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1964

Linear control system optimization using a model-based index of performance

Demetry, James S.; Titus, Harold A.

http://ndl.handle.net/10945/12557

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1964 DEMETRY, J.

LINEAR CONTROL SYSTEM OPTIMIZATION USING A MODEL-BASED INDEX OF PERFORMANCE

JAMES S, DEMETRY

Library _ U. S. Naval Post¢raduate School Monterey, California

LINEAR CONTROL SYSTEM OPTIMIZATION

USING A MODEL=BASED INDEX OF PERFORMANCE

keke ke Kk KKK KK

by

James S. Demetry

LINEAR CONTROL SYSTEM OPTIMIZATION

USING A MODEL-BASED INDEX OF PERFORMANCE

by

James S. Demetry 1]

Submitted in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

IN

ELECTRICAL ENGINEERING

United States Naval Postgraduate School

Monterey, California 1964

Library U. S. Naval Post¢raduate 8choal

Monterey, Cahfornia

LINEAR CONTROL SYSTEM OPTIMIZATION USING A MODEL-BASED INDEX OF PERFORMANCE by

James S. Demetry

This work is accepted as fulfilling the

Dissertation requirement for the degree

DOCTOR OF PHILOSOPHY

from the United States Naval Postgraduate School

= - ——

ABSTRACT

This paper deals with a method of optimizing the free coeffi- cients in the characteristic equation of a linear feedback control system. The optimization is carried out by minimizing an index of performance associated with the system's response to a given test disturbance.

The index of performance is the integral of a quadratic funce- tion of the system state variables. The structure of the index rests upon a logical interpretation of the regulator nature of the control problem. The index for an — order system contains n weighting factors whose values are determined from an ac order model system. This determination is such that the optimization of a completely free system will yield the model system. A syse- tem with fewer than n degrees of freedom in the state variable feedbacks may be optimized with respect to the free feedback co- efficients, yielding a system whose dynamic response to a given disturbance is, for this optimization scheme, a best approximation to that of the model.

Examples are presented for illustration of the salient features of the method. It is also shown by example that systems with closed-loop zeros may be optimized by this method.

The author wishes to thank Dr. Harold A. Titus for his guid-

ance in this investigation.

ii

i i = = = ee se => © Seems 664 > G& ae p —-a~ me —a wn la a ae ee ————i= tee

ek eee ce a t—_ ———emngn at £ di Mn er Se — ¢ eee ee | ee Seem © ere ree eee ae ae a eet ee =a @ ae eee «a ——— Le ——— a mm Ving SO A | Se ee ES aoe —_—— + (i) —) =| oe & —— ee eel —————s_ re oe = al, See! s et ee SS —eE— SS

—Stm CO OM) A a te |e ee eee eee Be © 2? 6 os © <2 a

—-_ = «§ — a? Ga ae ae

en me Dc

Chapter

III

IV

TABLE OF CONTENTS

INTRODUCTION

DERIVATION OF THE INDEX OF PERFORMANCE

ILLUSTRATIVE EXAMPLES

Example [

Example II

Example III

DISCUSSION AND CONCLUSION

A. Stability B. Cost Surface Selectivity Sensitivity to Plant Parameter Variations C. Potential Application in Adaptive Systems D. The Optimum System as a Function of Initial Conditions APPENDIX BIBLIOGRAPHY

at

Page

38

39

43

51

Figure

LIST OF ILLUSTRATIONS

Signal-flow graph for the transfer function of equation (1)

Feedback control system of Example I Signal-flow graphs, Example I

Root locations for Example I

Transient responses, Example I

Isometric projection of the cost surface in

Example I, as a function of the two variables a, and a,, for a model system characterized

by ( = 0.7 and w= 2.0 (a) = 4,0; a, = 2.8) Feedback control system of Example II Isometric projection of the cost subsurface

in Example II, as a function of the two var- iables a, and a,» for a model system char-

1 acterized by a, = 20.0, a, = 18.0, and a. = 7.8, and where the actual system is constrained by a, = ear

Root locations for Example II

Transient responses for Example ITI Feedback control system of Example III Signal-flow graph of equation (49)

