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Real-Valued Compact Genetic Algorithms for
Embedded Microcontroller Optimization
Ernesto Mininno, Francesco Cupertino, and David Naso
Abstract—Recent research on compact genetic algorithms
(cGAs) has proposed a number of evolutionary search methods
with reduced memory requirements. In cGAs, the evolution of
populations is emulated by processing a probability vector with
specific update rules. This paper considers the implementation of
cGAs in microcontroller-based control platforms. In particular, to
overcome some problems related to the binary encoding schemes
adopted in most cGAs, this paper also proposes a new variant
based on a real-valued solution coding. The presented variant
achieves final solutions of the same quality as those found by
binary cGAs, with a significantly reduced computational cost.
The potential of the proposed approach is assessed by means of
an extensive comparative study, which includes numerical results
on benchmark functions, simulated and experimental microcontroller design problems.
Index Terms—Compact genetic algorithms (cGAs), electric
drives, embedded systems, online optimization.
I. INTRODUCTION
C
OMPACT genetic algorithms (cGAs) [10] are evolutionary algorithms that mimic the behavior of conventional GAs by evolving a probability vector (PV) that describes
the hypothetic distribution of a population of solutions in the
search space. A cGA iteratively processes the PV with updating
mechanisms that mimic the typical selection and recombination
operations performed in a standard GA (sGA) until a stopping
criterion is met. Reference [10] showed that the cGA is almost
equivalent to a sGA with binary tournament selection and
uniform crossover on a number of test problems, and also
suggested some mechanisms to alter the selection pressure in
the cGA. The main strength of the cGA is the significant reduction of memory requirements, as it needs to store only the PV
instead of an entire population of solutions. This feature makes
them particularly suitable for memory-constrained applications
such as evolvable-hardware [26] or complex combinatorial
problems [2].
Since the cGA mimics the “order-one” behavior of sGAs, it
may be ineffective for optimization problems with higher order
building blocks. Recent research has investigated a number of
ways to address this fundamental issue. More specifically, [2]
applies the cGAs to the TSP problem, and proposes a variant
that employs an efficient local search operator to enhance the
speed of convergence and the quality of the solutions. Reference [1] discusses the analogies between cGAs and
-ES,
Manuscript received May 16, 2006; revised October 10, 2006.
The authors are with the Dipartimento di Elettrotecnica ed Elettronica,
Politecnico di Bari, 200 70125 Bari, Italy (e-mail: [email protected];
[email protected]; [email protected]).
Digital Object Identifier 10.1109/TEVC.2007.896689
and extends a mathematical model of ES [14] to cGAs obtaining
useful analytical performance figures. Moreover, [1] introduces
the concept of elitism, and proposes two new variants, with
strong and weak elitism respectively, that significantly outper-ES. Reference [26] suggests various
form both cGAs and
cGAs combining elitism, mutation (exploration of the neighborhood of the current individual), and resampling (periodic reevaluation of the fitness of the best known individual in case of noisy
or time-varying fitness). Reference [26] shows that some of the
proposed variants are extremely effective for both static and dynamic optimization problems, and can be implemented in commonly available field programmable gate array (FPGA) boards.
This paper shares the same fundamental motivations as the
work in [26]. In particular, extending our previous work on hardware-in-the-loop optimization [4], here we investigate the possibility of using a cGA as the main optimization engine for solving
online closed-loop control design problems. The typical task of
a GA in a control engineering application is to find the best
values for a predefined set of free parameters defining either a
process model or a control law. Updated surveys of this research
area can be found in [4] and [27]. Both surveys outline that the
number of hardware applications is extremely limited for various reasons. In such a context, the cGAs are particularly interesting not only for the mentioned reduction of memory (power
and space [26]) requirements, but also because their general
structure (the lack of a population, the absence of random selection and recombination) lends itself well to online implementations that overcome the typical problems related to the conventional population-after-population algorithmic scheme of sGA.
In our application, the design problem can be viewed as the
search for the optimal setting of a number of real-valued coefficients that specify the input–output difference equation of the
controller. While in our previous research [4], the main search
engine was a sGA executed on a host computer that continuously downloaded new controllers to the microprocessor running the feedback control law, here we focus on a family of
cGAs suitable to be directly executed on the floating-point microcontroller without the need of a host computer. This potential of cGAs appears particularly promising because embedded
microcontroller-based platforms are increasingly widespread in
various control and automation research areas, as mechatronics,
automotive, and rapid prototyping systems.
Implementing an effective cGA for the considered purpose is not trivial, because most microcontroller platforms
are programmable with object-oriented software based on
floating-point real-valued variables, whereas to date most
literature about cGAs (including all the references mentioned
above) concentrates on binary-encoding schemes. In a microcontroller implementation, the need to continuously perform
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Fig. 1. Normalization of two truncated Gaussian curves (with standard deviations ;
.
with a nonzero support only in the normalized interval
[01 +1]
binary-to-float coding and decoding operations may weaken,
to a variable extent, the advantages related to reduced memory
requirements offered by cGAs. To overcome this problem, this
paper proposes a variant of a cGA that works directly with
real-valued chromosomes. More specifically, for an optimization problem with real-valued variables, the real-valued cGA
2 matrix describing the mean and the standard
uses as PV a
deviation of the distribution of each gene in the hypothetical
population. New variants of the update rules are then introduced
to evolve the PV in a way that mimics the binary-coded cGA,
which in turn emulates the behavior of a sGA.
