Page 137 - Neural Network Modeling and Identification of Dynamical Systems
P. 137

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.
   132   133   134   135   136   137   138   139   140   141   142