•

..

NASA-CR-175369

19840009104

A Reproduced Copy

OF

•

Reproduced for NASA

by the

NASA Scientific and Technical Information Facility

LIBRARY COpy

..:.' !_ :1. ..' 1984

LANGLEY RESEARf:H CFNTER

LIBRARY, NASA

HAMPTON, VIRGINIA

FFNo 672 Aug 65
• DYNAMICS AND CONTROL OF

FORWARD SWEPT WING AIRCRAFT

•

Final Report

Grant No. NAG-I-305

Project Period: November 1.1982 - December 31.1983

Principal Investigators: Dr. David K. Schmidt

Dr. Terrence A. Weisshaar

School of Aeronautics and Astronautics

Purdue University

West Lafayette. IN 47907

NASA Tecnnical Officer: Mr. Jerry R. Newsom

NAOS. MS 243

loads and I\eroelasticity Division

NASA Langley Research Center

Hampton. VA 23565

February 10. 19£\4

(~ASA-~H-J75j69J U¥NAM1CS AhC CCHThOL Ck NSQ-17172 ~Ch~AUU ~WEPr WIN~ AlhCbAfl kinal U~poLt,

11OV. J:)8.d - 31 Dec. I'Jti3 (l;urtlu(~ Ulliv.)

43 p W": AOJ/l'Ir AO J C':;CL 0 Ie Ullcla::>

GJ/05 HJ274
Table of Contents

Page

1. Introduction

2. Personnel.

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

1

1

3. Summary. . . . . . . . . . . . . . . . . . . . . . . . . 2

Appendix A, "Aspects of Non-Zero Differential

Game Theory with Application to

Multivariable Control Synthesis" . . . . . . . . 3

Appendix B, "Optimal Linear Control Law Design Using OptiMum Parameter Sensitivity fI.nalysis.••.•••.••••.••.••••• 16
1. Introduction

This constitutes the final report on the research being performed

by the School of Aeronautics and Astronautics. Purdue University. for

the NASA langley Research Center under grant number NAG·1-305. The

area of research is multi·input/multi-output control synthesis techni·

ques n~tivated by applications such as forward-swept-win3 aircraft.

which exhibit significant rigid-body/aeroelastic modal coupling.

2. Con~nent on Personnel

A major portion of this research was perfor~ed by a doctoral

graduate student (Mr. Mike Gilbert). while the student was in resi·

dence at \.l1e langley Research Center. This was an important element

of this program, providing the opportunity for this \'lOrk to actually

proceed "in situ" with the graduate researcher interacting daily with

NASA's international experts 011 control of aeroelastic phenomena.

In addition to the research reported herein, the graduate student

actively participated in experimental evaluations of several candidate

control laws for stabilizing a dynamically scaled, statically unstable

wind tunnel model of a forward-s\vept-wi'1g aircraft tested in

langley's transonic dynamics test facility (TOT).

3. .Surrmary

In the early phase of this research. the potential of cooperative

game theory for multi-variable control ~ynthesis was briefly explored.

and a summary of key concepts are attached as Appendix A to this report.

The key conclusion of this survey was that if a multivariable (multiinput/multi-output) control law is to be synthesized to be cooperative.
2

this corresponds to a Pareto-optimal rather than a Nash solution to the

mathematical game, and the Pareto solution may be found via Linear

Quadratic Regulator control theory. On the other hand, if, for example,

more than one interacting controller is not being synthesized by the

same "designer." the optimal solution is the Nash equilibrium.

A second method of modern control law design was proposed that

addresses the problems of selecting the cost functional which the

control law is to optimize. and the lack of other useful design information. The approach recognizes that in any optimal control problem.

there are many more design parameters to be selected than the gains of

the optimal control law. These additonal design parameters may be part

of the system dynamics or they may be part of the optimal control

problem formulation. Either way. by calculating the sensitivity of

the dynamical system time and frequency domain performance to these

pa rameters. systematic ways of aHeri n9 these parameters to improve

the pe~formance of the control law can be developed. For example. if

the parameters are selected as the nominal values of the poles of a

compensator, the compensator design can be adjusted to provide better

performJnce. Similarly, if the parameters are entries of the optimal

control problem cost functional, then the control problem itself can

be altered to meet time domain design specifications. This technique

titled the "Optimum Parameter Sensitivity" approach is presented in

Appendix B of this report.
Appendix A

3
4

Aspects of Non-Zero Sum Differential

Game Theory with Application to Multivariable Control Synthesis

M.G. Gilbert

April 11.1983

Introduction:

Modern. optimal control theory and differential game theory (1-4)

were developed concurrently but indepenaently during the early 1960's.

Both are concerned with obtaining optimal control strategies (open or

closed loop) for n~lti-input systems describable by a set of differential

equations. Optimal control problems are characterized by a single controller

using a control law picked to minimize a single scaler objective function.

Non-zero sum differential games on the other hand involve several players

(controllers) each attempting to control the system to minimize it's own

cost function in the presence of the control actions of the other players.

Thus differential games are characterized by multiple controllers minimizing

multiple cost functions. leading to a vector minimization problem in

order to obtain optimal control strategies.

Modern optimal control theory has been studied extensively and has

found application in developing control laws for state regulation. terminal

guidance. and process control. Much less is known about differential

games. whose prima~ applications have included pursuit-evasion studies

and economics problems. It is generally recognized that optimal control

can be viewed as a subset of differential game theory. as will be apparent

in the linear, time invariant, game to be discussed in the next section.
5

control set U are needed. There are three widely accepted definitions

2}

B.u. 1) 1 1

m

x=Ax + L

;=1

of when a particular control law solution setU* is optimal in differential

system. Because such a control set cannot usually be found. definitions

of the sense ;n which a control set U* is optimal with regard to another

t:,

In general. it cannot be expected that a control strategy set U=

lUl' ..•• u

m} can be found which will absolutely minimize every player's

cost function in the presence of the other player's control actions. If

such a control set exists. it could be found by solving m uncoupled

optim~l control problems with each player independently controlling the

Note that the ith player may be penalized for the jth player's use of

control energy.

It is assumed that each player or controller picks his control ui to

minimize a quadratic cost function of the form

mathematically as

theory. The discussion which follows pertains particularly to linear.

time invariant. non-zero sum differential games which can be modeled

with optimal control theory than differential game theory; for that reason

this section will highlight some important features of differential game

Some Aspects of Dif~e~~Jial Ga~e Theory:

Generally, the average control law designer is much more familiar
6

This is because the min max solution assumes irrational behavior of the

4) * -1 T u. = -R .. B. S.x

1 11 1 1

incurring increased cost if all the other players use their own Nash controls.

Necessary conditions for the Nash Equilibrium solution for the game

gi ven by equations nand 2) are

where J * * i is given by 2, ul ' ... urn are the components of the Nash control

set e* * (u * l ' .. urn ), and ui is any control strategy other than the Nash

control for the i

th player. The Nash solution is a stable equilibrium

solution, since a player cannot deviate from his Nash control without

Of the three solutions. the Nash Equilibrium and Pareto-opt'imal are

the most useful for the development of control laws satisfying the game.

cally, the solution assumes the m-l other players are ignoring their own

costs and using their control to maximize the ith players cost. The game

then becomes zero sum (the ith player minimizing his cost and the m-l

players combined maximizing it). The solution is overly pessimistic

and may fail to be finite for the ith player even in \'Iell posed games.

