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Automated Synthesis of Optimal Controller Using Multi-Objective Genetic Programming For Two- Mass-Spring System

Conference Paper

Automated Synthesis of Optimal Controller Using Multi-Objective Genetic Programming For Two- Mass-Spring System

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There are much research effort in the literature using genetic programming as an efficient tool for design of controllers for industrial systems. In this paper, multi-objective uniform-diversity genetic programming (MUGP) is used for automated synthesis of both structure and parameter tuning of optimal controllers as a many-objective optimization problem. In the proposed evolutionary design methodology, each candidate controller illustrated by a transfer function, whose optimal structure and parameters, obtained based on performance optimization of each candidate controller. The performance indices of each controller are treated as separate objective functions, and thus solved using the multi-objective method of this work. A two-mass-spring system is considered to show the efficiency of the proposed method using performance optimization of open loop and closed loop control system characteristics. The results show that the proposed method is a computationally efficient framework compared to other methods in the literature for automatically designing both structure and parameter tuning of optimal controllers.
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978-1-4799-6743-8/14/$31.00 ©2014 IEEE
Automated Synthesis of Optimal Controller Using
Multi-Objective Genetic Programming For Two-
Mass-Spring System
Iman Gholaminezhad, Ali Jamali and Hirad Assimi
Dept. of Mechanical Engineering
University of Guilan, Rasht, Iran
i.gholaminezhad@gmail.com, ali.jamali@guilan.ac.ir, assimi@gmail.com
AbstractThere are much research effort in the
literature using genetic programming as an efficient tool
for design of controllers for industrial systems. In this
paper, multi-objective uniform-diversity genetic
programming (MUGP) is used for automated synthesis of
both structure and parameter tuning of optimal
controllers as a many-objective optimization problem. In
the proposed evolutionary design methodology, each
candidate controller illustrated by a transfer function,
whose optimal structure and parameters, obtained based
on performance optimization of each candidate controller.
The performance indices of each controller are treated as
separate objective functions, and thus solved using the
multi-objective method of this work. A two-mass-spring
system is considered to show the efficiency of the proposed
method using performance optimization of open loop and
closed loop control system characteristics. The results
show that the proposed method is a computationally
efficient framework compared to other methods in the
literature for automatically designing both structure and
parameter tuning of optimal controllers.
Index TermsMany-objective, Genetic programming, Optimal
controller design, Polynomial controller, Two-mass-spring.
I. INTRODUCTION
Genetic programming is a member of evolutionary
algorithms, which provides a powerful tool for dealing with
system engineering problems. Applications of Genetic
Programming (GP) in system identification and controller
design have been an attractive research area due to its unique
functionality to simultaneously optimize structure and
parameters of the problem to be solved and thus it has been
extensively used in control and system science field for many
years. GP evaluates performance measures of the controlled
systems by implicitly utilizing the information about
constraints and nonlinearities that the closed loop and open
loop simulation provides.
The first comprehensive effort in using genetic
programming for controller design, first carried out by its
inventor, John Koza. He well applied GP for automatic
synthesis of both topology and tuning of different types of
controllers in a parallel platform [1-4]. Although, the results
obtained by J. Koza were interesting and promising compared
to previously designated methods, however due to
computationally expensive nature of the method implemented
by J. Koza, many researchers have been tried to use genetic
programming in more effective approaches for design of
optimal controllers. In this way, R.A. Maher and M.J.
Mohamed proposed multiple basis function genetic
programming (MBFGP) for optimal design of both linear and
nonlinear control systems [5]. They also used the algorithm to
enrich the set of suboptimal state feedback controllers to
include controllers that have product time-state terms. B.
Kadlic, and I. Sekaj suggested an evolutionary-based design of
a controller of non-linear dynamic systems which is
constructed using interconnection and parametrisation of
simple building blocks using cartesian genetic programming
(CGP) in MATLAB [6].
A. Fukunaga et al. investigate the application of simulation-
based genetic programming to evolve controllers that perform
high-level tasks on a service robot [7]. T. Kobayashi et al.
proposed an efficient genetic programming algorithm for the
feedback controller design only using output information with
a new optimal inverse system concept [8]. P. Chen and Y. Lu
used a novel approach to automatic design of robust optimal
controller for interval plants with genetic programming based
on Kharitonov Theorem [9]. D. Gladwin et al. described a
genetic programming based automatic design methodology and
its applications to the maintenance of a stable generated
electrical output from a series-hybrid vehicle generator set [10].
