MINISTRY OF EDUCTION AND TRAINING
MINISTRY OF TRANSPORT
HO CHI MINH CITY UNIVERSITY OF TRANSPORT
TA VAN PHUONG
Dissertation Abstract
Designing Adaptive Tracking Controller For
Non-Linear MIMO Systems Using CMAC
Major: Automation and Control Engineering
Code: 9520216
Instructor 1: Assoc.Prof Dang Xuan Kien
Instructor 2: Dr. Ngo Thanh Quyen
Ho Chi Minh City, March 2019
Abstract
Designing control system for non-linear MIMO systems has attracted
many researchers for the recent decades. Due to the complex characteristics,
defining the dynamic model of the non-linear MIMO systems are invaluable
for the practical applications. Therefore, the model-based controllers cannot
satisfy the desired performances.
To cope with this problem, many advanced controllers have been
studied and applied for the non-linear MIMO systems such as Particle Swarm
Optimization (PSO), Fuzzy Logic Controller (FLC), Neural Network (NN),
Fuzzy Neural Network, and so on. By using these adaptive and intelligent
controllers, the performances have been achieved for the practical
applications. However, there exists disadvantages and shortcomings that need
to be improved such as online learning problems, selection number of fuzzy
rules, number of neurons and layers, the robustness of the system in the
presence of disturbances, noise, and uncertainties, and so on.
This thesis has proposed the CMAC, the recurrent CMAC, the
redundant recurrent CMAC, and the robust recurrent cerebellar model
articulation control system (RRCMACS) for the non-linear MIMO systems
to achieve the desired performances such as good tracking responses,
stability, robustness, disturbances attenuation, and noise rejection. The main
contributions of this dissertation are presented in Chapter 2, Chapter 3, and
Chapter 4. In Chapter 2, a traditional Cerebellar Model Articulation
Controller is represented to show the superior properties of the CMAC to
different intelligent controllers. Chapter 3 presents the factors affecting the
learning capability and efficiency of the CMAC and then some innovative
solutions are proposed to enhance the performance and the learning
effectiveness of the CMAC. Besides improving the CMAC, the redundant
solution is also proposed to maintain continuously the controling and
supervising process. A combination between the RCMAC and the robust
controller to form the robust RCMAC to achieve not only good tracking
response but also attenuate significantly the effects of the external
disturbances, and sensor noise is shown in Chapter 4. With this combination,
the stability and robustness of the control system are remarkably improved
during operation. Along with the main theorems, the experimental results
were also provided to prove the effectiveness of the proposed solutions.
Page 1
Chapter 1: INTRODUCTION OF NON-LINEAR MIMO SYSTEM
AND PROPOSED CONTROL SYSTEM
1.1 Introduction of the non-linear MIMO systems
and research problems
Most of the practical systems are non-linear systems. the non-linear
features of the system may come from the effects of dead-zone, hysteresis,
saturation, friction coefficients, cross-coupling, uncertainties, disturbances,
and noises [1]-[5].
Due to the effects of the non-linear characteristics, the dynamic model of
the practical systems cannot be completely obtained. Therefore, they must be
considered in designing the control systems. From the point of view in
designing control system, model-based controllers cannot achieve desired
performances for the non-linear MIMO systems [6]-[15].
To cope with the the non-linear characteristics and the uncertainties of the
system, Fuzzy Logic Controller (FLC) [16]-[23], Sliding Mode Controller
(SMC) [24]-[30], neural network (NN) [31]-[39] have been developed for the
non-linear system to achieve desired performances. However, these
controllers still exist shotcomings as flows
The performance of the FLC depended utterly on the selection of fuzzy
sets and the number of rules. However, there are not a specific method that
ensure the optimal selections of fuzzy sets and rules for the controllers so far.
To achieve good performance of the FLC for the practical applications, the
fuzzy sets and rules were mostly selected by trial and error.
