡𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑓((𝑠,𝑦(𝜏)=1𝑠,𝑥))d𝑠𝐸_𝛼(𝐴(𝑡−𝜏)^𝛼)²×(𝑡)×𝐻𝑥−𝐻_𝑡𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑓(𝑠,𝑥(𝑠))d𝑠×𝐸_𝛼𝐴(𝑡−𝜏)^𝛼. (23) Multiplying the above equation by 𝐸_𝛼(𝐴^∗(𝑡−𝜏)^𝛼)𝐻^∗ from the left and integrating from 𝜃 to 𝑡, we obtain ^𝑡_𝜃𝐸_𝛼(𝐴(𝑡−𝜏)^𝛼)²𝐸_𝛼𝐴^∗(𝑡−𝜏)^𝛼𝐻^∗𝑦(𝜏)=d𝜏_𝑡𝜃𝐸_𝛼𝐴^∗(𝑡−𝜏)^𝛼𝐻^∗𝐻𝑥(𝑡)𝐸_𝛼𝐴(𝑡−𝜏)^𝛼−d𝜏_𝑡𝜃𝐸_𝛼𝐴^∗(𝑡−𝜏)^𝛼𝐻^∗𝐻×^𝑡_𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑓(𝑠,𝑥(𝑠))d𝑠×𝐸_𝛼𝐴(𝑡−𝜏)^𝛼=d𝜏_𝑡𝜃𝐸_𝛼𝐴^∗(𝑡−𝜏)^𝛼𝐻^∗𝐻𝐸_𝛼𝐴(𝑡−𝜏)^𝛼(𝑡)−d𝜏𝑥_𝑡𝜃(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑓((𝑠×𝑠,𝑥))_𝑠𝜃𝐸_𝛼𝐴^∗(𝑡−𝜏)^𝛼𝐻^∗𝐻𝐸_𝛼𝐴(𝑡−𝜏)^𝛼()𝑥(𝑡)d𝜏d𝑠=𝑊𝜃,𝑡−^𝑡_𝜃(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑊(𝜃,𝑠)𝑓(𝑠,𝑥(𝑠))d𝑠. (24) If the matrix 𝑊(𝜃,𝑡) is invertible, that is, if the truncated linear system (19) and (20) is observable, then, from (24), we have 𝑥(𝑡)=𝑊⁻¹(𝜃,𝑡)^𝑡_𝜃𝐸_𝛼(𝐴(𝑡−𝑠)^𝛼)²×𝐸_𝛼𝐴^∗(𝑡−𝑠)^𝛼𝐻^∗𝑦(𝑠)d𝑠+𝑊⁻¹(𝜃,𝑡)^𝑡_𝜃(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×𝑊(𝜃,𝑠)𝑓(𝑠,𝑥(𝑠))d𝑠. (25) Now let 𝐺₁(𝑡,𝜃,𝑠)=𝑊⁻¹(𝜃,𝑡)𝐸_𝛼𝐴(𝑡−𝑠)^𝛼2×𝐸_𝛼𝐴^∗(𝑡−𝑠)^𝛼𝐻^∗,𝐺₂(𝑡,𝜃,𝑠)=𝑊⁻¹(𝜃,𝑡)𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑊(𝜃,𝑠). (26) Then, the following relation is obtained: 𝑥(𝑡)=^𝑡_𝜃𝐺₁()𝑦(𝑠)+𝑡,𝜃,𝑠d𝑠_𝑡𝜃(𝑡−𝑠)^𝛼−1𝐺₂(𝑡,𝜃,𝑠)𝑓(𝑠,𝑥(𝑠))d𝑠. (27) This equation represents the relation of the unknown state 𝑥 with the observed output 𝑦 over the interval [𝜃,𝑡]. Hence, we have the following result.

Theorem 9. The system (19) and (20) is globally (𝑎) observable at 𝑡 and (𝑏) completely observable, if the following conditions hold.
(i) There exists a constant 𝑐>0 such that 𝑑𝑒𝑡(𝑊(𝜃,𝑡))≥𝑐. (28)
(ii) (27) has a unique solution for any 𝑦 which is continuous on [𝜃,𝑡] (a) for some 𝜃<𝑡, in the case of an observable system at time 𝑡, and (b) for all 𝑡 and for some 𝜃<𝑡, in the case of a completely observable system.

