Design of minimum sensitivity systems

TRANSACTIONS IEEE

ON AUTOMATIC CONTROL,

VOL. AC-13, NO. 2, APRIL 1968

159

171 I. M. Gelfand and S. V. Fomin, Calczrlzrs of Vaariutions. Englewood Cliffs, X. J.: Prentice-Hall, 1963. IS1 M. Athansand F. C. Schweppe,“Gradientmatricesand

Michael Athans (S’58-M’61) was born in Drama, Greece, on May 3, 1937. H e received the B.S., MS., and Ph.D. degrees matrix calculations,” M.I.T. Lincoln Lab., Lexington, Mass., Tech. in electricalengineering from the UniverS o t e 1965-53, 1965,‘(unpublished). sity of California, Berkeley, in 1958, 1959, P I h.1. Xthans,Thematrixminimumprinciple,” Itzforntation and 1961, respectively. and Control, November 1967. From 1958 to 1961 he was a Teaching .Assistant at the University of California while pursuing his graduateeducation. From 1961 to 1964 he was employed by the M.I.T. Lincoln Laboratory, LexingDavid L. KleinmanwasborninNew ton, Mass.,where he conducted research York, X. Y., on January 4, 1912. H e on the theory and applications of optimal control. From 1963 to 1964 he was a part-time Lecturer in Electrical received the B.E.E. degree from the Engineering at iU.1.T.. Cambridge, LIass., where, in 1961, he was CooperUnion fortheXdvancement of Science andXrt, New York, N. Y . , in appointed .Assistant Professor and in 1966, .Associate Professor of 1962, and the S.W. degree in 1963 and Electrical Engineering. There he has contributed to the development the Ph.D. degree in 1967, both in elec- of undergraduate and graduate courses in the area of linear systems trical engineering, fromtheLlassachunetworks and optimal control. .At present, he is alsoa Consultant to the M.I.T. Lincoln Laboratory. His currentresearch interests involve setts Instituteof Technolop, Cambridge, Mass. The former degree was obtained the theor). and applications of optimal control, computational algounder a Sloan Foundation fellowship. rithms for the solution of optimization problems, and the use of the From 1963 t o 1967 heworked as a theory to design optimal waveforms in communications systems. He Research .Assistant in the M.I.T. Elecof the book O p t i m d is the author of several papers and coauthor Control (McGraw-Hill, 1966). tronic Systems Laboratory, Cambridge, where he studied problems Dr. Xthans is a member of Phi Beta Kappa, Eta Kappa T u , and in optimal control and stability theory. In February, 1967, he joined of the 1964 DonaldP.Eckman the staff of Bolt Beranek and SewmanInc., Cambridge, Mass. where Sigma Si.Hewastherecipient he hasbeen working on the applicationof optimization techniques to award. He has been a member of the 1966 J.ACC Program Com;\wards Committee,theG--ICAdministrative problems in manual control. He is also a part-time consultant to the mittee,theJXCC Committee, and Chairman of the Sonlinear and Optimal Systems GPS Instrument Company, Inc., Kewton, Mas. TechnicalCommittee.Heis now ProgramChairman of the 1968 Dr. Kleinman is a member of Tau Beta Pi, Eta Kappa N u , and Joint Xutomatic Control Conference. Sigma S i .

Design of Minimum Sensitivity Systems

Absiraci-A method for the design of linear time-invariant multivariableminimumsensitivitysystemsispresented.Themethod utilizes a quadratic form in the parameter-induced output errors as a performance index to be minimized, with the constraint that a prescribed nominal transfer function matrix be obtained. An essential ingredient in the procedure is the use of a comparison sensitivity matrix. Two advantages that follow from the useof the sensitivity matrix are: 1) Physical realization constraints

on the compensators may be included in the design. 2) The computational aspects of the problem arerelatively simple and may be carried out routinely using any parameter optimization algorithm.

Manuscript received October 4, 1967; revised January 3, 1968. This work was supported by the Joint Services Electronics Program underContract DAAB-07-67-C-0199, andbythe A.F. Office of Scientific Research under Grant AFOSR 931-67. W . R. Perkins and J. B. Cruz, Jr., are Kith the Department of ElectricalEngineeringandtheCoordinatedScienceLaboratory, University of Illinois, Urbana, Ill. R. L. Gonzales is with the Departmentof Electrical Engineering, Bradley University, Peoria, Ill.

