International Journal of Applied Mathematics

International Journal of Applied Mathematics ————————————————————– Volume 12 No. 1 2003, 73-85

STABILITY ANALYSIS OF COMPOSITE FRACTIONAL SYSTEMS Reyad El-Khazali1 , Shaher Momani2 § 1 Etisalat

College of Engineering Sharjah - UAE e-mail: [email protected] 2 Department of Mathematics and Computational Science Faculty of Science United Arab Emirates University P.O. Box 17550, Al-Ain, UAE e-mail: [email protected] Abstract: This paper introduces sufficient conditions for the asymptotic stability of linear time-invariant fractional composite systems. The Lyapunov second method is implemented to derive simple conditions in terms of the argument of the system matrix eigenvalues. Other simple sufficient conditions that relate the absolute values of the real and the imaginary parts of the composite system eigenvalues are also developed. Numerical examples are introduced to illustrate the main ideas of the results. AMS Subject Classification: 26A33 Key Words: fractional systems, fractional differential equations, Lyapunov stability 1. Introduction Consider the linear differential equation of order nα of the form: y (nα) + an−1 y ((n−1)α) + ... + a1 y (α) + ao y = u(t) Received:

May 8, 2003

§ Correspondence

author

(1)

c 2003 Academic Publications

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R. El-Khazali, S. Momani

with initial conditions y ((k−1)α) (to ) = Ck−1 ;

0 < α < 1 ; k = 1, ..., n ,

(2)

dα y(t) where to ∈ I = [to , to + T ] for some T > 0, y (α) ≡ D(α) y(t) ≡ , dtα and where Ck ∈ ℘ are arbitrary constants for k = 0, 1, ..., n − 1. The fractional differential equation (1) could represent a Newtonian or non-Newtonian motion of some physical systems such as diffusion processes, rheology, and composite fractional oscillation systems [10]. The existence and uniqueness of solutions has been investigated in [5]. It was shown that a unique solution to equation (1) exists, if u(t) and the coefficients ai ’s for i = 1, ..., n, are constants. This paper focuses on the stability of such type of fractional systems. The stability analysis is accomplished using the second method of Lyapunov by rewriting (1) as a set of linear differential equations each of fractional order α. It will be shown that the Lyapunov second method proves to yield new sufficient stability conditions for fractional systems similar to the conditions introduced in [4]. This paper is organized such that preliminaries and background are introduced next. Section 3 presents the main results of the paper. Several numerical examples are introduced in Section 4, while Section 5 presents conclusions and final remarks.

2. Background The following definitions are necessary to develop sufficient conditions for the asymptotic stability of the linear fractional differential equation described in (1). Definition 1. Let m − 1 < α < m, m ∈ N , the Riemann-Liouville fractional derivative of order α of any function f (t) is defined as follows [11, 12]: (α) Dt f (t)

dm 1 ≡ Γ(m − α) dtm

where Γ is the Gamma function.

t 0

f (q) dq , (t − q)α+m−1

(3)

STABILITY ANALYSIS OF COMPOSITE...

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dα f (t) denote the Riemann-Liouville dtα fractional derivative of f (t) of order α, the fractional derivative of Grunwald-Letnikov definition [12] is given by: (α)

Definition 2. Let Dt f (t) =

(α) Dt f (t)

≡ lim

 N  N α

N →∞ 

t

j=1

Γ(j − α) Γ(−α)Γ(j + 1)

(N − j)t f . N

Definition 3. (see [11]) If 0 < α < 1, the sequential fractional derivative of f (t) of order nα is given by: (nα)

Dt

(α)

(α) n times (α) .............. Dt f (t) ,

f (t) = Dt Dt

(4)

provided that D(α) f (t) exists. If f (t) = Cg(t) where C is a constant, dα f (t) (α) . then Dt f (t) ≡ C dtα The Lyapunov stability conditions for equation (1) can easily be developed by using Definition 3 to rewrite system (1) as a set of n fractional differential equations each of order 0 < α < 1. Suppose one (α) (α) (α) defines y ≡ x1 , x1 = x2 , x2 ≡ x3, ... , xn−1 ≡ xn , then system (α)

(1) yields xn = − an−1 x1 + an−2 x2 + ... + a0 xn + u(t) which defines a SISO linear time-invariant fractional system in the controllable canonical form. A typical MIMO fractional system takes the form: X (α) = AX + Bu(t) ,

