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Ind. Eng. Chem. Res. 2009, 48, 3052–3058
Incremental Modeling of Nonlinear Distributed Parameter Processes via Spatiotemporal Kernel Series Expansion Han-Xiong Li* and Chenkun Qi Department of Manufacturing Engineering & Engineering Management, City UniVersity of Hong Kong, Kowloon, Hong Kong, China
In this article, an incremental modeling approach is proposed to model nonlinear distributed parameter systems, with the help of the newly constructed spatiotemporal Volterra kernels. The complex spatiotemporal process is first decomposed into a series of spatiotemporal kernels, upon which the time-space separation can be further conducted with the spatial Karhunen-Loe`ve and temporal Laguerre basis function expansions. These two decompositions can gradually separate the nonlinear time/space coupled dynamics. Finally, the kernels in the spatiotemporal model are estimated from the experimental data incrementally, which can easily achieve satisfactory modeling performance. Simulations of two transport-reaction processes demonstrate the effectiveness of the proposed modeling approach. 1. Introduction Most physical and chemical processes, such as fluid processes,1 thermal processes,2 and convection-diffusion-reaction processes,3 are nonlinear distributed parameter systems (DPSs) because the inputs and outputs vary both spatially and temporally. They belong to a class of infinite-dimensional systems. In practice, finite- or low-dimensional process models are often required for many applications, such as system analysis, dynamics prediction, and control design. First-principles modeling based on physical or chemical knowledge of a process will often result in a set of partial differential equations (PDEs). Because of their infinite-dimensional nature, the PDEs are often reduced into a set of finite-dimensional ordinary differential equations for control design4 and dynamic optimization5 using the Galerkin method6 or the approximate inertial manifold method.3,7 However, this physics-based model might not be applicable in many practical situations because of incomplete process knowledge and complex boundary conditions. Then, parameter estimation or system identification from experimental data has to be used. The parameter estimation problem8-10 for known PDE structures is not considered here. The identification of PDE models has been studied.11-13 After recovering the PDE model from the spatiotemporal data, model reduction methods are still needed for practical applications. This article focuses on developing a low-dimensional nonlinear spatiotemporal modeling approach for unknown nonlinear DPS control. At present, finite-dimensional spatiotemporal model identification can be classified into local and global approaches. The local method assumes that the local dynamics is unchanged at different spatial locations and is determined by the neighborhood of the identified spatial location. Utilizing measurements in small spatiotemporal neighborhoods, local models can be established based on identification theories of the lattice dynamical system.14-17 Even though the local model at each spatial location might be simple, the global model in the overall spatial domain is of high dimensionality because it is determined by the number of spatial locations, which is often large. The idea of the global method comes from Fourier series expansion. A spatiotemporal variable (e.g., temperature) can be * To whom correspondence should be addressed. Tel.: 85227888435. Fax: 852-27888423. E-mail:
[email protected].
approximated by a finite number of spatial basis functions, and the corresponding temporal coefficients contain all essential information about the dynamics. In this way, the time-space coupled dynamics can be separated. Some identification methods, such as the nonlinear autoregressive with exogenous input (NARX) model with local spline basis functions,18 neural networks with global Karhunen-Loe`ve basis functions,19,20 and the Wiener model with global Karhunen-Loe`ve basis functions,39 have been studied. Local spline basis functions are not optimal in the sense that the model dimension is not the lowest given the modeling accuracy. Fortunately, the Karhunen-Loe`ve (KL) decomposition, also called proper orthogonal decomposition or principal component analysis,1,3,21,22 is one of the most popular approaches to finding principal spatial modes for the time-space separation. Among all linear expansions, the KL expansion is the most efficient in the sense that, for a given approximation error, the number of KL bases required and the model dimension are minimal. However, the neural KL method leaves difficulties for temporal model derivation and identification that could be difficult for complex spatiotemporal dynamics. Modern physics claims that the universe can be decomposed through tiny particles into the smallest string. Under this principle, any system dynamics can be decomposed first into smaller units (kernels) and then into bases. The modeling complexity will be gradually reduced through these two decompositions. A new spatiotemporal dynamics modeling methodology is thus proposed to realize this philosophy by extending traditional Volterra kernel series.23-26 First, a spatial domain is added to the traditional Volterra kernel that gives the kernel three-domain (3D) features. Because the Green’s function model (linear impulse response model)22,27-30 is a linear spatiotemporal system, the proposed model can also be considered as a high-order extension of Green’s function model. Second, to reduce the model dimension and the parametrization complexity, the spatiotemporal kernels are expanded onto spatial KL basis functions and temporal Laguerre basis functions with unknown coefficients. Finally, through incremental construction of these 3D kernels, this approach can satisfactorily model the complex nonlinear spatiotemporal system. This spatiotemporal Volterra model is useful for the model-based control of nonlinear distributed parameter processes because it is a direct extension of the linear impulse response model commonly used in process control.22,28,31,32
10.1021/ie801184a CCC: $40.75 2009 American Chemical Society Published on Web 01/29/2009
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3053
Figure 1. Input-output mapping of a 2D kernel model.
