adaptive neural network control by adaptive interaction

1

ADAPTIVE NEURAL NETWORK CONTROL BY ADAPTIVE INTERACTION

George Saikalis

Hitachi America, Ltd.

Research and Development Division

34500 Grand River Avenue

Farmington Hills, Michigan 48335

Feng Lin

Wayne State University

Department of Electrical and Computer Engineering

5050 Anthony Wayne Drive

Detroit, Michigan 48202

Abstract

In this paper, we propose an approach to adaptive neural network control by

using a new adaptation algorithm. The algorithm is derived from the theory of

adaptive interaction. The principle behind the adaptation algorithm is a simple but

efficient methodology to perform gradient descent optimization in the parametric

space. Unlike the approach based on the back-propagation algorithm, this

approach will not require the plant to be converted to its neural network

equivalent, a major obstacle in early approaches. By applying this adaptive

algorithm, the same adaptation as the back-propagation algorithm is achieved

without the need of backward propagating the error throughout a feedback

network. This important property makes it possible to adapt the neural network

controller directly. Control of various systems, including non-minimum phase

systems, is simulated to demonstrate the effectiveness of the algorithm.

Keywords: Adaptive Interaction, Adaptive Control, Neural Network Control, and

Back-propagation

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1. Introduction

Since their rebirth in 1980’s, neural networks have found applications in many

engineering fields, including control. For example, neural networks have been

used for system identification [1] [2] [3] and adaptive control [4] [5] [6] [7]. Neural

network controllers can control not only linear systems but also nonlinear

systems [8] [9] [10] [11] [12] [13]. Neural network control designs are divided into

two major categories: (1) the direct design where the controller is a neural

network [14] [15] and (2) the indirect design where the controller is not itself a

neural network, but uses neural networks in its design and adaptation [16] [17].

Issues such as robustness [18] and stability [19] have also been discussed.

Many books on neural network control have been published, including [20] [21]

[22] [23].

There are two major factors that contribute to the popularity of neural networks.

The first factor is the ability of neural networks to approximate arbitrary nonlinear

functions [24] [25]. This is important because in many cases control objectives

can be more effectively achieved by using a nonlinear controller. The second

factor is the capability of neural networks to adapt [25] [26]. In fact, the way for

neural networks to adapt is very natural. It requires no model building or

parameter identification. Such a “natural” adaptivity is rather unique among man-

made systems (but abundant in natural systems). It makes control design a much

easy job. For example, we all know how difficult it is to design a nonlinear

controller. However, if we can let a neural network controller to adapt itself, then

we can sit back and relax. (We know that this will make some people nervous, as

they will insist on the proof of stability.)

To adapt neural networks, many learning (or adaptation) algorithms have been

proposed, the two essential categories being the supervised learning and the

unsupervised learning [5] [25] [26]. Within each of these categories, there are

algorithms for feedback and feedforward neural networks. For the unsupervised

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learning applied to feedback networks, there is the Hopfield and Kohonen

approach among many others. For the unsupervised learning applied to

feedforward networks, there are the learning matrix and counterpropagation. For

the supervised learning applied to feedback networks, there are the Boltzmann

machine, recurrent cascade correlation and learning vector quantization. For the

supervised learning applied to feedforward networks, there are back-propagation,

time delay neural networks and perceptrons. These examples of learning

algorithms are by no means exhaustive; there are many others available in the

literature.

However, there is one main obstacle in the way to adapt neural network

controllers. That is, some most efficient adaptation algorithms such as back-

propagation algorithm cannot be applied directly to neural network controllers. To

use back-propagation algorithm, the system must consist of “pure” neurons. This

is because the back-propagation algorithm relies on a dedicated feedback

network to propagate the error back. No such network can be constructed if the

original system does not consist of pure neurons. However, the neural network

control system is “hybrid” because the plant to be controlled is usually not a

neural network. Therefore is it not possible to apply back-propagation algorithm

to adapt the controller directly.

To bypass this obstacle, people have tried to approximate the plant with a neural

network. But this may not always work because of the error in approximation. So,

what can we do? Fortunately, there is one adaptation algorithm proposed by

Brandt and Lin [27] that can do the same job as the back-propagation algorithm

but requires no feedback networks.

