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Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements
ISA Transactions xxx (xxxx) xxx
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Research article
Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements Mohammad-Reza Rahmani a , Mohammad Farrokhi b ,
∗
a
School of Electrical Engineering, Iran University of Science and Technology, Tehran 1684613114, Iran School of Electrical Engineering, Center of Excellence for Modeling and Control of Complex Systems, Iran University of Science and Technology, Tehran 1684613114, Iran b
highlights • • • •
Nonlinear dynamic system identification based on continuous-time fractional-order Hammerstein state-space model is proposed. Frequency-domain identification is used to identify the structural parameters. Joint state and parameter estimation is proposed. Lyapunov stability theory is used to develop time-domain algorithm.
article
info
Article history: Received 11 May 2017 Received in revised form 4 June 2019 Accepted 14 June 2019 Available online xxxx Keywords: Hammerstein model State estimation Fractional-order systems Neural networks
a b s t r a c t This paper introduces a continuous-time fractional-order Hammerstein state-space model with a systematic identification algorithm for modeling nonlinear dynamic systems. The proposed model consists of a radial-basis function neural network followed by a fractional-order system. The proposed identification scheme is accomplished in two stages. The structural parameters of the fractional-order system (i.e. the values of the fractional order and the degree of the denominator in the fractionalorder system) are estimated in the frequency domain. Then, the synaptic weights of the radial-basis function neural network and the coefficients of the fractional-order system are determined in the time domain via the Lyapunov stability theory, which guarantees stability of the given method and its convergence under a mild condition. Three examples are provided to show the effectiveness of the proposed method. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction The Hammerstein model was presented by the same-name German mathematician in 1930 [1]. The Hammerstein model consists of a Nonlinear Static Block (NSB) followed by a Linear Dynamic Block (LDB). It is a simple block-oriented model, which has been proven to effectively model a wide class of biological/engineering systems [2,3]. The NSB of the Hammerstein model is represented by different models such as polynomial functions [4], the weighted sum of known basis functions [5], Neural Networks (NNs) [6], and fuzzy models [7]. There is a vast literature on the discrete-time Hammerstein model, where the LDB is either Infinite Impulse Response (IIR) filter [8] or Finite Impulse Response (FIR) filter [9], while much less attention has been paid to the identification of Hammerstein models with continuous-time LDB. Continuous-time models can mimic the ∗ Corresponding author. E-mail addresses: [email protected] (M.-R. Rahmani), [email protected] (M. Farrokhi).
dynamic behavior of the dynamic system more effectively than the discrete-time models [10,11]. Moreover, since state-space models provide deep insights into the nature of the real-world systems, Hammerstein state-space modeling have attracted many researchers recently [12–14]. State-space models play an important role in system identification. Some of the distinguishing features of the state-space-based identification over the commonlyused input–output regression-based one are: (1) by the use of state-space realization of a system, the well-known Lyapunov stability theory can be conducted to achieve stable identification laws, which ensure boundedness of the estimated parameters [15, 16] and (2) provided that the system states are known, by the use of state-space models for the purpose of continuous-time system identification, the need for filtering techniques such as integral transform used in [17] is removed. However, the problem in the state-space modeling is due to the fact that in practice usually state variables of the system are unmeasurable, and only the system input–output pairs are available. To overcome this difficulty, two strategies have been suggested in literature: (1) in [16,18–20], the Hammerstein state-space modeling has been
https://doi.org/10.1016/j.isatra.2019.06.015 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
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accomplished under the assumption that the system states are known a priori ; (2) Hierarchical methods based on the cost function minimization problem, which commonly bring the system back to its discrete-time input–output representation [12,21]. Recently, two different strategies for the FOH state and parameter estimation has been proposed in the literature. The first strategy is based on the output-derivatives estimation of the FOH, wherein the system states are estimated from successive differentiating of the output signal [22]. However, the main drawback of this method is its inherent high sensitivity to the measurement noises. The second strategy is the observer-based state-space identification using an auxiliary model-based observer [23]. Although in [23] simultaneous state and parameter estimation is performed without the need for signal filtering techniques, the drawbacks of this method are: (1) it is sensitive to the initial conditions and (2) the overall stability of the algorithm has not been proven. In the present paper, on the other hand, the time-domain identification algorithm is proposed such that it is insensitive to the initial conditions and the global stability of the identification is ensured by using the Lyapunov stability theory. Fractional calculus is a branch of mathematics wherein the order of the conventional differential equation is extended to become a fractional number [24]. Recently, with the growing power of computers, Fractional-Order Systems (FOSs), whose describing differential equation is of fractional order, has been used in many engineering fields. During the last several decades, many realworld systems have been well characterized and identified using the FOS [25]. The Fractional-Order Hammerstein (FOH) model is a generalization of the Hammerstein models with the LDB being of the fractional order. To the best of the authors’ knowledge, there exist a few researches about identification of the continuous-time FOH models [26–28] and the discrete-time FOH systems [29,30]. A challenging issue encountered in fractional-order modeling is the estimation of the fractional orders that has been discussed in [27,28]. In these references, iterative time-domain algorithms have been proposed to estimate the fractional order under the assumption that the degree of the FOH is known a priory. In [27], two methods have been suggested to identify the fractional order: (1) a one-dimensional grid search, where the parameters of the FOH are estimated iteratively in the time domain at every point in the grid of the fractional order and (2) the fractional order identification based on the so-called P-type order learning law. The convergence of the latter method is guaranteed for small values of the learning rate, which may make the algorithm time consuming. In [28], a nonlinear optimization problem must be solved to estimate the fractional order. In this method, the parameters of the FOH are identified in the time-domain via subspace identification techniques. Since in every step of the nonlinear optimization, the parameters are identified via subspace identification, which depends on the fractional order, it may suffer from long computational time. In [22,23], identification of the fractional order using the frequency domain linear optimization based on the Best Linear Approximation (BLA) technique [31] has been developed. The main advantages of the aforementioned frequency-domain estimations are that they provide more accurate estimation of the fractional-order systems and they require less computational time as compared to the time-domain identification schemes. Nevertheless, these methods may suffer from the drawback that the amplitude variations of the input signal should be small enough in order to estimate the parameters with sufficient accuracy. This restriction has been removed in [22,23] for continuous-time FOH using frequency-domain identification. Based on the aforementioned literature review, the main contributions of this paper are two-folds. First, the accuracy of the frequency-domain identification is improved by acquiring the nonparametric frequency response of the LDS via the concept of
state-space modeling. Then, in order to identify the state matrices of the FOH, an adaptive observer, where the state variables and the state matrices of the FOH are estimated simultaneously, is developed. Besides, the convergence analysis of the proposed method is carried out and overall stability of the time-domain algorithm is ensured using the Lyapunov stability theory. It will be shown in simulating examples that the proposed Hammerstein model outperforms the traditional Hammerstein models with IIR filter as the LDB. This paper is organized as follows. Section 2 reviews basic concepts and the identification problem. Section 3 presents the proposed identification algorithm. Section 4 provides the illustrative examples. Finally, Section 5 concludes the paper. 2. Preliminaries and problem statement This section presents basic concepts related to the fractionalorder Hammerstein model. First, the principals of the fractional calculus, which are exploited in this paper, are given. Then, the nonlinear dynamic system identification problem is formulated as the proposed fractional-order Hammerstein modeling. 2.1. Fractional-order systems The uniform formula of a fractional-order integral is defined as [32] 1
−p
a Dt f (t) =
Γ (p)
f (τ )
t
∫
(t − τ )1−p
a
dτ
(1) −p
where f (t) is an arbitrary integrable function, a Dt is the fractional integral of order p > 0 on [a, t ], and Γ (·) stands for the Gamma function. For an arbitrary real number α , the Caputo derivative of fractional order α is defined by [33] −([α ]−α+1)
Dαt f (t) = Dt
{
d[α ]+1 dt
}
f (t) [α ]+1
(2)
where [α] denotes the integer part of α . The Laplace transform of an α -order derivative of signal f (t) is defined as [32] L {Dα f (t)} = sα F (s) −
n−1 ∑
sα−k−1 f (k) (0),
(n − 1 < α ≤ n)
(3)
k=0
where s, F (s), and f (k) (0) stand for the Laplace operator, the Laplace transform, and the kth derivative of f (t) at t = 0, respectively. A fractional-order linear system is represented by an equation or a system of differential equations characterized by real-derivative orders as [34] an y(t) + an−1 Dβ1 y(t) + · · · + Dβm y(t) = b0 Dα0 v (t) + b1 Dα1 v (t)
+ · · · + bn Dαn v (t)
(4)
where v (t) and y(t) are the input and output signals of the FOS, respectively, and differentiation orders β1 , . . . , βm and α0 , . . . , αn may be non-integer positive numbers. When all the differentiation orders are integer powers of the same fractional number, the system is called commensurate fractional-order system. In this manuscript, commensurate all-pole FOS of order α expressed by the following transfer function is used as the LDB in the Hammerstein model: 1 Y (s) = (5) G(s) = ∑n−1 V (s) snα + k=0 an−k skα The state-space realization of transfer function (5) is given as follows [34]
{
Dα x(t) = Ax(t) + bv (t) y(t) = cT x(t)
(6)
Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
M.-R. Rahmani and M. Farrokhi / ISA Transactions xxx (xxxx) xxx
where x(t) = [x1 (t) x2 (t) vector of state variables and
⎡ − a1 ⎢−a2 ⎢−a 3 A=⎢ ⎢ . ⎣ ..
1 0 0
− an [
0
c= 1
0 1 0
.. .
0
.. .
0 0
xn (t)]T is an n-dimensional
0 0 0⎥ ⎢0⎥ ⎢ ⎥ 0⎥ ⎥ , b = ⎢0⎥ , ⎥ ⎢.⎥ .. ⎦ ⎣.⎦
··· ···
⎤
1
..
