Outline Introduction Representation of Dynamical Systems Identification Model Example 1 Example 2 Example 3 Computational Intelligence Lecture 8: Identification Using Neural Networks Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2011 Farzaneh Abdollahi Neural Networks Lecture 8 1/30
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Outline Introduction Representation of Dynamical Systems Identification Model Example 1 Example 2 Example 3
Computational IntelligenceLecture 8: Identification Using Neural
Networks
Farzaneh Abdollahi
Department of Electrical Engineering
Amirkabir University of Technology
Fall 2011Farzaneh Abdollahi Neural Networks Lecture 8 1/30
Outline Introduction Representation of Dynamical Systems Identification Model Example 1 Example 2 Example 3
Introduction
Representation of Dynamical SystemsStatic NetworksDynamic Networks
I Behavior of systems in practice are mostly described by dynamical models.
I ∴ Identification of dynamical systems is desired in this lecture.
I In identification problem, it is always assumed that the system is stable
Farzaneh Abdollahi Neural Networks Lecture 8 3/30
Outline Introduction Representation of Dynamical Systems Identification Model Example 1 Example 2 Example 3
Representation of Dynamical Systems by Neural Networks1. Using Static Networks: Providing the
dynamics out of the network and apply staticnetworks such as multilayer networks (MLN).
I Consists of an input layer, output layerand at least one hidden layer
I In fig. there are two hidden layers withthree weight matrices W1,W2 and W3
and a diagonal nonlinear operator Γ withactivation function elements.
I Each layer of the network can berepresented by Ni [u] = Γ[Wiu].
I The I/O mapping of MLN can berepresented by y = N[u] =Γ[W3Γ[W2Γ[W1u]]] = N3N2N1[u]
I The weights Wi are adjusted s.t min afunction of the error between thenetwork output y and desired output yd .
Farzaneh Abdollahi Neural Networks Lecture 8 4/30
Outline Introduction Representation of Dynamical Systems Identification Model Example 1 Example 2 Example 3
Using Static Networks
I The universal approximation theorem shows that a three layers NN with abackpropagation training algorithm has the potential of behaving as auniversal approximator
I Universal Approximation Theorem: Given any ε > 0 and any L2
function f : [0, 1]n ∈ Rn → Rm, there exists a three-layerbackpropagation network that can approximate f within ε mean-squareerror accuracy.
Farzaneh Abdollahi Neural Networks Lecture 8 5/30
Outline Introduction Representation of Dynamical Systems Identification Model Example 1 Example 2 Example 3
Using Static Networks
I Providing dynamical terms to inject tostatic networks:
Outline Introduction Representation of Dynamical Systems Identification Model Example 1 Example 2 Example 3
I The model for identification purposes:
˙x = Ax + g(x , u)
I The identification scheme is based on the parallel configurationI The states of the model are fed to the input of the neural network.I an MLP with at least three layers can represent the nonlinear function g as:
g(x , u) = Wσ(V x)
I W and V are the ideal but unknown weight matricesI x = [x u]T ,I σ(.) is the transfer function of the hidden neurons that is usually considered
as a sigmoidal function:
σi (Vi x) =2
1 + exp−2Vi x− 1
I where Vi is the ith row of V,I σi (Vi x) is the ith element of σ(V x).
Outline Introduction Representation of Dynamical Systems Identification Model Example 1 Example 2 Example 3
Case Study: Simulation Results on SSRMSI The Space Station Remote Manipulator System (SSRMS) is a 7 DoF
robot which has 7 revolute joints and two long flexible links (booms).
I The SSRMS have no uniform mass and stiffness distributions. Most of itsmasses are concentrated at the joints, and the joint structural flexibilitiescontribute a major portion of the overall arm flexibility.
I Dynamics of a flexible–link manipulator
M(q)q + h(q, q) + Kq + F q = u
I u = [τT 01×m]T , q = [θT δT ]T ,I θ is the n × 1 vector of joint variablesI δ is the m × 1 vector of deflection variablesI h = [h1(q, q) h2(q, q)]T : including gravity, Coriolis, and centrifugal forces;I M is the mass matrix,
I K =
[0n×n 0n×m
0m×n Km×m
]is the stiffness matrix,
I F = diag{F1,F2}: the viscous friction at the hub and in the structure,I τ : input torque.
Outline Introduction Representation of Dynamical Systems Identification Model Example 1 Example 2 Example 3
Example 3 [3]
I A gray box identification,( the system model is known but it includessome unknown, uncertain and/or time-varying parameters) isproposed using Hopfield networks
I Consider
x = A(x , u(t))(θn + θ(t))
y = x
I y is the output,I θ is the unknowntime-dependantdeviation from the nominal valuesI A is a matrix that depends on the input u and the state xI y and A are assumed to be physically measurable.
I Objective: estimating θ (i.e. min the estimation error: θ = θ − θ).
Outline Introduction Representation of Dynamical Systems Identification Model Example 1 Example 2 Example 3
ReferencesA. Yazdizadeh and K. Khorasani, “Nonlinear system identification using adaptive timedelay neural netowrks,” in World Congress in Neural Networks, 1996.
H. A. Talebi, F. Abdollahi, R. V. Patel and K. Khorasani, Neural Network-based StateEstimation of Nonlinear Systems: Application to Fault Detection and Isolation.
Springer, 2009.
M. Atencia, G. Joya, and F. Sandoval, “Hopfield neural networks for parametricidentification of dynamical systems,” Neural Processing Letters, vol. 12 , pp. 143–152,2005.
K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systemsusing neural networks,” IEEE Trans. on Neural Networks, vol. 1, no. 1, pp. 4–27, March1990.
J. B. Cabrera and K. S. Narendra, “Issues in the application ofneural networks fortracking based on inverse control,” IEEE Trans. on Automatic Control, vol. 44, no. 11,pp. 2007–2025, Nov. 1999.
S. Mukhopadhyay and K. S. Narendra, “Disturbance rejection in nonlinear systems usingneural networks,” IEEE Trans. on Neural Networks, vol. 4, no. 1, pp. 63–72, Jan 1993.