(Billings and Peyton Jones, 1990, Worden and Manson, 2005, Swain and Billings, ...... EDWARDS, J. Y., ROBERTS, J. A. & FORD, N. J. (1997) A comparison of ...
Volterra Series Approximation of a Class of Nonlinear Dynamic Systems Using the Adomian Decomposition Method Yuzhu Guo, L.Z. Guo, S.A. Billings, Daniel Coca, and Z.Q. Lang Department of Automatic Control and Systems Engineering, The University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK
Revised 6 Jan. 2013 Revised 12 Mar. 2013 Revised 15 May 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 dynamic 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 dynamic 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 (Volterra, 1959) as a general description for a wide class of nonlinear dynamic 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 (Rugh, 1981b). Volterra series have played an important role in the modelling of nonlinear systems, both when the underlying system equations are known and when the
1
system is characterized only by the availability of input-output data. The identification of Volterra kernels has been reviewed by Billings (1980) and Korenberg and Hunter (1996). The properties and advantages of the Volterra series compared to implicit system models have been studied by George (1959). 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 a system admits a Volterra series, the convergence of the series expansion and the corresponding convergence region have been studied by many authors (Barrett, 1965, Helie and Laroche, 2011, Peng and Lang, 2007, Tomlinson et al., 1996, Boyd and Chua, 1985). The limitations of the Volterra series have been studied by Boaghe (2000) and new approaches have been derived to overcome these limitations including the multi-input Volterra series for sub-harmonic systems (Boaghe and Billings, 2003), the response spectrum map for bifurcations (Billings and Boaghe, 2001), and the piecewise Volterra models (Li and Billings, 2012). Computation of the Volterra series for an ordinary differential equation using successive approximations is well known. Bruni, DiPillo & Koch (1974) 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 (1976). An alternative approach was introduced by Gilbert (1977). However the calculation of the Volterra kernels in the time domain may not be easy especially using the Carleman technique (Krener, 1974) but some simple methods are available in the frequency domain (Billings and Peyton Jones, 1990, Worden and Manson, 2005, Swain and Billings, 2001). These computation methods have been summarised in Rugh’s book (1981a). Computation methods for other types of series expansions such as the Chen-Fliess series have also been studied (Bullo, 2002, Fliess et al., 1983). 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. The convergence of the obtained series expansion is also studied.
2
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 (Abbaoui and Cherruault, 1994, Abdelrazec, 2008, Abdelwahid, 2003, Adomian, 1988, Adomian, 1994a, Adomian and Rach, 1992, Adomian and Rach, 1996, Wazwaz, 2000, Adomian, 1996, Adomian, 1994b, Rack, 2008). 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 (Bellomo and Monaco, 1985, Rach, 1987, Wazwaz, 1998, Shawagfeh and Kaya, 2004, Wazwaz, 2007, Song et al., 2013, E. Babolian, 2012, Edwards et al., 1997, El-Sayed and Abdel-Aziz, 2003). 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 section 2. Section 3 shows that Volterra series approximation is a specialization of the Adomian decomposition with y0 = 0 . A class of nonlinear dynamic systems are considered and a new algorithm for computing Volterra kernels from ordinary differential equation descriptions is introduced. A Duffing oscillator is discussed in section 4 as an illustrative example to illustrate the new algorithm. Conclusions are finally drawn in section 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 1980’s. 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
(Bellomo and Monaco, 1985, Rach, 1987, Wazwaz, 1998,
Shawagfeh and Kaya, 2004, Wazwaz, 2007, Song et al., 2013, E. Babolian, 2012, Edwards et al., 1997, El-Sayed and Abdel-Aziz, 2003). 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 (Adomian, 1989)
3
∞
y ( t ) = ∑ λ n 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 ) = ∑ λ n An ( t , λ y1 , λ 2 y2 ,⋯ , λ n yn )
(2)
n =0
(
)
where An t , λ y1 , λ y2 ,⋯, λ yn are referred to as Adomian polynomials when the parameter λ = 1 . The 2
n
functions An ( t , y1 , y2 ,⋯ , yn ) only depend on y1 , y2 , …, and yn and no higher orders are involved. According to the original definitions by Adomian, the Adomian polynomials can be calculated by the following formulae.