In (27), time 𝜃 may not be necessarily fixed; therefore, 𝜃 can be replaced by 𝜏. After this change is made, expression (27) is substituted into (22). We obtain that 𝑥(𝜏)=𝐸_𝛼𝐴(𝑡−𝜏)^𝛼−1×^𝑡_𝜏𝐺₁()𝑦(𝑠)+𝑡,𝜏,𝑠d𝑠_𝑡𝜃(𝑡−𝑠)^𝛼−1𝐺₂(𝑡,𝜃,𝑠)𝑓(𝑠,𝑥(𝑠))d𝑠−^𝑡_𝜃(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝜏)^𝛼𝑓((𝑠=𝑠,𝑥))d𝑠_𝑡𝜏𝐺₃()𝑦(𝑠)+𝑡,𝜏,𝑠d𝑠_𝑡𝜏(𝑡−𝑠)^𝛼−1𝐺₄(𝑡,𝜏,𝑠)𝑓(𝑠,𝑥(𝑠))d𝑠for𝜏<𝑡, (29) where 𝐺₃(𝑡,𝜏,𝑠)=𝐸_𝛼(𝐴(𝑡−𝜏)^𝛼)⁻¹𝐺₁()𝐺𝑡,𝜏,𝑠₄(𝑡,𝜏,𝑠)=𝐸_𝛼𝐴(𝑡−𝜏)^𝛼−1×𝐺₂(𝑡,𝜏,𝑠)−𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼. (30) 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 ^𝐶𝐷^𝛼𝑥(𝑡)(𝑡)((𝑡,𝑦(𝑡)(𝑡),=𝐴𝑥+𝜖𝑓𝑡,𝑥))=𝐻𝑥 (31)(32) where 𝜖 is a scalar positive constant and there exists a constant 𝑘≥0 such that the nonlinear function 𝑓 satisfies the Lipschitz condition 𝑓𝑡,𝑥₁−𝑓𝑡,𝑥₂≤𝑘𝑥₁−𝑥₂. (33)

Theorem 10. The system (31) and (32) is globally (𝑎) observable at time 𝑡 and (𝑏) completely observable, if the following conditions hold.
(i) There exists a constant 𝑐>0 such that 𝑑𝑒𝑡(𝑊(𝜃,𝑡))≥𝑐. (34)
(ii) A positive constant 𝜖 satisfies 𝛼()𝜖<𝑡−𝜃^−𝛼𝑘(𝑡,𝜃) (35) (a) for some 𝜃<𝑡, in the case of an observable system at time 𝑡, and (b) for all 𝑡 and for some 𝜃<𝑡, in the case of a completely observable system.

Proof. A general solution 𝑥(𝑡) for (31) with initial condition 𝑥=𝑥(𝜏) is given by 𝑥(𝜏)=𝐸_𝛼𝐴(𝑡−𝜏)^𝛼−1×𝑥(𝑡)−𝜖^𝑡_𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑓(𝑠,𝑥(𝑠))d𝑠. (36) Just as (27) is derived from (22), the next equation is derived from (36): 𝑥(𝑡)=𝑊⁻¹(𝜃,𝑡)^𝑡_𝜃𝐸_𝛼𝐴(𝑡−𝑠)^𝛼2×𝐸_𝛼𝐴^∗(𝑡−𝑠)^𝛼𝐻^∗𝑦(𝑠)d𝑠+𝜖𝑊⁻¹(𝜃,𝑡)^𝑡_𝜃(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×𝑊(𝜃,𝑠)𝑓(𝑠,𝑥(𝑠))d𝑠. (37) Substituting (37) into (36), for every 𝜏∈[𝜃,𝑡], we have 𝑥(𝜏)=𝐸_𝛼𝐴(𝑡−𝜏)^𝛼−1×𝑊⁻¹(𝜃,𝑡)^𝑡_𝜃𝐸_𝛼𝐴^∗(𝑡−𝑠)^𝛼×𝐻^∗𝐸_𝛼(𝐴(𝑡−𝑠)^𝛼)²𝑦(𝑠)d𝑠+𝜖𝑊⁻¹(𝜃,𝑡)^𝑡_𝜃(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×𝑊(𝜃,𝑠)𝑓(𝑠,𝑥(𝑠))d𝑠−𝜖^𝑡_𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑓(𝑠,𝑥(𝑠))d𝑠. (38) Consequently, for