Volterra Series Approximation of a Class of Nonlinear ...

Nonlinear Dyn DOI 10.1007/s11071-013-0975-8

O R I G I N A L PA P E R

Volterra Series Approximation of a Class of Nonlinear Dynamical Systems Using the Adomian Decomposition Method Yuzhu Guo · L.Z. Guo · S.A. Billings · Daniel Coca · Z.Q. Lang

Received: 26 July 2012 / Accepted: 9 June 2013 © Springer Science+Business Media Dordrecht 2013

Abstract The relationship between the Adomian decomposition and the Volterra series is investigated and it is shown that the Volterra series can be considered as a specialization of the Adomian decomposition. Based on the relationship, the Volterra series can be calculated using an Adomian decomposition method whenever a convergent Volterra series representation exists. A class of nonlinear dynamical systems is considered and a new algorithm is introduced to compute the Volterra series representation for this class of nonlinear systems. The new method significantly simplifies the computation of the Volterra kernels and provides a new choice for the study of Volterra series of nonlinear dynamical systems. Keywords Nonlinear dynamical systems · Linear–analytic systems · Volterra series · Adomian decomposition method

1 Introduction The Volterra series has been extensively investigated and widely used since its introduction in 1887 [47] Y. Guo () · L.Z. Guo · S.A. Billings · D. Coca · Z.Q. Lang Department of Automatic Control and Systems Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD, UK e-mail: [email protected]

as a general description for a wide class of nonlinear dynamical systems. The Volterra series representation which provides an explicit description of the input–output relationship for the class of fading memory nonlinear systems is a powerful tool for the study of nonlinear dynamical systems. Frechet showed that a continuous functional can be approximated uniformly and to an arbitrary degree of precision by a sufficiently high, but finite order Volterra series [41]. Volterra series have played an important role in the modelling of nonlinear systems, both when the underlying system equations are known and when the system is characterized only by the availability of input–output data. The identification of Volterra kernels has been reviewed by Billings [15] and Korenberg and Hunter [33]. The properties and advantages of the Volterra series compared to implicit system models have been studied by George [29]. George argued that the Volterra series can be considered as an explicit model where the system response is expressed as a series of convolutions of the input and the Volterra kernels. Explicit models avoid the need to re-solve the system differential equations for each different input which can be complex when the system is nonlinear. Using the Volterra series complex nonlinear systems can also be studied by combining simpler subsystems. Another significant advantage of the Volterra series theory is that the Fourier transform of the Volterra kernels, which are called generalised frequency response functions, provide a very powerful tool for the study of the nonlinear systems in the frequency domain. When

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a system admits a Volterra series, the convergence of the series expansion and the corresponding convergence region have been studied by many authors [13, 20, 31, 37, 46]. The limitations of the Volterra series have been studied by Boaghe [18] and new approaches have been derived to overcome these limitations including the multi-input Volterra series for sub-harmonic systems [19], the response spectrum map for bifurcations [16], and the piecewise Volterra models [36]. Computation of the Volterra series for an ordinary differential equation using successive approximations is well known. Bruni et al. [23] first applied the successive approximation approach and derived the general form of Volterra kernels for bilinear state equation systems. These results have been extended to a very general class of linear–analytical systems by Brockett [22]. An alternative approach was introduced by Gilbert [30]. However, the calculation of the Volterra kernels in the time domain may not be easy especially using the Carleman technique [34] but some simple methods are available in the frequency domain [17, 45, 51]. These computation methods have been summarised in Rugh’s book [40]. Computation methods for other types of series expansions such as the Chen–Fliess series have also been studied [24, 28]. In the present study, a new approach to compute Volterra kernels using the Adomian Decomposition Method (ADM) will be developed. This method provides a new choice for the computation of Volterra kernels from an ordinary differential equation. Compared with the existing methods, the new method maintains the advantages of the Adomian decomposition method and provides a standard procedure for a wide class of systems. Adomian decomposition is a powerful tool for solving nonlinear equations, including algebraic, differential, and integral equations, and provides a bridge between analytic and numerical solutions [1, 2, 4, 5, 7– 11, 39, 49]. The Adomian decomposition method can be used to solve/approximate a wide class of nonlinear problems with only a few limitations and provides an infinite series approximation in an analytic form. The relationship between the Adomian decomposition method and other analytical and numerical methods has been investigated by several authors [12, 14, 26, 27, 38, 43, 44, 48, 50]. In this paper, the relationship between the Adomian decomposition and the Volterra series approximation is investigated. It will be shown

that the Adomian decomposition method does not necessarily produce a Volterra series and that the Volterra series is a specialised form of the Adomian decomposition under certain conditions. Although the Adomian decomposition method has previously been used for initial-value problems, it will be shown that it is also an ideal tool to obtain the input–output representations in a Volterra series form directly from ordinary differential equations. This paper is organized as follows: The principles of the Adomian decomposition method are briefly introduced in Sect. 2. Section 3 shows that Volterra series approximation is a specialization of the Adomian decomposition with y0 = 0. A class of nonlinear dynamical systems are considered and a new algorithm for computing Volterra kernels from ordinary differential equation descriptions is introduced. A Duffing oscillator is discussed in Sect. 4 as an illustrative example to demonstrate the new algorithm. Conclusions are finally drawn in Sect. 5.

2 Adomian Decomposition Method (ADM) The Adomian decomposition method (ADM) has been widely applied in many areas since the method was introduced in the 1980s. This method has some significant advantages over both analytical and numerical methods. These results are available in the literature where the Adomian decomposition method has been extensively compared with other methods [12, 14, 26, 27, 38, 43, 44, 48, 50]. The ADM technique decomposes a solution of a nonlinear functional equation in an infinite series of functions which rapidly converge to the solution as the number of terms n increases [6] y(t) =

∞

λn yn (t).