For the SMC, the chattering phenomena affects long life and responses of
the actuators, the selection of the boundary of uncertainties is a trade-off
between the stability and the chattering phenomena
The NNs remain several shortcomings such as all weights in the structure
of the neural network are updated each learning cycle, this is unsuitable for
the problems requiring real-time learning; the selection of the number of
neurons and hidden layers to achieve good performances is very difficult in
the practical applications.
Along with the development of the NNs, Cerebellar Model Articulation
Controller (CMAC) having the learning structure similar to the human brain
Page 2
has been studied from the 1970s [40]. The CMAC has been developing and
incorporating for the complex non-linear MIMO systems because of its
superior properties such as fast learning, good generation capability, and
simple computation [41]-[46]. The effectiveness of the CMACs rather than
NNs has been proved in the practical applications [47]-[48]. In the recent
works, the wavelet function and recurrent technique were utilized to improve
learning capability and dynamic response of the CMAC [49]-[51].
Although the above studies achieved good results in designing the
controller to cope with the high non-linear MIMO systems both in simulation
and experiment, however, the robustness of the system in the presence of the
disturbances and noise were not fully taken into account.
After studying the papers relating to the CMAC, the author proposed a
new methodology to design a control system for the non-linear MIMO system
basing on the Cerebellar Model Articulation Controller with the following
characteristics
i)
The control system is not dependence on the dynamic model.
Meanwhile, the stability and convergence error of the system can be
obtained in case the model cannot be exactly defined.
ii)
The control system has dynamic response capability and avoid the
local minimization during operation
iii)
The control system can deal with uncertainties, disturbances, and
change in parameters of the system to have good tracking response
iv)
The control system has redundant capability which maintains
continuously the control and supervisory process
v)
The control system guarantees the stability and robustness of the
system in the presence of the uncertainties, disturbances, and noise
1.2
Outline of the Dissertation
This dissertation is divided into five chapters. Chapter 1 mentions about
the non-linear MIMO system, scientific researches relating to the non-linear
system control, neccessory problems to research and proposed control
system. Structure of the traditional Cerebellar Model Articulation Controller
(CMAC) and its applications are shown in Chapter 2. Chapter 3 points out
the shortcomings of the traditional CMAC and proposes innovative solutions
to enhance the performances of the CMAC. The factors affect the robustness
Page 3
and the robust CMAC designing are provided in detail in Chapter 4. Finally,
the conclusion and future works are delivered in Chapter 5. Along with the
representation of research problems by theory, the simulation and
experimental results are also included to prove the effectiveness and merit of
the proposesd control system.
The organization of this dissertation is expressed in Fig 1.1 as follows.
Chapter 1
Introduction of Non-linear System and Proposed Control System
Chapter 2
Cerebellar Model
Articulation Controller
Chapter 3
Improved Cerebellar Model
Articulation Controller
Controlller CMAC
Chapter 4
Robust Cerebellar Model
Articulation Controller
Chapter 5
Conclusion and Future Works
Figure 1.1: Organization of the dissertation
Page 4
1.3
MIMO non-linear system including
uncertainties
In general, the dynamic equation of MIMO non-linear systems including
uncertainties, disturbances, and noise is described as flows
x n = F0 (x) + ΔF( x) + (G 0 (x) + ΔG (x))u + dn(x) F0 (x) + G 0 (x)u UD(x) (1.1)
y = x
where
y = x [x1 , x 2 ,..., x no ]T R
x [x T , x T ,..., x (n -1)T ]T R n is
u [u1 , u 2 ,..., u no ]T R
no
no
is
the
the
system
system
output
vector,
state
is the control input vector, F0 (x) R
vector,
n o n
in the
nomial non-linear function, G 0 (x) R no no is nomial gain matrix, ΔF(x) and
n n
n n
ΔG(x) are change in parameter of the F0 (x) R o , and G 0 (x) R o o
dn(x) = [dn1 ,dn 2 ,...,dn o ]T R
respectively.
disturbances
and
noise,
no
stands
for
UD(x) F(x) + G(x)u + dn(x) is
external
lumped
uncertainties, disturbances, and noise. The objective of the control system
synthesis is that
the output signals x can not only track the desired
trajectories xd R but also satisfy the robust performance in the presence
no
of the uncertainties, disturbances, and noise.