X(to ) = xo ,

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y(t) = CX , T ∈ n is the state, u ∈ where X T ≡ x1 , x2 , ... , xn−1 , xn m r is the control, and y ∈ is the output. The stability of the autonomous system in (5) is discussed in the sense of Lyapunov following the definition of internal stability [8]. Definition 4. The autonomous system (5) X (α) = AX; with X(to ) = xo is said to be: 1- Stable iff ∀ xo , ∃ K; ∀ t ≥ 0, X(t) ≤ K, 2- Asymptotically stable iff lim X(t) = 0. t→∞

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The following result introduces necessary and sufficient conditions for fractional system internal stability [4]. Theorem 1. Consider system (5). Let λk = rk eiθk be the kth −eigenvalue of the matrix A; k = 1, 2, ....., n. The homogeneous system (5) is: 1) Asymptotically stable iff |θk | > απ/2, ∀ k. 2) Stable iff either it is asymptotically stable, or the critically stable eigenvalues which satisfy |θk | = απ/2 have geometric multiplicity one. The asymptotic behavior of the autonomous system exhibits an ultra-slow decaying motion in the order of t−α as t → ∞, which is of a slower rate than any exponential decay [4]. The stability region is then defined outside the closed angular sector |θk | ≤ απ/2 as shown in Figure 1.

Figure 1: The stable region for the autonomous fractional system The second method of Lyapunov [3] is an alternative approach used to define stability regions for system (5). Even though it will lead to more restrictive conditions (sufficient) than the one defined in Theorem 1, this approach however, will pave the way to apply the theory of optimal control and the linear matrix inequality (LMI) techniques [14] on the fractional systems. Definition 3 formulates the successive differentiation property for fractional systems that can be used to define a “new” system of integer order. Such a system can be used to evaluate the rate of change of a typical Lyapunov energy function.


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The correlation between the state trajectories of both the fractional system (5) and that of the new system of integer order are not obvious. However, we will be able to show that the stability of the new system of the integer order implies the stability of the fractional system. The following definitions and theorems are introduced for completeness. Definition 4. A single-valued function V (X) which is continuous and has continuous partial derivative is said to be positive definite in some region Ω about the origin of the state space if V (0) = 0, ∀ 0 = X ∈ Ω. Theorem 2. (see [9]) Let A ∈ ℘n×n and Q be a positive definite matrix. There exists a positive definite matrix P, such that P A+AT P = − Q, if and only if the eigenvalues of A lie in the open left-half complex plane. Theorem 3. (see [9]) If the matrix A has the characteristic values , m m λ1 , λ2 . . . , λn , then Am has the characteristic values λm 1 , λ2 , ..., λn for m = 1, 2, ..., n.

3. Main Results Necessary and sufficient conditions for internal and external stability of the fractional autonomous system (5) are given in Theorem 1, [4]. However, they have not been linked to the Lyapunov stability in order to apply modern control techniques for control the systems of non-integer orders. The following results introduce sufficient stability conditions for the autonomous fractional composite system given by (5). Theorem 4. Consider the autonomous fractional system (5). Let 0 < α < 1. Define α ≡ 1/q, where q ∈ ℵ. Let λk = rk eiθk where λk be the kth -eigenvalue of the system matrix A ∈ ℘n×n . Then, the homogeneous system in (5) is globally asymptotically stable if: (2l ± 3)π (2l ± 1)π < θk < 2q 2q for k = 1, 2, · · · , n , and l = 0, 1, 2, ....

(6)

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Clearly, the principle solution is π/2q < θk < 3π/2q for all k when l = 0. Proof. Let condition (6) be satisfied. For simplicity, assume that A has distinct eigenvalues (the case for repeated eigenvalues is extended in the sense of Jordan-Canonical block). There exists a nonsingular transformation matrix, T , such that X ≡ T Z. Then, T −1 AT ≡ Λn , where Λn = diag (λ1 , λ2 , . . . , λn ) is a diagonal matrix of n distinct eigenvalues. By Definition 3, the homogeneous system (5) can be transformed to: Z (α) = Λn Z;

Z(to ) = T −1 X(to ) ,

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where Z T ≡ [z1 , z2 , ..., zn ]T . Consider the Lyapunov function V (Z) = Z T P Z, where P is a positive definite matrix and Z is the state solution of the homogeneous system (7). Evaluating the time derivative of V (Z), substituting successively from (7), and using Definition 3 yields, V˙ (t) = Z T { (Λqn )T P + P Λqn }Z .

