# Observability of Nonlinear Fractional Dynamical Systems

Authors: K. Balachandran; V. Govindaraj; M. Rivero; J. A. Tenreiro Machado; J. J. Trujillo
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study the observability of linear and nonlinear fractional differential systems of order0<α<1 by using the Mittag-Leffler matrix function and the application of Banach’s contraction mapping theorem. Several examples illustrate the concepts.


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## Body

## 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. Section2 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 orderα∈ℂ with n-1<α≤n, n∈ℕ, for a suitable function f, is defined as
(1)(CD0+αf)(t)=1Γ(n-α)∫0t‍(t-s)n-α-1fn(s)ds,
where fn(s)=dnf/dsn. In particular, if 0<α<1 then
(2)(CD0+αf)(t)=1Γ(1-α)∫0t‍(t-s)-αf′(s)ds.
For brevity, the Caputo fractional derivative D0+αC is taken as DαC.Definition 2.
The Miller-Ross sequential fractional derivative is defined as(3)(Dkαf)(x)=(DαD(k-1)αf)(x),
where k=1,…,n, (D0f)(x)=f(x) and Dα is any fractional differential operator, for example, it could be CDa+α.Definition 3 (see [25]).
The Mittag-Leffler matrix function for an arbitrary square matrixA is
(4)Eα,β(A)=∑k=0∞‍AkΓ(αk+β),α,β>0,Eα,1(A)=Eα(A)withβ=1.
Consider the fractional differential equation of order 0<α<1(5)CDαx(t)=Ax(t)+f(t),t∈[0,T]≔J,x(0)=x0,
where A is an n×n matrix and f(t) is a continuous function on J. The solution of (5) is given by [26]
(6)x(t)=Eα(Atα)x0+∫0t‍(t-s)α-1Eα,α(A(t-s)α)f(s)ds.
We use this solution representation to study the observability results.


## 3. Linear Systems

Consider the fractional order linear time invariant system(7)CDαx(t)=Ax(t),0<α<1,t∈J,
with linear observation
(8)y(t)=Hx(t),
where x∈ℝn, y∈ℝm,  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
(9)y(t)=Hx(t)=0,t∈J,
implies that
(10)x(t)=0,t∈J.Theorem 5.
The linear system (7) and (8) is observable on J if and only if the observability Gramian matrix
(11)W(0,T)=∫0T‍Eα(A*tα)H*HEα(Atα)dt
is positive definite.Proof.
The solutionx(t) of (7) corresponding to the initial condition x(0)=x0 is given by
(12)x(t)=Eα(Atα)x0,
and we have, for y(t)=Hx(t)=HEα(Atα)x0,
(13)∥y∥2=∫0T‍y*(t)y(t)dt=x0*∫0T‍Eα(A*tα)H*HEα(Atα)dtx0=x0*W(0,T)x0,
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 x0*W(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 x0*W(0,T)x0=0. Then, x(t)=Eα(Atα)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)=HEα(Atα)x0.Definition 6.
Then×n matrix function R(t) defined on J is an reconstruction kernel if and only if
(14)∫0T‍R(t)HEα(Atα)dt=I.Theorem 7.
There exists a reconstruction kernelR(t) on J if and only if the system (7) and (8) is observable on J.Proof.
If a reconstruction kernel exists and satisfying(15)∫0T‍R(t)y(t)dt=∫0T‍R(t)HEα(Atα)dtx0=x0
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 5(16)W(0,T)=∫0T‍Eα(A*tα)H*HEα(Atα)dt>0.
Let
(17)R0(t)=W-1(0,T)Eα(A*tα)H*,t∈J.
Then, we have
(18)∫0T‍R0(t)HEα(Atα)dt=W-1∫0T‍Eα(A*tα)H*HEα(Atα)dt=I,
so that (17) is a reconstruction kernel on J.