D.C. Dracopoulos and D. Effraimidis applied genetic
programming to a nonlinear high dimensional helicopter
hovering control problem and showed its efficiency compared
to neuro-evolutionary control design methods [11]. A.
Bourmistrova and S. Khantsis performed a Flight Control
System Design Optimization via Genetic Programming [12].
They present a methodology, which was developed to design a
controller that satisfies the objectives of shipboard recovery of
a fixed-wing UAV. I. Sekaj et al. performed different studies
on controller design based on genetic programming. In one
study they suggested three genetic programming-based
approaches for continuous-time process control design using a
network of interconnected and recurrent functions with
continuous-time or discrete-time elementary dynamic building
blocks [13]. J.M. Dolsma make use of genetic programming
for designing nonlinear controllers in his master thesis. He also
used genetic algorithm for optimization of GP parameters [14].
E. Alfaro-Cid et al. used application of Genetic Programming
for the Automatic Design of Controllers for a Surface Ship.
They beneficiate GP to evolve control strategies that, given the
current and desired state of the propulsion and heading
dynamics of a supply ship as inputs, and generate the
commanded forces required to maneuver the ship [15]. S.
Harding and J.F. Miller presented an Evolution of Robot
Controller Using Cartesian Genetic Programming. They used
integer-based version of the graph-based representation and
applied it to evolving an obstacle avoiding robot controller
[16]. R. Barate and A. Manzanera presented an automatic
design of vision-based obstacle avoidance controllers using
Genetic Programming. They used an off-line procedure to
automatically design algorithms adapted to the visual context
and used their controller for a robotic system which is able to
learn context dependent visual clues to navigate in its
environment [17]. In another study they suggested an evolving
vision controllers with a Two-Phase genetic programming
system using imitation [18].
The two-mass-spring system has been widely used as a
common control benchmark in the literature. C. I Marrison and
R. F Stengel applied stochastic robust design of two-mass-
spring system using MCS [19]. They have considered three
cost functions namely, stability, control effort and settling time.
W. Reinelt considered this benchmark for robust controller
design using its input constraints [20]. D. Henrion and M. L.
Overton maximizing the closed loop asymptotic decay rate for
the two-mass-spring system by finding a linearly fixed order
controller [21]. I. B. Tijani et al. performed robust H-infinity
controller synthesis using multi-objective differential evolution
algorithm for this control system [22].
This paper introduces a novel approach for structure
optimization and parameter tuning of optimal controllers using
multi-objective Uniform-diversity genetic programming
(MUGP) algorithm. Six performance requirements of a
candidate controller (stability, settling time, overshoot, rise
time, control effort, Integral time absolute error) are considered
as objective functions in an evolutionary optimization process.
However, several of the performance requirements for such
controller are in competition, which makes achieving all the
goals difficult. Such confliction is also shown in the value path
plot (which shows the distribution of obtained non-dominated
solutions). A two-mass-spring system, which is a fourth-order
benchmark, and has been studied extensively in the literature
using different methods considered as an illustrative examples
for evaluating the efficiency of such evolutionary design
process using MUGP algorithm.
II. MULTI-OBJECTIVE UNIFORM-DIVERSITY GENETIC
PROGRAMMING
Genetic programming is one of the optimization tools based
on evolutionary computation which known by its tree-base
structure. GP imitates the fundamental laws of Darwinian
theory and natural selection. It has the ability to express a
complex structure in a symbolic fashion. Each tree is a
combination of some nodes and branches constituted various
structures. These nodes consist of functions and terminals that
should be chosen according to the problem. The set of
functions includes operators, mathematical functions, condition
expressions and so on. The set of terminals includes variables
and constants.
In order to GP explores the searching space for better fitted
solutions, the principles of evolutionary operations like
crossover and mutation should applied on the initial population.
The crossover operation in GP brings variation into the
population by producing offspring that inherit parts of each
parent. It starts by selection of two parental expressions by the
fitness based selection methods and produces two offspring
expressions that consist of parts taken from each parent. In the
mutation operator the sub-tree rooted at the randomly selected
node is removed and replaced with a randomly generated sub-
tree.
