Swarm Control Designs Applied to a Non-Ideal Load

Transportation System

Fábio Roberto Chavarette2, Ivan Rizzo Guilherme1, Orlando Saraiva do Nascimento Junior2, Nelson José Peruzzi3, José Manoel Balthazar1

1UNESP - Univ Estadual Paulista, Department of Statistics, Applied Mathematics and Computation, PO BOX 178, 13500-230, Rio

Claro, SP, Brazil

2Ometto Herminio University Center at Araras - UNIARARAS, Engineering Center, Av. Dr. Maximiliano Baruto, 500, Jd. Universitário, 13607-339, Araras, SP, Brazil.

3UNESP - Univ Estadual Paulista , Department of Exact Science, Jaboticabal, SP, Brazil, 13484-900, Jaboticabal, SP, Brazil

chavarette@uniararas.br, ivan@rc.unesp.br, saraiva@uniararas.br, peruzzi@fcav.unesp.br, jmbaltha@rc.unesp.br

Abstract - In this paper, a load transport system in platforms is considered. It is a transport device and is modelled as an inverted pendulum built on a car driven by a DC motor. The motion equations were obtained by Lagrange’s equations. The

mathematical model considers the interaction between the DC motor and the dynamic system. The dynamic system was analysed and a Swarm Control Design was developed to stabilize the model

of this load transport system.

I. INTRODUCTION

To load, to move and to unload materials are usual activities that not add value to production and are expensive and decisive actions to the market competition. The load transference system in industry has the function to transport raw-material, semi-finished and finished products from storage stations to processing stations and vice versa.

The load transference system can be composed by module, tends to be automated and can uses robots to manipulate the materials. It is a complex activity and requires automatic communication between the several production cells. To realize the movement and positioning of the materials, the robotic system can be connected to a controller and each transport cell may be composed by a DC motor, a pneumatic system, a codification system and presence sensors.

The dynamic analysis and performance optimization of the load system may be done through computational simulations of a mathematical model automatic system. Simulating the operation of the load transport device through models aims to

determine instabilities, mechanical vibrations and the influences of auto-excitation of the system and the energy consumption.

The main problem in load transport device on platforms is the stabilization of the vibrations that may occur during the movement of materials due to the acceleration/deceleration or induced by the driving system. These effects generate vibrations in the system which may produce imprecision of the motions and/or cause impacts during the load transports. The vibrations could be minimized by the reduction of the velocities during the load transport, but it may decrease the productivity and efficiency.

The occurrence of mechanical vibrations during load transport is caused, basically, by: the dissipation (damping) and the excitation. The dissipation is produced for the friction and it is responsible for the reduction of the amount of kinetic and potential energy of the mechanical system. The excitation will depend on the characteristics of the dynamic system. If the excitation is influenced by system response, the excitation is called non-ideal [1]. The oscillations in the chaotic non-ideal dynamics depend, directly, of the excitation properties which are related with the energy source [2,3].

Lately, a significant interest in control of the chaotic non-ideal systems has been observed and many of the techniques have been discussed in the literature [6-8].

In this paper, an auto-excited oscillations model in load transport device will be considered. The chaotic model has a inverted pendulum fixed on a car that is driven by a DC motor. We controlled the chaotic systems using the Particle Swarm Optimization with optimization techniques. [9].

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This work is organized as follows: in section 2, we present the non-ideal mathematical model to a load transport device and analyze the dynamic system. In section 3, present Control Swarms technique. In Section 4, apply the Swarm Control for the non-ideal system. In section 5, present some remarks and finally, we list the main references used.

II. MATHEMATICAL MODEL AND THE DYNAMIC OF THE SYSTEM

Figure 1 presents a non-ideal physical model of a load device by a parametrically excited inverted pendulum fixed on a car that is driven by a DC motor.

Fig. 1. Complete Non-Ideal Physical System

The mathematical model considers the non-ideal load

transport system with periodic coefficients [4]. This problem was analyzed qualitatively by comparison of the stability diagrams for several motor torque constants and quantitatively by analysis of the Floquet multipliers. To analysis of the non-ideal periodic system was used Chebyshev polynomial expansion, the Picard iterative method and on Lyapunov-Floquet transformation (L-F transformation). Finally, non-ideal problem was controlled by Sinha’s Control technical [4,5].

