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126 3. NEURAL NETWORK BLACK BOX APPROACH TO THE MODELING AND CONTROL OF DYNAMICAL SYSTEMS
By the known criterial function (3.63), we can [2] Codrons B. Process modelling for control: A unified
now construct an optimality criterion (3.50)or framework using standard black-box techniques. Lon-
don: Springer; 2005.
(3.51) for the neurocontroller (and then we can [3] Narendra KS, Parthasarathy K. Identification and con-
build the ENC) for the case under considera- trol of dynamic systems using neural networks. IEEE
tion. For definiteness, let us use the guarantee- Trans Neural Netw 1990;1(1):4–27.
ing approach to evaluate the ENC efficiency for [4] Chen S, Billings SA. Neural networks for nonlinear dy-
the MDS (3.57)–(3.62)(see(3.52)), that is, the re- namic systems modelling and identification. Int J Con-
trol 1992;56(2):319–46.
quired criterion has the form
[5] Sjöberg J, Zhang Q, Ljung L, Benveniste A, Deylon B,
Glorennec PY, et al. Nonlinear black-box modeling in
F( ) = max f(λ,k). (3.66) system identification: A unified overview. Automatica
λ∈ ),
k=const 1995;31(12):1691–724.
[6] Juditsky A, Hjalmarsson H, Benveniste A, Deylon B,
In this case, for = ( 0 , 1 ), the neurocontroller Ljung L, Sjöberg J, et al. Nonlinear black-box modeling
in system identification: Mathematical foundations. Au-
0 implements the distribution function E(λ) =
tomatica 1995;31(12):1725–50.
1, ∀λ ∈ . [7] Rivals I, Personnaz L. Black-box modeling with state-
If the value of the maximum degree of nonop- space neural networks. In: Zbikowski R, Hint KJ, edi-
timality (3.57) for the obtained ENC is greater tors. Neural Adaptive Control Technology. World Sci-
than allowed by the conditions of the solved ap- entific; 1996. p. 237–64.
[8] Billings SA, Jamaluddin HB, Chen S. Properties of neu-
plication, it is possible to increase the number ral networks with applications to modelling nonlinear
N of the neurocontrollers i ,i = 1,...,N,in , dynamic systems. Int J Control 1992;55(1):193–224.
thereby decreasing the value of the index (3.66). [9] Chen S, Billings SA. Representation of non-
For example, let N = 3. Then linear systems: The narmax model. Int J Control
1989;49(3):1013–32.
[10] Chen S, Billings SA. Nonlinear system identification us-
= ( 0 , 1 , 2 , 3 ),
ing neural networks. Int J Control 1990;51(6):1191–214.
⎧ [11] Chen S, Billings SA, Cowan CFN, Grant PM. Practical
⎪1,λ 0 λ λ 12 ,
⎨ identification of narmax using radial basis functions. Int
E(λ) = 2,λ 12 <λ λ 23 , J Control 1990;52(6):1327–50.
⎪
⎩ [12] Kalman RE, Falb PL, Arbib MA. Topics in mathematical
3,λ 23 <λ λ k .
system theory. New York, NY: McGraw Hill Book Com-
pany; 1969.
To obtain a numerical estimate, we set a = [13] Mesarovic MD, Takahara Y. General systems theory:
b = 1, c 0 = 1, c 1 = 2, c 2 = 5, x 2 = 10, λ 0 = 0, Mathematical foundations. New York, NY: Academic
max
λ k = 1. It can be shown that in the given prob- Press; 1975.
lem k − ≈ 0.4, k + ≈ 0.9. Then in the considered [14] Dreyfus G. Neural networks: Methodology and appli-
cations. Berlin ao.: Springer; 2005.
case for the ENC with N = 1 the value of the
[15] Chen S, Wang SS, Harris C. NARX-based non-
index (3.66) in the problem (3.55)ofENCparam- linear system identification using orthogonal least
∗
∗
eter optimization will be F = F( ,k ) ≈ 0.42, squares basis hunting. IEEE Trans Control Syst Technol
and for N = 3 (i.e., with three “working” neu- 2008;16(1):78–84.
[16] Sahoo HK, Dash PK, Rath NP. NARX model based non-
rocontrollers and one “switching” in the ENC)
linear dynamic system identification using low com-
∗
F = F( ,k ) ≈ 0.28. plexity neural networks and robust H ∞ filter. Appl Soft
∗
Comput 2013;13(7):3324–34.
[17] Hidayat MIP, Berata W. Neural networks with ra-
REFERENCES dial basis function and NARX structure for mate-
rial lifetime assessment application. Adv Mater Res
2011;277:143–50.
[1] Billings SA. Nonlinear system identification: NARMAX [18] Wong CX, Worden K. Generalised NARX shunting neu-
methods in the time, frequency and spatio-temporal do- ral network modelling of friction. Mech Syst Signal Pro-
mains. New York, NY: John Wiley & Sons; 2013. cess 2007;21:553–72.