Identification of fractional Hammerstein system with delay

CHAPTER

Identification of fractional Hammerstein system with delay

22

Karima Hammara , Tounsia Djamaha , Ali Zemoucheb , Maamar Bettayebc a Department

of Control Engineering L2CSP, UMMTO, Tizi Ouzou, Algeria UMR CNRS 7039, University of Lorraine, Cosnes et Romain, France c Department of Electrical and Computer Engineering, University of Sharjah UAE and (CEIES) King Abdulaziz University, Jeddah, Saudi Arabia b CRAN

Chapter outline 1 Introduction....................................................................................... 441 2 Fractional calculus .............................................................................. 443 2.1 Definition of the fractional-order operator........................................ 443 2.2 Fractional-order state space model ............................................... 445 3 Problem definition ............................................................................... 446 4 Identification method............................................................................ 447 5 Simulation examples ............................................................................ 449 Example 1: Fractional commensurate case ........................................... 450 Example 2: Fractional noncommensurate example .................................. 451 6 Conclusion ........................................................................................ 459 References........................................................................................... 459

1 Introduction During the past decades, massive research studies have been devoted to the topic of time-delay systems, due to the emergence of delays in various dynamical processes such as data transmission in communication channels, propagation phenomena, energy transfer in diffusive processes, etc. [1], They can be described in most cases by a set of partial differential equations and they belong to the class of infinitedimensional systems. In these systems, the ignorance of the delays may induce oscillations, degradation of performances, and loss of stability especially for nonlinear systems, hence, some challenging issues are still pending in this research topic. Stability, Control and Application of Time-Delay Systems. https://doi.org/10.1016/B978-0-12-814928-7.00022-6 © 2019 Elsevier Inc. All rights reserved.

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CHAPTER 22 Identification of fractional Hammerstein system with delay

On the other hand, fractional-order systems are a class of time-delay systems suitable to describe phenomena on account of their history dependence and infinitedimensional structure. In fact, over the last decades, fractional calculus has attracted increasing interest and became a powerful tool for the compact modeling of real dynamical processes in various domains such as viscoelasticity, diffusion and wave propagation, traffic systems, heat transfer and biology, etc. [2–5]. The difficulty for these distributed delays’ systems is to fully describe them with a finite number of delays to a spatial extent and the fractional calculus allows to encapsulate all process delays. According to the discrepancy of natural dynamical processes, their modeling and identification is an important task for their control design, and fault diagnosis. A great number of contributions has focused on the identification of the classical integer case models [6–9], and relatively, less work considered the fractional-order systems [10–16], etc. In this chapter, identification of fractional nonlinear time-delay systems is addressed, based on the block-oriented Hammerstein structure. This class of nonlinear models allows the separation of the linear part and the nonlinear one into different interconnected simple blocks that are flexible enough to represent several pertinent nonlinear systems (Hammerstein, Wiener, Hammerstein-Wiener, etc.) [17–19]. The Hammerstein model consists of the cascade of a static nonlinear block with a linear block [20], where the nonlinear part may represent a nonlinear actuator or other nonlinear effects [21]. Some applications for industrial processes are heat exchangers [22], pH neutralization process [18], biological processes [19], etc. Identification of fractional nonlinear time-delay systems is complex since it requires the estimation of the parameters as well as the fractional orders, and a bad parameterization of the model may require a heavy computational load. Although some amount of contributions has been accumulated in the literature for the identification of the block-structured systems in the classical integer order [6–9], only few papers consider the fractional-order case [15,23–28]. However, in the previous studies, the fractional orders are often assumed set a priori and only the parametric estimation is performed or in other papers the study has been limited to the particular case of commensurate-order systems where the fractional orders are multiple of the same value. In Ref. [25], the parametric identification of a fractional Hammerstein system described by a recurrence equation is developed using an iterative method while, in Ref. [15], the Levenberg-Marquardt (L-M) algorithm is used to estimate the Hammerstein system parameters, where the linear part is a fractional transfer function; but, for both studies the fractional orders are assumed known. The subspace identification method based on the instrumental variables is reported in Ref. [24], and an iterative linear optimization algorithm with a Lyapunov method in Ref. [23]; however, they consider the commensurate case only. The main contribution of this chapter is the identification of a fractional Hammerstein time-delay system where its parameters as well as its fractional orders are estimated, for the general uncommensurate-order case, where the orders are completely independent. In this study, the Hammerstein system linear part is described by