. ··· ···
···
0
⎡ ⎤
.
.
0
1
]T
.
(7)
2.2. Fractional-order Hammerstein modeling
N ∑
wi ϕi (u(t)) = wT φ(u(t)) (8) i=1 ) ( 2 2 ϕi (u(t)) = exp − (u(t) − µi ) /σi [ ] w · · · wN T and φ(u(t)) where w = = [ ] T1 ϕ1 (u(t)) · · · ϕN (u(t)) are the weight and the radial-basis function vectors, respectively, and µi and σi represent centers and widths of ϕi (u(t)). It should be noted that the weighted sum of any nonlinear function could be used in replacement of the RBF. However, the RBF is selected as the NSB since it is a well-known function approximator, which is linear in parameters, i.e., its output weights. Thus, the state-space representation of the proposed fractional-order Hammerstein system can be given as Dα x(t) = Ax(t) + Bφ(u(t)), y(t) = x1 (t).
with B = bwT
Using the input–output data acquired from the system −1)ts {ϕ1 (u(t)), y(t)}t(M with sampling time ts at which the input– =0 output data are measured, the fractional order (α ) and the frac-
tional degree (n) of the LDB in the fractional-order Hammerstein model are estimated in the frequency domain using the data set {Φ1 (k), Y (k)}M k=1 , which is the Discrete Fourier Transform (DFT) (M −1)t of {ϕ1 (u(t)), y(t)}t =0 s . In this regard, a two-stage identification approach is used that includes the estimation of the nonparametric frequency response of the transfer function Gw (s) = w1 G(s) and then parameterization of the computed Gw (s). In the first √ step, the nonparametric frequency response Gw (jω), where j = −1, is estimated at different frequencies ωk (ωk = 2π k/Mts , k ∈ {1, . . . , M }) as follows [35]: Y (k) = Gw (jωk )Φ1 (k) + P(jωk )
The NSB of the proposed fractional-order Hammerstein model is considered to be of RBF type, which is given by
{
element of the set {1, . . . , N }. Herein, without loss of generality, i is set to one. Next, using the auxiliary signal-based Lyapunov stability theory, the weight vector (w) as well as the state matrix (A) are estimated on-line in the time domain. These steps will be explained in the followings. 3.1. Frequency-domain identification
It is well-known that a transfer function can be represented by different state-space realizations such as controllable form, observable form, etc. However, since in this paper, the identification problem is to estimate the parameters of the transfer function in (5) and the weights of RBF in (8), the state-space representation type in (5) is not important for the proposed method. Henceforth, we have selected this rare but effective representation of the state space just for the sake of time-domain identification of the Hammerstein system in (9) using the input–output data {u(t), y(t)}.
v (t) =
3
(9)
Therefore, the objective is to identify the fractional-order Hammerstein model described by (5) and (8) or equivalently, to estimate the parameters of Hammerstein state-space model in (9), including α , n, A, and w to mimic the input–output dynamic behavior of the nonlinear dynamic system at hand. This is accomplished using only the input–output measured data from the system. Herein, parameter N (the number of neurons in the hidden layer of the NN) is known or can be determined by trial and error. Moreover, the widths and centers of the RBF functions and in turn φ(u(t)) are selected such that the RBF covers the whole range of the input signal u(t).
where P(jωk ) contains the influence of the initial and final conditions of the experiment and tends to zero with respect to the main term Gw (jωk )Φ1 (k). To have one more degree of freedom and thereby to improve the identification efficiency, transfer function (5) is divided by an unknown constant a0 as follows Gn (s) =
G(s) a0
3. Identification of fractional-order Hammerstein model Using the input–output data obtained from the system, the identification algorithm is carried out in hybrid frequency-/timedomain. First, in the frequency-domain, the fractional order (α ) and the fractional degree (n) are estimated using the input– output pairs {ϕi (u(t)), y(t)}, where subscript i is an arbitrary
1
= a0
sn α
+ a0
∑n−1 k=0
an−k skα
(11)
In the second step, the objective is to estimate α and n via a grid search such that Gw (jωk ) is parameterized in the form of Gn (jω) within a predefined frequency interval. To do this, at first, a two-dimensional search space (α, n) ∈ { α, n| 0 < α ≤ 1, 1 ≤ n ≤ nmax }, where nmax is an arbitrary integer, is defined. Then, for any point on the search space, a constrained linear least squares is performed to identify the vector am = a0 [an an−1 · · · a1 1]T . Two constraints are considered for the optimization problem: (1) vector am must be real-valued and (2) the estimated transfer function Gn (jω) must be stable; i.e., the sα − poles of Gn (jω) have negative real parts. Consequently, for every point on the search space, the following constrained optimization problem: J = arg min (Ham − q)T (Ham − q)
a ⎧ m } ⏐= 0 ⎪ ⎨Im{a{ [ ]} n−1 ⏐ ∑ s.t. ⏐ n k ak p <0 ⎪ ⎩Real p ⏐⏐p ∈ roots p + k=0 ⎡ ⎤ 1 (jω1 )α · · · (jω1 )nα α nα · · · (jω2 ) ⎥ ⎢1 (jω2 ) H=⎢ .. .. ⎥ .. ⎣ .. ⎦, . . . .
1 Assumption 1. It is assumed that the system is a smooth timeinvariant dynamical system.
(10)
[ q=
(jωL )α 1
Gw (jω1 )
··· ···
(12)
(jωL )nα ]T 1 Gw (jωL )
should be solved to find am and the resultant cost function J. Eq. (12) shows that J is the sum of squares of errors between the nonparametric frequency response Gw (jω) and parametric one Gn (jω) at L different data points {ω1 , ω2 , . . . , ωL }, where n + 1 ≤ L ≤ M. Finally, the pair {α, n} that results in the minimum value of the cost function J is adopted as the estimated values for α and n.
Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
4
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As a remark on the identifiability, since the number of rows of matrix H in (12) is greater or equal to the dimension of the parameter vector am , for the perfect estimation of nonparametric frequency response Gw (jω) and the true values of α and n, the unknown vector am is estimated consistently using the cost function Jα,n = 0. Thus, if the number of the points in the 2-dimensional grid tends to infinity and the frequency-response estimation in (10) is accurate, the consistent estimates of α and n are obtained.
where P2 is an n × n − dimensional diagonal positive-definite matrix and
ˆ Dα w(t) = −P3 x˜ 1 (t)q(t), q(t) = q1 (t)
[
qj (t) =
ϕj (u(t)) , ∑ −2 kα + nk= 0 dn−k D
D(n−1)α
Once the structural parameters α and n are estimated, in the time-domain, the weight vector (w) and the state matrix (A) in (9) are identified via the Lyapunov stability theory. In this regard, the system is assumed to be of the form (9), whose state variables are immeasurable. Hence, the identification problem requires an adaptive observer for the fractional-order Hammerstein system in (9). First, the following lemmas should be considered that are needed in the sequel. Lemma 1. Let f (t) ∈ R be a continuous and derivable function. Then, for any t ≥ t0 t0 Dt
f (t) ≤ 2f (t)t0 Dt f (t),
∀α ∈ (0, 1] .
(13)
˜ φ(u) + r + a˜ x1 with B˜ = Bˆ − B, and a˜ = a − aˆ (19) Dα x˜ = Kx˜ + B Eq. (19) can be rewritten as an nth order scalar differential equation as Dnα x˜ 1 +
n ∑
˜ φ(u) + ki D(n−i)α x˜ 1 = w
n ∑ [
i=1
= D(n−1)α
D(n−i)α a˜ i x1 + ri
(
i=1
n ∑
[(
a˜ i x1 + ri
)
D(i−1)α
+
(20)
]
˜ φ(u) w nD(n−1)α
˜ =w ˆ − w. , w
Eq. (19) can be rewritten as
˜ Tq Dα x˜ = Kx˜ + d a˜ T z + w
(
n ∑
)
(21)
[(
a˜ i x1 + ri
)
D(i−1)α
+
˜ φ(u) w
]
nD(n−1)α
=
n ∑
di D(i−1)α
˜ Tq a˜ T z + w
[
Lemma 2. [Kalman–Yakubovich Lemma] For any vector [ ]T d = 1 d2 · · · dn and stable matrix K, if cT (sI − K)−1 d (I denotes identity matrix of appropriate dimension) is positive real, then there exist a positive-definite matrix P1 and a vector m such that [37]
By successive differentiation of (22), r = [r1 calculated as follows:
KT P1 + P1 K = −mmT , P1 d = c.
rk = −
(14)
In the fractional-order case, the transfer function cT (sI − K)−1 d in the above lemma is replaced with cT (sα I − K)−1 d. In the following theorem, the adaptive observer
ˆ φ(u(t)) + k − aˆ (t) x1 (t) + r(t), Dα xˆ (t) = Kxˆ (t) + B(t) yˆ (t) = xˆ 1 (t)
]
(15)
where r(t) = [r1 (t) · · · rn (t)]T is the auxiliary signal vector that will be defined later in (23), and
⎡ −k1 ⎢−k2 ⎢−k 3 K=⎢ ⎢ . ⎣ ..
1 0 0
.. .
−kn 0 ⎡ ⎤ k1
⎢.⎥ k = ⎣ .. ⎦ , kn
0 1 0
··· ··· 1
.. .
..
0 0⎥ 0⎥ ⎥ < 0, ⎥
⎤
.. ⎦ .