(1)

n=0

The nonlinearity g(t, y) in the nonlinear equation which is usually represented by a nonlinear operator on a Banach space and can be decomposed as g(t, y) =

∞

λn An t, λy1 , λ2 y2 , . . . , λn yn ,

(2)

n=0

where An (t, λy1 , λ2 y2 , . . . , λn yn ) are referred to as Adomian polynomials when the parameter λ = 1. The functions An (t, y1 , y2 , . . . , yn ) only depend on y1 , y2 , . . . , and yn , and no higher orders are involved.

Volterra Series Approximation of a Class of Nonlinear Dynamical Systems Using the Adomian Decomposition Method

According to the original definitions by Adomian, the Adomian polynomials can be calculated by the following formulae: ∞ 1 dn i g t, λ y (3) An = i n! dλn i=0

A0 = g(t, y0 ), A1 = y1 g (t, y0 ), 1 2 y g (t, y0 ), 2! 1

(4)

1 3 y g (t, y0 ) 3! 1 where g , g , g are the Fréchet derivatives of g(t, y) with respect to y. The Adomian decomposition method requires the nonlinear operator g(t, y) to be analytic with respect to y(t) near y0 (t). Many different approaches have been developed to calculate the Adomian polynomials [42, 49, 52], and the convergence of the Adomian decomposition has been intensively investigated [1, 3, 25, 32, 39]. Consider a general nonlinear dynamical system A3 = y3 g (t, y0 ) + y1 y2 g (t, y0 ) +

Ny(t) = u(t)

(5)

where N[·] is a nonlinear differential operator. A general approach is to split the nonlinear operator into two parts: Ny = Ly − Gy (for the convenience of expression, a subtraction is used instead of an addition), where L is a linear invertible differential operator and Gy = g(t, y) is an analytic function for both the independent variable t and the response of the system y(t). The nonlinear system can then be expressed as Ly = u(t) + g(t, y).

+

0

The solution of system (6) can then be expressed as t h(t, τ )u(τ ) dτ y(t) = y(0) + 0 t h(t, τ )g τ, y(τ ) dτ. (8) +

t

h(t, τ ) 0

∞

An (τ, y1 , y2 , . . . , yn ) dτ.

n=0

(9)

The Adomian decomposition solution of system (6) can then be computed recursively as t y0 = y(0) + h(t, τ )u(τ ) dτ, 0 t y1 = h(t, τ )A0 τ, y0 (τ ) dτ, (10) 0 ··· t h(t, τ )An (τ, y0 , y1 , . . . , yn ) dτ. yn+1 = 0

Setting y(0) = 0 and substituting (4) into (10) yields the first few terms in the Adomian decomposition of the zero-state response of system (6): t y0 = h(t, τ )u(τ ) dτ, 0 t y1 = h(t, τ )g(τ, y0 ) dτ, 0 t h(t, τ )y1 (τ )g (τ, y0 ) dτ, y2 = 0 t h(t, τ ) y2 (τ )g (τ, y0 ) y3 = 0

(11) 1 2 + y1 (τ )g (τ, y0 ) dτ, 2! t y4 = h(t, τ ) y3 (τ )g (τ, y0 ) 0

+ y1 (τ )y2 (τ )g (τ, y0 )

1 + y13 (τ )g (τ, y0 ) dτ, 3!

(6)

The invertible operator of L[·] is assumed to be of a convolution form t −1 h(t, τ )(·) dτ. (7) L [·] =

0

0

n=0

λ=0

where λ is a constant parameter. The first several Adomian polynomials can be calculated as

A2 = y2 g (t, y0 ) +

Substituting the Adomian decomposition (1) and (2) into (8) yields t ∞ yn (t) = y(0) + h(t, τ )u(τ ) dτ

.. . Remarks: (i) The split Ny = Ly − Gy of the nonlinear operator N [·] is not unique. Any appropriate split may lead to an Adomian decomposition. (ii) The Adomian polynomial decomposition in (2) and (3) can be referred tonas a parametric Taylor expansion of g(t, ∞ n=0 λ yn ) around the ‘point’ λ = 0, that is, y = y0 .

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(iii) The choice of y0 in (10) is not unique. Different y0 may correspond to different Adomian decompositions. Every Adomian gives a decomposition ∞ series expansion y(t) = n=0 yn (t) of the solution of the nonlinear system (6). These series may have different convergence rates because of the dependence of the yn ’s on y0 . However, all the series converge to the unique solution of the nonlinear system. This will be observed in the illustrative example in Sect. 4.

λn yn (t) by applying a scaled input λu(t), where λ is a scalar. Namely, the zero-state response of the sysn tem to an input λu(t) is y(t) = ∞ n=0 λ yn (t) which satisfies the following equation ∞

∞ n n L λ yn (t) = λu(t) + g t, λ yn (t) , (14) n=0

n=0

where an additional term y0 (t) = 0 is introduced for the convenience of the discussion so that ∞ ∞ y(t) = λn yn (t) = 0 + λn yn (t) n=1

3 Computing Volterra Kernels Using Adomian Decomposition

=

∞

n=1

λn yn (t).