2 Fractional calculus

a fractional state space model, and the static nonlinear block is a polynomial. Hence, the use of the fractional polynomial nonlinear state space (FPNLSS) model, which takes into account the nonlinear components is pertinent; it also avoid the presence of the coupled cross-products of the linear part and nonlinear part parameters that occurs for input/output (I/O) models. This way, the optimization problem has a better conditioning compared to the I/O counterpart and the computational effort is greatly reduced. The identification of the FPNLSS model is based on a nonlinear optimization approach, in occurrence the L-M algorithm, which combines the gradient descent and the Gauss-Newton methods. It requires the crucial sensitivity functions calculation which may be laborious and sometimes complex depending on the chosen model. In this aim, the parametric sensitivity functions can be developed as a multivariable FPNLSS model, reducing this way the computational effort of the approach. The chapter is organized as follows: in Section 2, some background on fractional calculus, necessary to understand the remainder of the chapter is provided. Section 3 presents the fractional Hammerstein model and derives the FPNLSS description. Section 4 formulates the fractional Hammerstein FPNLSS identification method, and the sensitivity functions calculation is developed. Numerical simulation examples are carried out in Section 5, and the estimator statistical properties are analyzed using Monte Carlo simulation. Finally, conclusions and perspectives are outlined in Section 6.

2 Fractional calculus Fractional calculus is a powerful tool applied in control and for the modeling of many physical processes. It is defined as the generalization of the differential operator dtd to noninteger values (real or complex) α, ˜ where the fractional-order differintegral operator is a Dαt˜ , with a and t the limits of the operation.

2.1 Definition of the fractional-order operator The mathematical definition of the differintegral operator of fractional order has been the subject of different approaches, the most used are the Riemann-Liouville (RL), the Grünwald-Letnikov (GL), and the Caputo’s (C) definitions. Although they are different in form, one can be transferred from each other under some conditions. Definition. The RL operator of order α˜ of a function g is defined as RL Dα˜ g(t) = a t

 1 g(τ ) dm t dτ ˜ Γ (m − α) ˜ dt a (t − τ )α−m+1

(1)

where Γ (x) is the Gamma function of x, for α˜ > 0, m − 1 < α˜ ≤ m, m ∈ N and for α˜ ≤ 0, m = 0.

443

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CHAPTER 22 Identification of fractional Hammerstein system with delay

Definition. The Caputo’s operator of order α˜ of a function g is defined as C Dα˜ g(t) = a t

 t 1 g(m) (τ ) dτ ˜ Γ (m − a) a (t − τ )α−m+1

(2)

for α˜ > 0, m − 1 < α˜ ≤ m, m ∈ N for α˜ ≤ 0, m = 0. Definition. The GL operator of order α˜ of a function g is defined as t

GL Dα˜ g(t) = lim a t h→0

h 1 

hα˜ j=0

(−1)j

  α˜ j

g(t − jh)

where h is the time increment, j is the number of samples, and

(3)

  α˜ is the Newton j

binomial term given by the following relation:    1 α˜ = α( ˜ α−1)...( ˜ α−j+1) ˜ j j!

for j = 0 for j > 0

(4)

These definitions show that fractional differintegral operators are global operators having a memory of all the past, which makes them suitable for modeling hereditary effects in most materials and processes. However, for numerical computation of the fractional derivative, the drawback of the infinite memory which requires an important number of terms is circumvented by considering a limited length memory denoted L. A good approximation can be obtained using the discrete GL difference operator α˜ with the assumption of a sampling interval h = 1, and initial time equal to zero. α˜ g(k) =

L  (−1)j j=0

  α˜ g(k − j) j

(5)

where Eq. (5) can be written under the form α g(k) = 

L 

β(j)g(k − j)

with β(j) = (−1)j

j=0

β(0) = 1 β(j) = β(j − 1) (j−1)(α−1) j

for j = 1, . . . , L

  α j

(6)

(7)

In this study, the discrete GL operator, which is the most adequate to simulate discrete fractional systems is used for numerical calculations.