An =
1 dn ∞ i g t , ∑ λ yi n ! d λ n i = 0 λ =0
(3)
where λ is a constant parameter. The first several Adomian polynomials can be calculated as
A0 = g ( t , y0 )
A1 = y1 g ′ ( t , y0 ) A2 = y2 g ′ ( t , y0 ) +
1 2 y1 g ′′ ( t , y0 ) 2!
A3 = y3 g ′ ( t , y0 ) + y1 y2 g ′′ ( t , y0 ) +
(4)
1 3 y1 g ′′′ ( t , y0 ) 3!
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 Adomain polynomials (Wazwaz, 2000, Zhu et al., 2005, Seng et al., 1996), and the convergence of the Adomian decomposition has been intensively investigated (Abbaoui and Cherruault, 1994, Abdelrazec and Pelinovsky, 2009, Cherruault et al., 1995, Hosseini and Nasabzadeh, 2006, Rack, 2008). Consider a general nonlinear dynamical system
4
Ny ( t ) = u ( t )
(5)
where N [ i ] is a nonlinear differential operator. A general approach is to split the nonlinear operator into two different 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 )
(6)
The invertible operator of L [ i] is assumed to be of a convolution form t
L−1 [ i] = ∫ h ( t ,τ )( i ) dτ
(7)
0
The solution of system (6) can then be expressed as t
t
0
0
y ( t ) = y ( 0 ) + ∫ h ( t ,τ ) u (τ ) dτ + ∫ h ( t ,τ ) g (τ , y (τ ) ) dτ
(8)
Substituting the Adomian decomposition (1) and (2) into (8) yields ∞
t
t
∞
0
0
n =0
∑ y ( t ) = y ( 0 ) + ∫ h ( t ,τ ) u (τ ) dτ + ∫ h ( t,τ ) ∑ A (τ , y , y ,⋯ , y ) dτ n =0
n
n
1
2
n
(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 0
(10)
⋯ t
yn +1 = ∫ h ( t ,τ ) An (τ , y0 , y1 ,⋯ , yn ) dτ 0
Setting y ( 0 ) = 0 and substituting equation (4) into (10) yields the first few terms in the Adomian decomposition of the zero-state response of system (6).
5
t
y0 = ∫ h ( t ,τ ) u (τ ) dτ 0 t
y1 = ∫ h ( t ,τ ) g (τ , y0 ) dτ 0 t
y2 = ∫ h ( t ,τ ) y1 (τ ) g ′ (τ , y0 ) dτ
(11)
0 t
1 y3 = ∫ h ( t ,τ ) y2 (τ ) g ′ (τ , y0 ) + y12 (τ ) g ′′ (τ , y0 ) dτ 2! 0 t
1 y4 = ∫ h ( t ,τ ) y3 (τ ) g ′ (τ , y0 ) + y1 (τ ) y2 (τ ) g ′′ (τ , y0 ) + y13 (τ ) g ′′′ (τ , y0 ) dτ 3! 0 ⋯
Remarks: i. The split Ny = Ly − Gy of the nonlinear operator N [ i ] is not unique. Any appropriate split may lead to an Adomian decomposition. ii. The Adomian polynomial decomposition in (2) and (3) can be referred to as a parametric Taylor expansion
of g t ,
∞
∑λ n =0
n
yn around the ‘point’ λ = 0 , that is, y = y0 .
iii. The choice of y0 in (10) is not unique. Different y0 may correspond to different Adomian decompositions. Every Adomian decomposition gives a series expansion y ( t ) =
∞
∑ y ( t ) of the solution of the nonlinear n =0
n
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 Section 4.
3. Computing Volterra Kernels Using Adomian Decomposition
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 dynamic system can be calculated using the Adomian decomposition method.
