4o. SBAI- Simpósio Brasileiro deAutomação Inteligente, São Paulo, Sp, 08-10de Setembro de 1999

GENETIC AlGORITHMS IN THE PROJECT OF A PO ANO A FUZZYCONTROllER FOR SATElLlTE ATTITUOE CONTROl

Cláudio Correacorrea@pgrad.inpe.br

Sandra Aparecida Sandrisandri@lac.inpe.br

Luiz Carlos Gadelha de Souzagadelha@dem.inpe.br

Applied Computing - INPElCAPLaboratory of Computing and Applied Mathematics - INPElLAC

Space Mechanics & Controls Division - INPElDMCBrazilian Nationallnstitute for Space Research - INPE

Abstract - In this paper we propose the use of genetic algo-rithms in the project of a PD controller and a fuzzy controller forsatellite attitude control. The simulated satellite model is basedon the French-Brazilian micro satellite, whose control system isitselfbased on the stabilization of3 axes, using 3 reaction wheelsas actuators. In the experiment, part of an orbit of the satelliteis used to test the fitness of each candidate. The performanceof the controllers are compared with a PD controller, projectcdpreviously using the poles allocation method.

Keywords: Genetic A1gorithm, Fuzzy Control, Satellite .

1 INTRODUCTIONThere exists nowadays a tendency to build small artificial satel-lites, aiming at guaranteeing a fast and simple means of reach-ing space. The tendency is to have these satellites equipped withhighly autonomous systems for altitude, maneuver and orbit con-trol, and to have them developed fastly and at very Iow costs(Guerra et alo 1997). Moreover, an augmentation of sateIlite au-tonomy has been long pursued with the goal of not only im-proving the performance, but especia1ly in order to reduce fuelconsumption.

Fuzzy logic is one of the most well-succeeded recent tech-nologies in the development of sophisticated controI systems(Driankov et alo1996, Lee 1990). With its employment, com-plex requirements can be implemented in simpler controllers,of easy maintenance and Iow cost. In what regards autonomy,fuzzy controllers are especia1ly suited when the model is subjectto uncertainty (Woodward 1996, Conway et ai. 1994, Guerra etai. 1997).

A fuzzy controller uses rules of the type I f < premíse >then < conclusion .>, which define controI actions in func-tion of some (usually ill-defined) intervals on which the statevariables may take their values. The definition of these intervals,modeled by fuzzy sets, constitute the main difficulty in the cre-ation offuzzy controllers. One way of dealing with this problemis leam the set of such parameters through the use of genetic al-gorithms (GA) (Husbands 1992). In our context, the quality ofeach candidate set can be assessed by verifying lhe performanceof the controller projected using that set.

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In this paper we present two altitude controllers, projected withthe use of genetic algorithrns, for the pointing phase of a reac-tion wheel artificial satellite: a PD and a fuzzy controller of theMandani type. The same fitness function was used in the geneticalgorithms in order to leam the parameters for both controllers.

The candidate sets in each experiment were tested using a previ-ously developed simulator (Souza and Silva 1999), for a modelsimilar to the French-Brazilian satellite, currently under devel-opment at the Brazilian National Institute for Space Research(INPE). This satelitte has its control system based on the stabi-lization of three axes and makes use of a proportional/derivativecontroller (PD), whose gains have been determined through thepoles allocation method , and whose sensor measures have beenprocessed by a Kalman filter (da Silva 1997). This PD controlleris used as basis of comparison to assess lhe quality ofthe resultsobtained by the GA-based controllers presented here.

This paper is divided as follows. In Section 2 we present somebasic concepts about genetic algorithms and fuzzy controllers.In Section 3 we present lhe modeI of lhe satellite used in ourapplications, and the original PD altitude controller projected forit in the pointing phase. In Section 4 we present the PD andlhe fuzzy control1ers projected using genetic algorithms for thesatellite modeI. Finally, Section 5 brings lhe conclusions.

2 BASICNOTIONS2.1 FuzzyContraiFuzzy controllers are based on fuzzy sets theory, which has beendeveloped since 1965 after the seminal works of Lotfi Zadeh(Zadeh 1965). Fuzzy controI techniques were fírst developedwith the works of E.H. Mamdani (Zadeh 1976, Mamdani andBaaklini 1975), and have been gaining increasing importanceover lhe years , being today the main application of fuzzy setstheory. The term fuzzy logic is usually employed in the controIfield to name the modeling of fuzzy pieces of information andthe inference mechanisms that act upon them.