Figure 2. Input-output mapping of a 3D kernel model.
is constructed as in eq 2, with ζ1,..., ζr, and x as spatial locations defined in the space Ω R
yˆ(x, t) )
∑∫
Ω
r)1
t
···
∫∑ Ω
t
∑ g (x, ζ ,...,ζ , τ ,...,τ )
···
r
τ1)0
1
τr)0
r
1
r
r
∏ u(ζ , t - τ ) dζ
This article is organized as follows: The construction of spatiotemporal kernels and the modeling methodology are described in section 2. In section 3, the implementation of nonlinear 3D kernel modeling is presented. Section 4 contains two simulation examples. Finally, a few conclusions are presented in section 5. 2. Construction of Spatiotemporal Kernels and Modeling Methodology The traditional Volterra series, which is a two-domain (2D) model, is popularly used to approximate nonlinear temporal systems, as shown in Figure 1.24,25 For example, the dynamics of a fading memory nonlinear system (FMNS), whose dependence on past inputs decreases rapidly with time, can be modeled by the Volterra series to any desired accuracy.23 The Rth-order Volterra model can be defined to have R kernels as shown in eq 1 (see the Appendix for detailed formulation) R
t
∑∑
r)1 τ1)0
t
···
∑
τr)0
r
gr(τ1,...,τr)
V
(2)
Now, the rth-order 3D kernel gr denotes the influence of the input u at locations ζ1,..., ζr and times t - τ1,..., t - τr on the output yˆ at location x and time t. For easy understanding, a simple example of a 3D kernel model for R ) 2 is given by
Figure 3. 3D kernel modeling philosophy.
yˆ(t) )
V
V
V)1
∏ u(t - τ ) V)1
V
(1)
where gr is the rth-order kernel, which denotes the influence of the input u at times t - τ1,..., t - τr on the output yˆ at time t and R is the number of the highest order. It is obvious that this 2D model (eq 1) is not able to deal with a spatiotemporal system. To reconstruct spatiotemporal dynamics, a 3D kernel is needed to have the spatiotemporal features. It can be developed by adding an extra spatial domain to the original 2D kernel. Then, a spatiotemporal Volterra model
t
yˆ(x, t) )
∫ ∑ g (x, ζ , τ ) u(ζ , t - τ ) dζ + Ω
1
1
1
1
1
1
τ1)0
t
t
∫ ∫ ∑ ∑ g (x, ζ , ζ , τ , τ ) u(ζ , t - τ ) u(ζ , t - τ ) dζ dζ Ω
Ω
2
1
2
1
2
1
1
2
2
1
2
(3)
τ1)0 τ2)0
For simplicity, we define the output corresponding to the rthorder 3D kernel as yˆr(x, t) )
∫
Ω
t
···
∫∑ Ω
τ1)0
t
···
r
∑ g (x, ζ ,...,ζ , τ ,...,τ )∏ u(ζ , t - τ ) dζ r
τr)0
1
r
1
r
V)1
V
V
V
(4) The 3D kernel model in eq 2 is actually a spatial extension of the traditional 2D kernel model (eq 1). With the spatiotemporal information provided by multiple actuators and sensors in the spatial domain, this model is capable of constructing the spatiotemporal dynamics because of its inherent spatiotemporal nature as illustrated in Figure 2. In practice, a few kernels could be sufficient to provide a satisfactory approximation for most spatiotemporal systems. The modeling capability will improve as the number of kernels increases, and the model complexity will increase at the same time. Because the kernel order R is not easily determined by trial and error, an incremental kernel construction is suggested to achieve a balance between model accuracy and model complexity. Then, a satisfactory model can be obtained more easily and effectively.
