Fuzzy Idle Speed Control

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FuzzyM odeling and Control of a spark I gnition Engine Idle M odeHazem Mohamed Mohamed* ,Said M unzir * ,Mohd Zaki Abdulmuin and Sayoud Hameida**

' Abstract In this paper a MIMO nonlineardiscrete time model of the idle mode of an SIengine is first developed using fuzzy logic modelin the form of a nonlinear radial basis functionexpansion. The inputs are the throttle angle andthe ignition advance and the outputs are the idlespeed and manifold pressure. The model isvalidated using nonlinear correlation tests forMIMO systems.Phase portrait analysis is used to study the globaldynamic behavior of the plant, in order to findperiodic motions and equilibrium points of thesystem. Optimal state trajectories are used asentries to a fuzzy logic expansion to approximatethe distorted control surface of the plant. Theresulting FA M or fuzzy controller smoothens thecontrol surface, and consequently, the transitionsof the state trajectories from a different initialcondition to another in its path to theequilibrium. Both the model and controllerparameters are optimized using a modem versionof an evolutionary algorithm, called covariancematrix adaptation-Evolution Strategies (CMA-ES).K eywords Idle mode, nonlinear control1.IntroductionIn engine control systems idle mode control is astand a lone control module and it's the mostfrequent encountered operating condition of SIengines especially in city driving conditions. Inthis mode the engine is running on its minimumspeed and minimum torque, suffering differentdisturbances (air condition, automaticstransmission shift, power steering, . etc), thesituation becomes worse due to significantparameter changes over the time life of theengine. The key factors to be considered for idlemode control [l ] are: engine speed set point,accessory load disturbances, control authorityand actuator limitation, available measurement,variations in engine characteristics over theentire operating range.*Process control and automation centerFacultyof Engineering -Universityof Malaya50603 Kuala Lumpur* *Faculty of Engineering-Universiti Multimedia63100Cyberjaya- Selangor- M alaysia

For wide ranges of speed changes and undersever loading condition the plant exhibits anonlinear behavior, in addition uncertainties intheir parametric and unparametric forms mayexists and linear approaches may fail to solve theproblem. As for nonlinear system theconventional control design methodologiesavailable are scares and mainly depend on themathematical model available, most of thesemethodologies require a linear in control ( x =f(x)+g(x) U, x is the state vector and U is thecontrol vector) mathematical form. In caseswhere such form is not available the onlyavailable approach to solve the problem is theunconventional approaches such as fuzzycontrol. Fuzzy controllers can compensate fornonlinearities, and give a robust stableperformance for the plant being controlled.2.Engine M odelTypical dynamic models of SI engines idle modeemploy t wo principle states and two controlinputs [l]. The two states are the engine speedand the manifold pressure, and the two controlinputs are the quantity of air breathed by theengine and ignition timing. In accordance, theplant can be viewed as a black box with twoinputs, throttle angle and spark timing, and thetwo outputs are engine speed and manifoldpressure. For identification purpose the engine isfirst calibrated to run near its idle mode underopen loop condition. Throttle angle wascalibrated to 6 degrees and ignition timing in therange of 8-10 degrees before TDC, air flow isonly allowed to be breathed through the intakemanifold. To excite the different modes of theplant and reveals engine nonlinearities, a seriesof random and variable magnitude input signalsare given to the engine, through an open ECU.Engine speed and manifold pressure responsesare then recorded. Both of the inputs and outputsof the plant interact with each other to yield thenonlinear discrete form:where:

i

y(t) = g( y"' ,u'-l ) + &(t) (1)~ ( t ) [ y d t ) , y2(0 , . . . , yq(t917, the outputvector, u(t) =[ l(t) , u2( t) , . . . ,U(?) ]* theinput vector, &(t)=[ E 2 ( t ) , ~ ~ ( t ). . . E y ( t ) ] 7 ' ,The residualsvector, g =[ l , g2,...,g, jr , a vector

