Computer Aided Design of Controllers for Large Systems

Copyright c" IFAC Computer Aided Design Indiana. USA 1982

PLfl'\ARY SESSION III

COMPUTER AIDED DESIGN OF CONTROLLERS FOR LARGE SYSTEMS E.

J. Davison*

Department of Electrical Engineering, University of Toronto, Toronto, Ontario , M5S lA4 , Canada

Abstract. A description of some problems arising in the computer-aideddesign of controllers for "large scale systems " is given, which do not normally arise in "normal" multivariable system problems. In particular, three separate classes of problems are described, which arise from the following types of situations: (1) Design of controllers for a multivariable system in which the controllers are constrained to be decentralized. (2) Design of controllers for a multivariable system which has a large number of high frequenc y modes present with little damping, e.g. in large flexible space structures. (3) Design of controllers for large scale interconnected systems in which there is great uncertainty about the model, e.g. in interconnected power systems. The problems which arise from these situations are particularly challenging to the designer. Many questions are raised; some tentative solutions ar e suggested. Keywords. Computer-aided-design; l arge systems; decentralized control; large flexible space vehicles; modelling. INTRODUCTION There has been a great deal of activity paid to the design problem for the multivariable servomechanism problem the last few years, e.g. see (1]-[3] . In this type of problem, an approximate mathematical model of the plant to be controlled is assumed to be known to the designer, and various constraints, specific to the particular plant being studied, must be considered in the problem formulation. Typically, the following type of conditions must be satisfied in obtaining a satisfactory solution to this servomechanism design problem: (1) The closed loop system should be a~ympto­ tically stable, with a desired "fast" transient behaviour. (2) Asymptotic tracking and regulation should occur for all reference signals and for all disturbances (measurable or unmeasurable) of a given class, and such that "low-interaction" between the control loops occurs. (3) Properties (1), (2) should occur in the presence of plant parameter perturbations i.e. the closed loop system behaviour

* This work has been supported by the Natural Sciences and Engineering Research Council Canada under grant #A4396 and by the Killam program of the Canada Council of Canada. 507

shou ld be "robust" in the l'resence of changes of plant parameters or plant dynamics. (4) Various constraints on integrity of the closed loop system, damping factor of the closed loop system, mag nitude of controller gains etc. may have to be imposed on the problem formulation. The general approaches suggested to solve this problem can be decomposed into three classes: (1) Frequency -domain design methods, e.g. see [1], [4]. (2) Graphical-type design methods, e.8. see [5], [6]. (3) Parameter optimization methods, e.g. see (7]- [10] . It is the purpose of this paper to outline some concerns associated with this servomechanism design problem, which do not normally arise in the "normal multivariable de sign problem", and II"hich are particularly challenging to the designer. In particu l ar, three classes of problems are described II"hich arise from the design of controllers for "large-scale systems". These problems may be classified as folloll"s: I. Design of controllers for a multivariable system in II"hich the contro ll ers are constrained to be decentralized [llJ. This will be called the decentralized servomechanism design problem.

508

E. J. Davison

Design of controllers for a multivariable system which has a large number of unknown high frequency modes present with little damping. This will be called the spill-over problem [12], [13]. Ill. Design of controllers for large scale interconnected systems in which there is great uncertainty about the augmented model. This will be called the large scale modelling problem [14]. 11.

THE GENERAL DESIGN PROBLHl In order to illustrate what we mean by the "normal servomechanism design problem", the following type of simplified problem description is given: Assume that the model of the plant to he regulated is described by the following linear time-invariant system: x

Ax

+

Bu

+

Ew

(1) Du + F'A' Y C x + 0 u + F :.;.; Ym m m m n where x ER is the state, u ERm are the control r inputs, y€R are the outputs to be regulated, rm Ym ER are the measurable outputs of the sys~ e . t he tem, wER are d'Istur b ances an d e=Y-Yref IS

Cx

+

r

error in the system, where Yref ER

.

.