2 Fractional calculus

2.2 Fractional-order state space model Let us consider the discrete fractional-order state space model, based on GL difference operator given in Ref. [29], described by the following equations: α x(k + 1) = Ax(k) + Bu(k)

(8)

y(k) = Cx(k) + Du(k)

where x(k) ∈ Rn represents the state vector; u(k) and y(k) ∈ R represent, respectively, the input and the output of the system; A ∈ Rn×n , B ∈ Rn×1 , C ∈ R1×n , and D ∈ R1 are the system matrices; α is the fractional orders vector; and α x is the fractional state variables vector as follows:  α = α1

α2

 αn ,

···

 α x = α1 x1

α2 x2

···

αn xn

T

The fractional orders are completely different and the system is called a fractional noncommensurate system or a generalized fractional system; in the special case where the state variables are differentiated to the same order α, ˜ the system is called a commensurate-order system with α˜ = α1 = α2 = · · · = αn

and

 α˜ x(k + 1) = α˜ x1 (k + 1)

...

T

xn (k + 1)

The simulation of the fractional state space model (8) is performed using the GL operator α x(k + 1) = Ax(k) + Bu(k) x(k + 1) = α x(k + 1) −

k+1 

β(j)x(k + 1 − j)

(9)

j=1

y(k) = Cx(k) + Du(k)

with β(j) the matrix of the elements βi (j) as follows: β(j) = diagonal[βi (j)]

for i = 1, 2, . . . , n

with βi (j) = (−1)j

  αi j

(10)

Using a limited memory L, Eq. (9) can be rewritten as α x(k + 1) = Ax(k) + Bu(k) x(k + 1) = α x(k + 1) − [β(1)x(k) + β(2)x(k − 1) + · · · + β(L)x(k − L)]

(11)

y(k) = Cx(k) + Du(k)

The fractional system in Eq. (11) is a system with multiple state delays; it will be used to describe the linear part of the Hammerstein system.

445

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CHAPTER 22 Identification of fractional Hammerstein system with delay

3 Problem definition Consider a fractional Hammerstein system shown in Fig. 1, which consists of a static nonlinear block (NL) followed by a fractional linear block (FL); the linear part is a fractional controllable fractional state space model equation (12) α x(k + 1) = A0 x(k) + B0 u˜ (k)

(12)

y˜ (k) = C0 x(k) + D0 u˜ (k)

where u(k) and y(k) are, respectively, the system input and the system output; y˜ (k) is the output of the linear part; and u˜ (k) is the nonlinear part output which is an intermediate variable. In this work, the nonlinear block is represented by a polynomial of order r with unknown coefficients pi (i = 1, 2, . . . , r). u˜ (k) = f (u(k)) =

r 

pi ui (k)

(13)

i=1

The system overall output is y(k) = y˜ (k) + v(k), where v(k) is the noise. Replacing u˜ (k) in Eq. (12) yields to the fractional Hammerstein model equation r 

α x(k + 1) = A0 x(k) + B0 y(k) = C0 x(k) + D0

pi ui (k)

i=1 r 

(14) pi ui (k) + v(k)

i=1

In order to normalize the Hammerstein model, the nonlinear part first coefficient p1 is set equal to 1 (p1 = 1). The Hammerstein model equation (14) is a state space model containing nonlinear terms, thus it is suitable to use the polynomial nonlinear state space (PNLSS) model which is the generalization of the state space model to nonlinear case; in our study, we extend it to the fractional case given by equations α  x(k + 1) = Ax(k) + Bu(k) + Eη(k) y(k) = Cx(k) + Du(k) + Fζ (k)

(15)

where the matrices A, B, C, and D describe the state space model linear part and the nonlinear part is represented by the matrices E ∈ R1×nη and F ∈ R1×nζ ; ζ (k) and η(k) are vectors containing the monomials expansion of u(k) and x(k), of degree 2 up to r. v(k) u(k)

FIG. 1 Hammerstein system.

NL

u˜(k)

FL

y˜(k)

+

y(k)

4 Identification method

Let us derive the relationship between the fractional Hammerstein system and the FPNLSS representation α x(k + 1) = A0 x(k) + B0 u(k) + B0

r 

pi ui (k)

i=2 r 

y(k) = C0 x(k) + D0 u(k) + D0

(16) pi ui (k) + v(k)

i=2

where A = A0

B = B0

C = C0 D = D0 E = [p2 B0 · · · pr B0 ] F = [p2 D0 · · · pr D0 ]

(17) (18)

The vectors ζ (k) and η(k) containing the powers of u(k) are equals under the form  ζ (k) = η(k) = u2 (k)

u3 (k)

···

ur (k)

T

(19)

The identification of the fractional Hammerstein system described by the FPNLSS (Eq. 15) is presented in the following section.

