In this article, we investigate an elastic inverted pendulum with hysteretic nonlinearity (a backlash) in a suspension point. Namely, problems of stabilization and the optimization of such a system are considered. An algorithm (based on a bionic model) which provides effective procedure for finding of optimal parameters is presented and applied to considered system. The results of numerical simulations, namely the phase portraits and the dynamics of Lyapunov function are also presented and discussed.
As known, the problem of the inverted pendulum plays a central role in the control
theory [
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 position) 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.
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).
A mathematical model of a similar mechanical system was considered in [
In the following consideration, we use the operator technique for the hysteretic
nonlinearities following the ideas of Krasnosel'skii and Pokrovskii [
2 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.
The detailed description of this operator as well as its properties is considered in the
book of Krasnosel'skii and Pokrovskii [
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).
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: 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: X EIl X gX 0, t , m M mX0, t mlX 0, t f t , g M mX 0, t MEI X 0, t f (t), l m f (t) Г X 0, t , Y t , L, F0 F , m pYt F.
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 e2 ), where a>0, k > 0 and l e1 X dl, 0l e2 X dl. 0 Here e1 is an average angle of the rod's deviation, and e2 is an average angular velocity of the rod.
As was mentioned above, the problem’s solution on stabilization of the elastic inverted 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 algorithms 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 "adaptive behavior." Main method of this line consists in the investigation of artificial organisms (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 [
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), respectively. 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 procedure for finding the coefficients a and k that lead to realization of the condition J a, k max .
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 obtained 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 criterion. In particular, we use the following Lyapunov function: V e12 e2 .
2 The phase trajectory of such a system together with the dynamics of Lyapunov function 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 velocity e2.
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. Also, the problem of the optimization for the system under consideration was analyzed. Main coefficients, namely a and k , that provide the solution of the optimization problem for the considered system were obtained using the so-called bionic algorithm.
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 pendulum with the hysteretic nonlinearity in the suspension point, we have used the Lyapunov criterion and the dynamics of the corresponding Lyapunov function has also been presented.
The work was partially funded by the Russian Foundation for Basic Research Grants (project №16-08-00312).