Decoupled Neural Network Reference Compensation Technique for a PD Controlled Two Degrees-of-Freedom Inverted Pendulum
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92 International Journal of Control, Automation, and Systems Vol. 2, No. 1, March 2004
Decoupled Neural Network Reference Compensation Technique
for a PD Controlled Two Degrees-of-Freedom Inverted Pendulum
Seul Jung and Hyun Taek Cho
Abstract: In this paper, the decoupled neural network reference compensation technique
(DRCT) is applied to the control of a two degrees-of-freedom inverted pendulum mounted on an
x-y table. Neural networks are used as auxiliary controllers for both the x axis and y axis of the
PD controlled inverted pendulum. The DRCT method known to compensate for uncertainties at
the trajectory level is used to control both the angle of a pendulum and the position of a cart si-
multaneously. Implementation of an on-line neural network learning algorithm has been imple-
mented on the DSP board of the dSpace DSP system. Experimental studies have shown success-
ful balancing of a pendulum on an x-y plane and good position control under external distur-
bances as well.
Keywords: Neural network, 2 DOF inverted pendulum, on-line learning, DSP system.
1. INTRODUCTION tries to move toward one direction to minimize a po-
sitional error while a controller for the pendulum tries
Inverted pendulum application using various con- to move in the opposite direction to minimize the
trol methods has been a typical example for advanced angle error. When this contradiction occurs, it is dif-
control education, as well as interesting research. ficult to decide the suitable control law. This is one
Control of an inverted pendulum has been considered reason that the conventional fixed PD controller can-
as a fascinating, but difficult problem to solve since not control both the angle of the pendulum and the
the system has very challenging characteristics such position of the cart concurrently. To tackle this prob-
as nonlinearity and a single-input multi-output struc- lem, suitable controllers’ gains for various cases
ture [1, 2]. Many successful results using the ad- should be considered.
vanced control theories for balancing the inverted Fuzzy algorithm is a good candidate for solving
pendulum using a cart have been reported throughout this type of problem, but the assessment of fuzzy
various literatures [1, 3]. Those successful results rules for fuzzification does not simply satisfy both
have been mainly focused on balancing the pendulum, objectives [7-9]. Several trial and error experiments
rather than on controlling the position of the cart. are required to obtain a certain satisfaction. Visual
Recently, successful control of both the angle and the feedback control for the inverted pendulum has also
position of the inverted pendulum has been demon- been proposed [9, 10]. Rather than solely depending
strated by practical experiments [4-6]. upon encoder signals, controlling the balance of the
The nature of controlling both the angle of the pendulum relies on visual feedback. The performance
pendulum and the position of the cart with a single of visual feedback control is dependent upon the ac-
input force has remained as an open nonlinear prob- curacy of the vision system.
lem to be overcome. The difficulty of controlling Neural network based control is another good can-
both the angle and the position of the inverted pendu- didate for this application. In our previous researches,
lum system comes from different dynamic movement control of both the angle and the position of the in-
patterns of the pendulum and the cart. For example, verted pendulum system has been successfully per-
let us consider the case that a controller for the cart formed on a large x-y table robot [4, 6]. However,
__________ due to the large size of our previous system, position
Manuscript received February 20, 2003; revised October 16, control and balancing of the inverted pendulum of
2003; accepted December 15, 2003. Recommended by Edito- both axes on the x-y plane failed. One of the reasons
rial Board member Jin Young Choi under the direction of Edi- for the failure was that the actuated motor could not
tor Jin Bae Park. This research has been supported by the Ko-
rea Research Foundation with the contract of KRF 2002-002-
generate enough torque for rapid movement of the
D000765. The authors would like to thank them for their kind axis due to the heavy weight of one side of the two
support. axes.
Seul Jung and Hyun Taek Cho are with the Department of In this paper, as an extension of our previous re-
Mechatronics Engineering, Chungnam National University, search [4, 6], control of the inverted pendulum on the
220 Gung-dong, Yuseong-gu, Daejeon 306-764, Korea (e- x-y table is revisited. Like the balancing of a stick on
mail: jungs@cnu.ac.kr, hyuntkc@empal.com).International Journal of Control, Automation, and Systems Vol. 2, No. 1, March 2004 93
a palm, a pendulum can freely move on the x-y plane. Connect
Panel
An x-y table is newly built and its size is relatively ds1003
reduced to generate faster movements. Neural net- Data
works are also used as auxiliary controllers to help PWM
the PD controller for the system to minimize the er- dSPACE dSPACE
rors of angles and positions of each axis. Differing ControlDesk
Computer
Expansion Box Motor
from the previous research, a decoupled neural net- Driver
Motor Power
work structure is used. Decoupled neural networks X-Y
Belt
Table Motor Power
structure means that two separate neural networks are
used for controlling each axis of the x-y table instead
of using a single neural network. Since the structure Z
of the x-y table is a more likely decoupled system, X
the use of two separate neural networks is suitable for
eliminating any coupling effects.
