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Outline Introduction Representation of Dynamical Systems Identiﬁcation Model Example 1 Example 2 Example 3
Neural Networks
Lecture 8: Identiﬁcation Using Neural
Networks
H.A Talebi
Farzaneh Abdollahi
Department of Electrical Engineering
Amirkabir University of Technology
H. A. Talebi, Farzaneh Abdollahi Neural Networks Lecture 8 1/32

Introduction
Representation of Dynamical Systems
Static Networks
Dynamic Networks
Identiﬁcation Model
Direct modeling
Inverse Modeling
Example 1
Case Study
Example 2
Numerical Example
Example 3

Engineers desired to model the systems by mathematical models.
This model can expressed by operator f from input space u into an output space
y.
System Identification problem: is finding ˆf which approximates f in desired
sense.
Identification of static systems: A typical example is pattern recognition:
Compact sets ui ∈ Rn
are mapped into elements yi ∈ Rm
in the output
Identification of dynamic systems: The operator f is implicitly defined by
I/O pairs of time function u(t), y(t), t ∈ [0, T] or in discrete time:
y(k + 1) = f (y(k), y(k − 1), ..., y(k − n), u(k), ..., u(k − m)), (1)
In both cases the objective to determine ˆf is
ˆy − y = ˆf − f ≤ , for some desired > 0.
Behavior of systems in practice are mostly described by dynamical models.
∴ Identification of dynamical systems is desired in this lecture.
In identification problem, it is always assumed that the system is stable

Representation of Dynamical Systems by Neural Networks
1. Using Static Networks: Providing the
dynamics out of the network and apply static
networks such as multilayer networks (MLN).
Consists of an input layer, output layer
and at least one hidden layer
In ﬁg. there are two hidden layers with
three weight matrices W1, W2 and W3
and a diagonal nonlinear operator Γ with
activation function elements.
Each layer of the network can be
represented by Ni [u] = Γ[Wi u].
The I/O mapping of MLN can be
represented by y = N[u] =
Γ[W3Γ[W2Γ[W1u]]] = N3N2N1[u]
The weights Wi are adjusted s.t min a
function of the error between the
network output y and desired output yd .

Using Static Networks
The universal approximation theorem shows that a three layers NN with a
backpropagation training algorithm has the potential of behaving as a
universal approximator
Universal Approximation Theorem: Given any > 0 and any L2
function f : [0, 1]n ∈ Rn → Rm, there exists a three-layer
backpropagation network that can approximate f within mean-square
error accuracy.

Providing dynamical terms to inject to
static networks:
1. Tapped-Delay-Lines (TDL): Consider (1)
for identiﬁcation (I/O Model)
y(k + 1) = f (y(k), y(k − 1), ..., y(k − n),
u(k), ..., u(k − m)),
Dynamical terms u(k − j), y(k − i) for
i = 1, ..., n, j = 1, ..., m is made by
delay elements out of the network and
injected to the network as input.
The static network is employed to
approximate the function f
∴ The model provided by the network
will be
ˆy(k + 1) = ˆf (ˆy(k), ˆy(k − 1), ...,
ˆy(k − n), u(k), ..., u(k − m)),

Considering State Space model:
x(k + 1) = f (x(k), x(k − 1), ..., x(k − n), u(k), ..., u(k − m)),
y(k) = Cx(k)

2 Filtering
in continuous-time networks the
delay operator can be shown by
integrator.
The dynamical model can be
represented by an MLN , N1[.], + a
transfer matrix of linear function,
W (s).
For example:
˙x(t) = f (x, u)±Ax,
where A is Hurwitz. Define
g(x, u) = f (x, u) − Ax
˙x = g(x, u) + Ax
Fig, shows 4 configurations using
filter.