In addition to common GP operators, a real-value alteration
operator executed for overcoming the problem of producing
variant integers in terminal nodes of syntax trees during the
evolutionary process. The used real-value alteration operator
executed on only one parental expression selected by the
fitness based selection method (tournament). The operator
continued by producing a random integer between intervals
[0,1] for each parental tree. If the integer is less than a
predetermine value PA the value of one or more of the selected
terminals changed based on the following equations,
𝑇
!,!=1𝜔𝛼+𝜔𝑇
!,! (1)
𝛼=𝑟𝑎𝑛𝑑 0,1𝑈𝐵
!𝐿𝐵
!+𝑈𝐵
! (2)
Where, PA is the alteration probability between interval
zero and one,!T
!,! is the selected terminal, T
!,! is the exchanged
terminal where 1𝑖𝑛𝑢𝑚!𝑜𝑓!𝑝𝑜𝑝 , ω is a real number
chosen randomly in 0 and 1 span, UB! and LB! are upper and
lower bound of the interval in which the new terminal value is
going to be produced and finally rand 0,1 is a random
number selected from normal distribution in the range of zero
and one.
After performing evolutionary operations, GP automatically
evolve individuals, which have better objective functions to
the next generation based on Pareto dominancy selection. In
order to better evolved individuals from the same Pareto front
to the next generation, the ε-elimination approach, which has
been proposed by some authors [23-25], used to get Pareto
optimal solutions of multi-objective problems more uniformly.
In the ε-elimination diversity approach, which is based on a
threshold value, all the clones or ε-similar individuals are
recognized and simply removed from the current population.
It should be noted that such ε-similarity must exist both in the
space of objectives and in the space of the associated design
variables. It is clear that two individuals can be similar in the
space of objective functions but may be different in the space
of design variables. Therefore, these individuals must be
preserved in the population. More detailed description of ε-
elimination can be found in [23-25].
And the final step of the algorithm is to verify if the
adopted stopping criterion was reached. Determining the
stopping (termination) criterion, for example, is usually a
simple process, e.g. stop after a certain number of generations
have elapsed or stop when a solution of a high enough quality
is found. In this work it is adopted a predefined number of
generations. The flowchart of the algorithm is illustrated in Fig.
1.
Fig. 1. Flowchart of the proposed MUGP algorithm
III. EVOLUTION OF OPTIMAL CONTROLLERS
Standard feedback control system shown in Fig. 2, is
consist of a dynamic controller system C(s) with unknown
structure, a dynamic continues-time controlled plant G(s),
reference signal, control error signal, controller variable,
disturbance and output signal. Closed loop as well as open loop
simulation of such control system gives some performance
characteristics of the whole system. These performance
measures consist of time-domain characteristics e.g. overshoot,
settling time, steady state error, energy consumed (controller
variable) and frequency domain behaviors e.g. stability and
sensitivity.
Fig. 2. Closed loop SISO system with plant G(s) and controller C(s)
The proposed method makes use of genetic programing for
evolution of polynomials, which are in fact the designed
controller for a continues-time process. Each syntax tree parsed
to an expression, which have constituted both numerator and
denominator of a transfer function and the fitness correspond to
each tree evaluated by simulating the whole control system.
Each expression tree provided one polynomial and then the
corresponding coefficients of numerator and denominator of
that tree structure calculated and substituted in the transfer
function. Transfer function is basically defined in s-plane
system. Hence, the polynomial controller is a rational function
in the complex variable that is, 𝑠=𝜎+𝑗𝜔,
𝐶𝑠=!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!
(3)
The evolved polynomial by GP provides important system
response characteristics without solving the complex
differential equation. Both open loop and close loop continues
simulation of the control system are necessary to obtained all
objective functions, which are overshoot, rise time, settling
time (the time for the output to settle within 5% units of the
input step value), integral time absolute error, maximum
energy consumed (controller variable) by close loop simulation
in time domain and stability which derive from open loop
simulation of the control system. The objective function
corresponds to stability defined by maximization of the
distance from the nearest point of Nyquist diagram to the
critical point (-1,0). It should be noted that instead of
maximization of stability in the evolutionary optimization
process, 1/stability is subjected to minimization. Therefore, all
objective functions tend to minimize as an optimization
problem. Other objective functions are simply achievable by
analyzing the transient response of the close-loop system in
time domain.