A nontrivial multivariable examp1e:illustrates the procedure. The design method may be viewed as the second part in a two-part proof a desired nominal cedure, where the first part is the determination transferfunction.Atwo-degree-of-freedomfeedbackstructureis used to realize an optimum or desired nominal closed-loop transfer matrix, as well a s a minimum in a sensitivity index.

I. INTRODKCTIOX

A

i\: I l ~ I P O R T A N Treasonforthe

use of feedback structures in control sg-;stem design is the possibi1it)- ofreducingundesirableparameter variation effects. In this paper, a design method yielding alow-sensitivitylineartime-invariantsystemispresented. The methodis a n extension of an approach suggested by Mazer[’l andfurtherdevelopedby Gonzales,I21and utilizes a sensitivity matrix and corresponding scalar sensitivity performance index involving parameter-induced errors introduced by Cruz and Perkins.r31 Inthisapproach,the designproblemis

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, APRIL 1968

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divided into two steps:

1) The nominal system transfer function matrix is specified. For example, the transfer function may result from the optimizationof an index. 2) A two-degree-of-freedom feedback structure realizing thisnominaltransfer-functionmatrixfor nominal parameter values is chosen t o optimize a scalar sensitivit): index for a specified system input. Physical realization constraints on the controller are included in this optimization. Manyimportant designproblemscanbeattacked meaningfully using this two-step approach. The introduction of the sensitivity matrix converts the problem t o a parameter optimization problem. A minimaxsolutiontothisproblemisproposed in this paper. The use of the sensitivity matrix results in an important separationof the computational problem.T h e maximization of theindexwithrespecttotheplant parameters is performed only once and is outside the loop performing the iterative minimization on the controller parameters. This separation vastlysimplifies t h e design. A nontrivial example illustrates the procedure in detail.

Fig. 1. kIultivariable feedback control system.

the scalar sensitivity index,t3l

J

=

Jome!(t,a > Q e ( t ,a ) d t ,

(41

where e(t, a) is theinverseLaplacetransform of E(s, a),Q is a positive definite weighting matrix, and the prime denotes the matrix transpose. Using ParseVal’s theorem, (4) becomes 1 ziJ-jz E’(--s, a ) Q E ( s , +jr.

J = -

4ds.

( 3

In ( 5 ) the parameter-induced error is evaluated for a specific system input R. Xotice that J is a functional of G and H. G and H are to be selected t o minimize ( 5 ) subject to certain restrictions. These restrictions depend on the details of the specified system being designed. 11. PROBLEM STATEMENT Typically, the following constraints may be imposed in Consider the linear time-invariant multivariable con- a realistic design. trol system shown in Fig. 1. The plant is characterized 1) G and H must represent parameter-independent by the transfer function matrix P ( s , a)which is rational (fixed) stable compensating s>-stems. in s, where s is the complex Laplace transform variable, 2) Itmaybedesirabletoavoiddifferentiation in and a is a plant parameter whose components are uneither G or H. known,buttime-invariant.Thecompensatingnet3) T h e final value of theparameter-inducederror works,representedbythetransferfunctionmatrices must be zero for (4) to bea meaningful measure of G ( s ) and H ( s ) , are parameter-invariant and rational in sensitivity. s. I t is supposedthatthesystemtransferfunction 4) The number of poles and zeros of G and H may matrix T is specified to be T,Lwhen the plant parambe further restricted for simplicity of design, deeters are at their nominal values Q! = a0 : sired asymptoticfrequencybehavior,avoidance

Tn=

of infinite gain, etc.

U s , Q!O)

Such constraints might be incorporated by a priori specification of the ordersof the numerator and denominator polynomialsin the elements of G and H. T h e The matrix T,, could be specified to obtain some desired problem is then reduced to the optimumselection of the time response, or t o optimize some performance index, coefficients in thesepolynomials. The maindifficulty, for example. however, is in the loss of freedom involved in so detailed In any physical realization of this system, a will difaspecification of G and H . How is i t known that a fer from a,,, and, thus, the output will differ from the smallervalue of (5) cannot be obtainedusingsome desired output. The output error induced by the paramother choice of G and H that are equally acceptable eter variation is from therealizabilityviewpoint?Moreover,thisapE ( s , a) = C(s, a01 as, 4 , (2) proach presents certain computational difficulties, as it involves the solution of simultaneousnonlinearalgewhere braic equations with complicated side constraints due a! = ACY. (3) to the requirement that G and H yield a specified ,Tn. These difficulties are reduced by the introduction of the A measure of the effects of this parameter variationis sensitivity matrix. =

[I4-P(s, ao)G(s)H(s)]-’P(s,Q!o)G(s).

