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Large Scale Control System Design
copyright © IFAC Theory and Application of Digital Control New Delhi . India 1982
LARGE SCALE CONTROL SYSTEM DESIGN E.
J. Davison
Department of Electrical Engineering, University of Toronto , Toronto, Ontario M5S lA 4, Canada
Abstract. An overview of some recent results obtained in the area of large scale control system design is made, in which an emphasis is placed on some of the more important differences occurring between the centralized and decentralized control problem. In particular, the following topics are discussed: (i) The large scale modelling problem. (ii) The sequential stability constraint problem. (iii) The local model problem. (iv) Some characterizations of interconnected decentralized systems. A numerical example of the load and frequency control problem for a power system is included to illustrate the results. Keywords. Large scale systems; decentralized control ; interconnected systems; model reduction; robust servomechanism problem; load and frequency control problem. INTRODUCTION
Some Differences Between Centralized and Decentralized Systems
Since the study of the multi variable control problem (both theoretical and design) is far from being complete, e . g. see Sain (1981), it perhaps is premature to consider surveying at this time the area of large scale control systems, which is a significantly more complex topic . In spite of this fact, the area of large scale control systems is a highly active area of research, e.g. see Ho, Mitter (1976), Saeks (1976), IEEE (1976), (1978), Davison (1977), IFAC (1976), (1979), (1982), and many significant results have been obtained.
There appears to be some fundamental differences between the usual multi variable control (centralized) problem and large scale control (decentralized) problems. The following gives some of these more important differences : 1. The Large Scale Modelling Problem. In this case, it is claimed that the modelling of a large scale system (typically consisting of a large number of interconnected subsystems) is particularly challenging, and is not just simply an extension of the modelling problem for a single subsystem.
The general question of what do we mean by a large scale system has no definite answer; generally, however the implication is that certain inputs and outputs of the system are naturally associated with' each other (typically through subsystems contained in the system), and that any realistic control strategy for the system must be constrainted to be decentralized (Wang, Davison (1973)) . This can be contrasted with the usual multivariable control approach, where there is an implicit assumption, at least, that all measurable outputs of the system can be used to control all manipulated inputs in the system .
2 . The Sequential Stability Constraint. In a decentralized control system, it is generally impossible to connect all decentralized controllers to a system simultaneously (due to lack of communications). This implies that the controllers must be connected to the system one at a time (in a sequential way) such that the resultant controlled system remains stable at all times. This type of constraint does not arise in the centralized control problem. 3. The Local Model Problem. Typically, each control agent of a large scale system possesses only a local model of the system, which may be different from each other and generally is incomplete. This differs from the centralized control problem where the control agent (single) has a complete knowledge of the model of the overall system .
This work has been supported by the National Research Council of Canada under Grant No. A4396 and by the Killam Program of the Council of Canada. 23
24
E. J. Davison
This paper shall examine some aspects of the above p~oblems. In doing so , it will be assumed that the reader is already acquainted with the decentralized stabilization problem (Wang, Davison (1973)), the concept of decentralized fixed modes (Wang, Davison (1973)) and the decentralized servomechanism problem (Davison (1976)). The paper is divided as follows: section 2 examines the Large Scale System Modelling Problem, section 3 examines the Sequential Stability and Local Model Problem, section 4 gives some characterizations of Interconnected Decentralized Systems and section 5 gives a numerical example of solving the Load and Frequency Control Problem for a power system using only local models. THE LARGE SCALE SYSTEM MODELLING PROBLEM Given a large scale system ar1s1ng from a large number of interconnected subsystems, the problem of how to model the large scale system is particularly challenging, i . e . a direct attack on modelling the large scale system by augmenting the interconnected subsystems together may require a model having an excessive number of states (e.g. n» lOOO) . It is clear that simplification methods, e . g. Davison (1966), 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 poss * ible 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 .
ing of 9 synchronous machines. It is desired to obtain a linearized mathematical model for this system about a given operating point (Davison, Tripathi (1978)). 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 (Tripathi, Davison (1978)). 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 as follows (see Table 1). TABLE 1
-0 . 12 -0.13
The following example illustrates this effect . An Example Showing the Large Scale System Modelling Problem Consider the power system of Figure 1 consist-
} steam prime mover
-0.36±j7.2
electrical
-2.5
t steam prime mover
-2.8
} electrical
steam prime mover
-3 . 3 -10 . 1 -28.2 -37.1 -101±j6.6
} steam prime mover
electrical
1
Assume now that a simplified 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 as follows (see Table 3) (Tripathi, Davison (1978)) : TABLE 2
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 model 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.
Eigenvalues of a Single Thermal Machine (12th Order Model) Connected to Infinite Bus
Eigenvalues of Reduced Single Thermal Machine Connected to Infinite Bus 7th order model
4th order model -0.12 -0.13
} steam prime mover
-0.25±j6.8}electrical
-0.12 -0 . 13
} steam prim€ mover
-0 . 54±j7 . 21 -2.7 . -28 electr1ca -40
J
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 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
Large Scale Control System Design
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.
open loop dominant eigenvalues of the composite system are given as follows (see Table 3). 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 approximaterylOx 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 simulation). 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 satisfactoryl TABLE 3
DJ 11 iUJ J System
(l Obt Order)
r
I :-o~ii :,---
i~('dlll:-("d Onkr Sy'.t l,'1.l LJ S IH~
.1:h
n~ ' Jl'r
~: i :I;~ Jc
~ ! , I,--- hill1..' ~·kdt'J
C-i:-.tll Ordl'r )
'---- - - - -00 :6,; ,- ._- .. _-0_--
!~('dlJLl' d n' l \ i (. , "
:"th O/",l! ' 1' Si n ;', \.' f.1:h..: hilll' ~hl
-o,o~.
- 0,0,
-O.OS9
-(), O:.,q
- O.O~,9
- 0 , 12 - 0, I ? -0 .1 :' -0 ,1 2 - 0,12 · 0,13 -0. 13 -D. LI -0 ,13
·U. I.' -0. I.?
-0, 1 2 -O .L? - 0,12 -0 . 1 ~ -Cl. I :! -0,13 - 0. L\ -0 ,1 3 - 0 .13 -0 .1 3
- 0. I" -0. 1 2
-0.12 -0. 13 C .13 - O. J j -0.1.' - O.Ll
;\ ·.~ tl' : 1
lbi!ll~
·o.oc,
-0, ]3
SEQUENTIAL STABILITY AND THE LOCAL MODEL PROBLEM It is desired to obtain a realistic way of solving the robust servomechanism problem for a decentralized system subject to the following constraints:
Open-Loop Dominant Eigenvalues of Power System Consisting of 9 Synchronous Machines (Original and Reduced Order Systems)
------ .---- -----
·0,0'.