Obviously, there are some constraints when using such
methodology for evolving transfer function polynomials. Like
for instance, some of the polynomials produced by GP do not
satisfy the order constraint of a transfer function (where the
order of numerator should be equal or less than denominator).
Such individuals eliminated from the population by a penalty
function. Furthermore, some constraints are considered for the
value of objective functions. In this way, the value of settling
time should not exceed from 15 seconds for each candidate
controller and the objective function correspond to stability
should not exceed from 2.
IV. TWO-MASS-SPRING SYSTEM
One of the common problems in controller design field is
the two-mass-spring system, which illustrated in Fig. 3.
Fig. 3. Block diagram of two-mass-spring benchmark problem
The state space equations of the system shown as follows,
𝑋!
𝑋!
𝑋!
𝑋!
=
0010
0001
𝐾𝑚!
𝐾𝑚!
𝐾𝑚!
𝐾𝑚!
0
0
0
0
𝑋!
𝑋!
𝑋!
𝑋!
+
0
0
1𝑚!
0
𝑢+
0
0
0
1𝑚!
𝑤 (4)
Where 𝑋!,𝑋!,𝑋!!and!𝑋! are the position of the first mass,
the position of the second mass, the velocity of the first mass
and the velocity of the second mass, respectively. If we
consider the position of the second mass (𝑋!) as the output of
the system, then the system transfer function can be illustrated
as follows,
!!(!)
!(!)
=!/!!!!
!!(!!!!(!!!!!)/!!!!) (5)
!!(!)
!(!)
=(!/!!)(!!!!/!!)
!!(!!!!(!!!!!)/!!!!) (6)
If the constants parameters of the system for multi-
objective optimization are set to its nominal values, 𝑚!=
𝑚!=1𝐾𝑔, 𝐾=1𝑁𝑚!, then based on equations 5 and 6, the
system transfer functions can be shown as follows,
!!(!)
!(!)
=!
!!(!!!!) (7)
!!(!)
!(!)
=(!!!!)
!!(!!!!) (8)
Ref [19] used stochastic robustness synthesis to find
compensators using Monte Carlo analysis for the two-mass-
spring system. They have considered three cost functions
namely, the probability of instability, probability of actuator
saturation and probability of settling time violation. And they
have suggested different controllers based on the sum of
weighted objective functions as follows,
𝑓=𝑤!𝑃
!+𝑤!𝑃
!+𝑤!𝑃!" (9)
Where 𝑃
!, 𝑃
!, 𝑃!" are probability of instability, probability
of actuator saturation and probability of settling time violation
respectively.
In this section, the MUGP algorithm used for many-
objective design of polynomial controllers. The algorithm
parameters for this case study are also identical to previously
simulated systems. However, the simulation performed for 30
seconds.
The six objective functions considered in this case study are
namely, the settling time (the time for the output to settle
within ±0.1 units of the final value) of the second mass
subjected to a unit impulse disturbance input, the control effort,
which applied to the first mass, the integral time absolute error
(ITAE), the peak value of output of the system defined as
Max!(x!), rise time, which is the time to reach Max!(x!) and
stability, which defined by maximization of the distance from
the nearest point of Nyquist diagram to the critical point (-1,0).
It should be noted that instead of maximization of stability in
the evolutionary optimization process, 1/stability is subjected
to minimization. Therefore all objective functions are subjected
to minimization in a multi-objective optimization procedure.
The closed loop transfer function of the system can be
illustrated as follows,
𝐺!" 𝑠=
(!!!!)
!!(!!!!)
!!!!"(!)!
!!(!!!!)
(10)
Where 𝐶!" 𝑠 is the transfer function of designed
controller. The MUGP parameters used in this section are also
given in TABLE. I.
TABLE I. GENETIC PROGRAMMING PARAMETERS
GP parameters
Values
Function set
+ - / *
Terminal set
S, rand
Initial population number
500
Number of iterations
250
Crossover probability
0.8
Mutation probability
0.5
Integer Alteration Probability
0.8
Max initial-pop tree length
5
Max crossover tree length
10
After 500 generations in a 6-objective optimization process,
466 non-dominated design points obtained from the total of
500 populations. Fig. 4 depicted the value path plot, which
gives important information about uniformity of obtained non-
dominated solutions as well as possible choices of trade-off
point.