4 Identification method Let us consider the fractional Hammerstein system described by the FPNLSS model α  x(k + 1) = Ax(k) + Bu(k) + Eη(k) y(k) = Cx(k) + Du(k) + Fζ (k)

(20)

The fractional linear part is assumed to be completely observable and controllable with matrices ⎡

0 ⎢ .. ⎢ A0 = ⎢ . ⎣0 a1

1

0

0 a2

... ...

⎡ ⎤ ⎤ 0 0 ⎢ .. ⎥ .. ⎥ ⎢ ⎥ .⎥ ⎥ , B0 = ⎢ . ⎥ , C0 = [c1 , c2 , . . . , cn−1 , cn ], D0 = [d] ⎣0⎦ ⎦ 0 1 an−1 an 1 (21)  ...

 and α = α1 α2 · · · αn being the fractional orders vector. The matrices linking the fractional Hammerstein model to the FPNLSS representation is given as follows: A = A 0 B = B 0 C = C0 ⎡ 0 0 0 ... ⎢ .. ⎢ E=⎢ . ⎣0 0 ... 0 p2 p3 . . . pr−1

D = D0 ⎤ 0 .. ⎥ .⎥ ⎥ , F = [p2 D0 0⎦ pr

···

pr D0 ]

(22)

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CHAPTER 22 Identification of fractional Hammerstein system with delay

The objective of this work is the matrices A, C, D, E, F coefficients estimation and the fractional orders vector α. Hence, the parameter vector θ to be estimated is  θ= a

c

d

p

  α = θ˜

 α ∈ R nθ

with  θ˜ = a

c

 p ,

d

p = [1, p2 , . . . , pr ],

a = [a1 , . . . , an ],

c = [c1 , . . . , cn ],

α = [α1 , . . . , αn ]

(23)

The identification procedure is based on a nonlinear optimization approach using L-M algorithm, which is a blend of two well-known optimization methods: the gradient method and the Gauss-Newton method. It is based on the calculation of the gradient and the Hessian by developing the parametric sensitivity functions [30]. The mean quadratic prediction error of the output evaluates cost function J(θ ), it is given by J(θ) =

K K 1  2 1  ε (k) = (y(k) − yˆ (k))2 K K k=1

(24)

k=1

where K is the samples number, ε(k) is the prediction error, and yˆ (k) is the estimated output. The updating equations of the algorithm are as follows: ⎧ ⎪ θ (i+1) = θ (i) − {[J  + λI]−1 J  }θ=θ ˆ (i) ⎪ ⎪   ⎪ ⎪ K K ∂ yˆ (k) ⎪ 2  ⎪ = −2 Jθ = − K k=1 ε(k) ∂θ ⎪ k=1 ε(k)σyˆ (k)/θ the Gradient K ⎪ ⎨   T   ∂ y ˆ (k) ∂ y ˆ (k) K 2 2 T Jθ = K k=1 ∂θ = K K k=1 σyˆ (k)/θ σyˆ (k)/θ the Hessian ⎪ ∂θ ⎪ ⎪ ⎪ ˆ (k) ⎪ ⎪ σyˆ (k)/θ = ∂ y∂θ the output sensitivity function ⎪ ⎪ ⎪ ⎩ λ : a tuning parameter for the convergence

(25)

The gradient J  and the Hessian J  are calculated based on the sensitivity functions ˜ they are obtained by differentiating the computation σyˆ (k)/θ . For the vector θ, FPNLSS with respect to each parameters of θ˜ 

α x(k + 1) = Ax(k) + Bu(k) + Eη(k)

y(k) = Cx(k) + Du(k) + Fζ (k)   ⎧ ∂x(k+1) α ⎪  = ∂A˜ x(k) + A ∂x(k) + ∂B˜ u(k) + B ∂u(k) + E ∂η(k) + ∂E˜ η(k) ⎪ ⎪ ˜ ∂ θ ∂ θi ∂ θ˜i ∂ θi ∂ θ˜i ∂ θ˜i ∂ θi ⎨ i ∂ yˆ (k) = ∂C˜ x(k) + C ∂x(k) + ∂D˜ u(k) + D ∂u(k) + F ∂η(k) + ∂F˜ η(k) ∂ θ˜i ∂ θi ∂ θ˜i ∂ θi ∂ θ˜i ∂ θ˜i ∂ θi i = 1, . . . , nθ˜