ˆ = B(t)
. ··· 0 ⎡ ⎤ aˆ 1 (t) ⎢ . ⎥ aˆ (t) = ⎣ .. ⎦ . aˆ n (t)
[
]
0 , ˆT w
Theorem 1. For the adaptive observer (15) of the system (9), limt →∞ x˜ (t) = 0 if the following updating laws are adopted for aˆ (t) ˆ and w(t): Dα aˆ (t) = P2 x˜ 1 (t)z(t), z(t) = z1 (t)
[
x1 (t)
(n−i)α
=
D D(n−1)α +
∑n−2 k=0
k−1 ∑
Dα a˜ i
dn−k Dkα
···
zn (t)
]T
, i ∈ {1, . . . , n}
with
n ∑
dj zj−k+i+1 +
j=k
n ∑
Dα a˜ i
i=k
k ∈ {2, . . . , n}
···
(22)
rn ]T can be
k−1 ∑
dj zj−k+i+1 ,
(23)
j=1
To obtain stable identification laws, consider the following Lyapunov function candidate: 1(
1 ˜ ˜ T −1 ˜ x˜ T P1 x˜ + a˜ T P− (24) 2 a + w P3 w 2 where P1 is a positive-definite matrix that satisfies (14). According to the Lyapunov stability theory for FOSs [38–40], if the identification laws are chosen such that the inequality Dα F < 0 holds, then the identification algorithm will be stable; i.e., limt →∞ x˜ (t) = 0. According to Lemma 1, Dα F satisfies the following inequality:
F =
0
(16)
i=1
]
r1 = 0,
Dα F ≤
is designed such that the prediction error x˜ (t) = xˆ (t) − x(t) tends to zero when the time t grows to infinity.
zi (t)
i=1
i=1
[
)]
if the following equality holds:
Proof. See [36].
{
(18)
j ∈ {1, . . . , N }
where P3 is an N ×N −dimensional diagonal positive-definite matrix.
i=1
α
with
Proof. From (9) and (15), by omitting the time index t for brevity, the error dynamic equation can be obtained as follows:
3.2. Time-domain identification
α 2
]T
qN (t)
···
)
1( α ) 1 ( α T) 1 α ˜ D x˜ P1 x˜ + x˜ T P1 D x˜ + a˜ T P− 2 D a 2 T −1 α ˜ P3 D w ˜ +w 2
(25)
Hence, the negativeness of the right-hand side of (25) implies that Dα F < 0 and this in turn, guarantees stability of the identification algorithm. Substituting (21) into the right-hand side of (25) yields Dα F ≤
1 2
˜ Tq x˜ T KT P1 + P1 K x˜ + x˜ T P1 d a˜ T z + w
(
)
(
)
1 α ˜ ˜ T −1 α ˜ + a˜ T P− 2 D a + w P3 D w
(26)
Now, if the following equations hold:
{
1 α ˜ a˜ T P− xT P1 da˜ T z, 2 D a = −˜
(27)
1 α ˜ T P− ˜ ˜ Tq w xT P1 dw 3 D w = −˜
then, (26) reduces to (17) Dα F ≤
1 2
( ) 1 x˜ T KT P1 + P1 K x˜ = − x˜ T Q1 x˜ < 0, 2
Q1 > 0
(28)
Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
M.-R. Rahmani and M. Farrokhi / ISA Transactions xxx (xxxx) xxx
5
since under the assumption that K is Hurwitz, for any positivedefinite matrix Q1 there exist a positive definite matrix P1 that satisfies KT P1 + P1 K = −Q1 [15]. Solving (27) for Dα a˜ and ˜ considering that system (9) is time invariant, leads to the Dα w following updating laws: Dα aˆ = P2 zx˜ T P1 d,
{
(29)
ˆ = −P3 qx˜ T P1 d Dα w
According to Lemma 2, equality x˜ T P1 d = x˜ 1 holds and thereby, the updating laws in (29) are reduced to those given by (17) and (18). From (28), it can be seen that x˜ = 0 and Dα F = 0 are equivalent and in turn, the updating laws in (17) and (18) result in limt →∞ x˜ (t) = 0. □ Theorem 2 (Convergence { } Analysis). If there exists no set of realvalued constants β i,j (i = 0, 1, . . . , n and j = 1, . . . , N) such that φ(u) in steady state is a solution of the following homogeneous differential equation: N n ∑ ∑ [ ] βi,j D(n−i)α ϕj (u) = 0
(30) 4. Simulating examples
j=1 i=0
ˆ are consistently identified. then, parameters aˆ and w T
T
˜ q Proof. It can be seen from (21) that when x˜ = 0, then a˜ z + w ˜ T q, the becomes zero. Using the Laplace transform of a˜ T z + w following relation can be obtained:
(
n ∑
) a˜ i D
(n−i)α
x1 (t) +
N ∑
˜ i ϕi (u)) = 0. (w
(31)
i=1
i=1
By expressing x1 in terms of u, (31) can be rewritten as follows:
(
n ∑
Fig. 1. System output y(t), model output yˆ (t) and error e(t) = y(t) − yˆ (t) for Example 1.
⎞ ( )⎛ N ) n ∑( ∑ ) (n−i)α (n − i) α ⎝ a˜ i D wj ϕj (u) ⎠ + ai D
i=1
j=1
(32)
(33)
The identification problem is to estimate vectors a = a1
j=1
or equivalently N n ∑ ∑ [(
a˜ i wj + ai w ˜ j D(n−i)α ϕj (u) = 0, a0 = 1, a˜ 0 = 0
)
]
Example 1 (Fractional-order Hammerstein System). Consider the system
⎧ 3 ∑ ⎪ ⎪ ⎪ ⎪ v = wi ϕi (u), wi = i, ⎪ ⎪ ⎪ i=1 ⎨ ) ( ϕi (u) =[ exp −0].32 (u[−](5i − 10))2 [, ] [ ] ⎪ ⎪ 5 − a1 1 0 a1 ⎪ ⎪ = , D0.7 x = x+ v, ⎪ ⎪ a2 10 − a2 0 1 ⎪ ⎩ y = x1 .
i=0
⎛ ⎞ N ∑ ( ) w ˜ j ϕj (u) ⎠ = 0, a0 = 1 ×⎝
The performance of the proposed fractional-order Hammerstein modeling is demonstrated using three simulating examples: (1) the stability and convergence properties of the time-domain identification algorithm is illustrated in Example 1; (2) the identification of a Continuous Stirred Tank Reactor (CSTR) is performed in Example 2, and (3) a real life nonlinear process, namely liquidsaturated steam heat exchanger is identified from its recorded input–output data.
j=1 i=0
which is of the form of (30) with βi,j = a˜ i wj + ai w ˜ j . Thus, (33) represents a condition upon φ(u) in the steady state; i.e., after the identification laws has forced x˜ to vanish. Two possibilities exist by considering the solution of (30): (1){either } the steady-state input φ(u) satisfies (30) for some values β i,j , or (2) coefficients { } β i,j are zero. Based on the assumption of the theorem, (1) cannot occur. Consequently, (2) is true. Since βi,j = a˜ i wj + ai w ˜ j, ˜ must be zero. This completes the proof. □ then a˜ and w Corollary 1. In the steady-state condition, the parameter estimates converge to their true values for every sinusoidal input u. Proof. On one hand, φ(u) is a Gaussian-function vector, which implies that it has infinite number of sinusoidal signals of distinct frequencies even if input u is a sinusoidal signal [41]. On the other hand, when α is set to one, which is the worst case, it can be deduced from (30) that φ(u) must have at least 0.5n + 1 distinct frequencies, which are not the solution of (30). Thus, φ(u) cannot be a solution of (30) for every sinusoidal input u unless a˜ = 0 ˜ = 0. When α is not equal to one, vector φ(u) that satisfies and w (30), is not Gaussian function of sinusoidal input unless a˜ = 0 ˜ = 0. □ and w
(34)
[
and w = w1
w2
[
w3
]T
a2
]T
using the measurements of u and y.
]T
[
]T
[
The parameters are selected as d = 1 3 and k = 6 8 such that cT (sα I − K)−1 d is a positive real transfer function. Then, the input–output filtered signals z and q are constructed using (17) and (18), respectively, as follows:
[ ] Z=
Z1 Z2
[ =T
s
0.7
L {ϕ1 (u)}
⎡
]
⎤
X1 , q = T ⎣L {ϕ2 (u)}⎦ , X1 L {ϕ3 (u)}
T =
1 s0.7
+3 (35)
Then, r is obtained using (23) and by trial and error. The suitable weighting matrices in (17) and (18) are determined as P2 = 107 I2 and P3 = 106 I3 . Following the procedure given in [42], the fractional-order operator D−0.7 is implemented and the sinusoidal input signal u(t) = 10(sin(t) + sin(2.5t)) is applied to the system in (34). The efficiency of the identification algorithm is measured in terms of the Root-Mean Square Error (RMSE), the Relative Error (RE), and the Variance Account For (VAF), which are defined as follows RMSE =
1 N
[ N ∑(
]1/2 i
i 2
yˆ − y
i=1
)
, RE =
[ ∑N ( i=1
yˆ i − yi
∑N ( i )2 i=1
y
)2 ]1/2 ,
Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
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Fig. 4. System/model output and the error for frequency-domain identification of CSTR.
Fig. 2. Real and estimated values of parameter a for Example 1.
Fig. 3. Real and estimated values of parameter w for Example 1. Fig. 5. (a) CSTR output, (b) the proposed modeling error and (c) the CFF-RLS method error.
Table 1 Performance indices of Example 2. Algorithms
Performance indices RMSE
RE
VAF
Linear modeling Hammerstein modeling in [43] Proposed modeling
0.7668 0.0013 4.8420e−04
0.0320 5.3243e−05 1.9852e−05
99.8852 99.9998 100
{ VAF = 100 max 0, 1 −
(
)}
var yˆ − y var (y)
(36)
where N, yi , and yˆ i are the number of data, the output of the system, and that of the model at the ith sample, respectively. In the case of perfect identification, the values of RMSE and RE are zero and the value of VAF is hundred. By constructing the adaptive observer in (15) using parameters aˆ and w ˆ , respectively obtained from (17) and (18), the results are shown in Figs. 1 to 3, where the stability and convergence properties of the time-domain algorithm can be observed. These figures show that the identified model mimics the input–output behavior of the system in (34) and the parameters a and w are identified consistently. Moreover, the performance indices RMSE=1.3925e−04, RE = 9.4599e−04, and VAF = 99.9996 show that the proposed identification algorithm identifies the system (34) with good accuracy.