-

+

(1)

SYSTEMS

PERKINS SENSITIVITY et d.:MINIMUM

161

I I I. DESIGNPROCEDURE Consider a n open-loopsystem,Fig. 2, havingthe same nominal plant input and the same nominal plant output as the closed-loop system of Fig. 1. Such systems I t hasbeen are called nominallyequivalent. that the parameter variation errors are related by as,

where

a>= SEOG, 4

,

(6)

Eob, a) = Cob, 4 - Co(s, ad

of the plant parameter deviations and the sensitivity numerator coefficients. Consequently, the optimization of (9) proposed here may be indicated by

J o = min

is the open-loop parameter-induced error, E(s, a) is as definedin (2), and where the sensitivity matrix S is given b y

{

@€a

max [ J ( &d a ) ] } .

(12)

daEa

For a single-input single-output sg.stem case, the sensitivitymatrix (8) becomesthefamiliarBodetransfer S = [I P ( s , c ~ ) G ( s ) H ( s ) ] - ' . ( 7) function sensitivity. If the plant contains only one parameter c y , theproblembecomesthatconsideredby Inthe following,onlydifferentiallysmallparameter Mazer.L1l This case simplifies considerably because the variations A a = d a will be considered. For this case, (9) ishomogeneous in scalar sensitivityindex Eo and E are differentially small. Thus, to first order, Thus, the optimum parameters 6 are independent of (6) becomes (dcy), and the minimum of J with respect t o 6 may be E(s, a ) = [ I P ( s , a o ) G ( s ) H ( ~ ) ] - ~ E oa). ( s , (8) found without first nlaximizing with respect to dor. T h e of severalvariableplant Therefore, the sensitivity matrix depends only on the morecomplicatedsituation parameters, but still single-input single-output. has nominal plant : been studied by Gonzales.['] 1 jThe procedure for the multivariable, multiparameter J = Eo'( -s, a)S'(-s)QS(s)Eo(s,a)& (9) case is illustrated in-detail in Section 11,'. The key to z?rj S - j s the simplicity of the procedure is (6), which expresses The sensitivity index J now may be regarded as a functhe closed-loop error as the output of a system whose tional of the sensitivity matrix S, to be minimized by transferfunctionmatrix is S and whose input is the choice of S. Recall that the input R is fixed. Once S is open-loop error of the structure of Fig. 2. The matrix S obtained, G and H can be found from (7) and (1) : is independent of d a , while the input Eo is independent of 6. Sensitivity modelsL61are employed to generate Eo. PG = S I T and The numerator coefficients in S are then adjusted iteratively until (9) is minimaxed, i.e., maximized over da and minimized over the numerator coefficients of S. In theevent (10) and (11) havenonuniquesolutions The procedure is relatively simple t o implement on a for G and H , the designer has additional freedom in the digital computer, as the example indicates. choice of compensation networks. Equations (10) and (11) alsoallowphysicalrealizabilityconditions on G IV. ,4 MULTIVARIABLE EXAMPLE and H to be incorporated easily into S , as will be illusIn this section, an example of control system design trated by the example in Section IV. using the procedure of Section I11 is considered. T h e However, S is not completely a t t h e designer's disproblem is t o design a noninteracting control system posal. ,4n examination of (8) and (1) reveals that the denominator polynomial of all entries in the matrix S for a rotating dc to ac converter. Thefield voltages for is the sameas the denominator polynomial of all entries the two machines are considered to be the two control in the specified systemtransferfunctionmatrix T,, inputs, and the speed and the rms generator voltage are namely, det [ I + P G H ] .Thus, only the numeratorpoly- outputs. The inputs will be unit step functions. This nomials in the matrix S are free. The minimum sensi- system with fixed parameters has been considered by tivity problem, then, has been expressed as one of pa- Peschon.[41A detailed derivation of the plant equations rameter optimization, the parameters being the coeffi- for variable parameters is given in the Appendix. Small cients in the numerator polynomials in S, and the per- variations about a staticoperatingpointareconsidered, and thus the plant may be described by linear difformance index optimized being the scalar sensitivity ferential equations: index (9). If the plant parameter variations are represented by 9, [5Z,02 - 'c'o'2]lyl [2@o']y2 d a E @ ,and if the S matrix numerator coefficients are represented by the vector p, with @E@, then the sensitivity index J may be regarded as a function J ( @ d , a)