0-7803- 6355- 8/00/10.00 Q 2000 IEEE

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2) , ... ,yi(t-nJ ] elayed output vector, u:-' =[ui(t-I) , u i ( t - 2 ) , . . .,ut (t-nJ 1,delayed inputvector, E , - [q(t-l),q(t-2 ),...,c(tne)],delayed error, 4=2, r=2. Combiningy"' ,U-' inthe regression vector 4 , Equation 1 can bewritten in the form:

/-I -

Y(0=g(4,B ) + 4 0 (2)(3)

The nonlinear function,g (1 can be expanded to:g(pY e)= a k g k (4 ,pkYyk)

Then: u(t>=xq&(t)+dt) (4)wheregk 'sare called basis function, (radial basisfunction is used here)$, is called the dilation orthe scale parameter yk is a transnationalparameter (centers in radial constructionM 4 )=K ( 1 4 -yk 11pk)) and a k are coordinateparameters, 4 is the input vector or theregressors vector Sjoberg et al(1995). Theparameters vector 4 is composed of theparameters setakpk , k .

3.TheFuzzy modelA fuzzy expansion for is used here due to itsgood approximation capabilities, whichcombines both of the advantages radial basisfimction of characterizing local properties andadvantages of sigmoidal nonlinearities incharacterizing global properities. of A fuzzymodel or a fuzzy expansion with the productoperator, center average def udier and Gaussianmembership hnction [2]takes the form:

The nonlinear termof previous expansion, which

Can corresponds to the nonlinear part of theexpansion of the functiong (@,p iYy i )hich inthis case will be in the form of a radial basis

function expansion, in this case a is a set oflinear parameters. The parameters to beoptimized in the previous are the centers of theradial basis function xi thespread factora:,and the set of the linear parametersa.4.M odel Parameters OptimizationEvolution strategise is an instant of stochasticsearch techniques that emulate evolution ofnatural biological processes. These techniqueshave demonstrated their ability to handle globaloptimization problems, of multimodal,discontinuous and noisy nature, without anyJacobian or Hessian needed, which make themqualified candidates for system identification andcontrol optimization problems. A general outlineof the ES algorithm is given below:t:=oInitializeEvaluate f (2 , o ) ) y . . . y f ( 2 p0))While T(P'" =0) doRecombine: P' ( t ) =r,(P(t))Mutatue: F(t>=r&, P' t))Evaluate: f(Zl ),.. ( z p

={il o),..., ip(0))E I'

Select P('+)=s,,((F)UQ)t=(t+l)Endz,i s an individual (search point, e) , P is thepopulation of individuals (search space), f(x) isthe objective fbnction (x' corresponds to 8)r, ,m , re recombination and mutationoperators, S,, is the selection operator. In ourcase here the objective variable is the vector ofparameters 8 which during the search take theform: 6(k+1)=6,ye,(k)+A 6 (7)where A 8 is a stochastic term called themutation. Mutation is done by adding anormally distributed random vector to B(k)(which is A 8 in the equation). The objectivefunction to be minimized here is given by:l Nmin( J )= CJ ~wg(qWY111 (8)0 I='A major part of the search process focus on theadjustment of the parameters of the normaldistributionof the mutation vector A B . This is

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what is called self-adaptation, which is apeculiarity of ESs, where the strategy parametersare dynamically adapted during the searchprocess. Parameters of the mutation distributionare called strategy parameters (different from theparameters to be optimized). A modem versionof ESs that make use of the history (evolutionpath) of successful (or selected) mutations andindependently of the problem dimension updatethe distribution parameters, is the covariancematrix adaptation Evolution strategies, CMA-ES. Given below the main equation of the CMA-ES: (e)(K)a(g)B(g)D(g) g+l)

W ' k1-@+I) =The object variable vector under optimizationD(g) s an nxn (n is the length of the vector8)diagonal matrix that scales the axis of thedistribution of the random vector z. Diagonalselements of D are the square roots of theeigenvalues of the covariance matrix (C) of z.B(') determines the new orientation of the plane(hyper-elliposid) containing the axis of thedistribution (asortof transformation matrix).z,is a random vector of expectation 0 and Ivariance.2-c('+l)s the covariance matrix of theB(g)D(g)~p+l)ormally distributed randomvector at generation (g+).