Fig. 6. Input–output data used for the identification of the heat exchanger.
Example 2 (CSTR Modeling). The CSTR is a well-known process in chemical industries, wherein the material and energy balance equations in dimensionless variables, under the assumption that
Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
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7
[
in turn, d is set to be 1
]T
8
10
in order to reach a positive
real transfer function cT (sα I − K)−1 d. The input–output filtered signals z and q are constructed as follows: Z1 Z2 Z3
[ ] Z=
T =
L {ϕ1 (u)} s2 X 1 = T ⎣ sX1 ⎦ , q = T ⎣L {ϕ2 (u)}⎦ , X1 L {ϕ3 (u)}
⎡
⎤
⎡
⎤
1 s2
(40)
+ d2 s + d3
The weighting matrices are selected as P2 = 105 I3 and P3 = 104 I3 . The identification results for the proposed fractional-order Hammerstein modeling are given in Table 1, which represent significant improvements from the linear model given by (38). For the sake of comparison, the RBF-IIR Hammerstein model, whose LDB is in the form of the commonly-used IIR filter, is identified with the recently proposed method, namely, Changing Forgetting Factor RLS (CFF-RLS) [43]. The NSB of the RBF-IIR Hammerstein model is described by (39) while its LDB is given by Fig. 7. System/model output and the error for the heat exchanger identification.
y(t) +
8 ∑
aˆ i y(t − i) = v (t) +
i=1
there is no time delay in the recycle stream, are described by [44, 45] dx1 (t) dt dx2 (t) dt
= −µx1 (t) + Da (1 − x1 (t)) exp
(
γ x2 (t) γ + x2 (t)
)
= −(µ + β )x2 (t)
(37)
+BDa (1 − x1 (t)) exp
(
γ x2 (t) γ + x2 (t)
)
+ β u(t)
where the state variables 0 ≤ x1 (t) ≤ 1 and x2 (t) correspond to the conversion rate of the reaction and the dimensionless temperature, respectively, and u(t) denotes the dimensionless coolant temperature. The constant parameters are given as µ = 1, β = 0.3, γ = 20, B = 1, and Da = 0.072. The input signal is a uniformly distributed random signal on the interval [95, 108], which changes every 3.5 s. Let ts = 0.1 s and M = 2400 for the estimation of the fractional order (α ) and fractional degree n. Performing the frequency-domain optimization, the minimum cost function is obtained for α = 1 and n = 3. To show the significance and validity of the frequency-domain identification, the following estimated linear dynamic model is utilized as an approximate model of the CSTR:
ˆ = G(s)
Yˆ (s) U(s)
=
1 0.
0059s3
+ 0.
1756s2
+ 3.5547s + 4.3325
(38)
Fig. 4 shows the linear modeling results of the system; the performance indices defined in (36) are shown in Table 1. The identification results show that the identified integer-order transfer function (38) can identify the dynamic behavior of the CSTR described by integer-order differential equations given in (37) at the expense of some bias. Once the parameters α and n are estimated, the parameters A and w of the fractional-order Hammerstein system in (9) is identified via the time-domain algorithm. The RBF is selected to be in the form
v=
3 ∑
( ) w ˆ i exp −0.1 (u − 95 − 6(i − 1))2
(39)
i=1
According to Lemma 2 the parameters k and d are selected such that cT (sα I − K)−1 d becomes a positive real transfer function. −1 T α α Thus by choosing the poles of [ c (s I − K) d]Tin terms of s to be −14, −15 and −16, k = 45 674 3360 is obtained and
8 ∑
bˆ i v (t − i)
(41)
i=1
The coefficients w ˆ i , aˆ i , and bˆ i of the RBF-IIR Hammerstein model are estimated based on the CFF-RLS method. The simulation results are shown in Fig. 5. The traditional Hammerstein model with 19 parameters and the performance indices are given in Table 1 cannot improve the identification performance as compared with the proposed fractional-order Hammerstein model with only 6 parameters. Example 3 (Heat Exchanger Identification). The heat exchanger is a benchmark for nonlinear system modeling and control system design. The input to the process is the liquid flow rate and the system output is the outlet liquid temperature. The identification data that is gathered from a real life system is shown in Fig. 6 [46]. By selecting ts = 1 s and M = 4000, the frequency-domain identification results in α = 0.7 and n = 2. The RBF is selected to be
v=
3 ∑
( ) w ˆ i exp 33.33 (u − 0.1 − 0.3(i − 1))2
(42)
i=1
[
]T
[
]T
1 3 , k = 6 8 , P2 = 104 I2 , and By choosing d = P3 = 103 I3 , the estimation results are illustrated in Fig. 7. Furthermore, performance indices RMSE = 0.3177, RE = 0.0033, and VAF = 96.36 show that the proposed identification algorithm identifies the heat exchanger efficiently. 5. Conclusion In this paper, nonlinear system identification based on the fractional-order Hammerstein model was proposed. The identification procedure includes a frequency-domain algorithm to estimate the fractional order and the number of state variables of the commensurate linear dynamic block and a time-domain identification to estimate the other parameters. The identification algorithm for the frequency domain is a constrained least squares problem while for the time domain is carried out using the input– output filtered signals. The stability and convergence properties of the algorithm were shown via the Lyapunov stability theory. Moreover, the results indicated that the proposed modeling can improve the nonlinear system identification more effectively than the traditional Hammerstein modeling. It should be noted that the proposed identification approach is not applicable to the Hammerstein models with non-commensurate linear dynamic
Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
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block. Although an all-pole fractional-order system is considered as the linear dynamic block in this paper, the idea can be generalized to include zero-pole fractional-order system. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Giri F, Bai E-W. Block-oriented nonlinear system identification. Springer; 2010. [2] Tang Y, Li Z, Guan X. Identification of nonlinear system using extreme learning machine based Hammerstein model. Commun Nonlinear Sci Numer Simul 2014;19:3171–83. [3] Gotmare A, Patidar R, George NV. Nonlinear system identification using a cuckoo search optimized adaptive Hammerstein model. Expert Syst Appl 2014;42:2538–46. [4] Liu G, Xu X, Wang F. Modeling of a class of nonlinear dynamic system. Sens Transducers 2014;169:153–8. [5] Hu H, Ding R. Least squares based iterative identification algorithms for input nonlinear controlled autoregressive systems based on the auxiliary model. Nonlinear Dynam 2014;76:777–84. [6] Hong X, Chen S, Harris C, Khalaf E. Single-carrier frequency domain equalisation for hammerstein communication systems using complex-valued neural networks. IEEE Trans Signal Process 2014;62:4467–78. [7] Zhao J, Ma X, Zhao S, Fei J. Hammerstein identification of supercharged boiler superheated steam pressure using Laguerre-Fuzzy model. Int J Heat Mass Transfer 2014;70:33–9. [8] Sun J, Liu X. A novel APSO-aided maximum likelihood identification method for Hammerstein systems. Nonlinear Dynam 2013;73:449–62. [9] Deng K, Ding F. Newton Iterative identification method for an input nonlinear finite impulse response system with moving average noise using the key variables separation technique. Nonlinear Dynam 2014;76:1195–202. [10] Zhai D, Rollins DK, Bhandari N, Wu H. Continuous-time hammerstein and wiener modeling under second-order static nonlinearity for periodic process signals. Comput Chem Eng 2006;31:1–12. [11] Garnier H, Young PC. The advantages of directly identifying continuoustime transfer function models in practical applications. Internat J Control 2014;87:1319–38. [12] Zhou L, Li X, Shan L, Xia J, Chen W. Hierarchical recursive least squares parameter estimation of non-uniformly sampled hammerstein nonlinear systems based on Kalman filter. J Franklin Inst B 2017;354:4231–46. [13] Wang X, Ding F. Joint estimation of states and parameters for an input nonlinear state-space system with colored noise using the filtering technique. Circuits Systems Signal Process 2015. [14] Ding F, Liu X, Ma X. Kalman state filtering based least squares iterative parameter estimation for observer canonical state space systems using decomposition. J Comput Appl Math 2016;301:135–43. [15] Fu Z-J, Xie W-F, Luo W-D. Robust on-line nonlinear systems identification using multilayer dynamic neural networks with two-time scales. Neurocomputing 2013;113:16–26. [16] Janakiraman VM, Nguyen X, Assanis D. A lyapunov based stable online learning algorithm for nonlinear dynamical systems using extreme learning machines. In: The 2013 international joint conference on neural networks (IJCNN). 2013, p. 1–8. [17] Wu W, Chen H-T. Identification and control of a fuel cell system in the presence of time-varying disturbances. Ind Eng Chem Res 2015;54:7141–7. [18] Wang S, Wang W, Liu F, Tang Y, Guan X. Identification of chaotic system using Hammerstein-ELM model. Nonlinear Dynam 2015;81:1081–95. [19] Salhi H, Kamoun S. A recursive parametric estimation algorithm of multivariable nonlinear systems described by Hammerstein mathematical models. Appl Math Model 2015;39:4951–62. [20] Rahmani M-R, Farrokhi M. Robust identification of neuro-fractional-order hammerstein systems. In: 2017 5th international conference on control, instrumentation, and automation (ICCIA). IEEE; 2017, p. 27–31.