+

+

+

+

and

From ( 6 ) , it can be shown that

where

where Uo is the open-loop plant input, yl

=

incremental speed output

(15)

yz = incremental rms voltage output

For this Droblem.

and zcl = incremental dc machine field voltage

a2 = incrementalacmachine field voltage.

(16)

The plant parameters subject to variation are the So, if (21) is to be satisfied for all dP, static operating valuesof the two normalized field voltlim S(s) = 0. (25) ages v o and d o , as indicated. Note that operating points 5 4 for nonlinear systems can be regarded as parameters in Thus B 5 = PI0 = 0 1 5 = B f O = 0. (26) linearized models for these systems.The nominal values of these normalized parameters are both unity, and the From (1 11, it can be seen that a reasonable bandwidth nominal plant transfer function becomes realizationconstraintis

-

lim S(s) T .+

=

f,

as this will ensure that the entries of H will increase with frequency no faster than the entriesof T decrease. Applying (27) t o (19) yields Acceptable closed-loop response is obtained using a nominal system transfer function matrix (see Peschon,[41 p. 101): 8

P1

= @I6 = 1

P6

= PI1 = 0.

(28)

1.

The formsof G and H may be checked using(10) and (11). The form of G is satisfactory, since each entry Tn(s)= contains more poles than zeros. If differentiation is per2 0 mitted, the structure of H resulting from (11) will be s2 2s 2 O satisfactory. If differentiation is not acceptable, further constraints among thep’s may be imposed to eliminate The nominal system is to be decoupled, with each chanwill be nel exhibiting second-order response having a damping theundesirableterms.Forthisexample,it assumed t h a t differentiation in H is acceptable. ratio of 0.707. T h e final realization constraints to be imposed conT h e first step in the design is t o form the sensitivity G and H compensators. T h e matrix S and to apply realization constraints forG and cern the stability of the H via (10) and (11). Noting that thepoles of the entries poles of the entries of H do not depend on the p’s, and of S are the sameas the poles of the entries of the nomi- thus the stability of H is not affected by the choice of p’s. The poles of the entriesof G are affected by the B’s, nal system transfer function matrix, s?

+ 4s + 8

+ +

where

D(s) = (s2

however, and a check for stability must be incorporated in the design procedure. Thus, the conditions of (26) and (28) result in the sensitivity matrix

+ 4s + 8)(s2 + 2s i2).

For thkjexistence of (4), lim e(t, YO, w:)

=0

(20)

for all vat vo‘. Thus, it is required that lim sE(s, DO, v i ) s-+o

S

S(s) =

t-+ m

=

0.

(s2+4s+8)(s2+2s+2) s3+Bzs2+fi3s+84

(21) *[1

~12s2+P13s+Plr

@is2+BSs+89

1.

s3+81is2+@~8s+B19

(29)

PERKINS et al.:MINIMUM SENSITIVITY SYSTEMS

163

Hence, there are 12 p’s available for adjustment. For this example, the plant parameter variations are . : given by the tolerance ranges

Compute Open-loop Sensitivity Functions

’-

I z‘O 5 1.2 0.8 5 z : ’ _< 1.2.

0.8

(30) Choose o value ior

Theactualminimaxcalculationiscarriedoutby computer.Thecalculationis simplified becausethe index J(Q, da) of (12) isconvex in da. Furthermore, because of (30), the plant parameter space is a convex polyhedron in Euclidean space (a rectangle in this twoparameter example). If this polyhedron is denoted by a and the set of its vertices by Bo, then it has been that and shown if 3 satisfies max J ( 5 , dol) = min max J ( @ ,d a ) ,

@€a dCYEao

d @€eo

then

B is the

& nnd for 2, Eqn. (29)

1

Compute gAand gB Uslng (6). E=SE_,

+-I

Select fl using Mlnlm~ation Subroutine

Compute JAand JE

s Form

(31)

J^= max (JA,Jg)

Fig. 3. Flow chart of minimax procedure.