25
I
(i) The controller synthesis be carried out in a sequential stable way (Davison, Gesing (1979)), i.e . the decentralized controllers are to be connected to the system one at a time (in a sequential way) such that the resultant controlled system remains stable at all times. The motivation for introducing this constraint is that it generally is impossible to connect all decentralized controllers to a system simultaneously (i.e. due to lack of communications, there will always be some time lag associated with connecting the controllers). (ii) Each control agent of the system possesses only a limited knowledge of the model of the system to be regulated, i.e. typically each agent of a large scale system possesses only a local model (Davison, Ozguner (1981)) of the system, which may be different from each other and generally is incomplete. In this case, it is assumed that there does not exist a central authority which possesses a knowledge of the complete mathematical model of the system, i.e. it is assumed that some central authority is not allowed to calculate the decentralized controllers of the system. Development The plant consisting of v control agents is assumed to be open loop asymptotically stable and to be described by the following linear time-invariant system:
On the other hand, modelling a single synchronous machine by a 7th order reduced order model produces a satisfactory composite system model. This implies that extreme caution must be exercised when modelling large scale systems; the fact that a ~ingle subsystem has a "good model", will not necessarily imply that the composite system has a "good" model representation.
The explanation of this effect is that some of the dominant oscillatory modes which describe the behaviour of the interconnected power system result from the interaction of the detailed "high frequency" electrical behaviour of the individual synchronous machines contained in the power system . Thus this implies that an apparent excessive amount of dynamic "high frequency behaviour" may have to be included in describing a subsystem's behaviour in order to accurately model the "low frequency behaviour" of a composite system. It is concluded that the modelling of very
v x = Ax
+
L
B.u.
i=l
1
Ew
+
1
C~x
+
D~u.
+
F~w
C.x
+
D.u.
+
F. w
1
1
1
111
1
, i=1,2, ... ,v
(1)
1
ref Yi - Yi m. where xER n is the state, U.ER 1 1
are the input and measurable output respectively of the i~h control station, w€R~ is a disr. 1, i=I,2, ... ,v is the error
turbance and e.ER 1 in the system.
Cv
D~lock diag(D l ,D 2 ,· .. ,Dv )
E. J. Daviso n
26
(3) y~, i=1,2, .. . , v must contain the output . 1y . 2 , ... , v respect 1ve .1 1=1,
( ref Yl ref Yref=
Yi'
~2
Contro ller Structu re
ref Yv
Assume that lemma 1 holds; then any robust contro ller which regula tes (1) must have the followi ng structu re :
and let
x. ,
(5) i=1,2, ... , v + Koi u. = Kil;. 111 r .p where I; .€ R 1_ is the output of the decent ralized sefvo-c ompen sator given by:
where
C~ [C~
o
+
* B.1e 1. , i=1,2, . . . , v
where
B~ ~ block diag(y , y , ... ,y)
o
<= [C~
r.1 matric es
o
I:!. where y€ RP is given by y = (0 0 .. . 0 1)' and C€RPxP is given by
o
The disturb ance w is assumed to sati s fy the followi ng equatio n:
C I:!. (2)
Cln l
where (Cl,Al~ is observ able. The referen ce input Yref 1S assumed to satisfy the following equatio n:
n2 = A2n 2
The followi ng prelim inary result is obtaine d for a solutio n to the robust decent ralized servom echanis m problem for (1) (Daviso n (1976); Lemma 1. A necess ary and suffici ent for there to exist a solutio n to the decent ralized servom echanis m problem is that the follow ing conditi ons all
conditi on robust for (1) hold:
(1) {Cm,A,S} has no unstab le decent ralized fixed modes with respec t to
K.
(2) The decent ralized fixed modes with respec t
the p s1stem s
A~Il'[:JI' J
1
- 15 2
- 15 3
jJ
(8)
i=l
(A-A . ), i.e.
1 P
(4)
lc:,[:
0
+-' • ••
where (C 2 ,A 2 ) is observ able . Let the minima l polyno mial of AI' A2 be denoted by hl(s), h 2 (s) respec tively and let the zeros of the least common multip le of h l(s), h 2 (s) (multi pliciti es include d) be given by:
Kof
r!!
1 0
where the coeffic ients cl ,c 2 , . . . ,c p are given by the coeffic ients of the polyno mial p IT
(3)
Yref = GC 2n 2
to
(7)
1
1
w
(6)
C.* I:!.= block diag(C ,C, .. . ,C) 1 r . matric es
[C~
C;
• . = C.* I;. 1;1 1 1
j=1,2, . .. ,p
do not contain Aj , j=1,2 , .. . ,p respec tively.
IT (A-A . )
i=l
1 (9)
and xi is the output of a de(ent ralized stabili zing compen sator which stabili zes the resulta nt closed loop system (Daviso n (1976) ). Defini tion (Daviso n, Gesing (1979)) . Assume that the decent ralized contro llers S., i=1,2, ... , v are applied to (1). Thefi if the resulta nt closed loop system obtaine d by applyin g the decent ralized contro llers S. , i=1,2, ... ,k to (1) is stable for k=1,2,.~., v , the contro llers S . , i=1,2, ... , v are called sequen tially staBle contro llers with respec t to contro l agent order (1,2, ... , v). Given (1), the robust decent ralized servomechanism problem with sequen tial stabil ity consis ts of solving the followi ng problem : Find a decent ralized contro ller for (1) so that: (i) There is a solutio n to the robust decentra lized servom echanis m problem . (ii) The contro ller synthe sis is carried out by applyin g a seri es of sequen tially stable contro llers with respec t to contro l agent order (1,2, .. . ,v).
Large Scale Control System Design Definition (Davison, Ozguner (1981)). Consider the plant (1) with w=O, Yref=O. Assume that the control j
j
A
Uj=K ';(KOX j ' j=1,2, .. . ,i - l, h : [2,3, ... , v ] (10) where';., x . are given by (5), are applied to J
J
(1), and let the minimal state realization of the closed loop system obtained by applying the controller (10) to the plant (1) for control agent No . i (with input u i ' output yi), be called the ith agent's local model of the plant (1) with respect to controllers (10) or more briefly the ith agent's local model of the plant. Then a synthesis procedure which solve s the decentralized robust servomechanism problem with sequential stability , in which each control agent is assumed to possess only a local model of the controlled plant , and in which there exists no central decision making authority, is called a local model decentralized robust servomechanism problem. The following assumptions are made in this problem.
27
~.