Pole and zero locations for Example III Transient responses for Example III Hypothetical curve of optimum gain k as a

function of plant pole p for the system of Figure 2

iv

Page

26 26 28 28 31 oy

36

@ => ene > ©

ets 55> =

Figure Page

16 An adaptive system using a digital com- 38 puter for surface search and optimization

17 Initial condition space, second-order 4l system 18 Model poles and optimum pole locus for 42

the conditions of Example I

Al Signal-flow graph for equations (A2) 43 A2 Signal-flow graph for equations (A5) 45 A3 Signal-flow graph for the Q, P equations 47

of the fourth-order system

. >_> = =D eeEp a= «¢ ——>_

°° — coo _ ewe ‘ ' nee ee *

_—_— « .

= ——-— —— = oa o

oem 6 Ga, "—=wy £4

1 9 on eet

CHAPTER I INTRODUCTION

The task of designing a feedback control system normally begins with a mathematical modeling of the open-loop system, process, or plant that is to be controlled. This modeling permits an analytical treat- ment whereby the response of the closed-loop system is determined for a given test input. This response is very likely, at first, to be unsdtis- factory with respect to a given set of specifications; compensation of the control system is then indicated.

Compensation is to some extent a matter of judgement; i.e., the choice of a compensation scheme depends upon such factors as cost, physical limitations of equipment, availability of certain items of information as readable signals, etc., all tempered by the experience of the designer.

It is now assumed that a compensation scheme has been chosen by the designer, and that the resulting compensated closed-loop transfer

%

function has no zeros” and may be written

k CLIF = a 29 (1)

s" a 5" a Ss a > n ds , a l

where the a's and k are functions of the various plant and compensator parameters. The signal-flow graph of Figure 1 represents the transfer

function of equation (1).

* This restriction will later be relaxed.

» == an —_<@ » —_ , —_ «= =) a a2 = = @ © &— —) es | os tae —_— <8) ee a as o = | eee Oe i i, ie lia ili a — oar a — ae a oe ea hae a © © SS -_ —_ = =e ate ) om @

i —=- =_—- << «— ee! ee Oe eee eet fee @ ee 6 ee —_ =—”- —_—— ——— 2 Se | + seminal ote <4 6 fee | —— gamle —! a @

—— 6 = a | ee @ a awl ems iy, le lal ll =—@ ) oa” 22 - 4.

ay tg ee -———— _ > —-_ ——s — a. a allie

6s ——- ©=§ (9° est ee *& GE en eh

a — nemo ee I

-a] Fig. 1 Signal-flow graph for the transfer function of equation (1)

It is at this point that the designer must translate the given time and/or frequency domain specifications into desired root values for the characteristic polynomial, or equivalently, the poles of the closed-loop transfer function. If all n coefficients in the character- istic equation were freely variable, the n poles could conceivably be placed at any pre-selected combination of positions. This, of course, is rarely the case, since some of the ais contain or represent fixed plant parameters. As a rule, one free parameter is required for each distinct pole that is to be placed in a pre-selected position, or conversely, as many poles may be arbitrarily placed as there are free coefficients in the characteristic equation.** Magnitude constraints and other physical limitations may further restrict the placement of poles to certain regions in the s-plane.

** A complex conjugate pair may have one or the other of its polar co- ordinates (zeta, omega) fixed by one free coefficient.

. 7 7 7

aie << <- a? ted baleen —_——_——— ——— ee

__—-— fe = <«<¢e + = —- =a? —_ eee e ew. = — — a

— Sy a

On the ability to pre-select some of the poles of the system rests the basis of the dominant mode method of design. The designer uses as many free coefficients as are available to place a corresponding number of the n poles in pre-selected positions, with the hope that these poles so placed will dominate the response of the system to any input. Whether or not they are indeed dominant depends upon where the remaining poles are found to lie, how they make their presence felt in the time or fre- quency domain, and upon the degree of extra-pole interference judged to be tolerable by the designer. It is to be noted that the presence of closed-loop zeros further complicates the dominance picture, very possibly destroying the hoped-for dominance of a carefully placed set of poles.

