Applications of iterative learning control tehniques on an inverted pendulum system.
Vacarescu, Cella Flavia
1. INTRODUCTION
The use of Iterative Learning Control algorithm aims to improve
performance of automatic control systems that perform a repeated
operation. This can be found in many industrial production processes,
robotics or chemistry, in which repetition is caused by mass production
or assembly lines. The basic ideas of ILC have been formulated for the
first time in a U.S. patent (Garden, 1971) and in a Japanese publication
in 1978 (Uchiyama, 1978). Various tests were performed with ILC and
concluded that ILC is truly a new method of control. In the first book
dedicated to ILC due to Arimoto (Arimoto et al., 1984), has been used a
derivative learning function (D) with continuous time. Proportional,
derivative and proportional-derivative functions are the most used types
of learning functions especially for nonlinear systems. Most known
learning algorithms use linear functions ranging in time (Lee et al.,
2000), (Moore et al., 2005), (Lee et al., 1997), and functions ranging
according to the iteration j (Moore, 1993), (Owens & Rogers, 2004).
2. THE "INVERTED-PENDULUM" SYSTEM
The inverted-pendulum system consists of a pendulum mounted on a
cart so that the pendulum can swing by itself vertically (fig. 2). The
cart is driven by a DC motor. To oscillate or swing the pendulum, the
cart is pushed back and forth on a track of limited length. Pendulum
stationary positions (vertical or down) are equilibrium positions in
which no force is applied. Generally, the problem is to bring the
pendulum in one of the equilibrium positions. It is preferable to do
this as soon as possible, with little vibration and without leaving the
angle or the speed grow too much.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
The pendulum can rotate verticaly around the axis fixed on the
cart. The cart is moving on a horizontal support (pair of rails) located
in the plane of rotation. In fig. 1, F, is the traction force (control
force) applied on the cart. Cart's mass is [m.sub.c], and
pendulum's mass is, [m.sub.p] The distance between the rotation
axis (OR) and system's mass center (CM) is noted with l and
system's inertia moment is denoted by J. The system's state is
the column vector x=col ([x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4])
where: [x.sub.1] is the cart's position (distance from the
support's center); [x.sub.2] is the angle between the upper
position and the line that connects the mass center with the momentary
position of the pendulum, measured counterclockwise reverse ([x.sub.2]=0
for its upper position); [x.sub.3] is the cart's speed and
[x.sub.4] is the angular velocity of the pendulum. The simulation model
of the pendulum-cart system is shown through a Simulink function and is
shown in fig. 2.
3. REAL TIME CONTROL PENDULUM-CART SYSTEM DESIGNED USING PID REGULATORS WITH ILC
A proportional-integral-derivative controller (PID controller) is a
generic control loop feedback mechanism (controller) widely used in
industrial control systems; a PID is the most commonly used feedback
controller. A PID controller calculates an "error" value as
the difference between a measured process variable and a desired
setpoint. The controller attempts to minimize the error by adjusting the
process control inputs. It will therefore be considered the
pendulum-cart system with two PID controllers (fig. 3), the first one to
adjust the angle and the second one to adjust the position.
Regulator's outputs are summed to achieve the final input (input U)
for the digital-analog converter, acting on the cart's movement.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
Therefore we implemented the PID controller in the Crane method,
optimized by using the ILC algorithm. "Crane control problem",
described below, is the simplest control problem, in terms of number of
input variables because its only one input. In this case the only
objective pursued is to establish a reference for the cart's
position, without the influence of other variables such as angular
velocity or pendulum's angle. ILC can be combined with a
conventional control structure in several ways: serial, parallel and
with current iteration. In the Crane mode, was applied, the learning
algorithm ILC in the serial version because in this situation the ILC
contribution acts directly on the reference. The Simulink diagram of the
controller is shown in fig. 4.
The output signal in fig. 4 is a vector containing the following
elements: pendulum's angle, cart's position, angular velocity,
cart's speed, cart's reference position, engine's control
input.
The override is defined by the relationship (1):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
It can be seen in fig. 5 that time to reach maximum value, tm, is
11.5 seconds and the adjustment time, [t.sub.r], is 12 seconds. So, the
override in this case is 5.5%.
The values for the quality indicators calculated above, are
characterizing the effect of introducing the ILC learning algorithm to
adjust the inverted pendulum system.
To observe the contribution brought by the ILC algorithm, in fig.
6, is presented the variation of the two signals: the reference (the
input) and cart's position (the output), without applying ILC
algorithm.
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
As it can be seen in fig. 6, the override does not increase very
much from the case when we included the ILC algorithm, but the
adjustment time [t.sub.r], is clearly bigger, going over 15 seconds.
4. CONCLUSION
The use of the ILC algorithm brings a clear improvement over the
situation presented in fig. 6. ILC learning algorithm leads to improved
empirical quality indicators and the override decreases during
iterations, but the disadvantage is that must be completed several
iterations to achieve this performance. A notable advantage of the ILC
algorithm is that it does not require a mathematical modelation of the
process on which is implemented. This has significance both for the
specific process but also in general, because there are many processes
whose mathematical models are very difficult if not impossible to
determin. The use of the ILC method on the presented system allows the
author to outline the simplicity in which the method can be applied on a
process. The contents of this paper aimed to provide the reader an
insight into important ideas, potential and limits of the ILC algorithm.
In conclusion, the results of the experiment conducted in this
paper, respond to the question "Why using ILC?". Answers
should be: ILC seeks exit using facilities without any knowledge about
the system's state, it has a simple structure (integration along
the iteration axis), it is a learning process based on memorizing
(requires to remember the error signals or input signals across the
whole period of time), requires little information about the process,
actually is almost a model independent method and this is a special
feature in implementation and finally the reference must be the same for
all iterations.
Indeed, the early results of the ILC include many results and
learning algorithms which exceed this exposure. Although is the
beginning of the third decade of active research and the ILC field does
not show any sign of a slowdown. The results are encouraging, especially
since the chosen process is a complex one because of the oscillating
character and of the shown nonlinearities.
5. REFERENCES
Arimoto, S.; Kawamura, S. & Miyazaki, F. (1984). Bettering
operation of robots by learning. Journal of Robotic Systems, Vol 1,
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Garden, M.; Park H. S. & Udiljak T. (1971). Learning control of
actuators in control systems. U.S. Patent US03555252, January 1971,
Leeds & Northrup Company, Philadelphia, USA
Lee, K.S.; Bien, Z. (1997). A note on convergence property of
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equivalence of causal LTI iterative learning control and feedback
control. Automatica, Vol 40, No. 5, 2004, pg. 895-898, ISSN 0005-1098
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