Contrary to what happens in conventional controI in which thecontrol algorithm is described analytically by aIgebraic or dif-ferential equations, in fuzzy control logical mies are employedin the control algorithm, obtained through the synthetization of

40. SBAI - Simpósio Brasileiro deAutomação Inteligente, SãoPaulo, SP, 08-10de Setembro de.1999

human experience, intuition and heuristics, in process control (da Silva 1997):(Zadeh 1965) . ç=tvA role in a fuzzy controlleris usually of the type: w= -Kp* , -J(D * (tv +Dk) +Tp(t) (1)

I f Xl = A l and Xz = A z ... and Xn = An then y = B

where lhe X i and y are respectively state and controllinguisticvariables, and lhe Ai 's and B are linguistic terms. A linguisticvariable is a 4-tuple (x, T(x), O,M), where x is lhe name oflhe variable, O is lhe domain of z, T(x) is a set of linguisticterms, i.e. a set of names of fuzzy sets, and M is a function thatassociates a fuzzy set in O to each term in T(x).

. A fuzzy controller is composed of a knowledge base , that con-tains lhe roles and lhe linguistic variable descriptions, and aninference engine, that yields a control action in function of lhevalue of lhe state varlables in a given moment of time.

Fuzzy controllers are highly adaptable and capable of incorpo-rating knowledge that many other systems are incapable of do-ing (Guerra 1998) . They are also versatile, especially when lhephysical model is very complex and of difficult malhematicalreproduction. In general, they are more useful in non-linear sys-tems , support very weIl perturbations and highly noisy plants,and are robust even in systems where uncertainty is intrinsicallypresent.

2.2 Genetic AlgorithmsGenetic algorithms are adaptive search strategies based on ahighly abstract model of biological evolution (Husbands 1992).They are primarily used in optimization problems for which oneaims to find not necessarily an optimal solution, but at least areasonably good one .

In these algorithms, individuaIs in a population (potential so-lutions) suffer a series of unary transformations (mutation) andof higher order (crossover). These individuals compete amongthemselves for survival; lhe most apt ones have better chances tobe chosen to pass their characteristics to the next generation. Af-ter some generations, lhe algorithm usually converges with lhebest individual representing a solution elose to lhe optimum.

The search for lhe solution involves an evaluation function (fit-ness) , which yields a grade for lhe performance of each individ-ual, according to aspects considered relevant to lhe problem athand.

1\ç= (tv+Dk) '1\

where çrepresents lhe estimation of rotation ç, tv represents lheangular speed and Tp(t) represents lhe perturbation torques. Dk

1\is given by Dk = bk- bk +n (0';), where bkrepresents lhe longperiodbias (derive) and nk lhe short period bias (noise).

The coordinate system for lhe satellite altitude control adopts anexternal referential, defined as:

• ZE oriented to lhe north poIe of the ecliptic;

• XE pointing to the sun position;

• YE pointing in such a way as to form a destrogerous system.

The following orbital data has been used:

• Altitude at apogee 1500 km;

• Altitude at perigee 400 km ;

• Inclination of 7°;

• Argument at perigee of 90° ;

• Longitude of lhe descending node of -90° ;

• Mean anomaly of -90° .

The satellite inertia moments for lhe 3 axes are :

• Ix = 7 kg.mê:

• Iv = Iz =10 kg.nr'.The project specifications are given by (da Silva 1997):

• Pointing precision: vi- = 0.5°, = v}.= 0.15°;

• Stability: vX =vy= vZ = 0.05 °.

In this paper, we have used a linear model for lhe sateIlite, onceit is in lhe normal mode of operate, where lhe atittud angles aresmall (da Silva 1997). The attitude controller developed for thissateIlite uses lhe pole allocation method, and consists of a PDcontrolIer for each axis, using lhe foIlowing equation:

Genetic algorithms have been used in many applications involv-ing fuzzy control (Karr 1994, Pham and Karaboga 1991, Kinzeletai. 1994, Aya and Costa J997); In lhe present work, lhe fitnessfunction of lhe GA for lhe pointing phase is a global measure oflhe performance of each solution in rel átíon to lhe simulation ofpart of an orbit under a certain amount of perturbations.

f(t) = Kp * e(t) +KD * (2)

3 SATELLlTE MOOEL ANO ORIGINAL POCONTROLLER

The French-Brazilian satellite model is a rigid bodymodel wherelhe inertia product is negIected. The controI system consistsof a set of gyro, star and sun sensors plus a PD controller inlhe pointing phase. The rotational dynarnics of lhe sateIlite canbe represented by the foIlowing differential equations system

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where e(t) is lhe signaI error and Ãe(t) = 1t (e(t)), Kp andKD are lhe proportional and derivalive gains.