⎪ ⎪ ⎪ ⎩

where

∂B ∂ θ˜i

= 0,

∂u(k) ∂ θ˜i

= 0, and

∂η(k) ∂ θ˜i

= 0.

(26)

(27)

5 Simulation examples

Eq. (27) is reduced to  ⎧   ∂A 0 x(k) + ∂E η(k) α [σ ⎪ ⎪  ] = Aσ + ⎪ x(k+1)/θ˜i x(k)/θ˜i ⎪ ∂ θ˜i ∂ θ˜i u(k) ⎨  x(k)  ⎪ σyˆ (k)/θ˜ = Cσx(k)/θ˜ + ∂C˜ ∂D˜ + ∂F˜ η(k) ⎪ ⎪ i i ∂ θi ∂ θi ∂ θi u(k) ⎪ ⎩ i = 1, . . . , nθ˜

(28)

Note that σx(k)/θ˜i = ∂x(k) and σyˆ (k)/θ˜i = ∂ yˆ (k) are, respectively, the state sensitivity ∂ θ˜i ∂ θ˜i function and the output sensitivity function. The overall sensitivity functions model can be written under the FPNLSS form 

α [σx(k+1)/θ˜ ] = As σx(k)/θ˜ + Bs us (k) + Es ηs (k) σyˆ (k)/θ˜ = Cs σx(k)/θ˜ + Ds us (k) + Fs ηs (k)

(29)

As = Diagonal block [A], Cs = Diagonal block[C],     Bs = ∂A˜ 0 , Ds = ∂C˜ ∂D˜ , ∂θ ∂θ ∂θ     ∂E ∂F , Fs = , Es = ∂ θ˜ ∂ θ˜  T us (k) = x(k) u(k) , ηs (k) = [u2 (k) · · · ur (k)]T .

(30)

where

The sensitivity functions with respect to the fractional orders α are calculated using the output Taylor series of order 1 with respect to each αi (i = 1, 2, . . . , n) [16] 

yˆ (k) yˆ (k, αi + δαi ) − yˆ (k, αi ) ≈ δαi ∂∂α = δαi σyˆ (k)/αi i i = 1, . . . , n

(31)

The overall sensitivity functions vector σyˆ (k)/θ of the model is expressed as  σyˆ (k)/θ = σyˆ (k)/θ˜

σyˆ (k)/α

T

Using the parametric sensitivity functions, the gradient Jθ and the Hessian Jθ are expressed as follows: 

 ε(k)(σyˆ (k)/θ ) Jθ = − K2 K  k=1 T Jθ = K2 K (σ k=1 yˆ (k)/θ )(σyˆ (k)/θ )

(32)

5 Simulation examples Two simulation examples are considered, the first of one is a commensurate-order fractional Hammerstein system and a noncommensurate-order system in the second

449

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CHAPTER 22 Identification of fractional Hammerstein system with delay

example. The input u(k) is a persistent excitation sequence of zero mean and unit variance, the disturbance v(k) is a white noise sequence of zero mean and the data length is K = 500. The first step is to obtain a good structure for each example. It is performed by the analysis the criteria evolution for different structures; the best structure corresponding to the smallest criterion. The identification is carried out in the absence of noise and in the presence of noise using the Monte Carlo simulations for different signal-to-noise ratios SNR = 34 dB and SNR = 25 dB.

Example 1: Fractional commensurate case The fractional commensurate Hammerstein system has its linear part and the nonlinear one as follows: u˜ (k) = f (u(k)) = u(k) + 0.75u2 (k) + 0.35u3 (k) α˜  x(k + 1) = A0 x(k) + B0 u˜ (k) y˜ (k) = C0 x(k) + D0 u(k)

(33) (34)

with ⎡

0 A0 = ⎣ 0 0.40

1 0 −0.10

⎤ 0 1 ⎦, −0.60

⎡ ⎤ 0 B0 = ⎣0⎦ , 1

(35)

C0 = [−0.20, −0.80, −0.70], D0 = [0.10].