Fig. 2. Details of the Sub Systems

The mathematical model to loads transport device problem

is split in two systems: primary system – car and motor; secondary system - inverted pendulum that represents the load

transported. The coordinates of the model are: θ , cx and sx , were θ is the angular displacement of the inverted pendulum,

cx car displacement and sx support displacement. The movement equations of the dimensionless system [4] are:

=′−−+′+′′

=′+−′−+′+

−′′+

+′′

=

−∆++′′+′′

32

2122

20

220

0..11

cos

0)(cos

cxx

ccxxn

sen

nx

seng

senm

Kx

p

γσφξφςφ

φφηαθθ

θθ

θω

λτθω

θθll

(1)

The set of parameters used in the numerical simulations of

the system are: 11 ==== cγσα ; 41.1=ξ ; 01.032 == cc ; πλ = ; 1.0=η ; 1=n ; 2.0=∆ ; 8.1=ς .

With these values the Floquet multipliers of the system are: 0.5765i 0.01562,1 ±=µ ; 0.2391i -0.62884,3 ±=µ ; 0.0610- 5 =µ and

0.3126- 6 =µ . The pair of complex eigenvalues with absolute values bigger

than one ( 12,1 >µ ) indicates that the system is unstable. Figure 3 shows that the displacement amplitude of the car and pendulum increase as t increases.

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Fig. 3. Non-Controlled System

The figure 4 and 5 illustrates the phase portrait of the car and

pendulum of the non-ideal chaotic system. The presence of chaos, due to auto-excitation of the system, is confirmed by positive Lyapunov exponents: 0.238071 =λ ; 0.153242 =λ and

0.0867023 =λ .

Fig. 4. Phase Portrait for the Pendulum

Fig. 5. Phase Portrait for the Car

III. CONTROL DESIGN

Chaos control problems consist to stabilize the dynamic chaotic of a system to an equilibrium point or a periodic orbit or to drive to a specific reference trajectory. Lately, a significant interest in control of the nonlinear systems, exhibiting chaotic behavior, has been observed and many of the techniques have been discussed in literature [6-8]. Among strategies of chaos control with feedback, the most popular is OGY (Ott-Grebogi-York) method [6]. This method uses the Poincaré map of the system. Recently, Sinha et al. [7], proposed a methodology based on the application of the Lyapunov-Floquet transformation, in order to solve this kind of problem. This method allows driving the chaotic motion to any desired periodic orbit or to a fixed point. It is based on linearization of the equations, which describes the error between the actual and desired trajectories. Another technique proposed by Rafikov and Balthazar in [8] uses the dynamic programming to the optimal control. These techniques can be applied to solve a wide range of problems. The Particle Swarm Optimization (PSO), is a technique based

on population stochastic optimization developed by Kennedy and Eberhart in 1995 [9]. A PSO consists of a number of individuals particles that refining their knowledge of the given search space. The PSO refines its search by attracting the particles to positions with good solutions. The Particle Swarm Optimization technique has ever since turned out to be a competitor in the field of numerical optimization. The PSO approach to nonlinear and control has been observed and discussed in the literature [10-12]. Here, we propose a method for control of chaotic load

transport system using the PSO algorithm. A PSO optimization algorithm was used for control the dynamic chaotic of the mathematical model of the load transport device. It is split in

137

two systems to stabilize the chaotic system to period orbit. The proposed method formulates the nonlinear system identification as an optimization problem in parameter space and then Particle Swarm Optimization are used in the optimization process to find the estimated values of the parameters. .