(15)

n=0

In this section, the underlying relationship between the Adomian decomposition and the Volterra series expansion will be investigated. Based on the close relationship between the Adomian decomposition and the Volterra series, a Volterra series representation of a nonlinear dynamical system can be calculated using the Adomian decomposition method. 3.1 From Volterra Series to Adomian Decomposition It will first be shown that for a nonlinear dynamical system that admits a Volterra series, the Volterra series expansion of the system response is an Adomian decomposition of the solution of the corresponding inhomogeneous ordinary differential equation. Consider the nonlinear dynamical system given in (6). Assume system (6) has a zero equilibrium and the zero-state response of the system can be expanded as a convergent Volterra series given in (12) and (13) when the amplitude of the input is small enough: y(t) =

∞

(12)

yn (t)

Because of the analyticity, the Maclaurin series expansion of the nonlinearity g(t, y) can be written as g(t, y) =

∞

(16)

i=1

Notice that g0 (t) = 0 since y(t) = 0 is an equilibrium point of the system when u(t) = 0. The Volterra series can then be obtained by equating the like terms of λn on both sides of (14) following the algorithm introduced in [35]. To apply the algorithm, the linear term g1 (t)y is subtracted on both sides of (14):

∞ ∞ n λ yn (t) − g1 (t) λn yn (t) L n=1

= λu(t) +

∞

gi (t)

i=2

n=1 ∞

i n

λ yn (t)

.

(17)

n=1

˜ = Define a new linear differential operator as L[·] L[·] − g1 (t)[·] and the associated inverse operator as t ˜ τ )(·)(τ ) dτ. h(t, (18) L˜ −1 [·] = 0

n=1

where yn (t) is given in a multi-dimensional convolution form as t t ··· hn (t, τ1 , τ2 , . . . τn )u(τ1 )u(τ2 ) · · · yn (t) = 0

gi (t)y i .

0

× u(τn ) dτ1 dτ2 · · · dτn .

(13)

The term hn (t, τ1 , τ2 , . . . τn ) is referred as to the nth order Volterra kernel. Term yn (t) in (12) is often known as the nth homogeneous subsystem of system (6). This is because the response of the nth homogeneous subsystem becomes

Equation (17) becomes ∞

n L˜ λ yn (t) n=1

= λu(t) +

∞ i=2

gi (t)

∞

i n

λ yn (t)

(19) .

n=1

The nth homogeneous subsystems can then be computed separately by equating the like term of λn as t ˜ τ )An (y1 , y2 , . . . , yn ) dτ, h(t, (20) yn (t) = 0


where A0 = 0, A1 = u, A2 = g2 y12 , (21)

A3 = 2g2 y1 y2 + g3 y13 , A4 = g2 y22 + 3g3 y12 y2 + g4 y14 , .. .

Redefine the nonlinearity on the right hand of (19) as ∞ i ∞ n g(t, ˜ y) = λu(t) + gi (t) λ yn (t) . (22) i=2

n=1

It can be shown that the obtained An satisfies the definition of the Adomian polynomial in (3) ∞ 1 dn i An = g ˜ t, λ y (23) . i n! dλn i=0

λ=0

This result shows that the obtained Volterra series is a specialisation of the Adomian decomposition of the following nonlinear dynamical system in which y0 (t) = 0: ˜ L[y] = g(t, ˜ y). (24) The Adomian decomposition is given in a formal form as

∞ ∞ ˜ yn (t) = An (y0 , y1 , . . . , yn ). (25) L n=1

n=0

Remarks: (i) Notice that the Adomian polynomial An (y0 , y1 , . . . , yn ) is actually not dependent on yn because g(t, ˜ y) does not include the term g1 (t)y which has been moved to the left hand side of the equation. Therefore, the yn can be recursively calculated using (20). The formal Adomian notations are used here, even though an accurate notation should be An (y0 , y1 , . . . , yn−1 ). (ii) Except for the choice of y0 (t) = 0, the calculation of yn (t) in (20) is different from the classical algorithm in (10), where An is used instead of An−1 . However, this does not prevent the Volterra series solution to be an Adomian decomposition since (25) and (23) always hold. 3.2 From Adomian Decomposition to Volterra Series The previous subsection shows that a Volterra series expansion is of an Adomian decomposition form and

the Volterra kernels hn (t, τ1 , τ2 , . . . τn ) are closely related to the Adomian polynomials. The relationships between Volterra kernels and Adomian polynomials will be further discussed below and a new method based on the Adomian decomposition method will be introduced to calculate Volterra kernels for a class of linear–analytic nonlinear systems. Reconsider the Adomian decomposition obtained in (11) in Sect. 2. It is easy to observe that following the classical Adomian decomposition method the obtained series expansion of the solution may not be a Volterra series. This is because the nth term of the Adomian decomposition is not necessarily an nth homogeneous operator on the input u(t). For example, assume the nonlinearity g(t, y) is of a polynomial form g(t, y) = y 3 + y 2 .

(26)

Substituting (26) into the second equation in (11) yields

3 τ t h(t, τ ) h(t, τ1 )u(τ1 ) dτ1 y1 (t) = 0

0

τ

+

h(t, τ1 )u(τ1 ) dτ1

2

dτ,

(27)

0

which is obviously not a homogeneous subsystem of the nonlinear system. This means the direct application of the Adomian decomposition method does not necessarily produce a Volterra series representation of nonlinear systems. The remaining part of this section will show that an Adomian decomposition could be a Volterra series by appropriately choosing the first term y0 (t) in the Adomian decomposition method. Consider a more general class of linear–analytic systems given as L(y) = f (t, y) + g(t, y)u(t).

(28)

The Volterra series representation for this kind of system has been intensively studied [22, 30]. It is assumed that system (28) admits a convergent Volterra series. A method based on the Carleman linearization technique has been employed to compute the Volterra kernels [34]. Using this method, a bilinear approximation of the linear–analytic system is computed first and then the Volterra series of the obtained bilinear system is calculated. However„ the Carleman linearization approach has inherent disadvantages not only because Carleman linearization approximation is not an accurate representation of the linear–analytic system

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but because computation of the linearization often involves complex vector operations. In this section the Adomian decomposition method is employed to obtain a Volterra series representation avoiding these disadvantages. From the knowledge of Volterra series, a Volterra series representation is an expansion of the response around the zero equilibrium state, whereas the Adomian decomposition expands the system response around y0 (t) according to Remark (ii) in the last subsection. However, Remark (iii) implies that the first term y0 (t) can be appropriately chosen according to the requirement of practical problems. Following this idea, the Volterra series will then be calculated using a slightly modified Adomian decomposition method. Start the Adomian decomposition method by setting y0 = 0 and the output can then be written as y(t) =

∞

yn (t) = 0 +

n=0

∞

yn (t).