The similarities between cGAs and evolutionary strategies
(ES) have already been pointed out in recent literature [1].
The analogies become even more evident in our case, as
the use of a probabilistic description of the population with
real-valued mean and standard deviation immediately recalls
-ES (e.g., [7] and [12]).
the well-known scheme of the
Furthermore, the real-valued cGA also has some important
relationships with other recently proposed evolutionary algorithms that incorporate probabilistic tools, such as continuous
population-based incremental learning (PBIL) [21], [22],
[29], or univariate marginal distribution algorithms (UMDAs)
[23], [25], [28]. The differences and analogies with respect to
these algorithms will be discussed after the description of the
real-valued cGA in Section II.
The behavior of both binary and real-valued cGAs is assessed
in three different ways. Section III considers a set of well-known
benchmark functions and compares the two cGAs (with binary
and real-valued coding schemes) with their standard population-based versions (i.e., binary and real-valued sGAs). Tests
also include noise-corrupted fitness measurements, to better emulate the typical conditions in which microcontroller optimization is performed. Section IV focuses on the feedback control of
a simulated second order model of a DC motor drive. The aim
of this case study is to provide an easily repeatable benchmark
for comparative purposes. Section V describes the experimental
application of the cGAs to design and optimize online the discrete anti-windup PI speed controller for a vector controlled in-
= 1 and = 10, respectively) so as to obtain a probability density function
duction motor (IM) drive subject to a time-varying load using
a torque-controlled brushless generator, mounted on the same
shaft. Conclusive remarks are finally drawn in Section VI.
II. COMPACT GAS WITH REAL-VALUED CODING
The cGA proposed in this paper combines the nonpersistent
elitist compact GA proposed in [1] with a real-valued chromosome representation. The real-valued coding has been widely
used in evolutionary algorithms (including GAs [12]) to avoid
the additional computational cost related to the binary-to-float
conversions. To use a real-valued coding in a cGA, a number of
modifications are necessary. For brevity, we will hereafter refer
to the nonpersistent elitist binary-coded cGA as bcGA, and to
the real-valued variant as rcGA. In order to avoid repetitive descriptions of known algorithms, we adopt here the same notation
used in [1], and focus only on the rcGA.
Let us consider a minimization problem in a -dimensional
hyper-rectangle ( is the number of parameters). Without loss
of generality, let us assume that parameters are normalized so
that each search interval is
. The estimate of the single
(binary) gene distribution in a cGA can be specified with a scalar
in [0,1] expressing the probability of finding a “0” or a “1” in
the single gene. As the rcGA uses real-valued genes, the distribution of the single gene in the hypothetical population must be
described by a probability density function (PDF) defined on the
normalized interval
. A variety of different PDFs could
be used for this purpose. We assume that the distribution of the
th gene can be described with a Gaussian PDF with mean
and standard deviation . More precisely, since the Gaussian
PDF is defined in
, we adopt a “Gaussian-shaped”
PDF defined on
whose height is normalized so that
its area is equal to one. Fig. 1 illustrates the effects of the normalization procedure on a Gaussian PDF with zero-mean and
standard deviations equal to 1 and 10, respectively. The height
normalization has a generally negligible computational cost, as
it is obtained by means of a simple numerical procedure that is
widely adopted in pseudorandom number generation, and more
MININNO et al.: REAL-VALUED COMPACT GENETIC ALGORITHMS FOR EMBEDDED MICROCONTROLLER OPTIMIZATION
205
Fig. 2. Real-valued variant of the nonpersistent elitist bcGA.
generally, in practical implementations of stochastic algorithms
(see, e.g., [3] and [15, Appendix D2]).
Therefore, the PV becomes in our case a
2 matrix specifying the two parameters of the PDF of each single gene. Thus,
we define
(1)
is the vector of the
where
mean values,
is the vector
of the standard deviations, and is the iteration index. Fig. 2
describes the rcGA with the typical pseudocode used in related
literature. The main differences with the bcGA regard the generation of the initial population and the update rule. The initial
population is generated by setting
and
, where
is a large positive constant (e.g.,
, see also Fig. 1). In
this way, after height normalization we obtain a PDF that approximates sufficiently well the uniform distribution
that is almost always used to generate initial individuals in realvalued GAs.
The update rule is designed so as to replicate the typical operations of a bcGA. As in [10], the idea is to mimic a steady-state
binary tournament selection. This operator simply replaces the
looser with a copy of the winner, and consequently also alters
the compact, probabilistic description of the population. In particular, while the mean is simply moved in the direction of the
winner with a step whose size depends on the population size ,
the standard deviation is subject to a less transparent effect, due
to its quadratic structure. Fig. 2 reports the update rule for the
standard deviation in a compact form, particularly convenient
for algorithm implementation, which can be directly obtained
with simple manipulations, as described in Appendix A.