Using centralized synth~sis (Davison, Goldenberg (1975) ; Dav1son, Ferguson (1981)) and a knowledge of agent No. l's local model of the plant, apply the servocompensator (6) with i=l to the terminals of control agent No. 1 and apply a stabilizing compensator vI =Kl ';l + Xl where ';1 is given by (6) and; is the output of a dynamic compensator so !hat the resultant closed loop system is stable and has a desired dynamic response, i . e. so that all non-fixed modes of agent No. l's local model of the controlled plant are shifted into ag o The resultant s ystem then has the property that YI is regulated.
Step 3. Repeat the centralized procedure of step 2 for each agent 2,3 , . . . ,v sequentially, using as control inputs v. , i=2,3, .. . , v respectively and a knowledg~ of the agent's local model of the controlled plant. The resultant system then has the property that Yl ' Y2' " ·,yv are all regulated and the closed loop system is stable with a desired dynamic response, i.e. the eigenvalues of the resultant closed loop system are all contained in a-, except for those fixed modes (if any) of g
(Cm,A , B) which lie outside of
a; .
Assumptions. 1. It is assumed that each control agent knows the disturbance/reference signal poles Al ,A 2 , .· . ,Ap' 2. It is assumed that each control agent has the same performance criterion, i.e . that the closed loop system be stable and that the eigenvalues of the resultant closed loop system all lie in a certain stable preassigned symmetric region of the complex plane denoted by a (except for any fixe~ modes of (Cm,A,B ) which may lie outside of agO
g
CHARACTERIZATIONS OF INTERCONNECTED DECENTRALI ZED SYSTEMS The following type of problem is now considered. Given a composite system consisting of v subsystems interconnected together, when does there exist a solution to the robust decentralized servomechanism prob l em for the composite s ystem, given that there exists a solution to the robust servomechanism problem for each subs ystem? In particular, assume that the ith subs ystem is described by:
v
The following main result is obtained : Theorem 1 (Davison , Ozguner (1981)). Consider the system (1); then there exists a solution to the local model decentralized robust servomechanism problem for (1) if and only if there exists a solution to the robust servomechanism problem fo? (1) (given by lemma 1) . Assume that theorem 1 holds; then the following synthesis procedure can be used to con struct a controller for (1) . Synthesis Procedure using Local Models (Davison, Ozguner (1981)). ~.
Apply the output feedback control
Am Am~ m m Ui=Kiyi+v i , i=1 , 2, ... , v , Yi=Yi-Diui sequentially to the plant (1), where K. are arbitrary non- zero matrices chosen "sm~ll enough" so as to maintain stability of the closed loop plant.
~ . =A.x . +B . u . +Eiw+ ~ A.. x. 1 1 1 1 1 j=l 1J J ~i
y.=C . x . +D . u . +F . w 11111 1
(11)
y~=C~x.+D~u . +F~w 11111 1
n. m. where x .E R 1 is the state, U.E R 1 is the in1 1 r . put, y . ER 1 is the output to be regulated, m1 m r. Y.E R 1 is the measurable output of the system 1
and that the disturbances, reference signals are described by (2), (3) respectively. In this case, the general interconnection matrix A. . is given by 1J A •. ~ . • K .. 1jJ •.
1J
1J 1J 1J
l:5j:5v ,
( 12)
where Kij denotes the interconnection gain connecting subsystems i and j. The following results are obtained:
28
E. J. Davison
Results Obtained
Yl
In
Theorem 2 (Oavison (1976)). In (11) , (12) assume that there exists a solution to the robust servomechanism problem for each subsystem of the composite system, i.e . that the following conditions are all satisfied for i=1,2, . . . , v :
~2
0
Yv
0
(i)
able.
(C~,A . ,B.) is stabilizable and detect111 A. -A . I
rank
(H)
1
[
)
B.] =n . +r . • 1
O.
C.
1
1
0
1
0 0 x
In :2
0 ... In
v
the following result, which gives a sufficient condition, in terms of the controllability of the subsystem components of (14), for the system (14) to have no decentralized fixed modes present, is obtained.
j=l ,2, ... ,po
1
Theorem 5 (Oavison, Ozguner (1981)). Consider the system (14) and assume that :
1
(iii) y~ contains Yi' then there exists a solution to the robust decentralized servomechanism problem for the composite system provided the interconnection gains K .. are "small enough".
(i) (A . . ,B.), i=1,2, .. . , v are all controllable. 11 1
1)
Theorem 3 (Davison (1976)). In (11), (12) assume that there exists a solution to the robust servomechanism problem for each subsystem of the composite system, and in addition assume that (C~,A.,B.), 111
i=1,2, . .. , v is controllable and observable; then there exists a solution to the robust decentralized servomechanism problem for the composite system for almost all (Oavison, Wang (1973)) interconnection gains K . .•
are all controllable.
1)
Assume now that the composite system (11) arises from interconnections consisting of input-output interconnections, i.e . assume that the interconnection matrix A. . =H . . K .. ljJ • . has the property that: 1) 1) 1) 1) H ..
1)
= B1.
1/! • . 1)
and that 0 . =0. obtained: 1
AI3] A 23
kl
J'
A3 3
0
0
82
0
0
(13)
= C. )
Then the following result is
Theorem 4 (Oavison (1979)). There exists a solution to the robust decentralized servomechanism problem for the composite system (11), (13) i f and only i f there exists a solution to the robust servomechanism problem for each subsystem of the composite system, i . e . if and only if for i=1,2, . .. , v: (C~,A . ,B.) is stabilizable and detect-
(i)
able .
111
AZ ,V _I
A . - A. I
(ii)
rank
1
[
Av_ l, v_ 1
j=1,2, ... ,p.
)
C.
1
(iii) the outputs Yi contain the outputs Yi . Consider now the following special case of (11) :
Bl x= ~2l
~22· · ·~2 v x+ 0
Av , V_1
,\
m
j
Av.. 2 , v-2 v -l, v-2
1''1),\1- 2
A AV-.! , \) - 1
v-l, v- 1 Av,v_ 1
A Av- 2, v
v- I, v Av , v
1 I(B v - 2
l'
are all controllable.
0 . .. 0
(v )
~2 u
v (14 ) is controllable .