Representation of Dynamical Systems by Neural Networks
2. Using Dynamic Networks: Time-Delay
Neural Networks (TDNN) [?] , Recurrent
networks such as Hopﬁeld:
Consists of a single layer network N1,
included in feedback conﬁguration and a
time delay
Can represent discrete-time dynamical
system as :
x(k + 1) = N1[x(k)], x(0) = x0
If N1 is suitably chosen, the solution of the
NN converge to the same equilibrium
point of the system.
In continuous-time, the feedback path has
a diagonal transfer matrix with 1/(s − α)
in diagonal.
∴ the system is represented by
˙x = αx + N1[x] + I

Neural Networks Identification Model
Two principles of identification problems:
1. Identification model
2. Method of adjusting its parameters based on
identification error e(t)
Identification Model
1. Direct modeling:
it is applicable for control, monitoring,
simulation, signal processing
The objective: output of NN ˆy converge to
output of the system y(k)
∴ the signal of target is output of the system
Identification error e = y(k) − ˆy(k) can be
used for training.
The NN can be a MLN training with BP, such
that minimizes the identification error.
The structure of identification shown in Fig
named Parallel Model

Direct Modeling
Drawback of parallel model:
There is a feedback in this
model which some times
makes convergence diﬃcult
or even impossible.
2. Series-Parallel Model
In this model the
output of system is
fed to the NN

Inverse Modeling
It is employed for the control
techniques which require inverse
dynamic
Objective is finding f −1, i.e.,
y → u
Input of the plant is target, u
Error identification is defined
e = u − û

Example 1: Using Filtering
Consider the nonlinear system
˙x = f (x, u) (2)
u ∈ Rm
: input vector, x ∈ Rn
: state vector, f (.): an unknown function.
Open loop system is stable.
Objective: Identifying f
Deﬁne ﬁlter:
Adding Ax to and subtracting from (2), where A is an arbitrary Hurwitz
matrix ˙x = Ax + g(x, u) (3)
where g(x, u) = f (x, u) − Ax.
Corresponding to the Hurwitz matrix A, M(s) := (sI − A)−1
is an n × n matrix
whose elements are stable transfer functions.

The model for identification purposes:
˙ˆx = Aˆx + ˆg(ˆx, u)
The identification scheme is based on the parallel configuration
The states of the model are fed to the input of the neural network.
an MLP with at least three layers can represent the nonlinear function g as:
g(x, u) = W σ(V ¯x)
W and V are the ideal but unknown weight matrices
¯x = [x u]T
,
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
where Vi is the ith row of V,
σi (Vi ¯x) is the ith element of σ(V ¯x).

g can be approximated by NN as
ˆg(ˆx, u) = ˆW σ( ˆV ˆ¯x)
The identiﬁer is then given by
˙ˆx(t) = Aˆx + ˆW σ( ˆV ˆ¯x) + (x)
(x) ≤ N is the neural network’s bounded approximation error
the error dynamics:
˙˜x(t) = A˜x + ˜W σ( ˆV ˆ¯x) + w(t)
˜x = x − ˆx: identiﬁcation error
˜W = W − ˆW , w(t) = W [σ(V ¯x) − σ( ˆV ˆ¯x)] − (x) is a bounded
disturbance term, i.e, w(t) ≤ ¯w for some pos. const. ¯w, due to the
sigmoidal function.
Objective function J = 1
2(˜xT ˜x)

Training:
Updating weights:
˙ˆW = −η1(
∂J
∂ ˆW
) − ρ1 ˜x ˆW
˙ˆV = −η2(
∂J
∂ ˆV
) − ρ2 ˜x ˆV
Therefore:
netˆv = ˆV ˆ¯x
netˆw = ˆW σ( ˆV ˆ¯x).
∂J
∂ ˆW
and ∂J
∂ ˆV
can be computed according to
∂J
∂ ˆW
=
∂J
∂netˆw
.
∂netˆw
∂ ˆW
∂J
∂ ˆV
=
∂J
∂netˆv
.
∂netˆv
∂ ˆV

