=Paper=
{{Paper
|id=Vol-1638/Paper78
|storemode=property
|title=Stabilization of elastic inverted pendulum with hysteresis
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper78.pdf
|volume=Vol-1638
|authors=Mikhail E. Semenov,Alexey M. Solovyev
}}
==Stabilization of elastic inverted pendulum with hysteresis ==
https://ceur-ws.org/Vol-1638/Paper78.pdf
Mathematical Modeling
STABILIZATION OF ELASTIC INVERTED
PENDULUM WITH HYSTERESIS
M.E. Semenov1, A.M. Solovyev2
1
Voronezh State University, Voronezh, Russia
2
JSC Concern “Sozvezdie”, Voronezh, Russia
Abstract. In this article, we investigate an elastic inverted pendulum with hys-
teretic nonlinearity (a backlash) in a suspension point. Namely, problems of
stabilization and the optimization of such a system are considered. An algo-
rithm (based on a bionic model) which provides effective procedure for finding
of optimal parameters is presented and applied to considered system. The re-
sults of numerical simulations, namely the phase portraits and the dynamics of
Lyapunov function are also presented and discussed.
Keywords: elastic inverted pendulum, hysteretic control, stabilization problem,
bionic algorithm.
Citation: Semenov ME, Solovyev AM. Stabilization of elastic inverted pendu-
lum with hysteresis. CEUR Workshop Proceedings, 2016; 1638: 650-657. DOI:
10.18287/1613-0073-2016-1638-650-657
Introduction
As known, the problem of the inverted pendulum plays a central role in the control
theory [1-11]. In particular, the problem of inverted pendulum (as a test model) pro-
vides many challenging problems to control design. Because of their nonlinear nature
pendulums have maintained their usefulness and they are now used to illustrate many
of ideas emerging in the field of nonlinear control [12]. Typical examples are feed-
back stabilization, variable structure control, passivity-based control, back-stepping
and forwarding, nonlinear observers, friction compensation, and nonlinear model
reduction. The challenges of control made the inverted pendulum systems a classic
tool in control laboratories. It should also be noted that the problem of stabilization of
such a system is a classical problem of the dynamics and the control theory. Moreo-
ver, the model of inverted pendulum is widely used as a standard for testing of the
control algorithms (for PID controller, neural networks, fuzzy control, etc.).
According to control purposes of the inverted pendulum, the control of inverted pen-
dulum can be divided into three aspects. The first widely researched aspect is the
swing-up control of the inverted pendulum [13-15]. The interesting and important
results on the time optimal control of the inverted pendulum were obtained in [13,
15]. In particular, in [15], the optimal transients (taking into account the cylindrical
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Mathematical Modeling Semenov ME, Solovyev AM. Stabilization…
character of the state space of the system under control) were built for different values
of the parameters and constraints on the control torque. The second aspect is the stabi-
lization of the inverted pendulum [16, 17]. The third aspect is the tracking control of
the inverted pendulum [18].
In practice, stabilization and tracking control are more useful for an application. A
backlash in the suspension point is a kind of hysteretic nonlinearity. The hysteretic
phenomena (especially in the form of control parameters) play an important role in
such a fields as physics, chemistry, biology, economics, etc. It should also be pointed
out that the hysteretic phenomena are insufficiently known in our days. The purpose
of this paper is investigation of the possible stabilization (in a vicinity of vertical posi-
tion) of the elastic inverted pendulum in the presence of a backlash in the suspension
point together with investigation of various aspects of such a dynamical system.
Problem
Let’s consider the model of stabilization of the inverted pendulum in the vicinity of
the vertical position. The pendulum is considered as an elastic rod which is hingedly
fixed on a cylinder. Motion of the cylinder is excited by the horizontal motion of a
piston (see the Fig. 1).
Fig. 1. Model of elastic inverted pendulum: geometry of the problem
A mathematical model of a similar mechanical system was considered in [19]. Inves-
tigation of the dynamics of an elastic inverted pendulum was carried out in [20-23].
Here x, y is the coordinates of the elastic rod with mass m and density ρ; the Ox
axis coincides with a tangent to rod's profile in the suspension point; θis an angle of
slope for the coordinates of a rod, and I is a centroidal moment of inertia of the rod's
section;
X , x is the Cartesian coordinate system connected with a considered mechanical
system (namely the X coordinate determines the position of the piston in a cylinder),
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Mathematical Modeling Semenov ME, Solovyev AM. Stabilization…
M is a mass of a cylinder with length L, and F is a force joined to the piston with mass
mp (such a force is treated as control).
Hysteretic nonlinearity
In the following consideration, we use the operator technique for the hysteretic non-
linearities following the ideas of Krasnosel'skii and Pokrovskii [24]. Output of the
backlash operator on the monotonic inputs can be described by the following expres-
sion:
X (t ) Г [ X 0 , L] Y (t )
L
0, Y (t ) X 0 2 ,
L L
Y (t ) , Y (t ) X 0 ,
2 2
L L
Y (t ) 2 , Y (t ) X 0 2 .
Here X0 is the initial position of the piston in a cylinder. Such an expression (action of
such an operator) can be illustrated by the Fig. 2.
Fig. 2. Dynamics of input-output relation for the backlash operator
The detailed description of this operator as well as its properties is considered in the
book of Krasnosel'skii and Pokrovskii [24].
Here X(t) is a displacement of the cylinder's center, and Y(t) is a displacement of the
piston in the horizontal plane (see Fig. 1).
Physical model
Let’s assume that the deviation y and angle θ are small, i.e., x x and the boundary
conditions that determine the curvature of the pendulum are:
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Mathematical Modeling Semenov ME, Solovyev AM. Stabilization…
y (0, t ) y (0, t ) 0,
y (l , t ) y (l , t ) 0.