0
J lO
0
Bv _ 1
Large Scale Control System Design then the system (14) has no decentralized fixed modes present. Remarks 1. The above results give a good deal of insight into the behaviour of interconnected systems in which local state feedback is allowed. It implies for example that a 2control agent interconnected system in which each subsystem is controllable and in which the composite system is controllable, can always be stabilized using local state feedback with dynamic compensation. 2. The following classification of some observations made in the literature can be made (the so-called Wang-Fessas-Ikeda counterexample (Wang (1978); Fessas (1979); Ikeda, Siljak (1979)) from theorem 5: (i) The "only if" part of Fessas's conjecture (Fessas (1979)) is false for v~2. (ii) The "if" part of Fessas's conjecture (Fessas (1979)) is true for v=2 and is false for v~3. NUMERICAL EXAMPLE : SOLUTION OF LOAD AND FREQUENCY CONTROL PROBLEM USING LOCAL MODELS The same power system example as discussed in section 2 is considered. It is now desired to find decentralized controllers to solve the "load and frequency control" problem for the 3-area system of Figure 1 for the case of any constant disturbances occurring in the . system. In this case, the 67th reduced order model of the interconnected system of section 2 was used, and it may be verified that theorem 1 holds for this example (Tripathi, Davison (1978); Davison, Tripathi (1980)). In particular, the following decentralized robust controller was obtained as a solution to the local model decentralized robust decentralized servomechanism problem so that the closed loop system is asymptotically stable with a desired transient response, and so that the so-called "area control errors" of area 1, 2, 3 are asymptotically regulated to zero, for all constant disturbances which may occur in the power system: t
Kl
Jo ACEl(T)dT t
K2
Jo ACE 2 (T)dT
(15)
t
liT.lng = K3 s
Jo ACE 3 (T)dT
where Kl=-O.OlS, K2=-0.01S, K =-0.019, where 3 ACEi, i=1,2,3 are the "area control errors" of areas 1,2,3 respectively, and lITing2' lIT ing5 , lITing8 are the inputs to the prime movers of the principle synchronous machines of area 1,2,3 respectively. Some typical closed loop responses of the reduced 67th power system and original power system controlled by (15) for the case of a step dis-
29
turbance in torque are given in Figure 3. It is seen that the controller (15) controls the power system so that the net power lIP2 (of area 2) and frequency 6w 2 (of area 2) of the system are satisfactorly regulated as predicted, and that the resultant closed loop responses based on the reduced model and the original system are quite satisfactory. CONCLUSIONS A brief discussion of some recent results obtained re the control of large scale systems is made in this paper. In particular, the decentralized servomechanism problem is examined under the following various conditions: (i) using simplified models to represent the plant, (ii) imposing a sequential stability c0nstraint, (iii) imposing a local model constraint, (iv) obtaining a characterization of the solvability of the decentralized servomechanism problem for interconnected systems. A numerical example of a load and frequency control problem using a simplified model and subject to a local model and sequential stability constraint, is included to illustrate the results. It is concluded that the control of large scale systems is still not well understood, and that many problems of both a theoretical and of a design nature still exist. REFERENCES Davison, E.J. (1966). A method for simplifying linear dynamic systems. IEEE Trans. on Automatic Control, vol. AC-ll, 93-101. Davison, E.J. (1976). The robust decentralized control of a general servomechanism problem. IEEE Trans. on Automatic Control, vol. AC-2l, No. 1, 16-24. Davison, E.J. (1977). Recent results on decentralized control of large scale multivariable systems. IFAC Symposium on Multivariable Technological Systems, Fredericton, Canada, June 1977, 1-10. Davison, E.J. (1979). The robust decentralized control of a servomechanism problem for composite systems with input-output interconnections. IEEE Trans. on Automatic Control, vol. AC-24, No.2, 325-327. Davison, E.J. and Ferguson, I. (1981). The design of controllers for the multivariable robust servomechanism problem using parameter optimization methods. IEEE Trans. on Automatic Control, vol. 26, No. 1, 93-110. Davison, E.J. and Gesing, W. (1979). Sequential stability and optimization of large scale decentralized systems. Automatica, vol. 15, 307-324. Davison, E.J. and Goldenberg, A. (1975). The robust control of a general servomechanism problem: the servo-compensator. Automatica, vol. 11, 461-471. Davison, E.J. and Ozguner, U. (19Sla). Characterizations of decentralized fixed modes for interconnected systems", 8th IFAC Congress, Kyoto, Japan, Aug. 1981, to appear.
30
E. J. Davison
Davison, E.J. and Ozguner, U. (1981b). Synthesis of the decentralized robust servomechanism problem using local models. Systems Control Report No. 8107, Dept. of Electrical Engineering, University of Toronto, March 1981; also 1980 Joint Automatic Control Conference, Aug. 1980, San Francisco, paper no. WAI-A. Davison, E.J. and Tripathi, N. (1978). The optimal decentralized control of a large power system: load and frequency control. IEEE Trans. on Automatic Control, vol. AS-23, No. 2, 312-325. Davison, E.J. and Tripathi, N. (1980). Decentralized tuning regulators: an application to solve the load and frequency control problem for a large power system. Large Scale Systems: Theory and Application, vo!. 1, No. 1,315. Davison, E.J. and Wang, S.H. (1973). Properties of linear time invariant multivariable systems subject to arbitrary output and state feedback. IEEE Trans. on Automatic Control, vol. AC-18, No. 1, 24-32. Fessas, P. (1979). A note on "an example in decentralized control systems". IEEE Trans. on Automatic Control, vol. AC-24, No. 4, 669. Ho, Y.C. and Mitter, S.K. (1976) editors. Directions in Large Scale Systems. Plenum Press, N.Y. IEEE (1976). Special issue on large-scale networks and systems. IEEE Trans. on Circuit Theory and Systems, vol. CAS-23, No. 12, Dec. 1976. IEEE (1978). Special issue on decentralized control. IEEE Trans. on Automatic Control, vol. AC-23, No. 2, April 1978. IFAC (1976). Proc. IFAC Symposium on Large Scale Systems Theory and Application. Udine, Italy, June 1976. IFAC (1979). Proc. IFAC Symposium on Large Scale Systems Theory and Application. Tolouse, France, June 1979. IFAC (1982). Proc. IFAC Symposium on Large Scale Systems Theory and Application. Warsaw, Poland, June 1982. Ikeda, ~1. and Siljak, D. (1979). Counterexamples to Fessas' conjecture. IEEE Trans. on Automatic Control, vol. AC-24, No. 4, 670. Saeks, R. (1976) editor. Large Scale Dynamic Systems. Point Lobos Press . Sain, M. (1981) editor. Special issue on "linear multivariable control systems". IEEE Trans. on Automatic Control, vol. AC-26, No. 1, 1981. Tripathi, N.K. and Davison, E.J. (1978). Th e automatic generation control of a multiarea interconnected system using reduced order models. Systems Control Report No. 7808, Dept. of Electrical Engineering, Universit y of Toronto, Nov. 1978; also IFAC Symposium on Computer Applications in Large Scale Power Sys tems . New Delhi, India, Aug. 16-18, 1979, 117-126. Wang, S.H. (1978). An example in decentralized control systems. IEEE Trans. on Automatic Control, vol. AC-23, No. 5, 938. \\fang, S.H. and Davison, E.J. (1973). On the stabilization of decentralized control sys".:ems. IEEE Tr'ans. on Automatic Control, vol. AC-18, 473-478.