As mentioned, this rcGA shares a number of similarities with
other types of evolutionary algorithms. All cGAs belong to a
larger class of methods known as estimation of distribution algorithms (EDAs), which have been extensively analyzed in recent literature [18]. In particular, among the EDAs that model
the hypothetical population with PDFs defined on continuous
intervals, the algorithms known as continuous-PBIL (
)
) share some common aspects
and continuous-UMDA (
algorithms (see [20] and [21] and the
with our rcGA. The
included references) are extensions of a population-based algorithm originally devised for binary search spaces. One specific
version of
uses a set of Gaussian distributions to model
the vector of problem’s variables [19]. During the iterations,
this PBIL algorithm only changes the mean values, while the
standard deviations are constant parameters that must be fixed
: 1) randomly
a priori. At each iteration, the Gaussianextracts a population of solutions using the current distribution
for each variable; 2) sorts the solutions according to their fitness;
and then 3) updates the means of the distributions with a simple
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TABLE I
SUMMARY OF CONFIGURATION PARAMETERS OF THE FOUR GAS
linear law (a combination of the best, the second best and the
worst individual). The rcGA in Fig. 2 could be viewed as a parwith
, improved by nonticular form of Gaussian
persistent elitism, and an adaptation law for the standard deviations. It is also interesting to note that other recent research about
confirms the advantages of standard deviextensions of
ation adaptation ([22] obtains promising results with a mechanism similar to the well-known “1/5 success rule”). Moreover,
the literature also suggests some other interesting possibilities
deserving further investigations (e.g., replacing Gaussian PDF
with histograms, or adapting the coefficient of the update rule
for the mean vector [21]), although the extension or adaptation of these ideas to our online microcontroller optimization
problem is not immediate, especially considering the significant limitations on the available computational time. Among the
[23], [25] also
many EDAs available in literature, the
uses Gaussian PDFs as probabilistic models of population distribution. There are many variants of this algorithm. The
proposed in [23] is among the most widely citd ones: it extracts
couples of individuals using the current PDFs, performs
tournaments, and uses the winners to estimate the parame, the general structure
ters of the new PDFs. When setting
of
may look quite similar to the rcGA in Fig. 2, but it
are
should be noted that: 1) the PDF update laws of the
quite different from those in Fig. 2 and 2) the
update
laws explicitly need a population size greater than one to work
properly. The main strengths of UMDAs lie in their amenability
to theoretical analyses, and in their capability to learn the interdependencies among the problem variables (by adopting more
complex multivariate PDFs). On the other hand, it should be
evident that implementing such a population-based
in our real-time environment could be even more prohibitive
than in the case of a sGA, due to the additional costs related
to the more complex update laws for multivariate PDFs (see remarks in [24]). Finally, as already pointed out in [1], cGAs can
-ES. Since ESs use
be viewed as a particular type of
Gaussian PDF to characterize the additive mutation, the simi-ES are even more
larities between our rcGA and the
apparent. The main difference between the two evolutionary algorithms can be seen in the meaning and actual use of parameters
and
. In
-ES, the generic gene
represents the value of the th element of the best solution known at
the iteration , rather than the center of the probabilistic func-
tion modeling the distribution of the fictitious population. Since
as affected by probabilistic uncertainty,
ES do not consider
using the same notation introduced in Fig. 2, the update law for
in a
-ES is
(i.e., the algorithm simply
replaces the gene of the parent with the one of the best-known
solution). On the contrary, it is evident in Fig. 2 that the rcGA
makes a much smoother move towards the winner, which also
depends on the parameter . Similar analogies can be observed
in the updates of the standard deviation
. In ESs, the paindicates the standard deviation of the mutation,
rameter
to geni.e., the random additive perturbations applied to
erate a new solution. In ESs,
is usually updated either with
heuristic rules that take into account the amount of successful
perturbations obtained over a predefined number of mutations
(e.g., the “1/5 success rule” [12]) or with self-adaptation (i.e.,
it becomes a further gene of the chromosome evolved by the
ES). The novel update rule proposed in Fig. 2 may be properly
placed in the first class of methods, although it still appears significantly different from the typical ES parameter update rules
(to our best knowledge there has been no previous effort in developing a mutation update rule in ESs inspired to binary cGAs).
III. NUMERICAL BENCHMARKS
Tuning and validation of the proposed rcGA are based on
a comparison with three other standard algorithms selected
from literature. The set of reference standards includes the
elitist bcGAs proposed in [1] and two population-based GAs.
The following subsections illustrate the configurations of the
reference algorithms, the suite of benchmark functions, and the
summary of numerical results, respectively.
A. Reference Algorithms for Comparison
Obtaining a significant comparison between GAs is always
difficult, as the relative results are strongly influenced by the
nontrivial configuration of the single algorithms. For this reason,
the set of reference algorithms includes two GA codes available
in the Genetic Algorithm Direct Search Toolbox (Version 1.0.3)
[17]. Essentially, the two standard algorithms differ for the encoding scheme and for the crossover operators. The configurations are summarized in Table I.
One fundamental difference between the population-based
and the compact GAs lies in the sizes of the actual or ficti-
MININNO et al.: REAL-VALUED COMPACT GENETIC ALGORITHMS FOR EMBEDDED MICROCONTROLLER OPTIMIZATION
tious populations used by the four algorithms. For all the GAs,
this parameter is chosen so as to make all the GAs approximately converge using the same number of fitness calls (which
is the stopping criterion for all our numerical and experimental
investigations). We have performed a number of preliminary
tests, obtaining results that agree with indications available in
recent literature [1], [10]). In particular, the standard GAs require smaller population sizes than the cGAs (about 2 to 4 times
in the case of nonpersistent elitist cGA, as also reported in [1]).