The proportional and derivative gains, for lhe X,Y and Z axes,are respectiveIy:

• Kpx =Kpy = 0.0263, Kpz = 0.0272;

• K Dx =KDy = 0.08, KDz = 0.1.

Eixo'"EIxoZ

;,...-............... ....... . .;,.,;,_.- -

Figure 2: Errors obtained with the use of the original PD con-troller.

40. SBAI - Simpósio Brasileiro de Automação Inteligente, São Paulo, Sp,08-10 de Setembro de 1999

4 GA PROJECTED ATTITUDE CON·TROLLER

In the following we present a PD and a fuzzy GA projected atti-tude controIlers for the satellite-pointing phase. We first presentthe simulation conditions used in both developments, and thenpresent the controllers themselves.

4.1 Simulation ConditionsIn order to effectively assess the quality of the controllers builtwith the GA's, severe conditions have been adopted for the satel- 'lite operation mode. In this sense, the magnitude order of theperturbations has been incremented in relation to those adoptedto assess the quality of the original PD controller (da Silva 1997).The perturbations due to atmospheric drag and solar radiationpressure have been multiplied by constants f3a = 10 and f3. =100 respectively, corresponding to the values necessary to reachthe same order of magnitude of the perturbations due to the mag-netic field, which had so far a predominant role over the othervalues (see Fig. I).

Figure 1: Perturbation torques multiplied by f3i.

Moreover, ali candidate solutions yiolating either a position or aspeed error project specification are disregarded:

• eliminate chromosomes for which Wi > vi' or (i > vi in atleast one ofthe axis i E {X, Y,Z}.

The performance of each GA was studied using different fitness .functions. The best results were obtained using an approxima-tion of the position error. The fitness function f is given by:

(f.'max(;( t)dt f.'max W;(t)dt)f = 1 - -61 E 'o + =.:':u.O_",--_vf vii E{ X ,Y,Z }

where tmax is the time taken by the simuIated orbits and the inte-gral Jt:m ax g(t)dt of a funtion g(t) is obtained though the use offunction trapz.m, available in Matlab (MathWorks 1992), whichcaIcuIates a numerical integral by the trapezoidal sum method.

The satellite state vectors at the initial moment represent a zeroerror condition, in what regards both pointing and speed . In thepresent work, an error elose to the limits imposed by the projectspecifications has been adopted for the initial position of the 3axes, with speed kept as in the original situation . The initialvalues for the 3 axes are: X = 0.35°, Y = -0.12° and Z =0.12°.

The attitude error relative to the control actions in the simulationof an orbit are shown in Fig. 2, calculated with the original PDcontrollaws (da Silva 1997).

In alI the tests, the same pararoeters for the GA's have beenmaintained: population composed of 30 individuals , probabil-ity of crossover Pc = 0.9, probability of mutation Pm = 0.033,sensibiIity of 3 decimal cases to detect fitness variation and, asIast stop criterium, a maximum number of 35 generations. Eachchromosome is represented by a bit vector, with each gene (pa-raroeter) occupying 20 bits. The total size of a chromosome de-pends on the number of codified pararoeters.

Each chromosome was evaIuated in relation to the simulation of1/4 of an orbit, corresponding to tmax = 1600 seconds.

4.2 GA CharacteristicsIn each GA used in this work, each chromosome in a given popu-lation is composed of a set ofpararoeters specifying a controller.The chromosomes for the PD controller contain the proportionaland derivative gains, and in the case of the fuzzy controller, theycontain the fuzzy terros employed by the roles .

4.3 GA Projected ControllersWe have used the simuIation conditions and the GA structuredescribed above to generate the gains of a PD controller, with acontrol law for each axis. Thus, 8 gains were codified in eachchromosome. The GA converged afier 23 generations; the re-suIts ofthis controller are shown in Fig. 3.