Alternatively, the Nash Equilibrium solution assumes rational behavior of the player~. The jth player's Nash Equilibrium control minimizes

his cost function when the other m-l players play their own Nash,controls.

Mathematically, this is stated as

. ' other m-l players when solvlng for the ith players control law. Specifigame theory. These are the Nash Equilibrium solution, the min max solution,

and the Pareto-optimal or non-inferior solution.

...
7

where 5i are solutions of the mcoupled Ricatti like equations

5)

[ . -1 -1 T S.B.R .. R..R.. B. S. J J JJ lJ JJ J J

-1 T -1 T ] -S.B.R.. B. S. - S.B.R .. B. S. 1 J JJ J J J J JJ J ,

T m S,. =-5.A - A5. - Q. - L , , 1 j=l

Unfortunately, these equations are difficult to solve and only a few

sufficient conditions for the existence of solutions to the equations are

known (Ref. 5). In the single player case (m=l), the equations 5) reduce

to the single Ricatti equation of optimal control theory.

The Nash solution, stated mathematically by 3), is an uncooperative

solution since each player is only concerned with minimizing his own cost,

and cares nothing about the other player's costs. The question then

becomes whether or not other solutions exist which simultaneously reduce

the costs of the players from their Nash costs. Pareto-optimal or noninferior solutions to the differential game may be solutions with that

property. In fact, Pareto-optimal solutiJns have the property that

or there is at least one ;=1, ... msuch that

6B)
78}

7A)

m

1, ... , m. and I a· = 1.

i =1 1

m

i=l •...• m. and La.= 1

i=l 1

C(("3~!~t.l .. F,\~~'.! ;.~J

OF POOR Q:.J"~L1rf

> 0

are the components of the Pareto-optimal control set

and u

l ••..• u

mare controls other than the ParetoThere are many solutions satisfying 6A), 68). some

!J m

where J(1/J) I a.J.(1)J) , a· > 0 If i = i=l 1 1 1

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

J(1jJ*) < J(1J;)

or

!J m

where J(1/J) = L a.J (lji) i =1 1 k

*ror the linear. quadratic cost game. it is believed all could be found.

however unrealistic it may be to do so.

8

players are faced with a negotiation in order to obtain a solution, or

else some level of cooperation among players must be enforced.

It has been shown (6, 7) that some* of the non-inferior solutions

to the vector optimization problem can be obtained by solving an m-l

parameter family of optimal control problems, if the cost functions

satisfy certain convexity requirements. Mathematically. this is stated

as follows. The control set 1J;*(ul

*.... , u

m*) is Pareto-optimal (noninferior) if

of which may have costs for each player lower than their corresponding

Nash costs. Selection of a particular non-inferior solution in all cases

involves trading off the costs of one player over another, thus the

where u * * l ' •..• u

m

~. *(U1*. ...• u

m*) ,

optimal controls.
9

which occurs at the intersection of two cost contours. The cross hatched

8)

9}

T B. S(a)x ,

where S(a) is given by

. T m m

[" ] -1

S = -SA - A S - I a.Q. + S I R I a .R .. B.TS

i=l ' , i=l Vi j=l J J' ,

referred to previously.

For the linear, quadratic cost games of equations 1) and 2) (the

Ji of 2) satisfy the necessary convexity requirements, see Reference 8),

the Pareto-optimal solutions are

ORIGiNAL l"J~~'G U

OF POOR QUAtrry

By 7A, the sets ~/* which give the unique minimum of any individual J

i

is Pareto-optimal. Note that selection of a set of ai' thereby defining

a particular solution, is equivalent to solving the negotiating problem

ul ' u2 control space, as shown in Figure 1.

The short-dash lines are the min~mizing control for one player

with the a. satisfying the conditions of 7A), i.e. (X" > O. J -

There is a geometrical interpretation, first given by (3), which

when the othel' plays every other control in his admissable control range.

The intersection of the two dash lines is the Nash Equilibrium solution,

area represents admissable controls u

l and u

2' which if played would

is useful in understanding the Nash and Pareto-optimal solutions. Consider a two player game, with contours of constant cost plotted ;n the

'-- .........,.,.....,~-~ - .............,.- ,.~'''-'~.~''"'-' .....,_;;op_....-.....;:s....,.... ' ",",.4,,,"".Z.........PO,"""'>'''l'',0''''."'''''''''-'",.... ,'_0Z4A1lIII.IltIlWlWlZ'""",,_,"!,,,"-":;"",WWlllll61"P!l!lllG$"'!IIIIIIIl$@!II\Il#a!Jllll~!lI'!I, 1411l111!,!IIIII!IllJ. . ,~,., . '-. '-'
10

Figure 1

-6.. ....... ;- .......

'"''' ... ~f:-- .,. J OmGiNri.L t "'_.~ :..~

OF peOR QUALII •

(,(.,

result in reduced cost (from the Nas!' cost) for both players. The longdash line, which follows the tangent points of the cost contours, are the

Pareto-optimal solutions. These are the minimal cost solutions for each

player. They require cooperation, since both players must trust the

other not to playa minimizing control in response to the first's use

able to cheating. For example, say player two uses a control on the

of control in the shaded area, and therefore these solutions are vulnertangency line in the cross-hatch region in anticipation of player one

using his control to arrive at solution A, but player one cheats, playing

his minimizing control resulting in solution B. This is the so called

-------

..
11

"prisoners clilemma" (1), which most differential games, particularly

linear games with quadratic costs, have. Note too what would happen if

the Nash solution were also non-inferior; the non-inferior solution

would be stable and invulnerable to cheating. This desirable situation

does not occur in general.

~iltion to Control La~l Development:

The last section discussed the types of solutions whi, occur in

differential games in general, and· linear, time invariant, quadratic

cost games in particular. There are ~any more important properties of

these solutions which were not discussed. Rather than devote much more

time to these important properties, the potential of differential game

theory to improve control law development for mu1ti-inputlinenr systems

will be discussed.

Optimal control· law synthesis for multi-input systems has in the

past been primarily performed using modern optimal control theory, that

is by minimizing a ~~~ scaler-valued quadratic cost function. Thi~ is

because modern control ~heory takes advantage Of powerful matrix 1.\2thods

for algaebraic manipulation, because most control system desigr.s have

in mind a single overall objective, and because of the difficulty of

obtaining solutions to vector-valued optimization problems. For example,

primary aircraft flight control systems have as their main objective

improvement of the aircraft handling qualities. In those cases where

several systems were desired, sayan automatic fl igl;t control system

and an active structural control syst<?m, the nature of the system plant
12

(aircraft) wa~ such that the systems might be designed separately. These

are cases where a control set uniquely minimized both cost functions.

