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Design of an eigenstructure assignment control using Genetic Algorithm

applied to a Ball and Beam system

Conference Paper · January 2015

DOI: 10.20906/CPS/COB-2015-0668

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 DESIG OF AN EIGENSTRUCTURE ASSIGNMENT CONTROLLER
USING GENETIC ALGORITHM APPLIED TO A BALL AND BEAM
SYSTEM
Lucas Niro
Marcio Aurelio Furtado Montezuma
Bruno Masaharu Shimada
Fabian Andres Lara-Molina
Edson Hideki Koroishi
Federal Technological University of Paraná, campus Conélio Procópio, Avenida Alberto Carazzai, 1640. 86300-000 Conélio Procópio
- PR - Brazil
lucasniro@gmail.com, montezuma@utfpr.edu.br, brunoshimada@gmail.com, fabianmolina@utfpr.edu.br

Lucia Valeria Ramos de Arruda

Universidade Tecnológica Federal do Paraná, Departamento de Pós Graduação e Pesquisa, Programa de Pós Graduação Em Engenharia
Elétrica e Informática Industrial. Av. Sete de Setembro,3165 Rebouças 80230901 - Curitiba, PR - Brasil
lvrarruda@utfpr.edu.br

Abstract. The present contribution aims at describing an optimal design procedure of a ball and beam control system
based on the so-called eigenstructure assignment control. The complete dynamic model of the ball and beam system
was obtained by the Lagrange-Euler equations. This model was simplified to obtain a one degree of freedom model.
The states variables and output of the system are feedbacked in the eigenstructure assignment controller. The gains of
the controller depend on the imposed eigenvalues of the system. The optimal design procedure consists in solving a
constrained nonlinear optimization problem to find the imposed eigenvalues. Therefore the optimal eigenvalues of the
controlled system should minimize the settling time of the response with a constrained control input (maximum alignment
angle of the beam). The optimization problem was solved by using a Genetic Algorithm. A prototype of the ball and beam
system was implemented by substituting the ball by a cart, additionally it was used a linear encoder and a wireless system
to measure the position of the cart along the beam. The control system was implemented with hardware-in-loop framework
using Matlab / Simulink in real-time. The experimental results show the improvement of the dynamic performance of the
system in terms of position accuracy and the transient response.
Keywords: Ball and Beam, Eigenstructure Assignment Control, Genetic Algorithm, Optimization.

1. INTRODUCTION

The ball and beam system is a classical framework used to study several control techniques as studied by Zhang et al.
(2015) and Andreeva et al. (2002). The typical configuration of the ball and beam system consists in a ball that rolls over
a long beam. Additionally, the beam is joined to a motor by means of a revolute joint, the motor modifies the inclination
of the beam. The current position of the motor is measured by a sensor, this allows to feedback the inclination of the
beam into a control system to regulate the position of the ball around a set point position of the beam (Sathiyavathi and
Krishnamurthy, 2013).
The ball and beam system is produced by several companies such as Quanserr , AM IRAr among others in view of
the popularity of ball and beam system. Furthermore, several authors proposed modifications in the system. Wieneke and
White (2011) studied assemble methods and different kinds of sensors. Lin et al. (2010) use magnetic actuator. Hasanzade
et al. (2008) use a camera to determine the position of the ball and Ruth et al. (2015) uses shape memory alloy as a sensor
and actuator to modify the angular position of the beam.
Some changes on ball and beam system are introduced in this contribution. The ball is substituted by a cart, this make
possible the use of an encoder sensor along the beam and an embed microcontroller in the cart. This microcontroller
decodes the encoder pulses and it transmits the actual position of the cart by Radio Frequency (RF) to the computer. The
prototype of the modified ball and beam system used in this contribution is presented in the Fig. 1.
On the other hand, ball and beam systems present a high degree of uncertainty, nonlinearity and instability increasing
the difficulties to design the controller. Some methods addressing control and optimization techniques have been devel-
oped to deal with the controller design difficulties (Oh et al., 2011). Among these works, Castillo et al. (2015) uses a
modified optimization technique based on an optimization method with colonies of ants to find the optimal membership
functions and rules of fuzzy controller; Mahmoodabadi et al. (2014) use particle swarm optimization to tune a decoupled
L. Niro, M. A. F. Montezuma, B. M. Shimada and F. A. Lara-Molina
Design of an Eigenstructure Assignment Controller Using Genetic Algorithm Applied to a Ball and Beam System

Figure 1. Cart and Cart system

sliding mode control.