(29)

n=1

Expressing the nonlinearities in (28) by Adomian polynomials yields ∞ ∞ L yn (t) = An (t, y0 , y1 , . . . , yn ) n=0

n=0

+

∞

Bn (t, y0 , y1 , . . . , yn )u

(30)

where An (t, y0 , y1 , . . . , yn ) and Bn (t, y0 , y1 , . . . , yn ) are Adomian polynomials of f (t, y) and g(t, y), respectively. The Maclaurin series expansions of the nonlinear analytic functions f (t, y) and g(t, y) around y0 = 0 can be written as ∞

fi (t)y i ,

i=2

g(t, ˜ y) = g(t, y)u =

∞

and B0 = g(t, 0) = g0 (t), B1 = y1 g (t, 0) = y1 g1 (t), 1 B2 = y2 g (t, 0) + y12 g (t, 0) = y2 g1 (t) + y12 g2 (t), 2! (33) 1 B3 = y3 g (t, 0) + y1 y2 g (t, 0) + y13 g (t, 0) 3! = y3 g1 (t) + 2y1 y2 g2 (t) + y13 g3 (t), .. . Notice that An (t, y0 , y1 , . . . , yn ) does not depend on yn . Therefore, the Adomian decomposition can be calculated as y0 = 0, t y1 = h(t, τ ) A1 (τ ) + B0 (τ )u(τ ) dτ, 0

.. .

yn =

(34) t

h(t, τ ) An (τ ) + Bn−1 (τ )u(τ ) dτ,

0

.. .

n=0

f (t, y) =

A0 = f (t, 0) = 0, A1 = y1 f (t, 0) = 0, 1 A2 = y2 f (t, 0) + y12 f (t, 0) = y12 f2 (t), 2! (32) 1 A3 = y3 f (t, 0) + y1 y2 f (t, 0) + y13 f (t, 0) 3! = 2y1 y2 f2 (t) + y13 f3 (t), .. .

(31) i

gi (t)y u.

i=0

Notice that f0 (t) is zero because of the assumption that the system has a zero equilibrium point and the linear part f1 (t)y has been moved to the left of (28). According to the formulae (3), the Adomian polynomials can be computed as

Applying an input λu(t), the first several terms of the Adomian decomposition are y0 = 0, t y1 = λ h(t, τ )g0 (τ )u(τ ) dτ, t0 y2 = h(t, τ ) y12 f2 (τ ) + λy1 g1 (τ )u(τ ) dτ 0 t t = λ2 h2 (t, τ1 , τ2 )u(τ1 )u(τ2 ) dτ1 dτ2 , t 0 0 y3 = h(t, τ ) 2y1 y2 f2 (τ ) + y13 f3 (τ ) 0 + λ y2 g1 (τ ) + y12 g2 (τ ) u(t) dτ t t t 3 =λ h3 (t, τ1 , τ2 , τ3 ) 0

0

0

× u(τ1 )u(τ2 )u(τ3 ) dτ1 dτ2 dτ3 , .. .

(35)


where h2 (t, τ1 , τ2 ) t h(t, τ )f2 (τ )h(τ, τ1 )h(τ, τ2 ) = 0

× g0 (τ1 )g0 (τ2 )δ−1 (τ − τ1 )δ−1 (τ − τ2 ) dτ + h(t, τ1 )h(τ1 , τ2 )g0 (τ2 )g1 (τ1 )δ−1 (τ1 − τ2 ) (36) and

(i) Solve the associated linear equation L[y](t) = v(t) to obtain the impulse response function h(t, τ ) of the linear system; (ii) Set the first term of the Adomian decomposition y0 = 0; (iii) Compute A1 (A1 = 0) and B1 according to formulae (3); (iv) Compute y1 following formulae (34); (v) Repeat steps (iii) and (iv) to compute y2 , y3 , . . . ; (vi) The obtained i yi is the Volterra series; (vii) The Volterra kernels can be obtained by rearranging the multiple integrals like (35)–(37).

h3 (t, τ1 , τ2 , τ3 ) t h(t, τ )f2 (τ )h2 (τ, τ1 , τ2 )h(τ, τ3 ) =2 0

× g0 (τ3 )δ(τ − τ1 )δ(τ − τ2 )δ(τ − τ3 ) dτ t + h(t, τ )f3 (τ )h(τ, τ1 )h(τ, τ2 )h(τ, τ3 ) 0

× g0 (τ1 )g0 (τ2 )g0 (τ3 )δ(τ − τ1 )δ(τ − τ2 ) × δ(τ − τ3 ) dτ + h(t, τ3 )h2 (τ3 , τ1 , τ2 )g1 (τ3 )δ(τ3 − τ1 ) × δ(τ3 − τ2 ) + h(t, τ3 )h(τ3 , τ2 )h(τ3 , τ1 )g0 (τ1 ) × g0 (τ2 )g2 (τ3 )δ(τ3 − τ1 )δ(τ3 − τ2 ).

(37)

The notation δ−1 (·) represents the Heaviside step function. It is easy to observe that the first few terms which have a standard Volterra series form are homogeneous with respect to the input. Abbaoui and Cherruault [1] showed that Adomian polynomials An can be expressed with the following explicit formulae:

n 1 (k) An = g (t, y0 ) yp1 · · · ypk , k! p +···+p =n k=1

n ≥ 1.