With the ilicreasing complexity of aircraft design has corne instances

where several control systems could no~ be designed separately, the

Grumman forward-swcpt-wing (FSW) demonstrator being an example. Because

of the coupled rigid-body and structural dynamics of the FSW configuration, the separate flight and structural control systems of that

aircraft have interacted unfavorably with each other. This is a case

where an integrated approach, satisfying two different. broad, overall

objectives and making use of differential game theory solutions might

lead to better control laws for both systems.

Of course the question remains as to what is meant by better? All

differential game theory has promised is that the individual cost

functions, which mayor may not have physical significance. will be

minimized taking into account the control action of the othel'. The

important physical properties of these solutions, like closed~loop

eigenvalue locations, robustness properties, frequency response, etc .•

remain unknO\'Jn. On the plus side h.wever, there are now more solutions

(control laws) to evaluate which at least have known properties among

several scaler parameters than there are with an optimal control solution.

There exists at least one eXilmple \~here a Nash equilibrium solution

(while pot explicitly labeled as such) was proposed for control system

development (9). The methodology attempted to account for the control

action of the aircraft pilot in the design of an aircraft longitudinal
,

.,

•

i

.

I

13

stability augmentation system. The pilot was assumed to be an optimal

regulator. leading to a two player. linear. quadratic cost differential

game. Properties of the solution like eigenvalue lecation were not

investigated as the scaler cost of the pilot was directly related to

the Cooper-Harper handling qualities rating scale. the primary objective of the methodology being the best (lowest) Cooper-Harper rating

possible. This example does serve however as an incentive to investigate the potential of differential game theory for control law design.

since it was advantageous to use the theory in this case.

QuesJions:

The previous section has vaguely called for an "investigation" of

differential game theory in the context of multi-input control system

development. Some \'Jork is known to have been done in this area (10).

The author is puzzled though that apparently a lot more has !)ot been

done. Either there exists unknown work which dispels any advantage of

game theory in these situations. or else they have never been considered

due to their complexity. the lack of a need to integrate several systems,

or whate~er. If the latter is the case. then enough questions arise

about the differential games solutions and their applications to advise

many investigations. Some questions are listed here.

1) Only a few sufficiency conditions for the existence of Nash

Sollltions are known, given in terms of the norms of defined

matrices when either a) the system Amatrix has a prescribed

degree of stability, or b) the solutior. to certain auxillary

I

lIp;... r - t • t
14

control problems exist (5). Are there conditions on the

weighting matrices of the cost functions (like observability

and controllability concepts from optimal control) which

detennine directly if a Nash solution exists. rather than

through the solution of an auxillary problem?

2) Are there weighting matrices in the cost functions such that

the Nash solution is also non-inferior? Such a solution would

have many desirable properties. In addition. the mathematical

question of when a set of coupled Ricatti like equations could

be solved by a single Ricatti equation would be answered.

3) Are there enough relationships between the individual cost

functions and the physical properties of the closed-loop system.

i.e. eigenvalue locations. frequency response. etc .• to justify

considering Pareto-optimal solutions over the Nash solutions?

If so. can the problem be further reduced to a simple modern

control problem with pre-defined structure? Or is the Paretooptimal solution just another way of picking weighting matrices

in modern control problem cost functions. similiar to methods

used in Bryson and Ho and Kwakernaak and Sivan?

4) What kind of solution is a linear. quadratic cost. multi-input

modern control problem in the context of differential gar.1es?

Are the individual controls in the problem in a Nash Equilibrium.

are they cooperating in a Pareto-optimal s~nse. or are they

in a "prisoners dilemma" situation?
""

15

1. Starr. A.W .• and Ho. V.C .• "Nonzero-Sum Differential Games",

Journal of Optimization Theory and Applications, Vol. 3,

No.3, 1969.

2. Starr. A.W., and Ho, V.C., "Further Properties of Nonzero-Sum

Differential Games", JOTA, Vol. 3, No.4, 1969.

3. Ho, V.C., "Differential Games, Dynamic Optimization, and

Generalized Control Theory", JOTA, Vol. 6, No.3, 1970.

4. Leitmann, G., "Cooperative and Non-Cooperative Many Players Differential Games", Monograph of Courses and Lectures No.

190, CISM, published by Springer-Verlag, 1974.

5. Papavassilopoulos, G.P., Medanic, J.V., and Cruz, J.B.• "On

the Existence of Nash Strategies and Solutions to Coupled Riccati Equations in Linear-Quadratic Games", JOTA, Vol. 28,

No. 1,1979.

6. Zadeh, L.A., "Optimality and Non-Sealer-Valued Perfonnance

Criteria", IEEE Transactions on Automatic Control, pg. 59,

1963.

7. Klinger, A., "Vector-Valued Performance Criteria", IEEE

TrJnsactions on Automatic Control, pg. 117,1964.

8. Bryson, A.E., and Ho, V.C., "Applied Optimal Control", Book

published by Hemisphere Publishing Corporation. 1975.

Distributed by Halstead Press.

9. Schmidt, D.K.• "Optimal Flight Control Synthesis via Pil~t

Modeling", Journal of Guidance and Control. Vol. 2, No.4,

July-August 1979.

10. Innocenti, M., "Cooperative Pilot-Optimal Augmentation System Synthesis for Complex Flight Vehicles", Ph.D. Thesis,

Purdue University, Aug. 1983.
't\'

.. '~" l

r;:: •

'. ;

Appendix B

16
•

:

. ,----t-.

OPTII'!(IL LlNlAf\ CON1ROL LAI4 DESIGN USING OF-'T IMUM

PARAMETER SENSITIVITY ANALYSIS

by

Michael G. Gilbert

•

A Propo~al for Doctoral Dissertation Research
.;

I - HlTHOIiUC T I ml

The use of feedback is a well known and effective means

of al ten' i I1g the dynarni cs of a ~ystem in order' to improve

errors, ~nd meet

perfurm~nce specifications. The design mclhQd~ u5~d to

stabillty, reduce sensitivity to model

Cl~~ssical methuds provide system3tlc design infor-rn~·tion th~,t

the

Modern

to refl £Ie: t

in nat.ure.

law5 can be broadly

classical

The disadvantage of modern methods is the

They ar'e 1ess ttl.:!." l~a".:iY to LIse if the

into two c~tegaries,

on the other hand, determi ne high or'ck'r contr'ClI

S'lst~m states.

methods,

laws easily, and they optimize a quadratic functional of the

difficulty of writing the cost functional

possibly with thelr own dynamics,

is u':.>E.'d by the de'3iqncr lo develop ;:;ingle loop cuntrolleni;,

spec if i (:<:It ions.

sy~tem is rnulti·,inp·..lt/mLllti-o'_ltput

cl..ls'3ifled

develop these clo5ed-Ioop control /

;

,~

performance specifications qf the system, and the la.,ck of

systematic redesign information if the original deslgn is

ullsatisfllctory.

d(~v(~loped I~hich dddr-e~5es the problems of \~rltiIl9 th~'? cost

functional which the control law is to optimize,

A method of modern control law de~.>ign has been
dc~sign p.w""me1e,.-s 10 be selected than the gains of the •

OR1GlrML F',l\·~r~ ;~

OF POOR QUALITY

I ..\C l.: 0 f

'"\

-

The method reL:uqIl1::e!l

therE! ",..e many more

optil1l~l cOflt,..ol law. These add i ti onal' desi gn pC\ran.e1 ers ma.y

be p~rt of the system dynC\mics or th~y may be part of the

optimal control problem formulation. Either way, by

calCUlating the sensitivity of the dynamical system time

domain p~rformance to these parameter~, systematIc ways of

",,1 tering th~?c.:H? parameters to improve tho perform;lnc~ of the

control l3w can be de~eloped. For e:< Clmp 1e, the

par _,mot.el·s eire sel ected as the nomi nal va.l t..les of the pol ~'S

of a compE:.'n~c.,tor, the con,pE.:nsator desi gn can be adjLlsted to

prOVide better perf~r·mance. SimilC\rly, if the parameters

are ent.l-ies of the opt1m<",1 control problem cost fLlnetional,

then the control ~roblern itself can be altered to meet time

domain design specifications.

