Optimal sensitivity design techniques to design feedback control system for large, flexible booste

Rillings, J. H.

Roy, R.

English

NASA Technical Reports Server

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Rensselaer Polytechnic I n s t i t u t e  
Troy, New Ynrk  
FINALF3RlRT PARTC 
GRANT no. NGR-33-018- 
NATIONAL AERONAUTICS 
AND SPACE ADMINISTRATION 
SENSITIVITY DESIGN TECHNIQUE 
FOR OFTIMAL CONTROL 
Submitted on behalf of 
Rob Roy 
Associate Professor of E l e c t r i c a l  Engineering 
Co-Investigator: J.  H. Ri l l ings  
October 1967 
https://ntrs.nasa.gov/search.jsp?R=19680009798 2020-03-12T09:27:28+00:00Z
Introduction 
This report  describes a f e a s i b i l i t y  study conducted t o  investigate the 
poss ib i l i t y  of using optimal s ens i t i v i ty  design techniques t o  design a feed- 
back control  system for  a large, f lexible  booster. The vehicle t o  be controlled 
i s  described i n  NASA MSFC "Model Vehicle No. 2". The design objective i s  t o  
maintain the  vehicle i n  the immediate neighborhood of a predefined nominal 
t r a j ec to ry  i n  s p i t e  of disturbances act ing on the vehicle. 
are both external, such as random wind gusts, and in t e rna l  such as changes i n  
vehicle parameters. 
of the vehicle within design l imits,  preferably a s  small a s  possible. 
These disturbances 
An addi t ional  design objective i s  t o  keep the  bending 
The nominal t r a j ec to ry  of the vehicle i s  described by a set of nonlinear 
d i f f e r e n t i a l  equations. However, small perturbations of the vehicle 's  motion 
about t he  nominal can be adequately characterized by a set of l inear  incre- 
mental d i f f e r e n t i a l  equations with time varying coeff ic ients .  
t o  calculate  these coefficients were obtained from the  "Model Vehicle No. 2" 
data  package. 
The data needed 
1 
The Problem 
The vehicle t o  be controlled i s  a large f l ex ib l e  booster of the Saturn 
type. 
The control  objectives can be divided in to  categories -- r i g i d  body objectives 
and bending body objectives.  
Control i s  exerted by gimballing four of the eight  propulsion engines. 
The object of the r i g i d  body control  i s  t o  maintain the vehicle a s  close 
as possible t o  i t s  nominal t ra jec tory  i n  s p i t e  of random wind gusts.  
f irst  approximation it i s  assumedthat the  necessary parameters of the f i c -  
t i c i o u s  r i g i d  body can be measured exactly (c f .  Fisher) .  
A s  a 
2 
-2 - 
Unfortunately, because the vehicle i s  f lex ib le ,  t h i s  r i g i d  body is  non- 
ex is ten t .  
t r o l l a b l e )  torque applied t o  one end. 
control  torque for  the  r i g i d  body exci tes  bending modes i n  t h e  vehicle. 
ignored, these osc i l l a t ions  could reach a point where the  vehicle exceeds 
i t s  e l a s t i c  limits and i s  destroyed. 
vent t h i s .  
experimental determinations of the values of t he  parameters involved i n  the 
bending i s  d i f f i c u l t .  Thus, the  successful control  s t ra tegy  must limit the  
bending when the  parameters are known only approximately and change with time. 
For s implici ty  only the first bending mode i s  considered and i t s  equation 
i s  assumed t o  be that of a l i nea r  o sc i l l a to r .  
Rather t he  vehicle behaves l i k e  an unsupported beam with a (con- 
Gimballing the engines t o  provide a 
If 
A successful control  s t ra tegy  must pre- 
Because of the  large s ize  of the vehicle, however, full scale  
The r i g h t  hand side of the equation i s  the  engine exc i ta t ion  of the 
Combining t h i s  f irst  bending mode with the  r i g i d  body descr ipt ion mode.  
gives the  vehicle descr ipt ion i n  terms of f ive  state variables.  