This contribution proposes an optimal design method to tune an eigenstructure assignment controller using Genetic
Algorithms (GA). Consequently, an optimization problem is proposed to select the eigenvalues of the controller aiming at
minimize the settling time of the cart for a desired set-point position. This optimization problem was solved by numerical
simulations considering the complete model of the ball and cart system. The performance of the controlled ball and
cart system is evaluated experimentally with the gains obtained with the optimal eigenvalues. The experimental results
indicated an improvement of the dynamical performance of the ball and cart system.
This paper has five sections. In section 2 it is presented the classical equations of ball and beam system, the complete
model is simplified and it is adapted to the ball and cart proposes. The section 3 presents the eigenstructure assignment
controller. In section 4, the optimal design method to tune the control system based on an optimization problem is
introduced. The numerical and experimental results are shown in section 5. Finally, the conclusion are presented in the
section 6.

2. BALL AND BEAM MODELLING

The dynamic model of the ball and beam system is obtained from the Langrange-Euler approach considering the
generalized coordinates x and α. x is the ball position and α is the angular position of the beam as shown at Fig. 2. In
this case, equation (1) represents the dynamics of the ball and equation (2) models the dynamics of the beam.

Figure 2. Typical Ball and Beam System

[(Jb /R2 ) + m]ẍ − mg sin α − mxα̇2 = 0 (1)

2
(mx + J)α̈ + 2mxẋα̇ − mg cos α = τ (2)
 23rd ABCM International Congress of Mechanical Engineering (COBEM 2015)
December 6-11, 2015, Rio de Janeiro, RJ, Brazil

Because of the ball was repalced by a cart in our modified system, the term Jb , which represents the moment of inertia
of the ball, is neglected. Considering small angles for α, some terms can be canceled, so the simplified model of the
system is presented in eq. (3) and eq. (4).

mẍ − mgα = 0 (3)

2
(mx + J)α̈ + 2mxẋα̇ − mg = τ (4)

Additionally, it is considered that the torques produced by the cart weight and moment of inertia of the beam are small
compared with the torque of the motor. Therefore, the eq. (4) can be ignored. Consequently, the dynamic of the system is
summarized to eq. (5).

ẍ = gα (5)

Where g is the acceleration of the gravity. Applying the space states formulation to eq. (5), considering α is an input
control, the position of the cart x as the variable to be controlled. With x1 = x and x2 = ẋ1 , thus:
      
ẋ1 0 1 x1 0
= + α (6)
ẋ2 0 0 x2 9810
 
  x1
y= 1 0 (7)
x2

3. EIGENSTRUCTURE ASSIGMENT CONTROLLER

The open-loop system is represented by the eqs. (8) and (9). The state vector x has nth-order.

ẋ = Ax + Bu (8)
 
E
y = Cx = x (9)
F

where y is the p × 1 output vector, w = Ex is a m × 1 vector. It is required that y tracks an input p × 1 reference vector
r. According to D’Azzo and Houpis (1995) the design method consists in the addition of a comparator vector and an
integrator which satisfies the eq. (10).

ż = r − w = r − Ex (10)

D’Azzo and Houpis (1995) present the state feedback control law to be used here which is:

 
  x
u = K1 x + K2 z = K1 K2 (11)
z

A block diagram representing the feedback control system is shown in Fig. 3. The control system consisting of the plant
dynamics and output equations given by eqs. (8) and (9) and the control law given by eq (10) and eq (11)

Figure 3. Eigenstructure assignment control system

L. Niro, M. A. F. Montezuma, B. M. Shimada and F. A. Lara-Molina
Design of an Eigenstructure Assignment Controller Using Genetic Algorithm Applied to a Ball and Beam System

This control law assigns the desired closed loop eigenvalues spectrum if and only if the system modeled by matrices
(Ā, B̄) is controllable. It has been shown that this condition is satisfied if (A, B) is a controllable pair and
 
B A
Rank =n+p (12)
0 −E

To (A, B) be controllable it is necessary to satisfy the controllability condition of eq. (13).

A2 B An−m B = n
 
Rank(Mc ) = Rank B AB ... (13)

Satisfying the conditions of eq. (12) and eq. (13), it is guaranteed that the control law of eq. (14) can be synthesized
such that the closed-loop output tracks the input set-point. In that case the closed-loop state equation is:

      
ẋ A + BK + 1 BK2 x 0
= + r (14)
ż −E 0 z I

The feedback matrix must be selected so that the eigenvalues are in the left-half plane for the closed-loop plant matrix
of eq. (14). The state feedback is applied to assign the closed loop eigenvalue spectrum.
The Ker(S(λi )) imposes constraints on the eigenvector υi that may be associated with the assigned eigenvalue λi .
Ker(S(λi )) identifies a specific subspace, and the selected eigenvectors υi must be located within this subspace. In
addition, the selected eigenvectors must be linearly independent so that the inverse matrix V−1 exists (Montezuma,
2010).
The synthesis of state-feedback control assumes that all the states x are measurable or that they can be generated from
output. In many practical control systems it is physically or economically impractical to install all the transducers which
would be necessary to measure the states (D’Azzo and Houpis (1995)). The ability to reconstruct the plant states from
output requires that all the states be observable. The necessary condition for complete observability is given by Eq. (15).