## 4. Nonlinear Systems

Consider the nonlinear system described by the fractional differential equation(19)CDαx(t)=Ax(t)+f(t,x(t)),t∈J,
where f is an n vector and is continuous on t∈J, with linear observation
(20)y(t)=Hx(t),
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
(21)x(t)=Eα(A(t-τ)α)x(τ)+∫τt‍(t-s)α-1Eα,α(A(t-s)α)f(s,x(s))ds.
We can rearrange
(22)x(τ)=[Eα(A(t-τ)α)]-1×[x(t)-∫τt‍(t-s)α-1Eα,α(A(t-s)α)f(s,x(s))ds],
and y(τ) is given by
(23)y(τ)=[Eα(A(t-τ)α)]-1×[∫τt‍Hx(t)-H×∫τt‍(t-s)α-1Eα,α(A(t-s)α)f(s,x(s))ds],y(τ)=1[Eα(A(t-τ)α)]2×[∫τt‍(t-s)α-1Eα,α(A(t-s)α)f(s,x(s))dsHx(t)-H×∫τt‍(t-s)α-1Eα,α(A(t-s)α)f(s,x(s))ds]×Eα(A(t-τ)α).
Multiplying the above equation by Eα(A*(t-τ)α)H* from the left and integrating from θ to t, we obtain
(24)∫θt‍[Eα(A(t-τ)α)]2Eα(A*(t-τ)α)H*y(τ)dτ=∫θt‍Eα(A*(t-τ)α)H*Hx(t)Eα(A(t-τ)α)dτ-∫θt‍Eα(A*(t-τ)α)H*H×(∫τt‍(t-s)α-1Eα,α(A(t-s)α)f(s,x(s))ds)×Eα(A(t-τ)α)dτ=∫θt‍Eα(A*(t-τ)α)H*HEα(A(t-τ)α)dτx(t)-∫θt‍(t-s)α-1Eα,α(A(t-s)α)f(s,x(s))×(∫θs‍Eα(A*(t-τ)α)H*HEα(A(t-τ)α)dτ)ds=W(θ,t)x(t)-∫θt‍(t-s)α-1Eα,α(A(t-s)α)W(θ,s)f(s,x(s))ds.
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
(25)x(t)=W-1(θ,t)∫θt‍[Eα(A(t-s)α)]2×Eα(A*(t-s)α)H*y(s)ds+W-1(θ,t)∫θt‍(t-s)α-1Eα,α(A(t-s)α)×W(θ,s)f(s,x(s))ds.
Now let
(26)G1(t,θ,s)=W-1(θ,t)[Eα(A(t-s)α)]2×Eα(A*(t-s)α)H*,G2(t,θ,s)=W-1(θ,t)Eα,α(A(t-s)α)W(θ,s).
Then, the following relation is obtained:
(27)x(t)=∫θt‍G1(t,θ,s)y(s)ds+∫θt‍(t-s)α-1G2(t,θ,s)f(s,x(s))ds.
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 constantc>0 such that
(28)det(W(θ,t))≥c.
(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
(29)x(τ)=[Eα(A(t-τ)α)]-1×[∫τt‍G1(t,τ,s)y(s)ds+∫θt‍(t-s)α-1G2(t,θ,s)f(s,x(s))ds-∫θt‍(t-s)α-1Eα,α(A(t-τ)α)f(s,x(s))ds]=∫τt‍G3(t,τ,s)y(s)ds+∫τt‍(t-s)α-1G4(t,τ,s)f(s,x(s))dsforτ<t,
where
(30)G3(t,τ,s)=[Eα(A(t-τ)α)]-1G1(t,τ,s)G4(t,τ,s)=[Eα(A(t-τ)α)]-1×[G2(t,τ,s)-Eα,α(A(t-s)α)].
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(31)CDαx(t)=Ax(t)+ϵf(t,x(t)),(32)y(t)=Hx(t),
where ϵ is a scalar positive constant and there exists a constant k≥0 such that the nonlinear function f satisfies the Lipschitz condition
(33)∥f(t,x1)-f(t,x2)∥≤k∥x1-x2∥.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 constantc>0 such that
(34)det(W(θ,t))≥c.
(ii) A positive constantϵ satisfies
(35)ϵ<α(t-θ)-αk(t,θ)
(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 solutionx(t) for (31) with initial condition x=x(τ) is given by
(36)x(τ)=[Eα(A(t-τ)α)]-1×[x(t)-ϵ∫τt‍(t-s)α-1Eα,α(A(t-s)α)f(s,x(s))ds].
Just as (27) is derived from (22), the next equation is derived from (36):
(37)x(t)=W-1(θ,t)∫θt‍[Eα(A(t-s)α)]2×Eα(A*(t-s)α)H*y(s)ds+ϵW-1(θ,t)∫θt‍(t-s)α-1Eα,α(A(t-s)α)×W(θ,s)f(s,x(s))ds.
Substituting (37) into (36), for every τ∈[θ,t], we have
(38)x(τ)=[Eα(A(t-τ)α)]-1×[W-1(θ,t)∫θt‍Eα(A*(t-s)α)×H*[Eα(A(t-s)α)]2y(s)ds+ϵW-1(θ,t)∫θt‍(t-s)α-1Eα,α(A(t-s)α)×W(θ,s)f(s,x(s))ds-ϵ∫τt‍(t-s)α-1Eα,α(A(t-s)α)f(s,x(s))ds].