1

Although nonlinearities of the form f (t, y) and g(t, y) are used here, it should be emphasized that the ADM can be applied to a more general class of systems where g(t, y, y , y , . . . , y (k) ) can be an analytical function of y(t) and the associated derivatives. Examples can be found in [49]. The new algorithm to calculate the Volterra series of the linear–analytic system (28) with nonlinearities (31) can be summarized as follows:

k

(38)

Combining (38) and (34), it can be shown by induction that yn (t) will be the nth homogeneous term in Volterra series if the terms yp1 , yp2 , . . . , and ypk are the p1 , p2 , . . . , and pk -th homogeneous Volterra series terms, respectively. Repeating the recursive processes in (35) yields a Volterra series representation of the nonlinear system. The obtained Volterra series is equal to the Volterra series computed using the Caleman linearization method because of the uniqueness of the Volterra series. Theorem 2.5.2 (uniqueness theorem for Volterra series) in [21] states that any two Volterra series for the same input–output relation are equal.

3.3 Calculation Volterra Series for More General Systems This subsection will show that the new method introduced in the previous subsection can be extended to more general nonlinear systems without difficulty. Consider the following nonlinear systems: L(y) = f (t, y) +

K

gk (t, y)uk (t),

(39)

k=1

where K is an integer of a limited value. The right hand side of the equation has a limited number of terms. Assume system (39) admits a convergent Volterra series. It will be show that the Volterra series can be calculated using an Adomian decomposition method. Applying formulae (2) and (3), the nonlinearities in (39) can be decomposed as a summation of the Adomian polynomials f (t, y) =

∞

An (t, y1 , y2 , . . . , yn−1 )

(40)

n=0

and gk (t, y) =

∞ n=0

Bk,n (t, y1 , y2 , . . . , yn ).

(41)

Y. Guo et al.

The output of the system can then be written as follows according to (7), t y(t) = y(0) + h(t, τ )f (τ, y) dτ 0

t

+

h(t, τ ) 0

K

gk (τ, y)uk (τ ) dτ.

(42)

yn (t)

n=1

=

t

h(t, τ ) 0

+ ×

K k=1 0 ∞

4 An Illustrative Example

k=1

Substituting the Adomian decomposition (1), the Adomian polynomials (40), (41), and the zero initial condition y(0) = 0 into (42) yields ∞

mian decomposition ∞ n=1 yn (t) gives a Volterra series representation of the system. The Volterra kernels can then obtained by comparing (44) with the canonical form of the Volterra series (13).

∞

An (t, y1 , y2 , . . . , yn−1 ) dτ

n=0 t

h(t, τ )

Bk,n (t, y1 , y2 , . . . , yn )uk (τ ) dτ.

(43)

n=0

Using the Adomian decomposition method, the sequence can recursively be calculated as y0 = 0, t y1 = h(t, τ ) A1 (τ ) + B1,0 (τ )u(τ ) dτ, 0 t h(t, τ ) A2 (τ ) + B1,1 (τ )u(τ ) y2 = 0 + B2,0 u2 (τ ) dτ, t y3 = h(t, τ ) A3 (τ ) + B1,2 (τ )u(τ ) 0 + B2,1 u2 (τ ) + B3,0 u3 (τ ) dτ, (44) .. . t yK = h(t, τ ) AK (τ ) + B1,K−1 (τ )u(τ ) 0 + B2,K−2 u2 (τ ) + · · · + BK,0 uK (τ ) dτ, .. . t yn = h(t, τ ) An (τ ) + B1,n−1 (τ )u(τ ) 0 + B2,n−2 u2 (τ ) + · · · + BK,n−K uK (τ ) dτ. Applying an input λu(t), it is easy to observe that yn (t) is the output of the nth homogeneous subsystems of a Volterra series representation. That is, the Ado-

The discussion in the previous section has shown that Volterra series cannot be obtained by using the Adomian decomposition method in a classical way for general nonlinear dynamical systems. However, for some special cases the Adomian decomposition produces a Volterra form representation of the solution. In this section, a nonlinear Duffing oscillator is used to illustrate the algorithm introduced in Sect. 3. Both the classical ADM and the new introduced algorithm will be used to compute the Volterra kernel of the Duffing equation, and results show that both methods produce the same Volterra series representations. The Duffing equation has been used as a prototype in the study of the Volterra approximation of nonlinear dynamical systems and has been well investigated. It is well known that the Duffing oscillator admits a Volterra series expansion around the zero equilibrium point when the input is small enough. The convergence radius of the Volterra series has been discussed using numerical methods by several authors [37, 46]. Since the Duffing equation has only a third-order nonlinearity, all the even-order Volterra kernels are zero. Consider the Duffing oscillator model subject to an exogenous input with small amplitude. ⎧ 2 dy d y ⎪ ⎪ ⎪m 2 (t) + c (t) + k1 y(t) + k3 y 3 (t) = u(t), ⎪ ⎨ dt dt (45) y(0) = 0, ⎪ ⎪ ⎪ ⎪ ⎩ dy (0) = 0. dt Splitting the linear and nonlinear parts of the Duffing equation yields dy d 2y (t) + c (t) + k1 y(t) = u(t) − k3 y 3 (t). (46) 2 dt dt The impulse response function of the associated linear system is 1 −σ (t−τ ) e sin ω0 (t − τ ) , (47) h(t, τ ) = h(t − τ ) = ω0 where σ = c/(2m) and ω0 = 4mk1 − c2 /(2m) when c2 − 4mk1 < 0. m


The solution can then be written as t y(t) = y(0) + h(t − τ )u(τ ) dτ t 0 h(t − τ )k3 y 3 (τ ) dτ. −

where h3 (t, σ1 , σ2 , σ3 ) t h(t − τ )h(τ − σ1 )h(τ − σ2 )h(τ − σ3 ) = −k3 0

(48)

0

Define the Adomian decomposition as y(t) =

∞

0

(49)

yn (t)

n=0

and f (y) = −k3 y 3 (t) =

∞

An (t, y1 , y2 , . . . , yn ).