1.4 The proposed control system
For the high-order system, the sliding error manifold was defined
[2],[29]-[30] to reduce the order of variables during designing and
computating the control system. The sliding error manifold has the following
form
S en-1 + Ke
(1.2)
Therein, e xd x and e e, e,... , en-1 are the tracking error and error
T
vector of the system, respectively. Derivative both sides of S and
combination with the dynamic equation (1.1), yields
Page 5
S en + Ke = xnd - xn + K(e) = x nd - Fo (x) - G o (x)u - UD(x) + K(e)
(1.3)
In case the nominal functions Fo (x) R no x n , Go1 (x) R no xno , and the
lumped of external disturbances, uncertainties, and noise UD(x) are exactly
known, an ideal sliding mode (ISM) controller is designed to guarantee the
stability of the system as follows [49],[52]-[53].
(1.4)
uISM Go1 (x) xdn - Fo (x) UD(x) K(e) + ηsgn(S)
However, for the complex high non-linear systems, the external
disturbances, uncertainties, and noise UD(x) can’t be defined, measured or
estimated exactly in practical applications. Consequently, the u ISM can’t
satisfy the stability and robust performance of the system.
To cope with the drawbacks of the model-based controllers, many modern
controllers have been developed such as Fuzzy Logic Controller (FLC),
Sliding Mode Controller (SMC), Neural Networks (NNs), and Cerebellar
Model Articulation Controller (CMAC) [17-51].
Although the above studies achieved impressive results in designing the
controllers to cope with the high non-linear MIMO systems, the dynamic
response and robust specifics of the system in the presence of the
uncertainties were not totally mentioned.
In this research, a robust recurrent cerebellar model articulation control
system (RRCMACS) is proposed for the non-linear MIMO system. Therein,
the RCMAC is designed to imitate the ideal controller to minimize error
surface and the H robust controller is designed to attenuate the effects of the
uncertainties acting on the system to achieve the robustness performance of
the system during operation.
The total control system is described in (1.7) and a block diagram of the
RRCMACS is depicted in Fig. 1.2.
(1.7)
uRRCMACS = u ISM - u RC - u RCMAC
Page 6
xd
n
uISM = G -1
0 (x) xd - Fo (x) - UD(x) + K(e) - ηsgn(S)
η w , η m , ησ , η w r
xd +
x-
e
S
Learning
Rules
en-1 + Ke
u ISM
+
ˆ σ,
ˆ m,
ˆ w
ˆr
w,
-
MIMO Nonlinear
System
RCMAC
UD(x ) u RCMAC
S
Robust
Controller
u RC -
+
u RRCMACS
Figure 1.2: Block diagram of the proposed control system
The block diagram of the proposed control system includes three main
control parts as follows
The ideal sliding mode controller is used when the uncertainties are
exactly known.
The RCMAC is the main controller, it is used to learn the uncertainties,
UD(x) to minimize the error sliding surface, S . With the selection of
the appropriate learning rate, the error sling surface will tend to zero
by learning capability of the RCMAC.
The robust controller u RC guarantees the robustness of the system in
the presence of the uncertainties during operation.
Page 7
Chapter 2: THE CEREBELLAR MODEL ARTICULATION
CONTROLLER (CMAC)
2.1
Introduction to the CMAC
Cerebellar model articulation controller (CMAC) is a neural network
model proposed by Albus [54]–[56]. The CMAC with its fast learning and
good generation capability has been studied and implemented to identify and
control the non-linear systems [57]-[59]. Based on its superior properties, the
CMAC is unnecessary to require much prior knowledge of the system.