3054 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
Figure 4. Incremental construction of 3D kernel series.
The philosophy of the 3D Volterra modeling approach is shown in Figure 3. The nonlinear spatiotemporal system is first decomposed into spatiotemporal kernels, upon which the time-space separation is carried out using spatial KL basis functions and temporal Laguerre basis functions. These two decompositions gradually separate the nonlinear time/space coupled dynamics. To achieve satisfactory modeling performance, these kernels can be easily reconstructed through an incremental approach as shown in Figure 4.
N,L is the system output measured at N spatial where {y(xi,t)}i)1,t)1 locations and L lengths of time. Note that the predicted output, N,L {yˆ1(xi,t)}i)1,t)1, of the first-order kernel can be estimated from M,L at M spatial eq 7 using the pointwise input data {u(xi,t)}i)1,t)1 locations; then, expression 8 can be minimized using a linear optimization method. Then, the prediction error of the firstorder kernel, e1(x,t) ) y(x,t) - yˆ1(x,t), is used as the new desired output for training of the second-order kernel. Finally, e2(x,t),..., eR-1(x,t) can be obtained by the same approach for training of the rest of kernels up to Rth order. The kernel order R will increase until a satisfactory performance is achieved. If M and N are very large, then the optimization of expression 8 could involve a complex computation. In this case, it can be approximately solved in the temporal domain. Using the Galerkin method,3,6 the time/space separation of the input and output can be further performed. Projecting eq 7 onto the output n leads, in this case, to the expression basis functions {φi(x)}i)1
n
m
Ω
h
r
i)1
h
Ω
l
∑
gr(·) ) m
m
∑∑
l
l
r
∑ ∑ ··· ∑
···
i)1 j1)1
jr)1 k1)1
kr)1
θij(r)1...jrk1...kr
∏ φ (x) ψ (ζ ) φ (τ ), i
js
s)1
s
ks
s
r ) 1,..., R (5) (r)
where θij1...jrk1...kr ∈ R (i ) 1,..., n; j1,..., jr ) 1,..., m; k1,..., kr ) 1,..., l) are unknown constant coefficients of the rth-order kernel and n, m, and l are the dimensions of the corresponding bases. For example, the second-order kernel is expanded as g2(·) ) n
m
m
l
(2) ij1j2k1k2φi(x)
i)1 j1)1 j2)1 k1)1 k2)1
ψj1(ζ1) ψj2(ζ2) φk1(τ1) φk2(τ2) (6)
Note that r
r
r
Π js Π kV ) Π jsks
s)1 V)1
s)1
Then, substituting eq 5 into eq 4 gives
∫
Ω
t
···
∫∑ Ω
t
n
m
m
l
∑ ∑∑ ∑ ∑ ···
···
···
τ1)0
l
τr)0 i)1 j1)1
jr)1 k1)1
r
∑θ
kr)1
r
θij(r)1...jrk1...kr
∏V
jsks(t),
s)1
h ) 1,..., n
t
Vjk(t) )
∑ u (t - τ) φ (τ), j
k
τ)0
uj(t - τ) )
∫ ψ (ζ) u(ζ, t - τ) dζ j
Ω
(j - 1,..., m;k ) 1,..., l)
n are orthonormal, this results in Because the bases {φi(x)}i)1 an n-dimensional model
m
yˆri(t) )
m
q
q
r
∑ ···∑ ∑ ··· ∑ θ
j1)1
jr)1 k1)1
kr)1
(r) ij1...jrk1...kr
∏V V)1
jVkV(t),
i ) 1,..., n (9) (1)
where yˆir(t) ) ∫Ωyˆr(x,t) φi(x) dx. Now parameters θij1jrk1 of the first-order kernel can be estimated with the least-squares method for solving the minimization
l
∑∑∑ ∑ ∑θ
yˆr(x, t) )
···
jr)1 k1)1
where
Motivated by Fourier expansion, the spatiotemporal kernels are assumed to be expanded onto spatial output bases {φi(x)}ni)1, m l , and temporal bases {φi(t)}i)1 as spatial input bases {ψi(x)}i)1 follows n
l
i
j1)1
kr)1
3. Implementation of Nonlinear 3D Kernel Modeling
m
∫ φ (x) yˆ (x, t) dx ) ∑ ∫ φ (x) φ (x) dx ∑ · · · ∑ ∑
(r) ij1...jrk1...kr
∏ φ (x) ψ (ζ ) φ (τ ) u(ζ , t - τ ) dζ i
s)1
js
s
ks
s
s
s
s
(7)
The kernels can be estimated using an incremental identification algorithm as shown in Figure 4. First, the unknown parameters θ(1) ij1k1 of the first-order kernel are estimated by minimizing the cost function N
min
L
∑ ∑ |y(x , t) - yˆ (x , t)|
2
i
i)1 t)1
1
i
(8)
L
min
∑ |y (t) - yˆ (t)| , i
i 1
2
i ) 1,..., n
(10)
t)1