(50)

n=0

Apply the classical ADM algorithm (10) by setting y0 (t) =

× δ−1 (τ − σ1 )δ−1 (τ − σ2 )δ−1 (τ − σ3 ) dτ, h5 (t, σ1 , σ2 , σ3 , σ4 , σ5 ) (54) t = −3k3 h(τ − σ4 )h(τ − σ5 )h3 (t, σ1 , σ2 , σ3 )

t

h(t − τ )u(τ ) dτ.

(51)

0

The Adomian polynomials can then be calculated by employing formulae (3) A0 = −k3 y03 , A1 = −3k3 y02 y1 , A2 = −3k3 y0 y12 − 3k3 y02 y2 , A3 = −k3 y13

(52)

− 6k3 y0 y1 y2 − 3k3 y02 y3 ,

× δ−1 (τ − τ1 )δ−1 (τ − τ2 )δ−1 (τ − τ3 ) × δ−1 (τ − σ4 )δ−1 (τ − σ5 ) dτ where δ−1 (·) represents the Heaviside step function. Following the Adomian decomposition method in (10), the next terms can recursively be computed. Using the classical ADM, the terms y0 , y1 , y2 , . . . form a series expansion ∞ n=0 yn (t) of the solution of system (46). However, the obtained series is not of a Volterra series form since the nth term is a (2n + 1)th but not an nth homogeneous subsystem. For example, the first three terms y0 , y1 , and y2 correspond to the first, third, and fifth order homogeneous subsystems in a Volterra series representation. The new algorithm is now be applied to obtain the Volterra series by setting y0 (t) = 0. Consider the Duffing equation (46). The new Adomian decomposition is ∞ yn (t). (55) y(t) = 0 + n=1

And the system is of a linear–analytic form with f (t, y) = −k3 y 3 ,

.. .

g(t, y) = 1.

(56)

The Adomian polynomials are The first two terms of the Adomian decompositions are t y1 (t) = h(t − τ )A0 (τ ) dτ 0 t h(t − τ )k3 y03 (τ ) dτ =− 0 t t t = h3 (t, σ1 , σ2 , σ3 )u(σ1 )u(σ2 ) 0

0

0

× u(σ3 ) dσ1 dσ2 dσ3 , (53) t y2 (t) = − h(t − τ )3k3 y02 (τ )y1 (τ ) dτ 0 t t t t t h5 (t, σ1 , σ2 , σ3 , σ4 , σ5 ) = 0

0

0

0

0

× u(σ1 )u(σ2 )u(σ3 )u(σ4 ) × u(σ5 ) dσ1 dσ2 dσ3 dσ4 dσ5 ,

A0 = −k3 y03 = 0, 3 A1 = − k3 y02 y1 = 0, 2! 3 A2 = − k3 y02 y2 − k3 y0 y12 = 0, 2! 3 A3 = − k3 y02 y3 − k3 y0 y1 y2 − k3 y13 = −k3 y13 , 2! (57) 3 2 2 2 A4 = − k3 y0 y4 − k3 y0 y2 − 3k3 y1 y2 2! = −3k3 y12 y2 = 0, 3 A5 = − k3 y02 y5 − 2k3 y0 y2 y3 − 3k3 y1 y22 − 3k3 y12 y3 2! = −3k3 y12 y3 and B0 = u, B1 = B2 = B3 = B4 = B5 = 0.

(58)

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Fig. 1 The first few Volterra kernels obtained using the new method: (a) h1 (t, τ1 ), (b) h3 (t, τ1 , τ2 , τ3 ), τ2 = 4, τ3 = 2, (c) h5 (t, τ1 , τ2 , τ3 , τ4 , τ5 ), τ2 = 4, τ3 = 3, τ4 = 2, τ5 = 1 with the system parameters m = 1, c = 1.4, k1 = 1, and k3 = 4

y4 = 0, t y5 = h(t, τ )(A5 + B5 ) dτ 0 t h(t, τ ) −3k3 y12 y3 dτ = 0 t t t t t h5 (t, τ1 , τ2 , τ3 , τ4 , τ5 ) =

Substituting (57) and (58) into (34) yields y0 = 0, t y1 = h(t, τ )(A1 + B0 ) dτ 0 t h(t, τ )u(τ ) dτ, = 0

0

y2 = 0, t y3 = h(t, τ )(A3 + B2 ) dτ 0 t h(t, τ ) −k3 y13 dτ = 0 t t t = h3 (t, τ1 , τ2 , τ3 )u(τ1 ) 0

0

0

0

0

× u(τ5 ) dτ1 dτ2 dτ3 dτ4 dτ5 , where h3 (t, τ1 , τ2 , τ3 ) t = −k3 h(t, τ )h(τ, τ1 )h(τ, τ2 )h(τ, τ3 )

0

× u(τ2 )u(τ3 ) dτ1 dτ2 dτ3 ,

0

× u(τ1 )u(τ2 )u(τ3 )u(τ4 )

0

(59)

× δ−1 (τ − τ1 )δ−1 (τ − τ2 )δ−1 (τ − τ3 ) dτ,

Volterra Series Approximation of a Class of Nonlinear Dynamical Systems Using the Adomian Decomposition Method Fig. 2 Volterra series approximation of the output of system (45) with a parameter setting m = 1, c = 1.4, k1 = 1, and k3 = 4

h5 (t, τ1 , τ2 , τ3 , τ4 , τ5 ) (60) t h(t, τ )h3 (τ, τ1 , τ2 , τ3 )h(τ, τ4 )h(τ, τ5 ) = −3k3 0

