Upload
t-a
View
215
Download
3
Embed Size (px)
Citation preview
A Comparison Study between Static and Dynamic
Recurrent Neural Networks Based Adaptive
Control Of Nonlinear Multivariable Systems T. A. Al-Zohairy
Community collage in ALRiyadh
King Saud University
ALRiyadh, Kingdom of Saudi Arabia
Abstract – This paper considers the problem of real time adaptive control of nonlinear multivariable systems. Two neural networks techniques are presented to solve the
problem mentioned above. The first technique combines the
ability of a single-layer feedforward neural network for
modeling purposes and a linear control law to design the controller. The second technique combines the ability of dynamic recurrent neural network for modeling purposes
and a linear control law to design the controller. In this paper, we consider that the state of the system is accessible. A
comparison between the simulation results for the above two techniques are presented to complete the study.
I. INTRODUCTION
There has been considerable interest in the past few
years in exploring the applications of artificial neural
networks for identification and adaptive control of
dynamical systems [1]-[7].
It has been realized by systems theorists that most real
dynamical systems are nonlinear [6]. However,
linearizations of such systems around the equilibrium
states yield linear models which are mathematically
tractable.
From [9], it was found that the discrete-time linear multivariable system with m inputs and m outputs
described by the state equations
)()(
)()()1(
kCxky
kBukAxkx
=
+=+ (1)
where nkx ℜ∈)( is the state vector at instant k,
m)k(u ℜ∈ is the input vector, and mky ℜ∈)( is the
output vector, can be represented by
)k(uE)k(xAC)dk(y
)k(uE)k(xAC)dk(y
)k(uE)k(xAC)dk(y
md
mmm
2d
222
1d
111
m
2
1
+=+
+=+
+=+
LLL
(2)
where BACE idii
1−= and id is the relative degree of the
ith output iy . In the following we will see that the form
(2) is important for the control developments given in
sections II and III. The adaptive linear controller for the linear
multivariable system of the form (2) is given by
).()()()(
)(12
1
12
1
kFrkGxkrEkx
AC
AC
AC
Eku
mdm
d
d
+=+
= −−
L
(3)
In the following we will show how (3) can be used as an
adaptive linear controller for a nonlinear multivariable
system.
Consider a discrete-time nonlinear multivariable system
S be described by the state equations
S:x(k) Cy(k)
k)]f[x(k), u()1x(k
=
=+ (4)
where ,)k(y ),k(u mℜ∈ and n)k(x ℜ∈ are the input,
output, and state, respectively, at time t and ∞∈Cf .
Consider the reference input )k(r is defined as
mTm21 )]k(r , ),k(r ),k(r[)k(r ℜ∈= L is specified and
the state variables are accessible. The control problem in
this paper is to determine the input )k(u so that
)k(yi follows the reference input )k(ri . Note that we
consider that f is unknown and only input, output state
variables are known.
In this paper, two control strategies are proposed, the
first combines a single-layer feedforward network model
with self tuning indirect adaptive control and the second
combines a dynamic recurrent neural network model with
self tuning indirect adaptive control. The proposed control
structures are based on linearization of both single-layer
feedforward and dynamic recurrent network models at every operating point. A standard linear state space model
of the form (2) is derived in both strategies and a state
space feedback decoupling controller of the form (3) is
applied.
With simultaneous online training of both single layer
feedforward neural network and recurrent neural network
with control synthesis the resulting two algorithms are
indirect adaptive control laws.
Some prior information concerning the system is
required to determine the controller which are the order of
the system and relative degree id of each output iy . For
known system some additional conditions are required to
determine the controller as given in [8] which are:
1) the linearized system SL is controllable observable and
can be decoubled using state feedback.
2) the plant satisfies the minimum phase condition.
The paper is organized as follows: In section II two
neural network architectures used for the identification
process is given. Section III study the linearization of the
nonlinear process obtained from the neural network. In
section IV the algorithms that used to solve the control
problem is stated in details. Section V gives the simulation
study on a real example for the two methods mentioned
above. A comparison between simulation results for the
two presented methods is given in section VI and the
conclusion of our study is shown in section VII.
II. NETWORK ARCHITECTURES
The network architectures used in this paper depends on
the concept of relative degree of a nonlinear multivariable
system S given in (4).