Rank(Mo ) = Rank CT A T CT (AT )2 CT ... (AT )n−m CT = n

 
(15)
ŷ = Cx̂ (16)

The purpose of this section is to present methods of reconstructing the states from the measured output by a dynamical
system which is called an observer. The reconstructed state vector x̂ may then be used to implement a state-feedback
control law u = Kx̂. The basic method of reconstructing the states is to simulate the state and output of the plant. These
equations are simulated with the same input u applied to the physical system. The states of the simulated system and of the
physical system will then be identical only if the initial condition of both be the same. However, the physical plant may be
subjected to unmeasurable disturbances which cannot be applied to the simulation, therefore, the difference between the
actual plant output y and the simulation output ŷ is used as another input in the simulation equation. Thus, the observer
state and output equations became:

x̂˙ = Ax̂ + Bu + L(y − ŷ) (17)

where L is the n × p observer matrix, the synthesis of L can be seen in D’Azzo and Houpis (1995).
The eigenvalues of (A-LC) are usually selected so that they are to the left of the eigenvalues of A. Thus, the state
observer rapidly approaches the plant states. The representation of the physical plant represented by Eqs. (8) and (9) and
the observer represented by Eqs. (16) and (17) are shown in Fig. 4.

4. Optimization problem

The optimization problem of eq. (18) aims at finding the eigenvalues λi of the control system that minimize the settling
time of the controlled ball and beam system response. The objective function is the settling time te which is function of
the eigenvalues λ = λ1 λ2 λ3 of the controlled system. A constraint is imposed in the control to assure that the
revolute joint α not exceed the maximum reachable angle. Moreover the three eigenvalues λ should be different to ensure
the stability of the eigenstructure assignment controller.
 23rd ABCM International Congress of Mechanical Engineering (COBEM 2015)
December 6-11, 2015, Rio de Janeiro, RJ, Brazil

Figure 4. Control system diagram with state observer.

min te (λ)
λ
subjected to
|α| ≤ αmax
λ1 6= λ2 6= λ3
(18)
The optimization problem of eq. (18) is nonlinear and constrained since the response of the controlled system depends
of the controller gains which in turn depends on the eigenvalues λ of the as stated in eq. (14). The control action which
is the inclination of beam is constrained to avoid the saturation of the motor, thus |α| ≤ αmax . Heuristic techniques have
been used in order to solve this sort of optimization problems(Lara-Molina et al., 2014; Koroishi et al., 2014), specifically
Genetic Algorithms have been used in several optimization problems of engineering (Lara-Molina et al., 2011).

4.1 Genetic Algorithm

Genetic Algorithms (GAs) are heuristic search algorithms based on the mechanism of natural selection and natural
genetics initially proposed by Holland (Holland, 1992). GAs are high performance and robust optimization methods to
solve engineering problems.
In general, a genetic algorithm has four basic characteristics: i) A genetic representation of solutions to the problem;
ii) A way to create an initial population of solutions; iii) Selection of the population for next generation, an evaluation
function rating solutions in terms of their fitness; iv) Genetic operators that alter the genetic ascendants during reproduc-
tion. A fluxogram of a standard genetic algorithm is presented in Fig. 5.

Figure 5. Fluxogram of a standard Genetic Algorithm.

GAs start with an initial set of random solutions, this set of solutions are called the population. Each individual
L. Niro, M. A. F. Montezuma, B. M. Shimada and F. A. Lara-Molina
Design of an Eigenstructure Assignment Controller Using Genetic Algorithm Applied to a Ball and Beam System

of the population, which is a chromosome, represents a potential solution to the problem. The encoding is a genetic
representation of the chromosome. In the evaluation, a measure of fitness is assigned to each individual. Individuals called
parents are selected, the parents contribute to the population at the next generation. Some individuals of the population
suffer genetic operations to create new individuals through stochastic transformations. There are two types of genetic
operations: crossover and mutation. Crossover creates new individuals by combination of the parts of two parents; and
mutation creates new individuals by randomly altering chromosome characteristics to guarantee genetic diversity in the
population. New individuals of the population are called offspring. A new population is formed by selecting the more fit
individuals from the present population and the offspring population. After successive iterations called generations, the
algorithm converges to the best individual, which hopefully represents an optimal solution to the problem.

5. Results

The simulations of the model were implemented using MATLAB. After some preliminary simulations, the tuning
parameters used in the GA optimization algorithm are presented in Table 1.

Table 1. Parameters used in the GA.