In this work, a single fitness function has been employed to ob-tain optimized PD and fuzzy controller. In order to simplifythe evaluation, only variables ( and w are used in the function .

.The same GA parameters specification and fitness function wereused to develop a fuzzy controIler of the Marodani type, usingthe fuzzy control tooIbox available in Matlab.

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Sat pDlntEI.oXEI.oVEI.oZ

; ,

Set potn.EixoXBx:oYElxoZ

40. SBAI - Simpósio Brasileiro deAutomação Inteligente, SãoPaulo, SP, 08-10deSetembro de'1999lã

Figure 3: Errors obtained with the use of a PD controIler opti -mizedbyaGA

Figure 5: Errors obtained with lhe useof a fuzzy controller optí-mizedbyaGA

A fixed set of9 rules was built through the observation ofthe PDcontroller. The chromosomes were used to encode the centers ofthe fuzzy terrns appearing in the premises and conclusion of themIes .

The fuzzy sets appearing at the extremities of their domains aretrapezoidal and the remaining ones are triangular. AlI fuzzy setsare symmetrical in relation to O, and therefore, only the posi-tive centers bad to be learned. To'confer more smoothness tothe control surface, neighboring fuzzy sets were superposed.These restrictions do not limit the model, and have been fre-quently used in fuzzy controIlers (Driankov et ai. 1996, Heideret ai. 1995, Larsen 1981, Sanchez et ai. 1997).

errors are nevertheless always under lhe specifications designo

We can also see that the errors yielded by CN-GA are smallerthan in PD-PA and PD-GA for both position and speed, in aIlsenses.

Fig. 6 brings the output torques of PD-PA, PD-GA and CN-GA.We can observe lhe answer of PD-GA presents an increase insmoothness and a decrease of oscillations in relation to PD-PA.However, lhe torques of PD-GA are about 10 times greater thanthose yielded by PD-PA. This can be explained by the reductionof position and speed errors presented by PD-GA in relation toPD-PA (see Fig.s 2 and 3).

To further reduce the number of parameters to be learned wehave used a single control for alI the axes. The surface of bestMamdani fuzzy controIler obtained, with 3 fuzzy terms for eachinput variable (error and error derivative) and 3 fuzzy terms theoutput variable (torque), is depicted in Fig. 4. The results of thiscontroIler are shown in Fig. 5.

The fuzzy controIler CN-GA presents lhe 'best features. Itpresents a satisfactory reaction at lhe initial moment of simu -lation (where lhe errors are greater) without provoking oscilla-tions, and a1so maintains lhe errors lower than the other con-troIlers.

4.5 Robustness AnalysisIn order to verify the robustness of lhe projected controllers, onesatelli te orb it was simulated, under lhe same conditions of op-timizations, but with a 30% variation at the Y axis inertial mo-

Table 1: Integral of lhe errors in relation to the 3 axes

Figure 4: Fuzzy controlIer output torque.

4.4 ComparisonTable 1 below brings a comparison of the performance of theoriginal controller (PD-PA) and lhe GA projected controllers(pD-GA and CN-GA) in terms of lhe sum of lhe errors yieldedby each controller during the simulation of a complete orbit. Wecan see that the errors relative to PD-GA are smaller than theones related to PD-PA, in total terms. That does not always hap-pen, however, in what concerns lhe axes taken individually. The

ABSCLUTE VALUE CF THE INTEGRAL(Trapezoidal method)

ErrorsController Position

AxisX AxisY AxisZ TotalPD-PA 78.1237 84.5526 31.1589 193.8354PD-GA 43.1680 12.6005 7.1152 62.8839CN-GA 17.3987 16.5945 7.0337 41.0269Controller AngularSpeedPD-PA 0.9383 0.3276 0.2972 1.5632PD-GA 0.6149 0.4463 0.3818 1.4432CN-GA 0.3584 0.1260 0.1280 0.6125

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40. SBAI- Simpósio Brasileiro deAutomação Inteligente, SãoPaulo, SP, 08-10 de Setembro de 1999

Slt'•EJxoXEixoVEIxoZ

SetpolntEluXEI.oYElloZ

I, I

Ii

Figure 7: Angular errors of the PD-PA controller and 30% vari-ation in Y inertial moment.

.",- :.