minimax solution, that is,

of the quantities at the two vertices are denoted by superscripts -4and B. doEao For this example, the minimax solution was obtained For this example, only two vertices, rather than four, using a computer procedure devised by Salmon.[51 T h e needbeconsideredbecause of thesymmetry of the minimizationschemeimplementedwasproposed by toleranceranges(30),coupledwiththequadraticnaRosenbrock. f i l ture of J with respect t o da. T h e main virtues of using the sensitivity matrix S in T h e basic procedure, then, is to compute the maxi- this procedure are twofold. First, to computee,, eo need mum with respect to da of J ( @ ,dol) for the vertices be computed only once for each daE Bo, since this caldm€ Q o , and then minimize with respect t o @. The com- culation is done outside the iterative loop for @.Only S putation of J and the resulting minimization could be is varied as @ changes. The second advantage is that the done for the closed-loop system of Fig. 1. The computa- minimization with respect t o G and H is reduced t o a tion is considerably simplified, however,if the open-loop parameter minimization with respect t o @. error is calculated first, and then related to closed-loop Fortherotatingconverterexample,the optimum error,andhence J , by the sensitivity matrix S. The values of the P;’s of (29) are: open-loop error, t o first order, is p2 = 2.588 plz = - 0.2915

~ ( 5=) min pa

m a s J ( 0 , da).

(32)

(33)

p3

= 2.855 = 0.2888

y2 is the rms generator voltage. where yl is the speed and However.

pi

=

/3s =

0.5194 1.297

ps = 1.391 (34)

+

r 3 . 4 1 ~ ~15.1s‘ H=l

+ 23.7s + 16

-k 2) ~ ( 0 . 2 9 2 ~0.775s ~ i0.733)

L

S(s2 4-2s

+

2(s2

+ 4s f 8)

014

pis

=

- 0.7754

=

- 0.7332

=

2.490

=

2.643

019 =

0.6319.

The corresponding compensators are

+ 1.297s + 1 . 3 9 ) l + 2s + 2) I 3.51~ +~15.49 + 23.4s f 16 -s(O.519s2

The partial derivatives appearing in(34) are open-loop and sensitivity can and be readily computed from sensitivity models. f61 The iterative computational procedure is shown as a flon7 chart in Fig. 3. The values

8(S2

2(s2 f 4s

4-8)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, APRIL

164

1968:

where

+ 2s + 2)(0.25~4+ 3 . 2 4 ~ + 5 . 7 8 2 ~+~4.305s - 0.125) g12(s) = - 2(s2 + 4s + 8)(1.29s3 + 4.50s2 + 6.69s + 3.22) gll(s) = - 8(s + 2)(s2+ 2s + 2)(s34-3.01s’ + 3.94s + 2.02) g’?(s) = 2(s2 + 4s + 8)(s3 + 2 . 6 4 ~+~ 2.08s - 0.444)(s + 2) A(s) = s6 + 5 . 0 8 ~+~1 2 . 1 + ~ ~1 5 . 6 ~ +~1 1 . 7 ~ ~ + 4.60s (38)+ 1.20. gll(s)

= -

q s 2

Response curves for step inputs are shown in Fig. 4. Responses are shown for several combinations of parameter variations, andfor nominal parameter values. For comparisonpurposes,responsesareshown in Fig. 5 for a two-degree-of-freedom design suggested by Peschon[41for achievingthesamenominaltransfer function matrix. Peschon uses t h e configuration of Fig. 1 with

The responses of Fig. 4 clearly exhibit less departure from nominal than do the responses of Fig. 5.

V. CONCLUSIONS In this paper,a design procedure for a minimum sensitivity feedback sy-stem was proposed. The sensitivity index used is a quadratic form in the output error induced by plant parameters which are subject to variation.By using a comparisonsensitivitymatrix,the computation of the closed-loop error for a specified system input can be carried out by first computing the open-loop error, which depends on the plant parameter variations alone. The minimization with respect to the controllers reduces t o a parameter optimization problem involving the comparison sensitivity matrix alone. The separation of the plant parameter variations from the control parameter perturbations greatly simplifies (b) the computation of the minimum sensitivity controllers. Fig. 4. (a) kIinimumsensitivitysystemoutput y l ( t ) for unit step The method was applied to a multivariable dc to ac input u 1 and unit step input ZL? of dc to ac converter examplefor se\.eral plant parameters. The four plots correspond to (VO= 1 0, converter example. The system has two plant paramv0’=1.0), ( ~ 0 = 0 . 8 ,ro’=O.S), (zo=1.2,, el0’=1.2), and (00=0.8, eters subject to variation. Based on the design procevo‘ = 1.2). One division of-the horizontal scale corresponds t o one normalized timeunit.Ten d;visions of the verticalscale cor-. dure, 1 2 controlparameters werechosen t o achieve respond t o oneunitoutput. (b) Mihimumsensitivitysystem minimum sensitivity. output y?(t) for inputs,. parameters, and scales as in (a).