IS a gIven

set-point. Assume that the plant (1) is open loop asymptotically stahle, and for simplicity, assume that the disturbances w and setpoint Yref are constant. It is desired to find a controller for (1) so that: i) The closed loop system is asymptotically stable. ii) Asymptotic regulation occurs, i.e. n r lim e(t)=O, ~~ RJ , Vy f f R , Vx(O) ER re t-+
sists of finding a controller for (1) which satisfies conditions (i)-(v). The following result is obtained: Proposition 1 [9]: There exists a solution to the servomechanism design problem for (1) if and only if the following conditions are all satisfied: (A

B',

DJ =n+r, i.e. m~r and the trans-

1.

rankle

2.

mission zeros of (C,A,~,D) exclude the origin. The output y can be measured, i.e. YCYm'

The following generic result follows from [IS]. Corollary 1: Assume that the plant (1) is open loop asymptotically stable and that the outputs y can be measured. Then there exists a solution the servomechar.isn design problem for "almost all systems" if and only if the number of inputs is at least equal to the number of outputs to be reZl~';} ted (i. e. m~r) . Thus this result states t!.at under mild conditions, for systems of the tyre (1), one should be able to solve the servomechanism design problem for "most systems". 2.1

Controller Design for tLe Servomechanism Design Problem

Assume that Proposi tion 1 !lolds. Then a controller which solves the design problem consists of two elements: i)

A "servo-compensator" [IS;; in the case of constant disturbances and set-points, this consists of the following set of integrators: 11

=

e

(2)

ii) A "stabilizing-compensator" [IS] described by u = KO ~ + KI1 + KlYm

t

(3)

A O~ + AlYm + A211

It is to be noted that the stabilizing compensator is not unique, and that an "optimal stabilizing compensator" consists of state feedback [9], i.e. (4)

The "optimal choice" of compensator parameters KO' K, Kl , AO' AI' A2 can be carried out by using either frequency domain concepts; e.g. see [1], [4], graphical design methods, e.g. see [5], [6] or parameter optimization methods, e.g. see [7]-[10]. If the parameter optimization method of [9] is used, the performance index: J=E

C

(6z'Q6z+).16u'R6u)d T,

(5)

Q>O, R>O, ).1> 0 where 1'+0 as defined in [9] is minimized with respect to the compensator parameters KO' K, K ; A ' l O AI' A2 of (3), subject to the constraints out-

509

CAD of Controllers for Large Systems lined in condition (v). The following example gives an illustration of this design method. Example 1: A pressurized head box described by the following equations: x =

l -0.395

0.01145'J' 0 x

l-0.011 +

1 .038'J" 0 u

10.03362 \,0.000966

~lx

y

+

+

Ew

(6)

Fw

+

is to be regulated against constant disturbances such that the output tracks constant set-points. In this case, it can be verified that Proposition 1 holds and on assuming an output proportional term is used for the stabilizing compensator, the following robust controller is obtained:

a centralized controller configuration is to be used in the design problem. In the case of large scale system problems, it may often be unrealistic or uneconomical to consider such a configuration, and so one is obliged to study the servomechanism design problem from a decentralized control configuration viewpoint. Much attention has been recently devoted to the problem of decentralized control, e.g. see [11], [17]-[20], but the emphasis of the results obtained is related to structural, stability or existence type results and not to design. In this case, the plant (1) is assumed to be open loop asymptotically stable and to be described by: \!

rt

u = KOY

+

KI J

o

-8

the following results

t = 0.565 op closed loop = (-2.2 ±J·2.2 -44 ±J'62) elgenvalues ' 1 28.3 -4560 I 1750 -lOOOO'J' (KO,Kl)opt= -85.3 150 1 -5590 340

J

l

It may be verified that excellent tracking/ regulation occur. If it is in addition desired that the c losed loop system have a " gain-margin" of (2, 0.2) [16], (i.e. the closed loop system should remain stable if the "gains" of the plant are changed by a factor of 2 or 0.2), the following results are obtained on minimizing (5) subject to this constraint: J

0.901

opt closed loop . 1 = (-1.4 ±j1.5, -5.3, -437) elgenva ues • ) [2320 245 6690 281 J' ( KO,K l opt= -504 -820 -1960 -1580

It is seen that in this case, the gains of the controller are reduced and the time response of the closed loop system is slower compared to the case when this gain-margin constraint is not imposed. A description of some classes of problems are now described in which the design of controllers is not as straight forward as has previously been described. These classes of problems all arise from a "large-scale system" type of problem consideration. CLASSES OF PROBLH1S WHICH ARE PARTICULARLY CHALLENGING FOR THE DESIGN PROBLEM 3.1

The Decentralized Servomechanism Design Problem

The previous problem description has assumed

+

,.

L B.1 u.1 i=l

Ew

+

y.

Cix

+

D.u.

+

F. w

m Yi

Cmx

+

Dmu.

+

Fmw

1

(Y-Yref)d T

and on minimizing the performance index (5) with Q=I, R=I, u=lO are obtained [9]:

= Ax

x

1

1

1

1

1

1

(7) }i=1,2, ...

, \!