[21] Wang X, Ding F. Recursive parameter and state estimation for an input nonlinear state space system using the hierarchical identification principle. Signal Process 2015;117:208–18. [22] Rahmani M-R, Farrokhi M. Rahmani m-r farrokhi m nonlinear dynamic system identification using neuro-fractional-order Hammerstein model. Trans Inst Meas Control 2017;0142331217734301. [23] Rahmani M-R, Farrokhi M. Identification of neuro-fractional hammerstein systems: a hybrid frequency-/time-domain approach. Soft Comput 2017;1–10. [24] Oldham KB, Spanier J. The fractional calculus: Integrations and differentiations of arbitrary order. New York: Academic Press; 1974. [25] Narang A, Shah S, Chen T. Continuous-time model identification of fractional-order models with time delays. IET Control Theory Appl 2011;5:900–12. [26] Aoun M, Maltl R, Cois O, Oustaloup A. System identification using fractional Hammerstein models. In: 15th IFAC world congress. 2002. [27] Zhao Y, Li Y, Chen Y. Complete parametric identification of fractional order hammerstein systems. In: IEEE international conference on fractional differentiation and its applications (ICFDA). 2014, p. 1–6. [28] Liao Z, Zhu Z, Liang S, Peng C, Wang Y. Subspace identification for fractional order hammerstein systems based on instrumental variables. Int J Control Autom Syst 2012;10:947–53. [29] Hammar K, Djamah T, Bettayeb M. Fractional hammerstein system identification using particle swarm optimization. In: 2015 7th international conference on modelling, identification and control (ICMIC). 2015, p. 1–6. [30] Ivanov D. Identification discrete fractional order hammerstein systems. In: 2015 international siberian conference on control and communications (SIBCON). 2015, p. 1–4. [31] Pintelon R, Schoukens J. System identification: A frequency domain approach. John Wiley & Sons; 2012. [32] Podlubny I. Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. 1999. [33] Li Y, Chen Y, Podlubny I. Mittag–Leffler Stability of fractional order nonlinear dynamic systems. Automatica 2009;45:1965–9. [34] Djamah T, Mansouri R, Bettayeb M, Djennoune S. State space realization of fractional order systems. In: 2nd mediterranean conference on intelligent systems and automation (CISA). Zarzis, Tunisia: AIP Publishing; 2009, p. 37–42. [35] Pintelon R, Schoukens J, Rolain Y. Frequency-domain approach to continuous-time system identification: Some practical aspects. In: Identification of continuous-time models from sampled data. Springer; 2008, p. 215–48. [36] Aguila-Camacho N, Duarte-Mermoud MA, Gallegos JA. Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul 2014;19:2951–7. [37] Kudva P, Narendra KS. Synthesis of an adaptive observer using Lyapunov’s direct method. Internat J Control 1973;18:1201–10. [38] Delavari H, Baleanu D, Sadati J. Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dynam 2011;67:2433–9. [39] Duarte-Mermoud MA, Aguila-Camacho N, Gallegos JA, Castro-Linares R. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simul 2015;22:650–9. [40] Wang F, Yang Y. Fractional order Barbalat’s lemma and its applications in the stability of fractional order nonlinear systems. Math Model Anal 2017;22:503–13. [41] Er-Wei B. Frequency domain identification of Hammerstein models. IEEE Trans Automat Control 2003;48:530–42. [42] Poinot T, Trigeassou J-C. Identification of fractional systems using an output-error technique. Nonlinear Dynam 2004;38:133–54. [43] Tang Y, Han Z, Wang Y, Zhang L, Lian Q. A changing forgetting factor RLS for online identification of nonlinear systems based on ELM–Hammerstein model. Neural Comput Appl 2016;1–15. [44] Yan Z, Wang J. Robust model predictive control of nonlinear affine systems based on a two-layer recurrent neural network. In: The 2011 international joint conference on neural networks (IJCNN). 2011, p. 24–9. [45] Juang C-F, Chung I-F. Recurrent fuzzy network design using hybrid evolutionary learning algorithms. Neurocomputing 2007;70:3001–10. [46] De Moor B, De Gersem P, De Schutter B, Favoreel W. DAISY: A database for identification of systems. J A 1997;4–5.
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Research article
Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements Mohammad-Reza Rahmani a , Mohammad Farrokhi b ,
∗
a
School of Electrical Engineering, Iran University of Science and Technology, Tehran 1684613114, Iran School of Electrical Engineering, Center of Excellence for Modeling and Control of Complex Systems, Iran University of Science and Technology, Tehran 1684613114, Iran b
highlights • • • •
Nonlinear dynamic system identification based on continuous-time fractional-order Hammerstein state-space model is proposed. Frequency-domain identification is used to identify the structural parameters. Joint state and parameter estimation is proposed. Lyapunov stability theory is used to develop time-domain algorithm.
article
info
Article history: Received 11 May 2017 Received in revised form 4 June 2019 Accepted 14 June 2019 Available online xxxx Keywords: Hammerstein model State estimation Fractional-order systems Neural networks
a b s t r a c t This paper introduces a continuous-time fractional-order Hammerstein state-space model with a systematic identification algorithm for modeling nonlinear dynamic systems. The proposed model consists of a radial-basis function neural network followed by a fractional-order system. The proposed identification scheme is accomplished in two stages. The structural parameters of the fractional-order system (i.e. the values of the fractional order and the degree of the denominator in the fractionalorder system) are estimated in the frequency domain. Then, the synaptic weights of the radial-basis function neural network and the coefficients of the fractional-order system are determined in the time domain via the Lyapunov stability theory, which guarantees stability of the given method and its convergence under a mild condition. Three examples are provided to show the effectiveness of the proposed method. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction The Hammerstein model was presented by the same-name German mathematician in 1930 [1]. The Hammerstein model consists of a Nonlinear Static Block (NSB) followed by a Linear Dynamic Block (LDB). It is a simple block-oriented model, which has been proven to effectively model a wide class of biological/engineering systems [2,3]. The NSB of the Hammerstein model is represented by different models such as polynomial functions [4], the weighted sum of known basis functions [5], Neural Networks (NNs) [6], and fuzzy models [7]. There is a vast literature on the discrete-time Hammerstein model, where the LDB is either Infinite Impulse Response (IIR) filter [8] or Finite Impulse Response (FIR) filter [9], while much less attention has been paid to the identification of Hammerstein models with continuous-time LDB. Continuous-time models can mimic the ∗ Corresponding author. E-mail addresses: [email protected] (M.-R. Rahmani), [email protected] (M. Farrokhi).
dynamic behavior of the dynamic system more effectively than the discrete-time models [10,11]. Moreover, since state-space models provide deep insights into the nature of the real-world systems, Hammerstein state-space modeling have attracted many researchers recently [12–14]. State-space models play an important role in system identification. Some of the distinguishing features of the state-space-based identification over the commonlyused input–output regression-based one are: (1) by the use of state-space realization of a system, the well-known Lyapunov stability theory can be conducted to achieve stable identification laws, which ensure boundedness of the estimated parameters [15, 16] and (2) provided that the system states are known, by the use of state-space models for the purpose of continuous-time system identification, the need for filtering techniques such as integral transform used in [17] is removed. However, the problem in the state-space modeling is due to the fact that in practice usually state variables of the system are unmeasurable, and only the system input–output pairs are available. To overcome this difficulty, two strategies have been suggested in literature: (1) in [16,18–20], the Hammerstein state-space modeling has been
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Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
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accomplished under the assumption that the system states are known a priori ; (2) Hierarchical methods based on the cost function minimization problem, which commonly bring the system back to its discrete-time input–output representation [12,21]. Recently, two different strategies for the FOH state and parameter estimation has been proposed in the literature. The first strategy is based on the output-derivatives estimation of the FOH, wherein the system states are estimated from successive differentiating of the output signal [22]. However, the main drawback of this method is its inherent high sensitivity to the measurement noises. The second strategy is the observer-based state-space identification using an auxiliary model-based observer [23]. Although in [23] simultaneous state and parameter estimation is performed without the need for signal filtering techniques, the drawbacks of this method are: (1) it is sensitive to the initial conditions and (2) the overall stability of the algorithm has not been proven. In the present paper, on the other hand, the time-domain identification algorithm is proposed such that it is insensitive to the initial conditions and the global stability of the identification is ensured by using the Lyapunov stability theory. Fractional calculus is a branch of mathematics wherein the order of the conventional differential equation is extended to become a fractional number [24]. Recently, with the growing power of computers, Fractional-Order Systems (FOSs), whose describing differential equation is of fractional order, has been used in many engineering fields. During the last several decades, many realworld systems have been well characterized and identified using the FOS [25]. The Fractional-Order Hammerstein (FOH) model is a generalization of the Hammerstein models with the LDB being of the fractional order. To the best of the authors’ knowledge, there exist a few researches about identification of the continuous-time FOH models [26–28] and the discrete-time FOH systems [29,30]. A challenging issue encountered in fractional-order modeling is the estimation of the fractional orders that has been discussed in [27,28]. In these references, iterative time-domain algorithms have been proposed