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-u= 
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-2p 7 
-3- 
The numerical values Of these t i m e  varying coeff ic ients  are obtained i n  tabu- 
lar form from R e f .  1. 
Sens i t i v i ty  Design Method 3 4 5  ’ ’ 
The t r a j ec to ry  of an optimal control system i s  sens i t ive  t o  changes 
i n  the vehicle parameters. Thus, it is  important t o  develop a control  strat-  
egy which compensates for  the  e f f ec t  of these changing parameters, preferably 
without having t o  t rack  them. The control system should be designed s o  t h a t  
t h e  vehicle t r a j ec to ry  approximates the nominal t r a j ec to ry  corresponding t o  
the  nominal parameter values, i n  spi te  of var ia t ions of these parameters from 
their  nominal values. 
t i ons  t o  the standard optimal control problem mkes t h i s  possible. 
Adjoining a sens i t i v i ty  index and s e n s i t i v i t y  equa- 
To consider the  problem i n  more concrete terms, it i s  necessary t o  make 
t h e  following assumptions, a l l  of which are met by the  f l ex ib l e  booster pro- 
blems : 
1. 
2. 
3- 
It 
The process has a 
k = f(x, p, 
where 
f i n i t e  number of state equations of the  form: 
9, t )  x( to> = c 
x: n-dimensional s t a t e  vector 
p: r-dimensional control vector 
q: m-dimensional parameter vector with nominal value q . 0 
The parameters appear as coeff ic ients  i n  the  d i f f e r e n t i a l  equation. 
Any parameter var ia t ion  does not change the number of state variables.  
is a l s o  assuned that  f s a t i s f i e s  the  continuity conditions which 
guarantee a unique solut ion x ( t >  
parameter q are specified.  
with x ( to )  = c once the control  p and the  
-4- 
Since closed loop control  i s  desired, p can be wr i t ten  as 
B = B(x,t) 
For a given control  s t ra tegy  l e t  the nominal t r a j e c t o r y  corresponding 
t o  go be 
x = x 0 (t,B0,qo) 
Then the  t r a j ec to ry  deviation due t o  the  parameter var ia t ion  
9 = 9, + b  9 i s  given by 
The magnitude of the deviation can be defined i n  terms of 11x11 , the  Euclidean 
norm of x. Expanding t h i s  deviation i n  a Taylor Ser ies  gives - 
where 
Jy, = 
t h a t  i s  a s ingle  parameter. 
i n  t he  Taylor Ser ies  can be 
For small perturbations,  the higher order terms 
neg,lected, giving - 
n 
ax, - * * - 
To simplify the mathematics a t  t h i s  point, q i s  assumed t o  be a scalar ,  
If an upper bound on the parameter var ia t ion  b q ( t )  i s  known o r  assumed, 
then a bound can be found for &x, the  f i rs t  order t r a j ec to ry  dispersion. 
-5- 
For su i tab ly  small &q, then, Il&\\ can be l imited by l imit ing 
At a point on the  perturbed t ra jec tory  (l$l> = %e+ At) the 
s t a t e  equations become 
Expanding t h i s  i n  a taylor  Series about (xn, q,) - 
+ higher order terms 
where f an i t s  der ivat ives  are evaluated a t  (x 0,qo ). Again, f o r  small 
$q = (q - qo) the  higher order terms a re  negligible,  leaving - 
where s x  is  the  f i r s t  order approximation t o  A x .  
I f  6q i s  a (known) bound on the parameter var ia t ion which yields  the 
maximum t r a j ec to ry  dispersion b x, then, l e t t i n g  z ( t )  4s xllas above, the  
s e n s i t i v i t y  equation can be written as 
This i s  a l inear  d i f f e ren t i a l  equation defined on the nominal t ra jec-  
-6- 
tory.  
limited, the nominal t ra jec tory  and nominal control  must be chosen t o  l i m i t  
z ( t ) .  
cont ro l  problem. 
minimize t h e  sum of the  or ig ina l  index of performance J and an index of sen- 
s i t i v i t y  J subject t o  the  constraints of the vehicle and the  s e n s i t i v i t y  
If t h e  t r a j ec to ry  dispersion due t o  parameter var ia t ion  i s  t o  be 