Using Brandt-Lin algorithm, the errors required for adaptation is inferred from

local information in such a way that the error back-propagation is done implicitly

rather than explicitly. As a result, Brandt-Lin algorithm can be implemented in a

simple and straightforward manner without using feedback network.

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Mathematically, however, it can be shown that Brandt-Lin algorithm is equivalent

to the back-propagation algorithm.

Furthermore, Brandt-Lin algorithm can be applied to arbitrary systems, including

hybrid systems as we are dealing with in neural network control systems. This is

because Brandt-Lin algorithm is derived from a theory of adaptive interaction that

is applicable to a large class of systems. For example, it has been applied to self-

tuning PID controllers [28] and parameter estimation [29].

Using Brandt-Lin algorithm, we can adapt a neural network controller directly

without approximating the plant by a neural network. This not only eliminates the

error in approximation, but also significantly reduces the complexity of design.

The rest of the paper will be organized as follows. In Section 2, we will introduce

the theory of adaptive interaction and review Brandt-Lin algorithm for adaptation

in neural networks. In Section 3, we will propose our adaptive neural network

controller and apply Brandt-Lin algorithm to derive the adaptation law for the

controller. Simulation results will be presented in Section 4.

2. Theory and Background of Interactive Adaptation

The proposed adaptation algorithm is based on a recently developed theory of

adaptive interaction [27]. A general adaptation algorithm developed in the theory

of adaptive interaction is applied to adapt the system coefficients. Depending on

the application and configuration of the algorithm, the adjusted coefficients can

be neural network weights, PID gains or transfer function coefficients. To apply

the algorithm to a control system, the only information needed about the plant is

its Fréchet derivative. Furthermore, it will be shown that the Fréchet derivative

can be approximated by a certain constant. This will make the algorithm robust to

system uncertainties and changes and hence can be applicable to a large class

of systems.

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The theory of interactive adaptation considers N subsystems called devices.

Each device (indexed by n ∈ Ν := {1,2,…,N}) has an integrable output signal yn

and an integrable input signal xn. The dynamics of each device is described as a

causal functional:

Ν∈Υ→Χ n,:F nnn

where Χn and Υn are the input and output spaces respectively. Therefore, the

relation between input and output of the nth device is given by:

Ν∈== n)],t(x[F)t)(xF()t(y nnnnn o

where ° denotes functional composition.

The interactions among devices are achieved by connections. Figure 1

shows a graphical illustration of devices and their connections. The set of all

connections is denoted by C.

Figure 1: Devices and their connections

In this paper, the following notations are used to represent relations between

devices and connections:

prec is the device whose output is conveyed by connection c,

postc is the device whose input depends on the signal conveyed by c,

In = { c : prec = n } is the set of input interactions for the nth device, and

On = { c : postc = n } is the set of output interactions for the nth device.

Device 1 Device 2

Device 3

Device 4 Device 5

C1

C2 C3 C4

PreC1 PostC1

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We assume linear interaction among devices and external signal un(t), that is,

∑∈

Ν∈α+=n

cIc

precnn n),t(y)t(u)t(x

where αc are the connection weights.

With this linear interaction, the dynamics of the system is described by

∑∈

∈α+=n

cIc

precnnn Nn],)t(y)t(u[F)t(y

The goal of the adaptation algorithm is to adapt the connection weights αc so the

performance index E(y1, ….,yn, u1, ….,un) as a function of the inputs and outputs

will be minimized. To present the algorithm, we must first introduce the Fréchet

derivatives [30]. As described in [30], let T be a transformation defined on an

open domain D in a normed space X and having range in a normed space Y. If

for a fixed x∈D and each h∈X there exists δT(x;h)∈Y which is linear and

continuous with respect to h such that

lim

0||h|| → 0||h||

||)h;x(T)x(T)hx(T||=

δ−−+

then T is said to be Fréchet differentiable at x and δT(x;h) is said to be the

Fréchet differential of T at x with increment h. In our case, T(x)=Fn(x)and δT(x;h)=

hxFn o)(′ , where )(xFn′ is the Fréchet derivative.

The adaptation algorithm is given in the following theorem [27]. For the sake of

simplicity, the explicit reference to time is removed.