∂J
∂netˆw
=
∂J
∂˜x
.
∂˜x
∂ˆx
.
∂ˆx
∂netˆw
= −˜xT
.
∂ˆx
∂netˆw
∂J
∂netˆv
=
∂J
∂˜x
.
∂˜x
∂ˆx
.
∂ˆx
∂netˆv
= −˜xT
.
∂ˆx
∂netˆv
and
∂netˆw
∂ ˆW
= σ( ˆV ˆ¯x)
∂netˆv
∂ ˆV
= ˆ¯x
∂ ˙ˆx(t)
∂netˆw
= A
∂ˆx
∂netˆw
+
∂ˆg
∂netˆw
∂ ˙ˆx(t)
∂netˆv
= A
∂ˆx
∂netˆv
+
∂ˆg
∂netˆv
.
Which is dynamic BP. Modify BP algorithm s.t. the static
approximations of ∂ˆx
∂netˆw
and ∂ˆx
∂netˆv
(˙ˆx = 0)

Thus, ∂ˆx
∂netˆw
= −A−1
∂ˆx
∂netˆv
= −A−1 ˆW (I − Λ( ˆV ˆ¯x))
where
Λ( ˆV ˆ¯x) = diag{σ2
i ( ˆVi ˆ¯x)}, i = 1, 2, ..., m.
Finally
˙ˆW = −η1(˜xT
A−1
)T
(σ( ˆV ˆ¯x))T
− ρ1 ˜x ˆW
˙ˆV = −η2(˜xT
A−1 ˆW (I − Λ( ˆV ˆ¯x)))T ˆ¯xT
− ρ2 ˜x ˆV
˜W = W − ˆW and ˜V = V − ˆV ,
It can be shown that ˜x, ˜W , and ˜V ∈ L∞
The estimation error and the weights error are all ultimately bounded [?].

Series-Parallel Identiﬁer
The function g can be approximated
byˆg(x, u) = ˆW σ( ˆV ¯x)
Only ˆ¯x is changed to ¯x.
The error dynamics
˙˜x(t) = A˜x + ˜W σ( ˆV ¯x) + w(t) where
w(t) = W [σ(V ¯x) − σ( ˆV ¯x)] + (x)
only deﬁnition of w(t) is changed.
Applying this change, the rest remains
the same

Using Dynamic BP Without Static Approximation
∂J
∂ ˆW
=
∂J
∂˜x
.
∂˜x
∂ˆx
.
∂ˆx
∂netˆw
.
∂netˆw
∂ ˆW
= −˜xT
.
∂ˆx
∂netˆw
.σ( ˆV ˆ¯x)
∂J
∂ ˆV
=
∂J
∂˜x
.
∂˜x
∂ˆx
.
∂ˆx
∂netˆv
.
∂netˆv
∂ ˆV
= −˜xT
.
∂ˆx
∂netˆv
ˆ¯x
and dw =
∂ˆx(t)
∂netˆw
˙dw =
∂ ˙ˆx(t)
∂netˆw
= Adw +
∂ˆg
∂netˆw
= Adw + 1 (4)
dv =
∂ˆx(t)
∂netˆv
˙dv =
∂ ˙ˆx(t)
∂netˆv
= Adv +
∂ˆg
∂netˆv
= Adv + ˆW (I − Λ( ˆV ˆ¯x)) (5)

Using Dynamic BP Without Static Approximation
Finally
˙ˆW = η1(˜xT
dw )T
(σ( ˆV ˆ¯x))T
− ρ1 ˜x ˆW
˙ˆV = η2(˜xT
dv ))T ˆ¯xT
− ρ2 ˜x ˆV
In learning rule procedure, ﬁrst (4) and (5) should be solved then the
weights W and V is updated

Case Study: Simulation Results on SSRMS
The Space Station Remote Manipulator System (SSRMS) is a 7 DoF
robot which has 7 revolute joints and two long flexible links (booms).
The SSRMS have no uniform mass and stiffness distributions. Most of its
masses are concentrated at the joints, and the joint structural flexibilities
contribute a major portion of the overall arm flexibility.
Dynamics of a flexible–link manipulator
M(q)¨q + h(q, ˙q) + Kq + F ˙q = u
u = [τT
01×m]T
, q = [θT
δT
]T
,
θ is the n × 1 vector of joint variables
δ is the m × 1 vector of deflection variables
h = [h1(q, ˙q) h2(q, ˙q)]T
: including gravity, Coriolis, and centrifugal forces;
M is the mass matrix,
K =
0n×n 0n×m
0m×n Km×m
is the stiffness matrix,
F = diag{F1, F2}: the viscous friction at the hub and in the structure,
τ: input torque.