The dominant mode method places emphasis on a limited number of the system's closed-loop poles in the hope that they will dominate the System dynamics. It is shown herein that more attention might pro- fitably be directed toward placing all the system's poles in such a way that no group of poles is necessarily dominant, while all the poles collectively give system dynamics closely emulating those of the specifications. Such a procedure would start by establishing an ideal or model system of the form of Figure 1. This model system would be of the same order as the actual system, and the a, coefficients would be selected on the basis of the given specifications and their trans- lation to suitable pole values. A similar diagram, drawn for the actual system, would show some of the feedback coefficients as functions of fixed system parameters, the remainder as variable functions of the

3

aS <= >

—a -_=—_ & 4

— oe —_ «& ——. — «= as — -— ase

a rrr i re ll

io = & : *- = = —_, re ee ee ee ee ce — = = ae © (| [= = aa =e —_ crits etsy, Lh ming | cme Tl A “<6

ss - <_- Oe _ ee ———— — &— —— © =) oo de ee =| www ee —_ 6) <= © met ee, ee ¢ ee

oe mmee c——7 — EE = =a «ss a ™ @

cs & = - <= ~~ —y CX @ —— _-4 —-_— — - Fe el

xe ——- & @ =] eee ee Le (Se — oe at — <a -. —— et - = a i as

imme: 42> «cp GE «& am a Og ae 9

- “a ——- = « Gaga =

free plant or compensator parameters. The presence of fixed parameters in the actual system effectively constrains the eventual design to cer- tain definite regions in the s-plane. The design proceeds by selecting values of the variable coefficients in the constrained, actual system in such a way that it closely approximates the ideal model in its dy- namic behavior in response to a given disturbance. This selection is to be made through the mechanism of a cost function, or index of per- formance, the fundamental structure of which is based upon intuitive judgement and some mathematical requirements. The index is so fash- ioned that it contains n undetermined weighting factors, where n is

the order of the system being considered. To derive an algorithm for the determination of these factors for all systems of order n, an optimization, or minimization of the index is first carried out in such a way that the n weighting factors are forced to be functions

of the n model feedback coefficients, the a's.

The actual ach order system is once again considered. The index becomes, at this point, a function of some fixed system parameters, some variable parameters whose values are to be optimized, the n weighting factors whose values are determined by the designer's choice of a model system, and the n state variables as they respond to a given disturbance. If n parameters in the actual system were free, minimization of the index would drive those parameters to the model values, giving the desirable result of identical model and actual sys- tems. If m (m< n) of the actual coefficients were fixed, minimization

of the index with respect to the remaining n-m free coefficients would

4

no = ese arr ee &. id ee @ een Gee § =) a! « -_— = ——* == ¢a

- Cmte ill _— a a

nes «4% = =—= © Gene t —_= & «&

im & = —— — TT | i 4 os F oS

—a a «~ se me © ll? Ge == (fee

ee

a=a«~9' ——_— —_ Sap es a—— + —- ee — — oo eee — em oo LE —-@<-@e—41¢¢ = 8 a <= oo Se

yield a system optimum with respect to an index of performance not en- tirely arbitrary, but based in good measure upon the pole locations of an ideal model system. In this sense, then, the system so designed is a "best" approximation to the model.

In the literature dealing with the optimum design of linear feed- back control systems, much reference is made to arbitarily selected in- dices of performance, many of which will either not allow the selection of finite values for certain parameters, or will not allow optimization with respect to more than one system parameter; e.g., integral square error. Furthermore, optimization with respect to such indices often seems too much like an end in itself, rather than a means to the end of procuring system dynamics that will meet given time and frequency domain specifications.

Some attempts have been made to overcome the deficiencies mentioned

2

above. Rekasius”~ has proposed an index based on a model differential equation of order less than that of the actual system being designed. This approach somewhat restricts the flexibility available to the de- signer in constructing a satisfactory model. The present method also allows optimization with respect to an index of performance; a single measure, to be sure, but one which incorporates such time and frequency domain specifications as bandwidth, rise time, and peak overshoot through the correlation of model dynamic response to model pole location and the dependence of the index's weighting factors on these pole lo-

cations. Furthermore, the model and actual systems are necessarily of

the same order.