1

i.e. the plant has v control agents associated m· In this case, u. eR 1 is the input . 1 ri to the lth control agent, Yi ER is the out-

with it.

put to bemregulated at the ith control agent m

r·

and Yi ER

1

is the measurable output of the

ith control agent.

The error in the ith con-

trol agent is now given by ei=Yi-y~ef' where i

Yref is the ith agent's set -point. A decentralized controller for (7) is now constrained to have the structure: u.

i m KCY i

t·1

'\~ E: i

1

1

+

KI E: i

+

+

li m " 1Yi

+

Ki i 2Yref fl

i

i

'2Yref

(8) i=I,2, ... , v

It is now desired to solve the same servomechanism design problem as described before with the additional constraint that the controller be decentralized, i.e. it is desired to find a decentralized controller for (7) so that condit ions (i) - (v) of the "General Design Problem" are satisfied. This problem is called the decentralized servomechanism design problem for (7). The following result is obtained: Defini tion: C* = 1

C* 2

C

\!

[:~

Let 0 I

rl

0

01

0

o! 01

rcm

0

0

lo

0

Ir

ICm 1 \!

0

0

lo

0

, 0 ... I r J v

1 2

01 2 0

II

E. J. Davison

510

Proposition 2 [21): There ex ist s a solution to the d ecen tralized servomechanism d es ign problem for (7) if and only if the following condit ions are all satisfied:

L

fl:jl '~~~-~ ,I-I::· ':, '~2 ° l,< ° 1

:

,Cr :

2.

II:

?2

llO

°

does

not have a decentralized f ixed mode [11) at 0 . The output Yi can be measured at the ith m

control agent's station, i.e. Yi cYi'

iii ) G =(2,0 .5 )

senso r 2a in constraint (i.e. the resultant closed loop sys t em should remain s table if the gain of a sensor changes by a factor of 2 or )5)

i v) integrity (wi th ( 1=0.25)

i nt eg rit y cons tra int ( i .e. t he eigenva lue s "-1 of the closed loop system satisfy the cons trai nt Re( A ' )~E l for the case of a seAsor failure in output Yl or Y2)

5

In thi s case, the foIlowin ~ results are obtained from [21]:

Control l e r Structure [11], [21]: Assume that Proposit i on 2 holds; then any contro ll e r ~hich solves the decentra l ized servomechanism design prob l em must have the follow ing struc ture:

i=] 2

1"',

J

1

,i ,

o"' i

"i

J

•

••

, \)

(9)

+

~h ere ~,=e. is a decentra l ized servo-compen1 1

= 2.05 opt c lo sed l oop e l. genva 1ues = (- 1. 6C ±jO.98, -4.13, -1 7.1) J

[o k~

~

~ i ~

.1

m .

i)

.

i i)

In this case, it is not obvious how th e f requency domain or graphical design methods can be extended to find th e "optimal choice" of compensator parameters to so l ve the prob l em. 1I0\\:ever, the parameter opt imi za t ion method can sti ll be used to determine compensa tor parameters. For exampl e, if the parameter optimization method of [9] is used, then the performance index (5) can be minimized with respect to t h e compensato r parameters Ki'

,.0

~i'

,. 1

~i'

, ~, " , ~, i= I ,2, .. . , '; of (9), subject t o

the co nstraints of co ndi ti on (v). Th e following examp l es give an illustr a tion of this design me th od . Examp le 2 : Conside r t he same prob l em as given in example 1, in ~hich the controller is no~ cons t rained to ha ve the following decentra li zed structure : lUl l =

r. k~

l u2 ·I

1,

°

1

°0'1 )'1 1 + k 2'

I}' 2 ·I

rkl

°

1 J, t

i' O k 2 '!

°

rYl-y~efJ' .y 2 - y re f

It is desired to minimize the ~erformance index (5) ~ith Q=I, R=I, ~ =10 - subject t o the follo~ing constraints : i)

I O~ 02 ? ? lk l - +k 2 +ki +k;

ii)

~

? 0.5

~ 2000

gain constra int

damping factor const r aint ( i.e. the eigcnvalues ~ i of the closed loop system satisfy the c onstraint I Re( ~ .) 1~~! I m( ~ .) ! , i = 1 , 2 ~ .. , , \. ) 1

0 J"

kO 2 I 0 k2

= [' - 714 (]

op t

°

-1860

o

10 9

It may be verified that all constrai nt s are satisfied; i n particular , t! ,c active con straints are given as fol lo ws:

sator, and where the decentra li zed stabi li zi ng compensator C, i =" oo, i +, l Y1 I S not unIq ue.