where yi(t) ) ∫Ωy(x,t) φi(x) dx. Because m and n can be smaller than M and N, the optimization in expression 10 can be solved more easily than that in expression 8. The remaining kernels can be estimated in a similar way. In this study, the bases are designed as follows: n are identified at N (1) The spatial output bases {φi(x)}i)1 spatial locations using the Karhunen-Loe`ve decomposition1,3,21,22 N,L from the system output y(xi,t)}i)1,t)1 . Then, one can interpolate the eigenfunctions to locations where the data are not available. Because these bases can be ordered according to their importance to the system,1 only the first few spatial bases can be used to represent the dominant spatial modes. This is why we chose the Karhunen-Loe`ve method. The small value of n is determined using the energy method.1 m (2) The spatial input bases {ψi(x)}i)1 are often determined from the a priori physical knowledge about the distribution of the control action in the spatial domain. Approximation using other functions (e.g., splines) can also be used when the a priori knowledge is not available. The dimension m is just the number of independent inputs.
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3055 l {φi(t)}i)1 33
(3) The temporal bases can be selected as oneparameter Laguerre functions, two-parameter Kautz functions,34 or generalized multiparameter orthonormal basis functions35 according to the system complexity. Here, the Laguerre functions φi(t) } √2ξ
di-1 i-1 -2ξt eξt (t e ), (i - 1)! dti-1
i ) 1, 2,..., l;ξ > 0
were chosen for the development, because of their simplicity of use (only one adjustable time scaling factor ξ). Of course, ξ should be properly designed to achieve significant modeling performance. The optimal selection method is proposed for the temporal kernel system.36 For this nonlinear spatiotemporal kernel modeling, the time scaling factor ξ and the dimension l are selected using the cross-validation technique. Note that one advantage of the expansion onto spatial Karhunen-Loe`ve basis functions and temporal Laguerre basis functions is that the number of parameters to be estimated in the spatiotemporal Volterra model can be small, which will largely reduce the modeling complexity. Moreover, spatial Karhunen-Loe`ve basis functions can lead to a low-dimensional model, which is very attractive for practical control. The incremental kernel modeling algorithm can be summarized as described in algorithm 1: Algorithm 1. Step 1: Compute the spatial output bases n by the Karhunen-Loe`ve method from the measured {φi(x)}i)1 N,L . output {y(xi,t)}i)1,t)1 Step 2: Determine the spatial input bases {ψi(x)}mi)1 and select l as Laguerre functions. Set y(x,t) the temporal bases {φi(t)}i)1 as the desired output and R ) 1. (r) Step 3: Set r ) R. Obtain the unknown parameters θij1...jrk1...kr of the rth-order kernel using the least-squares estimation as in expression 8 or 10. Step 4: Simulate the model in eq 2 with R kernels. If satisfactory performance is achieved, the modeling ends. Otherwise, set the error of the Rth-order kernel, er(x,t) ) y(x,t) - yˆr(x,t), as the new desired output for training of the next kernel. Set R ) R + 1 and repeat steps 3 and 4. Remark 1. The dimension of this spatiotemporal Volterra model could be further reduced with nonlinear model reduction methods, e.g., the singular perturbation-based approximate inertial manifold method.3,7 Using this model, some control design methods, e.g., model predictive control developed for known PDEs,37 could be applied to work for unknown processes. Because unmodeled dynamics often exist, robust control design might be useful3 because the control performance and stability might depend on the obtained model for the system. 4. Numerical Simulations 4.1. Catalytic Rod. The catalytic rod reactor is a typical convection reaction process in the chemical industry. The mathematical model that describes the spatiotemporal evolution of the rod temperature consists of the following parabolic partial differential equation (PDE)3 ∂y(x, t) ∂2y(x, t) ) + βT(x)[e-γ⁄(1+y) - e-y] + βu[u(x, t) - y(x, t)] ∂t ∂x2 subject to the boundary and initial conditions y(0, t) ) 0,
y(π, t) ) 0
y(x, 0) ) y0(x) where y(x,t), u(x,t), βT, βu, and γ denote the temperature in the reactor, the temperature of the cooling medium, the heat of
Figure 5. Measured output.
