The Minimum Total Heating Lander By The Maximum Principle Pontryagin

DangThiMai Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 12( Part 2), December 2014, pp.77-80
www.ijera.com 77|P a g e
The Minimum Total Heating Lander By The Maximum Principle
Pontryagin
DangThiMai*, Nguyen Viet Hung
*(Department of Basic Science, University of Transport and Communications,Lang Thuong, Dong Da, Ha Noi,
Viet Nam,)
**(Le Quy Don Technical University)
ABSTRACT
The article will research a lander flying into the atmosphere with the flow velocity constraint, i.e. the total load
by means of minimizing the total thermal energy at the end of the landing process. The lander’s distance at the
last moment depends on the variables selected from the total thermal energy minimum. To deal with the problem
weapplyPontryagin maximum principle and scheme Dubovitskij Milutin.
Solvingboundaryusingtheparameterandthesolutionobtained inthe choiceof variables. The results of simulations
performed on Matlab.
Keywords-maximum principle; control; the overload; total heat; minimum.
I. Introduction
Research on the problem of choosing an angle
of launch of the flying object which is reducing
velocity in atmospheric conditions under which it is
taken into account the minimizing of total heat flow
with the load limits of aircraft equipment. The total
heat output of the device is the integral form of the
following:
1
3 2
0
T
Q CV dt  ..1((1)
Required to deteminea control )(tCy , which
minimizes Q(T) (1) under the following restrictions:
)2(,
2
,
2
22 V
qN
G
S
qCCn yx



(
2maxmin
, yxoxyyy kCCCCCC 
(3)
)4(,,
)(
, 2
2
00 mgG
HR
R
gge H


 

)5(,sin,sin  VHg
m
S
qCV x 

)6(,cos 









V
g
HR
V
mV
S
qCy
)7(.
cos
HR
RV
L


 
Where

n - full overload, q - speed pressure,
 - atmospheric density, V - velocity of the
vehicle,  - path angle, H- height, L - the remote,
G- the weight of the machine, m – mass, 0g -
acceleration due to gravity on the surface of the
planet, R - the radius of the planet, xC - the drag
coefficient, yC - lift coefficient, S - characteristic
area apparatus,
NCCCkC yyxo ,,,,,,, maxmin
0  - constants.
For the system (1) - (7) the initial conditions:
)8(.)0(,)0(
)0(,)0(
00
00
LLHH
VV

 
and conditions and limitations:
    ,,,)( 111 HTHTVTV  
,)( aTL  T - not fixed. (9)
where a – parameter.
II. Application of Maximum principle in
the regular case.
Let lander comes from the initial state (8) in a
washed-position (9) in an optimal way in the sense
of minimizing the total amount of heat under the
assumption that the optimal trajectory regularity
condition [3,4]. In the above problem, the regularity
condition is equivalent to
.,0 Nn
C
n
y






(10)
In this case, the maximum principle is as follows:

 QPLPVPHPP QLVH
, (11)
RESEARCH ARTICLE OPEN ACCESS

 NntL 

 )(1  , (12)
H
P
V
PP HV








 
,,


.,
Q
P
L
P QL





  (13)
Here  t - the Lagrange multiplier, which is
determined from the condition of Bliss
[3, 4].
  0






yy C
n
t
C
 . (14)
 - Pontryagin function, 1L - Lagrange function.
QLHV PPPPP ,,,, - corresponding conjugate
variables. For inequality constraints (2) satisfies the
complementary slackness.
   0

Nnt . (15)
Since the system (1)- (7) is autonomous and the
descent no restrictions, the Pontryagin function (11)
is identically zero, ie
   ,,,,,,0,, LHVxCuuxP yy 
.
 qLHV PPPPPP ,,,, . (16)
Conjugate variable  tPQ normalized by the
condition:
 tPQ -1. (17)
The initial conditionsfor the system (13)
andareunknownparameters of the problem.
Condition  tPQ  -1 and   0,,  uxP is
essentially determined by threefree parameters:
      321 0,0,0 CPCPCP LV  (18)
since  0HP is determinedfrom the
condition   0,,  uxP .
In this case,the number of controlledfunctionsat the
end ofthe trajectory(9)coincides with the numberof
free parametersof the problem (1) - (9), (11), (12),
because the time Tis not fixedandis a free parameter.
According to the principleof maximumcontrol
programchosen from the condition:
min)(max  TQkhiП
yC
(19)
We write downthe partPontryaginfunction(11),
which clearly depends onthe control  tCy
m
SVC
P
m
VSC
P x
V
y
22
2
0

  . (20)
)(tCy can takecontrolnot onlylimitvalues(3), but
alsoan intermediate, which is determined from the
condition
maxmin0
2
,0 yyy
V
y
y
CCC
VkP
P
C
C


   . (21)
Wecalculatethreevaluesof the function 0 in (20)
     
 yyy CCC 03
max
02
min
01 ,, .
and
 321
max
0 ,,max  . (22)
Equation (22) determine the nature ofthe optimal
controlproblemof Pontryagin, ie provided
that Nn 

.Solution of the problemis greatly
simplifiedif the rightend of the trajectoryis
controlled bythe condition:
  1HTH  . (23)
In this case,the solution of (1) - (9) is determined by
theboundary conditions
  11 )(, VTVT  ,   aTL  . (24)
Anddepends on threearbitrary constants 21, CC and
3C .
Thus, the initialproblem is reduced toa three-
parameterproblem (1) - (9), (13), (18), (24), and the
optimal controlis determinedat each point t ofthe
maximum principle(24).
III. Restriction onoverload
Inthe taskdifficulty of determiningthe
geometry ofthe optimal trajectoryis the identification
ofpointscoming off the disabled Nn 