0 l kl

When output sensor Yl fails, the eigenva lu es of the r es ultant c lo sed loop system a re given by (0 . 249±j l.2 8,0,-0 . 89)

,,{~ 2 + k ~2

+ ki + k; = 1999.6

In this case, it can be seen that the decen tra liz ed case is onl y "slightly worst" than the centra li zed case of examp l e 1, a nd is a much simpler con tro l configur ation to implement. 3.2

The Spill- Over Problem

I n this case, the same se rvomechanism design problem as described previously is to be considered except that the open loop plant is no longer assumed to be asym pt o tic al l y stable. In par ti cu lar, i t is assumed that: i)

The order of the plant is finite but arbitrarily l arge. i i) The e i genvalues of the open l oop pl ant are distributed along th e imaginary axis inc l uding the origin. ii i) Th e "l ow freque ncy behaviour" of t he pl ant is knolm, but th e "high frequency behav iour " is unk nown. In parti cular, it is assumed that the plant's model is given by: (

,

(EL 1 + iBLL + I '" lBH ! Efl J

1 I" L o 1rXL I I',xH)I I 0 ) H'11"xHI r; L

(x

Y

(CL

CH)

r

(C~,C~)

'I

I' LI + F..J IX

Ym

l

l',

H"

X

" I

L[ AB'

+ F

0J

m

(l0)

5 11

CAD of Controllers for Large Systems n

where xLER

n

L

H

, xHER ,where n H is arbitrarily m m large, where CL' CL' AL, BL is known and CH' CH' All' BH is unknown. Here the eigenvalues

x y

of A , AH are symmetric imaginary values, L where the eigenva lu es of AH ar,e a ll larger in

This type of plant description typically occurs in modelling large flexible space structures (LFSS) [12], [13], [22], in which the eigenva lue s of the system at the origin correspond to the "rigid-body modes" of the vehicle, and the imaginary eigenva lu es of the system correspond to the "flexible-body modes" of the spacecraft. Genera ll y the l ow frequency flexible modes of the vehicle can be modelled, but very little is known about the high frequ e ncy modes of the system.

If rot _~2 ot k +

+

rOjw

BJ

O)x

(B' rB'

Ym

absolute magnitude than the eigenvalues of AL' The eigenvalues of AL may include a multiplicity at the origin.

[

o

la

Fw

+

1

B' , x

+

(12)

[:l~

and based on (12 ), the following generic result can be established: Proposition 3 [22]: There exists a so lut ion to the robust decentralized servomechanism spill-over problem for almost all systems ( 12), if and only if the number of control inputs is at least equal to the number of rigid bodi modes of ( 12 ). Assuming that a solution exists, the follow ing controller will so l ve th e servomechanism problem for (12) : rt

u = KOY

In this case, an " approximate mode l" of the plant ( 10 ) may be assumed to be given by: XL

\xL

+

Y

CL XL m CL XL

+

Fw

+

Fmw

Ym

BLu

+

KIy

+

K

10 (y - Yref)U T

(13)

and in thi s case, the "optimal" con troll er parameters KO' Kl , K can be found hy minimiz-

ELw (11 )

and a contro ll er can be designed to satisfy the problem statement based on (1 1). However it is essential that the controller so obtained should have the property that when applied to (10) (in whi ch nil' CH' C~, AH, BH are unknown), the resultant c l osed loop sys tem shou ld not be unstable. This is ca ll ed the "spill-over problem" in the space vehicle con trol l iterature. If a contro ll er exists to solve the robust servomechanism problem for (1 1) and which so lv es t~e spill-over problem for (10), then there is said to be a solution to the robust servomechanism spillover problem for (10). It is clear that this extra constraint imposed on the controller design problem is a very demanding one, and that, in general, it is very difficult to obtain existence conditions for a contro ll er solution to exist. 3.2.1

+

ing the performance index (5) as was done previou s ly. Example 3: As an illustration of the results which can be achieved using this a pproach, a LFSS model developed at Purdue [23] was considered. It consists of a r ec tangular plate with no edge constraints, and contains a uniformly distributed elastic co ntinuum with a rigid body located at the center of the "pl ate". It is assumed that there are five point force actuators, co-located with five displacement and di s plac ement-rat e sensors, lo cated at the four c orners and at the center of the rigid body. The sys tem equations are mouelled by ( 12 ) where n=200, m=r=S. The numerical va lues of a ll parameters used may be found in [23]. In this case , some of th e dominant natur a l frequencies of the open loop system arc given in Table 1, and the fixed modes of th e system are given in Table 2. On minimizing the performance index (5) , the following c l osed loop eigenvalues of the system were obtained on using the controller ( 13 ) (see Ta b le 3).