= 1- . g +c pF+')(p!,K+'))T

p:'+) is the evolution path of the strategy,c,determine the cumulating time for p, , c,"normalize the variance of p, , C, to alignc, ( ) ( ~ ~ + ' )nd zf+') variances,

i=lrate of covariance matrix C.

4-dg)s the step size in generation g, dg)is also updated according to :

d, 2 1-damping factor which determine thechange rate of a, , i ns the expectation of a(0,I ) normally distributed random vector. Detailsof the algorithm can be found in [3] and[4]. Foroptimizing 0 a population A of 16 individualsis initialized, the number of parents = A /2.A11986 evaluaiotns of the objective fbnction wasenough to estimate the parameters of the plantmodel. For the controller parameters theevaluations number increases to (at least100n2A imes) due to the distorted nature of thesurface to be optimized.

5.M odel validationModeling results can be seen in Fig.1, whereboth the input and output signals are shown. ANA RM AX first order model structure(nu=1,ny=l,nE1=1,nE2=1 , for eachsub-model of the MIMO model was foundreasonable to represent the dynamics of theprocess, this can be realized from the correlationtests results shown in Fig. 2The regressor vectorwhere ul( k- ) ,the first input (throttle angle),u 2 k-1), the second input (spark timing).y, k- ) , the first output (enginepeed), y2 k- ), the second output(manifo1dpressure).A fuzzy expansion of 9 terms was ableto represent the plant dynamics as shown in Fig.1 .To check for model structure adequacy,nonlinear correlation tests [ 5 ] are applied tocheck for the correlation between modelresiduals and both the input and the output. The

is:4r=[ u , ( k-11,u* ( k-by , k-1),y,(k - ) 1

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two correlation functions used:

where:


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[2] Wang, L.X. 'I Adaptive Fuzzy systems andControl 'I, Prentice-Hall Inc., Englewood CliffsN.J .,lst edition 1994.[3] Hansen,N. and Ostermeier, A .(2000).ompletely Derandomized Self-A daptation inEvolution Strategies. Evolutionary Computation,Special Issue on Self-A daptation[4] Hansen, N. and A. Ostermeier (1997)."Convergence properties of evolution strategieswith the derandomized covariance matrixadaptation:The@U \lambda)-CM A-ES'.In EUFIT97, 5th Europ.Congr.on IntelligentTechniques and Soft Computing, Proceedings,Aachen, pp.650-654. Verlag Mainz,WissenschaftsverlagThrottle angle

14

12

10

a

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4 I7 850spe7-----l

0. 2I I0 200 400 600 800 1000

Fig. 1 Modeling results, ---- real date, predicted output-

[ 5 ] S.A. Bi ll ings and Q.M. Zhu, "M odelValidation Tests for Multivariable Nonlinear[5] S.A. Bill ings andQ. M hu, "M odelValidation Tests for M ultivariable NonlinetirModels Including Neural Networks", Int. J . ofControl, Vol. 62, No. 4, 1995, pp. 749-766.[6] Smith S.M, Nokleby, R B. and Comar, D J"A computational approachfuzzy logic ontrollersdesign and analysis using cll state space method.In K andel (Eds),Fuzzy control systems),pp397-427 CRC Press 1994

Ignition timingo r I3025201510

0 850Manifvld PressureI '

0.8

0.6

0.4

0.2010 200 400 600 800

0 . 5 t \10 1 5- 0 . 0 5 ~

Fig2 Correlation tests

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000 , I I I I I I00

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80 I I I I I I

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1000 I I I I I I

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0 0 0 0 0 0 0

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0 200 400 600 800 1000 1200 140C

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Fig.4Speed and pressure responses under f uzzy control (no load)

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Fuzzy Idle Speed Control

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