6
3.1 From Volterra Series to Adomian Decomposition
It will first be shown that for a nonlinear dynamic 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 equation (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 ) = ∑ yn ( t )
(12)
n =1
where yn ( t ) is given in a multi-dimensional convolution form as t
t
0
0
yn ( t ) = ∫ ⋯ ∫ hn ( t ,τ 1 ,τ 2 ,⋯τ n ) u (τ 1 ) u (τ 2 )⋯ u (τ n ) dτ 1dτ 2 ⋯ dτ n
(13)
The term hn ( t ,τ1 ,τ 2 ,⋯τ n ) is referred as to the n -th order Volterra kernel. Term yn ( t ) in (12) is often known as the n -th homogeneous subsystem of system (6). This is because the n response of the n -th homogeneous subsystem becomes λ yn ( t ) by applying a scaled input λu ( t ) , where ∞
λ is a scalar. Namely, the zero-state response of the system to an input λu ( t ) is y ( t ) = ∑ λ n yn ( t ) which n =0
satisfies the following equation
∞ ∞ L ∑ λ n yn ( t ) = λ u ( t ) + g t , ∑ λ n yn ( t ) n=0 n=0
(14)
where an additional term y0 ( t ) = 0 is introduced for the convenience of the discussion so that ∞
∞
∞
n =1
n =1
n=0
y ( t ) = ∑ λ n yn ( t ) = 0 + ∑ λ n y n ( t ) = ∑ λ n yn ( t )
(15)
Because of the analyticity, the Maclaurin series expansion of the nonlinearity g ( t , y ) can be written as ∞
g ( t , y ) = ∑ gi ( t ) y i
(16)
i =1
7
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 λ n on both sides of (14) following the algorithm introduced in (Ku and Wolf, 1966). To apply the algorithm, the linear term g1 ( t ) y is subtracted on both sides of equation (14). ∞ ∞ ∞ ∞ L ∑ λ n yn ( t ) − g1 ( t ) ∑ λ n yn ( t ) = λu ( t ) + ∑ gi ( t ) ∑ λ n yn ( t ) n =1 i =2 n=1 n=1
i
(17)
Define a new linear differential operator as Lɶ [ i ] = L [ i] − g1 ( t ) [ i ] and the associated inverse operator as t
Lɶ−1 [ i] = ∫ hɶ ( t ,τ )( i )(τ ) dτ
(18)
0
Equation (17) becomes ∞ ∞ ∞ Lɶ ∑ λ n yn ( t ) = λu ( t ) + ∑ gi ( t ) ∑ λ n yn ( t ) n=1 n =1 i =2
i
(19)
The n -th homogeneous subsystems can then be computed separately by equating the like term of λ n as t
yn ( t ) = ∫ hɶ ( t ,τ ) An ( y1 , y2 ,⋯ , yn ) dτ
(20)
0
where
A0 = 0 A1 = u A2 = g 2 y12
(21)
A3 = 2 g 2 y1 y2 + g 3 y13 A4 = g 2 y22 + 3 g3 y12 y2 + g 4 y14 ⋯ Redefine the nonlinearity on the right hand of equation (19) as ∞
∞ gɶ ( t , y ) = λu ( t ) + ∑ gi ( t ) ∑ λ n yn ( t ) i=2 n=1
i
(22)
It can be shown that the obtained An satisfies the definition of the Adomian polynomial in (3)
8
An =
1 dn ∞ i gɶ t , ∑ λ yi n ! d λ n i = 0 λ =0
(23)
This result shows that the obtained Volterra series is a specialisation of the Adomian decomposition of the following nonlinear dynamic system, in which y0 ( t ) = 0 .
