# Stability Analysis of Neural Networks-Based System Identification

Authors: Talel Korkobi; Mohamed Djemel; Mohamed Chtourou
Journal: Modelling and Simulation in Engineering
 (2008)

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## Abstract

This paper treats some problems related to nonlinear systems identification. A stability analysis neural network model for identifying nonlinear dynamic systems is presented. A constrained adaptive stable backpropagation updating law is presented and used in the proposed identification approach. The proposed backpropagation training algorithm is modified to obtain an adaptive learning rate guarantying convergence stability. The proposed learning rule is the backpropagation algorithm under the condition that the learning rate belongs to a specified range defining the stability domain. Satisfying such condition, unstable phenomena during the learning process are avoided. A Lyapunov analysis leads to the computation of the expression of a convenient adaptive learning rate verifying the convergence stability criteria. Finally, the elaborated training algorithm is applied in several simulations. The results confirm the effectiveness of the CSBP algorithm.


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## Body

## 1. Introduction

The last few decades have witnessed
the use of artificial neural networks (ANNs) in many real-world applications
and have offered an attractive paradigm for a broad range of adaptive complex
systems. In recent years, ANNs have enjoyed a great deal of success and have
proven useful in wide variety pattern recognition feature-extraction tasks.
Examples include optical character recognition, speech recognition, and
adaptive control, to name a few. To keep the pace with the huge demand in diversified
application areas, many different kinds of ANN architecture and learning types
have been proposed to meet varying needs as robustness and stability.The area of system identification has
received significant attention over the past decades and now it is a fairly
mature field with many powerful methods available at the disposal of control
engineers. Online system identification methods to date are based on recursive
methods, such as least squares, for most systems that are expressed as linear
in the parameters.During the past few years, several authors [1–3] have
suggested neural networks for nonlinear dynamical black-box modelling. The
problem of designing a mathematical model of a process using only observed data
has attracted much attention, both from an academic and an industrial point of
view. Neural models can be used either as simulators or as models.Recently, feedforward neural networks
have been shown to obtain successful results in system identification and
control [4]. Such neural networks are static input/output mapping schemes that
can approximate a continuous function to an arbitrary degree of accuracy.
Results have also been extended to recurrent neural networks [5, 6].Recent results show that neural
network technique seems to be very effective to identify a broad category of
complex nonlinear systems when complete model information cannot be obtained. The Lyapunov approach has been used directly to
obtain stable training algorithms for continuous-time neural networks [7–9].
The stability of neural networks can be found in [10, 11]. The stability of
learning algorithms has been discussed in [6, 12].It
is well known that conventional identification algorithms are stable for ideal
plants [13–15]. In the presence of disturbances or unmodeled dynamics, these
adaptive procedures can go to instability easily. The lack of robustness in
parameters identification was demonstrated in [10] and became a hot issue in
1980s. Several robust modification techniques were proposed in [13, 14]. The weight-adjusting
algorithms of neural networks are a type of parameters identification; the normal-gradient
algorithm is stable when neural-network model can match the nonlinear plant
exactly [6]. Generally, some modifications to the normal-gradient algorithm or
backpropagation should be applied, such that the learning
process is stable. For example, in [12, 16], some hard restrictions were added to
the learning law, and in [11], the dynamic backpropagation has been modified
with NLq stability constraints.The paper is organized as follows. Section2 describes the
neural identifier structure considered in this paper and the usual
backpropagation algorithm. In Section 3 and through a stability analysis, a
constrained adaptive stable backpropagation algorithm (CSBP) is proposed to provide stable
adaptive updating process.
Three simulation examples give the effectiveness of the suggested algorithm in Section
4.