Tentative Approach to the Problem

If one exami nes in detail the structure of a particular plant model corresponding to ( 10), then one may be able to obtain some insight into this type of problem. For examp l e, if one assumes (10) describes a LFSS with co located sensors and actuators with zero damping, and if one allows position and rate sensors, then the LFSS can be described by:

It is seen that, except for the fixed modes, the eigenvalues of the closed loop system have be en considerably damped as desired. 3.3

Large Sca l e

~fodelling

Problem

This type of problem is also associated with plant modelling and occurs typically when a large sca le system consisting of a large number of interconnec ted subsystems, is to be modelled. The problem of how to model such a large scale system is particularly chal lenging, i.e. a direct attack on model ling the l arge scale system by augmenting the interconnected subsystems to gether may require a model having an excess iv e num ber of

E. J. Davison

512

states (e.g. n»lOOO). It is clear that simplification methods, e.g. [24], could be used to reduce such a composite model to produce a reasonable simplified model, but the development of the composite model itself would be unrealistic for very large scale systems. A question which immediately suggests itself to solve this large scale system model problem therefore is the following: would it not be possible to simplify the models of the subsystems themselves in some appropriate way and then consider augmenting the simplified subsystem models together to form an approximate simplified composite system model? In fact this type of approach is used (at least implicitly) in actual engineering practice. It will be shown that unfortunately this type of modelling procedure may lead to quite erroneous results. This is because the dominant time behaviour of the composite system ~odel may arise, in fact, from the interaction of the "high frequency effects" of the interconnected subsystems, and if the "high frequency" effects of the subsystems are ignored in modelling the composite system, then the behaviour of the composite system model may differ considerably from the exact composite system model. The implication of this result is that the effective modelling of very large scale systems may require constructing quite high dimensional models for the subsystem models and that great caution should be taken when developing large scale composite models.

7th order models are "satisfactory" approximate reduced models of the original system. In this case, a composite model of the interconnected system of Figure 1 may be directly obtained on using the simplified models of the synchronous machine; on doing this, the open loop dominant eigenvalues of the composite system are given in Table 6. It is seen that . some of the dominant eigenvalues of the resultant simplified composite system, using the 4th order model, agree well with the exact composite system, but that the dominant oscillatory modes are approximate~ 10x slower than the correct dominant oscillatory modes. This implies that the resulting simplified composite model would have an exaggerated oscillatory mode present in its dynamic response compared to the exact system (verified by simul tat ion) . Thus it is concluded that even though the reduced 4th order model of a single synchronous machine is satisfactory, the resulting model obtained by grouping together 9 synchronous machines is not satisfactory! On the other hand, modelling a single synchronous machine by a 7th order reduced crder model produces a satisfactory composite system model. This implies that extreme caution must be exercised when modelling large scale systems; the fact that a single subsystem has a "good model", will not necessarily imply that the composite system has a "good" model representation.

The following example illustrates this effect. 3.3.1

An Example Showing the Large Scale System Modelling Problem

Consider the power system of Figure consisting of 9 synchronous machines. It is desired to obtain a linearized mathematical model for this system about a given operating point [25]. In this case, it is assumed that a model of the individual synchronous machines is known; in particular that a single thermal machine is described by a 12th order model and a single hydraulic machine by a 10th order model [14]. In obtaining these models, an attempt at including the more important dynamic processes associated with a synchronous machine was made, e.g. damper winding, governor action, prime mover action, exciter etc. is included. The eigenvalues of the resulting linearized model of a single thermal synchronous machine obtained in this case are given in Table 4.

It is concluded that the modelling of very large systems is a very delicate affair - it may well be that the only reliable way of modelling is to use high dimensional models in the subsystem representation. 3.3.2

Example Showing Controller Design Based on Subsystem Simplification of a Large Scale System

Given the power system of Figure 1, it is desired to find decentralized controllers to solve the "load and frequency control problem" (25] for the system, based on modelling a single synchronous machine by a 7th order reduced order model. In this case, the interconnected power system is described by a 67th model and the following controller is found to solve the decentralized servomechanism design problem [14] rt .:iTing = K 2 1 10 ACE I ( t )d "rt

Assume now that a simplifjed model of the above single machine has been obtained (of order 4th, 7th say respectively) so that the eigenvalues of the simplified single thermal machine are now given in TRble 5 [14]. Some typical output simulations of the exact and simplified 4th, 7th order model synchronous machines are given in Figure 2. It can be seen that at this point both the 4th and