Lɶ [ y ] = gɶ ( t , y )
(24)
The Adomian decomposition is given in a formal form as
∞ ∞ Lɶ ∑ yn ( t ) = ∑ An ( y0 , y1 ,⋯ , yn ) n =1 n=0
(25)
Remarks: i. Notice that the Adomian polynomial An ( y0 , y1 ,⋯, yn ) is actually not dependent on yn because gɶ ( 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 equation (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 ,τ 2 ,⋯τ n ) 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 equation (11) in section 2. It is easy to observe that following the classical Adomian decomposition method the obtained series expansion of the solution is not a Volterra series. This is because the n -th term of the Adomian decomposition is not necessarily an n -th homogeneous operator about the input u ( t ) . 9
For example, assume the nonlinearity g ( t , y ) is of a polynomial form
g ( t , y ) = y3 + y 2
(26)
Substituting (26) into the second equation in (11) yields 3 2 τ τ y1 ( t ) = ∫ h ( t ,τ ) ∫ h ( t ,τ 1 ) u (τ 1 ) dτ 1 + ∫ h ( t ,τ 1 ) u (τ 1 ) dτ 1 dτ 0 0 0 t
(27)
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 (Brockett, 1976, Gilbert, 1977). 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 (Krener, 1974). 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 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 the 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 Adomain decomposition method. Start the Adomian decomposition method by setting y0 = 0 and the output can then be written as
10
∞
∞
n =0
n =1
y ( t ) = ∑ yn ( t ) = 0 + ∑ yn ( t )
(29)
Expressing the nonlinearities in (28) by Adominan polynomials yields ∞ ∞ ∞ L ∑ yn ( t ) = ∑ An ( t , y0 , y1 ,⋯ , yn ) + ∑ Bn ( t , y0 , y1 ,⋯ , yn ) u n =0 n =0 n=0
(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 ∞
f ( t , y ) = ∑ fi ( t ) y i i=2
(31)
∞
gɶ ( t , y ) = g ( t , y ) u = ∑ g i ( t ) y u i
i =0
Notice that f 0 ( 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 equation (28). According to the formulae (3), the Adomian polynomials can be computed as
A0 = f ( t , 0 ) = 0
A1 = y1 f ′ ( t , 0 ) = 0 1 2 y1 f ′′ ( t , 0 ) = y12 f 2 ( t ) 2! 1 A3 = y3 f ′ ( t , 0 ) + y1 y2 f ′′ ( t , 0 ) + y13 f ′′′ ( t , 0 ) = 2 y1 y2 f 2 ( t ) + y13 f3 ( t ) 3! ⋯ A2 = y2 f ′ ( t , 0 ) +
(32)
and
B0 = g ( t , 0 ) = g 0 ( t )
B1 = y1 g ′ ( t , 0 ) = y1 g1 ( t ) 1 2 y1 g ′′ ( t , 0 ) = y2 g1 ( t ) + y12 g 2 ( t ) 2! 1 B3 = y3 g ′ ( t , 0 ) + y1 y2 g ′′ ( t , 0 ) + y13 g ′′′ ( t , 0 ) = y3 g1 ( t ) + 2 y1 y2 g 2 ( t ) + y13 g 3 ( t ) 3! ⋯ B2 = y2 g ′ ( t , 0 ) +
(33)
11
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
(34)
⋯ t
yn = ∫ h ( t ,τ ) ( An (τ ) + Bn −1 (τ ) u (τ ) ) dτ 0
⋯ Applying a input λu ( t ) , the first several terms of the Adomian decomposition is
y0 = 0 t
y1 = λ ∫ h ( t ,τ ) g 0 (τ ) u (τ ) dτ 0 t
y2 = ∫ h ( t ,τ ) ( y12 f 2 (τ ) + λ y1 g1 (τ ) u (τ ) ) dτ 0
=λ
t t
∫ ∫ h ( t ,τ ,τ ) u (τ ) u (τ ) dτ dτ
2
2
1
2
1
2
1
(35)
2
0 0
(
t
)
y3 = ∫ h ( t ,τ ) 2 y1 y2 f 2 (τ ) + y13 f3 (τ ) + λ ( y2 g1 (τ ) + y12 g 2 (τ ) ) u ( t ) dτ 0
=λ
t t t
3
∫ ∫ ∫ h ( t , τ ,τ 3
1
2
,τ 3 ) u (τ 1 ) u (τ 2 ) u (τ 3 ) dτ 1dτ 2 dτ 3
0 0 0
⋯ where t
h2 ( t ,τ 1 ,τ 2 ) = ∫ h ( t ,τ ) f 2 (τ ) h (τ ,τ 1 ) h (τ ,τ 2 ) g 0 (τ 1 ) g 0 (τ 2 ) δ −1 (τ − τ 1 ) δ −1 (τ − τ 2 ) dτ 0
(36)
+ h ( t ,τ 1 ) h (τ 1 ,τ 2 ) g 0 (τ 2 ) g1 (τ 1 ) δ −1 (τ 1 − τ 2 ) and
12
t