× δ−1 (τ − τ1 )δ−1 (τ − τ2 )δ−1 (τ − τ3 ) × δ−1 (τ − τ4 )δ−1 (τ − τ5 ) dτ. Substituting (47) into (60) yields the third and fifth Volterra kernels. Figure 1 shows the first few Volterra kernels when m = 1, c = 1.4, k1 = 1, and k3 = 4. Using the obtained kernels, the output of the nonlinear system can be synthesised by a truncated Volterra series as yˆN =

N

yn ,

homogeneous subsystems of the Volterra series representation. All these terms compose the Volterra series. Comparing (59) with (53), series (53) is a subseries of (59) and both series converge to the same results. Since the obtained Adomian series is exactly the Volterra series, the new method naturally inherits the advantages of the Adomian decomposition method. A significant advantage of the new algorithm is that the (n + 1)th term in the Volterra series can recursively calculated using only the previous terms as shown in (10). The illustrative example shows that the processes to calculate the Volterra series are straightforward and easy to carry out.

(61)

n=1

where yn is the nth term of the Volterra series which is given in (13). The Duffing oscillator (45) was simulated with a step input where the strength of the step signal was set as 0.1. The simulated output and the synthesised output using the obtained Volterra series are shown in Fig. 2. Only up to third order approximations are given since the higher order multiple integrals are computationally intensive. However, even the first few approximations show that the synthesised outputs using the truncated Volterra series approach the real output when the order N increases. Using the new algorithm, the obtained nth term yn (t) in the Adomian decomposition is the nth order

5 Conclusions Adomian decomposition (ADM) and Volterra series are two kinds of commonly used approximation methods in the investigation of nonlinear dynamical systems. For the nonlinear systems which admit a convergent Volterra series representation, the relations between these two approaches are studied in this paper. It has been shown that the Volterra series approximation is a specialization of Adomian decomposition when the first term y0 (t) of Adomian decomposition is chosen to be zero. The straightforward relationship between Volterra series and ADM provides a new choice for calculating

Y. Guo et al.

Volterra kernels using ADM which significantly simplifies the procedure of deriving a Volterra series representation from a differential equation form model. A new algorithm has been given in the paper to calculate the Volterra series for a wide class of nonlinear dynamical systems. Although not all nonlinearities are discussed in the paper, such as the nonlinearities including the derivatives of the output, it is believed that the Adomian decomposition based method can be used to calculate the Volterra series for more nonlinear dynamical systems through applying some simple mathematical operations. Furthermore, the generality of the Adomian decomposition method makes it possible to extend the Volterra series approximation to partial differential equations. This is likely to be important for the investigation of nonlinear distributed parameter systems and this will be studied in a further paper. Acknowledgements The authors gratefully acknowledge support from the UK Engineering and Physical Sciences Research Council (EPSRC) and the European Research Council (ERC). The authors also thank the reviewers for the helpful comments.

References 1. Abbaoui, K., Cherruault, Y.: Convergence of Adomian’s method applied to differential equations. Comput. Math. Appl. 28, 103–109 (1994) 2. Abdelrazec, A.: Adomian decomposition method: convergence analysis and numerical approximation. Department of Mathematics, McMaster University, Hamilton, Ontario (2008) 3. Abdelrazec, A., Pelinovsky, D.: Convergence of the Adomian decomposition method for initial-value problems. Numer. Methods Partial Differ. Equ. 27, 749–766 (2009) 4. Abdelwahid, F.: A mathematical model of Adomian polynomials. Appl. Math. Comput. 141, 447–453 (2003) 5. Adomian, G.: A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 135, 501–544 (1988) 6. Adomian, G.: Nonlinear Stochastic System Theory and Application to Physics. Kluwer Academic, Dordrecht (1989) 7. Adomian, G.: Solution of physical problems by decomposition. Comput. Math. Appl. 27, 145–154 (1994) 8. Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston (1994) 9. Adomian, G.: Solution of coupled nonlinear partial differential equations by decomposition. Comput. Math. Appl. 31, 117–120 (1996) 10. Adomian, G., Rach, R.: Inhomogeneous nonlinear partial differential equations with variable coefficients. Appl. Math. Lett. 5, 11–12 (1992)