the system (31) and (32) to be observable, it is sufficient that the inverse of 𝑊(𝜃,𝑡) exists, and the solution of (38) exists and is unique. If we assume that there exist solutions 𝑥₁,  𝑥₂ of (38), for a given 𝑦 such that 𝑥₁≠𝑥₂, then, using (33), we have 𝑥₁(𝜏)−𝑥₂(𝜏𝐸)≤𝑘𝜖_𝛼𝐴(𝑡−𝜏)^𝛼−1×^𝑡_𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑥₁−𝑥₂𝐸d𝑠+𝑘𝜖_𝛼𝐴(𝑡−𝜏)^𝛼−1𝑊⁻¹(×𝜃,𝑡)_𝑡𝜃(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑥₁−𝑥₂𝐸d𝑠≤𝑘𝜖_𝛼𝐴(𝑡−𝜏)^𝛼−1×^𝑡_𝜃(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑥₁−𝑥₂𝐸d𝑠+𝑘𝜖_𝛼𝐴(𝑡−𝜏)^𝛼−1𝑊⁻¹(×𝜃,𝑡)_𝑡𝜃(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑥₁−𝑥₂≤𝜖d𝑠𝛼(𝑡−𝜃)^𝛼𝑘₁(𝑡,𝜃)𝑥₁−𝑥₂+𝜖𝛼(𝑡−𝜃)^𝛼𝑘₂(𝑡,𝜃)𝑥₁−𝑥₂, (39) where 𝑘₁(𝑡,𝜃)=max_{𝜃<𝜏<𝑠<𝑡}𝐸_𝛼𝐴(𝑡−𝜏)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑘𝑘,₂(𝑡,𝜃)=max_{𝜃<𝜏<𝑠<𝑡}𝐸_𝛼𝐴(𝑡−𝜏)^𝛼−1𝑊⁻¹(×𝐸𝜃,𝑡)_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼𝑊(𝜃,𝑡)𝑘. (40) From this, there exists a 𝑘(𝑡,𝜃) such that 𝑥₁(𝜏)−𝑥₂(𝜏≤𝜖)𝛼𝑘(𝑡,𝜃)(𝑡−𝜃)^𝛼𝑥₁−𝑥₂, (41) where 𝑘(𝑡,𝜃)=𝑘₁(𝑡,𝜃)+𝑘₂(𝑡,𝜃). If 𝜖 satisfies the inequality 𝜖𝛼𝑘(𝑡,𝜃)(𝑡−𝜃)^𝛼<1, (42) it follows that 𝑥₁=𝑥₂ on [𝜃,𝑡]. 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 𝛼=0, (7) becomes an algebraic equation and so 𝛼 must be greater than zero. When 𝛼=1, (7) and (19) become 𝑥̇(𝑡)(𝑡),𝑥̇(𝑡)(𝑡)((𝑡.=𝐴𝑥=𝐴𝑥+𝑓𝑡,𝑥)) (43) The stability of these systems are related to the eigenvalues of the matrix 𝐴 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 ^𝐶𝐷^2𝛼₀₊𝑥(𝑡)+𝑥(𝑡)=0,0<𝛼<1,𝑡∈[0,1]. (44) Let us introduce the auxiliary variables 𝑥₁(𝑡)=𝑥(𝑡) and 𝑥₂(𝑡)=^𝐶𝐷^𝛼₀₊𝑥₁(𝑡). Then, ^𝐶𝐷^𝛼₀₊𝑥₁(𝑡)=^𝐶𝐷^𝛼₀₊𝑥(𝑡)=𝑥₂(𝑡),_𝐶𝐷^𝛼₀₊𝑥₂(𝑡)=^𝐶𝐷^2𝛼₀₊𝑥(𝑡)=−𝑥₁(𝑡), (45) and, therefore, problem (44) can be expressed as ^𝐶𝐷^𝛼₀₊𝑥(𝑡)=𝐴𝑥(𝑡), where 𝐴=⁰¹₋₁₀ and 𝑥(𝑡)=^𝑥1_𝑥^(𝑡)_2(𝑡). Suppose that observation for the system (44) is 𝑦(𝑡)=𝑥₁(𝑡)=(1/2)𝑒^−𝑡+(1/2)𝑒^−𝑡erfi(√𝑡). Let us take 𝛼=1/2. We pose the problem of computing 𝑥₁(0) and 𝑥₂(0). The Mittag-Leffler matrix function, for