h3 ( t ,τ 1 ,τ 2 ,τ 3 ) = 2 ∫ h ( t ,τ ) f 2 (τ ) h2 (τ ,τ 1 ,τ 2 ) h (τ ,τ 3 ) g0 (τ 3 ) δ (τ − τ 1 ) δ (τ − τ 2 ) δ (τ − τ 3 ) dτ 0 t
+ ∫ h ( t ,τ ) f3 (τ ) h (τ ,τ 1 ) h (τ ,τ 2 ) h (τ ,τ 3 ) g 0 (τ 1 ) g 0 (τ 2 ) g 0 (τ 3 ) δ (τ − τ 1 ) δ (τ − τ 2 ) δ (τ − τ 3 ) dτ (37) 0
+ h ( t ,τ 3 ) h2 (τ 3 ,τ 1 ,τ 2 ) g1 (τ 3 ) δ (τ 3 − τ 1 ) δ (τ 3 − τ 2 )
+ h ( t ,τ 3 ) h (τ 3 ,τ 2 ) h (τ 3 ,τ 1 ) g0 (τ 1 ) g 0 (τ 2 ) g 2 (τ 3 ) δ (τ 3 − τ 1 ) δ (τ 3 − τ 2 ) The notation δ −1 ( i ) 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 & Cherruault (1994) showed that Adomian polynomials An can be expressed as the following explicit formulae:
1 (k ) g ( t , y0 ) ∑ y p1 ⋯ y pk k =1 k ! p1 +⋯+ pk = n n
An = ∑
, n ≥1
(38)
Combining (38) and (34), it can be shown by induction that yn ( t ) will be the n-th homogeneous term in Volterra series if the terms y p1 , y p2 , …, and y pk 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 the paper (Boyd et al., 1984) states that any two Volterra series for the same input-output relation are equal. Although nonlinearities of the form f ( t , y ) and g ( t , y ) are used here it should be emphasized that the
(
k ADM can be applied to a more general class of systems where g t , y , y ′, y ′′,⋯ , y ( )
)
can be an analytical
function of y ( t ) and the associated derivatives. Examples can be found in (Wazwaz, 2000). The new algorithm to calculate the Volterra series of the linear-analytic system (28) with (31) can be summarized as follows: 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 Adomain decomposition y0 = 0 ;
13
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
∑y
i
is the Volterra series;
i
vii. The Volterra kernels can be obtained by rearranging the multiple integrals like (35) ~(37).
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 K
L ( y ) = f ( t , y ) + ∑ gk ( t, y ) u k ( 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 formula (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 ∞
g k ( t , y ) = ∑ Bk ,n ( t , y1 , y2 ,⋯ , yn )
(41)
n =0
The output of the system can then be written as follows according to (7), t
t
K
0
0
k =1
y ( t ) = y ( 0 ) + ∫ h ( t ,τ ) f (τ , y ) dτ + ∫ h ( t ,τ ) ∑ g k (τ , y ) u k (τ )dτ
(42)
Substituting the Adomian decomposition(1), the Adomian polynomials (40), (41), and the zero initial condition
y ( 0 ) = 0 into (42) yields ∞
t
∞
K
t
∞
∑ y ( t ) = ∫ h ( t ,τ ) ∑ A ( t , y , y ,⋯ , y )dτ + ∑ ∫ h ( t ,τ ) ∑ B ( t , y , y ,⋯, y )u (τ ) dτ k
n =1
n
0
n=0
n
1
2
n −1
k =1 0
n =0
k ,n
1
2
n
(43) 14
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
y2 = ∫ h ( t ,τ ) ( A2 (τ ) + B1,1 (τ ) u (τ ) + B2,0u 2 (τ ) ) dτ 0 t
y3 = ∫ h ( t ,τ ) ( A3 (τ ) + B1,2 (τ ) u (τ ) + B2,1u 2 (τ ) + B3,0u 3 (τ ) ) dτ
(44)
0
⋯ t
yK = ∫ h ( t ,τ ) ( AK (τ ) + B1, K −1 (τ ) u (τ ) + B2, K − 2u 2 (τ ) + ⋯ + BK ,0u K (τ ) ) dτ 0
⋯ t
yn = ∫ h ( t ,τ ) ( An (τ ) + B1,n −1 (τ ) u (τ ) + B2,n − 2u 2 (τ ) + ⋯ + BK , n − K u K (τ ) ) dτ 0
Applying an input λu ( t ) , it is easy to observe that yn ( t ) is the output of the n -th homogeneous ∞
subsystems of a Volterra series representation. That is, the Adomian decomposition
∑ y ( t ) gives a Volterra n =1
n
series representation of the system. The Volterra kernels can then obtained by comparing equation (44) with the canonical form of the Volterra series (13).