Theorem:

For the system with dynamics given by

∑∈

∈α+=n

cIc

precnnn Nn],yu[Fy

assume that the connection weights αc are adapted according to

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CcyxFy

E

yxFdy

dE

xFdy

dE

ccc

cpost ccss

s

ss

s

prepostpostOs post

postpostpostpost

postpostpost

ssc ∈′∂

∂−

′

′

= ∑∈

,][)][

][

( oo

oo

o

&& γααα

-------- (1)

where γ > 0 is the adaptation coefficient. If (1) has a unique solution for cα& , c∈C

(that is, the Jacobian determinant must not be zero in the region of interest), then

the performance index E(y1, ….,yn, u1, ….,un) will decrease monotonically with

time and the following equation is always satisfied:

Cc,ddE

cc ∈

αγ−=α&

It is important to note that if Fn and E are instantaneous functions, then the

functional composition ° can be replaced by multiplication. Equation (1) will then

be simplified to:

c

ccc

cpostc

c

cc

postprepostpost

Osss

post

prepostpostc y

E.y].x.[F.

y

y].x.[F

∂∂′γ−αα

′=α ∑

∈

&& -------- (2)

The above equations can be applied to a very general class of systems, including

neural networks, as shown below.

A neural network will be decomposed in multiple devices as described in Figure

1. Figure 2 shows a graphical representation for a simple neural network.

Figure 2: A simple neural network

x1

x2

Σ

Σ

Σ

σ(.)

Log-Sig

Log-Sig

w1

w2

w3

w4

w5

w6

r1

r2

p3

p4

r3

r4

p5

σ(.)

σ(.) r5

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Here we use the notation commonly used in neural networks as follows.

n is the label for a particular neuron;

s is the label for a particular synapse;

Dn is the set of dendritic (input) synapses of neuron, n;

An is the set of axonic (output) synapses of neuron, n;

pres is the presynaptic neuron corresponding to synapse, s;

posts is the postsynaptic neuron corresponding to synapse, s;

ws is the strength (weight) of synapse, s;

pn is the membrane potential of neuron, n;

rn is the firing rate of neuron, n;

γ is the direct feedback coefficient for all neurons;

fn is the direct feedback signal; and

σ is the sigmoidal function; xe1

1)x( −+

=σ .

Mathematically, the neural network and adaptation algorithm are described as

follows.

∑∈

=n

sDs

presn rwp

)p(r nn σ=

If we denote

∑∑∈∈

==φnn As

ssAs

2sn www

dtd

21 & -------- (3)

then by applying the adaptation law in (2), the weight adaptation becomes:

)f)p((rwssss postpostpostpres γ+−σφ=& -------- (4)

Equations (3) and (4) describe Brandt-Lin algorithm for adaptation in neural

networks. As shown in [27], it is equivalent to back-propagation algorithm but

requires no feedback network to back-propagate the error.

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3. Adaptive Neural Network Controller

We now apply Brandt-Lin adaptation algorithm to neural network control. The

proposed closed loop configuration of a neural network control system is shown

in Figure 3.

Figure 3: Neural network based control system

To be more specific, the neural network controller have two inputs e1 and e2. e1 is

the error between the set point and the plant output and e2 is a delayed signal

based on e1.

The reason for introducing e2 is as follows. Since the neural network controller is

itself a memory-less device, in order for control output to depend not only on the

current input (error in our case), but also on past inputs, some delayed signals

must be introduced. In this paper, we will consider only one simple delayed

signal. However, in principle, multiple delayed signals can be introduced (that is,

the neural network controller will have more than two inputs). Hence, the

configuration of the neural network controller is further described in Figure 4.

Neural Network Controller

Input Excitation Signal

Proposed Neural Network

Adaptation Algorithm

Σ error-

+

W1 W2 | | Wn

Plant Gp(s)

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Figure 4: Neural network controller

If we use the simple neural network with two hidden neurons as in Figure 2, then

the neural network controller is shown in Figure 5. More sophisticated neural

network can be used to improve the performance.

Figure 5: Adaptive neural network controller configuration

In Figure 5, we propose two ways to configure he output stage of the controller:

1) tangent sigmoid at the output and 2) constant gain output.

The reason behind the tangent sigmoid (tan-sig) is the ability to provide a dual

polarity signal to the output. Based on simulation results, the simple constant

gain output will also work and often provide a better result.