Parameter GA
Max. Generation number 100
Encoding Type Real
Population size 30
Crossover probability 0.5
Mutation rate 0.08

In order to find the optimal set of the eigenvalues λ to determine the gains of the controller, the following methodology
was used. First, the optimization problem of eq. (18) was solved with different constraints in the control action αmax
minimize the setting time in the regulation of the position of the cart at 100mm. The solutions for different constraint in
the control action αmax are presented Tab. 2. It was verified that increasing the constraint αmax the magnitude of the
optimal eigenvalues λ increases too.

Table 2. Optimal λ obtained with several values of αmas .

αmax (rad) λ1 (rad/s) λ2 (rad/s) λ3 (rad/s)

0.005 1.356942668 1.477015601 1.552328977
0.01 1,913269581 2.060862589 2.231463561
0.015 2.443027943 2.538950259 2.601279831
0.02 2.811474014 2.973049064 2.973876922
0.025 3.203312615 3.231752491 3.351833777
0.027 3.186871169 3.331554309 3.442817997

Figures 6 and 7 show the convergence of the optimization after 9 generations. The Fig. 6 shows the settling time versus
generations, the settling time is minimized after 9 generation.
Figure 7 presents the behavior of the eigenvalues along the generations. The results indicates that the magnitude of
the eigenvalues increases to minimize the setting time. After 9 generations, the eigenvalues reach their optimal values
that minimize the setting time. As seen in Fig. 7 the GA was executed by 100 generations to avoid an early stoping of the
optimization.
In order to validate the method, the optimal eigenvalues λ were used in the real plant. All eigenvalues of Tab. 2
were tested in the real plant. It was verified that the system diverges for a control action α equal or superiors to 0.028rad
(1.6Âř). The system exhibits an acceptable behavior for αmax > 0, 02rad. Nevertheless, for αmax < 0, 02rad the system
exhibits a considerable steady error because the control system can not compensate the friction between the cart and the
beam. However, the last eigenvalues λ in Tab. 2 exhibits goods results specially the last one that executes the regulates
the system with the minimum setting time.
The optimal solution for αmax = 0.027 rad of Tab. 2 were used in the prototype. The Fig. 8 shows experimental results
to regulate set-point position of the cart at 100mm. The experiment was performed 4 times to illustrate the repeatability
of the system and the small steady state error are shown in Fig. 8. Additionally, it can be seen that the simulation response
is very closed to the experimental results.
The constraints imposed in the optimization problem limit the control action, therefore the selected eigenvalues do not
saturate the motor. Additionally they minimize the time to stabilize the system in a desired set-point. The experiments in
 23rd ABCM International Congress of Mechanical Engineering (COBEM 2015)
December 6-11, 2015, Rio de Janeiro, RJ, Brazil

3.5
Fitness Function

Settling Time (s)

3

2.5

1.5

0.5

0
0 10 20 30 40 50 60 70 80 90 100

Generations
Figure 6. Fitness Function

4
GA Variables
Eigenvalues (rad/s)

3.5

2.5

1.5

h
0.5 1
h2
h3
0
0 10 20 30 40 50 60 70 80 90 100

Generations
Figure 7. Design variables: eigenvalues λ

Transient Response
120

100

80
Position (mm)

60
Set Point
Simulation
40
Real 1
20
Real 2
Real 3
0
Real 4

ï20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Time (s)

Figure 8. Position control: simulation and experimental results

the prototype show the efficiency of the design method. Additional benefits were obtained by using this design method,
the steady state error was reduced considerably. Using imposed eigenvalues was obtained a steady error of 5 mm. Using
the optimal design method to find the optimal eigenvalues the steady error was reduced to 0.5 mm.
The necessary time to solve the optimization process is smaller than 2 minutes using Desktop Computers Core I7.
This shows that the design method to tune the control system is not computationally intensive.
 L. Niro, M. A. F. Montezuma, B. M. Shimada and F. A. Lara-Molina
Design of an Eigenstructure Assignment Controller Using Genetic Algorithm Applied to a Ball and Beam System

6. Conclusion

An optimal design method to tune an eigenstructure assignment controller using Genetic Algorithms (GA) was de-
scribed in this contribution. This optimal design method was evaluated by means of numerical simulation and experi-
mental test in the prototype to verified that the the purposed methodology enhance the dynamic performance of the ball
and beam system. Additional benefits were obtained by using this design method, e.g. the steady state error was reduced
considerably.
The next step of the work will enhance the dynamic model of the system to approximate simulation to the real dynamics
of the prototype. Additionally, a nonlinear model of the friction between the cart and the beam will be also considered.
Nonlinear control techniques using fuzzy approaches will be considered to enhance the efficiency of the control system.

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8. RESPONSIBILITY NOTICE

The authors are the only responsible for the printed material included in this paper.

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