This i s  done by incorporating the  sens i t i v i ty  masure i n t o  the  optimal 
To do t h i s  the optimal control  problem i s  reformulated t o  
equation. 
The augmented optimal 
determine 
where 
subject  t o  
control  problem thus becomes 
where f and g a re  posit ive def in i te  functions of x,v,p, and z. 
0 0 
Upon inspection it i s  found tha t  t h i s  problem does not admit t o  a solu- 
t i o n  as there i s  no r e l a t ion  which y i e l d s  the  s t r u c t u r a l  information necessary 
t o  construct p*(t). Specif ical ly  
specif ied.  
t he  form of a feedback control  l a w .  This completes the  specif icat ion of the 
problem and allows it t o  be solved. The designer, however, must be judicious 
i n  h i s  choice of feedback structure,  since the  resu l t ing  "optimal" closed 
loop system i s  then optimal only with respect t o  the spec i f ic  type of feed- 
back s t ruc ture  specified. 
&? i s  unknown. The problem i s  under- 
Therefore, the  designer imposes t h i s  s t r u c t u r a l  information i n  
b x  
Two rather  general s t ruc tures  a re  considered below. 
-7- 
Cascade Gains 
Consider the  control  system shown i n  Fig. 1. This consis ts  of a s e t  
of cascade gains w i t h  u n i t  feedback around the plant.  The control  l a w  f o r  
t h i s  case i s  
p(x,t) = K(v - X) 
where K is  the  vector of feedback gains and v i s  the  desired state response. 
The gains are t o  be chosen t o  reduce the  s e n s i t i v i t y  of t he  nominal t ra jec-  
t o ry  t o  parameter var ia t ions and t o  keep the vehicle a s  close t o  t he  nominal 
t r a j ec to ry  as possible. 
Using t h i s  control, the  plant equation becomes - 
and the s e n s i t i v i t y  equation i s  
Thus, the  optimal control problem becomes 
min (J + J) where 
K 
subject  t o  the  s t a t e  and sens i t i v i ty  equations. 
The solut ion of t h i s  problemby standard parameter optimization methods 
gives a set  of feedback gains K which i s  r e a l l y  a t rade off between the two 
object ives  of the problem. Two ta rge ts  a re  being aimed a t  wi th  only one arrow. 
The next section improves on t h i s  s i tua t ion .  
Feedback Gains with Autonomous Input 
Consider t he  system of Fig. 2. This time control i s  exerted by a set 
-8 - 
of feedback gains and a pre-computed autonomous input (or  p r e f i l t e r ) .  
To solve t h i s  problem it i s  necessary t o  combine control  s igna l  and 
parameter optimization. 
F i r s t ,  parameter optimization is  used t o  determine the  feedback gains 
t h a t  o f fe r  t h e  grea tes t  protection against t r a j ec to ry  perturbations due t o  
parameter changes. Then, constraining the  feedback gains t o  be constant, 
t h e  autonomous input i s  selected t o  keep the  vehicle as near the desired 
t r a j ec to ry  as possible while fur ther  minimizing parameter s e n s i t i v i t y  i f  
possible.  Thus, t he  control  l a w  i s  of the  form: 
where K i s  the constant feedback matrix and u ( t )  the autonomous input - 
The nominal process equation i s  now 
2 = f(x, u - Kx, qo , t )  x ( to )  = c 
The s e n s i t i v i t y  equation i s  
act,, = 0 
Assuming t h a t  t he  index of performance integrand can be wri t ten as 
f o  (x, v, B, t>  = f l ( X , V , t )  + f , ( B , t )  
Le.that no cross-product terms appear, the problem can be stated i n  the 
subject  t o  the  state and sens i t i v i ty  equations. 
I n  order t o  adapt t h i s  problem t o  standard optimization techniques it 
can be r e s t a t ed  as two re la ted  problems. 
-9- 
1. 
2. 
Results 
Feedback Gain Selection. 
equations. 
Autonomous Input Selection: 
equations and k = 0. 
The cascade gain configuration was solved on a d i g i t a l  computer. Since 