## 2. Preliminaries

The main concern of this section is to introduce the
feedforward neural network, which is the adopted architecture, as well as some
concepts of backpropagation training algorithm. Consider the following
discrete-time input-output nonlinear system:(1)y(k+1)=F[y(k)⋯y(k−n+1),u(k)⋯u(k−m+1)]. The neural model for the plant can be
expressed as (2)ym(k+1)=F^(Y(k),θ), where Y(k)=(y(k),y(k−1),…,y(k−n+1),u(k),u(k−1),…,u(k−m+1)), and θ=[W,V]T is the weight parameter vector for the neural model.A typical multilayer feedforward
neural network is shown in Figure1, where Ij is
the jth hidden neuron input, Oj is the jth hidden neuron output i,j, and k indicate neurons, wij is the weight between neuron i and neuron j,
while  vj is
the weight between neuron j and output neuron. For all neurons, the
nonlinear activation function is defined as (3)f(x)=11+e−x.
The output ym(k) of the considered NN is (4)Ij=∑i=1n+m+1xiwij,Oj=f(Ij),j=1,…,N,Ik=∑j=1NOjvj,ym(k)=f(Ik),k=1. Training
the neural model is to adjust the weight parameters so that it emulates the
nonlinear plant dynamics. Input-output training is obtained from the operation
history of the plant.Figure 1
Feedforward neural model.Using the gradient descent, the weight connecting neuroni
to neuron j is updated as(5)wij(k+1)=wij(k)−ε⋅∂J(k)∂wij(k),vj(k+1)=vj(k)−ε⋅∂J(k)∂vj(k), where J(k)=(1/2)[y(k+1)−ym(k+1)]2, and ε is the learning rate. The partial derivatives are calculated with respect to the vectors of
weights W and V, (6)∂J(k)∂vj(k)=f′(Ij)⋅(y(k+1)−ym(k+1))Oj,∂J(k)∂wij(k)=f′(Ij)[∑j=1Lf′(Ij)(y(k+1)−ym(k+1))vj]xi.    Backpropagation
algorithm has become the most popular one for training of the multilayer
perceptron [1]. Generally, some modifications to the normal-gradient algorithm
or backpropagation should be applied, such that the learning
process is stable. For example, in [12, 16], some hard restrictions were added in
the learning law, and in [11], the dynamic backpropagation was modified with
stability constraints.The research on modified algorithms
of feedforward neural networks is becoming a challenging field. These researches involve the development of
heuristic techniques, which arise out of a study of the distinctive performance
of standard backpropagation algorithm. These heuristic
techniques include such ideas as varying the learning rate [17], using momentum
[18], and rescaling variables [19].