The fractional order α˜ = 0.3.  θ = 0.40

−0.10

−0.60

−0.20

−0.80

−0.70

0.10

0.75

0.35

0.30



(36)

Before estimating the system parameters, the best structure is chosen based on several tests for different orders n and r. The evolution of the criteria J is shown in Fig. 2 and the obtained values of J are listed in Table 1. The results show that the orders n = 3 and r = 3 correspond to the best structure with J ≈ 1e − 31. Based on the best structure, the presented method is applied for the estimation of the parameter vector θ . Fig. 3 plots the simulation results for the noise-free case, the error is null and the estimated output overlaps with the data. In the presence of noisy measurements, a Monte Carlo simulation is performed for 50 sets of computer realizations for SNR = 34 dB and SNR = 25 dB. The results are summarized in Table 2, where the estimated parameters mean value is recorded with a satisfactory criterion accuracy (J ≈ 10−4 for SNR = 34 dB) and (J ≈ 10−1 for SNR = 25 dB). Figs. 4 and 5 show, respectively, the simulated versus the estimated outputs. It can be concluded that the obtained models show a perfect adequacy with the data.

5 Simulation examples

6

n = 3, r = 2 n = 2, r = 3 n = 3, r = 3

5

Criterion J

4

3

2

1

0

0

2

4

6

8 10 12 14 Number of iterations

16

18

20

22

FIG. 2 Evolution of the criteria versus the number of iterations for the commensurate example.

Table 1 Structure test results of the commensurate example Structure

n=2 r=3

n=3 r=2

n=3 r=3

J

0.670

0.433

7.4e − 31

The statistical performance of the estimator is analyzed on a Monte Carlo simulation, for different amount of noise and a good efficiency of the optimization method is obtained.

Example 2: Fractional noncommensurate example The Hammerstein linear part is a noncommensurate fractional state space model of order n = 2, with the fractional orders vector α = [0.4 0.6]. The linear part matrices are given here: 

0 A0 = −0.37

 1 , −0.58

C0 = [−0.10, −0.20],

  0 B0 = 1

D0 = [0.10]

The nonlinearity is described by the following polynomial of order r = 3: u˜ (k) = f (u(k)) = u(k) + 0.5u2 (k) + 0.25u3 (k)

(37)

451

CHAPTER 22 Identification of fractional Hammerstein system with delay

10

Simulated Estimated

5

Output

0 –5

Zoom –10 –15 0

50

100

150

200

250

300

350

400

450

500

110

120

130

140

150

160

170

180

190

200

200

250

k

5 0 –5 –10 100

(A) 4

× 10–1

3 2

Prediction error

452

1 0 –1 –2 –3 –4 –5

(B)

0

50

100

150

300

350

400

450

500

k

FIG. 3 Identification results for the commensurate example for the noise-free case. (A) Simulated and estimated outputs; (B) prediction error.

Table 2 Monte Carlo simulation results of the commensurate example a2

a3

c1

c2

c3

a1

0.400 −0.100 −0.600 −0.200 −0.800 −0.700 0.100 0.749 0.350 0.320 1.5e − 4

25 dB

0.397 −0.105 −0.586 −0.206 −0.792 −0.709 0.098 0.736 0.344 0.314 0.103

Exact values 0.400 −0.100 −0.600 −0.200 −0.800 −0.70

d

p2

p3

α˜

SNR 34 dB

J

0.100 0.750 0.350 0.300 −

5 Simulation examples

20

Output

0 Zoom

–20

Simulated Estimated

–40 0

100

200

5

300

k

400

500

0 –5 –10 –15 100

110

(A)

120

130

140

150

160

250

300

170

180

190

200

0.03 0.02 0.01 0 –0.01 –0.02 –0.03

(B)

05

0

100

150

200

350

400

450

500

k

FIG. 4 Identification results for the commensurate example for SNR = 34 dB. (A) Simulated and estimated outputs; (B) prediction error.