t h i s  was a f e a s i b i l i t y  study, weighting matrices were not selected t o  give 
the  bes t  possible resul ts ,  r a the r  t he  f i r s t  weights t r i ed  t h a t  gave reason- 
able results were used. With more time invested i n  t h i s  select ion r e s u l t s  
should be be t t e r .  
Figure 3 shows pi tch e r ro r  and normalized bending for  s ens i t i v i ty  
weighting equal t o  zero. 
t i o n  solut ion.  
vehicle f o r  nominal bending frequency, wo. 
response f o r  a -20% perturbation i n  bending frequency. 
case are not only much greater,  but they exhibi t  diverging osc i l la t ions .  
This corresponds t o  the usual  parameter optimiza- 
In  both graphs the sol id  l i n e  indicates  the response of the 
The dotted l i n e  indicates the  
The er rors  i n  t h i s  
Figure 4 shows the same quant i t ies  with the  sens i t i v i ty  weighting in-  
creased t o  10,000. 
the  results fo r  nominal bending frequency are  s l i g h t l y  worse than those above. 
However fo r  t he  -20% perturbation i n  cu the  r e s u l t s  a re  c lear ly  grea t ly  improved 
and i n  f ac t ,  qu i te  acceptable. 
Since the  s t a t e  and control  weighting was not changed, 
-10- 
This study has proven the f e a s i b i l i t y  of using optimal s e n s i t i v i t y  
design techniques t o  design a control system fo r  a large,  f l e x i b l e  booster. 
Work i s  current ly  underway t o  use the techniques on a more sophis t icated 
vehicle model where, f o r  instance, s t a t e  information is  not d i r e c t l y  ava i l -  
able .  Further development of the Feedback Gain Autonomous Input Technique 
i s  a l s o  underway and t h i s  is expected t o  give even b e t t e r  r e s u l t s  than the  
Cascade Gain Configuration. 
CASCADE GAINS SYSTEM 
FIGURE 1 
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FEEDBACK GAINS- PREFILTER SYSTEM 
FIGURE 2 
0.2 
- 
-0.2. 
NORMALIZED BENDING 
SENSITIVITY WTG = 0 - 
.08 rTCH (radians)  ANGLE 
FIGURE 3 
SENSITIVITY WTG = lo5  1 
4 PITCH ANGLE 
FIGURE 4 
DEFINITION OF SYMBOLS 
F 
K 
1 C P  
M 
M1 
N' 
43 
R '  
U 
v 
V 
X 
X 
y(x,) 
Z 
ot 
ocw 
P 
4 
f , 
@I 
Total  thrust of Booster 
Amplifier gain 
Distance from engine gimbal t o  vehicle center of gravi ty  
Distance from vehicle center of gravi ty  t o  center of pressure 
Total  vehicle mass 
Generalized mass of f irst  bending mode 
Aerodynamic force 
(unknown) parameter 
Control th rus t  
Autonomous input 
Reference input 
Vehicle veloci ty  
S ta t e  vector 
Drag Force 
(output of p r e f i l t e r )  
Normalized displacement a t  engine gimbal 
Sens i t iv i ty  vector 
Attack angle 
Wind induced a t t ack  angle 
Engine gimbal angle 
Atti tude angle 
F i r s t  bending mode damping 
F i r s t  bending mode amplitude 
F i r s t  bending mode frequency. 
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REFERENCES 
1. "Model Vehicle No. 2 fo r  Advanced Control Studies" NASA- 
MSFC In te rna l  Document. 
2. Fisher, E. E. "An Application of the  Quadratic Penalty Function Cri ter ion 
t o  the  Determination of a Linear Control for  a Flexible Vehicle" AIAA 
Journal, Vol 3 ,  No. 7 1965, pp. 1262-1267. 
3 .  Tomovic, R. Sens i t i v i ty  Analysis of Dynamic System, McGraw-Hill, New 
York, 1963. 
4. Dougherty, H. "Synthesis of Optimal Feedback Control Systems Subject t o  
Modelling Inaccuracies", Ph.D. Thesis, R.P.I., Troy, New York, February 1966. 
5. Pagurek, B. "Sensi t ivi ty  of the Performance of Optimal Control Systems 
t o  Plant  Parameter Variations" IEEE Trans. on Automatic Control, Vol. 
AC-10, No. 2, Apri l  1965, pp. 178-180. 


Sensitivity design technique for optimal control, part C Final report

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