the given matrix 𝐴, is given by 𝐸_1/2𝐴𝑡^1/2=𝑒^−𝑡𝑒^−𝑡erfi√𝑡𝑒^−𝑡erfi√𝑡𝑒^−𝑡. (46) The observability Gramian matrix for this system is 𝑊(0,1)=¹₀𝐸_1/2𝐴^∗𝑡^1/2𝐻^∗𝐻𝐸_1/2𝐴𝑡^1/2=d𝑡₁₀𝑒^−2𝑡𝑒^−2𝑡erfi√𝑡𝑒^−2𝑡erfi√𝑡𝑒^−2𝑡erfi√𝑡²=.d𝑡0.4323320.3096700.3096700.286423 (47) Here, 𝑊(0,1) is nonsingular, and then its inverse exists: 𝑊⁻¹(0,1)=10.2535−11.0857−11.085715.4768. (48) The reconstruction formula gives 𝑥(0)=¹₀𝑅(𝑡)𝑦(𝑡)=d𝑡₁₀𝑊⁻¹(0,1)𝐸_1/2𝐴^∗𝑡^1/2𝐻^∗𝑦(𝑡)=1d𝑡2¹₀10.2535𝑒⁻¹−11.0857𝑒^−𝑡erfi√𝑡−11.0857𝑒⁻¹+15.4768𝑒^−𝑡erfi√𝑡×𝑒^−𝑡+𝑒^−𝑡erfi√𝑡=.d𝑡0.5000070.499998 (49) Then, we conclude that 𝑥₁(0)=0.500007 and 𝑥₂(0)=0.499998.

Example 2. Consider that the sequential nonlinear fractional differential equation is ^𝐶𝐷^2𝛼₀₊𝑥(𝑡)(𝑡)=1−𝑥𝑥²[].+1,0<𝛼<1,𝑡∈0,2 (50) Let us introduce the following auxiliary variables 𝑥₁(𝑡)=𝑥(𝑡) and 𝑥₂(𝑡)=^𝐶𝐷^𝛼₀₊𝑥₁(𝑡). Then, ^𝐶𝐷^𝛼₀₊𝑥₁(𝑡)=^𝐶𝐷^𝛼₀₊𝑥(𝑡)=𝑥₂(𝑡),_𝐶𝐷^𝛼₀₊𝑥₂(𝑡)=^𝐶𝐷^2𝛼₀₊𝑥(𝑡)=𝑥₁(𝑡)+1𝑥²₁.+1 (51) Therefore, problem (50) can be expressed as ^𝐶𝐷^𝛼₀₊𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡)), where 𝐴=⁰¹₁₀,  𝑓(𝑡,𝑥(𝑡))=⁰_1/(𝑥²₁₊₁₎ and 𝑥(𝑡)=^𝑥1_𝑥^(𝑡)_2(𝑡). Suppose that observation for the system (50) is 𝑦(𝑡)=𝑥₂(𝑡). Let us take 𝛼=3/4. The Mittag-Leffler matrix function for the given matrix 𝐴 is 𝐸_3/4𝐴𝑡^3/4=𝑁₁(𝑡)𝑁₂(𝑡)𝑁₂(𝑡)𝑁₁(𝑡), (52) where 𝑁₁(𝑡)=(1/2)[𝐸_3/4(𝑡^3/4)+𝐸_3/4((−𝑡)^3/4)] and 𝑁₂(𝑡)=(1/2)[𝐸_3/4(𝑡^3/4)−𝐸_3/4((−𝑡)^3/4)]. The observability Gramian matrix for this system, 𝑊(0,2)=²₀𝐸_3/4𝐴^∗𝑡^3/4𝐻^∗𝐻𝐸_3/4𝐴𝑡^3/4=d𝑡₂₀𝑁²₂(𝑡)𝑁₁(𝑡)𝑁₂(𝑡)𝑁₁(𝑡)𝑁₂(𝑡)𝑁²₁(𝑡)=,d𝑡10.00611.244111.244112.9504 (53) is positive definite. Then, 𝑊⁻¹(0,2) exists, and the nonlinear function 𝑓(𝑡,𝑥(𝑡)) satisfies the condition (33) with constant 𝑘=2. 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 (𝑎) observable at time 𝑡 and (𝑏) completely observable.

A final remark noting that 𝑥₂ 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 0<𝛼<1. 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.