Figure 6. Predicted output of the second-order model.
reaction, the heat-transfer coefficient, and the activation energy, respectively. The process parameters are often set as βT ) 50,
βu ) 2,
γ)4
The temperature u(x,t) of the cooling medium can be controlled by four actuators, u(t) ) [u1(t), · · · , u4(t)]T with the spatial distribution ψ(x) ) [ψ1(x), · · · , ψ4(x)], i.e., u(x,t) ) ψ(x) u(t), where ψi(x) ) H[x - (i - 1)π/4] - H(x - iπ/4) [H( · ) being the standard Heaviside function] and ui(t) ) 1.1 + 10 sin(t/10 + i/10) (i ) 1,..., 4). In the simulation, the finitedifference solution of the PDE with a sufficient number of discretization points is assumed to be the true output of the system. Both the finite-element method and the finite-difference method provide accurate solutions of the PDE and similar modeling results in our case; for simplicity, finite-difference method is used here. Twenty-two sensors are used for output measurement. The sampling time ∆t is 0.01, and the simulation period is 5. The incremental modeling algorithm is used to establish the 3D kernel model. The detailed formulation can be easily obtained based on algorithm 1 and section 3. For more details on the Karhunen-Loe`ve method, Laguerre functions, and leastsquares method, see related references.3,33,38 Numerical computation shows that the first four Karhunen-Loe`ve spatial bases can capture the dominant behavior of the system. The temporal bases φi(t) are chosen as Laguerre functions with the time scaling factor ξ ) 4.05 and the dimension l ) 3. The measured output y(x,t) is shown in Figure 5. The predicted output yˆ(x,t) and the prediction error e(x,t) ) y(x,t) - yˆ(x,t) of the second-order model with the first two 3D kernels are presented in Figures 6 and 7, respectively. It is obvious that the second-order model can reproduce the spatiotemporal dynamics of the original system very well. For this case, the first two kernels are enough for a good approximation because the higher-order kernel has a
3056 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
Figure 7. Prediction error of the second-order model. Figure 9. Measured output.
Figure 8. SNAE(t) of the first-, second-, and third-order models.
Figure 10. Predicted output of the second-order model.
lower influence on the modeling performance, as shown in Figure 8, where SNAE(t) ) [∫|e(x,t)| dx]/(∫ dx) is the spatial normalized absolute error. Note that, as shown in Figures 7 and 8, the model error is enlarged during the initial period. This is because the initial condition of the system is not very close to the working point corresponding to the given input signals. In general, for a nonlinear system, we can obtain an accurate nonlinear model only at a certain working point. The model error will increase when the system state is moving away from the working point. To achieve a similar performance, about 22 local models or 2D kernel models are required because the spatial domain is discretized into 22 spatial points. 4.2. Tubular Reactor with Chemical Reaction. The tubular reactor is a typical convection-diffusion-reaction system. The following equations provide a dimensionless model to describe a nonlinear tubular chemical reactor with both diffusion and convection phenomena and a nonlinear heat-generation term22
Le, Da, γ, η, µ, Ti, and Ci denote the Peclet number (energy), Peclet number (mass), Lewis number, Damko¨hler number, activation energy, heat of reaction, heat-transfer coefficient, inlet temperature, and inlet concentration, respectively. This system of equations can be classified as nonself-adjoint, parabolic PDEs with Neumann boundary conditions. The two PDEs are coupled, and for practical reasons, only temperature measurements are available. The parameter values used in this study are
1 ∂2T ∂T 1 ∂T 1 ) + ηC exp γ 1 + ∂t Peh ∂x2 Le ∂x T µ[Tw(x, t) - T(x, t)]