I.a - Bacl.:gro'_lnd

The concE'pt of sensitivity of optimum solutions to

problem parameters has recently been developed by Sobieski,

Barthelemy, and Riley [ll, and Barthelemy and Sobieski [2,3l

in the field of parameter optimization theory. This concept

differs from the tradition~l idea of sensitivity analyses in

that the results ar~ used in the design process rather than

thf.! per-ftlrm,;\lIce assussrnc'r,l stage. Wtlat thi s means
""I teri ng pr·~'3p.I ected parameters of the probl em (parameter's

development, sensitivity analyses were used to determine

.....,

to this

and Riley

uncontrollable

Prior

Barthelemy,

The optimal sen~itivity

to arbitrary,

optimization>.

iu used to redefine lhe or1ginal

ORIGINAL PAGTZ [8

OF POOR QUALITY

changes

pertl.\rb.:ltions in ,;;ystem parameters.

derivatives calculated by Sobieski,

per- formance

wplimi=atiun probl~m so as to improve performance by

not selected by the

I

w~rc obt~ined by differentiating the necessary conditions of

opt. i mal i ty for non'-l i nei1r programming problems with

inequality c:onstr.:\ints. One of the most interesting

~pplicatjons of the optimal sensitivity results has b~en in

tht? development of multi-level optimization schemes for

large structure.parameter optimi:ation problems (Sobieski,

James, and Davi [4]).

Sensitivity analyses are commonly used in both

r.:l.:l'5sical and moder"n control theories to assess the effl:cts

uneontroled variations in the system parameters will have on

sY~itC!m performance, primarily in sti:\bility margins. Methods

have been daveloped by var i OLIS r·eseull'"chers (e. g. Yedavalli

and ~'I::el ton [5]) for designing control systems which are

insensitive to parameter variations. Only recently, time

response scnsitivit)' W~~ irltrod'~ced as a means of assessing

system perfcrmanc~ changes to parameter variations by

Sch<..'.ecllter of these appl1cations of
OutP'.I.t feedtJ~".t:k theory \'laS developed by Levi ne and Athans

linear n~gulators (Ha.rvey and Stein (7) and Stein (8).

design.

selecting

4

the design

methcJds for

include

reduced order compensator

These

Then also e:: i st sever al

objective~, be~ides the commonly used

ORIGINAL PACt:: ~~

OF POOR QUALITY

system

a.nd frequency shaped linear regulator

output feedhack theory,

Other a.re~s af control system design worth ffientioning

design,

intuition of the designer.

weighting matrice5 on the ba.S1S of asymptotic: properties of

prwces~ howev~r.

defin!n!) t.he 1,l(J<:h~t'n contrlJl theory problem '1>0 as to meet

certain

./

/ ,.

•

(9] and solves the problem of feeding back fewer outputs of

the system than there a.re states. Reduced order compens3tor

design is receiving attention because it recognizes the

actual contr"ol la,,, str'_lct'..'re that is usually imple,nented on

working hardware. One method of

compensators has been

designing reduced order

parameter optimization methods

(Mukhopadhyay, Newsom, a.nd Abel [10]). Frequency shaping

techniques of linear regUlator design were developed by

Gupta [Ill, by making the weighting m~trices of the optimal

c:ontrol problem functions of frequency. These a.reas of

control theory are noted because they have the potentlal to

m",.I.I? use of optimal sensitivity analyses as will be

1Jt'" !! ..... ~.Jnted in thi S pi.\per.
tlw

First.

TCoIylar

improve

of

solved.

squ-",re,lo

Thi~ is to be

is a connection

or me ..,n

problem being

ORKilNl-\L ri\GE .7.;i

Of POOR QUALIlY

selected ~~ part

The result

contr 01

These sensitivitIes are then to b~

5unsltivity to problem parameters to

021 thc:.·r·

opt I rr.'.\l

this connection, it is possible to specify changes in

i df:~a 0+

thl~ ',;c?llslti'vlty of the net.:~~sscH-Y condition for the conlr··ol

tllnc resporl:-;e is comp,-'.tE!d .:\s a funct.Ion of the sensli:lvit'y'

series, .3I1d.the optimal control pt·oblem resolvC?d, leaclin~J to

du,n)i.n

With

the

optimi~·:",tIUrl pt·CJCI~5S.

bet\oJeen

p~u"'a/llL'ter·s of the problem not

to be optlm",l is obtained, and then the st=f.sitivity [Jf the

done by COillpo.d:lnq th~ ~,ensIt.lvity Clr derivative of the time

develop and usc sy~lematic deSIgn data to further

which speLify the optimal

contrcll~d system time dom~in performance.

used to r '.'def i ne the par "".rn€.'t~rs

the designer selected parameters using a first order

of thE~ necess;:>ry cnndltion.

/

:/

..

a two stage optimal control design process.
.. 1 1- t'1l: THUDI)I IJI.),'

ORIGINAL PAG£ IS

OF POOR QUALITY

This ~~clion pre~ents a discussion uf tile LISE- oi

parameter 5~nsltlvity analysis in the dDv~lapment of lin~ar

for eIther

the time domC'in

nlethod

sensitivity r'esults <;,"d

it systemat i c

The fall owi ng sLtbsect i or: loJi.ll t.hen

In this section,

Tile derlvation of the sensitivity of these

pa.racnetlH'

ttle first sLibsection, follol;lcd b't a Y"t?'11'-':W of

linea.,... regL\IDtor t~l(?ory.

t i mt,! ru'.=,ponsI.?s to a.rbi trary parameters of the d y'n~ml c·~l

regul~tor theory tu give

S·/St.~lh or to p.:>.rame+':'cr"" of the control 1 aw f orm'..112.t 1 on wlil

r~gulalor control law••

,,",re ur intert=~i...

sldte, oulput, ~ncl control responses, both absolute and RMS,

d,:?f i r1 i ng th·:? linei:u- regular problem or <J.ltering the syste.n

t.::!',namic::. to c.<c.hieo'vc de;"ired time domain respunses. This

s'..:ction ,..,ill tllen tH::! conclLlded by a discl_Ission of pra.ctical

OLltput, ":•. nd

the case of

st.ate,

f (Jr'

ti.me drJmain

1~1 \'>1 f ot'lnul dt ion,

TI1~? sen'3i t, i."i ty of

II.a - Paramet~r Sensitivity of Time Domain Re~pon~e5

tim8 domain d':?sigr1 ct'iter'ia.

aspDCt.s of this methodology, as it applies in

\
The dflr 1 Vi-It i Dn begi r'5 by consi deri ng the state SfJac.e model

system is d~rlved 1n this ~ub~uctjon•

•

.. i nv~.r i Clnt dyrt'-'l.mi c:.;>1

ORlGiNf,L PAm::,: i::]

OF pooR QUALlTY 7

of the system in the form

;< = A" + Elu + Dv

(11. U

wher"-? A is def i ned as

an

the

is a

(11.2)

is

(11.2)

Tho m.:>.tri-:.es A,

disturb~nce vector WhlCh

Assuming thiJ.t a linear, full-·st~.te

control vector, y is an I-dimen~ional output

is an n-dimensional state vector, U

and D are constant coeffici~nt matrices of

The closed-loop system given by eqn.