Fig. 4. Value path plot of non-dominated solutions obtained by the method of
this work for the two-mass-spring system
Eq. 11, shows the evolved transfer function of the chosen
trade-off controller,
𝐶!" 𝑠=!!.!""!!!!".!"#!!!".!""
!.!!"!!!!.!"#!!!!".!"#!!!!"#.!""!!!!"#.!"!!!".!"" (11)
Interestingly, the MUGP algorithm captured the transfer
function structure for the two-mass-spring system, which has
been considered widely in the literature [19]. It also fulfills the
common constraint on parameters of the controller transfer
function, which have been used in the literature [19]. The
designed controller has three more poles than zeros, resulting
in good high-frequency attenuation. Fig. 5 depicted the impulse
response of the designed controller using MUGP algorithm
with that of suggested by Ref [19] based on different weighted
coefficients.
The corresponding response of control effort applied to the
first mass with respect to time shown in Fig. 6. The Nyquist
plot of the designed controller illustrated in Fig. 7. It should be
noted that this controller has 13 closed loop poles.
Fig. 5. Closed loop response of 𝑥! to a disturbance impulse for the designed
optimal controller by MUGP and other suggested controllers by Ref [19]
Fig. 6. Closed loop response of the actuator (variation of control effort) for
designed controllers in Fig. 5
Fig. 7. The Nyquist diagram of the open loop control system correspond to
designed controller by MUGP
The more comparison of the obtained controller by MUGP
algorithm with that of Ref [19] is illustrated in TABLE. II. The
controller corresponds to point P2 in [19] could be considered
as the trade-off solution of that study. It can be seen that the
suggested controller of this work dominated all objective
functions of controller P2.
TABLE II. COMPARISON OF OBJECTIVE FUNCTIONS OF DESIGNED
CONTROLLERS IN FIG. 5 FOR THE TWO-MASS-SPRING SYSTEM
Objective
functions
MUGP
P1 by Ref
[19]
P2 by Ref
[19]
P3 by Ref
[19]
ITAE
9.86
16.15
10.89
5.95
Control
effort
0.402
0.59
0.46
0.82
Stability
1.032
1.00
1.09
1.49
Max!(x!)
1.50
2.08
1.59
1.00
Rise time
5.33
5.48
5.41
1.87
Settling
time
11.54
14.40
12.10
10.10
V. CONCLUSION
This paper presented an approach for automatically
synthesis of optimal controllers based on a transfer function
representation method using multi-objective uniform-diversity
genetic programming (MUGP) algorithm. The proposed
methodology is based on the optimization of both structure and
parameters of given a transfer function polynomials, which
considered as a candidate controller in a many-objective
optimization framework. It has shown that the proposed
algorithm well performed for such many-objective
optimization problem based on the value path plot and other
illustrative figures e.g. impulse response of the system, the
Nyquist diagram. The comparison of the designed controllers
with that of obtained by other methods in terms of objective
functions values showed that the proposed methodology using
the MUGP algorithm is a computationally efficient approach
for design of optimal controllers for complex dynamical
systems.
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... CGP has been applied to the problem of water turbine control design and nonlinear Duffing oscillator (Balandina 2017). In Gholaminezhad et al. (2014), a multi-objective GP was proposed for the deterministic controller design problem in the time domain. In Hu et al. (2018), an automatic crowd control framework to generate control strategies based on Pareto multi-objective optimization of genetic programming was proposed. ...
... Neglecting uncertainties in the optimal design of a controller can lead to unstable solutions which are infeasible in practice (Jamali et al. 2013(Jamali et al. , 2014. Therefore, we aim to design a robust controller to reduce performance variation of a controller in a noisy environment. ...
... (3) inspired from differential evolution mutant operator (Jamali et al. 2014;Gholaminezhad and Jamali 2016). The adaptive parameter F is changed according to Eq. (4), and its variation versus g is shown in Fig. (5). ...