165

PERKINS et af.: MINIMUM SENSITIVITY SYSTEMS APPENDIS

InthisAppendix,thelinearizedequationsforthe rotating converter are obtained. The converter consists of a dc and an ac machineconnected in tandem, as shown in Fig. 6. The following nomenclature is introduced.

Dc Ifackine I.'= armature voltage I=armature current E!= field voltage i = field current w = angular shaft velocity R =armature resistance r =field resistance Motor torque: T = k l i l , neglecting saturation in the dc machine. Back emf: e, = kliw, neglecting saturation. Armature and field inductances are negligible.

-4c Machine 1;' I' v'

= rms = rnls

generator voltage load current = field voltage 'i = field current Rl = load resistance 7' = field resistance I' = field inductance Back emf constant = kz (neglect saturation). Torque constant= ka (neglect saturation). Armature inductance and resistance are negligible. J=mornent of inertia of both rotors. Rotor mechanical damping and shaft springiness are negligible. Using the above assumptions and nomenclature,

V

=

'I =

I R f kliw

ri

(44) (45)

The variables v and v ' are designated as inputs, and w and V' as outputs. V is to be fixed, a constant. Other variables are to be eliminated. Thus, (45) becomes

(b) Fig. 5. (a) Peschon system output pl(t) for inputs, parameters, and scales as in Fig. 4(a). (b) Peschon system output y,(t) for inputs, parameters, and scales as in Fig. 4(a).

Similarly, (43) becomes

resulting operating found. Suppose, be point canfor example, that the desired nominal operating point is given b y

I'

I

2'0

=

1.0

wo =

vo'

=

1.0

vo'

io = 1.0

f V -

io'

Fig. 6 . Dc t o ac r o w converter.

1.0

= 1.0

lo' = 1.0.

= 1.0

From (41) through(45), T h e nonlinear differential equations (46) and (47) describe the converter in terms of inputs v and v', outputs V' and a. Kext, consider operation about the static operating point

v

=

Vo = R + k1

1 =r

w = wg

cg

V'

= z;'

=

vo'

v = vo

(48)

Hence,

where subscript the 0 denotes value the of variable the at the operating point. Equations (46) and (47) show that the operating point values are related by

Vo=R+l

r = l r'

=

1

and will yield thedesirednominaloperatingpoint. If Vof (50) R=0.2, for example, then V 0 = 1.2. Two further mak')(tjo chine parameters, static affect operatthe not which do specified independently. Suppose Sndl-signal variations about this operating point areing point, may be defined by J = 1, 1' = 0.5. (57)

vo' = -. If

y1 = w - wo y 2

=

Vf-

261 = I:

v ;

212

=

- 2'0

vr - ?o'.

(51)

Linearization of (46) and (47) about the operating point (48), using the relations (49), (50), and (51), yields the small-signal equations

143th thenlachineparameterschosen as in (56) and (57), (52) and (53) become, for m y operating point,

+

[jtli?

-

v L] wo?

2v;

yl

+[ TI ya =

[6 - l O ~ ~ ~ w (58) ~]z~~

and

and

-

I' [ G I 9 2

- [2]91+

[&]Y. -

[Z]

y1 = U?.

[Z]Y1

=

up.

For the example treated in Section IV, oo and v: are selected as plant parameters. Hence,VO'and w o must be expressed in terms of erg and vi. This is done using (49) (j3) and (50) withmachineparametersasin (56) and (57).