A = A _. BG

then the closed-loop dynamics of the system are given by

approprlate dimensions.

B, C,

fe2dback control la\1 e::ists and is speci.fled as Ll = ·-G:~,

may be either deterministic or stochastic.

vector, and v is a k-dimensional

y - C:<

Ill-dimensional

where :<

-function of the original model dynamics matrix A, input

matrix B, and the gain matrix G. Each of these matrices may

themselv~s be functions of parameters which may vary due to

J natural process of the system, may be imprecisely known,

~)1' may be:' specifif1d by the de~,igner. For p::ample, the

f:.'ntrles of t1w A matxi:? indY be subject t.o llIod.?linq UITOr,.,
t.ht!

ORIGINAL PAGE rsJ

OF POOR QUALrN

/

/ •

•

(JL1tPI_Ii: •

the closed-loop sy~t.C''11 (II.:?; in tttu

foiLowin9 m"",Il11er.

Thl! t.im'.! res\Junse of thE' '_losed"loop system <II. 21 due

tu U,H d1:.itlltb;:\IlLI.! ... is wf211 known and is given 1n term,;; of

<11.4)

,) f the

matri::

transition

and

(t ,rr) Dv (t) d't:

) nt~~CJr<.',l (1:?] <),5

.~ :";J~ :: ( t ) ;~ (t. • t ,.) +) _1 -~ ~+

, "0

~'1IH!re -:l}.(t:..t\) !''''

In

to calculate th~ :;ensitivity of the state tu arbitrary

p=U'il'flet12r",. it is cd COLlrSfJ possible t.o diferE:'ntiatE..' eqr••

<I1.4i wi th respE..'ct to lIlo'::.e parclmeters, but bec:=-.usu the: A

mo.tTi:; of eqn. (II.:D 15 in yeneral not symmeLt· ie, the

difer-cnti~tion will prove difficult. The aiternat1vQ 1S to

differentiate eqn. (11.:2i first, and then the

F'roceed11l9

• O·", ,JA "".# _(1"

.. .. -I- A--

'<lr '-'p ~p

\'lll~'rf! p l~ th':.? arUllrar'y pc:,t·AmE,~ter. end It, has beef'

':lnrj

1ndepcnclant
1'3

will

<11.7)

<11.6)

(11. :-j)

(11.6)

of

eqn.

For' thco ",ystem contl"als,

CRluIN,'"·.~. FM.l' :'J

OF POOR QU~Lrrv

in1tial state of the system will also be

J

-l -

-a A

+ il (t , "tl - -,d\:) d 1:

t. ~p

•

to problems parameters is obtained using

tho

to th<.tt of <11.4), wllh cqn. (1I.~) driven by the

5L![)·:.it.ivlty

to c~lculate these qu~ntit\es ~s well, since both

The i\bclVt:.! resL'l ts gi ve the s(m~i t i V1 ty of the sti\t.e to

tit•.'! sensitlvity

In

problem par-atnetcr<3; if the senslt.ivity of the system OLltPLlts

LI •• (3:: i\5

.and the senSl t i vi tv of the system OLlt.puts to Ule pari\mc:ter '3

or cont.t'ols .are uf lnterest, tho reSLllts of eqn. (11.6) can

i\r~ lin~ar functions of the state.

is

dY dC .....

<1"

"" --':< + C-- (11.8)

Jp dp ~[)

reduce to th~ c~lculation of the integral term only.

·..»t:~tl: of tilt? ~1')'3~d-loop uyc.:.t~m;

./

/

•
/ ORIGlNt\t P;~G~~ f~

OF POOR QUALITY

1')

'lint· Il".:.oP':'fl'.;I: \.~ e,f lllt.t!r",l,. the>

c I '.l'JL·I.l-l oop

• eql\~tlon'S. Ttlt-, c.o"'.:-r 1 <~nc:E.' r c~pon~e of tt,l'.' ~y~.t(tnl gi ven by

eqn. lIr.:!i t.o t.rH~ dl~tUI'b~IKU v iv. [13]

• - - T T X c AX 4 XA + DVO ;X<t O) = X~ <11.9)

",her,! X i'i dufllll~d ~'j ttw !;)'stem cov.."ri",nt:e m~tri:,:, A is the

"'UC lnr' , or 1.hl~ ~quc\,fe UH.turb",rlcQ if v is deterministic <i 1

s l!n 1 t i \' it." 0 f tt,e uy~t.i·111 cova, 1 ",rICe to the par'am~t'2t- p 1 S

'\ X· o dX c: "A--

ap it p

-T .jA

x--

dp ( I I • 1(I j

wher-e it hat> again been ~~,~uff,ed th':'lt the dlstLwbc3nce and the

way it enter'S thl? ~'y'stem C.'re i ndc.'rH.mdent uf the param';?t<~r's.

Noti ce that eqn. <II. 1t) hC\s lhe sC\me form as eqn. <I I. 9) ,

and cont..:\ins the system covarian'::e (~::plicitly.

If the closed-loop control law yields an asymptotically

~t<3ble systl?m (all the elgenvaluus of (11.2) strictly in the

<11.9) , for

I: h I"! C <;.'.'.:;e of

(ll.tli
:~

;

~.' :

_;.ox 'd)(..T ~A (I = (\._- .. --A of --~ +

~p dp ap

... , .. fI: .. :t

~.t·""l r··.....;,r. ,-.; OH~i"'l ':/<1..' \Tt

Of pOOR QJA\.

~qu"'tl0" (1!.Il) WIll

[6] gIven by

dAT

x--Cp

1 1

<11.1::>

;.

\ j.

,~

Ii

,

Th~ ~ensit1vitv of the control and output covarlances

~ra ubt~ined from the state covariance sensitivity 1n the

follm'l1f,CJ mi\nner. The control cQv,~riance is, in tel-ms c,f

( I 1 • 1.:. )

[hfftH'lmtlating eqn. <11.13) WIth respect to the p~'r,-l.llletL'r p

gives

';)U dG T J X T = -"XG + G"-G ..

dP dP :>p

<11.14)

The sensItivity of the output covariance (13J 1~ ubtaincd

v = CXCT (11.15)

..lnd

~y de "3 X ....