Mathematically, the input-output relations of neurons are as follows:

Σ Neural Network Controller

Gp(s)

Delay

r e1 y +

-

e2

e1

e2

Σ

Σ

Σ

Log-Sig

+/- Sig

A

Log-Sig

w1

w2

w3

w4

w5

w6

r1

r2

p3

p4

r3

p5

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r1 = e1 and r2 = e2

p3 = w1r1 + w2r2 and p4 = w3r1 + w4r2

r3 = σ(p3) and r4 = σ(p3)

p5 = w5r3 + w6r4

Let

E = 21e = ( r – y )2 = r2 – 2.y.r + y2

Then

1e.2)yr.(2y.2r.2yE

−=−−=+−=∂∂ .

Apply Brandt-Lin algorithm of Equations (3) and (4), we have

)p(..e)0.)p((rw 3313311 −σφ=γ+−σφ=&

)p(..e)0.)p((rw 3323322 −σφ=γ+−σφ=&

)p(..e)0.)p((rw 4414413 −σφ=γ+−σφ=&

)p(..e)0.)p((rw 4424424 −σφ=γ+−σφ=&

where 553 ww &=φ and 664 ww &=φ .

The adaptation law for w5 and w6 is more complicated as it is linked to the plant

to be controlled. By Equation (2), since Opostc is empty, we have

)e.2.(r].u.[F.w 13post5 c−′γ−=&

If the Fréchet derivative is approximated by a constant that will be absorbed in γ,

then the above expression is approaximated by

135 e.r.w γ=&

Similarly,

146 e.r.w γ=&

The constant γ is considered as the adaptation rate or learning rate. It will be

varied to analyze the rate of adaptation of the neural network controller.

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4. Simulation results

4.1. Matlab/Simulink model

To demonstrate the theory described previously, software simulation has been

performed. The MatLab/Simulink model is shown in Figure 6.

Figure 6: Simulink model of the adaptive neural network controller

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4.2. Effects of initial weights

This section covers the investigation of the effect of the initial weights on the

convergence of the algorithm. The following elements are set during the

simulation.

Plant: )474.2s)(526.21s(s

76.88)s(G

++=

Input Signal: Amplitude: 10

Type: Sinewave

requency: 0.01 Hz

Output Stage: Tangent Sigmoid

Learning Rate: γ=10

The results of four simulations with different initial weights are shown in Figures

7-10 and summarized in Table 1. It is observed that te initial weights must have

opposite signs in the hidden units of the neuron connection link.

Table 1: Effects of the initial weights on adaptation

Figure Number Initial Weights Results (500s)

Figure 7 W1=-100, W2=100, W3=100

W4=-100, W5=-100, W6=100

Adapted

Figure 8 W1=-1, W2=1, W3=1

W4=-1, W5=-1, W6=1

Adapted

Figure 9 W1=1, W2=1, W3=1

W4=1, W5=1, W6=1

Not Adapting

Figure 10 W1=100, W2=100, W3=100

W4=100, W5=100, W6=100

Not Adapting

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Figure 7

Figure 8

15

Figure 9

Figure 10

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4.3. Effects of learning rates

This section covers the effects of the learning rates on the adaptation. The

following elements are set during simulation.

Plant: )474.2s)(526.21s(s

76.88)s(G

++=

Input Signal: Type: Sinewave

Amplitude: 5

Offset: 5

Frequency: 0.01 Hz

Output Stage: Tangent Sigmoid

Initial Weights: W1=-100, W2=100, W3=100

W4=-100, W5=-100, W6=100

The results of three simulations with different learning rates are shown in Figures

11-13 and summarized in Table 2. It is observed that the larger the learning rate,

the faster the algorithm will adapt. However, if the learning rate is too large, the

output may not be robust and may lead to the system breaking-up. Also, the

weights converge to local minima depending on the different learning rates.

Table 2: Effects of the learning rate on the adaptation algorithm

Figure Number Learning Rate Results (500s)

Figure 11 γ=100 Adapted



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Figure 11

Figure 12

18

Figure 13

4.4. Effects of input frequency with output gain versus tan-sigmoid

This section covers the effect of changing the output stage from a tan-sigmoid to

a constant gain. The following elements are set during simulation.