—- (4% <4) @ © = =m O~ ee | come i hen — Ss OO ee —— mm OO ee eee & eee ———— ee ee ee ee * @ > ake — 99 -SEi= oe = eee == (he & aT ee --—ae ae *oO™, « —_—_—_ —_ — ©. | eae ——— oe” ~ = em ae! CO we Se 6 he See aes ==

The chapters that follow will show the main steps in the selection of the model-based index of performance and the derivation of the algo- rithm by which the n weighting factors are determined. The algebraic details of the derivation are given in the appendix. The index is generalized for the ie order system with no zeros in the closed-loop transfer function. Included are several examples of the use of the index in system design, for demonstration of the salient features of the method. [It is also shown by example that the index may be used

in the design of systems including zeros.

——— = « - *&® 4 se oe -. _ tt, ee eects, illian, Aeiiaiaml. > i ——s ee i el le, ean ; “

-— co GS coe ee ee cor el

— es te § =e ——— aT

—_ ig ee ere et Seer ——“<— = 2c ——a eng E>

—tmm Gt ——_-

CHAPTER II

DERIVATION OF THE INDEX OF PERFORMANCE

The index of performance, or cost fuNeeLon”, that is to link the actual and model system behavior will by choice be a quadratic integral

functional of the form

co s={ ('Q)x + Tu) at (2) O where x and u are as shown in Figure 1. A quadratic form is chosen simply because it admits of some mathematical treatment that would be very difficult if not impossible had another form been chosen. The

differential equation of the system of Figure 1 may be written in

vector matrix form as X= GX + u (3)

where G is given by

0 1 G = Jus NON \ ‘“ a (4) 0 from a eel m<n (5a) and ree (5b)

For convenience, system response will be initiated by initial conditions

Pa”

=> &° Gf ££ ° lay

2 a _ a 2 Se _ ¢ . : sa —-— io ‘* swords 7 —_— ee —— ——— TF a, ~~

| —/# «!

_ fee en - 4

——— ee

rather than by external inputs, resulting in a u matrix of the form

1 7489 ++ eee a Equation (3) may now be written X= GX + AX O x (7)

The transpose of equation (6) may be substituted into equation

(2) to give j= f | x7(Q, + A’Q,A) x | dt

Lo)

Sf xToe at (8)

Oo

Except for the form of the oF and Q, matrices, equations (2) through

(8) completely describe the model system and its associated cost func Cleat The evaluation of J for an arbitrary set of initial conditions is

to be carried out using vector matrix methods. Consider a Liapunov func-

tion of the systen;

vix) 2 xT px (9)

Where P is a positive-definite, symmetric matrix. V(x) possesses the

2 following properties , provided the system is stable;

-—i

ss 7s

Sex Se “5 —— * ae oe es oe

ome 7T

Ow ———-_ - a

LE Be © Em 4S he | ime me me me eee eG’ in|; a; Ss eee | meio @- @ as ones ees oe

«1 otm

_————_— < - -_—- ee 4 o@= — «- — a er

a) V(x) >0, x #0 = 0, x= 0 “ry — AV(x) b) V(x) = | 0, S00 (10) = Oj;exaeen0

c) V(x) + © as [x| —~ ©

d) v(x) ~ 0 as t ~@ Differentiation of equation (9) with respect to time yields V(x) = x Px + x’ Px (11) Using the transpose of equation (7) in equation (11) gives v(x) = xl (Fp + PF) x (12)

To this point, the P matrix and the V(x) function associated with it

are not defined. This definition is established by

V(x) x' (FIP + PF) x

Wr

x? (-Q) x (13)

Equation (13) is now integrated as follows: =) Leo) T [ Vexae = - | x7ox ae (14) Oo O The right-hand side of the preceeding equation is immediately recog- nized as the negative of the cost function, equation (8). Evaluating

the left-hand side of (14) and substituting limits yields::

y= - vix(@) | + v}x(0) | (15)