For experiments, the newly designed smaller x-y Y
table with the dSpace DSP system is implemented. Fig. 1. The overall system.
Interface between the robot and the DSP system has
been implemented to drive the motors of each axis.
On-line neural computation algorithm is developed
and implemented on the DSP board of the dSpace
system to achieve real time control. Successful re-
sults of maintaining balance of the pendulum and
position control of the cart on the x-y plane have
been obtained by the proposed control algorithm. v
mg
h
2. SYSTEM STRUCTURE Ux
2.1. Overall system structure
The overall system structure is shown in Fig. 1.
The system consists of three parts: a controller, a x
pendulum on an x-y table, and actuators. The control
component includes a computer with the dSpace DSP Fig. 2. Inverted pendulum model.
board and interface. The DSP board is used to calcu-
late the neural network learning algorithm in an on- and horizontal forces are given as follows:
line fashion. The main body includes an inverted
x + Lθx cos θ x − Lθx2 sin θ x ) ,
h = m (
pendulum and an x-y table. The size of the x-y table (2)
is 0.8m × 0.9m. As shown in Fig. 1, a 2-DOF in-
v − mg = − mLx sin θ x − mLθx2 cos θ x ,
verted pendulum is mounted on the x-axis of the ro-
bot. The x-axis moves along the y-axis with an LM where m is the mass of a pendulum and x is the
Guide. Two axes are actuated by two DC motors
displacement in the x axis.
through timing belts. Belt tension is often considered
For a cart,
as an uncertainty.
The neural network control algorithm is imple- x = ux − h ,
M x (3)
mented on the DSP board and the board generates
PWM signals to the motor drivers. where M x is the mass of a cart in the x axis.
Solving for θx and x by combining (1), (2) and
2.2. Inverted pendulum
(3) yields the dynamic model of an inverted pendu-
The dynamics of an inverted pendulum can be
lum shown as follows:
modeled in x and y directions separately. The dy-
namic equation for the x axis is x + mL cos θ xθx − mL sin θ xθx2 + b x x = u x ,
( M x + m )
Jθx = v sin θ x − hL cosθ x + bθ x θ x , (1) (4)
where J is an inertia of a pendulum, θ x is the
( J + mL2 )θx + mL cos θ x
x − mgL sin θ x − bθ x θx = 0 .
angle of x axis, L is the length of a pendulum, bθ x is
(5)
friction constant, and v and h are vertical force
and horizontal force vectors, respectively. Vertical We know from equations (4) and (5) that the94 International Journal of Control, Automation, and Systems Vol. 2, No. 1, March 2004
a two degrees-of-freedom pendulum, the combined
two separate control inputs are required.
The control input u x for an x axis is formed by add-
ing two controllers’ outputs from an angle error and a
displacement error. The errors are defined as follows:
eθ = θ d − θ , ex = xd − x , (10)
where eθ is the angle error and ex is the positional
error.
The PD controller is now formed as
_ _ _
u x = u θ x + u px , (11)
Fig. 3. PD controlled pendulum system.
_
system has a single input u x and two outputs x u θ x = k pθ x ⋅ eθ + k d θ x ⋅ eθ ,
and θ for an x axis. _
(12)
Combining (4) and (5) yields u px = k ppx ⋅ e x + k dpx ⋅ e x ,
M ′( mgLsθ x + bθ x θx ) − mLcθ x ( mLsθ xθx2 − bx x + u x ) _ _
θx = where u θ x and u px are nominal control inputs for
M ′J ′ − m 2 L2 c 2θ x
(6) a pendulum and for a cart of the x axis, respectively.
_ _
J ′( mLsθ xθx2 − bx x + u x ) − mLcθ x ( mLgsθ x + bθ x θx ) The control input u θ y and u p y for y axis can
x=
be represented in a similar way. The difficulty of con-
M ′J ′ − m 2 L2 c 2θ x
trolling the pendulum angle and the cart position si-
(7) multaneously comes from different configurations
where _ _
u θ and u p .
M ′ = M x + m, J ′ = J + mL2 , sθ = sin θ , cθ = cosθ .