Figure 1 : A Thre ," · \,'""
OC
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Figure 2: Sing le The]'",;,l S::nchronous ~1achine Output Kesponses for a Step Function Disturbance in Electrical Torque (a) Original 12th order system, (b) 4th order reduced system. (c) 7th order reduced system. .....(.)
(r8d 1a n .!-/se c )
-0 .004
1 11
-0006
Figure~:
~1~Jj
l)utI'ut Response of Power SystcrJ with Load and Frequency Controller (15) Connected for Case of Step Function Disturbance in Torque of ~lachine 5. (a) Results obtained using original power system model. (b) Results obtained using reduced power system model. Clo~cd·Lo0;)
-
LARGE SCALE CONTROL SYSTEM DESIGN E.
J. Davison
Department of Electrical Engineering, University of Toronto , Toronto, Ontario M5S lA 4, Canada
Abstract. An overview of some recent results obtained in the area of large scale control system design is made, in which an emphasis is placed on some of the more important differences occurring between the centralized and decentralized control problem. In particular, the following topics are discussed: (i) The large scale modelling problem. (ii) The sequential stability constraint problem. (iii) The local model problem. (iv) Some characterizations of interconnected decentralized systems. A numerical example of the load and frequency control problem for a power system is included to illustrate the results. Keywords. Large scale systems; decentralized control ; interconnected systems; model reduction; robust servomechanism problem; load and frequency control problem. INTRODUCTION
Some Differences Between Centralized and Decentralized Systems
Since the study of the multi variable control problem (both theoretical and design) is far from being complete, e . g. see Sain (1981), it perhaps is premature to consider surveying at this time the area of large scale control systems, which is a significantly more complex topic . In spite of this fact, the area of large scale control systems is a highly active area of research, e.g. see Ho, Mitter (1976), Saeks (1976), IEEE (1976), (1978), Davison (1977), IFAC (1976), (1979), (1982), and many significant results have been obtained.
There appears to be some fundamental differences between the usual multi variable control (centralized) problem and large scale control (decentralized) problems. The following gives some of these more important differences : 1. The Large Scale Modelling Problem. In this case, it is claimed that the modelling of a large scale system (typically consisting of a large number of interconnected subsystems) is particularly challenging, and is not just simply an extension of the modelling problem for a single subsystem.
The general question of what do we mean by a large scale system has no definite answer; generally, however the implication is that certain inputs and outputs of the system are naturally associated with' each other (typically through subsystems contained in the system), and that any realistic control strategy for the system must be constrainted to be decentralized (Wang, Davison (1973)) . This can be contrasted with the usual multivariable control approach, where there is an implicit assumption, at least, that all measurable outputs of the system can be used to control all manipulated inputs in the system .
2 . The Sequential Stability Constraint. In a decentralized control system, it is generally impossible to connect all decentralized controllers to a system simultaneously (due to lack of communications). This implies that the controllers must be connected to the system one at a time (in a sequential way) such that the resultant controlled system remains stable at all times. This type of constraint does not arise in the centralized control problem. 3. The Local Model Problem. Typically, each control agent of a large scale system possesses only a local model of the system, which may be different from each other and generally is incomplete. This differs from the centralized control problem where the control agent (single) has a complete knowledge of the model of the overall system .
This work has been supported by the National Research Council of Canada under Grant No. A4396 and by the Killam Program of the Council of Canada. 23
24
E. J. Davison
This paper shall examine some aspects of the above p~oblems. In doing so , it will be assumed that the reader is already acquainted with the decentralized stabilization problem (Wang, Davison (1973)), the concept of decentralized fixed modes (Wang, Davison (1973)) and the decentralized servomechanism problem (Davison (1976)). The paper is divided as follows: section 2 examines the Large Scale System Modelling Problem, section 3 examines the Sequential Stability and Local Model Problem, section 4 gives some characterizations of Interconnected Decentralized Systems and section 5 gives a numerical example of solving the Load and Frequency Control Problem for a power system using only local models. THE LARGE SCALE SYSTEM MODELLING PROBLEM Given a large scale system ar1s1ng from a large number of interconnected subsystems, the problem of how to model the large scale system is particularly challenging, i . e . a direct attack on modelling the large scale system by augmenting the interconnected subsystems together may require a model having an excessive number of states (e.g. n» lOOO) . It is clear that simplification methods, e . g. Davison (1966), 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 poss * ible 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 .
ing of 9 synchronous machines. It is desired to obtain a linearized mathematical model for this system about a given operating point (Davison, Tripathi (1978)). 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 (Tripathi, Davison (1978)). 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 as follows (see Table 1). TABLE 1
-0 . 12 -0.13
The following example illustrates this effect . An Example Showing the Large Scale System Modelling Problem Consider the power system of Figure 1 consist-
} steam prime mover
-0.36±j7.2
electrical
-2.5
t steam prime mover
-2.8
} electrical
steam prime mover
-3 . 3 -10 . 1 -28.2 -37.1 -101±j6.6
} steam prime mover
electrical
1
Assume now that a simplified 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 as follows (see Table 3) (Tripathi, Davison (1978)) : TABLE 2
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 model 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.
Eigenvalues of a Single Thermal Machine (12th Order Model) Connected to Infinite Bus
Eigenvalues of Reduced Single Thermal Machine Connected to Infinite Bus 7th order model
4th order model -0.12 -0.13
} steam prime mover
-0.25±j6.8}electrical
-0.12 -0 . 13
} steam prim€ mover
-0 . 54±j7 . 21 -2.7 . -28 electr1ca -40
J
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 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
Large Scale Control System Design
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.
open loop dominant eigenvalues of the composite system are given as follows (see Table 3). 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 approximaterylOx 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 simulation). 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 satisfactoryl TABLE 3
DJ 11 iUJ J System
(l Obt Order)
r
I :-o~ii :,---
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~: i :I;~ Jc
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:"th O/",l! ' 1' Si n ;', \.' f.1:h..: hilll' ~hl
-o,o~.
- 0,0,
-O.OS9
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- 0 , 12 - 0, I ? -0 .1 :' -0 ,1 2 - 0,12 · 0,13 -0. 13 -D. LI -0 ,13
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;\ ·.~ tl' : 1
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SEQUENTIAL STABILITY AND THE LOCAL MODEL PROBLEM It is desired to obtain a realistic way of solving the robust servomechanism problem for a decentralized system subject to the following constraints:
Open-Loop Dominant Eigenvalues of Power System Consisting of 9 Synchronous Machines (Original and Reduced Order Systems)
------ .---- -----
·0,0'.