## 3. Stability Analysis and CSBP Algorithm  Formulation

In the literature,
the Lyapunov synthesis [4, 5] consists of the selection of a positive function candidate V which leads to the computation of an
adaptation law insuring its
decrescence,  that is, V˙≤0 for continuous systems and ΔV(k)=V(k+1)−V(k)≤0 for discrete-time systems. Under these
assumptions, the function V is called Lyapunov function and guarantees
the stability of the system. Our objective is the determination of a
stabilizing adaptation law ensuring the stability of the identification scheme
presented in what follows and the boundness of the output signals. The stability of the learning process in an
identification approach leads to a better modelling and a guaranteed reached performance. The proposed learning rule is the backpropagation
algorithm adopting a constrained learning rate. Satisfying such condition,
unstable phenomena during the learning process are avoided. This problem has been treated in the literature of neural
identification so this work is considered as a solution for extended problems.
The originality of this work consists of the constraints themselves. In fact, a
choice of the learning rate with respect to the proposed constraints ensures an
efficient stable identification which is not the case adopting an arbitrary
learning rate especially when it does not belong to the specified stability
domain. In the proposed one and through the original calculation results, the
learning rate is iterative and computed instantaneously with respect to the
elaborated constraints. The following assumptions are made
for system (1).Assumption 1.
The unknown nonlinear functionF(⋅) is continuous
and differentiable.Assumption 2.
System outputy(k) can be measured and its initial values are assumed to be in a compact set Ω0.Theorem 1.
The
stability in Lyapunov sense of the identification scheme is guaranteed for a
learning rate verifying the following inequality:(7)0≤ε≤2tr(Jθ⋅θ)‖Jθ‖2, where Jθ=[∂J/∂W(k),∂J/∂V(k)]
denotes the gradient of J with respect
to θ.Proof.
Considering
the Lyapunov function(8)VL(k)=tr(θT(k)⋅θ(k)), where tr(⋅) denotes the matrix trace operation, θ(k)=θ(k)−θ*, and  θ* denotes the
optimal value of the weight vector parameters.
The
computation of theΔVL(k) expression leads to (9)ΔVL(k)=VL(k+1)−VL(k), where (10)VL(k+1)=tr(WT(k+1)W(k+1))+VT(k+1)V(k+1),VL(k)=tr(WT(k)W(k))+VT(k)V(k). The
adopted adaptation law is the gradient algorithm. We have (11)θ(k+1)=θ(k)−ε⋅Jθ−θ*,W(k+1)=W(k)−ε∂J∂W(k)−W*,V(k+1)=V(k)−ε∂J∂V(k)−V*,   where the partial derivatives are expressed as (12)∂J∂W(k)=∂J∂ym(k+1)∂ym(k+1)∂W(k),∂J∂V(k)=∂J∂ym(k+1)∂ym(k+1)∂V(k). The partial derivatives are given through (13)[∂J∂Wij(k)]=[ym(k+1)(1−ym(k+1))⋅e(k)⋅∑j=1NVj1⋅Oj⋅(1−Oj)]xi,[∂J∂Vj(k)]=[ym(k+1)(1−ym(k+1))⋅e(k)⋅Oj]. Let A and B be
defined as follows: (14)A=tr(WT(k+1)W(k+1))−tr(WT(k)W(k)),B=VT(k+1)V(k+1)−VT(k)V(k). The ΔVL(k) expression is calculated as (15)ΔV(k)=A+B=tr(ε2[∂J∂WT(k)∂J∂W(k)]−2ε∂J∂W(k)W(k))+(ε2[∂J∂VT(k)∂J∂V(k)]−2ε∂J∂V(k)V(k))=(ε2∑i,j[∂J∂Wij(k)]2−2εtr(∂J∂Wij(k)WT(k))+(ε2∑j[∂J∂Vj(k)]2−2ε∂J∂V(k)VT(k)))=ε2(∑i,j[∂J∂Wij(k)]2+∑j[∂J∂Vj(k)]2)−2ε(∂J∂V(k)VT(k)+tr(∂J∂W(k)WT(k)))≤ε2(∑i,j[∂J∂Wij(k)]2+∑j[∂J∂Vj(k)]2)−2ε(∂J∂V(k)VT(k)+tr(∂J∂W(k)WT(k)))≤α⋅ε2−2⋅β⋅ε, where (16)α=∑i,j[∂J∂Wij(k)]2+∑j[∂J∂Vj(k)]2=‖Jθ‖2,β=∂J∂V(k)VT(k)+tr(∂J∂W(k)WT(k))=tr(Jθ⋅θ). The
stability condition ΔVL(k)≤0 is satisfied only if (17)α⋅ε2−2⋅β⋅ε≤0. Solving
this ε, second-degree equation leads to the establishment of the result
presented in (7); ΔV(k)≤0 if ε satisfies the following condition:  (18)0≤ε≤εs,  where  εs=2tr(Jθ⋅θ)/∥Jθ∥2.
Using the expressions ofJθ and θ, we obtain (19)εs=2[tr(H1⋅WT(k)+H2⋅VT(k))]D+∑j([ym(k+1)(1−ym(k+1))⋅e(k)⋅Oj])2, where H1 denotes {[ym(k+1)(1−ym(k+1))⋅e(k)⋅∑j=1NV(j,1)⋅Oj⋅(1−Oj)]xi}j∈[1⋯m]i∈[1⋯n], H2 denotes {[ym(k+1)(1−ym(k+1))⋅e(k)⋅Oj]}j∈[1⋯m], and  D denotes ∑i,j([ym(k+1)(1−ym(k+1))⋅e(k)⋅∑j=1NV(j,1)⋅Oj⋅(1−Oj)]xi)2.
The
previous result is useful in the case of a backpropagation adaptation law
adopting the same learning rate for training in all the neural network
architecture. An extension is made in the next section. This extension consists
in the fact of
considering two different constrained learning rates which improve the
efficiency of the first elaborated algorithm.Theorem 2.
LetεI¯=[εW,εV]T be the learning rates for the tuning
parameters of the neural identifier θ=[W,V]T and let λ  be
defined as (20)λ=[λ1,λ2]T,whereλ1=∂ym(k)∂W(k),λ2=∂ym(k)∂V(k). Then asymptotic convergence is
guaranteed if the learning rates are chosen to satisfy (21)εW≺2(λ1max⁡)2,εV≺2(λ2max⁡)2, where (22)λ1max⁡=‖∂ym(k)∂W(k)‖2=max⁡([∂ym(k)∂W(k)]T⋅[∂ym(k)∂W(k)]),λ2max⁡=‖∂ym(k)∂V(k)‖2=max⁡([∂ym(k)∂V(k)]T⋅[∂ym(k)∂V(k)]).Lemma 1.
If the learning rates are chosen asεW=εV=ε, then one has the
convergence condition (23)ε≺1(λ1max⁡)2+1(λ2max⁡)2.Proof.
Considering
the Lyapunov function(24)VL(k)=12e2(k), where (25)e(k)=y(k)−ym(k). The
computation of the ΔVL(k) expression leads to (26)ΔVL(k)=VL(k+1)−VL(k),ΔVL(k)=12(e2(k+1)−e2(k))=Δe(k)[e(k)+12Δe(k)], where Δe(k)=[∂e(k)/∂θ(k)]TΔθ(k).
The
expression ofΔW(k) is given by (27)Δθ(k)=θ(k+1)−θ(k),ΔWI(k)=εI¯⋅e(k)⋅∂ym(k)∂θ(k). Substituting
the expression of Δe(k) in ΔVL(k), we have (28)ΔVL(k)=[∂e(k)∂θ(k)]TεI¯e(k)∂ym(k)∂θ(k)⋅{e(k)+12[∂e(k)∂θ(k)]TεI¯e(k)∂ym(k)∂θ(k)}=−[∂ym(k)∂θ(k)]TεI¯e(k)∂ym(k)∂θ(k)⋅{e(k)−12[∂ym(k)∂θ(k)]TεI¯e(k)∂ym(k)∂θ(k)}=−[∂ym(k)∂θ(k)]TεI¯e2(k)∂ym(k)∂θ(k)⋅{1−12[∂ym(k)∂θ(k)]TεI¯∂ym(k)∂θ(k)}=−e2(k)⋅{12[∂ym(k)∂V(k)]TεV[∂ym(k)∂V(k)]×(2−[∂ym(k)∂V(k)]TεV[∂ym(k)∂V(k)])+12[∂ym(k)∂W(k)]TεW[∂ym(k)∂W(k)]×(2−[∂ym(k)∂W(k)]TεW[∂ym(k)∂W(k)])}=−e2(k)⋅{12εV‖[∂ym(k)∂V(k)]‖2×(2−εV‖[∂ym(k)∂V(k)]‖2)+12εW‖[∂ym(k)∂W(k)]‖2×(2−εW‖[∂ym(k)∂W(k)]‖2)},ΔVL(k)≤0, so (29)2−εW‖∂ym(k)∂W(k)‖2≤0,2−εV‖∂ym(k)∂V(k)‖2≤0. Finally,
when we define the matrix norm ∥⋅∥2 by (30)‖ρ‖2=max⁡(ρT⋅ρ), the
theorem results are established.
The
stability conditionΔVL(k)≤0 is satisfied only if (31)εW≺2(λ1max⁡)2,εV≺2(λ2max⁡)2, where (32)λ1max⁡=‖∂ym(k)∂W(k)‖2=max⁡([∂ym(k)∂W(k)]T⋅[∂ym(k)∂W(k)]),λ2max⁡=‖∂ym(k)∂V(k)‖2=max⁡([∂ym(k)∂V(k)]T⋅[∂ym(k)∂V(k)]),λ1max⁡=‖∂ym(k)∂W(k)‖2=max⁡([∂ym(k)∂W(k)]T⋅[∂ym(k)∂W(k)]),λ2max⁡=‖∂ym(k)∂V(k)‖2=max⁡([∂ym(k)∂V(k)]T⋅[∂ym(k)∂V(k)]).Remark 1.
Through
simulations, learning rates are chosen belonging to the defined learning rates
stability range to prove the effectiveness of the proposed CSBP algorithm. The learning rate which guarantees convergence corresponds to(33)εW=2ϕ+(λ1max⁡)2,εV=2ϕ+(λ2max⁡)2, where ϕ is a small value guarantying the convergence
stability condition.