Consequently, it can be considered as an intelligent controller that suits many
practical non-linear systems [60]-[61]. The superior properties of the CMAC
to NNs were proved in the references [62]-[64].
2.1.1
Block diagram of the proposed control system and
Structure of the CMAC
The CMAC with its fast learning and good generalization capability
places an important role in learning the unknown uncertainties, UD(x) to
minimize the error sliding surface. The block diagram of the proposed control
system is depicted in Fig 2.1 and the structure of the CMAC is shown in Fig.
2.2. The controller includes input space S , association memory space A ,
receptive field space R , weight memory space, and output spaces O . The
signal propagation in the CMAC is presented as follows [48], [65].
Page 8
xd
n
uISM = G -1
0 (x) xd - Fo (x) - UD(x) + K(e) - ηsgn(S)
η w , η m , ησ
xd +
S
e
x-
Learning
Rules
e n -1 + Ke
u ISM
+
ˆ σ
ˆ m,
ˆ
w,
-
MIMO Nonlinear
System
CMAC
UD(x ) u CMAC
S
Compensator
Controller
ηB
u CC -
+
u CMACS
Figure 2.1: Block diagram of the proposed control system using CMAC
Input
Space S
Association MeReceptive
mory Space A Field Space R
Weight Memory
Space W
Output
Space O
O1
S1
μ1k
bik
w1k
w jk
Sn i
μ ik
Layer 1
nk
Figure 2.2: The structure of the CMAC
Page 9
Oj
In this research, the error-sliding surfaces are considered as input
variables of input space, S = [S1 S2 S3 ...Sni ] . Each input variable Si can be
quantized into n e discrete regions corresponding to the working space.
The activation degrees of input variables to each layer is calculated by the
Gaussian function as follows
(2.1)
(S - m ) 2
μ ik = exp - i 2 ik
2σik
where μ ik is the activation degrees of the input variable Si to layer k .
m ik and ik are mean and variance of Gaussian at the layer k , respectively.
The activation degrees μ ik of input variables at each layer are overlapped
and stored in the receptive field space at blocks b k . The method of
overlapping and storing data for two inputs and 7 elements n e = 7 is shown
in Fig. 2.4. The blocks formed by the data overlapping of the input variables
are called hypercubes. Each activated element in each block or layer becomes
a firing element, thus, the weight of each block or layer can be obtained. The
content of kth hypercube can be defined as follows:
b k S, mk ,σ k =
(2.2)
n
μ
ik
(Si )
i=1
where i = 1, 2,..., n i and k = 1, 2,..., n k
In the structure of the CMAC, the content of each hypercube
corresponding to each specific value in the weight memory space and can be
described as follows:
(2.3)
v jk (S) = w jk b k (S, m k , σ k )
where w jk is the weight of the kth hypercube corresponding to the jth
output.
Page 10
S1
7
6
5
4
State
(3,3)
3
2
1
1
2
3
4
5
6
7
S2
Layer 1
Layer 2
Layer nki
Figure 2.4: Store data in the receptive field space of the CMAC
The output of the CMAC is the algebraic sum of the activated weights
correspongding to the hypercube. The output of the controller can be
expressed as follows:
w11 , w12 ,..., w1nk b1 (S)
w , w ,..., w
21
22
2nk b 2 (S)
= w jk b k =
j=1 k =1
w j1 , w j2 ,..., w jnk b nk (S)
nj
u CMAC
nk
(2.4)
, n j and k = 1, 2,..., n k are the number of outputs, and
where j = 1, 2,
layers, respectively
2.1.2
The cost function and learning rules of the CMAC
The effects of the uncertainties UD(x) in (1.4) is learned by the CMAC
u CMAC with approximation error ε as follows:
UD(x)
(2.5)
1
uCMAC (S, w kj , mik , ik ) ε
G 0 (x)
In this study, the square of the error sliding surface STS is selected as an
error function. The controller aims to learn to minimize this function. By
combination (1.1) to (1.4), (1.7), and (2.5), the error of the system is described
as bellow
STS ST F (x) ST G (x)(u u
u ) ST (xn - UD(x) + K(e)) (2.6)
0
0
ISM
CMAC
CC
Page 11
d
The gradient descent algorithm is applied to learn to minimize the error
function of the system [41], [48]. The error function depends on three
parameters w, m, σ . The learning rules of these parameters are described by
(2.7) to (2.9). The parameters of the controller are updated by (2.10)-(2.12).