25
I
(i) The controller synthesis be carried out in a sequential stable way (Davison, Gesing (1979)), i.e . the decentralized controllers are to be connected to the system one at a time (in a sequential way) such that the resultant controlled system remains stable at all times. The motivation for introducing this constraint is that it generally is impossible to connect all decentralized controllers to a system simultaneously (i.e. due to lack of communications, there will always be some time lag associated with connecting the controllers). (ii) Each control agent of the system possesses only a limited knowledge of the model of the system to be regulated, i.e. typically each agent of a large scale system possesses only a local model (Davison, Ozguner (1981)) of the system, which may be different from each other and generally is incomplete. In this case, it is assumed that there does not exist a central authority which possesses a knowledge of the complete mathematical model of the system, i.e. it is assumed that some central authority is not allowed to calculate the decentralized controllers of the system. Development The plant consisting of v control agents is assumed to be open loop asymptotically stable and to be described by the following linear time-invariant system:
On the other hand, modelling a single synchronous machine by a 7th order reduced order model produces a satisfactory composite system model. This implies that extreme caution must be exercised when modelling large scale systems; the fact that a ~ingle subsystem has a "good model", will not necessarily imply that the composite system has a "good" model representation.
The explanation of this effect is that some of the dominant oscillatory modes which describe the behaviour of the interconnected power system result from the interaction of the detailed "high frequency" electrical behaviour of the individual synchronous machines contained in the power system . Thus this implies that an apparent excessive amount of dynamic "high frequency behaviour" may have to be included in describing a subsystem's behaviour in order to accurately model the "low frequency behaviour" of a composite system. It is concluded that the modelling of very
v x = Ax
+
L
B.u.
i=l
1
Ew
+
1
C~x
+
D~u.
+
F~w
C.x
+
D.u.
+
F. w
1
1
1
111
1
, i=1,2, ... ,v
(1)
1
ref Yi - Yi m. where xER n is the state, U.ER 1 1
are the input and measurable output respectively of the i~h control station, w€R~ is a disr. 1, i=I,2, ... ,v is the error
turbance and e.ER 1 in the system.
Cv
D~lock diag(D l ,D 2 ,· .. ,Dv )
E. J. Daviso n
26
(3) y~, i=1,2, .. . , v must contain the output . 1y . 2 , ... , v respect 1ve .1 1=1,
( ref Yl ref Yref=
Yi'
~2
Contro ller Structu re
ref Yv
Assume that lemma 1 holds; then any robust contro ller which regula tes (1) must have the followi ng structu re :
and let
x. ,
(5) i=1,2, ... , v + Koi u. = Kil;. 111 r .p where I; .€ R 1_ is the output of the decent ralized sefvo-c ompen sator given by:
where
C~ [C~
o
+
* B.1e 1. , i=1,2, . . . , v
where
B~ ~ block diag(y , y , ... ,y)
o
<= [C~
r.1 matric es
o
I:!. where y€ RP is given by y = (0 0 .. . 0 1)' and C€RPxP is given by
o
The disturb ance w is assumed to sati s fy the followi ng equatio n:
C I:!. (2)
Cln l
where (Cl,Al~ is observ able. The referen ce input Yref 1S assumed to satisfy the following equatio n:
n2 = A2n 2
The followi ng prelim inary result is obtaine d for a solutio n to the robust decent ralized servom echanis m problem for (1) (Daviso n (1976); Lemma 1. A necess ary and suffici ent for there to exist a solutio n to the decent ralized servom echanis m problem is that the follow ing conditi ons all
conditi on robust for (1) hold:
(1) {Cm,A,S} has no unstab le decent ralized fixed modes with respec t to
K.
(2) The decent ralized fixed modes with respec t
the p s1stem s
A~Il'[:JI' J
1
- 15 2
- 15 3
jJ
(8)
i=l
(A-A . ), i.e.
1 P
(4)
lc:,[:
0
+-' • ••
where (C 2 ,A 2 ) is observ able . Let the minima l polyno mial of AI' A2 be denoted by hl(s), h 2 (s) respec tively and let the zeros of the least common multip le of h l(s), h 2 (s) (multi pliciti es include d) be given by:
Kof
r!!
1 0
where the coeffic ients cl ,c 2 , . . . ,c p are given by the coeffic ients of the polyno mial p IT
(3)
Yref = GC 2n 2
to
(7)
1
1
w
(6)
C.* I:!.= block diag(C ,C, .. . ,C) 1 r . matric es
[C~
C;
• . = C.* I;. 1;1 1 1
j=1,2, . .. ,p
do not contain Aj , j=1,2 , .. . ,p respec tively.
IT (A-A . )
i=l
1 (9)
and xi is the output of a de(ent ralized stabili zing compen sator which stabili zes the resulta nt closed loop system (Daviso n (1976) ). Defini tion (Daviso n, Gesing (1979)) . Assume that the decent ralized contro llers S., i=1,2, ... , v are applied to (1). Thefi if the resulta nt closed loop system obtaine d by applyin g the decent ralized contro llers S. , i=1,2, ... ,k to (1) is stable for k=1,2,.~., v , the contro llers S . , i=1,2, ... , v are called sequen tially staBle contro llers with respec t to contro l agent order (1,2, ... , v). Given (1), the robust decent ralized servomechanism problem with sequen tial stabil ity consis ts of solving the followi ng problem : Find a decent ralized contro ller for (1) so that: (i) There is a solutio n to the robust decentra lized servom echanis m problem . (ii) The contro ller synthe sis is carried out by applyin g a seri es of sequen tially stable contro llers with respec t to contro l agent order (1,2, .. . ,v).
Large Scale Control System Design Definition (Davison, Ozguner (1981)). Consider the plant (1) with w=O, Yref=O. Assume that the control j
j
A
Uj=K ';(KOX j ' j=1,2, .. . ,i - l, h : [2,3, ... , v ] (10) where';., x . are given by (5), are applied to J
J
(1), and let the minimal state realization of the closed loop system obtained by applying the controller (10) to the plant (1) for control agent No . i (with input u i ' output yi), be called the ith agent's local model of the plant (1) with respect to controllers (10) or more briefly the ith agent's local model of the plant. Then a synthesis procedure which solve s the decentralized robust servomechanism problem with sequential stability , in which each control agent is assumed to possess only a local model of the controlled plant , and in which there exists no central decision making authority, is called a local model decentralized robust servomechanism problem. The following assumptions are made in this problem.
27
~.
Using centralized synth~sis (Davison, Goldenberg (1975) ; Dav1son, Ferguson (1981)) and a knowledge of agent No. l's local model of the plant, apply the servocompensator (6) with i=l to the terminals of control agent No. 1 and apply a stabilizing compensator vI =Kl ';l + Xl where ';1 is given by (6) and; is the output of a dynamic compensator so !hat the resultant closed loop system is stable and has a desired dynamic response, i . e. so that all non-fixed modes of agent No. l's local model of the controlled plant are shifted into ag o The resultant s ystem then has the property that YI is regulated.