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Figure 2: Stability regions of system (5) due to the sufficient condition (6) Since λqk = rkq eiqθk is the kth -eigenvalue of Λqn , then e (λqk ) = cos(qθk ), ∀k = 1, 2, · · · , n. Since condition (6) is satisfied by hypothesis, then all eigenvalues of Λqn lie in the open left half-plane. By Theorem 2, there exists positive definite . matrices, P and Q such that −Qq ≡ Λqn P + P Λqn < 0. Consequently, V (Z) < 0, ∀Z = 0. This proves rkq


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that the “new ” system z(t) ˙ = Λq Z(t)is asymptotically stable. Finally, since X = T Z and Z(t) → 0, as t → ∞. Then, system (5) is asymptotically stable. Figure 2 shows the stability regions for system (5). Clearly, this region is a subset of the stability sector shown in Figure 1. Clearly, the lines (with angles ±π/2q) that define the unstable sector correspond to the marginally stable boundaries for system (5); i.e., on those lines, the eigenvalues of A are complex conjugate pairs of the form λk = σk ± i βk where σk > 0, while the eigenvalues of Aq are all pure imaginary; λqk = (σk ± i βk )q = ±iw. The fractional system (5) exhibits an oscillatory motion on those lines, and the system could be stable or marginally stable with even when σk > 0.This is an intuitive distinction between the actual fractional system defined in (5) and the “new” system with integer derivative. Remark 1. Observe that as q → ∞ (i.e. α → 0) the region of stability defined in Figure 2 converge to the region of stability defined in Figure 1. In other words, the sufficient condition of stability (6) becomes almost necessary and sufficient one in this case. Remark 2. Notice that the choice of α ∈ (0, 1) where α = 1/q; q ∈ ℵ does not restrict the results of condition (6). For example, if α = p/q where p , q ∈ ℵ, one can always define a new linear timeinvariant fractional system (LTIF) with fractional derivative equals to α ˆ = 1/q and proceeds as stated in Theorem 4. It is interesting to realize that condition (6) is not the only sufficient condition for fractional system stability. Theorem 5 introduces different stability regions that are still a subset of the region defined in Fig. 1. Theorem 5. Let 0 < α = 1/q < 1 where q ∈ ℵ, and consider system (5). Let λk be the kth -eigenvalue of A ∈ ℘n×n . The homogeneous system is globally asymptotically stable if: e (λqk ) ≤ 0;

∀k = 1, 2, · · · , n .

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Proof. The proof follows directly from Theorem 4. Corollary 1. Consider system (5). Let λk = σk + iβk be the kth eigenvalue of A. The system is asymptotically stable if:

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1- |σk | < |βk |; for α = 1/2; √ 2- |σi | < 3 |βi | ; σi > 0 for α = 1/3 √ |σi | > 3 |βi |

(10) ;

σi < 0

Proof. The proof follows directly from Corollary 1. Figures 3a and 3b show the regions of asymptotic stability of the fractional system for α = 1/2 and α = 1/3 which are clearly subset of the stability regions defined by Theorem (4).

Figure 3: The stable sectors for system (5) when: a) α = 1/2 and b) α = 1/3 Remark 3. Similar regions of stability can be defined for α = 1/4, 1/5, 1/6, · · · , etc. It is clear that in all such cases, the region of stability is a subset of the stability sector defined in Theorem 4.

4. Numerical Examples Example 1. Consider the fractional differential equation for t ≥ 0: dα y(t) dy(t) + a1 + a2 y(t) = u(t), y(0) = C , (11) dt dtα where a1 and a2 are positive constants and C is an arbitrary constant and where 0 < α < 1.