Finally, the number of bits used for the parameter encoding in
binary GAs was chosen so as to have a sufficient resolution to
converge reasonably close to the known optima. Finally, the parameters defining the elitist mechanism in the two cGAs were
chosen empirically.
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4) Michalewicz Function:
(5)
This is a multimodal test function ( local optima). The
defines the “steepness” of the valleys or edges.
parameter
leads to more difficult search. The function values
Larger
for points in the space outside the narrow valleys give very
little information on the location of the global optimum. In our
, which places the global minimum in
tests, we use
[12].
5) Rosenbrock’s Function (Also Known as Dejong’s Second
Function):
B. Benchmark Functions
(6)
The comparison is based on five well-known benchmark
functions selected from GA literature for their different characteristics. For each function, we consider two cases: in the first
one, fitness measurements are noise-free, while in the other one
they are subject to additive Gaussian noise with zero-mean and
(such a noise amplitude is chosen
standard deviations
to reproduce the typical uncertainty affecting control performance measures). The number of free parameters for each
. The use of relatively small size problems is
function is
motivated by the interest in evaluating the performances of the
algorithms in noisy search spaces having the same size of the
control design problems tackled on the microcontroller board.
The five benchmark functions are defined as follows.
1) Rastrigin’s Function:
(2)
This benchmark is highly multimodal due to the cosine modulation that produces regularly distributed local minima. The global
[16].
optimum is the origin with
2) Circle Function:
(3)
this is a multimodal function described in [13]. It has many local
optima (minima) that are located on concentric circles near the
global optimum (the origin).
3) Schwefel’s Function:
(4)
This is a deceptive function in which the global minimum is
geometrically distant, over the parameter space, from the next
best local minima. This may cause the search algorithm to move
toward the wrong direction. The global minimum is located at
,
with
.
This is a standard test function in optimization theory. Due to a
long narrow valley present in this function, gradient-based techniques may require a large number of iterations before the global
,
is found.
minimum
C. Summary of Results: Quality of Final Solutions
Each algorithm is run 1000 times to obtain average performances. The results are summarized in Figs. 3–7. The figures
draw the histograms of the distribution of the best solutions
found in each of the 1000 replications. In particular, to give an
immediately interpretable index of performance, the distribution of final solutions is viewed in terms of the normalized distance (with respect to the actual width of the search space) from
the true optimum.
The analysis of the single benchmarks highlights a number of
interesting results. For the Rastragin function, all the algorithms
are relatively unable to escape the local minima. Especially for
the bGA and rcGA, the distribution of the final solutions tends
to become looser as the noise level is increased, but the overall
performance of the four algorithms can be considered approximately equivalent. Local minima have a significant influence on
the final performance of the GAs also for the Schwefel benchmark. In this case, it is interesting to note that both cGAs outperform the population-based algorithms, being able to find the true
optimum in about 40%–50% of the noise-free runs and about
20%–30% in the noisy case, respectively. The Schwefel benchmark is particularly troublesome for the rGA, which only occasionally (or never in the case of high noise) converges to the
true optimum. In contrast, the results of the rcGA are particularly interesting, as they also slightly outperform the bcGA in
both noise-free and noisy cases. Similar results are obtained on
the Circle function, for which the performance of the rcGA in
the noise-free case is particularly remarkable. The cGAs exhibit
a relatively better ability to overcome multimodality also in the
Michalewicz benchmark, although this time the bcGA is largely
the best performing algorithm, being able to hold a remarkable
result (over 60% of runs ending in global optimum) also in the
noisy case. Both bGA and rcGA are able to get closer to the
optimum of the Rosenbrock’s function within the allowed 1000
fitness calls. The advantage of these algorithms becomes even
more evident in the noisy case.
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Fig. 3. Distribution of the final best solution found by the four compared GAs in each run for the Rastragin function. The distance from optimum is
normalized with respect to the width of the search interval. For all the algorithms, only few runs end sufficiently close to the true optimum, independently
from the presence of noise. On the other hand, bGA and rcGA seem to be slightly more sensitive to noise.
A joint analysis of the results on the five benchmarks
suggests that, in low-dimensional optimization benchmarks,
the two cGAs exhibit, in general, very satisfactory results,
sometimes even more accurate and less sensitive to noise than
those of the sGAs. On the other hand, it is also quite foreseeable that the relative performance of the algorithms observed
in these low-dimensional cases may dramatically change for
larger size problems involving higher order building blocks,
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209
Fig. 4. Distribution of the final best solution found by the four compared GAs in each run for the Schwefel function. The distance from optimum is normalized with
respect to the width of the search interval. The cGAs are evidently more effective than their population-based correspondents.
as often remarked in technical literature on cGAs. However, the analysis of the performance of the rcGA on larger
size optimization problems is not among the scopes of this
manuscript.
IV. MICROCONTROLLER DESIGN ON A SIMULATED PLANT
As previously mentioned, the cGAs have an algorithmic
structure that lends itself readily to online applications in which
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Fig. 5. Distribution of the final best solution found by the four compared GAs in each run for the Circle function. The distance from optimum is normalized with
respect to the width of the search interval. The rcGA yields in this case the most remarkable performance, with over 400–800 runs ending in the true optimum regardless of the amount of noise.
fitness measures are performed uninterruptedly on a hardware
process. Such a case is examined in the experimental investigations described in the remainder of this paper, which are all
performed on a dSPACE 1104 microcontroller board based on
a 250 MHz Motorola Power PC microprocessor. The board,
specifically designed for measurement and control application,
is equipped with several AD and DA converters, and is directly
programmable in the Matlab/Simulink object-oriented language.