The use of diagonal controller gains can decouple
For the y axis, we have a similar dynamic equation as two axes, but there still are coupled effects such as
M y ( mgLsθ y + bθ y θy ) − mLcθ y ( mLsθ yθy2 − b y x + u y ) the coriolis force and other unknown nonlinear terms.
θy = As a result the linear controller cannot cope with sys-
M y J ′ − m 2 L2 c 2θ y tem parameter variations well enough, resulting in
(8) the failure of position control of the cart [6]. Exten-
sive simulation studies have been done.
J ′( mLsθ yθy2 − b y x + u y ) − mLcθ y ( mLgsθ y + bθ y θy ) In order to improve the control performance, two
y=
neural networks are used. Since the neural network is
M y J ′ − m 2 L2 c 2θ y
nonlinear it is a good candidate for nonlinear system
(9) control. A decoupled neural network structure for
Note that M y includes the mass of an x axis be- controlling each axis separately helps the system to
be more decoupled. Fig. 5 indicates the decoupled
cause the y axis carries the x axis at all times. neural network structure for a single axis.
3. PD CONTROL OF INVERTED PENDU- 4. REFERENCE COMPENSATION TECH-
LUM NIQUE FOR NN CONTROL
In this section, the non-model based PD control is In this section, one of the on-line learning algo-
presented. It is known that a PD control can stabilize rithms for neural network control is presented. The
the second order system. Even though we have de- algorithm called the reference compensation tech-
rived the dynamic equations in the previous section, nique has been proposed and it has shown good per-
it is very difficult to obtain the exact dynamic equa- formances in the robot position control [11, 12].
tions including uncertain nonlinear terms. Therefore, This scheme is identical to the feedback error
ignoring the system dynamics, the most uncompli- learning method in that it performs inverse dynamic
cated method is to use the simple linear PD control- control, but it is also different in that compensation is
lers as the main controllers. done without modifying pre-fixed linear controllers.
Fig. 3 depicts the PD control structure of controlling This control scheme is depicted in Fig. 4. The basic
the inverted pendulum in one axis. In order to controlInternational Journal of Control, Automation, and Systems Vol. 2, No. 1, March 2004 95
NN Compensator PD Controller Pendulum System
qr ε τ q
qd Neural Network
Controller Σ Controller Plant θd
θd
ν
Φ θr uθ f
θ
θ (t ) Σ Σ K Pθ Σ Σ Cart
θ
θr &
P
θ (t − 1) F.D Σ Σ K Dθ Pole
ε Σ
Neural uP P
Network Pr
P (t ) Σ Σ K PP Σ
Pr
P (t − 1) F.D Σ Σ K DP
Pd
Pd
Fig. 5. Neural network control block diagram for a sin-
Fig. 4. Reference compensation technique scheme. gle axis.
concept of this scheme is that the NN controller acts .
as the inverse of the system under PD control so that Φ x = k pxφ px + kdx φ px
the system response q tracks the desired response Also note that the compensating signal Φ x
qd with minimal distortion. Neural networks are solves the sign problem between uθ x and u px .
placed in front of the closed loop controlled system Since compensation is performed at the trajectory
as pre-filters as seen in Fig. 4. Neural network out- level, those compensating signals are amplified
puts are added to reference trajectories. Added terms through controller gains so that the magnitudes of
are subtracted by output signals to generate error sig- those pre-filtered signals are small compared with
nals ε . The errors are multiplied by controller gains. ones in other auxiliary type controllers [5]. In the
Therefore, they eventually shape the reference in- paper [5], the neural network is used to adjust PID
put trajectory qr in such a way that the output er- controller gains. The goal of minimizing error is the
ror ε is minimized to zero. same with the proposed RCT algorithm, but the dif-
Our proposed control block diagram is shown in ference comes from the controlling structure. The
Fig. 5. The inverted pendulum system is controlled RCT algorithm is known to have the advantage of
by PD control and neural network control. outer loop control without modifying the internal
From Fig. 5, the PD controller with compensating control structure [11, 12].
signals from a neural network forms the following
control inputs: 5. LEARNING ALGORITHM
.
uθ x = k pθ (eθ + φθ ) + kdθ (eθ + φ θ ) , (13) The neural network structure for a single axis is
shown in Fig. 6. A two layered feed-forward structure
. is used. Input patterns are the combination of position
u px = k px (ex + φ p ) + kdx (ex + φ p ) . (14) and angle errors. For experiments, 9 hidden units are
used. Selection of the number of hidden units is
. . based on trial and error. Delayed states are used as
The derivatives φ θ and φ p of neural network inputs of a neural network to give dynamics into the
outputs φθ and φ p are obtained by the finite differ- neural network. Two neural network outputs are used,
one for compensating an angle error and the other for
ence method, respectively. a position error are used for a single axis.