6Ting

5

K

2

J

o

ACE 2 ( : )d :

(14 )

rt

I

6Ting 8 = K3 ACE 3 ( 1)d t )0 where ACE , i=1,2,3 are the "area control i errors" of areas 1, 2, 3 respectively and 6Ting , 6Ting , 6Ting are the inputs to the 2 5 8 prime movers of the principle synchronous

CAD of Controllers for Large Systems machines of areas 1, 2, 3 respectively . On minimizing the performance index (5), the following "optima l" gains are obtained for (14): K =-0.018, K =-0.018, K3 =-0.019. l 2

[8) [9)

Some typical closed loop responses of the reduced 67th order power system and the original lOlst order power system , controlled by (14) for the case of a step function disturbance in torque are given in Figure 3.

[lO)

It is seen that the controller (14) controls the power system so that the net power 6P2 (of area 2) and frequency 6w (of area 2) of 2 the sys tem are satisfactorily regulated as predicted, and that the resultant c losed loop r esponses based on the reduced mod e l and th e original sys tem are quite satisfactory.

[11)

CONCLUSIONS A brief description of some problems arIsIng in the computer-aided design of controllers for large scale systems is made, which are particularly c hallenging to the designer. In particular, the following three classes of ~roblems a re considered: ( 1) The decentralized servomechanism design problem - this involves the si tuati on i n which the controllers are constrained to be decentralized. (2) The spi ll-over problem - thi s involves the case when a plant has a large number of high frequency modes present with little damping pres ent. (3) The l a rge scale system modellin g problem - thi s involves the case of modelling a larg e scale sys t em consisting of a large number of interconnected subsystems. Many questions are raised re the natur e of these problems, and some tentati ve so lution s are suggested as waY 3 to resolve them.

[12)

[131

[14]

(15)

[1 6)

REFERENCES [1) [2)

[3) [4) [5)

[6) [7)

Rosenbrock , H.11., Computer-Aided Control Design, New York, Academic, 19 74. Sai n, I~.K., Peczkowski, J.L., 11elsa, J.L., Alternatives for Linear 11ultivar i ab l e Control, Na tional Engineering Consortium, Inc., Chicago, 1978. IE EE Trans. on Automatic Control, Specia l Issu e on Lin ear riultivariable Control, vo l. AC-26 , No. 1, February 198 1. MacFarlane, A.G.J., Belletrutti, "The Char acter istic Locus Design I!ethod", Automatica, vol. 9, 1973, pp. 575-588. Horowitz, 1.11., "A Syn thesi s Theory for Linear Time-Varying Feedback Systems with Plant Uncer tainty", I EEE Trans. on Automatic Control, vol. AC-20, 19 75, pp. 454-464. Acke rmann, J .E., "A Robust Control System Design", Proc. 1979 Joint Automatic Control Conference, pp. 877-883. Polak, E., Trahan, R., "An Algorithm for Computer Aided Design Problems", 1976 IEEE CDC, pp. 537-542.

[17) [18) [19)

[20)

[21)

[22)