Step 3. Repeat the centralized procedure of step 2 for each agent 2,3 , . . . ,v sequentially, using as control inputs v. , i=2,3, .. . , v respectively and a knowledg~ of the agent's local model of the controlled plant. The resultant system then has the property that Yl ' Y2' " ·,yv are all regulated and the closed loop system is stable with a desired dynamic response, i.e. the eigenvalues of the resultant closed loop system are all contained in a-, except for those fixed modes (if any) of g
(Cm,A , B) which lie outside of
a; .
Assumptions. 1. It is assumed that each control agent knows the disturbance/reference signal poles Al ,A 2 , .· . ,Ap' 2. It is assumed that each control agent has the same performance criterion, i.e . that the closed loop system be stable and that the eigenvalues of the resultant closed loop system all lie in a certain stable preassigned symmetric region of the complex plane denoted by a (except for any fixe~ modes of (Cm,A,B ) which may lie outside of agO
g
CHARACTERIZATIONS OF INTERCONNECTED DECENTRALI ZED SYSTEMS The following type of problem is now considered. Given a composite system consisting of v subsystems interconnected together, when does there exist a solution to the robust decentralized servomechanism prob l em for the composite s ystem, given that there exists a solution to the robust servomechanism problem for each subs ystem? In particular, assume that the ith subs ystem is described by:
v
The following main result is obtained : Theorem 1 (Davison , Ozguner (1981)). Consider the system (1); then there exists a solution to the local model decentralized robust servomechanism problem for (1) if and only if there exists a solution to the robust servomechanism problem fo? (1) (given by lemma 1) . Assume that theorem 1 holds; then the following synthesis procedure can be used to con struct a controller for (1) . Synthesis Procedure using Local Models (Davison, Ozguner (1981)). ~.
Apply the output feedback control
Am Am~ m m Ui=Kiyi+v i , i=1 , 2, ... , v , Yi=Yi-Diui sequentially to the plant (1), where K. are arbitrary non- zero matrices chosen "sm~ll enough" so as to maintain stability of the closed loop plant.
~ . =A.x . +B . u . +Eiw+ ~ A.. x. 1 1 1 1 1 j=l 1J J ~i
y.=C . x . +D . u . +F . w 11111 1
(11)
y~=C~x.+D~u . +F~w 11111 1
n. m. where x .E R 1 is the state, U.E R 1 is the in1 1 r . put, y . ER 1 is the output to be regulated, m1 m r. Y.E R 1 is the measurable output of the system 1
and that the disturbances, reference signals are described by (2), (3) respectively. In this case, the general interconnection matrix A. . is given by 1J A •. ~ . • K .. 1jJ •.
1J
1J 1J 1J
l:5j:5v ,
( 12)
where Kij denotes the interconnection gain connecting subsystems i and j. The following results are obtained:
28
E. J. Davison
Results Obtained
Yl
In
Theorem 2 (Oavison (1976)). In (11) , (12) assume that there exists a solution to the robust servomechanism problem for each subsystem of the composite system, i.e . that the following conditions are all satisfied for i=1,2, . . . , v :
~2
0
Yv
0
(i)
able.
(C~,A . ,B.) is stabilizable and detect111 A. -A . I
rank
(H)
1
[
)
B.] =n . +r . • 1
O.
C.
1
1
0
1
0 0 x
In :2
0 ... In
v
the following result, which gives a sufficient condition, in terms of the controllability of the subsystem components of (14), for the system (14) to have no decentralized fixed modes present, is obtained.
j=l ,2, ... ,po
1
Theorem 5 (Oavison, Ozguner (1981)). Consider the system (14) and assume that :
1
(iii) y~ contains Yi' then there exists a solution to the robust decentralized servomechanism problem for the composite system provided the interconnection gains K .. are "small enough".
(i) (A . . ,B.), i=1,2, .. . , v are all controllable. 11 1
1)
Theorem 3 (Davison (1976)). In (11), (12) assume that there exists a solution to the robust servomechanism problem for each subsystem of the composite system, and in addition assume that (C~,A.,B.), 111
i=1,2, . .. , v is controllable and observable; then there exists a solution to the robust decentralized servomechanism problem for the composite system for almost all (Oavison, Wang (1973)) interconnection gains K . .•
are all controllable.
1)
Assume now that the composite system (11) arises from interconnections consisting of input-output interconnections, i.e . assume that the interconnection matrix A. . =H . . K .. ljJ • . has the property that: 1) 1) 1) 1) H ..
1)
= B1.
1/! • . 1)
and that 0 . =0. obtained: 1
AI3] A 23
kl
J'
A3 3
0
0
82
0
0
(13)
= C. )
Then the following result is
Theorem 4 (Oavison (1979)). There exists a solution to the robust decentralized servomechanism problem for the composite system (11), (13) i f and only i f there exists a solution to the robust servomechanism problem for each subsystem of the composite system, i . e . if and only if for i=1,2, . .. , v: (C~,A . ,B.) is stabilizable and detect-
(i)
able .
111
AZ ,V _I
A . - A. I
(ii)
rank
1
[
Av_ l, v_ 1
j=1,2, ... ,p.
)
C.
1
(iii) the outputs Yi contain the outputs Yi . Consider now the following special case of (11) :
Bl x= ~2l
~22· · ·~2 v x+ 0
Av , V_1
,\
m
j
Av.. 2 , v-2 v -l, v-2
1''1),\1- 2
A AV-.! , \) - 1
v-l, v- 1 Av,v_ 1
A Av- 2, v
v- I, v Av , v
1 I(B v - 2
l'
are all controllable.
0 . .. 0
(v )
~2 u
v (14 ) is controllable .