## 4. Simulation Results

In this section, two discrete time systems are considered to
demonstrate the effectiveness of the result discussed below.


### 4.1. First-Order System

The considered system is a famous
one in the literature of neural adaptive control and identification. The system
is described by the following recurrent equation [2]: (34)y(k+1)=y(k)1+y(k)2+u(k)3. For the neural model, a three-layer NN was selected with two inputs,
three hidden and one output nodes. Sigmoidal activation functions were employed
in all the nodes.The weights
are initialized to small random values. The learning rate is evaluated at each
iteration through (21). It is also recognized that the training performs very
well when the learning rate is small.As input signal, a sinusoidal one is
chosen in which the expression is defined by(35)u(k)=0.5⋅cos⁡(0.05kπ+π5). The
simulations are realized in the two cases during 120 iterations. Two learnning rate values are fixed in
and out of the learning rate range presented in (7).Simulation
results are given through Figures2 and 3.Figure 2
Evolution of the system output and
the neural model output (ε∈stabilitydomain).Figure 3
Evolution of the system output and
the neural model output (ε∉stabilitydomain).Figures2 and 3 show that if the learning rate belongs to the range
defined in (7), the stability of the identification scheme is guaranteed. It is
shown through this simulation that the identification objectives are satisfied.
Out of this
variation domain of the learning rate, the identification is instable and the identification
objectives are unreachable.