to estimate the fractional order under the assumption that the degree of the FOH is known a priory. In [27], two methods have been suggested to identify the fractional order: (1) a one-dimensional grid search, where the parameters of the FOH are estimated iteratively in the time domain at every point in the grid of the fractional order and (2) the fractional order identification based on the so-called P-type order learning law. The convergence of the latter method is guaranteed for small values of the learning rate, which may make the algorithm time consuming. In [28], a nonlinear optimization problem must be solved to estimate the fractional order. In this method, the parameters of the FOH are identified in the time-domain via subspace identification techniques. Since in every step of the nonlinear optimization, the parameters are identified via subspace identification, which depends on the fractional order, it may suffer from long computational time. In [22,23], identification of the fractional order using the frequency domain linear optimization based on the Best Linear Approximation (BLA) technique [31] has been developed. The main advantages of the aforementioned frequency-domain estimations are that they provide more accurate estimation of the fractional-order systems and they require less computational time as compared to the time-domain identification schemes. Nevertheless, these methods may suffer from the drawback that the amplitude variations of the input signal should be small enough in order to estimate the parameters with sufficient accuracy. This restriction has been removed in [22,23] for continuous-time FOH using frequency-domain identification. Based on the aforementioned literature review, the main contributions of this paper are two-folds. First, the accuracy of the frequency-domain identification is improved by acquiring the nonparametric frequency response of the LDS via the concept of
state-space modeling. Then, in order to identify the state matrices of the FOH, an adaptive observer, where the state variables and the state matrices of the FOH are estimated simultaneously, is developed. Besides, the convergence analysis of the proposed method is carried out and overall stability of the time-domain algorithm is ensured using the Lyapunov stability theory. It will be shown in simulating examples that the proposed Hammerstein model outperforms the traditional Hammerstein models with IIR filter as the LDB. This paper is organized as follows. Section 2 reviews basic concepts and the identification problem. Section 3 presents the proposed identification algorithm. Section 4 provides the illustrative examples. Finally, Section 5 concludes the paper. 2. Preliminaries and problem statement This section presents basic concepts related to the fractionalorder Hammerstein model. First, the principals of the fractional calculus, which are exploited in this paper, are given. Then, the nonlinear dynamic system identification problem is formulated as the proposed fractional-order Hammerstein modeling. 2.1. Fractional-order systems The uniform formula of a fractional-order integral is defined as [32] 1
−p
a Dt f (t) =
Γ (p)
f (τ )
t
∫
(t − τ )1−p
a
dτ
(1) −p
where f (t) is an arbitrary integrable function, a Dt is the fractional integral of order p > 0 on [a, t ], and Γ (·) stands for the Gamma function. For an arbitrary real number α , the Caputo derivative of fractional order α is defined by [33] −([α ]−α+1)
Dαt f (t) = Dt
{
d[α ]+1 dt
}
f (t) [α ]+1
(2)
where [α] denotes the integer part of α . The Laplace transform of an α -order derivative of signal f (t) is defined as [32] L {Dα f (t)} = sα F (s) −
n−1 ∑
sα−k−1 f (k) (0),
(n − 1 < α ≤ n)
(3)
k=0
where s, F (s), and f (k) (0) stand for the Laplace operator, the Laplace transform, and the kth derivative of f (t) at t = 0, respectively. A fractional-order linear system is represented by an equation or a system of differential equations characterized by real-derivative orders as [34] an y(t) + an−1 Dβ1 y(t) + · · · + Dβm y(t) = b0 Dα0 v (t) + b1 Dα1 v (t)
+ · · · + bn Dαn v (t)
(4)
where v (t) and y(t) are the input and output signals of the FOS, respectively, and differentiation orders β1 , . . . , βm and α0 , . . . , αn may be non-integer positive numbers. When all the differentiation orders are integer powers of the same fractional number, the system is called commensurate fractional-order system. In this manuscript, commensurate all-pole FOS of order α expressed by the following transfer function is used as the LDB in the Hammerstein model: 1 Y (s) = (5) G(s) = ∑n−1 V (s) snα + k=0 an−k skα The state-space realization of transfer function (5) is given as follows [34]
{
Dα x(t) = Ax(t) + bv (t) y(t) = cT x(t)
(6)
Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
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where x(t) = [x1 (t) x2 (t) vector of state variables and
⎡ − a1 ⎢−a2 ⎢−a 3 A=⎢ ⎢ . ⎣ ..
1 0 0
− an [
0
c= 1
0 1 0
.. .
0
.. .
0 0
xn (t)]T is an n-dimensional
0 0 0⎥ ⎢0⎥ ⎢ ⎥ 0⎥ ⎥ , b = ⎢0⎥ , ⎥ ⎢.⎥ .. ⎦ ⎣.⎦
··· ···
⎤
1
..
. ··· ···
···
0
⎡ ⎤
.
.
0
1
]T
.
(7)
2.2. Fractional-order Hammerstein modeling
N ∑
wi ϕi (u(t)) = wT φ(u(t)) (8) i=1 ) ( 2 2 ϕi (u(t)) = exp − (u(t) − µi ) /σi [ ] w · · · wN T and φ(u(t)) where w = = [ ] T1 ϕ1 (u(t)) · · · ϕN (u(t)) are the weight and the radial-basis function vectors, respectively, and µi and σi represent centers and widths of ϕi (u(t)). It should be noted that the weighted sum of any nonlinear function could be used in replacement of the RBF. However, the RBF is selected as the NSB since it is a well-known function approximator, which is linear in parameters, i.e., its output weights. Thus, the state-space representation of the proposed fractional-order Hammerstein system can be given as Dα x(t) = Ax(t) + Bφ(u(t)), y(t) = x1 (t).
with B = bwT
Using the input–output data acquired from the system −1)ts {ϕ1 (u(t)), y(t)}t(M with sampling time ts at which the input– =0 output data are measured, the fractional order (α ) and the frac-
tional degree (n) of the LDB in the fractional-order Hammerstein model are estimated in the frequency domain using the data set {Φ1 (k), Y (k)}M k=1 , which is the Discrete Fourier Transform (DFT) (M −1)t of {ϕ1 (u(t)), y(t)}t =0 s . In this regard, a two-stage identification approach is used that includes the estimation of the nonparametric frequency response of the transfer function Gw (s) = w1 G(s) and then parameterization of the computed Gw (s). In the first √ step, the nonparametric frequency response Gw (jω), where j = −1, is estimated at different frequencies ωk (ωk = 2π k/Mts , k ∈ {1, . . . , M }) as follows [35]: Y (k) = Gw (jωk )Φ1 (k) + P(jωk )
The NSB of the proposed fractional-order Hammerstein model is considered to be of RBF type, which is given by
{
element of the set {1, . . . , N }. Herein, without loss of generality, i is set to one. Next, using the auxiliary signal-based Lyapunov stability theory, the weight vector (w) as well as the state matrix (A) are estimated on-line in the time domain. These steps will be explained in the followings. 3.1. Frequency-domain identification
It is well-known that a transfer function can be represented by different state-space realizations such as controllable form, observable form, etc. However, since in this paper, the identification problem is to estimate the parameters of the transfer function in (5) and the weights of RBF in (8), the state-space representation type in (5) is not important for the proposed method. Henceforth, we have selected this rare but effective representation of the state space just for the sake of time-domain identification of the Hammerstein system in (9) using the input–output data {u(t), y(t)}.
v (t) =
3
(9)
Therefore, the objective is to identify the fractional-order Hammerstein model described by (5) and (8) or equivalently, to estimate the parameters of Hammerstein state-space model in (9), including α , n, A, and w to mimic the input–output dynamic behavior of the nonlinear dynamic system at hand. This is accomplished using only the input–output measured data from the system. Herein, parameter N (the number of neurons in the hidden layer of the NN) is known or can be determined by trial and error. Moreover, the widths and centers of the RBF functions and in turn φ(u(t)) are selected such that the RBF covers the whole range of the input signal u(t).
where P(jωk ) contains the influence of the initial and final conditions of the experiment and tends to zero with respect to the main term Gw (jωk )Φ1 (k). To have one more degree of freedom and thereby to improve the identification efficiency, transfer function (5) is divided by an unknown constant a0 as follows Gn (s) =
G(s) a0
3. Identification of fractional-order Hammerstein model Using the input–output data obtained from the system, the identification algorithm is carried out in hybrid frequency-/timedomain. First, in the frequency-domain, the fractional order (α ) and the fractional degree (n) are estimated using the input– output pairs {ϕi (u(t)), y(t)}, where subscript i is an arbitrary
1
= a0
sn α
+ a0
∑n−1 k=0
an−k skα
(11)
In the second step, the objective is to estimate α and n via a grid search such that Gw (jωk ) is parameterized in the form of Gn (jω) within a predefined frequency interval. To do this, at first, a two-dimensional search space (α, n) ∈ { α, n| 0 < α ≤ 1, 1 ≤ n ≤ nmax }, where nmax is an arbitrary integer, is defined. Then, for any point on the search space, a constrained linear least squares is performed to identify the vector am = a0 [an an−1 · · · a1 1]T . Two constraints are considered for the optimization problem: (1) vector am must be real-valued and (2) the estimated transfer function Gn (jω) must be stable; i.e., the sα − poles of Gn (jω) have negative real parts. Consequently, for every point on the search space, the following constrained optimization problem: J = arg min (Ham − q)T (Ham − q)
a ⎧ m } ⏐= 0 ⎪ ⎨Im{a{ [ ]} n−1 ⏐ ∑ s.t. ⏐ n k ak p <0 ⎪ ⎩Real p ⏐⏐p ∈ roots p + k=0 ⎡ ⎤ 1 (jω1 )α · · · (jω1 )nα α nα · · · (jω2 ) ⎥ ⎢1 (jω2 ) H=⎢ .. .. ⎥ .. ⎣ .. ⎦, . . . .
1 Assumption 1. It is assumed that the system is a smooth timeinvariant dynamical system.
(10)
[ q=
(jωL )α 1
Gw (jω1 )
··· ···
(12)
(jωL )nα ]T 1 Gw (jωL )
should be solved to find am and the resultant cost function J. Eq. (12) shows that J is the sum of squares of errors between the nonparametric frequency response Gw (jω) and parametric one Gn (jω) at L different data points {ω1 , ω2 , . . . , ωL }, where n + 1 ≤ L ≤ M. Finally, the pair {α, n} that results in the minimum value of the cost function J is adopted as the estimated values for α and n.
Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
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As a remark on the identifiability, since the number of rows of matrix H in (12) is greater or equal to the dimension of the parameter vector am , for the perfect estimation of nonparametric frequency response Gw (jω) and the true values of α and n, the unknown vector am is estimated consistently using the cost function Jα,n = 0. Thus, if the number of the points in the 2-dimensional grid tends to infinity and the frequency-response estimation in (10) is accurate, the consistent estimates of α and n are obtained.
where P2 is an n × n − dimensional diagonal positive-definite matrix and
ˆ Dα w(t) = −P3 x˜ 1 (t)q(t), q(t) = q1 (t)
[
qj (t) =
ϕj (u(t)) , ∑ −2 kα + nk= 0 dn−k D
D(n−1)α
Once the structural parameters α and n are estimated, in the time-domain, the weight vector (w) and the state matrix (A) in (9) are identified via the Lyapunov stability theory. In this regard, the system is assumed to be of the form (9), whose state variables are immeasurable. Hence, the identification problem requires an adaptive observer for the fractional-order Hammerstein system in (9). First, the following lemmas should be considered that are needed in the sequel. Lemma 1. Let f (t) ∈ R be a continuous and derivable function. Then, for any t ≥ t0 t0 Dt
f (t) ≤ 2f (t)t0 Dt f (t),
∀α ∈ (0, 1] .
(13)
˜ φ(u) + r + a˜ x1 with B˜ = Bˆ − B, and a˜ = a − aˆ (19) Dα x˜ = Kx˜ + B Eq. (19) can be rewritten as an nth order scalar differential equation as Dnα x˜ 1 +
n ∑
˜ φ(u) + ki D(n−i)α x˜ 1 = w
n ∑ [
i=1
= D(n−1)α
D(n−i)α a˜ i x1 + ri
(
i=1
n ∑
[(
a˜ i x1 + ri
)
D(i−1)α
+
(20)
]
˜ φ(u) w nD(n−1)α
˜ =w ˆ − w. , w
Eq. (19) can be rewritten as
˜ Tq Dα x˜ = Kx˜ + d a˜ T z + w
(
n ∑
)
(21)
[(
a˜ i x1 + ri
)
D(i−1)α
+
˜ φ(u) w
]
nD(n−1)α
=
n ∑
di D(i−1)α
˜ Tq a˜ T z + w
[
Lemma 2. [Kalman–Yakubovich Lemma] For any vector [ ]T d = 1 d2 · · · dn and stable matrix K, if cT (sI − K)−1 d (I denotes identity matrix of appropriate dimension) is positive real, then there exist a positive-definite matrix P1 and a vector m such that [37]
By successive differentiation of (22), r = [r1 calculated as follows:
KT P1 + P1 K = −mmT , P1 d = c.
rk = −
(14)
In the fractional-order case, the transfer function cT (sI − K)−1 d in the above lemma is replaced with cT (sα I − K)−1 d. In the following theorem, the adaptive observer
ˆ φ(u(t)) + k − aˆ (t) x1 (t) + r(t), Dα xˆ (t) = Kxˆ (t) + B(t) yˆ (t) = xˆ 1 (t)
]
(15)
where r(t) = [r1 (t) · · · rn (t)]T is the auxiliary signal vector that will be defined later in (23), and
⎡ −k1 ⎢−k2 ⎢−k 3 K=⎢ ⎢ . ⎣ ..
1 0 0
.. .
−kn 0 ⎡ ⎤ k1
⎢.⎥ k = ⎣ .. ⎦ , kn
0 1 0
··· ··· 1
.. .
..
0 0⎥ 0⎥ ⎥ < 0, ⎥
⎤
.. ⎦ .
ˆ = B(t)
. ··· 0 ⎡ ⎤ aˆ 1 (t) ⎢ . ⎥ aˆ (t) = ⎣ .. ⎦ . aˆ n (t)
[
]
0 , ˆT w
Theorem 1. For the adaptive observer (15) of the system (9), limt →∞ x˜ (t) = 0 if the following updating laws are adopted for aˆ (t) ˆ and w(t): Dα aˆ (t) = P2 x˜ 1 (t)z(t), z(t) = z1 (t)
[
x1 (t)
(n−i)α
=
D D(n−1)α +
∑n−2 k=0
k−1 ∑
Dα a˜ i
dn−k Dkα
···
zn (t)
]T
, i ∈ {1, . . . , n}
with
n ∑
dj zj−k+i+1 +
j=k
n ∑
Dα a˜ i
i=k
k ∈ {2, . . . , n}
···
(22)
rn ]T can be
k−1 ∑
dj zj−k+i+1 ,
(23)
j=1
To obtain stable identification laws, consider the following Lyapunov function candidate: 1(
1 ˜ ˜ T −1 ˜ x˜ T P1 x˜ + a˜ T P− (24) 2 a + w P3 w 2 where P1 is a positive-definite matrix that satisfies (14). According to the Lyapunov stability theory for FOSs [38–40], if the identification laws are chosen such that the inequality Dα F < 0 holds, then the identification algorithm will be stable; i.e., limt →∞ x˜ (t) = 0. According to Lemma 1, Dα F satisfies the following inequality:
F =
0
(16)
i=1
]
r1 = 0,
Dα F ≤
is designed such that the prediction error x˜ (t) = xˆ (t) − x(t) tends to zero when the time t grows to infinity.
zi (t)
i=1
i=1
[
)]
if the following equality holds:
Proof. See [36].
{
(18)
j ∈ {1, . . . , N }
where P3 is an N ×N −dimensional diagonal positive-definite matrix.
i=1
α
with
Proof. From (9) and (15), by omitting the time index t for brevity, the error dynamic equation can be obtained as follows:
3.2. Time-domain identification
α 2
]T
qN (t)
···
)
1( α ) 1 ( α T) 1 α ˜ D x˜ P1 x˜ + x˜ T P1 D x˜ + a˜ T P− 2 D a 2 T −1 α ˜ P3 D w ˜ +w 2
(25)
Hence, the negativeness of the right-hand side of (25) implies that Dα F < 0 and this in turn, guarantees stability of the identification algorithm. Substituting (21) into the right-hand side of (25) yields Dα F ≤
1 2
˜ Tq x˜ T KT P1 + P1 K x˜ + x˜ T P1 d a˜ T z + w
(
)
(
)
1 α ˜ ˜ T −1 α ˜ + a˜ T P− 2 D a + w P3 D w
(26)
Now, if the following equations hold:
{
1 α ˜ a˜ T P− xT P1 da˜ T z, 2 D a = −˜
(27)
1 α ˜ T P− ˜ ˜ Tq w xT P1 dw 3 D w = −˜
then, (26) reduces to (17) Dα F ≤
1 2
( ) 1 x˜ T KT P1 + P1 K x˜ = − x˜ T Q1 x˜ < 0, 2
Q1 > 0
(28)
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5
since under the assumption that K is Hurwitz, for any positivedefinite matrix Q1 there exist a positive definite matrix P1 that satisfies KT P1 + P1 K = −Q1 [15]. Solving (27) for Dα a˜ and ˜ considering that system (9) is time invariant, leads to the Dα w following updating laws: Dα aˆ = P2 zx˜ T P1 d,
{
(29)
ˆ = −P3 qx˜ T P1 d Dα w
According to Lemma 2, equality x˜ T P1 d = x˜ 1 holds and thereby, the updating laws in (29) are reduced to those given by (17) and (18). From (28), it can be seen that x˜ = 0 and Dα F = 0 are equivalent and in turn, the updating laws in (17) and (18) result in limt →∞ x˜ (t) = 0. □ Theorem 2 (Convergence { } Analysis). If there exists no set of realvalued constants β i,j (i = 0, 1, . . . , n and j = 1, . . . , N) such that φ(u) in steady state is a solution of the following homogeneous differential equation: N n ∑ ∑ [ ] βi,j D(n−i)α ϕj (u) = 0
(30) 4. Simulating examples
j=1 i=0
ˆ are consistently identified. then, parameters aˆ and w T
T
˜ q Proof. It can be seen from (21) that when x˜ = 0, then a˜ z + w ˜ T q, the becomes zero. Using the Laplace transform of a˜ T z + w following relation can be obtained:
(
n ∑
) a˜ i D
(n−i)α
x1 (t) +
N ∑
˜ i ϕi (u)) = 0. (w
(31)
i=1
i=1
By expressing x1 in terms of u, (31) can be rewritten as follows:
(
n ∑
Fig. 1. System output y(t), model output yˆ (t) and error e(t) = y(t) − yˆ (t) for Example 1.