The following subsections describe two different benchmarks
for binary and real-valued cGAs.
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211
Fig. 6. Distribution of the final best solution found by the four compared GAs in each run for the Michalewicz function. The distance from optimum is normalized
with respect to the width of the search interval. The bcGA is the best performing algorithm for this benchmark, in which both cGAs outperform their populationbased equivalents.
A. Measuring Actual Computational Costs
Implementing a conventional population-based GA directly
on a microcontroller is not simple, for various reasons. As men-
tioned, a sGA requires the memory necessary to store the current population and an additional memory to perform selection,
crossover, and mutation operations (and binary to real coding/
decoding operations if present). Moreover, since the selection,
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Fig. 7. Distribution of the final best solution found by the four compared GAs in each run for the Rosenbrock’s function. The distance from optimum is normalized
with respect to the width of the search interval. The bGA and rcGA are the best performing algorithm for this benchmark.
crossover and mutation operations are concentrated at the end
of an iteration, the computational cost is evenly distributed over
time. Such an algorithmic scheme causes peaks of computational requirements that may overload the microcontroller at
every end of GA iteration. This, in turn, may generate con-
trol delays that are particularly undesirable in closed-loop feedback schemes. All the cGAs can be effectively used to overcome these limitations, and in particular, the rcGA proposed in
this paper is even “lighter” in terms of actual computational requirements on a typical microcontroller. To assess the amount
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Fig. 8. Block diagram of the controlled plant with the reference PI controller.
TABLE II
EXECUTION TIME OF ONE ITERATION OF CGAS ON DS1104 BOARD
Fig. 9. Schema of the cGA-based optimization.
of benefits offered by the rcGA in this respect, we implemented
the bcGA and the rcGA on the microcontroller, and measured
the average time needed by each algorithm to perform all the
numerical operations in one single iteration (from step 2 to step
4 of the pseudocode in Fig. 2), except for the computation of
the fitness. For completeness, we mention that the fitness function for this benchmark is the Michalewicz function described
above (with a progressively increased number of parameters),
but it should be clear that the considered hardware configuration makes the result independent of the particular objective
function. Unless explicitly specified, all the configuration parameters of the cGAs are the same as those used in the previous
tests and summarized in Table I. Moreover, in the bcGA, each
real-valued variable is converted in a 16 bit binary-coded string.
Table II summarizes the results of these tests, showing the ratio
of the iteration execution times of rcGA and bcGA. As expected,
the rcGA is about 3 to 5 times faster, depending on the number of
parameters. The difference is particularly significant in our control applications: the rcGA permits the use of shorter sampling
times for the same parametrized law (or the implementation of
more complex control laws at the same sampling time), or the
use of less-performing microcontroller for the same embedded
control application.
B. Online Design of a PI Controller for a Simulated Plant
To provide repeatable results on a case study commonly available in technical literature [6], [11], this section focuses on the
online optimization of the proportional and integral gains for a
discrete-time anti-windup PI controller applied to a nonlinear
plant. As described in Fig. 8, the controlled plant is a linear
second-order transfer function with an input saturation when
. Fig. 8 also
the control signal exceeds the range
shows the reference PI controller which can be obtained using
the classical, model-based “Symmetrical Optimum” design approach [11].
The overall scheme of the cGA-based design is illustrated in
Fig. 9. The differential equations describing the controlled nonlinear plant are discretized with Euler method and a sampling
step equal to 3 ms, and the resulting difference equations, as
well as the cGA and the PI controller, are run simultaneously by
the microcontroller board, while an external PC is used for monitoring purposes. The speed reference is a square wave having
amplitude equal to 0.5 p.u. and period equal to 0.6 s. The objective function (hereafter cost function) to be minimized by the
cGAs is the integral of the absolute error (IAE) calculated comparing the simulated DC motor speed response with the output
of the reference model (in Fig. 8) to the same reference input.
The parameters optimized by the cGAs are the two Proportional
and Integral controller gains. The search is bounded in [0 3.5]
and [0 10] for proportional and integral gain, respectively. At
each speed step, the current IAE value (that is, the cost function
to be minimized) is passed to the cGA and the integrator is reset
to zero. The cGA depicted in Fig. 9 is then executed every 0.3 s.
The execution time of one full iteration (from step 2 to step 4 in
Fig. 2 except for the cost function measurements) of the rcGAs
for the bcGA.
is 65 , while this time raises to 180
The main configuration parameters of the cGAs are selected
according to preliminary tuning experiments. The population
size is set equal to 200 for both algorithms. The stopping criterion is 5000 cost function evaluations for all the tests. Moreover, the maximum length of inheritance is set equal to 100.
Under these general settings, we execute three tests with slightly
differing configurations. The first and simplest test (hereafter
test a) considers a linear reference model (no actuator saturation) and a noise-free simulation. The second test (test b) is
still performed with a linear reference model but the output of
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Fig. 10. Comparison of the simulated speed responses of the reference controller (model-based symmetrical optimum, continuous line) and the speed response
of a PI controller obtained with the rcGA (dotted line) and with the bcGA (dashed line). The perfect overlap of the three responses is also emphasized on the
right-hand side on an expanded view of the overshoot.