It can be shown that if the vector relative degree exists for the multivariable system S in some neighborhood Ω of
the equilibrium point ( 0u ,0 ==x ), the nonlinear system
S can be expressed as
)]( ),([)(
)]( ),([)(
]u(k) ),([)(
222
111
kukxdky
kukxdky
kxdky
mmm Ψ=+
Ψ=+
Ψ=+
LLL (5)
Assume that the unknown nonlinear multivariable
system to be considered is expressed by (5) where
,)( ),( mdkyku ℜ∈+ and nkx ℜ∈)( . The two architectures
are described as follows:
A. Single-layer Feedforward Network and its Dynamics
The single-layer feedforward network architecture used
in this case is shown in Fig. 1, in which the input layer is
composed of )1mn( ++ nodes for n states, m inputs all at
time k and an additional node for input bias its value is always 1 and the output layer of m nodes for the m outputs
of the system at times m21 dk , ,dk ,dk +++ L . The
interconnection matrices are n,myxW ℜ∈ and 1m,myuW +ℜ∈ , respectively among the output-state nodes
and output-input nodes.
Fig. 1. The neural network structure.
The dynamics of the network is described by the following
equations:
][ˆ)(ˆ jjj Sdky Ψ=+ (6)
∑∑+
==
+=1m
1
yuj
n
1i
iyxjij )k(uw)k(xwS
l
ll (7)
The activation functions Ψ are hyperbolic tangent given
by (8)
+
−
−
ℜ∈+
−=Ψ b a, )(ˆ
asas
asas
ee
eebs (8)
Then the neuro model for the system (5) can be expressed
as
]u(k) ),([ˆ)(ˆ kxdky jj Ψ=+ (9)
B. Recurrent Network Architecture and its Dynamics
The recurrent network architecture taken in this paper is
that given in [10] in which a recurrent network is
composed of n neuron units, and m external inputs.
The dynamics of each neuron unit is described by
∑ ∑∈ ∈
+=+U X
xk
ukk twtwts
l l
llll )(x )(u )1( (10)
)1(ˆ)1(ˆ +=+ tsftx kk (11)
In the above equation xkwl
represents the weight of
connection from the thl neuron to the kth neuron, ukwl
represents the weight of the connection from the thl input
neuron to the kth neuron, )(tsk is an intermediate variable
and f is a nonlinear function, an hyperbolic tangent,
given by (12):
+−
−
ℜ∈+
−= b a, )ˆ(ˆ
ˆˆ
ˆˆ
xaxa
xaxa
ee
eebxf (12)
This equation can be rewritten, similarly to a nonlinear
state space model
Wu(t), ),(ˆˆ)1(ˆ txftx =+ (13)
Fig. 2 shows a recurrent network with three neuron units
and two external inputs. For this particular example the
neuron outputs are given as follows with 2 ,1 ,0=l .
)( )( )(
)( )( )( ˆ)1(
352413
322110
tuwtuwtuw
txwtxwtxwftx
lll
llll
++
+++=+ (14)
where )(3 tu is an additional input for bias its value is
always 1.
Fig. 2. Example of a recurrent network with three neuron units and two inputs.
The epochwise backpropagation through time algorithm
[10] is used to train a recurrent network.
III. LINEARIZATION OF THE NEURAL MODELS
The linearized models for the two nonlinear neural
models (9) and (13) can be obtained by computing the
derivatives from the outputs with respect to the inputs of
the network (x(k),u(k)).
A. Linearization of the Single-Layer Feedforward Network
The linearized system for the single layer network is
given by
u(k) K)k(x H)dk(y jj +=+ (15)
where the two matrices H and K are of orders )nm( × and
)1mm( +× respectively which are defined by
ˆ)(
ˆΨ′=
∂
Ψ∂= yxW
kxH (16)
ˆW)k(u
ˆK yuΨ
Ψ′=
∂
∂= (17)
B. Linearization of the Recurrent Network
The linearized system for the recurrent network is given
by
u(t)K )(ˆ )1(ˆ +≈+ txHtx (18)
where the two matrices H and K are of orders )( nn× and
)1( +× mn respectively which are defined by
)(
fW
kx
fH x ′=
∂
∂= (19)
)(
fW
ku
fK
u ′=∂
∂= (20)
Experimentally, the linearized systems (15) and (18) are
similar to the linear system given in (2).
IV. ALGORITHMS DESCRIPTION
Fig. 3 shows the adaptive control structure when we use a single-layer and recurrent networks which consists of
1) the system (5).
2) a single-layer feedforward network or a recurrent
network which estimate ] , , ,[ m21 ΨΨΨΨ L= or
] , , ,[ 21 mffff L= respectively.