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This fractional differential equation corresponds to the Basset problem [2]. This equation represents a classical problem in fluid dynamics where the unsteady motion of a particle accelerates in a viscous fluid due to the gravity force. Many authors [10, 12] have introduced analytical solutions to (11). For example when a1 = 1 and a2 = −2 the following analytical solution using the Laplace transform technique introduced in [12] is clearly unstable for u(t) = 1; i.e., y(t) =

√ √ C (2 e4t erf c(2 t) + et erf c(− t) , 3

(12)

where erfc(t) is the error function of t. Numerical solutions of (12) for different values of a1 , a2 and α are introduced in [6] using the generalized fractional finite difference method. To gain more inside to the numerical solution of (12), when α = 1/2, which corresponds to the Basset problem, the system is described in the state space model as:

(1/2) x1 0 0 1 x1 + u(t) . (13) = (1/2) −a x 1 −a 2 1 2 x2 For a1 = 1and a2 = −2, the eigenvalues of the coefficient matrix are λ1 = −2and λ2 = 1. Clearly, λ2 = 1 lies outside the region of the asymptotic stability shown in Figure 3a. Moreover, the eigenvalues of A2 are λ21 = 4 and λ22 = 1 and that coincides with the analytical solution given by (12).

Figure 4. System response for a2 = 1, and a) a1 = 1.0 b) a1 = −1.0, c) a1 = 1.5, and d) a1 = −1.5 Figure 4 presents the step response when α = 1/2 for different values of a1 and a2 and for a step size h = 0.005. Notice that for case 3 when a1 = 1.5 and a2 = 1, the eigenvalues of A are inside the stability region defined by Figure 3a. Example 2. Consider the following second-order fractional differential equation:

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d2 y(t) dα y(t) + a + a2 y(t) = f (t), y(0) = c1 , y(0) ˙ = c2 , where 1 dt2√ dtα 2S ηρ , and a2 = K. a1 = M This fractional differential equation is called the composite fractional oscillation equation [1]. It represents the motion of a large plate of the surface S and a mass M in a Newtonian fluid of viscosity ηand density ρ. The plate is hanging a massless spring of a stiffness K and a function f (t) represents the external force. An analytical solution was introduced in [1] for α = 3/2, while a general numerical solution for 0 < α < 2 was given in [6]. Let α = 3/2, the corresponding state-space fractional system can be defined as:     

(1/2)

x1 (1/2) x2 (1/2) x3 (1/2) x4



 0     0 = 0  −a2

1 0 0 −a1

0 1 0 0

 x1 0  x2 0   1   x3 0 x4





 0   0   +   f (t) .   0  1

(14)

The simulation results are shown in Figure 5 for a unit step input, for different cases of a1 and a2 and a step size h = 0.0067.

Figure 5. System response for a) a1 = −0.1, a2 = 2.,


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b) a1 = 0.5, a2 = 0.75, and c) a1 = 0.5, a2 = 1.5 It is clear that when a1 = a2 = 0.5, two eigenvalues of the system matrix (14) lie outside the stability region defined in Figure 3a and yet the system exhibits a stable motion. This emphasizes that condition (6) is only a sufficient one. Notice also that case (b) exhibits an unstable motion since two eigenvalues of A2 are at λ21,2 = 0.1878 ± 0.6726i which has positive real parts.

5. Conclusion The stability of fractional systems was discussed using the second method of Lyapunov. Sufficient conditions are introduced that guarantee the stability of a composite fractional differential equation of the form given in (1). One, simply, can verify stability by evaluating the argument of the eigenvalues of the system coefficient matrix similar to the conditions stated in [4], or by evaluating the real part of eigenvalues of A1/α for all rational integer numbers. The matrix A1/α represents a ‘new” matrix of an equivalent system derived by successively differentiating (5) q = 1/α times. The simplicity of utilizing the sufficient conditions introduced by (6) or (10) becomes clear when one desires to develop an optimum controller to the fractional system or when implementing the LMI theory on the virtual “new” or equivalent system. Thus, the well known feedback

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control techniques can be applied easily on the new system to place the eigenvalues of A1/α inside the different stable sectors depending on the value of α or simply checking e(A1/α) ). The only withdraw of this method is the limitation imposed on α being rational integer number; i.e., α = m/q, where m and q are both integers. The Lyapunov stability method when α is irrational is open for further investigation.

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[10] F. Mainardi, Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wein-New York (1997). [11] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus, John Wiley and Sons, Inc., New York (1993). [12] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York (1974). [13] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Sci., Amsterdam (1993). [14] S. Boyed, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM Studies in Applied Math. (1994).

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