Fig. 11. Comparison of average bcGA and rcGA evolution, in test a (linear, noise free), b (linear, noisy), and c (nonlinear, noisy), respectively. To emphasize the
differences in terms of computational costs, the horizontal axis reports the actual computation time used by each cGA.
the DC motor is artificially corrupted at each sample time with
of the rated
a uniform random number ranging between
motor speed. Moreover, the end of the evaluation of each individual (at every 0.3 s), a second uniform random number beof the measured value of the IAE is also added
tween
to the cost function, to further increase the amount of uncertainty on the simulated cost measurement. In particular, the two
sources of noise simulate: 1) the inherently noisy measurement
of the rotor speed due to transducer inaccuracy and interferences
with other devices and 2) the imperfect repeatability of each
experiment, which is a typical challenge of experimental hardware-in-the-loop optimization processes. Finally, in addition to
the noise sources of the test b, the third test (test c) considers an
range, and a PI
actuator with saturation outside the
regulator with an antiwindup algorithm. This last test better approximates the real-time experiment on an actual electric drive.
Each test is repeated 100 times and the average results are summarized in Table III.
Table III highlights two interesting results. It can be immediately noted that the rcGA outperforms the bcGA in terms of both
average cost function values, and smallest standard deviations.
Moreover, considering the final values for controllers parameters, it can be also noted that the rcGA is able to converge to
the gains of the reference controller also in the noise-corrupted
and nonlinear cases. Fig. 10 compares the speed response of the
reference control loop with the response of one PI obtained by
TABLE III
AVERAGE RESULTS FOR SIMULATED PI DESIGN (NOTATION STD
INDICATES STANDARD DEVIATION)
the rcGA. The differences between the two controllers are indistinguishable, even in the expanded view of transient responses
[Fig. 10(b)].
The differences between the two cGAs in terms of speed of
convergence and computational requirements are illustrated in
Fig. 11. It can be noted that the rcGA obtains a significantly
lower average cost value in test a, while the average cost values
for tests b and c are almost equivalent, although the rcGA is
much faster in converging towards the reference values. Another
important advantage of rcGA over bcGA is the execution speed.
As evidenced by Fig. 11, both algorithms converge after about
2000 fitness calls, but the rcGA needs about 36% of the computational requirement of a bcGA. Fig. 11 clearly highlights that
the rcGA has a more “turbulent” explorative search due to the
MININNO et al.: REAL-VALUED COMPACT GENETIC ALGORITHMS FOR EMBEDDED MICROCONTROLLER OPTIMIZATION
215
Fig. 12. Induction motor drive block diagram.
combination of nonpersistent elitism with a real-valued approximation of the probabilistic gene distribution.
V. EMBEDDED MICROCONTROLLER OPTIMIZATION
This section extends the ideas and investigations illustrated in
the previous sections to an experimental bench for the feedback
control of a vector-controlled IM drive [4]. Fig. 12 illustrates
the overall schema of the system [11]. The current controllers
are not subject to optimization, and are the same for all the experiments. The plant is a IM loaded using a torque controlled
brushless generator, mounted on the same shaft. The IM name, current
plate parameters are as follows: voltage
, power
, speed
,
, inertia
, pole pairs
torque
, and torque constant
.
In this case study, the microcontroller runs the cGAs, the
control laws and a supervision algorithm that interrupts the experimentation of badly performing solutions. The various modules have different sampling times. In particular, controller and
sampling time, while
stability supervisor run with a 200
the cGAs have a 0.75 s sampling time. Each run is stopped
after about 6 min of search, but both algorithms converge in
nearly 3 min. According to preliminary investigations for algorithm configuration, in this case study we set the population
, and the maximum length of inheritance
size
for both cGAs . We also remark that the execution time of a
complete iteration of the vector control scheme (the four controllers in Fig. 12 and necessary coordinate transformations) is
for
about 100 , and one iteration of the cGA requires 65
for the bcGA. Since both cGAs
the rcGA and about 180
are executed at every 3750 iterations of controller difference
equation, the increase of computer cost due to the introduction
of the optimization algorithms is negligible, as it amounts to
of the cycle time for the
rcGA and 0.048% for the bcGA.
Performing experiments directly on hardware raises the fundamental issue of the risks related to unstable or badly performing individuals, generated due to the inherently random
exploration of the GAs, especially during the early iterations.
Testing such controllers is not necessary and often dangerous,
since it may cause hardware damage. The use of linear controllers, combined with the availability of an accurate linear
model of the plant makes possible, at least in principle, the
analysis of stability prior to the actual implementation of each
solution. Obviously, due to the various unknown parameters
and nonlinearities neglected in this process, the information on
closed-loop poles does not yield a fully reliable prediction of
each controller’s stability. However, as foreseeable, controllers
with unstable model-based closed-loop poles seldom provide
acceptable performances on the real motor. This information
can be profitably used to define appropriate bounds of the search
space for the two gains, but the theoretical guarantees of closedloop stability on the actual hardware remain a matter of accuracy
of the available model. In order to overcome the need of an accurate model, and simultaneously guarantee a sufficient level of
“safety” during the genetic search, in this paper we have adopted
an heuristic strategy that effectively overcomes the limitations
of model-based analysis. Instead of computing the cost function at the end of each evaluation interval, as usually done in
online tuning approaches, we compute each index contributing
to the total cost function online, i.e., updating its value at each
time sample of the experiment. The online value of each cost
term is constantly monitored, and whenever it exceeds a predefined threshold the current experiment is immediately stopped.