ni
(2.7)
STS
STS uCMAC
w kj ηw
ηw
ηw ST G 0 (x) μ ik (Si )
w kj
uCMAC w kj
i=1
mik ηm
STS
STS uCMAC μ ik
ηm
mik
uCMAC μ ik mik
ni
ηmST G 0 (x)w kj ( μ ik (Si )
i=1
ik ησ
ST S
ik
ησ
2(Si mik )
ik2
ST S u CMAC μ ik
u CMAC
ni
μ
ησ ST G 0 (x)w kj (
μ ik
ik (Si )
i=1
(2.8)
(2.9)
ik
2(Si mik ) 2
ik3
w kj (t 1) w kj (t) w kj
(2.10)
mik (t 1) mik (t) mik
(2.11)
ik (t 1) ik (t) ik
(2.12)
2.2
The Estimated Boundary Compensator Controller.
The most useful property of CMAC is to lean the non-linear part UD(x)
, via learning capability to minimize the sliding error surface (SES). With an
appropriate learning rates selection, the SES will tend to zero as a specific
period of time. To guarantee the stability of the system, the controller must
have the capability to maintain the SES nearby zero during operation.
According to sliding mode theory, a compensator controller u CC is designed
for this objective as follows
u CC G 01 (x)Bsgn(S)
(2.13)
where B is the error boundary
With the compensator controller u CC in (2.13), the stability of the system
will be guaranteed in case the approximation error ε in (2.5) is bounded by
Page 12
an error boundary B . The selection error boundary B is a trade-off between
the stability of the system and chattering phenomenon at control output. As
the error boundary is smaller than ε the system will be unstable, otherwise
the control output will be chattering when the error boundary is too large to
ε . Therefore, this parameter is estimated in this study and represented as
follows
(2.14)
ˆ
B = B- B
where B̂ is the estimated value of the error boundary B .
Substituting (1.7) into (1.1) yields
xn F0 (x) G 0 (x)(u ISM u CMAC u CC ) UD(x)
(2.13)
where u ISM is represented in (1.4) excepting for UD(x) and u CMAC is given
in (2.5). On the basis of some straightforward manipulations, the error
equation of the system can be obtained as follows
(2.14)
S xn + K(e) = G (x)u
0
CC
The Lyapunov function is chosen dependence on two variables s and B
as follows
(2.15)
1
B2
V(S, B) = S2
2
2ηB
where ηB is the learning rate of the bound error. Differentiating (2.15) with
respect to time, results in equation (2.16). If the adaptive rule is chosen as
(2.17), the stability of the system can be guaranteed in the Lyapunov-like
Lemma sense.
(2.16)
BB
BB
BB
ˆ
V(S, B) = STS +
= ST ε - Bsgn
S + = ST ε - Bˆ S 1 +
ηB
ηB
ηB
(2.17)
ˆ = -B
ˆ = -η S
B = B-B
B
1
2.3
Experiments and results
To show the effectiveness of the CMAC in reality, the controller is
applied to control the pressure and water level in the tank
Page 13
The Pressure Control Model (PCM) and the Water Level Control Model
(WLCM) are described in Fig 2.5 and Fig 2.6, respectively.