= _·-XCr .. C""C' +

~p Op

<11.10)

II.b - Optimal LInear Regulator Theory

1 S prt:~(.'ntl!d fur llJ(~ infinitl! lIme, LUI\~"t."d,t
Tlli,:> ~,i 1 1

",itll scnsitn/ity

ORIGINAL PAOF. m

OF POOR QUALIrv

the fli-'::t

C(h~f f i L: 1'!( ,I.• I 'J 1 1

lie u<:i(.'d III

!./'

I

"

..

• reslJl ts lo dcvt!l op .;~ m~tlloojolog':I of control l~w de'.:oiqn to

meet lime dom~in responue critDri~.

Con""ider ag.:.\ir. the open-loop dyni\mic.:\l system given by

!:itated as a quadr~tic.: fLlnctlon of the states. and cont.rols,

LD

ThE;.'

The

the cJesign can be! generall::E..'d to

Sl\ch that the utc:,les of the '.:J)'::.lr.:m

by using th~ output equAtion in (11.1).

It 1~ desir~d to find a ~late feedbat~ control

the objective of

is the standard linedr. quadratic regulator.

l,.w. t:h.J.t. IS. c;l cuntr'ol 1J01icy \~hich is i.\ h.l.netion Df the

uptimal soll.Jtion for lhis probloln, when the objective i~

-.it'\'1 <:10;:'0 to ::eru In the presence of the dislurb"mce v. In

fact.

eqn. (I I • 1 ) •

regul~tor problem is formulat~d as follows.

A scaler cost funct.ion",l of the outputs and the

t _

\0 -

controls is formed as

J(y Q'y + U F\l.\}dt

()

<11.17)

"Iher r~ O' and R are designer ~elected weighting matrIces

whIch reflect the importanc(~ of tho responses to ti,e

the R ,".:11-1"1:: must be positlvu dullnitL' (14),

f LHH.: i~ 1 un cd
..

~:

I

!

!

I

t

/ OklGIfMt PN.~~: r~7

OF POOR QUALlTf

i\;.~ ","!,~lf_?1lI ·;t;.t.e~; by d(~finillg t1 - C Q'C, : ., J - l .. Q.. + \.1 nU) d t

.'"

(1J.18~

(II.19a~

~.t.'.t f\.,nction.:.,l of eqn. <11.18) yield the equations [14)

-, T

:_1 •• • R B F' ( t. ):: (l )

P .. '-f'A - ATF' + F'BR' 1'.1 F' -- (I ; P (l tl) I: (I ( 1 1 • 19b )

J f tll'.:- /0:1 tr· 1 :: p ~\ 1 r' ( A , ,---Q ) i~ completely observabl~ [ 15] ,

tirE! malrl:: dlfferE.'ntial Riccati equation gl~en by

<JI.19b) will have a steady-state solution for P which IS

'-Jlvcn b'l

<I I • 2(:)

In i:11Q slUctdy-state case, thl? optimal control pol iev ber.::C1mf~s

tIm':..' lllvariant, tile gain matri:: d~flned as G:= h: D f' i:-,

con:::,t~llt , Clnd the closf!d--loop sy·.:;t~',\'1 dynamics is gi\'en bv

l

r

I,

f

"

eqn. (11.2).

II.e - RUYll13tor Dc,5ign USing Parameter Sen!:>itivity

m'2thnduloqy of cllntrol law de~ign to meet t i 11Il..! (J(Jm~ j n

This method ",duress£..'c:., the "'<'1111 dri.:w~hacl: of LQ

pI '\ 'i Ii 1 c: .J I
ORIGlrJAL F':,(:;: r9

OF POOR QUALI,."

sel ~(.. t Hit] (.'It:.'mc,,t!:o elf the Q' ",nd R matr ices as pari1mf.·t cor ... ,

the p3r~rr..~tt!r· ",mH,itivity resLllt~ of 'subsection 11.", c,;,>,n be

the ""y~le,". For e::am~let by

u~ed to duiinf! the uptimal LQ problem to meet the time

domai n ":1" i tOt' 1 a. The fact that a physical relatlonuhlP

between the desig"~r selected ~'Jeights and the time domain

resp'::>ns'.::'s carl ho t.?~;t~blished 15 seen by e:·:tending the

parameter sen~iljYlty results.

To begl n, n::"c~d 1 th::\t both the absol '_Ite time response

(l 1. l ... ) , and the covariance sensItivity,

eqn. (11.10), ",ru 1\.HIc.tJons of the partial derivative of tlte

cll)s'.?d-It:Jop d·yn.:..rnics matr°i>: A with respect to the par'C'm,:ter

P of i rlterest. Fell' tile C:i:lse of the optimal LQ conlrol law

the partial derivative can be calculated, Llsing eqn. (11.3).

Thus

~A 4A ~B ;)G

= --8 B-- (11.21>

Ip dP dp dP

Now, the partIal derivative of the gain matrix G with

respect to the paramlter p is

d G ~R-I l: _Idt/ _I 1 dP = ---8 P + R -- P + R 8 --

dP dP ~p )p

calculated as

<11.22)

An e:,pressiorl for the p"wtlal derivative of the steady,osta.te

matri:: f~icccJ.ti cqui.\ticlrI (t!qn. I r •20 ) can be oblained by

~v1diny and ~llbtrdct-irHJ 2F'['F< 8 F' t.o eqn. (II. 2(1) t.\..l Clf.?t

-I T· T -I T -I I ({j .. DR [l F·') f'l f-' (A .• [If, II f:') -t Q ,~ F'[ln £1 f'
...,

ORIGINAL PAm: t~

OF POOR QUALITY

1'3

(J 1. ':::.) with respect to p, canc~ling

t~rms, and t- C!p I c.\C i rl',l r('BTp by G gives

• cW 'dr' f;)Q ~A d A

0 - ?i-- + --A +{ --. + --p + p--

~p (tp L~p ;)p ()p

... Fi:-'

(I I. 24) ("0., ,. r3 \

- F' ~~R B + D-- BT + BR'';~ 'OT) p}

dp

With eqn.·~ <11.21>, ( I I • 22), a.nd ( I 1.24) , it is now

possible to ~a.lcul~te the snnsitivity of the absolute time

response to parameters of the optimal LQ regulator problem

Llsing eqtl. <11.6), and the sensitivity of the sta.te

coval"iance to the parametE.'rs using eqn. (11.10).

and OLltput ab'iOl1l l.lte ti me re';;ponse sensitivities

calculated from eqn.s (11.7) and (11.8) ollce the c?bsoILlte

state ~ensit\vity is obtc.\lned. Control and

coval"iClnce sensitiviti~s <.\re obtClincd from eqn.s <11.14) Clnd

(11.16) once the state covariance s~nsitivity 15 calculated

from eqn. (11.1(1). These sensitivities are the ptl)sicc.l

linls between the d~signer selected parameters of the LQ

design problem and the actual time respun5es of the

•

closed-loop system.