" d.'.. . :

,;', : r; '. - . -, .. . '.::. .', .; :/r·:..:<:.:....... . ,..::; ",.:.'::.' .

. ....,. . .' . -;..;S." _ _, . .... . l· : .

...,.:..::/;: ,..;.. '

:·.•.;.. .. o,rig,net ..'-:-, .t.:

ment. Table 2 shows the results of the angular errors for positionand speed, obtained through the simulation of one orbit:

Figure 6: Output torques: a) PD-PA b) PD-GA c) CN-GA.

Table 2: Integral ofthe errors, with a 30% variation ofthe Y axisinertial moment

INTEGRAL OF THE ERRORS WITH 30% VARIATIONY AXIS INERTIAL MOMENT

ErrorsController Position

AxisX AxisY AxisZ TotalPD-PA 78.1237 120.1006 31.1589 229.3834PD-GA 43.1680 17.5652 7.1152 47.5528CN-GA 17.3987 23.1204 7.0337 41.0269Controller AngularSpeedPD-PA 0.9383 0.3339 0.2972 1.5696PD-GA 0.6149 0.6149 0.3818 1.4436CN-GA 0.3584 0.1286 0.1280 0.6151

Figures 7,8 and 9 correspond to these simulations. We can ob--serve, in a graphical manner, the efficiency of each controller inthe task of maintaining the pointing under this new condition.

The reactions of the controllers, in relation to the Y axis massvariation, are similar, and all the controllers presented an in-crease of errors in relation to the Y axis,

However, we can observe, through the graphics and the resultsin Table 2, that the stability characteristics necessary during thenominal operation regime remain practicalIy unaltered. With the

Figure 8: Errors obtained with the PD-GA controIler and 30%variation in Y inertial moment.

exception of the original PD controller, which, similar to the pre-vious simulations presents a certain dificult to maintain the errosdose to zero at the end of one orbit simulation, alI the othercontrollers maintained the pointing, even with this change in theinertial moment.

Among lhe projected controllers, we can observe that the secondMarndani model is slightly more sensible to variations, as canbe varified through the' error results in axes X and Z, whichalso suffered a small alteration. However, contrary to whathappenned in the Y axis, the angular error values of positionand speed in these others axes are even smaller when comparedto the values presented in Table 1.

It should be noted, that to maintain the error results inside theproject especifications is not a complex task, even under the testcircumstances, since the model sattellite has a relatively simplestructure, and neither flexible panels nor other appendices havebeen considered.

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Setpolsatélite artificial durante a fase de transferência orbital e

-- ElxoZ apontamento. Msc Thesis IJIlPE-6397-TDII613. INPE - S.J. Campos.

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---_.......__......__..... ..,;,._-

Aya, J. C. C. and O. L. V. Costa (1997). In: Otimização de con-troladores nebulosos usando algoritmos genéticos. Proc -.SBAI'97. Vitória. pp. 207-212.

We have presented a PD and a fuzzy controller for attitude con-trol for the pointing phase of a satellite model, whose parametershave been obtained through the use of a genetic algorithm.

We envisage the development of a more complete fitness func-tion. A hypothesis to be considered is to take the variance of theoutput signal of the controller, in order to augment the sensibilityin what regards oscillations. Predictions of the energy consumeddue to lhe controller action could also be taken into account inthe function .

We have shown that the controllers thus obtained are robust, andthe results produced validate the use of genetic algorithms as anoptimization tool to treat similar problems as those addressed inthis work.

5 CONCLUSIONS

Figure 9: Errors of the first CN-GA model obtained with a 30%variation in Y inertial moment.

REFERENCES

The satellite model used here is a very simple one for whichother kinds of controllers alsopresent very good results. How-ever, the framework proposed here may be very usefuI in sit-uations for which other controllers do not yield good resultsor are toa difficult to build . The project reported in this work(Correa 1999) also included the criation of a user-friendly in-terface that can be used for prototyping and for educational pur-poses.

Conway, D., R. Sperling, D. Folta, K. Richon and R. DeFazio(1994) . Automated maneuver planning using afuzzy logiccontroller. Proc. Guidance: Navigation and ControI Conf..San Diego-CA

Correa, Cláudio (1999). Uso de algoritmos genéticos na con-strução de controlador nebuloso para o .controle de atitudede um satélite artificial durante a fase de apontamento.INPE/CAP. São José dos Campos.

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