0
J lO
0
Bv _ 1
Large Scale Control System Design then the system (14) has no decentralized fixed modes present. Remarks 1. The above results give a good deal of insight into the behaviour of interconnected systems in which local state feedback is allowed. It implies for example that a 2control agent interconnected system in which each subsystem is controllable and in which the composite system is controllable, can always be stabilized using local state feedback with dynamic compensation. 2. The following classification of some observations made in the literature can be made (the so-called Wang-Fessas-Ikeda counterexample (Wang (1978); Fessas (1979); Ikeda, Siljak (1979)) from theorem 5: (i) The "only if" part of Fessas's conjecture (Fessas (1979)) is false for v~2. (ii) The "if" part of Fessas's conjecture (Fessas (1979)) is true for v=2 and is false for v~3. NUMERICAL EXAMPLE : SOLUTION OF LOAD AND FREQUENCY CONTROL PROBLEM USING LOCAL MODELS The same power system example as discussed in section 2 is considered. It is now desired to find decentralized controllers to solve the "load and frequency control" problem for the 3-area system of Figure 1 for the case of any constant disturbances occurring in the . system. In this case, the 67th reduced order model of the interconnected system of section 2 was used, and it may be verified that theorem 1 holds for this example (Tripathi, Davison (1978); Davison, Tripathi (1980)). In particular, the following decentralized robust controller was obtained as a solution to the local model decentralized robust decentralized servomechanism problem so that the closed loop system is asymptotically stable with a desired transient response, and so that the so-called "area control errors" of area 1, 2, 3 are asymptotically regulated to zero, for all constant disturbances which may occur in the power system: t
Kl
Jo ACEl(T)dT t
K2
Jo ACE 2 (T)dT
(15)
t
liT.lng = K3 s
Jo ACE 3 (T)dT
where Kl=-O.OlS, K2=-0.01S, K =-0.019, where 3 ACEi, i=1,2,3 are the "area control errors" of areas 1,2,3 respectively, and lITing2' lIT ing5 , lITing8 are the inputs to the prime movers of the principle synchronous machines of area 1,2,3 respectively. Some typical closed loop responses of the reduced 67th power system and original power system controlled by (15) for the case of a step dis-
29
turbance in torque are given in Figure 3. It is seen that the controller (15) controls the power system so that the net power lIP2 (of area 2) and frequency 6w 2 (of area 2) of the system are satisfactorly regulated as predicted, and that the resultant closed loop responses based on the reduced model and the original system are quite satisfactory. CONCLUSIONS A brief discussion of some recent results obtained re the control of large scale systems is made in this paper. In particular, the decentralized servomechanism problem is examined under the following various conditions: (i) using simplified models to represent the plant, (ii) imposing a sequential stability c0nstraint, (iii) imposing a local model constraint, (iv) obtaining a characterization of the solvability of the decentralized servomechanism problem for interconnected systems. A numerical example of a load and frequency control problem using a simplified model and subject to a local model and sequential stability constraint, is included to illustrate the results. It is concluded that the control of large scale systems is still not well understood, and that many problems of both a theoretical and of a design nature still exist. REFERENCES Davison, E.J. (1966). A method for simplifying linear dynamic systems. IEEE Trans. on Automatic Control, vol. AC-ll, 93-101. Davison, E.J. (1976). The robust decentralized control of a general servomechanism problem. IEEE Trans. on Automatic Control, vol. AC-2l, No. 1, 16-24. Davison, E.J. (1977). Recent results on decentralized control of large scale multivariable systems. IFAC Symposium on Multivariable Technological Systems, Fredericton, Canada, June 1977, 1-10. Davison, E.J. (1979). The robust decentralized control of a servomechanism problem for composite systems with input-output interconnections. IEEE Trans. on Automatic Control, vol. AC-24, No.2, 325-327. Davison, E.J. and Ferguson, I. (1981). The design of controllers for the multivariable robust servomechanism problem using parameter optimization methods. IEEE Trans. on Automatic Control, vol. 26, No. 1, 93-110. Davison, E.J. and Gesing, W. (1979). Sequential stability and optimization of large scale decentralized systems. Automatica, vol. 15, 307-324. Davison, E.J. and Goldenberg, A. (1975). The robust control of a general servomechanism problem: the servo-compensator. Automatica, vol. 11, 461-471. Davison, E.J. and Ozguner, U. (19Sla). Characterizations of decentralized fixed modes for interconnected systems", 8th IFAC Congress, Kyoto, Japan, Aug. 1981, to appear.
30
E. J. Davison
Davison, E.J. and Ozguner, U. (1981b). Synthesis of the decentralized robust servomechanism problem using local models. Systems Control Report No. 8107, Dept. of Electrical Engineering, University of Toronto, March 1981; also 1980 Joint Automatic Control Conference, Aug. 1980, San Francisco, paper no. WAI-A. Davison, E.J. and Tripathi, N. (1978). The optimal decentralized control of a large power system: load and frequency control. IEEE Trans. on Automatic Control, vol. AS-23, No. 2, 312-325. Davison, E.J. and Tripathi, N. (1980). Decentralized tuning regulators: an application to solve the load and frequency control problem for a large power system. Large Scale Systems: Theory and Application, vo!. 1, No. 1,315. Davison, E.J. and Wang, S.H. (1973). Properties of linear time invariant multivariable systems subject to arbitrary output and state feedback. IEEE Trans. on Automatic Control, vol. AC-18, No. 1, 24-32. Fessas, P. (1979). A note on "an example in decentralized control systems". IEEE Trans. on Automatic Control, vol. AC-24, No. 4, 669. Ho, Y.C. and Mitter, S.K. (1976) editors. Directions in Large Scale Systems. Plenum Press, N.Y. IEEE (1976). Special issue on large-scale networks and systems. IEEE Trans. on Circuit Theory and Systems, vol. CAS-23, No. 12, Dec. 1976. IEEE (1978). Special issue on decentralized control. IEEE Trans. on Automatic Control, vol. AC-23, No. 2, April 1978. IFAC (1976). Proc. IFAC Symposium on Large Scale Systems Theory and Application. Udine, Italy, June 1976. IFAC (1979). Proc. IFAC Symposium on Large Scale Systems Theory and Application. Tolouse, France, June 1979. IFAC (1982). Proc. IFAC Symposium on Large Scale Systems Theory and Application. Warsaw, Poland, June 1982. Ikeda, ~1. and Siljak, D. (1979). Counterexamples to Fessas' conjecture. IEEE Trans. on Automatic Control, vol. AC-24, No. 4, 670. Saeks, R. (1976) editor. Large Scale Dynamic Systems. Point Lobos Press . Sain, M. (1981) editor. Special issue on "linear multivariable control systems". IEEE Trans. on Automatic Control, vol. AC-26, No. 1, 1981. Tripathi, N.K. and Davison, E.J. (1978). Th e automatic generation control of a multiarea interconnected system using reduced order models. Systems Control Report No. 7808, Dept. of Electrical Engineering, Universit y of Toronto, Nov. 1978; also IFAC Symposium on Computer Applications in Large Scale Power Sys tems . New Delhi, India, Aug. 16-18, 1979, 117-126. Wang, S.H. (1978). An example in decentralized control systems. IEEE Trans. on Automatic Control, vol. AC-23, No. 5, 938. \\fang, S.H. and Davison, E.J. (1973). On the stabilization of decentralized control sys".:ems. IEEE Tr'ans. on Automatic Control, vol. AC-18, 473-478.
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