To illustrate the use of optimal parameter sensitivity

analysis in the design of optimal linear regulators, the

following Situation is considered. Assume that a dynamical

bu cuntruii f;~d by feedl ng b"'cl. all the sy~t~m stales,
10

I 16

ll1Cludll"J ~t.:itus wllich are used in the model to repre~,'nt

• the dyrh~.ml cc:. uf the control ac: tuators • A qu~dratic cost

fUllction of the form given by eqn. <11.18) is written, which

WE".·i ghts the states and control s. The opti mal 1 inear

regulator problem is solved (eqn.s 11.19), and the system

state covariance response to a random input calculated from

eqn. II.ll. If the RMS response of one or more of the

stat~s i~ unacceptable to the designer, sensitivity an3lysis

can be used to change the design, in the following manner.

A sel of one or more parameters is selected, say for example

the time constC'l1ts of the actuators a~d elements of the

st~\tt:? \'leighting matri:: Q. The sensitivity of the

unaccepti... bl Po stab2 response to these param~ters is

calculated using eqn.s (11.24) and <11.12), and the

unacceptable response expanded in first order Taylor series

about the nominal value of one of the parameters. This

expansi~n is then used to make a change in the nominal value

of the parameter such that lhe state response is improved.

The optimal linear regulator problem is then resolved with

the new value of the parameter. The result is an optimal

control law which more nearly meets the state response

•

..

criteria of tho designer •
ORIGINAL Pt.G~ i~

OF POOR QUALITY

In the previou9 subtiections, the idea of par"ameter

... scns\tivity was disclJ'Zsed, LQ reg'_llator theor"y reviewed, and

the means of using parameter sensitivity as a redstiign tool

in optimal control theory presonted. This subsection will

discuss potential ewtensions of the use of op+':' imal

sensitivity analysis, consider the practical aspects of

computing the sensitivities, and give the reasoning behind

the ~ssumptions and restrictions that were m~de during the

course of the development.

The use of opt.imal sensitivity analysis in other kinds

of optimal control design ~rocesses is readlly apparant, -.

particulary in the three areas mentioned in subsection J • a.

The first of these was the output feedbac~ problem, where a

number of system outputs, less than the number of states,

this prublf.:uTI, since the necessary conditiuns fa,' the optim~l

are fedback. Optimal sensitivity results can be derived for

can be used in the same manner for this problem as for the

solutinn are Lnol'IO (9]. Thus optimal sensitivity analysis

using t:-H! sensitivity results for the OLltPLlt feedbc\ck to

i

t

f L' 11 state feedback problem. Perhaps more appealing is

This cCluld be done b'y'
I'

I

18

cOf?ffi(;lt~"t~;, (~! til,:: COlllplm<:>ator, vlhich dofine it's dynamics,

emu 1.I:inq output feedback theory to find

9",.i ns for the the compensator, sensitivity

analy~i5 could be used to iterate on a $olution for the

'::\nfJther possible e}:tension is to the frequency

shaping problem of Gupta [11] , where the sensitivity

C)naly',;,is cOt.tld p(~rh,).p~; be used to define the elements of the

freqt.ll2flCY dep~rlu",nt ~j€:'i ght i ng matri c.:es.

As far· of sensitivities is

concerned, it ~huuld be apparant that for each differant

scalur il1t.ere~t, a set of vee to..-1m=,tr 1::

equatiDns must b~ solved. At first, this seems like a large

computational burden, sin~e these kinds of equations can be

difficult and e;·.pt.·nsive to solve. Fortunately however, the

coeff~cients of tha time response equations and their

a~sociatcd 5~nsltivity equations are exactly the same, and

further they are ind~penuent of the parameters. Thi s mea.ns

that onCl! the cuefficient matrices have been factored to

solve tha covariancu equation for example, they need not be

f",ctored a.gain. R.:~ther , each new equation can be solved

..'1nd back substitution of new

• r i ght-hand'-o;:i de. In ilddition, a method simi liar to that

.. Llsed in [14 J to c.:~l cuI ate grad ient.s f.Jf cost fl.lnct lOriS m-olY be

to thiS an""lysis tpc.:hnique, further simplifying the

bLWd(;:'II. Similarly for tha ~bsolute time
"

/

/

/

:.

ORIGINAL Pfl,G~ 1$

OF POOR QUALITY 19

r·,,-·:.pl.ms'~ equ':4lil.'fl::', th\? st.~tl." tr",nsitiol1 matri:: needs to be

comput~d unly once for c~ch control law.

seml-definit.e i:\nd the R m.:>tri:: be positive definite insllre

Most of the assumptions and restrictions made during

thi~ development are related to solving the LQ regulator

problem. The restrictions that the 0' matrix be positive

t.hat .., solLltion e::ists that will stabilze the system if it

i!:i initially unsti:\ble, if the matri:< pair <A,B> 1S at least

Riccatl equation, the ~election of parameters from lhe 0 and

In Ii 'Jht of til!::' desire for a s'immetric solution to the

be

dnd the matrix pair (A ,e) is at least

The requiremant that the pair

R matrices sho~ld be made in such a way so as to ma~e sure Q

and R are always symmetric matrices•

symmetric solution to the matrix Riccati equation exists.

completely observable is desirable so that a steady-state,

detec.tit'tle.

st..-\bi 1 i;:i~ble,

• r

,

;0-

..
In ~r:(.tlorl II, the idea of ~ensitivity of optimum

solution~ to probl~m parameters was adapted to optimal

conlrol problem!:i. A derIvation of optimal sensitivity

anal ysi "i for th•.:!sE' t ypc? prabi ems was pre!iented to ill ustra.te

that the ffi£'thodology COLlI d be used to aid the desi gn of

line.:>.r fuedb~c~ control laws. This section will propos~ a

systematIc inve~tiqation of optimal !:ienstitivlty analysis a~

desi ,:~n tool in optimal control design. This investigation

hDS as an objeLtive the development of a research progr2m

satisfying the r~!quirements of a doctoral dissertation.

III.d - Propus~d Research Program

Section II presented the initial development of a

desIgn methodology based on optimal sensitiVIty analysis.

Clearly t.his d(!velopment was incomplete in the sense thC't

only thu infinIte time, constant coefficient 1 i near

regulator problem was addressed. As the first step in a

logic",l development of a methodology, a broader class of

formuL:\ti"n of optimal sf.!nsitivit'l C\n<llysl~ in .. To tlli s t:md,

This would include
/

/ ,

./

/ ORIGINAL P,:'.,CE '3

OF POOR QUALI'N :':1

.~:

~'.

to

Height i nCJ':..., ,mel to the t i Il,e varyi ng S~~1lI dynC'.llIi cs case~.•

Cert,Hnl'! tl1~ method must: ~lso bf~ e;:tendod tel c'-lnsidc'''' C,"~5es

",I ,erE' lhE' fC""dlJ<.H'::l.. Vi.~r i ~bl!i!s are contamin,",ted wi th noi sea

Thus th~ optimal sQnsitivity of linear-qu~dr3tIc-G3us5ian

CLOG) regulators would be a result of this studY.