### 4.2. Second-Order System

An example is used to illustrate the effectiveness of the
proposed constrained updating
law. Consider a nonlinear discrete time plant(36)y(k)=50tanh[ϕ(k−1)]+0.5u(k−1), where ϕ(k−1)=2.102(((24+y(k−1))/3)y(k−1)−8(u2(k−1)/(1+u2(k−1)))y(k−2)).The
process dynamic is interesting. In fact it has the behaviour of a first-order lowpass filter for input
signal amplitude about 0.1, the behaviour of a linear second-order system in
the case of small amplitudes (0.1<|u|<0.5),
and the behaviour of a nonlinear second-order system in the case of great input
amplitudes (0.5<|u|<5) [20].For the neural model, a three-layer NN was selected with three inputs,
three hidden and one output nodes. Sigmoidal activation functions were employed
in all the nodes.The weights are initialized to small
random values. The learning rate parameter is computed instantaneously. As
input signal, a sinusoidal one is chosen which the expression is defined by(37)u(k)=0.5⋅cos⁡(0.005kπ+π3). The
simulations are realized in the two cases during 120 iterations. Two learning
rates values are fixed in and out of the learning rate range presented in (7).Simulation
results are given through Figures4 and 5.Figure 4
Evolution of the system output and
the neural model output (ε∈stabilitydomain).Figure 5
Evolution of the system output and
the neural model output (ε∉stabilitydomain).The simulation results, through Figures4 and 5, show
that a learning rate arbitrarily chosen out of the predefined stability domain leads to an unstable
identification of the considered system; however, a specified learning rate
belonging to the range verifying stability condition ensures the tracking capability and the
stability of the identification schemeExample 1 : (identification of semiconductor manufacturing process).
This example illustrates the advantage and effectiveness of
our approach-on-line self-tuning property (stability). We consider here the
SISO simple first-order linear process of the form [21] (38)y(k+1)=ϕy(k)+α+βu(k)+N(k+1), where α and β are process parameters, ϕ is
the autoregressive coefficient, and N denotes the noise term that
follows an ARMA process: (39)N(k)=1−cz−11−wz−1⋅r(k). In this simulation, r(k) is
a uniform distribution and the system parameters are chosen as (40)α=2,β=2,ϕ=0.1,c=0.7,w=1.0,r(k)=1−2⋅rand(k). Here,
the current output of the plant depends on four previous outputs and four
previous inputs. In this case, the feedforward neural network, with four input
nodes for feeding the appropriate past values of y(k) and u(k) were used. In this paper, only four values are fed into the FFNN to determine
the output ym(k). In training the FFNN, we used 100 epochs. The testing input signal is used to determine the identification results and is given by (37).
The
weights are initialized to very small random values. The learning rate
parameter is calculated each iteration.
The
simulations are realized in the two cases. Two learning rates values are fixed
in and out of the learning rate range presented in (21) (see Figures 6
and 7).
In order to compare the performances
of different learning rate,
rules are chosen in and out the learning rate stability range.
Adopting an adaptive constrained
learning rate inside the stability domain, faster convergence, stability, and
tracking capability are guaranteed.Figure 6
Evolution of the system output and
the neural model output (ε∈stabilitydomain).Figure 7
Evolution of the system output and the neural
model output (ε∉stabilitydomain).


## 4.1. First-Order System

The considered system is a famous
one in the literature of neural adaptive control and identification. The system
is described by the following recurrent equation [2]: (34)y(k+1)=y(k)1+y(k)2+u(k)3. For the neural model, a three-layer NN was selected with two inputs,
three hidden and one output nodes. Sigmoidal activation functions were employed
in all the nodes.The weights
are initialized to small random values. The learning rate is evaluated at each
iteration through (21). It is also recognized that the training performs very
well when the learning rate is small.As input signal, a sinusoidal one is
chosen in which the expression is defined by(35)u(k)=0.5⋅cos⁡(0.05kπ+π5). The
simulations are realized in the two cases during 120 iterations. Two learnning rate values are fixed in
and out of the learning rate range presented in (7).Simulation
results are given through Figures2 and 3.Figure 2
Evolution of the system output and
the neural model output (ε∈stabilitydomain).Figure 3
Evolution of the system output and
the neural model output (ε∉stabilitydomain).Figures2 and 3 show that if the learning rate belongs to the range
defined in (7), the stability of the identification scheme is guaranteed. It is
shown through this simulation that the identification objectives are satisfied.
Out of this
variation domain of the learning rate, the identification is instable and the identification
objectives are unreachable.