⎞ ( )⎛ N ) n ∑( ∑ ) (n−i)α (n − i) α ⎝ a˜ i D wj ϕj (u) ⎠ + ai D
i=1
j=1
(32)
(33)
The identification problem is to estimate vectors a = a1
j=1
or equivalently N n ∑ ∑ [(
a˜ i wj + ai w ˜ j D(n−i)α ϕj (u) = 0, a0 = 1, a˜ 0 = 0
)
]
Example 1 (Fractional-order Hammerstein System). Consider the system
⎧ 3 ∑ ⎪ ⎪ ⎪ ⎪ v = wi ϕi (u), wi = i, ⎪ ⎪ ⎪ i=1 ⎨ ) ( ϕi (u) =[ exp −0].32 (u[−](5i − 10))2 [, ] [ ] ⎪ ⎪ 5 − a1 1 0 a1 ⎪ ⎪ = , D0.7 x = x+ v, ⎪ ⎪ a2 10 − a2 0 1 ⎪ ⎩ y = x1 .
i=0
⎛ ⎞ N ∑ ( ) w ˜ j ϕj (u) ⎠ = 0, a0 = 1 ×⎝
The performance of the proposed fractional-order Hammerstein modeling is demonstrated using three simulating examples: (1) the stability and convergence properties of the time-domain identification algorithm is illustrated in Example 1; (2) the identification of a Continuous Stirred Tank Reactor (CSTR) is performed in Example 2, and (3) a real life nonlinear process, namely liquidsaturated steam heat exchanger is identified from its recorded input–output data.
j=1 i=0
which is of the form of (30) with βi,j = a˜ i wj + ai w ˜ j . Thus, (33) represents a condition upon φ(u) in the steady state; i.e., after the identification laws has forced x˜ to vanish. Two possibilities exist by considering the solution of (30): (1){either } the steady-state input φ(u) satisfies (30) for some values β i,j , or (2) coefficients { } β i,j are zero. Based on the assumption of the theorem, (1) cannot occur. Consequently, (2) is true. Since βi,j = a˜ i wj + ai w ˜ j, ˜ must be zero. This completes the proof. □ then a˜ and w Corollary 1. In the steady-state condition, the parameter estimates converge to their true values for every sinusoidal input u. Proof. On one hand, φ(u) is a Gaussian-function vector, which implies that it has infinite number of sinusoidal signals of distinct frequencies even if input u is a sinusoidal signal [41]. On the other hand, when α is set to one, which is the worst case, it can be deduced from (30) that φ(u) must have at least 0.5n + 1 distinct frequencies, which are not the solution of (30). Thus, φ(u) cannot be a solution of (30) for every sinusoidal input u unless a˜ = 0 ˜ = 0. When α is not equal to one, vector φ(u) that satisfies and w (30), is not Gaussian function of sinusoidal input unless a˜ = 0 ˜ = 0. □ and w
(34)
[
and w = w1
w2
[
w3
]T
a2
]T
using the measurements of u and y.
]T
[
]T
[
The parameters are selected as d = 1 3 and k = 6 8 such that cT (sα I − K)−1 d is a positive real transfer function. Then, the input–output filtered signals z and q are constructed using (17) and (18), respectively, as follows:
[ ] Z=
Z1 Z2
[ =T
s
0.7
L {ϕ1 (u)}
⎡
]
⎤
X1 , q = T ⎣L {ϕ2 (u)}⎦ , X1 L {ϕ3 (u)}
T =
1 s0.7
+3 (35)
Then, r is obtained using (23) and by trial and error. The suitable weighting matrices in (17) and (18) are determined as P2 = 107 I2 and P3 = 106 I3 . Following the procedure given in [42], the fractional-order operator D−0.7 is implemented and the sinusoidal input signal u(t) = 10(sin(t) + sin(2.5t)) is applied to the system in (34). The efficiency of the identification algorithm is measured in terms of the Root-Mean Square Error (RMSE), the Relative Error (RE), and the Variance Account For (VAF), which are defined as follows RMSE =
1 N
[ N ∑(
]1/2 i
i 2
yˆ − y
i=1
)
, RE =
[ ∑N ( i=1
yˆ i − yi
∑N ( i )2 i=1
y
)2 ]1/2 ,
Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
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Fig. 4. System/model output and the error for frequency-domain identification of CSTR.
Fig. 2. Real and estimated values of parameter a for Example 1.
Fig. 3. Real and estimated values of parameter w for Example 1. Fig. 5. (a) CSTR output, (b) the proposed modeling error and (c) the CFF-RLS method error.
Table 1 Performance indices of Example 2. Algorithms
Performance indices RMSE
RE
VAF
Linear modeling Hammerstein modeling in [43] Proposed modeling
0.7668 0.0013 4.8420e−04
0.0320 5.3243e−05 1.9852e−05
99.8852 99.9998 100
{ VAF = 100 max 0, 1 −
(
)}
var yˆ − y var (y)
(36)
where N, yi , and yˆ i are the number of data, the output of the system, and that of the model at the ith sample, respectively. In the case of perfect identification, the values of RMSE and RE are zero and the value of VAF is hundred. By constructing the adaptive observer in (15) using parameters aˆ and w ˆ , respectively obtained from (17) and (18), the results are shown in Figs. 1 to 3, where the stability and convergence properties of the time-domain algorithm can be observed. These figures show that the identified model mimics the input–output behavior of the system in (34) and the parameters a and w are identified consistently. Moreover, the performance indices RMSE=1.3925e−04, RE = 9.4599e−04, and VAF = 99.9996 show that the proposed identification algorithm identifies the system (34) with good accuracy.
Fig. 6. Input–output data used for the identification of the heat exchanger.
Example 2 (CSTR Modeling). The CSTR is a well-known process in chemical industries, wherein the material and energy balance equations in dimensionless variables, under the assumption that
Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
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7
[
in turn, d is set to be 1
]T
8
10
in order to reach a positive
real transfer function cT (sα I − K)−1 d. The input–output filtered signals z and q are constructed as follows: Z1 Z2 Z3
[ ] Z=
T =
L {ϕ1 (u)} s2 X 1 = T ⎣ sX1 ⎦ , q = T ⎣L {ϕ2 (u)}⎦ , X1 L {ϕ3 (u)}
⎡
⎤
⎡
⎤
1 s2
(40)
+ d2 s + d3
The weighting matrices are selected as P2 = 105 I3 and P3 = 104 I3 . The identification results for the proposed fractional-order Hammerstein modeling are given in Table 1, which represent significant improvements from the linear model given by (38). For the sake of comparison, the RBF-IIR Hammerstein model, whose LDB is in the form of the commonly-used IIR filter, is identified with the recently proposed method, namely, Changing Forgetting Factor RLS (CFF-RLS) [43]. The NSB of the RBF-IIR Hammerstein model is described by (39) while its LDB is given by Fig. 7. System/model output and the error for the heat exchanger identification.
y(t) +
8 ∑
aˆ i y(t − i) = v (t) +
i=1
there is no time delay in the recycle stream, are described by [44, 45] dx1 (t) dt dx2 (t) dt
= −µx1 (t) + Da (1 − x1 (t)) exp
(
γ x2 (t) γ + x2 (t)
)
= −(µ + β )x2 (t)
(37)
+BDa (1 − x1 (t)) exp
(
γ x2 (t) γ + x2 (t)
)
+ β u(t)
where the state variables 0 ≤ x1 (t) ≤ 1 and x2 (t) correspond to the conversion rate of the reaction and the dimensionless temperature, respectively, and u(t) denotes the dimensionless coolant temperature. The constant parameters are given as µ = 1, β = 0.3, γ = 20, B = 1, and Da = 0.072. The input signal is a uniformly distributed random signal on the interval [95, 108], which changes every 3.5 s. Let ts = 0.1 s and M = 2400 for the estimation of the fractional order (α ) and fractional degree n. Performing the frequency-domain optimization, the minimum cost function is obtained for α = 1 and n = 3. To show the significance and validity of the frequency-domain identification, the following estimated linear dynamic model is utilized as an approximate model of the CSTR:
ˆ = G(s)
Yˆ (s) U(s)
=
1 0.
0059s3
+ 0.
1756s2
+ 3.5547s + 4.3325
(38)
Fig. 4 shows the linear modeling results of the system; the performance indices defined in (36) are shown in Table 1. The identification results show that the identified integer-order transfer function (38) can identify the dynamic behavior of the CSTR described by integer-order differential equations given in (37) at the expense of some bias. Once the parameters α and n are estimated, the parameters A and w of the fractional-order Hammerstein system in (9) is identified via the time-domain algorithm. The RBF is selected to be in the form
v=
3 ∑
( ) w ˆ i exp −0.1 (u − 95 − 6(i − 1))2
(39)
i=1
According to Lemma 2 the parameters k and d are selected such that cT (sα I − K)−1 d becomes a positive real transfer function. −1 T α α Thus by choosing the poles of [ c (s I − K) d]Tin terms of s to be −14, −15 and −16, k = 45 674 3360 is obtained and
8 ∑
bˆ i v (t − i)
(41)
i=1
The coefficients w ˆ i , aˆ i , and bˆ i of the RBF-IIR Hammerstein model are estimated based on the CFF-RLS method. The simulation results are shown in Fig. 5. The traditional Hammerstein model with 19 parameters and the performance indices are given in Table 1 cannot improve the identification performance as compared with the proposed fractional-order Hammerstein model with only 6 parameters. Example 3 (Heat Exchanger Identification). The heat exchanger is a benchmark for nonlinear system modeling and control system design. The input to the process is the liquid flow rate and the system output is the outlet liquid temperature. The identification data that is gathered from a real life system is shown in Fig. 6 [46]. By selecting ts = 1 s and M = 4000, the frequency-domain identification results in α = 0.7 and n = 2. The RBF is selected to be
v=
3 ∑
( ) w ˆ i exp 33.33 (u − 0.1 − 0.3(i − 1))2
(42)
i=1
[
]T
[
]T
1 3 , k = 6 8 , P2 = 104 I2 , and By choosing d = P3 = 103 I3 , the estimation results are illustrated in Fig. 7. Furthermore, performance indices RMSE = 0.3177, RE = 0.0033, and VAF = 96.36 show that the proposed identification algorithm identifies the heat exchanger efficiently. 5. Conclusion In this paper, nonlinear system identification based on the fractional-order Hammerstein model was proposed. The identification procedure includes a frequency-domain algorithm to estimate the fractional order and the number of state variables of the commensurate linear dynamic block and a time-domain identification to estimate the other parameters. The identification algorithm for the frequency domain is a constrained least squares problem while for the time domain is carried out using the input– output filtered signals. The stability and convergence properties of the algorithm were shown via the Lyapunov stability theory. Moreover, the results indicated that the proposed modeling can improve the nonlinear system identification more effectively than the traditional Hammerstein modeling. It should be noted that the proposed identification approach is not applicable to the Hammerstein models with non-commensurate linear dynamic
Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.
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Please cite this article as: M.-R. Rahmani and M. Farrokhi, Fractional-order Hammerstein state-space modeling of nonlinear dynamic systems from input–output measurements. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.015.