4. An Illustrative Example
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 dynamic systems. However for some special cases the Adomian decomposition produces a Volterra form approximation of the solution. In this section a nonlinear Duffing oscillator is used to illustrate the algorithm introduced in section 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 dynamic systems and has been well investigated. It is well known that the Duffing oscillator admits a Volterra series approximation 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 (Tomlinson et al., 1996, Peng and Lang, 2007). Since the Duffing equation has only a third-order nonlinearity all the even-order Volterra kernels are zero. 15
Consider the Duffing oscillator model subject to an exogenous input with small amplitude.
d2y dy 3 m dt 2 ( t ) + c dt ( t ) + k1 y ( t ) + k3 y ( t ) = u ( t ) y ( 0) = 0 dy (0) = 0 dt
(45)
Splitting the linear and nonlinear parts of the Duffing equation yields
m
d2y dy t + c ( t ) + k1 y ( t ) = u ( t ) − k3 y 3 ( t ) 2 ( ) dt dt
(46)
The impulse response function of the associated linear system is
h ( t ,τ ) = h ( t − τ ) = where σ = c ( 2m ) and ω0 = 4mk1 − c 2
1
ω0
e
( 2m )
−σ ( t −τ )
sin (ω0 ( t − τ ) )
(47)
when c 2 − 4mk1 < 0 .
The solution can then be written as t
t
y ( t ) = y ( 0 ) + ∫ h ( t − τ ) u (τ ) dτ − ∫ h ( t − τ ) k3 y 3 (τ ) dτ 0
(48)
0
Define the Adomian decomposition as ∞
y ( t ) = ∑ yn ( t )
(49)
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 t
y0 ( t ) = ∫ h ( t − τ ) u (τ ) dτ
(51)
0
The Adomian polynomials can then be calculated by employing formulae (3)
16
A0 = − k3 y03 A1 = −3k3 y02 y1 A2 = −3k3 y0 y12 − 3k3 y02 y2
(52)
A3 = −k y − 6k3 y0 y1 y2 − 3k y y 3 3 1
2 3 0 3
⋯ The first two terms of the Adomian decompositions are t
t
0
0
y1 ( t ) = ∫ h ( t − τ ) A0 (τ ) dτ = − ∫ h ( t − τ ) k3 y03 (τ ) dτ t t t
= ∫ ∫ ∫ h3 ( t , σ 1 , σ 2 , σ 3 ) u (σ 1 ) u (σ 2 ) u (σ 3 ) dσ 1dσ 2 dσ 3 0 0 0
(53)
t
y2 ( t ) = − ∫ h ( t − τ ) 3k y (τ ) y1 (τ ) dτ 2 3 0
0 t t t t t
= ∫ ∫ ∫ ∫ ∫ h5 ( t , σ 1 , σ 2 , σ 3 , σ 4 , σ 5 ) u (σ 1 ) u (σ 2 ) u (σ 3 ) u (σ 4 ) u (σ 5 ) dσ 1dσ 2 dσ 3dσ 4 dσ 5 0 0 0 0 0
where
h3 ( t , σ 1 , σ 2 , σ 3 ) t
= − k3 ∫ h ( t − τ ) h (τ − σ 1 ) h (τ − σ 2 ) h (τ − σ 3 ) δ −1 (τ − σ 1 ) δ −1 (τ − σ 2 ) δ −1 (τ − σ 3 ) dτ 0
(54)
h5 ( t , σ 1 , σ 2 , σ 3 , σ 4 , σ 5 ) = −3k3 × t
∫ h (τ − σ ) h (τ − σ ) h ( t ,σ , σ 4
5
3
1
2
, σ 3 ) δ −1 (τ − τ 1 ) δ −1 (τ − τ 2 ) δ −1 (τ − τ 3 ) δ −1 (τ − σ 4 ) δ −1 (τ − σ 5 ) dτ
0
where δ −1 ( i ) 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
∑ y (t ) n =0
n
of the solution of
system (46). However, the obtained series is not of a Volterra series form since the n -th term is a ( 2n + 1) -th but not an n -th homogeneous subsystem. For example, the first three terms y0 , y1 , and y2 a 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 17
∞
y ( t ) = 0 + ∑ yn ( t )
(55)
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
A0 = − k3 y03 = 0 3 k3 y02 y1 = 0 2! 3 = − k3 y02 y2 − k3 y0 y12 = 0 2! 3 = − k3 y02 y3 − k3 y0 y1 y2 − k3 y13 = − k3 y13 2! 3 = − k3 y02 y4 − k3 y0 y22 − 3k3 y12 y2 = −3k3 y12 y2 = 0 2! 3 = − k3 y02 y5 − 2k3 y0 y2 y3 − 3k3 y1 y22 − 3k3 y12 y3 = −3k3 y12 y3 2!