Equations (58) and (59) then become If the desired operating point values are specified, some of the machine parameters canbe determined; 01, conversely, if some machine parameters are given, the

9, + [5c02 - 2';~]y+ l [2v;]y2

=

-6

5v02

+

8012

PERKINS et ul.: M I NSYSTEMS I M U M SENSITIVITY

167

Jose B. Cruz, Jr. (S’56-MI’57-SM’61F’68) was born in Bacolod City, Philippines, onSeptember 17, 1932. H e received the B.S.E.E. degree (su~mnucum These equations are used as the plant model in (13) lazldej from the University of the Philippines, Diliman, in 1953; the S.M. degree and (14). of from the Massachusetts Institute Technology, Cambridge, &lass., in 1956; REFEREKES and the Ph.D. degree from the Univer[‘I \V. 11. LIazer:“Specifications of the linear feedback system sity of Illinois, Urbana,in 1959; allin sensitivity function, I R E Trans. Automatic Control, vol. -IC-5, pp. electrical engineering. 85-93, June 1960. From 1953 to 1954, he taught at the I- of California,Berkeley. He is also a Research Professor at [(I J. Peschon, “Multivariable and timeshared systems,“ in Dis- the Coordinated Science Laboratory and an &Associateof the Center ciplines afzd Tecl~niguesof Systmrs Control, J.Peschon,Ed. Sew for Advanced Studies, University of Illinois. H e was a n Associate York: Blaisdell Publishing Co., 1965, ch. 3. ON CIRCUIT THEORY from 1962 Editor of the IEEE TRANSACTIOXS Is] D. M . Salmon,“Minimaxcontrollerdesign,”Coordinated Science Lab., Univesity of Illinois, Urbana, Rept. R-358, Jull- 1967. to 1964. Since 1966 he has been the Chairman of the Linear Systems P. Kokotovit. “Method of sensitiritJ; points in the investiga- Committee of the G-AC. He is coauthor (with 31. E. Van Valkention and optimization of linear control systems’! (in English), Az~to- burgj of the book Introdtlctor>l Sign& and Circuits (\Valtham, mation and Remote Coziro!. vol. 25, pp. 1512-1518. December 1964. >,lass.: Blaisdell Publishing Co., 1967). C7] H. H. Rosenbrock. “-An automatic procedure for finding the Dr. Cruz is a member of the .American Association of University greatest or least \-due of a function,’ Compz&l J.: vol. 3? pp. 174Professors, Eta Kappa Nu, Phi Kappa Phi, and Sigma Xi. He is 181,October 1960. listed in Awzeriraz M e n of ScB~-zceand Who’s W7zo i n the Mid.aest.

William R. Perkins (S’55-$1’61) was born in Council Bluffs, Iowa, on September 1, 1934. H e received t h e A B . degree (cum lazldej in engineering andapplied physics from Harvard University, Cambridge, AIass., in 1956; and the 1I.S. and Ph.D. degrees from Stanford Vniversity, Stanford, Calif., in 1957 and 1961, respectively. During the summer months of 1957 and 1958 he worked on systems problems at the Sylvania Reconnaissance Systems Laboratories,Mountain \-iew,Calif. -It Stanford he \\-asa Research .Assistant in the Department of Electrical Engineeringfrom 1957 to 1960, and served as Instructor in 19591960. Since February. 1961, he has been at the University of Illinois, Vrbana! where heis currentl>--1ssociate Professor of Electrical Engineering and Research .‘issociate Professor in the Coordinated Science Laborator)-. Dr. Perkins is a member of the American Association of Unirersity Professors and the American Xssociation for the -4dvancement of Science.

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Richard L. Gonzales (S’57-IL1’60) was born in Oak Park, Ill., on September 10, 1933. He received the B.S.E.E. and 1I.S.E.E. degreesfrom Bradley University,Peoria, Ill,, in 1960 and 1963, respectivel?-, and the 1’h.D. degree in electrical engineering from the L-niversity of Illinois in 1966. He served in the U.S. -Air Force from 1953 to 1957 and was employed by .AutomaticElectricLaboratories,lnc. as a Research Engineerin 1960. SinceSeptember, 1960, he has been a member of the faculty a t Bradley L-niversity where he is presently an Associate I n 1965 and 1966, he \vas Professor of ElecrricalEngineering. alvarded a Sational Science Foundation Science Faculty Fellowship to complete his dissertationresearch at theUniversity of Illinois Coordinated Science Laboratories in the field of control system sensitivit?-. He has also been an analog simulation consultant for Caterpillar Tractor Company. Dr. Gonzales is a member of Sigma Si, Sigma Tau, Eta Kappa S u , and Sigma Phi Delta.