3) a controller realized by the linearization of the neural
model.
Fig. 3. Adaptive control structure when we use single layer network and
recurrent network.
A. Algorithm When single Layer Network Is Used
The neural model for the unknown system (5) can be
expressed as (9)
]u(k) ),k(x[ˆ)dk(y jj Ψ=+
where )dk(y jj + is the jth output of the neural network
and Ψ is the estimate of the functons m21 , , , ΨΨΨ L . The
training algorithm used guarantees that
.min)]dk(y)dk(y[
m
1j
2jjjj =+−+∑
=
(21)
The objective function J is defined as follows:
∑=
=m
1j
2j )k(e
2
1J (22)
where
)dk(y)dk(y)k(e jjjjj +−+= . (23)
The elements of the gradient of J with respect to both yxW
and yuW is given by the following two equations
ijjyxji
x].S[.eW
JΨ ′−=
∂
∂ (24)
ijjyuji
u].S[.eW
JΨ ′−=
∂
∂ (25)
The weight changes may be made along the negative
gradient of of J by means of the equations
yxji
yxji
W
JW
∂
∂−= η∆ (26)
yuji
yuji
W
JW
∂
∂−= η∆ (27)
Once the weights are changed, the linearization process
start using the process stated in section III. Then the
linearized system of the form (18)
u(k) K)k(x H)dk(y jj +=+
is obtained and the state feedback control low can be
defined as (3).
Then equation (3) obtained can now be used in computer
program for real time control.
A pesudocode outline of the algorithm when a single layer
is used:
Select initial weights
Select the state
Select the values of the desired set points r(k)
u(k) = any random value (very small) or equal to zero.
Repeat up to approximately no change occur in the control signal
produce )dk(y jj + using (9);
find J using (22);
update the weights using (24), (25), (26) and (27);
find the linearized model (15);
compute new control signals u(k+1) using (3);
)1k(u)k(u += ;
B. Algorithm When Recurrent Network Is Used
The neural model for the unknown system (4) can be
expressed as (14)
Wu(t), ),(ˆˆ)1(ˆ txftx =+
Where W is the weight matrix obtained by training the
network using BTT algorithm [10].
Using the linearization process given in section III the
linearized system of the form (18)
u(t)K )(ˆ )1(ˆ +≈+ txHtx
can be obtained.
Using the matrices H, K, and C consider that the relative
degree of the multivariable nonlinear system under study is
known, the linear decoupling system of the form (2) can be
obtained. Using the linear decoupling system (2) if the
desired output vector is denoted by )(tr , it follows that the
desired )(tu can be expressed as a linear combination of
)(tr and )(ˆ tx as given in (3). Now equation (3) can now be
used in computer program for control implementation.
A pesudocode outline of the algorithm when a recurrent
network is used:
Select initial weights
Select the state
Select the values of the desired set points r(t)
u(t) = any random value (very small) or equal to zero.
Repeat up to approximately no change occur in the control
signal
produce )1(ˆ +tx using (14);
update the weights using BTT algorithm [10];
find the linearized model (18);
compute new control signals u(t+1) using (3);
)1k(u)k(u += ;
V. SIMULATION RESULTS
Note that the learning rate used for the two networks used
in this section takes the value 0.001. The values of both a
and b in the activation function takes the values 1 and 8
respectively and random initial weights in the
range 0.1) ,1.0( − .
The plant considered here is a third-order system given
in [8] which is described by the state equations
)k(x)k(y );k(x)k(y
)k(u)]k(x2sin[3)1k(x
)k(x1
)k(x)]k(x4sin[1)k(x)1k(x
)k(u])k(x1
)k(x2)k(x
)k(u])k(x1
)k(u)k(x5.12[)]k(xsin[)k(x9.0)1k(x
2211
213
23
3332
221
11
121
11211
==
+=+
+++=+
++
++
++=+
where T
321 )]k(x ),k(x ),k(x[)k(x = represents the state,
T21 )]k(u ),k(u[)k(u = the input, and
T21 )]k(y ),k(y[)k(y = the output, at instant k.