Figure 2.5: Structure of the Tank Pressure Control Model
Figure 2.6: Structure of the Water Level Control Model
The PCM and the WLCM are non-linear and time-varying parameter
systems [67]–[69]. The precise dynamic equation of these systems are very
difficult to define exactly. In this study, the dynamic model of the PCM and
the WLCM are obtained by the identification toolbox of the Matlab as bellow
(2.22)
y(t) + 0.051y(t) + 0.001y(t) + UD(t) = 0.004u(t)
y(t) + 0.000321y(t) + 3.474e -15y(t) + UD(t) = 0.1796u(t)
(2.24)
where u(t) and y(t) are control signal of the inverter and the pressure or
water level in the tank, respectively. UD(t) is lumped uncertainties due to
change in parameters, disturbances, noises, and error in linearization. The
dynamic equation of these systems can be rewritten in the state equation form
as bellow
(2.23)
x = F0 (x) + G 0 (x)u + UD(x)
.
.
T
T
T
y = x; x = x x
Page 14
where . F0 (x) ., G 0 (x) are nominal parameters of the system, and x is the
state variable.
The parameters of the CMAC are initialized as follows
n i 1; n k 7; j 1; ηw = ηm = ησ = 0.5
, K1 = 0.001, K2 = 0.001
, m 0.6 0.4 0.2 0 0.2 0.4 0.6
w 0.1
0.1 0.1 0.1 0.1 0.1 0.1
0.3
0.3 0.3 0.3 0.3 0.3 0.3
The experimental results of the PCM and the WLCM are shown in Fig.
2.7 and Fig 2.8, respectively.
a.
Tracking response of the pressure in the tank (Kpa)
b.
Tracking Error (Kpa)
c. Control Effort (Volt)
Figure 2.7: Experimental results of CMAC for The Pressure Control
Model due to periodic step command.
Page 15
a.
Tracking response of the water Level in the tank (cm)
b. Tracking Error (cm)
c. Control Effort (Volt)
Figure 2.8: Experimental results of CMAC for The Water Level
Control Model due to periodic step command.
The experimental results of the WLCM showed the impressive properties
of the CMAC as bellow
Although the dynamic equation of the WLCM is obtained by the
identification tool of the Matlab, the tracking responses of the system
can be achieved exactly in reality.
The stability of the system can be guaranteed in the presence of the
uncertainties UD(x) .
2.4
Conclusion.
This chapter introduces the CMAC including the structure, the cost
function, and learning rules. Besides, the compensator controller is combined
with the CMAC to maintain the stability of the system in the presence of the
uncertainties. The experimental results are also provided to prove the
Page 16
effectiveness of the proposed control system. However, the CMAC should be
improved to cope with dynamic response and local minimum problems.
Chapter 3: IMPROVED CEREBELLAR MODEL ARTICULATION
CONTROLLER
3.1
Drawbacks of the traditional CMAC and proposed
improvements
The cerebellar model articulation controller (CMAC) has proposed since
the 1970s by Albus [54]-[56]. It has been considered as an intelligent
controller which has a computational ability as the human cerebellum. The
structure of the CMAC is the same with a non-fully connected perceptron
network. It uses less memory area than other advanced controllers by
overlapping data storing in the receptive field space. Several advantages of
the CMAC are fast learning property, good generalization capability, and
ease of hardware implementation [59]. Therefore, the CMAC has been used
for identification and control of the complex non-linear systems [62]-[64].
The CMAC, however, has several drawbacks as follows
For the traditional CMAC, the step or triangular function was used as
the activation function for input variables. However, these functions
do not have differential abilities. Consequently, the learning
effectiveness of the controller is not good.
The structure of the CMAC was straightforward form, therefore, it
was suitable to the static systems
The controller did not design the redundant solution, consequently, it
could not maintain continuously the control and supervisory process.