513

l1ayne, D.Q., "Computcr-,\ided Design of Control Systems via 0;-otir:lization", 1979 JACC, pp. 371 -3 74. -Davison, E. J., Ferguson, I.J., "Design of Controllers for thc r~l ti va riable Robust Servomechanism Problem Using Parameter Optimization ;~ethods", I EEE Trans. on Automatic Con trol, vol. AC-26, No. 1, 1981, pp . 93-110. Davison, E.J., Fergllson, I.J., "Design of Controllers for t:,e j'ultivariable Robust Servomechanism Problem Using Parameter Optimization riethods - Some Case Studie s", 2nd Ir:AC \'; orkshop on Control App li cations of Xonlinear Programming and Optimization, Munich, Germany , Sept. 15-1 7, 1 ~8C , pp. 61-73. Davison, E . J ., "The !'.oLust Decentralized Control of a Ge neral Servomechanism Problem", IEE E Tran s . on Automatic Control, vol. AC-21, No. 1, 1976, pp. 16-24. Balas, 1.1. J., "Some A~rroaches to the Control of Large Space Structures", AAS Annual Rocky ~lountain Guidance a nd - Control Conference, ~eystone, Colorado, ~la rc h 10 - 13, 1978. Gran, R., Ros s i, M., " t. Survey of the Large Structures Control Problem", 1979 I EEE Control and ~ecision Conferenc~ pp . 1002-1007. Tripathi, f.i.K., Davison, E .. J., "The Automatic Generation Control of a Multiarea Interconnected Sys tem Using Reduced Order ~fod els", IFAC Symposium on Computer Applications in Lar2 c Scale Power System~ New De lhi, India, 1 ~7C' , Aug . 16-18, pp. 11 7-126. Davison, E.J., Go ld enberg, A., "The Robust Control of a Genera l Se rvomechanism Problem: The Servo Compensator", Automatica, vo l. 11, 19 75, pp. 46 1-471. Davison, E.J., Copeland, B., "Des i gn of the Robust Servomechanism Contro ll er Subject to Specified Gains/Phase Margin Constrai nt s", Dept. of Electrical Engineering, Systems Contro l Report No. 82 02, June 1982. 110. Y.C., ~fitt e r, S .K. (ed itors), Directions in Large Sca l e Systems, Pl enum Press, N.Y ., 1976. IEEE Trans . on Automat i c Control, Specia l I ssue on Decentralized Control, vo l. AC-23, ;\0. ~, April 19 78. Pro c. I FAC Symposium on Large Scale Systems Theory a nd App li cation, Udine , Italy, Ju ne 19 76 ; Tou l ouse, France, June 1979; \\arsalo.·, Poland, .July 1983. IEEE Tran s. on Ci rcui t Theory and Sys tems, Specia l Issue on Large Scale Networks and Systems, vol . CAS-23, \0. 12, December 1976. Davi son, E.J., Chang, T., "The Des i gn of Decentralized Controllers for the Robust Servomechanism Problem Using Parameter Optimization ~lethods", 1982 Automatic Control Conference, Ju ne 1982, Arlington, Vi rginia, to appear. ~est-Vuko v ich, G., Davison, E.J . , Hughes, P.C. , "The Decentral ized Control of Large Flexible Space Structure s ", 20th IEEE CDC, San Diego, Dec . 1981, pp. 949-953.

514

E. J. Davison

[23] Ilablani,ll.B., Skelton, R.E., "Generic Model of a Large Fl exible Space Structure for Control Concept Evaluation", submitted to J. Guidance and Control. [24] Davison, E.J., "A Method for Simplifying Linear Dynamic Systems", IEEE Trans. on Automatic Cor.trol, vol. AC-ll, 1966, pp. 93-101. . _ [25] Davison, E.J ., Tripathi, N. K., "The Optimal Decentralized Control of a Large Po~er System: Load and Frequency Control", IEEE Trans. on Automatic Control, vol. AC-23, Ko. 2, 1978, pp. 312-325. Table 1:

14t.

48 of the 100 Frequencie s

~:

113 .

1.1320E-02 5'11 .8 447£-02 1 6 3.1936E-02 7' 3.9320E-02 5.8471£-02 1 8: , 9'16.6701£-02 10 . 7.2993[-02

110.-=.0-148[-0;;' 1 ~.6985E-0; Table 2:

1.3203E-01 1.5507E-01 1.6438E-Ol 1.8918 E-0 1 2. 14 27£-01 2.2272E-01 18 i ~9. 38783E-01 _ 0 . 3.9136E-01 3 928(,E-Ol 21.1 -I 2)67E-Ol :!2. 4.3: 03 £ -01 23. 14. IS. 16. 17.

'1 1

, 24.

1

I

I

1

S~~tem ~atural

j

1

25.! 26. 27. 28'1 29 . 30 ' 1 31· 32 . 33. 34' 11 35.

E-01 I 5.6604 5.6986E-01 5.8664E-01 6.6187E-01 7.222 9E -01 7.8044E-01 7.83(,2E-Ol 8.5397£-01 8 . 0868r: - 01 9 . 3997[-01 1. 22H7

1 37. 38. 139. 40. 41. 42 . 143 . 44 . 145. 46. 47.

2

1.2622 11 2.5341 3.1148 3.1506 3.1603 3.4258 3.4885 1 3.4897 3.9308 3.9609 4.2036

1

±jO.0730 ±jO.132

±jO.722

±jO.i22

-0.B73 x lO- ±jO.0261 1 -0.133 ±jo .lll

I i

1-0.01B7 ±jO.240 -0.41B XIO- 3 ±jO.500

I

-0.01BO ±jO.504 -0.662 XIO- 3 ±jO.7B2

I

-0.0139 ±jl .30 -0 .II O±j1.36

I

-0.653 x lO- 2 ±jl.92

I

1

1

1

1

-0.05B ±j3.11 -0.llB ±j3.B2

=jO .570

Eigenva lu es of a Sing l e Thermal 11achine l l2th Order ~10de1) Connected tc Inf in i te Bus " J