For a stable system, x(~) ~ 0, and from the properties given in equations

~ wile

| ”, - «al

ee a. T ale dl bolle

— —_—_- 9 ———_- ——- de « eth ee . some

Oi eee <t & eer

o -

7 -*@ @) - a 5 ——_————— a Ge li, 7

aa © SS = s * —_———— ae) — _—el_”_~_”’”’—»h_ Dl (= &——=—«

ws oa? F

(10), v [0] = 0. Therefore

y= v |*(0) | = x(0)"P x(0) (16)

Equation (16) reveals that the cost incurred by the system in response to a set of initial conditions is a function only of those initial conditions and of the P matrix. The P matrix in turn is a function of the system's F matrix and of the Q matrix, as given in equation (13). There remains, then, the task of fashioning the Q matrix to fulfill the mission set forth in the introduction and above. The following requirements, some mathematical and others intuitively logical, are to be considered in the selection of the Qv and Qw matrices: a) Q must be a square, nxn, symmetric matrix. b) Q is to be such that all states are included in J. This is a logical consequence of the regulator nature of the problem. Since all states are to be driven to zero, they should all contribute in some measure to the cost, the subsequent mini- mization of which yields an optimum transition to the origin of the state space. e) Qy and Q, are to be such that together they introduce n | weighting factors, whose values will ultimately depend upon the n feedback coefficients of the model system, or, equi- valently, upon the model poles. d) Q is to be such that J includes the “control effort" vari- able u. This will act as a constraint to prevent results

calling for infinite system parameters. It also insures that

10

SEE © — & - ‘cm + > - ~~ oe _ — = - =» om a |

(f- ae -_-—_—_ =a

——-—se =e == oF ine al eheegat

SS —EE Me c= — Er > a

—— — = - © aa at (ee

i min i mmm Camm Ogg

-_ —_ — ——s = <a & eee @= em apy @ i i i ~~ 9 Nall PO! - Se4. ee Ce eam «eF Se 6 ee ee ee ee —_—_~ = SS ce i a). tn &

—— ccm: me oem me ie cme Fe

yo © 4

=>: — ——— es tear e- 1 A A VT _—s _——E——@ = 4 ia

the optimum control is a linear function of the system states“, In the light of the requirements stated above, the following con-

figurations are suggested for the x and Q, matrices:

2 nl Ry Rs

and (17)

Q,, = \ 4; A is a sealar

where a, through a and )} constitute the n weighting factors. These

combine by equation (8) to give

2 1 + ay Aaa, ercccesece- 2,4, a ac Se ue:aeenanaaaca es a a Aaya) M + ra, Aaa, . Rh Q= ~ (18) [ ~~ | = | | ~ | ~ ) N | Aa 8) asveannnaaneaanannaaaacaa a, 4 a

The P matrix may now be evaluated literally, from equation (13). This is done for the second and third-order systems in the appendix, with results included for the fourth and fifth-order models as well. The

|

_ te 1

“he es,

— 66 et eee ee ey _s & «lat ©

Lo al Ve @

a —— | >

- i :

»_o——— & aCe ee eens a _ Ka 2 eee a

_

: oe |

second-order results are presented here for illustration.

ae. P29 gt ( oh Pri to + 95h, TL Wye oo wee 1 2 1 q 11 ries lh lily! Sh, —

= 3 (4,,+ “a ~ Po 2a, Io9 a. /

a aia Y 4 Pi tae + a +o +5 1 2 J, 2 1 Aay Pio = Po) = Za, om (20) Gost Sg 2 1e2 2

Equation (16) is now written in scalar form to show the relationship

between J, the initial conditions, and the elements of the P matrix. y= x7(0) p,, + 2p, x,(0) x,(0) + x2(0) p,. (21) 1 ll 12. =] 2 2 22

What follows is the key point of the entire derivation. The values

of Oo and ) must be found so that, for the unconstrained system, the

minimization of J with respect to a, and aj gives resulting optimum

values of ay and ao that are, independently of the initial conditions, equal to the model values; i.e., it is necessary to find the elements

a, and \ as functions of the desired coefficients ay and a. so that the adjustment of the actual coefficients to the desired values will

simultaneously minimize the J of equation (21) for any set of initial

12

MT te <« = «€ Gee beara 1S te m= es) GH)

— VW —_—- = -

2

conditions.