This allows us to detect unstable (or highly unsatisfactory) solutions well before the involved signals reach potentially dangerous values. In case a monitored index exceeds the prescribed
threshold, the value of the cost is multiplied by a penalty factor
and assigned to the individual, and the algorithm proceeds with
another experiment. In this way, the evolutionary search is never
interrupted until the terminating condition occurs. On average,
only 2%–3% of the experiments are prematurely interrupted due
to bad performance, and mostly in the first iterations.
The design of the experiment for cost measurement also plays
a fundamental role in the success of the evolutionary design. In
this section, the speed reference signal is a square wave having
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IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 12, NO. 2, APRIL 2008
Fig. 13. Evolution of cost function and proportional gain over time in a sample run: currently tested individual is in thin line, current elitist in thick line. (a) Cost
function bcGA. (b) Cost function rcGA. (c) Proportional gain bcGA. (d) Proportional gain rcGA. (Figures about the evolution of the integral gain are very similar,
and omitted for brevity.)
amplitude equal to 1 p.u. (between
) and period
equal to 1.5 s. After 0.5 s from every change of the reference
signal, a step change of load torque (from 0% to 70% of motor
rated torque) is applied, in order to also evaluate the overall disturbance rejection. The profiles of both motor speed and control action (the current isqref) are acquired during the experiment, and used to compute the cost function, which essentially
takes into account three criteria, i.e., the tracking performance,
the disturbance rejection, and the smoothness of the control action. In mathematical terms, the optimization problem is formulated as the minimization of a cost function obtained as the sum
of two terms
TABLE IV
AVERAGE RESULTS FOR EXPERIMENTAL PI DESIGN
(NOTATION STD INDICATES STANDARD DEVIATION)
(7)
where
is the duration of the experiment for cost function
measurement, and the two functions and are defined as
if
otherwise
(8)
and
if
otherwise
(9)
The function takes into account the tracking performance in
the first part of the experiment, and the disturbance rejection
in the time interval in which the step load torque is applied.
More specifically, the tracking error is considered only in the
time intervals in which the current feed is lower that a predefined threshold (the tracking error due to controller saturation
compares
cannot be further reduced). The second function
the control action filtered by a first-order linear filter with time
Fig. 14. Comparison of controllers obtained with rcGA (first row of figures)
and linear model-based design (second row). From left to right, the figure show
the overall speed response on one cost function evaluation experiment (a) and
(c), and the corresponding control action (b) and (d).
constant
(indicated in (9) as
), with the unitself. As smoother control actions
filtered actual action
give lower values of the integral of , this index is intended to
MININNO et al.: REAL-VALUED COMPACT GENETIC ALGORITHMS FOR EMBEDDED MICROCONTROLLER OPTIMIZATION
217
Fig. 15. (a) Comparison of controllers obtained with rcGA, (b) bcGA, and (c) linear model-based design: expanded view of the steady-state time interval of one
cost function evaluation experiment when the load torque is applied. The expanded view of the symmetrical-optimum model-based design (c) also reports the
closed-loop response obtained on the approximated model used for design purposes. It can be noted that in spite of model accuracy (the modeling error seems
negligible prior to the application of the load), the final result is clearly less effective than the PI obtained with cGAs.
penalize controllers with an excessively oscillatory control action, which may cause stresses for the IM producing vibrations,
acoustic noise, and extra losses. At the end of the experiment
, the measured cost is passed to the cGA, and a
interval
new experiment is started.
The search interval is [0 1.5] for the proportional gain and
[0 20] for the integral gain. The stopping criterion is the maximum number of cost function calls chosen equal to 500 (i.e., the
whole experiment lasts 375 seconds, although all the algorithms
converge in about half of this interval). The average results are
summarized in Table IV. Also, in this case, the rcGA is able to
obtain better PI controllers, in spite of the higher standard deviation of controller gain (noise, actuator saturation, and other
nonlinearities in this case make the considered cost function less
sensitive to small variations of the controller gains).
Fig. 13 shows the evolution of the cost function and proportional gain in two typical runs of a bcGA and a rcGA. The effects
of nonpersistent elitism are clearly visible in the evolution of
both algorithms, which have a remarkably similar behavior. To
provide a quantitative comparison with other available design
strategies, Fig. 14 compares the result of a PI controller optimized by the rcGA with a PI designed with the IM model-based
symmetrical-optimum approach. The design technique can be
summarized as follows (further details can be found in [4] and
[9]). The gains of the current controllers are selected so as to
achieve a first-order closed-loop response with time constant
. In order to design the speed controller, the
equal to
plant is approximated with a first-order system having time con, i.e., the sum of all the delays found in
stant
the speed control loop (current control , speed low-pass filter
, and delays due to the digital implementation of the con). Once the first-order model is defined, it can be
trol scheme
easily shown [9] that the open loop transfer function reduces to
the following:
(10)
and is the complex Laplace transform variwhere
able. Then, according to the design theory of the symmetrical
optimum [9], the gains of the PI can be obtained with the following formula:
(11)
The differences between the model-based PI and the
cGA-based PI are illustrated in Figs. 14 and 15. In particular,
Fig. 15(c) compares the closed-loop responses obtained using
the same model-based controller in simulation and during one
experiment on the actual hardware. The similarity between
the two responses confirms that the mathematical model used
for the symmetrical optimum design is fairly accurate. Nevertheless, the model based procedure obtains a less satisfactory
result (larger overshoot, larger effects of the load torque). It has
to be mentioned that it is a fairly common practice in all PID
model-based design methods to perform a final hand-tuning
session directly on the controlled hardware and refine the
performance of the closed-loop system. Thus, it is reasonable
to believe that the symmetrical-optimum PI could be made
more satisfactory with hand-tuning refinements. On the other
hand, this possibility does not make the result of the cGA less
significant, since the latter is obtained with a fully automated
model-free procedure in less than 3 min.