1. Particle Swarm Optimization

The Particle Swarm Optimization (PSO) algorithm is a

computational simulation inspired in social and biological algorithm. This technique has low computational cost and information sharing innate to the social behavior of the composing individuals. The individuals, also called particles, flow through the multidimensional search space looking for feasible solutions of the problem. The position of each particle in this search space represents a possible solution whose feasibility is evaluated using an objective function. Sometimes, PSO is related to the Evolutionary Computation

(EC) techniques, basically with Genetic Algorithms (GA) and Evolutionary Strategies (ES), but there are significant differences between those techniques. Particle Swarm Optimization technique has, ever since, turned out to be competitor in the field of numerical optimization. The Particle Swarm Optimization refines its search by

attracting the particles to positions with good solutions, using the best solution ( ip

r) found by the particle that far and the best

solution found so far considering all the particles ( gpr

). In each interaction, a particle i having position ix

r have its

velocity ivr updated in the following way:

( ) ( )( )igiiiiii xpxpvwvrrrrrrrr

−+−+Χ= 21 ϕϕ (2) where Χ is known as the constriction coefficient described in [13], w is the inertia weight, ip

r

is best solution found by the particle that far and the gp

r

best solution found so far considering all the particles, and 1ϕ

rand 2ϕ

rare random values

different for each particle and for each dimension. The position of each particle ix

r is updated during the execution of iteration.

This is done by adding the velocity vector to the i position vector:

iii vxxrrr

+= (3) The velocity parameters setting that determines the

performance of the Particle Swarm Optimization to a large extent. This process is repeated until the desired result is obtained or a certain number of iterations is reached or even if the solution possibility is discarded.

Fig. 6. Time History (Noncontrolled and Controlled) for the Pendulum System

Fig. 7. Time History (Noncontrolled and Controlled) for the Car System

Fig. 8. Phase Portrait (Noncontrolled and Controlled) for the Pendulum System

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Fig. 9. Phase Portrait (Noncontrolled and Controlled) for the Car System

IV. CONTROL SWARM APPROACH

The proposed algorithm formulates the nonlinear system

identification as an optimization problem in parameter space, and then adaptive Particle Swarm Optimizations are used in the optimization process to find the estimation values of the parameters. The algorithm is used for control the behavior chaotic of the

nonlinear dynamics model (3). The goal of this control synthesis is find the estimation values of the parameters, to drive the orbit of the system to a periodic orbit. We apply the Particle Swarm Optimization algorithm for the

load transportation system (1), to reduce the chaotic behavior of this nonlinear system to a period orbit. The Figures 6 and 7 showed the behavior controlled and uncontrolled of the system (1). In comparing, non-controlled system (see Figure 3) with of numerical results of PSO (Figure 8, 9) we can verify that controlled orbit generated by PSO approach, has small diameter.

Algorithm of the Particle Swarm Optimization Create and initialize an nx-dimensional swarm, S, through the

system of equations (1) and shown in the projection of the phase space (figure 4 and 5). // S.xi i is used to denote the position of particle i // in swarm S. // S.yi i is used to denote the best position of particle i // in swarm S. // S. y , is used to denote the global best position of // particle i in swarm S.

repeat

for each particle i = 1,….,S.ns do

// set the personal best position if f(S.xi) < f(S.yi) then S.yi = S.xi; end

// set the global best position // if f (S.yi) < f(S. y ) then

S. y = S.yi;

end

end

for each particle i=1,…,S.ns do

update the velocity using equation (2); update the position using equation (3); end

until stopping condition is true;

V. CONCLUSION

In this paper, it is considered a non-ideal system whose

application is interesting for load transport problem. The auto-excitations in load transport produce vibrations to system reducing the performance. We applied the Particle Swarm Optimization technique to control non-ideal dynamic system. This control allows reduction of the vibrations of the chaotic non-ideal system to a desired period orbit. The Fig 6-9, illustrates the effectiveness of the control algorithm for load transport problem.

Comparing the numerical results of PSO with the non controlled system (Fig. 6 and 7), we can verify that control orbit generated by PSO approach (Fig. 8 and 9) has small diameter.

The spent with energy during the load transport tends to increase because the auto-excitation increases the vibrations of the system and, consequently, the cost of production may increase too, becoming the final product more expensive. Thus, the control of vibrations of the system is fundamental to minimize the expenses with energy.

REFERENCES

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