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
(39)∥x1(τ)-x2(τ)∥≤kϵ|(Eα(A(t-τ)α))-1|×∫τt‍(t-s)α-1|Eα,α(A(t-s)α)||x1-x2|ds+kϵ|(Eα(A(t-τ)α))-1||W-1(θ,t)|×∫θt‍(t-s)α-1|Eα,α(A(t-s)α)||x1-x2|ds≤kϵ|(Eα(A(t-τ)α))-1|×∫θt‍(t-s)α-1|Eα,α(A(t-s)α)||x1-x2|ds+kϵ|(Eα(A(t-τ)α))-1||W-1(θ,t)|×∫θt‍(t-s)α-1|Eα,α(A(t-s)α)||x1-x2|ds≤ϵα(t-θ)αk1(t,θ)∥x1-x2∥+ϵα(t-θ)αk2(t,θ)∥x1-x2∥,
where
(40)k1(t,θ)=maxθ<τ<s<t|(Eα(A(t-τ)α))-1Eα,α(A(t-s)α)|k,k2(t,θ)=maxθ<τ<s<t|(Eα(A(t-τ)α))-1W-1(θ,t)|×|Eα,α(A(t-s)α)W(θ,t)|k.
From this, there exists a k(t,θ) such that
(41)∥x1(τ)-x2(τ)∥≤ϵαk(t,θ)(t-θ)α∥x1-x2∥,
where k(t,θ)=k1(t,θ)+k2(t,θ). If ϵ satisfies the inequality
(42)ϵαk(t,θ)(t-θ)α<1,
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
(43)x˙(t)=Ax(t),x˙(t)=Ax(t)+f(t,x(t)).
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(44)CD0+2αx(t)+x(t)=0,0<α<1,t∈[0,1].
Let us introduce the auxiliary variables x1(t)=x(t) and x2(t)=D0+αCx1(t). Then,
(45)CD0+αx1(t)=CD0+αx(t)=x2(t),CD0+αx2(t)=CD0+2αx(t)=-x1(t),
and, therefore, problem (44) can be expressed as CD0+αx-(t)=Ax-(t), where A=[01-10] and x-(t)=[x1(t)x2(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
(46)E1/2(At1/2)=(e-te-terfi(t)e-terfi(t)e-t).
The observability Gramian matrix for this system is
(47)W(0,1)=∫01‍E1/2(A*t1/2)H*HE1/2(At1/2)dt=∫01‍(e-2te-2terfi(t)e-2terfi(t)e-2terfi(t)2)dt=(0.4323320.3096700.3096700.286423).
Here, W(0,1) is nonsingular, and then its inverse exists:
(48)W-1(0,1)=(10.2535-11.0857-11.085715.4768).
The reconstruction formula gives
(49)x-(0)=∫01‍R(t)y(t)dt=∫01‍W-1(0,1)E1/2(A*t1/2)H*y(t)dt=12∫01‍(10.2535e-1-11.0857e-terfi(t)-11.0857e-1+15.4768e-terfi(t))×(e-t+e-terfi(t))dt=(0.5000070.499998).
Then, we conclude that x1(0)=0.500007 and x2(0)=0.499998.Example 2.
Consider that the sequential nonlinear fractional differential equation is(50)CD0+2αx(t)-x(t)=1x2+1,0<α<1,t∈[0,2].
Let us introduce the following auxiliary variables x1(t)=x(t) and x2(t)=CD0+αx1(t). Then,
(51)CD0+αx1(t)=CD0+αx(t)=x2(t),CD0+αx2(t)=CD0+2αx(t)=x1(t)+1x12+1.
Therefore, problem (50) can be expressed as CD0+αx-(t)=Ax-(t)+f(t,x-(t)), where A=[0110],  f(t,x-(t))=(01/(x12+1)) and x-(t)=[x1(t)x2(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
(52)E3/4(At3/4)=(N1(t)N2(t)N2(t)N1(t)),
where N1(t)=(1/2)[E3/4(t3/4)+E3/4((-t)3/4)] and N2(t)=(1/2)[E3/4(t3/4)-E3/4((-t)3/4)]. The observability Gramian matrix for this system,
(53)W(0,2)=∫02‍E3/4(A*t3/4)H*HE3/4(At3/4)dt=∫02‍(N22(t)N1(t)N2(t)N1(t)N2(t)N12(t))dt=(10.00611.244111.244112.9504),
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 thatx2 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 order0<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.



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