The total control input for one axis is the sum of Nonlinear function of hidden unit is tangent hyper-
(13) and (14). bolic function.
_ _
u x = uθ x + u px + Φ x . (15) 1 − exp(− x)
f ( x) = . (17)
1 + exp(− x)
The new control input is actually the addition of a
neural network compensating signal to nominal con- From (4), (5), and (15), we have the following
trol input u x . These compensating terms compen- closed loop error equation:
sate for any dynamic uncertainties that are not mod- _ _
eled. u θ x + u p x = f (θ, θ , θ ,
x , x , x ) − Φ x . (18)
Define the neural network output as
Define the training signal v with nominal con-
Φ x = Φθ x + Φ px , (16) trol inputs as
.
where Φθ x = k pθ φ θ x + kdθ φθ x and96 International Journal of Control, Automation, and Systems Vol. 2, No. 1, March 2004
bj yields the gradient as
Oj
Oi
1
bk ∂E ∂E ∂v ∂v
1
= =v
θ (t ) ∂w ∂v ∂w ∂w
Ok (22)
θ (t − 1) ∂Φ ∂Φ ∂Φ px
φθ = −v = −v( θ x + ),
θ (t − 2 ) ∂w ∂w ∂w
p (t ) ∂Φθ x ∂φ ∂Φ px ∂φ px
φp where = k pθ θ x and = k px .
p (t − 1) ∂w ∂w ∂w ∂w
p (t − 2 ) By using the gradient function in (22) the back-
w jk propagation algorithm can be used. The weight
wij
change is formed as
Fig. 6. The structure of neural network for a single ∂E
∆w(t ) = η v + α∆w(t − 1) . (23)
axis. ∂w
Weights are updated as
w(t + 1) = w(t ) + ∆w(t ) , (24)
where η is a learning rate and α is a momentum
coefficient.
Neural network weights are updated at every sam-
pling time. Since rapid sampling time can be
achieved with the help of DSP hardware technology,
real time control of a neural network becomes possi-
ble. Even on-line learning and control of the system
can be performed. This means that no a priori learn-
ing before control action is required.
In the next section, experimental studies of neural
network control are presented.
6. EXPERIMENTAL SETUPS
The overall system structure is shown in Fig. 7.
The whole system consists of three parts: an x-y table
robot, the dSpace DSP system, and a PC. The DSP
board is used to implement the neural network algo-
Fig. 7. Real inverted pendulum system. rithm in real time, which requires a huge calculation
time. Numerical values of neural network outputs
_ _ calculated by the DSP board are converted to PWM
v = uθ x + u px . (19) signals for motor drivers.
The DC motors driven by its own drivers com-
Then, at the convergence v = 0 , from equation manded from the DSP board mounted actually actu-
(18) an ideal neural network output becomes the in- ate each axis of the x-y table. Encoders mounted on
verse dynamics of the system. each axis sense rotations and those sensed data are
sent to the encoder board. An encoder board collects
Φ ≅ f (θ, θ, θ , x, x , x) . (20)
data and converts them into digital values that can be
So, ultimately, the inverse dynamic control can be used as a feedback in the control loop. Table 1 shows
achieved. This is the difference in the control struc- the specifications of actuators.
ture from those of other PID tuning methods [5]. Table 1. Specification of actuators.
The objective function is defined to minimize the
error v as Devices Specifications
1 2 DC motor 24V/70W
E = v . (21)
2 Encoder 2000 counts/turn
In order to use the steepest decent algorithm, the
Gear ratio 5:1
gradient should be obtained. Differentiating (21)International Journal of Control, Automation, and Systems Vol. 2, No. 1, March 2004 97
Movement of pendulum on x-y plane y axis pendulum angle error
0.08 0.025
0.07
0.02
0.06
0.05 0.015
Angle error of y axis (rad)
0.04
y axis (m)
0.01
0.03
0.02 0.005
0.01
0
0
-0.01 -0.005
-0.02
-0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
-0.01
x axis (m) 0 10 20 30 40 50 60
time (sec)
Fig. 8. Movement of pendulum on x-y plane.
Fig. 10. Y axis angle error.
x axis pendulum angle error
0.02 control to neural networks after stabilizing the system
by PD controllers.