The linearized system is controllable, observable, and is
of minimum phase. Hence, a nonlinear decoupling
controller exists for the nonlinear plant in a neighborhood of the origin. Our objective is to realize this nonlinear
controller using neural networks to make the two outputs
)k(y1 and )k(y2 follow two independent reference
signals )k(y*1 and )k(y*
2 , respectively. We assume that
)k(r)dk(y 11*1 =+ and )k(r)dk(y 22
*2 =+ are
specified at instant k. Hence, the control problem is to
determine a control input u(k) so that
)2,1i( 0)k(r)dk(ylim iiik ==−+∞→ .
A. Simulation Results When A Single-Layer Network Used
Using the network architecture and Algorithm given in
sections II and IV respectively it was found that:
The state input output neural model obtained in this case
is given by:
]. u w u w
u w x w x w x w[ˆ)2k(y
]. u w u w
u w x w x w x w[ˆ)1k(y
2yu121
yu11
0yu102
yx121
yx110
yx102
2yu021
yu01
0yu002
yx021
yx010
yx001
+
++++=+
+
++++=+
Ψ
Ψ
where 1u2 = is the additional input used for bias.
After a weight change obtained using (26) and (27) the linearization process started using the method given in
section II which in our example is given by
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
+
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
+
+
2
1
0
2
1
1
1
0
1
2
0
1
0
0
0
2
1
0
2
1
1
1
0
1
2
0
1
0
0
0
2
1
u
u
u
u
ˆ
u
ˆ
u
ˆ
u
ˆ
u
ˆ
u
ˆ
x
x
x
x
ˆ
x
ˆ
x
ˆ
x
ˆ
x
ˆ
x
ˆ
)2k(y
)1k(y
ΨΨΨ
ΨΨΨ
ΨΨΨ
ΨΨΨ
(28)
Now to find a control low that match our purpose, we
will use a control low similar to that given in (3). Then we
replace )1k(y1 + and )2k(y2 + by the desiresd
prespecified setpoints )k(r1 and )k(r2 . Then we have to
find a control signals 1u and 2u in terms of the states and
desiresd prespecified setpoints. The process is shown in the
following equations:
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
+
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
2
1
0
2
1
1
1
0
1
2
0
1
0
0
0
2
1
0
2
1
1
1
0
1
2
0
1
0
0
0
2
1
u
u
u
u
ˆ
u
ˆ
u
ˆ
u
ˆ
u
ˆ
u
ˆ
x
x
x
x
ˆ
x
ˆ
x
ˆ
x
ˆ
x
ˆ
x
ˆ
)k(r
)k(r
ΨΨΨ
ΨΨΨ
ΨΨΨ
ΨΨΨ
(29)
. ˆ
ˆ
u
ˆ
ˆ
ˆ
ˆ
x
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
)(
)(
2
2
1
2
0
1
0
1
1
0
1
1
0
0
0
2
1
0
2
1
1
1
0
1
2
0
1
0
0
0
2
1
u
u
u
u
uu
uu
x
x
xxx
xxx
kr
kr
∂
Ψ∂
∂
Ψ∂
+
∂
Ψ∂
∂
Ψ∂
∂
Ψ∂
∂
Ψ∂
+
∂
Ψ∂
∂
Ψ∂
∂
Ψ∂
∂
Ψ∂
∂
Ψ∂
∂
Ψ∂
=
because 1u 2 = then the controller can be given as
.
x
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ)(
ˆ)(
ˆ
ˆ
ˆ
ˆ
u
2
1
0
2
1
1
1
0
1
2
0
1
0
0
0
2
12
2
01
1
1
1
0
1
1
0
0
0
1
0
∂
Ψ∂
∂
Ψ∂
∂
Ψ∂
∂
Ψ∂
∂
Ψ∂
∂
Ψ∂
−
∂
Ψ∂−
∂
Ψ∂−
∂
Ψ∂
∂
Ψ∂
∂
Ψ∂
∂
Ψ∂
=
−
x
x
xxx
xxx
ukr
ukr
uu
uu
u (30)
equation (30) represent the state feedback control low used
which is similar to that given in (3).
Once the control signals are obtained using (30) it can
be used instead of the old value.
The processes of the training, linearization, and
obtaining the control signals are continued until no change
occur in the control signals.
The performance of the proposed method for controlling
the system given in this section was tested with the sinusoidal reference inputs given by
+
=
+
=
20
k2sin75.0
30
k2sin75.0)k(r
10
k2sin75.0
50
k2sin75.0)k(r
1
1
ππ
ππ
(31)
is shown in figures 4, 5, and the control signals are shown
in Fig. 6.