The measures indicated above are carried out by first taking the

gradient of J with respect to ay and ay:

OP OP OP Gel aug 11 12 2 22, 0 x, (0) Sale 2x, (0) x4 (0) 7 x (9) = 0 1 1 1 (22) oP OP OP Oe «2 ue 11 12 2 (oj a 3a, = x, (0) 3a, “= 2x, (0) x. (0) 3a, * x, (0) Sa, 7 0

The initial conditions will be assumed to be generally non-zero and completely independent. The only possible solution to equations (22)

must then be of the form

— il. Oc i, jeden (23)

eS (24) ai and as = 2a, =o (25) al

Similar developments have been carried out for the third and fourth-order systems, some details of which are included in the ap- pendix. At this point, a recursion formula for the structure of the a's in an Age order model was postulated on the basis of second, third, and fourth-order model calculations. From this recursion

formula, equation (26), a prediction was made for the fifth-order

13

|

oe - —_— a kt —) -

—- SS <= wt es.) Ga @ eee ee ee cee cee eee ce | TEE | i —_— _ —— aT, gare : ee Oe ee

case, and was proved correct by extremely lengthy but straightforward algebra. This successful prediction of the fifth-order q's has been

accepted as sufficient proof of the validity of the recursion formula.

n-1 l 2 k . al a — 2) (= Tega” ae (26) 1 k=1 and

1 ea 4)

where n is the order of the model, andl s isn. By definition,

a = |] (27)

14

CHAPTER III

ILLUSTRATIVE EXAMPLES

EXAMPLE I

Consider the second-order feedback control system of Figure 2. The amplifier gain k is the only variable parameter, and is to be selected so that the system's transient response to a given disturbance is a best approximation to the model, or desired response. The model

response has been chosen by the designer as the response characterized

Fig. 2 Feedback control system of Example I

by closed-loop roots of damping ratio @ = 0.7 and natural frequency Eee 2.0 radians per second. An inspection of the system root locus for 0 <= k s » makes it immediately apparent that the ideal root lo- cation is unattainable. One may therefore turn to minimization of a performance index, as suggested in the previous pages, as a means of selecting k.

The system of Figure 2 is for present purposes best represented by the state variable signal-flow graph of Figure 3(a). The model system of Figure 3(b) reflects the desired but unattainable char-

acteristic equation coefficients for the specified root locations.

15

—« —— — a oe

is

—_

——- — ee gg gm —_ «= -_——— ee ce eG seem ml

2 « —— -— — (ke tom” « ——_—-_- -— a = — ——_ Sane =a

Fig. 3 Signal-flow graphs, Example I. (a) actual, (b) model. From characteristic equation s* 4 a,8 + a) = 0

For convenience, let the system be disturbed by the initial conditions x, (0) = 1.0, x, (0) = 0, and r(t) = 0. The cost function of equation

(21) then reduces to seenO) p.. = 28 meet? Py een eo)

where Pay is given, for this second-order system, by equation (19). The Q matrix is now to be evaluated. Substituting the model coef-

ficients, a, = 4.0 and a, = 2.8 into equations (24) and (25) gives

Zz a, = -0.01 and (29) ] ae

These values are then substituted in equation (18), yielding

2 k k Cl+i) 16 Q= (30) k _ 3) To (-0.01 + 35 where a, = k and a, = 1.0 are the actual system coefficients. It is

1 2

16

— —— | aA faa al

to be noted that the model coefficients appear in Q. only through the quantities a, and )}. Whenever else in the Q matrix the ais appear, they are the actual system coefficients, either fixed and known, or functions of the parameters that are to be determined as optimun. Equations (19), (28), and (30) combine to give eee

J= -0.00k+ S + =

1 ; + 5 (31)

Minimizing J with respect to k gives

ae wie @ See a 0.005 5 + Te 0 (32) 2k Equation (32) has the real solution k= 2.03 (33)

Figure 4 shows the system root locus and compares the model and jw

actual root locations,

Model ~Ly» 4 on = 220 = 1.425 n n

Fig. 4 Root locations for Example I

Figure 5 compares the model and actual transient responses, as

simulated on the analog computer.