Also in s·?ctioll II, a. d~5ign methl.Jdology, L'.sing optImal

sensitivity alt~ly:.;is, "Ias suggested. This method too n~eds

a.nd e:·: erc i sed on ,~.ppropt"! ate,

problems. pr ob 1 em I-IOU I d

control of flight dynaml cs .;l.nd

s tr- LIe t ur <,'.1 dynamic.s of a highly coupled, fle::ible "'i .... c:ri.~ft. .•

With thi:.; e::amplt~, it is e,:pected that the m(?thCJd ''Jill prove

the designe.... wi 11 be afforded mo.... e

inform~tion with whlc.h to select

c:>ystem<:'.t i c mi:~nroet-.

design parameters in an

int"Ht.:.'st!ng optlm..::d St'nsltlvlt,

analYSIS would be in the are~ of .... educed-order dynamic

compcnsc'. tor s. One previous method for solving 1:11£''''.£' type

probl~ms d~~ been d i r C!c t

con':roJ

•

Ttle·n,
/ • I

"

ORlGINP.t. I.":~:·'::: t~

OF POOR QUAU((

-,,..,

...-

COlllpeno;:,.:.tur,

the! st.,;.ltc-s of t.h ... l":olnlJnll~a.tor·. the <;;ensi t i vi ty of the systnrn

response to the elements of the comp8n~ator could be

Thesa sensitivities could then be used to

redefine the compensatur.

As a fina.l area of proposed research, the s~nsitivity

of frequency dom~in r0ponses to parameters of the optim~l

control lc\\'/ formul.:\tion o;:,tloL\ld be investigated.

would help I~st.~.bll·c.>h a r·(:!l~.tionship between these 'v'ari61bI8'=

and the fr'equel'lcy lespan5£' of the system. In the course af

this stUdy, it may b~ necessary to consider the fr~quency

shaping q'_'i:'dr~tic: fanTls of Gupta [llL
ORIG\NJl.L ~.:..'.~:: j~

OF POOR 0: ·I·.~l~\

[1J Soui ...!o::;;:(;:.,rl~'l .- SOblP~ll, J., Bartheleml, J. F., i'.nd

F'i.lp'y', I.. r'I.: "Sens! tlvity of Optimum Solutions of

F't"otdf!11l f',,:,ri'm€::lct'o.;;", AIAA JOllrnal, Vol • .20, No.9, pg.

1~91.

[:!] BC:lrthelem',' , J. F. M., and Soiuszc::ansl:i - Soules-Ii, J.;

IIExtr.polat1011 of Optimum Dosign Based on S~nsitivity

DurlvatiVE!S", ilIAA JOllrnal, Vol. 21, No.5, pg. 79.,.

[:~.] f1 ..,-thelelll'".J. ··F. 11., and Sobie5=~;:",nsl i-Sobieski,

J.; "OpllOltliTI SEm~itivity Der1.vat.ives of Obiuctive

FUI1'.:l:101l;' in No.)"l1.nt;?:;>r Progr'arnming", AIAA JO'..WI1.31., Vol.

~1, NC!. 6, pg. 91::,.

(4] StJbi",,~:::c::.;\n~l.i ... Sol!lesl:i, J., J",ml:!s. B., "'rid Duvi, A.;

"Stt'.,ctl.ll· ..'l Optimi::·:>t.ion by MLIltilevel Deco,;)positlon",

AI{\~' F'"'per 8·';'··('U-';.2--Cr-, 24th SOH Conference, L.:.I.e T.:>.hoe,

1·I'":!v~ldc"., 11.:.oy 1 (';8·-';,.

[5] Ycdi< h·l 11 , R. !.:., and SIf:lton, R. E.; "Cont!'ollE:l-

[j.;?::.ign fo.... F'-:.u'ametel- Sensitivity Reduction in Linear

r,cqIJI<:ltnrs", Optimal Cent-ral Applic",tions ",nd Method:',

Va 1. 3, 198:".

(t.. ] Sch,o.>l:!c.ht'.Jr·, D. B. ; "Closl2d-·Loop Control

Sen3;tlvity to P~rameter Vdriations",

GUidance, Control, and Dynamics, Vol. 6,

399.

F'erfor-mance

Jou.... n""] of

t.Jo. 5, pg.

(7] }li3I-vey, C. A•• c".nd Stein, G.; "(luadr",tic' Weights for

Asymptotic Reg'.llator Properties", IEEE Transactions on

Automatic Control, Vol. AC-23, No.3, pg. 378.

(8] ~;t~in, G.; "Gem.:r·ali;::ed QuadratiC Weiqhts fa....

Asymptotic net]uli'.twr F-'l-operti~s", IEEE Tr",n<;;2lctions on

Aul:om~t1.c Control, Vol. ac-~4, No.4, pq. ~S~.

.jtt~~f_~t~~
•

/

/

ORir.;:r:rL ;~.'.~'~: ',:

OF POOR ~;.jH~i ;-"

['// LL"/JIi'.:. 'oJ. ~; ••..•nd IH.h;>n5, 11.; .. Un the Df..!lermlnatlon

u·f U·,'..., Uptll1lc.' J Cur' .... t ..... nt.. O"It..put F cl:.'dlJ;;~ck G",i n;j fot'"

Ll t ",','/ M,.'lll '!::l.I~ 1 ~b 1 c 5y~;te/ll:;" , IEEE Tran!.:i ..\ct i OilS Un

A'.".UO'i.'tlC- CCJr,t,'ol, VeIl. AC-15, No·. 1, pg. 44.

(to] M'.t1hop ...'dh'I"IV, V., NI~I~S(Jm, J. R., and Abel, I.; "A

11ethod for OLJtaining Heduced'Order Control LC'ws for

Hlqh-Order ~;ystL!ms Using Optimization Techniq'_'t?s", NASA

TF'-18'l6, i~LlqLlst, 1981.

(11] GLlpt .... , IJ. I·., "F'-E4,-,ency'-ShClped Cost Functional5:.:

E::.";-!I\~irJI1 ·wf Lll1f::'e,r-O'_'adratic-GcLlssian Desi(Jn 11ethods" ,

JOL'rn",.l oi Gtl.d .."mce Clnd Control, Vol. ::!., No.6, pg.

S::,cl.

[ 1:] b'oq =' n • W.

Puh 1 1 st ,{.!/' ' .. ,

L. ••

In'.= •

110DEI·!'o!

N(JIJ (or k •

cmnr·:OL 1HEor;:y ,

N. Y. 19~14

[1~.) rl'"y5un, i'\. E., ..<TId Ho, Y. C.; AF'F'LIED OPTIMAL CO!'HROL,

H.:.l'Zit:~._Hj Fr'..':s, N'21~ 'r'orl·, N. V. 1975

(14) 1\~':"E'/n~~I·. H., :;l.nd Slv<Jn, r..:.; LINEAR

S'{S 1 U"\S. l·Ji 1 '?y - 1n tet" sc i ence, New Vorl:,

OPTIMAL CONTROL

N. V. 1972

CIS] Ande,-son, [I. D. 0., ~nd t1Qore, J.

CONTROL, Prentice-H~ll, 1971

B. ; L I NEAR OPT I 11AL

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