[(
)]
1 ∂2C ∂C 1 ∂C ) - Da · C exp γ 1 ∂t Pem ∂x2 ∂x T
[(
)]
subject to the boundary conditions
{ {
∂T ) Peh[Ti(t) - T(x, t)] ∂x x ) 0, ∂C ) Pem[Ci(t) - C(x, t)] ∂x ∂T )0 ∂x x ) 1, ∂C )0 ∂x The state variables, T and C, are the dimensionless temperature and concentration, respectively. The parameters Peh, Pem, -
Peh ) 5.0, Pem ) 5.0, Le ) 1.0, Da ) 0.875, γ ) 15.0, η ) 0.8375, µ ) 13, Ti ) 1, Ci ) 1 . The wall temperature, Tw, is the manipulated variable, for which there are three actuators distributed along the length of the reactor, i.e., u(x,t) ) ψ(x) u(t), where the temporal input is u(t) ) [u1(t), · · · , u3(t)]T with ui(t) ) 1.0 + 0.2 sin(t/10 + i/10) and the spatial distribution is ψ(x) ) [ψ1(x), · · · , ψ3(x)] with ψi(x) ) H[x - (i - 1)π/3] - H(x - iπ/3). There are 61 temperature sensors located at different positions along the reactor length. The sampling time ∆t is 0.0005, and the simulation period is 0.4. The incremental modeling algorithm is used to obtain the 3D kernel model with up to three kernels. The first five Karhunen-Loe`ve spatial bases contain most of the dominant behavior of the system. The temporal bases φi(t) are chosen as Laguerre functions with the time scaling factor ξ ) 5.05 and the dimension l ) 5. The measured output y(x,t) is shown in Figure 9. The predicted output yˆ(x,t) and prediction error e(x,t) ) y(x,t) yˆ(x,t) of the second-order model with the first two 3D kernels are illustrated in Figures 10 and 11, respectively, which show very good agreement between the original system output and the model output. As shown in Figure 12, the third-order kernel has less of an effect on the modeling performance. To obtain
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3057 r
Π u(t - τV) ) u(t - τ1) u(t - τ2) · · · u(t - τr)
V)1
Then, the kernels can be expressed as t
yˆ1(t) )
∑ g (τ ) u(t - τ ) 1
τ1)0 t
yˆ2(t) )
1
1
t
∑ ∑ g (τ , τ ) u(t - τ ) u(t - τ ) 2
1
2
1
2
τ1)0 τ2)0
l t
Figure 11. Prediction error of the second-order model.
yˆR(t) )
t
t
∑ ∑ ∑ g (τ , τ , · · · , τ ) u(t - τ ) u(t - τ ) · · · u(t - τ ) ···
τ1)0 τ2)0
R
1
2
R
1
2
R
τR)0
Literature Cited
Figure 12. SNAE(t) of the first-, second-, and third-order models.
similar performance, 61 local models or 2D kernel models are required in this case. 5. Conclusions Modeling of nonlinear spatially distributed dynamical systems is very important in physical and chemical engineering. Based on the traditional Volterra kernel series, an incremental 3D kernel series has been constructed for modeling complex nonlinear spatiotemporal systems. This nonlinear spatiotemporal kernel series model can be considered as a spatial extension of the traditional Volterra series and a high-order extension of the Green’s function model. The kernels are functions of time and space variables that can be estimated incrementally in the spatiotemporal or temporal domain using linear optimization. The simulations demonstrate the effectiveness of the proposed modeling method and its potential application to a wide range of nonlinear spatiotemporal systems. Acknowledgment The authors would like to thank the associate editor and the anonymous referees for their valuable comments and suggestions. The work was partially supported by grants from RGC of Hong Kong SAR (CityU: 116406 and 117208), and a grant from NSF China (50775224). Appendix: Detailed Formulation of Volterra Model Equation 1 For easy understanding, the Rth-order Volterra model (eq 1) can be expressed as yˆ(t) ) yˆ1(t) + yˆ2(t) + · · · + yˆR(t) where the terms yˆ1(t), yˆ2(t),..., yˆR(t) correspond to the first-order, second-order,..., Rth-order kernels, respectively. Note that
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ReceiVed for reView August 1, 2008 ReVised manuscript receiVed December 10, 2008 Accepted December 19, 2008 IE801184A