-2
-1
0
1
2
0 30 60 90 120 150 180 210 240 270 300
k
r1 &
y1
-2
-1
0
1
2
0 30 60 90 120 150 180 210 240 270 300
k
r2 &
y2
Fig. 4. Performance of linear controller when the plant equations are unknown; solid line is the desired output, and dotted line is the actual
output of the plant.
-2
-1
0
1
2
0 30 60 90 120 150 180 210 240 270 300
k
r1 &
yL
1
-2
-1
0
1
2
0 30 60 90 120 150 180 210 240 270 300
k
r2 &
yL
2
Fig. 5. Performance of linear controller on a linearized model; solid line is
the desired output, and dotted line is the output of the linear model.
-1
-0.5
0
0.5
1
0 30 60 90 120 150 180 210 240 270 300
k
u1
& u
2
Fig. 6. The required control signals; solid line represent 1u and dotted
line represent 2u .
B. Simulation Results When A Recurrent Network Used
Using the network architecture and the algorithm, given
in sections II and IV respectively, with an epoch size
3=T it was found that after a weight change obtained
using the BTT training algorithm the linearization process
started using the method given in section III which in our
example at 1=t is given by
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
+
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
02
01
00
02
12
01
12
00
12
02
11
01
11
00
11
02
10
01
10
00
10
02
01
00
02
12
01
12
00
12
02
11
01
11
00
11
02
10
01
10
00
10
12
11
10
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
u
u
u
u
f
u
f
u
f
u
f
u
f
u
f
u
f
u
f
u
f
x
x
x
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
x
x
(32)
The linearization process for t = 2, 3 can be obtained easily
by the same method.
In case of t = 1, 2, 3, 1221202 === uuu is the additional
input used for bias. Then the corresponding output at t =1
is given by
=
12
11
10
11
10
ˆ
ˆ
ˆ
0 1 0
0 0 1
ˆ
ˆ
x
x
x
y
y
(33)
The output at t = 2, 3 can be obtained.
The linear decoupling system in this case at time 1=t is
given by
[ ] [ ]
+
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
02
01
00
020100
02
01
00
02
12
01
12
00
12
02
11
01
11
00
11
02
10
01
10
00
10
10 E1 E1 1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
0 0 1ˆ
u
u
u
E
x
x
x
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
y
[ ] [ ]
+
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
02
01
00
121110
02
01
00
02
12
01
12
00
12
02
11
01
11
00
11
02
10
01
10
00
10
11 E1 E1 1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
0 1 0ˆ
u
u
u
E
x
x
x
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
y
(34)
Where
00
1000
ˆ1
u
fE
∂
∂= .
01
1001
ˆ1
u
fE
∂
∂= .
02
1002
ˆ1
u
fE
∂
∂= .
00
12
02
11
00
11
10
11
00
10
00
1110
ˆ
ˆ
ˆˆ
ˆ
ˆˆ
ˆ
ˆ1
u
f
x
f
u
f
x
f
u
f
x
fE
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂= .
01
12
02
11
01
11
10
11
01
10
00
1111
ˆ
ˆ
ˆˆ
ˆ
ˆˆ
ˆ
ˆ1
u
f
x
f
u
f
x
f
u
f
x
fE
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂= .
02
12
02
11
02
11
10
11
02
10
00
1112
ˆ
ˆ
ˆˆ
ˆ
ˆˆ
ˆ
ˆ1
u
f
x
f
u
f
x
f
u
f
x
fE
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂= .
The linear decoupling system (34) can be represented in
the following form
+
∂
∂
∂
∂
∂
∂
=
02
01
00
121110
020100
02
01
00
121110
02
10
01
10
00
10
11
10
E1 E1 1
E1 E1 1
ˆ
ˆ
ˆ
A A
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
u
u
u
E
E
x
x
x
A
x
f
x
f
x
f
y
y
(35)
Where
00
12
02
11
00
11
01
11
00
10
00
1110
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
x
f
x
f
x
f
x
f
x
f
x
fA
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂= .
01
12
02
11
01
11
01
11
01
10
00
1111
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
x
f
x
f
x
f
x
f
x
f
x
fA
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂= .
02
12
02
11
02
11
01
11
02
10
00
1112
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
x
f
x
f
x
f
x
f
x
f
x
fA
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂= .
From the system (35) the controller can be represented as
∂
∂
∂
∂
∂
∂
−
−
−
=
−
02
01
00
121110
02
10
01
10
00
10
1201
0200
1
1110
0100
01
00
ˆ
ˆ
ˆ
A A
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
1
1
E1 1
E1 1
x
x
x
A
x
f
x
f
x
f
Er
Er
E
E
u
u
(36)
The linear decoupling systems at times 2=t and 3=t can
be obtained in a similar fashion.