To enhance the learning ability of the CMAC, the Gaussian and Wavelet
functions were used to define the active degrees of the input variable at
association memory space [70]. Besides, the recurrent and redundant
techniques were also studied and combined with the CMAC to improve
dynamic response ability and maintaining the control and supervisory process
of the system. All the proposed improvements above will be presented in
detail in the following parts.
3.2
Wavelet Cerebellar Model Articulation Controller (WCMAC)
According to the Wavelet functions that is depicted in Fig 2.3. It is clear
that the Wavelet has more differential ability than the step and Gaussian
Page 17
functions. This characteristic is very useful in case the computation of the
controller regards to derivative. Consequently, the Wavelet function is
combined with the CMAC to form the Wavelet CMAC (WCMAC) to
improve the learning effectiveness of the controller. The structure of the
WCMAC is the same as the CMAC in Fig 2.2 excepting the Wavelet function
was used instead of Gassian function at Association memory space. The
signal propagation of the WCMAC is presented as follows [48].
For the WCMAC, the Gaussian function was replaced by the Wavelet
function in the association memory space. Therefore, the activation degrees
of input variables to each layer is computed as follows
(3.1)
(S - m ) 2
(S - mik )
μ ik = - i
exp - i 2 ik
σik
2σik
With the cost function is selected in the same way as the CMAC in chapter
2, the learning rules are performed by the back-propagation algorithm as
bellow
(3.2)
ST S
ST S u WCMAC
w kj ηw
ηw
w kj
u WCMAC w kj
ni
ηw s T G 0 (x)( μ ik (Si ))
i=1
mik ηm
S S
S S u WCMAC ik
ηm
mik
u WCMAC μ ik mik
T
T
ηmST G 0 (x)w kje
ik ησ
ST S
ik
ησ
Si -mik
2 ik
)2
((1 (
Si - mik
ik
) 2 ) / ik )
ST S u WCMAC ik
u WCMAC
(
ησ ST G 0 ( x)w kj e
(
Si mik
2 ik
)
2
(((
μ ik
Si - mik
ik
(3.3)
(3.4)
ik
) 2 1) / ik )
The back-propagation algorithm was used to learn and update the
parameters of the WCMAC. However, the weakness of this algorithm is that
it may get stuck in local minima points [71].
To avoid getting stuck in a local minimum, the standard back-propagation
algorithm is modified by adding momentum term ρ and proportional term
Page 18
υ into calculating new parameters of the controller [72]. The momentum
term places an important role to prevent the learning algorithm from falling
in local minima and the proportional term speeds up the convergence as the
activation function having a flat slope. The new update rules of the WCMAC
as follows
(3.5)
w kj (t) w kj (t) + ρΔw kj (t -1) + υ(x - x d )
m kj (t) m kj (t) ρΔm kj (t -1) + υ(x - x d )
(3.6)
kj (t) kj (t) ρΔσ kj (t -1) + υ(x - x d )
(3.7)
3.3
Recurrent Cerebellar Model Articulation Controller
(RWCMAC)
The CMAC and WCMAC place an important role in learning the
uncertainties UD(x) to minimize the error sliding surface [49]. However,
these controllers depend only on direct inputs, hence, it is suitable for static
problems [27]. In this investigation, the recurrent technique is incorporated
into the CMAC and the WCMAC to form the RCMAC to adapt the dynamic
problems [73]-[74]. The structure of the RCMAC is depicted in Fig 3.1. It is
obvious that the RCMAC has the same structure as the CMAC and the
WCMAC excepting the feeback term in the association memory space A.
Input
Space S
Association Memory Space A
μ i(k -1)
w rik
Si
Z
Receptive
Field Space R
1
Weight Memory
Space W
Recurrent Unit
μ ik (k)
Sri
Output
Space O
O1
S1
μ1k
bik
w1k
w jk
Sn i
Oj
μ ik
Layer 1
nk
Figure 3.1: The structure of the RCMAC
The signal propagation in the RCMAC is presented as follows [47]-[51]
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