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Optimal design of controllers without considering uncertainty in the plant dynamics can induce feedback instabilities and lead to obtaining infeasible controllers in practice. This paper presents a multi-objective evolutionary algorithm integrated with Monte Carlo simulations (MCS) to perform the optimal stochastic design of robust controllers for uncertain time-delay systems. Each potential optimal solution represents a controller in the form of a transfer function with the optimal numerator and denominator polynomials. The proposed methodology uses genetic programming to evolve robust controllers. Using GP enables the algorithm to optimize the structure of the controller and tune the parameters in a holistic approach. The proposed methodology employs MCS to apply robust optimization and uses a new adaptive operator to balance exploration and exploitation in the search space. The performance of controllers is assessed in the closed-loop system with respect to three objective functions as (1) minimization of mean integral time absolute error (ITAE), (2) minimization of the standard deviation of ITAE and (3) minimization of maximum control effort. The new methodology is applied to the first-order and second-order systems with dead time. We evaluate the performance of obtained robust controllers with respect to the upper and lower bounds of step responses and control variables. We also perform a post-processing analysis considering load disturbance and external noise; we illustrate the robustness of the designed controllers by cumulative distribution functions of objective functions for different uncertainty levels. We show how the proposed methodology outperforms the state-of-the-art methods in the literature.
... Genetic Programming (GP), introduced by Koza [63], is regarded as a powerful approach which has extended from GAs, with completely new specifications and features. Different studies have so far used GP in various optimization problems such as circuit design, the system identification, neural networks, modeling, and controller design [64][65][66]. Further, it is a supervised machine learning method that searches program space instead of data space. The programs created by GP are expressed by using a functional programming language and demonstrated as tree structures. ...
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... As a result, many naturebased methods, including ant colony, particle swarm optimization, artificial immune system, evolutionary algorithms, and so forth, were suggested by researchers (Coello et al., 2007). Among the several direct algorithms in solving optimal problems, genetic programming (GP) has proven robust for its capabilities to symbolically present both topology and mathematical details of the solutions (Assimi et al., 2017;Bruns et al., 2019;Brameier and Banzhaf, 2001;Devarriya et al., 2019;Li et al., 2006;Gholaminezhad et al., 2014;Jamali et al., 2016;Koza, 1989Koza, , 1990Koza, , 1994aPoli et al., 2008). In the study conducted by Maher and Mohamed (2013), the efficacy of the GP algorithm in obtaining optimal control solutions of the nonlinear problems was assessed. ...
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... Inspired by Darwin's natural selection concept in the mathematical platform, Genetic Programming (GP) is a meta-heuristic as well as evolutionary optimization approach capable of solving optimization problems [36][37][38][39]. GP, which is an improved version of the Genetic Algorithm (GA), was introduced by Koza in 1992 and so far has been used in many optimization problems such as the system identification, controller design, modeling, neural networks and circuit design [40][41][42]. The GP algorithm uses computer programs as members of its population and displays them with a tree structure. ...
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The development of modern engineering systems has introduced increasing levels of complexity and uncertainty over time. Combined with the design philosophy of engineering itself, this has given rise to many studies addressing the simple or multi-objective optimization problems present in these complex systems. Although conventional approaches can be applied to engineering optimization depends largely on the nature of problem, but they suffered to provide some quick and reasonable feedback to designers and cannot be challenging to further possible problems. Nevertheless, heuristic approaches that apply mixtures of different exploratory with or without traditional search and optimization methods are proposed to solve such complex problems. This chapter briefly provides the conventional optimizations and basic knowledge about the most widely implemented heuristic optimization techniques, as well as their application in optimization problems in mechanical engineering systems. It also presents the genetic programming that searches the space of possible computer programs which is extremely fit for solving the complex problem in truss structure design and optimization of mechanical engineering. Genetic programming employs tree structure of computer programs as individuals in its initial population, which gets evolved through generations by the algorithm operators to reach the optimum solution. To prove the ability of the genetic programming to solve complex mechanical engineering problems, a case study in design of truss with discrete design variables will be examined. In this example, genetic programming employs to find the optimum topology and discrete cross-section sizes of 10-bar truss problem which is a nonlinear problem subjected to different constraints such as the stability, maximum allowable stress and displacement in the truss nodes, and critical buckling load. As results and in comparison, with other state-of-art approaches, genetic programming finds a lighter truss structure with fewer elements because it could be constructed a tree-based expression to explore the search space.
... This operator has been used for different purposes by some authors of this paper. [55][56][57] ...
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... The goal of this work is to combine ideas of both fields in order to find robust controllers for nonlinear systems with non-obvious system structure. While most publications on evolutionary controller design focus on optimizing general performance measures using multi-objective [10] or weighted sum approaches [4], robustness is rarely considered. However, in most industrial applications, robustness of a given control structure with respect to uncertainties is essential. ...
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