Once the control signals are obtained using (36) it can be
used instead of the old value.
The process of the training, linearization, and obtaining the
control signals are continued until no change occur in the
control signal.
The performance of this method for controlling the
system was tested with the sinusoidal reference inputs
given by (31) is shown in figures 7 and 8.
VI. COMPARISON BETWEEN SINGLE-LAYER AND
RECURRENT CONTROLLERS In this section, we will make a comparison between the
two models given in section V to solve the control problem
mentioned in section I.
From the results obtained during the simulation, it is
clear that, recurrent controller needs intensive
computations compared with the single-layer controller for
computation of u(k).
In comparison between the two models presented in
this paper it was found that:
1) the single-layer model use a simple training algorithm
while the recurrent model is computationally intensive. 2) the single-layer model is more suitable for real time
problems than that of recurrent model.
3) the performance given by the single-layer model is more
accurate than that for the recurrent model the root mean
square error for the single-layer network equal 0.3 610−×
and for the recurrent network equal 0.000006 which give
an advantage for a single-layer controller .
VII. CONCLUSION
In this paper we study the ability of both a single layer
and recurrent models to find a real time adaptive
controller. From our study it was found that the two
models used are very important in the field of adaptive control of nonlinear dynamical systems and they supply us
with very acceptable results. Also it was found that the
effectiveness of the single-layer strategy was validated in
real time tracking of a multivariable system than that of the
recurrent strategy because of its simple architecture and
lowest of computationally used in the learning process.
Future work will include a study the adaptive control
problem using radial-Basis function networks. The
comparison between the two mentioned models with the
radial-Basis function model will also be made.
-2
-1
0
1
2
0 30 60 90 120 150 180 210 240 270 300
t
r1 &
y1
-2
-1
0
1
2
0 30 60 90 120 150 180 210 240 270 300
t
r2 &
y2
Fig. 7. Performance of the recurrent neural network controller when the state variables are accessible, but the plant equations are unknown; solid line is the desired output, and dotted line is the actual output of the plant.
-1
-0.5
0
0.5
1
0 30 60 90 120 150 180 210 240 270 300
t
u1
& u
2
Fig. 8. The required control signals; solid line represent 1u and dotted
line represent 2u .
VIII. REFERENCES
[1] O. Patrick and N. Sadegh, "control of discrete-time systems via online learning and estimation," International conference on advanced intelligent mechatronics, California, USA, 24-28 July, 2005. [2] M. I. Jordan and R. A. Jacobs, “Learning to control an unstable system with forward modeling,” Advances in Neural Inform. Processing Syst. Vol. 2, pp. 324-331, 1990.
[3] K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Tran. Neural Networks, vol. 1, pp. 4-27, 1990.
[4] A. U. Levin and K. S. Narendra, “control of nonlinar dynamical systems using neural networks: Controllability and stabilization,” IEEE Tran. Neural Networks, vol. 4, no. 2. pp. 192-205, 1993.
[5] A. U. Levin and K. S. Narendra, “Control of nonlinear dynamical
systems using neural networks Pare II: Observability, Identification,
and control,” IEEE Tran. Neural Networks, vol. 7, no. 1, pp. 30-42, 1996. [6] K. S. Narendra and S. Mukhopadhyay, “Adaptive control of dynamical using neural networks and approximate models,” IEEE Tran.
Neural Networks, vol. 8, no. 3, pp. 475-485, 1997. [7] T. A. Al-zohary and A. M. Wahdan, "Adaptive Control for nonlinear multivariable systems using a single layer neural network and conventional linear controller," the 11th Mediterranean conference on control & automation, Greek, 2003. [8] K. S. Narendra and S. Mukhopadhyay, “Adaptive control of nonlinear multivariable systems using neural networks,” Neural Networks, vol. 7, no. 5, pp. 737-752, 1994. [9] P. L. Falb and W. A. Wolovich, “Decoupling in the design and synthesis of multivariable control systems,” IEEE Tran. On Automatoc control, vol. 12, pp. 651-659, 1967. [10] R. J. Williams and D. Ziposer, "A learning algorithm for continually
running fully recurrent neural networks," Neural Computation, vol. 1, no. 2, pp. 270 – 280, 1989.