## 4.2. Second-Order System

An example is used to illustrate the effectiveness of the
proposed constrained updating
law. Consider a nonlinear discrete time plant(36)y(k)=50tanh[ϕ(k−1)]+0.5u(k−1), where ϕ(k−1)=2.102(((24+y(k−1))/3)y(k−1)−8(u2(k−1)/(1+u2(k−1)))y(k−2)).The
process dynamic is interesting. In fact it has the behaviour of a first-order lowpass filter for input
signal amplitude about 0.1, the behaviour of a linear second-order system in
the case of small amplitudes (0.1<|u|<0.5),
and the behaviour of a nonlinear second-order system in the case of great input
amplitudes (0.5<|u|<5) [20].For the neural model, a three-layer NN was selected with three inputs,
three hidden and one output nodes. Sigmoidal activation functions were employed
in all the nodes.The weights are initialized to small
random values. The learning rate parameter is computed instantaneously. As
input signal, a sinusoidal one is chosen which the expression is defined by(37)u(k)=0.5⋅cos⁡(0.005kπ+π3). The
simulations are realized in the two cases during 120 iterations. Two learning
rates values are fixed in and out of the learning rate range presented in (7).Simulation
results are given through Figures4 and 5.Figure 4
Evolution of the system output and
the neural model output (ε∈stabilitydomain).Figure 5
Evolution of the system output and
the neural model output (ε∉stabilitydomain).The simulation results, through Figures4 and 5, show
that a learning rate arbitrarily chosen out of the predefined stability domain leads to an unstable
identification of the considered system; however, a specified learning rate
belonging to the range verifying stability condition ensures the tracking capability and the
stability of the identification schemeExample 1 : (identification of semiconductor manufacturing process).
This example illustrates the advantage and effectiveness of
our approach-on-line self-tuning property (stability). We consider here the
SISO simple first-order linear process of the form [21] (38)y(k+1)=ϕy(k)+α+βu(k)+N(k+1), where α and β are process parameters, ϕ is
the autoregressive coefficient, and N denotes the noise term that
follows an ARMA process: (39)N(k)=1−cz−11−wz−1⋅r(k). In this simulation, r(k) is
a uniform distribution and the system parameters are chosen as (40)α=2,β=2,ϕ=0.1,c=0.7,w=1.0,r(k)=1−2⋅rand(k). Here,
the current output of the plant depends on four previous outputs and four
previous inputs. In this case, the feedforward neural network, with four input
nodes for feeding the appropriate past values of y(k) and u(k) were used. In this paper, only four values are fed into the FFNN to determine
the output ym(k). In training the FFNN, we used 100 epochs. The testing input signal is used to determine the identification results and is given by (37).
The
weights are initialized to very small random values. The learning rate
parameter is calculated each iteration.
The
simulations are realized in the two cases. Two learning rates values are fixed
in and out of the learning rate range presented in (21) (see Figures 6
and 7).
In order to compare the performances
of different learning rate,
rules are chosen in and out the learning rate stability range.
Adopting an adaptive constrained
learning rate inside the stability domain, faster convergence, stability, and
tracking capability are guaranteed.Figure 6
Evolution of the system output and
the neural model output (ε∈stabilitydomain).Figure 7
Evolution of the system output and the neural
model output (ε∉stabilitydomain).


## 5. Conclusion

To avoid unstable phenomenon during the learning
process, constrained stable backpropagation algorithm is proposed (CSBP). A
stable adaptive updating process
is guaranteed. A Lyapunov analysis is made in order to extract new updating formulations which
contain a set of inequality constraints. Both
the convergence rate and the tracking capability of the CSBP algorithm are
mainly determined by the learning rate. For a larger learning rate, one has the
faster convergence but the poorer tracking capability; while for a smaller
learning rate, one gets
the slower convergence but the better tracking capability. With The CSBP
algorithm, faster convergence, stability, and tracking capability are
guaranteed. The applicability and the effectiveness
of the approach presented are proved through simulation examples.



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*Source: 343940-2009-01-11.xml*