The function X ( x , t ) describes the behavior of the pendulum's profile in the time and
shows the deviation of the pendulum's points relative to the vertical axis; X , x are
the coordinates of the pendulum's profile, and X (0, t ) s(t ) is a displacement of the
suspension point in the horizontal plane.
The coordinate system transformation in the matrix form is given by
X cos sin y X (0, t )
x sin cos x 0 .
Let us construct the physical model of the considered mechanical system taking into
account a backlash in the suspension point of the elastic rod. In order to do this, we
use the Lagrange formalism.
Making the same transformations (Lagrange formalism, variational principle and
Tailor’s expansion), the system of equations which describes the dynamics of the
system under consideration has the following form:
EIl
X m X gX 0, t ,
M m X 0, t mlX 0, t f t ,
g
M m X 0, t X 0, t f (t ),
MEI
l m
f (t ) Г X 0, t , Y t , L, F0 F ,
m p Y t F .
Stabilization
Let us consider the problem of control of the pendulum using the feedback principles,
i.e., the force which affects the piston can be presented by the following equality:
F k sign(a e1 e 2 ),
where a>0, k > 0 and
l
e1 X dl,
0
l
2 X dl.
e
0
Here e1 is an average angle of the rod's deviation, and e2 is an average angular veloci-
ty of the rod.
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Mathematical Modeling Semenov ME, Solovyev AM. Stabilization…
Optimization problem
As was mentioned above, the problem’s solution on stabilization of the elastic invert-
ed pendulum in the vicinity of the upper position is consisted in search of the optimal
values for coefficients a and k .
In order to solve the optimization problem in the system under consideration, we use
the bionic adaptation algorithms because the hysteretic peculiarities in the considered
pendulum's model lead to some difficulties in use of the classical optimization algo-
rithms due to nondifferentiability of the functions in the system of equations.
Such algorithms are the part of the line of investigation that can be called as an "adap-
tive behavior." Main method of this line consists in the investigation of artificial or-
ganisms (in a form of a computer program or a robot) that can be named as animats
(these animats can be adapted to the environment). The behavior of animats emulates
the behavior of animals.
Actual line of investigation in the frame of the animat approach is emulation of
searching behavior of animals [25, 26]. Let us consider the bionic model of adaptive
searching behavior on the example of caddis-flies larvae or Chaetopteryx villosa. The
main schema of searching behavior can be characterized by the two stages:
Motion in chosen direction (conservative tactics);
Random change of motion direction (stochastic searching tactics).
We consider this model for the simple case of maximum search for the function of
two variables. Let us describe the main stages of the considered model:
1. We consider an animat which is moved in the twodimensional space x, y. Main
purpose of animat is maximum search for the function f (x, y).
2. Animat is functioned in discrete time t = 0, 1, 2, … Animat estimates the change of
current value of f (x, y) in comparison with the previous time
f t f t f t 1.
3. Every time animat moves so its coordinates x and y change by Δx(t) and Δy(t), re-
spectively.
4. Animat has two tactics of behavior:
a) conservative tactics;
b) stochastic searching tactics.
In that way, we can use the proposed algorithm for searching the optimal control in
the stabilization problem of the elastic inverted pendulum. Taking into account the
reasoning presented above, we can apply the presented algorithm to functional J(a,k)
where the coefficients a and k determine the character of control of the mechanical.
Due to the fact that the presented bionic algorithm is used to maximum search of the
function of two variables, we will consider minimization of the functional as proce-
dure for finding the coefficients a and k that lead to realization of the condition
J a, k max .
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Simulation results
Characteristics and initial conditions for the mechanical system under consideration are:
m = 1 kg; M = 10 kg; l = 1 m; ρ = 1,04 kg/m; E = 210·109 Pa; I = 0,087 kg·m2; α=
0.06°; L = 0,01 m; mp = 1 kg.
In the searching process for optimization (using the bionic algorithm), we have ob-
tained the following values of coefficients: a = 8,4 and k = 1,39.
In order to estimate the stability of the considered system, we use the Lyapunov crite-
rion. In particular, we use the following Lyapunov function:
V e12 e 22 .
The phase trajectory of such a system together with the dynamics of Lyapunov func-
tion in time (namely in discrete time which corresponds to the difference scheme) are
presented in the Fig. 3. In this figure, the integral angle e1 and integral angular veloci-
ty e2.
Fig. 3. Phase trajectory (top panel) and dynamics of Lyapunov function (bottom panel)
Conclusions
In this paper, we have considered the stabilization problem of the elastic inverted
pendulum under hysteretic control in the form of a backlash in the suspension point.
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Mathematical Modeling Semenov ME, Solovyev AM. Stabilization…
Also, the problem of the optimization for the system under consideration was ana-
lyzed. Main coefficients, namely a and k , that provide the solution of the optimiza-
tion problem for the considered system were obtained using the so-called bionic algo-
rithm.
All the results on stabilization of the system under consideration have obtained using
the corresponding numerical methods based on the difference scheme. The results of
numerical simulations show that the considered system eventually tends to the stable
state both in the case of the absence of a backlash and in the case of its presence.
These facts were presented in the form of the corresponding phase portraits for the
considered system. Moreover, in order to estimate the stability of the elastic pendu-
lum with the hysteretic nonlinearity in the suspension point, we have used the Lya-
punov criterion and the dynamics of the corresponding Lyapunov function has also
been presented.
Acknowledgement
The work was partially funded by the Russian Foundation for Basic Research Grants
(project №16-08-00312).
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