.
Note that the total overload (2) has two components
xn and yn . The first is called a longitudinal
overload, and the second - normal.
22
0
2
0
2
,
2
,
2
yxxxyy nnnC
mg
SV
nC
mg
SV
n 


 .
Instead of limiting (2), we introduce a new
restriction
  0,, 11  uxNnnNnn xyxy  (25)
With an appropriate choice 1N ofthe inequality(25) is
known to besatisfiedconstraint (2). This factfollows
from
  22
1 yxxy nnnnN  . (26)
equal signoccurs when 0yC .
We now compute the derivative of  ux, (25)
follow yC
 yy
y
kCsignC
mg
SV
C
2
2 0
2


  . (27)
In this case, the Lagrange multiplier  t for
limiting   0, ux (25) is determined by the
formula

 
 yy
yV
kCsignCV
gVCkP
P
t
2
2
2 0










 . (28)
IV. Necessary optimality conditions inthe
irregular case
Now consider the casewhenthe optimal
trajectorycontains an interval, when Nn 

,
andin this intervalat some point 0



yC
n
.
The set of pointsdefinedby the equations:
Nn
C
n
y






,0 . (29)
following [3], we call irregularpoints.For the
problem 0



yC
n
at 0yC . For thegiven
problemwecanusethe resultsof A. I. Dubovistkij and
A. A. Miliutin [3, 4]. According to [3, 4] in the
presence ofirregular pointsconjugatesystemof
equations are
 
 
,
,
,
0
0
H
V
L
Q
P
n nd
P t
H H dt H
n nd
P t
V V dt V
P
P












 

      
  
      
  


(30)
Here-  t Lagrange multiplier-a
d
dt
 generalized
function. For theseobjectsare madecomplementary
slackness condition
   0, 0y
d
t n N C
dt

   

.(31)
From (29) it follows that in theirregular
point(28)andthe conjugate variableswill
experienceracingon the values of
H
n



 and
V
n



 when 0 . This is theessential difference
between thecaseofirregular, wherethe conjugate
variablesare continuous functionsfor
mixedclassconstraints[3, 4].
Besidesthe conditions (29) - (31) on the
optimal trajectoryshould bethe conditionsof
integrabilityof the Lagrange multipliersand the
normalization condition(non-triviality condition of
the maximum principle).
V. Example and numerical result
For more details of the problem, we solve the
problem of finding the minimum total heating of
space shuttle [7], constants and the boundary
conditions are:
min
yC =-0.5; 6.0max
yC ;
50000
m
S 12 
kgkm ,
3
0 10.3769.2 
 kgkm
3
,
kmR 2.6371 ; 88.00 xC ,
k = 0.5 ;
23
0 10.8.9 
 kmsg ,
C = 20; N = 4 ; 145.0 1
km ,
  0 = -1.25(deg);V(0) =0.35kms
1
,
H(0)= 100(km);L(0)= 0(km).
The resultreceivedby using Matlab:
Figure 1.Height H [km]
Figure1illustratesthe shuttle'saltitudeover time, we
seethatthe heightHdecreases rapidly
from100kmdownto40kmover a period[0, 200s].
Figure 2. Lift coefficient yC
In Figure2 illustratesthe state of
thecontrolvariableschangeover time.
t [s]
H [km]
t [s]
yC

Figure 3. Full overload n
Figure 4.Velocity V[km/s]
Figure4 showsthe velocity ofthe
shuttlealsodroppedsignificantly duringthisperiod.
Figure 5. Path angle [deg]
Figure5describestheorbitalinclinationangle.
Figure 6. Remote L [km]
Figure6illustrates thedistanceofthelandingshipsover
time. Wefound thatthedistanceL(t) does
notincreasemuchafter200speriod.
Figure 7. Totalheat Q
Figure7, in theinterval[0, 200s] the
totalamountofsurfacetemperatureincreaseand
stabilizethe shipduring the periodclose
tolanding[200-720s]. According
tooursimulationsonthe heatatthe surfaceof
thevesselcan be consideredto have
beenminimizedduringlanding.
VI. Conclusion
Three parameterboundary value (24), the
problem is solvedfor a fixed value a. Next, we can
choosethe desired value a from the minimumof the
minimumvalue of the functional(1).The boundary
value problemwas solved by thecontinuation of
solutionsto the parameter[6].
REFERENCES
[1]. A.E. Bryson and Y.-C. Ho, Applied Optimal
Control, Rev. Printing (Hemisphere, New
York, 1975).
[2]. Ярошевский В.А. Вход в атмосферу
космических летательных аппаратов. — М.:
Наука, 1988.
[3]. Афанасьев А.П., Дикусар В.В., Милютин
А.А., Чуканов С.В. Необходимое условие в
принципе максимума. — М.: Наука, 1990.
[4]. Дикусар В.В., Милютин А.А.
Количественные и качественные методы в
принципе максимума — М.: Наука, 1989.
[5]. O. von Struk and R. Bulirsch. Direct and
indirect methods for trajectory optimization.
AnnalsofOperationsResearch 37(1992)357-373
[6]. Дикусар В. В, Кошька М, Фигура А.
Продолжение решений в прикладных задач
оптимального управления. М.,МФТИ. 2001.
[7]. M. H Breitner and H. Josef Pesch. Reentry
trajectory optimization under atmospheric
uncertainty as a differential Game. Advances in
dynamic games and applications, 1994.
t [s]
n
t [s]
V [km/s]
t [s]
[deg]
t [s]
L [km]
Q
Q
q [km]
t [s]

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