A1 = − A2 A3 A4 A5
(57)
and
B0 = u B1 = B2 = B3 = B4 = B5 = 0
(58)
Substituting (57) and (58) into (34) yields
y0 = 0 t
t
0
0
y1 = ∫ h ( t ,τ )( A1 + B0 ) dτ = ∫ h ( t ,τ ) u (τ ) dτ y2 = 0 t
t
0
0
y3 = ∫ h ( t ,τ )( A3 + B2 ) dτ = ∫ h ( t ,τ ) ( −k3 y13 ) dτ t t t
= ∫ ∫ ∫ h3 ( t ,τ 1 ,τ 2 ,τ 3 ) u (τ 1 ) u (τ 2 ) u (τ 3 ) dτ 1dτ 2 dτ 3
(59)
0 0 0
y4 = 0 t
t
0
0
y5 = ∫ h ( t ,τ )( A5 + B5 ) dτ = ∫ h ( t ,τ ) ( −3k3 y12 y3 ) dτ t t t t t
= ∫ ∫ ∫ ∫ ∫ h5 ( t ,τ 1 ,τ 2 ,τ 3 ,τ 4 ,τ 5 ) u (τ 1 ) u (τ 2 ) u (τ 3 ) u (τ 4 ) u (τ 5 ) dτ 1dτ 2 dτ 3 dτ 4 dτ 5 0 0 0 0 0
18
where
h3 ( t ,τ 1 ,τ 2 ,τ 3 ) t
= −k3 ∫ h ( t ,τ ) h (τ ,τ 1 ) h (τ ,τ 2 ) h (τ ,τ 3 ) δ −1 (τ − τ 1 ) δ −1 (τ − τ 2 ) δ −1 (τ − τ 3 ) dτ 0
(60)
h5 ( t ,τ 1 ,τ 2 ,τ 3 ,τ 4 ,τ 5 ) = −3k3 × t
∫ h ( t ,τ ) h (τ ,τ ,τ 3
1
2
,τ 3 ) h (τ ,τ 4 ) h (τ ,τ 5 ) δ −1 (τ − τ 1 ) δ −1 (τ − τ 2 ) δ −1 (τ − τ 3 ) δ −1 (τ − τ 4 ) δ −1 (τ − τ 5 ) dτ
0
Substituting (47) into (60) yields the 3rd and 5th Volterra kernels. Figure 1 shows the first few Volterra kernels when m = 1 , c = 1.4 , k1 = 1 , and k3 = 4 .
(a)
(b)
(c) Figure 1 The first few Volterra kernels obtained using the new method. (a) h1 ( t ,τ 1 ) (b) h3 ( t ,τ 1 ,τ 2 ,τ 3 ) τ 2 = 4 ,
τ 3 = 2 (c) h5 ( t ,τ 1 ,τ 2 ,τ 3 ,τ 4 ,τ 5 ) τ 2 = 4 , τ 3 = 3 , τ 4 = 2 , τ 5 = 1 with the system parameters 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 19
N
yˆ N = ∑ yn
(61)
n =1
where yn is the n-th 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 3rd 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.
Figure 2 Volterra series approximation of the output of system (45) with a parameter setting m = 1 , c = 1.4 , k1 = 1 , and k3 = 4 Using the new algorithm, the obtained n -th term yn ( t ) in the Adomian decomposition is the n -th order 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 obtaind 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 20
equation (10). The illustrative example shows that the processes to calculate the Volterra series are straightforward and easy to carry out.
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 is 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 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 dynamic 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 dynamic systems through applying some simple mathematical operations. Furthermore, the generality of the Adomain 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.
Acknowledges
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.
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