7

XN

Model

Fig. 5 Transient responses, Example I

Further insight into the method and means by which the optimum value of k was found may be obtained by investigation of the cost function associated with two free coefficients in Example I. If both coefficients in the characteristic equation are allowed to be free,

the Q matrix becomes, for the same model values of a, = 4.0 and

1 a, = 220, oS aa l 1 2 (1+ oe 16 Qz (34) aa ne 12 t eZ a ( 0.01 + te The cost function from equations (19), (28), and (34) is now a a ae 2 l l 1 J=—— - 0.005 4+ So + DD (35) 2a, a. 2a, 32a.

Figure 6 is a plot of the surface generated by equation (35). This surface is, for the initial conditions of the example, uniquely asso- ciated with the values of the model coefficients, a= 4.0 and

18

a 000 ee

—<_ — —- « —— mam a Om GE

A _e — «eam ge Pig > —— i =. =a = O—i —exwnumnwn ee =f a at

ini t= — es « a sem ¢

ri ¢ | ar, ae, é = = — | ail” snail: oe

_—

< (=, > qaupamy T°

a, = 2.8, and displays its minimum J . .. at these very values. min min

This is a consequence of the way in which the Q matrix was constructed,,. and of the manner in which the weighting factors were derived; com- plete freedom to vary all coefficients will always yield the model sys- tem. In the case of Example I, where a constraint is placed on one

of the two coefficients in the form of a, = 1.0, it is not possible to drive to the absolute minimum on the cost surface. The constraint is equivalent to passing the plane a, = 1.0 through the cost surface.

The optimum value of a, = k is then that value which achieves a min-

imum J on the intersection of the plane and surface. From Figure

in

6, it is seen that this value is k= 2.03, in agreement with equation

(33).

19

=< aa ~ ~~ = <a a = au» * = = e

4

- e *—— « ie

—_—_—_— & ™ 50) eo ss > —_> & ——_—_———- & — | —_— a cite = = = =a

Fig. 6 [Isometric pro-

jection of the cost

surface in Example I,

as a function of the two variables ay and

for a model sys-

tem characterized by

an

(a, = 4.0, = == Zao)

EXAMPLE II

The third-order feedback control system of Figure 7 is to be

compensated using rate feedback as shown. The gains h and k are

Fig. 7 Feedback control system of Example IIL

to be. selected so that the system is optimum with respect to a model having a complex conjugate root pair with wae 2.0 radians per sec- ond and © = 0.7, and the third root at s = -5.0. Inspection shows that the model poles have been placed in a region of the s plane made unattainable by the constraints of the compensation scheme chosen.

The model root locations correspond to the characteristic equation

6° 4 7.85> + 18st = 0 (36)

The actual system is characterized by the equation 3 Z s + 2s + (h+ l)s a k= 0 (37)

Equation (37) shows that 2 in the actual system is constrained to the value 2.0, and that there exists complete freedom to vary the ay and a, coefficients.

21

amit =—=—=2> +04;¢% === se "> 6 a oe eel ee ee oe a = ie a A F ' —— —, a “gi? E> = —_ e ii Ti) i ily a

* —— —

~ wel _ ¢-& — ——iir |?) = _— ——=- ta Meet tar

- =

To show the single cost surface associated with three degrees of coefficient freedom is, of course, graphically impossible. The constraint a, = 2.0, however, is effectively a plane which intersects the hypersurface in such a way as to yield a subsurface dependent

upon a, and a, alone, and hence graphically representable. Equi-

valently, a family of surfaces giving J as a function of ay and a,

with a., the family parameter could be drawn. The surface corres-

ponding to a, = 7.8 would display the absolute minimum of all sur- faces, while the surface drawn for a, = 2.0 would display a minimum

greater