http://english.sohu.com/20050729/n226492517.shtml

A joint PD control is applied to stabilize the closed-loop system
boundedness of the signal x(t) is assured.
For a two link flexible manipulator
x = [θ1... θ7
˙θ1... ˙θ7 δ11 δ12 δ21 δ22
˙δ11
˙δ12
˙δ21
˙δ22]T
The input: u = [τ1, ..., τ7]
A is defined as A = −2I ∈ R22×22
Reference trajectory: sin(t)
The identifier:
Series-parallel
A three-layer NN network: 29 neurons in the input layer, 20 neurons in the
hidden layer, and 22 neurons in the output layer.
The 22 outputs correspond to
7 joint positions
7 joint velocities
4 in-plane deflection variables
4 out-of plane deflection variables
The learning rates and damping factors: η1 = η2 = 0.1, ρ1 = ρ2 = 0.001.

Simulation results for the SSRMS: (a-g) The joint positions, and (h-n)
the joint velocities.

Example 2: TDL
Consider the following nonlinear
system
y(k) = f (y(k − 1), ..., y(k − n))
+b0u(k) + ... + bmu(k − m)
u: input, y:output, f (.): an
unknown function.
Open loop system is stable.
Objective: Identifying f
Series-parallel identiﬁer is applied.
β = [b0, b1, ..., bm]
Cost function: J = 1
2 e2
i where
ei = y − yh,

Consider Linear in parameter MLP,
In sigmoidal function.σ, the weights of first layer is fixed V = I:
σi (¯x) = 2
1+exp−2¯x − 1
Updating law: w = −η( ∂J
∂w )
∴ ∂J
∂w = ∂J
∂ei
∂ei
∂w = −ei
∂N(.)
∂w
∂N(.)
∂w is obtained by BP method.
Numerical Example: Consider a second order system
yp(k + 1) = f [yp(k), yp(k − 1)] + u(k)
where f [yp(k), yp(k − 1)] =
yp(k)yp(k−1)[yp(k)+2.5]
1+y2
p (k)+y2
p (k−1)
.
After checking the stability system
Apply series-parallel identifier
u is random signal informally is distributed in [−2, 2]
η = 0.25

Numerical Example Cont’d
The outputs of the plant and the model after the identiﬁcation procedure

Example 3 [?]
A gray box identiﬁcation,( the system model is known but it includes
some unknown, uncertain and/or time-varying parameters) is
proposed using Hopﬁeld networks
Consider
˙x = A(x, u(t))(θn + θ(t))
y = x
y is the output,
θ is the unknowntime-dependantdeviation from the nominal values
A is a matrix that depends on the input u and the state x
y and A are assumed to be physically measurable.
Objective: estimating θ (i.e. min the estimation error: ˜θ = θ − ˆθ).

At each time interval assume time
is frozen so that
Ac = A(x(t), u(t)), yc = y(t)
Recall Gradient-Type Hopefield
C du
dt = Wv(t) + I
the weight matrix and the bias
vector are defined:
W = −AT
c Ac, I = AT
c Acθn −AT
c yc
The convergence of the identifier is
proven using Lyapunov method
It is examined for an idealized
single link manipulator
¨x = −g
l sinx − v
ml2 ˙x + 1
ml2 u
assume A = (sinx, ˙x, u) and
θn + θ = (−g
l , − v
ml2 , 1
ml2 )

References
K.S. Narendra and K. Parthasarathy, “Identiﬁcation and control of dynamical systems
using neural networks,” IEEE Trans. on Neural Networks, vol. 1, no. 1, pp. 4–27, March
1990.

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