Plant: )474.2s)(526.21s(s

76.88)s(G

++=


Amplitude: 10



W4=-100, W5=-100, W6=100

The results of four simulations with two different frequencies are shown in

Figures 14-17 and summarized in Table 3. It is observed that it is easier for the

controller to adapt if the input frequency is low. Also, a constant gain output

provide better adaptation at higher input frequencies.

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Table 3: Effects of the input frequency and output stage on adaptation

Figure Number Input Frequency Output Stage Results (500s)

Figure 14 0.01 Hz Tan-sigmoid Adapted

Figure 15 0.01 Hz Gain=0.001 Adapted

Figure 16 0.1 Hz Tan-sigmoid Not Adapted

Figure 17 0.1 Hz Gain=0.001 Adapted

Figure 14

Figure 15

20

Figure 16

Figure 17

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4.5. Effect of different plants

To further validate the adaptation algorithm, the neural network based adaptive

controller is applied to different plants.

Plants: )5)(10(

1000)(2

++=

sssG

)100)(5(

5000)(3

++=

ssssG

)100)(5)(1(

5000)(4

+++=

ssssG


Amplitude: 10

Output Gain: 0.001


W4=-100, W5=-100, W6=100

We change the learning rate and input frequency for these plants and see how

high the frequency can be increased. The results of five simulations are shown in

Figures 18-22 and summarized in Table 4. It is observed that the input frequency

can be increased to 10 Hz for G2(s). With third order plants G3(s) and G4(s), a

maximum of 1 Hz input signal is possible. Note that G3(s) is open loop unstable.

Table 4: Effects of input frequency and learning rate on G2(s)

Figure Number Plant Input Frequency Learning Rate Results (500s)

Figure 18 G2(s) 0.01 Hz γ=10 Adapted

Figure 19 G2(s) 0.01 Hz γ=100 Adapted

Figure 20 G2(s) 10 Hz γ=100 Adapted



22

Figure 18

Figure 19

23

Figure 20

Figure 21

24

Figure 22

4.6. Application to non-minimum phase systems

A non-minimum phase system has either a pole or a zero in the right-half of the

s-plane. Since it is well known that it is difficult to apply adaptive control to non-

minimum phase systems, we decide to test the following non-minimum phase

system.

Plants: )5s)(1s(

500)s(6G

+−=


Amplitude: 10


The results of four simulations with different frequencies, output stage and initial

weights are shown in Figures 23-26 and summarized in Table 5. It is observed

that the weight adaptation does occur. The adaptation convergence depends on

two factors: (1) Frequency of the input signal and (2) the magnitude of the initial

weights. The constant gain (= 0.001) is required when dealing with large initial

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weights and higher frequency. It was found that the tangent sigmoid is suited

when the initial weights are small and the input frequency is low.

Table 6: Effects of the input frequency, output stage and initial weights on G6(s).

Figure

Number

Input

Frequency

Output

Stage

Initial Weights Results (500s)

Figure 23 0.01 Gain=

0.001

W1=-100, W2=100, W3=100

W4=-100, W5=-100, W6=100

Adapted

Figure 24 1 Gain=

0.001

W1=-100, W2=100, W3=100

W4=-100, W5=-100, W6=100

Adapted

Figure 25 0.01 tan-sig W1=-1, W2=1, W3=1

W4=-1, W5=-1, W6=1

Adapted

Figure 26 1 Gain=

0.001

W1=-1, W2=1, W3=1

W4=-1, W5=-1, W6=1

Adapted

Figure 23

26

Figure 24

Figure 25

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Figure 26

5. Conclusion

The application of theory of adaptive interaction to adaptive neural network

control results in a new direct adaptation algorithm that works very well.

Simulation results show the following characteristics of the algorithm.

- Learning works well with a variety of second and third order plants.

- Controlled plants can be open loop stable or unstable.

- Maximum input frequency depends on the plant order.

- For higher input frequencies and large initial weights the output stage

with a constant gain works better.

- The initial weights must be non-zero and have alternating polarity.

- Faster learning rates are required for higher input frequencies.

- Adaptation is applicable to both minimum phase and non-minimum

phase plants

This new approach does not require the transformation of the continuous time

domain plant into its neural network equivalent. Another benefit for applying the

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proposed algorithm is that it does not require a separate feedback network to

back propagate the error. The adaptation algorithm is mathematically isomorphic

to the back-propagation algorithm.

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adaptive neural network control by adaptive interaction

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