generalized gradient algorithm

7/28/2019 Generalized Gradient Algorithm

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1 1 6 6 J . GUIDANCE J . GUIDANCEAppendix

B y expanding the A matr ix with the B matrix as its firstcolumn, the following approximate expressions for the zerosmay be der ived (neglect ing unimpo rtant contr ibutions)

CW.)/2]a, - K0/(2 Mc)[CL - (c/iy)\Cm +C m / 2 ] ( A l )

a2 0 (A2)Equations (Al) agree with th e well-known approximations forth e short-period mode (a a, un ), thus resulting in Eqs. (2).With regard to Eqs. (A2), th e following phugoid approxima-tions resulting from expanding th e A matrix m a y be consid-ered

(A3)Combining these expressions with Eqs. (A2), th e correlationdescribed by Eqs. (2) results.

References'Roskam, J., "Airplane Flight Dynamics an d Automatic FlightControls," Roskam Aviation and Engineering Corp., Lawrence, KS,P t s . I, II, 1979/82.2Brockhaus, R., Flugregelung II , R. Oldenbourg Verlag, Munich,1979.3Powers, B. G., "Phugoid Characteristics of a YF-12 Airplane withVariable-Geometry Inlets Obtained in Flight Tests at a Mach Numberof 2.9," N ASA TP 1107, 1977.4 Berry, D. T., "Longitudinal Long-Period Dynamics of AerospaceCraft ," Proceed ings of the AIAA Atmospher ic Flight MechanicsConference, AIAA, Washington, D C, 1988, pp. 254-264.5Gilyard, G . B., an d Burken, J. J. , "Development and Flight TestResults of an Autothrottle Control System at Mach 3 Cruise," NAS A

Restorat ion Algor i thm ( S G R A ) ,1 which can solve general tra-jectory opt imiza t ion prob lems . Goh and Teo2 proposed aunif ied control parameter iza t ion approach that conver ts anoptimal control problem into a parameter optim ization prob-l e m . These algorithms employ a sequence of cycles, each cyclehaving two phases. One phase improves the performance in-dex while the other reduces the constraint violations.In th is N ote , a generalized gradient is found that improvesth e performance index and reduces the constraints at the sametime.

Problem StatementT he general problem that will be considered has the fol low-

ing form:Pmin / = < [ J C ( 1 ) , 7 T ] + L(*, H , T T , r) dr (1)J o

subject to*=/(*, W , T T , T ) (2 )

an d jc(0) is given with th e following constraints:1/>[JC(1) ,7T] =0 (3)

S(x, u , T T , T) = 0 (4)where u is the m x 1 control vector, x is the n x 1 state vector,T T is the p x 1 parameter vector, r is the normalized time in(0,1), ^ is the q x 1 terminal constraint vector, and S is ther x 1 path equality constraint vector. A meaningful problemrequires: r


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NOV.-DEC. 1990 E N G I N E E R I N G NOTES 1167Subst i tu t ing Eq. (7) into Eq. (5), one obtains

Hr-HuDuST)dT\dv

[(H x + X ^ - H D uS x)5 x + // M6*w] dr (8)Choose

HUDUSX and6 i r = - (H v - H UDUSV) dry

(9)(10)

Then we would like to choose d*u such that H ud *u 0, S0 is the pathconstraint vector evaluated at the current step, an d d S =Sxdx +Su5u + S V 6 7 T .There foreSx5x = 0 (13)

The same arguments are carried out as before until selectionof 6 * M , which has to satisfy H u6*w


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1 1 6 8 J. G U I D A N C E J. G U I D A N C EFor the convenience of programming, th e following quanti-ties are defined:

(m x r )(m x n)(m xp)(m x 1)(m x m )( i x n)( i x p )(n x n )(n xp)

12 ) Compute th e generalized gradients:

T T L,-Jx Jx ~ J

where 1 % , = Ini and 7,^ = /,. N o w , the algorithm can be statedstep by step as follows;1) Guess the initial control an d parameter vectors: M ( T ) , T T .2) In tegrate forward: x = '/'(*, M , T T , r) with Jt(0) given.3) Compute th e gradients: Sx,S u ,S vJx,f u,f V9 \I /x,\l/^ an d 7F.If 7F


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NOV.-PEC. 1990 E N G I N E E R I N G N O T E S 1169

i.o

0. 4

0.01.0

0.0

-1.0

1.0

0.4

0.01.0

0 . 4 -

0.0Normalized Time 0.0 -5

Fig. 1 Problem with bounded control .

T he init ial guesses of the controls an d final tirne are:

In th e current implementation, th e steepest descent schemeis employed in th e feasibility phase. T he program parametersare: K ltf = 0.2, K pf = 0.2, K u o = 0.05, K po = 0.05, p = 0.6, = 0.1, ' e/ = 0.0001, and e0 = O.OQl, where K uf , K pf denoteth e stepsizes for control an d parameter, respectively, in thefeasibility phase, whereas Kuo, K po are in the optimizationphase.T he optimal minimum t ime is found to be t* = 2.0023. Thiscompares well with th analytical solution obtained in Ref . 3(t f = 2.000). The basic feature of a "bang-bang*' control isclearly demonstrated. T he results are plotted in Fig. 1.

T he computer program is coded in C language. Details ofth e algorithm an d more examples can be found in R e f . 6.

ConclusionA generalized gradient algorithm has been proposed an dverified for t rajectory optimization problems with possiblepath equality constrain ts , terminal equal i ty constra ints , andcontrol parameters. The generalized gradient was shown toimprove th e performance index an d reduce th e constraints atth e same t ime.Acknowledgments

This research was supported b y N A S A Grant 19 1 from th eN A S A A m es Research Center , monitored by H einz Erzberger .T he authors would like to thank Jiangang Dai and Jinyi Chenfor helpful discussions.References

'Miele, A ., "Gradient Algor i thms for the Optimization of D y nam i cSystems," Co n t r o l an d Dynamic Sys t ems , Advances in Theory an dAppl icat ion, edited by C. Leondess, Vol. 16, Academic, New Y o r k ,1980.2Goh, C. J., an d Teo, K . L., "Control Parametr iza t ion : A UnifiedApproach to Optimal Control Problems with General Constraints,"Automatica, Vol. 24, N o. 1, 1988, pp. 3-18.3Bryson , A. E ., an d H o , Y . C., Applied Optimal Co n t r o l , Hemi-sphere, Washington, DC, 1975.4Lee, A., "Opt imal Landing of a Helicopter in Autorotat ion ,"Ph.D. Thesis , Dept. of Aeronautics and Astronautics, StanfordUniv . , S t an fo rd , CA , July 1986.5Jacobson, D . ' H . , an d Lele, M, M., "A Transformation Techniquefo r Op t ima l Control Problems with a State Variable Inequality Con-st rain t ," IEEE Transactions o n Automatic C on t ro l , Vol. AC-14, N o.5, 1969.

6Zhao, Y ., "Opt imimal Control of an Aircraft Flying Through aDow nbur s t , " P h . D . Thesis, Stanford University , Stanford, CA, June1989.DownloadedbyBeihangUniversity(CNP

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generalized gradient algorithm

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