~

Open-Loop Dominant Eigenvalues of Power System Consisti ng of 9 Synchronous Machines (Original and Reduced Order Systems) RedUCed Order

Original System (IOlst order)

steam prime mover

steam prime mover

-2 .8

e l ect ri ca l

-3.3

steam prime mover

-10.1

steam prime mo " er

-28.2 -37.1 -101 ±j6.6

~th

Eigenvalues of Reduced Sing le Thermal Machine Connected to Infinit e Bus order model

7th order model

-1.2 -1.3:j7.1 -1.~ :j6.7

I

1- 0.12 ~ steam prime 1-0.12 ~ steam prime 1-0 .13 J mover -0.13 ) mover I , i -0 . 25 ± j6.8 ~ e1ectrical l - 0.54 ± j7.2 ·~· I -2. 7 l' 1 1-28 e ectrlca 40

I

I 1

-0 . 12 -0.12 -0 .12

1-

!

I

I=~ :g

=~: ;;

-0 .13 -0.13 1-0.13 -0 .13 -C.13 1-0.ls±jO.19 -0.40

=~::~:·S.l} do~inant l-o.IO! -0.91 !~S.1

Table 5:

-0.05~

=~:;;

-1 .2

-0.05 -0.05 -0.059 -0.12 -0.12 -0.12

' -0.05

-0.13 -0.13 -0.13 -0.13 -0.1:1 -0.15 ±jO.22 -0.47

electrica l

Reduced Order Syste'l': Using 7th Order Single Mac h ine Model (62nd Or;er)

1- 0 . 05

-0.05 -0.05 -0.059 -0.12 -0.1 2 -0 .12

I

S~'stl:m

Using 4th Order Single Machine Mvdel (33th Order)

-0.36 ±j7.2 1 e l ectrical -2.5

flexible body modes

-O.474±j3.41

~jO . 214

-0.12

~ correspond to

-0.030 ±j2.03

Decentralized Fixed Holies

±jO.0730 ±jO.132 tjO.2 14 =jO . 570

-0.13

correspond to fixed modes

-0.2BO XIO- 2±jO '[ - 0.976 x lO- 2±jO -0.1l3±jO l [ correspond to - 0 . 119 ±jO -0.269 ±jO ri gid body Modes 1-0 .559±jO -0.353 XIO- 2±jO.0155

Table 6 : Tab le 4:

I

1

Centralized and Decentralized Fixed 110des of ~;)"stt' m ----,--------~lodes

Typical Eigenva lu es of Closed Loop System with Stabi lizing Compensator App li ed

-0.204 x lO- 4 ±jO.073 -0.733 x lO- 4 ±jO.132 -0.322 XIO- s±jO.2 14 - 0.B67 x lO- 4 ±jO.570 -0 .B30 x lO- 4 ±jO.722

4.82-lUI-~~ 23~_~-l_8_'..L-4_._2_4_%--,

r-r.entralized Fixed

r-

Table 3:

osclllatory ! modes

I

do~inan t

dO~i!'l.a~t

~0.12~ 6.4 OS':ll~atot'y :

-1.0: s.c} -1.i! 5 . 0 oscllla .. ory

-O.IS! 6.5

-1 .3! 1).7

6.S}

1-0.24= 4.8 1-0.24: 4.7 :

I

-0.13 -0.13 - 0 .13 -0 .13 -0.13 -0.15±jO.19 -0.41

.

~oaE:S

j

-1.4: 6.9 -1.4: 7.1 -1.6: 6.2

modes

CAD of Controllers for Large Systems

",.,.,

515

-- - ...... -

, ,, ,! T1\'=--=:....::.=:....~TT o.e+JC)..53

··.., ,

O.1+jCl

I

Figure 1:

0._

A Three-Area Int erconn ec t ed Sys tem

Av i 'r~l.n"' •• c.l

0.002

0 ·0007

66 CD.w.)

Figure 2 :

o

40

60

80

t (sec .)

40

60

80

t (sec.)

-0.002 .0.""

(radians/sec)

-0.004

-0.006

o

20

.o.P. (p.u.)

-0.002

-0.004

-0.006

Figure 3:

Cl osed-Loop Output Response of Power Sys t em with Load and Frequency Con troll er ( 14) Connected for Case of Step Function Dis turb ance in Tor que of Machine 5. (a) Results obtai ned using origina l power system model. (b) Results ob t ai ned using reduced power system model .

Single Th ermal Synchronous Machine Output Responses for a Step Func tio n Disturbance in r l ectrical Torque (a) Origina l 12t11. order sys tem. (b) 4th order reduced system. (c) 7t h order reduced system.