0.015
For the neural network structure, 9 hidden units for
0.01
each neural network are used. Learning rate η x =
Angle error of x axis (rad)
0.0011, η y = 0.001 and momentum α x = 0.15, α y =
0.005
0.05 are optimized. These constant values are opti-
0
mized by trial and error basis. We found that the
learning rate is the most sensitive variable to per-
-0.005 formance. Selecting a larger learning rate gives faster
convergence of errors as well as occasional instabili-
-0.01 ties. The overall sampling time is 1 KHz.
-0.015
0 10 20 30 40 50 60 7. EXPERIMENTAL RESULTS
time (sec)
Fig. 9. X axis angle error. Initially the pendulum is located at (0, 0) on the x-
y plane. The pendulum is well maintained at the ini-
The weight of the pendulum is 0.35kg without a tial position until there is an external hit. Then the
weight and 0.5Kg with a weight at the top. movement of the pendulum is affected by an external
There are two encoders mounted on the cart to disturbance by the hand. The pendulum is required to
measure the rotational angles of both axes. The in- move back to the initial position.
verted pendulum can fall in any direction while a cart Fig. 8 shows the movement of the pendulum on the
can move on the x-y plane. The movement of the cart x-y plane after following hits.
is measured by encoders attached to the motor joint. We can see that there have been two disturbances
The x-y table is actuated by DC motors through tim- by hitting the pendulum in diagonal directions. The
ing belts. Since the x axis cart moves on the top of y pendulum has kept returning to the initial position
axis, there are coupling effects, and this leads to the while maintaining the balance of the angle of the
reason that control of the y axis is more difficult. pendulum shown in Fig. 9. Traveling distances are
For the controller gains, k dθ x = 0.8, k pθ x = 6, from - 10.5 cm to 6 cm in the x axis and from -1.5 cm
to 4 cm in the y axis. The pendulum angle errors for
k dpx = -0.6, k ppx = -0.5 for x axis and k dθ y = 1.4, both axes are shown in Figs. 9 and 10 for each direc-
tion. Large overshoots by disturbances are observed
k pθ y = 8.5, kdpy = -0.95, k ppy = -0.8 for y axis are at approximately both 13 seconds and 29 seconds and
used. The PD gains are optimized by trial errors basis. they settled down quickly.
However, gain values are small enough to maintain The positional errors are shown in Figs. 11 and 12.
stability so that neural networks are allowed to per- Comparing performances of two axes' movements,
form the most of control. We found from experiments the x axis position error is less than 2 mm at steady
that if the PD gains are set too large, performance is state, but that of the y axis is about 2 cm. The y axis
even worse. As such, it is better to leave the keeps oscillating with a small bound.98 International Journal of Control, Automation, and Systems Vol. 2, No. 1, March 2004
Position error of pendulum on x axis Even though balancing of the pendulum was quite
0.12
successful, controlling the cart position shows small
0.1 oscillatory errors in the y axis. One of the reasons
0.08 might be the location of optical encoders. Shown in
0.06
Fig. 7, encoders are mounted at the axis of rotating
Position error of x axis (m)
rod, not directly at the motor axis.
0.04
Another possibility is the velocity estimation. Sim-
0.02 ple velocity estimation using the finite difference
0 might not provide good estimation for slow move-
-0.02
ment of the pendulum. Finally, the y axis is actuated
by one side and not the other, which leads to the mis-
-0.04
alignment that causes oscillation. This yields further
-0.06 nonlinear characteristics such as backlash. These are
-0.08 left as future works to be performed.
0 10 20 30 40 50 60
time (sec)
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Fig. 11. X axis position error.
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Seul Jung received the B.S. degree in
Electrical & Computer Engineering
from Wayne State University in 1988,
and the M.S and Ph.D. degrees from
the University of California, Davis in
1991 and 1996, respectively. After a
post doctoral position at the AHMCT
(Advanced Highway Maintenance and
Construction Technology) of Univer-
sity of California, Davis, he joined the Chungnam National
University in 1997. He is currently an Associate Professor
at the Department of Mechatronic Engineering, Chungnam
National University. His research interests include a hard-
ware implementation of intelligent controllers, intelligent
emotional engineering, and intelligent robotic systems.
Hyun-Taek Cho received the B.S.
and M.S. degrees in Mechatronics
Engineering from Chungnam National
University, in 2000 and 2003, respec-
tively. He is now a Researcher in the
Department of Intelligent Precision
Machines at KIMM. His research
interests involve neural network con-
trollers and DSP systems.You can also read