11. Adomian, G., Rach, R.: Modified Adomian polynomials. Math. Comput. Model. 24, 39–46 (1996) 12. Babolian, E., Vahidi, A.R., Azimzadeh, Z.: A comparison between Adomian’s decomposition method and the homotopy perturbation method for solving nonlinear differential equations. J. Appl. Sci. 12, 793–797 (2012) 13. Barrett, J.F.: The use of Volterra series to find region of stability of a non-linear differential equation. Int. J. Control 1, 209–216 (1965) 14. Bellomo, N., Monaco, R.: A comparison between Adomian’s decomposition methods and perturbation techniques for nonlinear random differential equations. J. Math. Anal. Appl. 110, 495–502 (1985) 15. Billings, S.A.: Identification of non-linear systems—a survey. IEE Proc. Part D, Control Theory Appl. 127, 272–285 (1980) 16. Billings, S.A., Boaghe, O.M.: The response spectrum map, a frequency domain equivalent to the bifurcation diagram. Int. J. Bifurc. Chaos Appl. Sci. Eng. 11, 1961–1975 (2001) 17. Billings, S.A., Peyton Jones, J.C.: Mapping nonlinear integrodifferential equations into the frequency domain. Int. J. Control 52, 863–879 (1990) 18. Boaghe, O.M.: Volterra series modelling limitations. The Department of Automatic Control & Systems Engineering, The University of Sheffield (2000) 19. Boaghe, O.M., Billings, S.A.: Subharmonic oscillation modeling and MISO Volterra series. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50, 877–884 (2003) 20. Boyd, S., Chua, L.O.: Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Trans. Circuits Syst. 32, 1150–1161 (1985) 21. Boyd, S., Chua, L.O., Desoer, C.A.: Analytical foundations of Volterra series. IMA J. Math. Control Inf. 1, 243–282 (1984) 22. Brockett, R.W.: Volterra series and geometric control theory. Automatica 12, 167–176 (1976) 23. Bruni, C., Dipillo, G., Koch, G.: Bilinear systems: an appealing class of “nearly linear” systems in theory and applications. IEEE Trans. Autom. Control 19, 334–348 (1974) 24. Bullo, F.: Series expansions for analytic systems linear in control. Automatica 38, 1425–1432 (2002) 25. Cherruault, Y., Adomian, G., Abbaoui, K., Rach, R.: Further remarks on convergence of decomposition method. Int. J. Bio-Med. Comput. 38, 89–93 (1995) 26. Edwards, J.Y., Roberts, J.A., Ford, N.J.: A comparison of Adomian’s decomposition method and Runge Kutta methods for approximate solution of some predator prey model equations. Manchester Center of Computational Mathematics (1997) 27. El-Sayed, S.M., Abdel-Aziz, M.R.: A comparison of Adomian’s decomposition method and wavelet-Galerkin method for solving integro-differential equations. Appl. Math. Comput. 136, 151–159 (2003) 28. Fliess, M., Lamnabhi, M., Lamnabhi-Lagarrigue, F.: An algebraic approach to nonlinear functional expansions. IEEE Trans. Circuits Syst. 30, 554–570 (1983) 29. George, D.A.: Continuous nonlinear systems. Tech. Rep., MIT (1959)

Volterra Series Approximation of a Class of Nonlinear Dynamical Systems Using the Adomian Decomposition Method 30. Gilbert, E.: Functional expansions for the response of nonlinear differential systems. IEEE Trans. Autom. Control 22, 909–921 (1977) 31. Helie, T., Laroche, B.: Computation of convergence bounds for Volterra series of linear-analytic single-input systems. IEEE Trans. Autom. Control 56, 2062–2072 (2011) 32. Hosseini, M.M., Nasabzadeh, H.: On the convergence of Adomian decomposition method. Appl. Math. Comput. 182, 536–543 (2006) 33. Korenberg, M.J., Hunter, I.W.: The identification of nonlinear biological systems: Volterra kernel approaches. Ann. Biomed. Eng. 24, 250–268 (1996) 34. Krener, A.J.: Linearization and bilinearization of control systems. In: Kokotovic, P.V., Davidson, E.S. (eds.) 12th Allerton Conference on Circuits and Systems, Montecello, Illinois (1974) 35. Ku, Y.H., Wolf, A.A.: Volterra–Wiener functionals for the analysis of nonlinear systems. J. Franklin Inst. 281, 9–26 (1966) 36. Li, L.M., Billings, S.A.: Piecewise Volterra modeling of the Duffing oscillator in the frequency-domain. Mech. Syst. Signal Process. 26, 117–127 (2012) 37. Peng, Z.K., Lang, Z.Q.: On the convergence of the Volterraseries representation of the Duffing’s oscillators subjected to harmonic excitations. J. Sound Vib. 305, 322–332 (2007) 38. Rach, R.: On the Adomian (decomposition) method and comparisons with Picard’s method. J. Math. Anal. Appl. 128, 480–483 (1987) 39. Rack, R.C.: A new definition of the Adomian polynomials. Kybernetes 37 (2008) 40. Rugh, W.J.: Nonlinear System Theory. Johns Hopkins University Press, Baltimore (1981) 41. Rugh, W.J.: Nonlinear System Theory: the Volterra/Wiener Approach. Johns Hopkins University Press, London (1981)

42. Seng, V., Abbaoui, K., Cherruault, Y.: Adomian’s polynomials for nonlinear operators. Math. Comput. Model. 24, 59–65 (1996) 43. Shawagfeh, N., Kaya, D.: Comparing numerical methods for the solutions of systems of ordinary differential equations. Appl. Math. Lett. 17, 323–328 (2004) 44. Song, J., Yin, F., Cao, X., Lu, F.: Fractional variational iteration method versus Adomian’s decomposition method in some fractional partial differential equations. J. Appl. Math. 2013, 1–10 (2013) 45. Swain, A.K., Billings, S.A.: Generalized frequency response function matrix for MIMO non-linear systems. Int. J. Control 74, 829–844 (2001) 46. Tomlinson, G.R., Manson, G., Lee, G.M.: A simple criterion for establishing an upper limit to the harmonic excitation level of the Duffing oscillator using the Volterra series. J. Sound Vib. 190, 751–762 (1996) 47. Volterra, V.: Theory of Functionals and of Integral and Integro-Differential Equation. Dover, New York (1959) 48. Wazwaz, A.-M.: A comparison between Adomian decomposition method and Taylor series method in the series solutions. Appl. Math. Comput. 97, 37–44 (1998) 49. Wazwaz, A.-M.: A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl. Math. Comput. 111, 33–51 (2000) 50. Wazwaz, A.-M.: A comparison between the variational iteration method and Adomian decomposition method. J. Comput. Appl. Math. 207, 129–136 (2007) 51. Worden, K., Manson, G.: A Volterra series approximation to the coherence of the Duffing oscillator. J. Sound Vib. 286, 529–547 (2005) 52. Zhu, Y., Chang, Q., Wu, S.: A new algorithm for calculating Adomian polynomials. Appl. Math. Comput. 169, 402–416 (2005)