VI. CONCLUSION
This paper proposed a variant of an effective elitist cGA
working with a real-valued PV. In a wide set of simulated and
experimental investigations, the variant exhibited satisfactory
behavior, with reduced computational requirements and repeatable and generally accurate results, comparable to those of
other state-of-art compact and population-based GAs. As in the
case of population-based GAs, a real-coded implementation of
a cGA is an interesting tradeoff between simplicity of the code,
interpretability of the PVs and chromosomes, and accuracy of
the final solutions.
The efficacy of the proposed algorithm is confirmed by a realworld application in the context of embedded microcontroller
optimization. On the proposed platform, the effects of the actual and unknown high-order phenomena and nonlinearities are
fully accounted in the cost measurement, and the final controller
is ready-for-use with known performance. The cGAs require a
very low computational cost, and therefore can be directly implemented on the same processor running the control algorithm,
overcoming the typical limitations related to the conventional
population-after-population algorithmic scheme of sGA.
The presented research leaves many directions open for further investigations. In principle, the cGAs seem suitable to also
solve more challenging online tuning problems, e.g., optimizing
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IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 12, NO. 2, APRIL 2008
(A3)
a nonlinear interpolator such as a neural network. However, further research is necessary to investigate the potential limitations
related to the typical “order-one” behavior of cGAs in these
larger size optimization problems. Another promising direction
regards the simultaneous optimization of the four controllers operating in the IM. Also, such a simultaneous design problem is
particularly challenging because of the unpredictable interactions between the various controllers.
APPENDIX
Derivation of the Update Rules: Let us consider a population of individuals and focus on one generic gene . Let us
group the values of the gene in the population in the vector
, where is a
and
the
generic iteration index. Let us indicate with
mean and standard deviations of
. Let us assume that
is subject to a binary tournament selection involving the generic
individuals and , where index corresponds to the winner
and to the looser. The vector of genes after the tournament
becomes
. The
can be immediately computed as
mean value of
(A1)
obtaining the straightforward update rule for the mean introduced in Fig. 2. Analogously, the standard deviation after tournament is defined as
(A2)
Substituting (A1) in (A2), we obtain equation (A3) at the top
of the page. Expanding the squared terms in brackets, and after
some straightforward manipulation, the latter equation can be
rewritten as
(A4)
which is the update rule for the standard deviation used in the
rcGA in Fig. 2.
ACKNOWLEDGMENT
The authors wish to thank the anonymous reviewers, the
associate editor, and the editor-in chief for their valuable
suggestions.
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Ernesto Mininno was born in Bari, Italy, on December 1977. He received the M.Sc. and Ph.D.
degrees in electrical engineering from the Politecnico di Bari, Bari, in 2002 and 2007, respectively.
Since December 2003, he has been with the Italian
National Research Council (CNR) and Sintesi-scpa,
working in the field of industrial automation. He
is also currently working with the Converters,
Electrical Machines, and Drives Research Team at
the Polytechnic of Bari. His main research interests
focus on intelligent control of electrical machines
and real-time control systems for industrial robotics
219
Francesco Cupertino was born in Italy on December 1972. He received the Laurea degree and
the Ph.D. degree in electrical engineering from the
Technical University of Bari, Bari, Italy, in 1997 and
2001, respectively.
From 1999 to 2000, he was with the PEMC
Research Group, University of Nottingham,
Nottingham, U.K. Since July 2002, he has been an
Assistant Professor with the Electrical and Electronic
Engineering Department, Technical University of
Bari. He teaches several courses in electrical drives
at the Technical University of Bari. His main research interests cover the
intelligent motion control, design, and fault diagnosis of electrical machines.
He is the author or coauthor of more than 50 scientific papers on these topics.
David Naso received the Laurea degree (Hons.) in
electronic engineering and the Ph.D. degree in electrical engineering from the Polytechnic of Bari, Bari,
Italy, in 1994 and 1998, respectively.
He was a Guest Researcher with the Operation
Research Institute, Technical University of Aachen,
Aachen, Germany, in 1997. Since 1999, he has been
an Assistant Professor of Automatic Control and
Technical Head of the Robotics Laboratory at the
Department of Electric and Electronic Engineering,
Polytechnic of Bari. His research interests include
computational intelligence and its application to control, industrial automation
and robotics, numerical and combinatorial optimization, and distributed control
of manufacturing systems. He is author of 78 journal and conference papers on
these topics.