453

CHAPTER 22 Identification of fractional Hammerstein system with delay

10 Simulated Estimated 0

Output

454

Zoom

–10

k

–20 0

50

100

150

200

250

300

350

400

450

500

110

120

130

140

150

160

170

180

190

200

300

350

5

0

–5

–10 100

(A) 1 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1 0

(B)

50

100

150

200

250

400

450

500

k

FIG. 5 Identification results for the commensurate example for SNR = 25 dB. (A) Simulated and estimated outputs; (B) prediction error.

5 Simulation examples

Table 3 Structure test results of the noncommensurate example Structure

n=2 r=2

n=2 r=3

n=3 r=3

J

0.132

1.2e − 33

1.1e − 07

1.4 n = 3, r = 2 1.2

n = 2, r = 2 n = 2, r = 3

Criterion J

1 0.8 0.6 0.4 0.2 0

0

2

4

6

8

10 12 14 Number of iterations

16

18

20

22

FIG. 6 Evolution of the criteria versus the number of iterations for the noncommensurate example.

The parameters vector to be estimated is as follows:  θ = −0.37

−0.58

−0.10

−0.20

0.10

0.50

0.25

0.40

0.60



(38)

The first step is the examination of the best structure. The obtained values of J for differents n and r ((n = 2, r = 2), (n = 2, r = 3), and (n = 3, r = 3)) are tabulated in Table 3 and the evolution of the different criteria is depicted in Fig. 6. The obtained results show that the best structure is recorded for the exact orders (n = 2, r = 3). The obtained results for the noise-free data and in the presence of noise, that is, SNR = 34 dB and SNR = 25 dB are shown, respectively, in Figs. 7–9. The values of the criterion J and the mean of the parameters using Monte Carlo simulations for 50 runs of noisy data are tabulated in Table 4. On the basis of the presented results, the following conclusions can be drawn: •

In the absence of noise, from the plot illustrated in Fig. 7, we can conclude that the prediction error is null  ≈ 10−16 ) and the estimated output overlap with the simulated one.

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CHAPTER 22 Identification of fractional Hammerstein system with delay

2

0

Output

456

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–2

Simulated Estimated

–4 0

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k

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× 10

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(B)

0

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k

FIG. 7 Noise-free identification results of the noncommensurate example. (A) Simulated and estimated outputs; (B) prediction error.

•

Figs. 8 and 9 depict the obtained results for noisy data. It shows that the prediction errors are very low and the estimated output and the simulated one overlap. • The mean values of the parameters and the errors are given in Table 4. It show that all the parameters are recovered in each case and the errors are very low. On the basis of these numerical simulations, we can conclude that the developed method efficiency is confirmed in the presence of noise and for noise-free data.

5 Simulation examples

2 Simulated Estimated

Output

1 0

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–2 100 10 –3

5

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–10

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0

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k

FIG. 8 Identification results for the noncommensurate example for SNR = 34 dB. (A) Simulated and estimated outputs; (B) prediction error.

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CHAPTER 22 Identification of fractional Hammerstein system with delay

2 1

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Estimated

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0.2 0.15 0.1 0.05 0 –0.05 –0.1 –0.15 0

(B)

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FIG. 9 Identification results for noncommensurate example for SNR = 25 dB. (A) Simulated and estimated outputs; (B) prediction error.

References

Table 4 Monte Carlo simulation results of the noncommensurate example SNR

a1

34 dB

−0.370 −0.580 −0.100 −0.200 0.100 0.500 0.250 0.435 0.538 2.8e − 6

a2

c1

c2

d

p2

p3

25 dB

−0.370 −0.579 −0.099 −0.199 0.100 0.502 0.252 0.419 0.447 0.004

α1

α2

J

Exact values −0.370 −0.580 −0.100 −0.200 0.100 0.500 0.250 0.400 0.600 −

6 Conclusion This chapter presents a novel identification method for the class of fractional nonlinear block-structured systems with time-delay. The fractional Hammerstein system is considered. The flexible polynomial nonlinear state space representation is used to describe the Hammerstein model. This allows a better parameterization of the model and reduces greatly the computational load. The robust L-M algorithm is developed for the parameters estimation of the Hammerstein model as well as its fractional orders. The proposed approach is based on the on parametric sensitivity functions computation, which are implemented as an FPNLSS model. Various simulations show the good performance of the proposed algorithm for the commensurate and the noncommensurate systems. In future work, it is worthwhile to investigate other block-oriented structures identification such as Wiener, Hammerstein-Wiener, etc.

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