- Email: [email protected]
Sensitivity of large-scale control systems
Sensitivity of Large-scale Control Systems-t by M.E.SEZERS and D.D. SILJAK School of Engineering, Uniuersity
of Santa
Clara, Santa
Clara,
CA 95053,
U.S.A. ABSTRACT-The purpose of this paper is to consider sensitivity of suboptimal decentralized control schemes for large-scale systems composed of interconnected subsystems. It is shown that sensitivity of suboptimal systems with respect to distortions in the local control laws can be expressed in terms of the classical notions of the gain and phase margin measured directly by the degree of suboptimality of the decentralized control. This establishes the degree of suboptimality as an index of both the system performance regarding the optimality criterion and the robustness to uncertainties in the interconnections among the subsystems as well as distortions of the individual control laws, thus raising further the confidence in suboptimal designs of large-scale systems.
I. Introduction Over the years, control
systems evolved from simple servomechanisms to sophisticated control and estimation schemes for multivariable systems. The increase of complexity has been a consequence of the emerging new technologies that involved a large number of components related in nonsimple ways to each other in order to achieve high performance under stringent design constraints. One of the essential roles of feedback has been to achieve a satisfactory control of processes having parameters which are either not known exactly due to modeling errors, or are varying in time during operation. In classical control theory (1, 2), sensitivity of feedback control systems to parameter uncertainty has been considered by frequency domain methods whereby the gain and phase margin have served to measure the ability of the closed-loop system to withstand gain and phase changes in the open-loop dynamics. Once it has been shown (3) that optimal linear-quadratic regulators satisfy a frequency domain condition, the same sensitivity measures have been shown (3) to apply to optimal control systems designed to meet time domain performance criteria. This opened up a real possibility to establish the gain and phase margin, as well as the tolerance to nonlinearities, as performance characteristics of the optimal multivariable feedback systems (5), thus increasing considerably the applicability of the linear-quadratic control design. A further increase of complexity has appeared in large-scale systems comtThe research reported herein was supported Under Grant No. ECS-8011210.
by the National
SOn leave from the Electrical Engineering Department, University,
Science Foundation
Middle East Technical
Ankara, Turkey.
0 The Franklin lnslitute 00160032/81/090179-19$02.00/O
179
M. E. Sezer and D. D. Siijak
posed of interconnected subsystems, which is characterized by multiple controllers associated with individual subsystems. This nonclassical information and control structure constraint leads to decentralized control laws that have to be implemented by local feedback loops. The decentralized control schemes have been proven to be effective in handling the modeling uncertainties in the interconnections among the subsystems, which are invariably present in the control problems of large-scale systems (6). It has also been shown that the concept of suboptimality (7) provides a suitable framework for generating the decentralized control laws that are insensitive to perturbations in the interconnection structure. The purpose af this paper is to consider sensitivity of the suboptimal decentralized schemes and establish the robustness of the schemes in terms of the classical sensitivity measures of the gain and phase margin. In this way, the dual purpose of the sensitivity analysis is established since the tolerable uncertainties are quantitatively characterized in both the plant and the control structure. The plan of the paper is as follows: In Section II, we first summarize briefly the suboptimal design approach for generating decentralized feedback laws for large-scale systems, and then investigate the sensitivity properties of the resulting closed-loop systems. We show that suboptimal systems can tolerate distortions in the local controls, characterized as insertion of memoriless nonlinearities or stable linear systems into the feedback loops, in pretty much the same way as optimal systems. In particular, when the subsystems constituting the overall system have single input, this result has the same interpretation of sensitivity as provided by gain and phase margin. In this section, we also show that once a suboptimal decentralized control is obtained, simple gain adjustments in each local feedback loop restore optimality, thus verifying that suboptimal systems having similar sensitivity properties as optimal systems is not a coincidence. In Section III, we broaden our sensitivity analysis by formulating the concept of connective suboptimality, and characterize a class of large-scale systems for which a connectively suboptimal decentralized control always exists. We show that for that class of systems a choice of performance criterion can be made, which leads to an almost optimal decentralized feedback scheme, and a closed-loop system that has almost the same sensitivity properties as optimal systems. In Section IV, we examine the sensitivity of the suboptimal design of a system composed of overlapping subsystems to demonstrate the use of suboptimality approach in the sensitivity analysis of large-scale systems under other types of structural constraints on control than decentralization. Finally, the Appendix contains the proofs of the propositions stated in the paper. II. Interconnected Systems: We consider a system described as
Gain and Phase Margin
S composed
of N interconnected
Si : ii = AiXi+ 5 Ai$j + Biui, i = 1) 2, . . . 9 NT j=l
180
subsystems
Si
(1)
Journal of The Franklin Institute Pergamon Press Ltd.
Sensitivity of Large-scale where xi(t) is Si, and Ai, A, that the pair compact form
Control Systems
an ni vector-the state of Si, ui(t) is an mi vector-the input of and Bi are constant matrices of appropriate dimensions such (A, Bi) is controllable. For convenience, we describe S in a as S:i = (A, + A,)x + Bu
(2)
AD = diag{A,, AZ,. . . , u = (UT, UT,. . . , u;)~, where x = (XT,XT,. . . ,xi)‘, AN}, B = diag {B,, I??, . . . , BN}, and Ac = (Aij)NxN To obtain a decentralized control for the system S, we consider the decoupled subsystems Sp:~;=Aixi+Biui,
i= 1, 2, . . . . N,
which are obtained from Sj by setting A, = 0 in (1). The collection be described as
(3) of Sq can
S,:l=A~+Bu, where x, u, A, and B are as defined for S of (2). With the decoupled SD we associate a quadratic cost J(x,, u) =
I0
(4) system
s (x’Qx + uTRu)dt,
where x0=x(O), Q = diag {Q,, Qz, . . . , QN} is a symmetric positive semidefinite matrix with blocks Qi as constant ni X ni matrices such that the pair (A,, Q”‘) is observable, and R = diag {R,, I?*, . . . , RN} is a symmetric positive definite matrix with blocks Ri as constant mi X mi matrices. The cost J in (5) can be considered to be the sum of N costs
JiCxiO,
ui)
=
I
r (XyQiXi + u;rRiui)dt, i = 1, 2, . . . , N,
0
(6)
associated with the decoupled subsystems ST of (3) where x,~ = Xi(O), i = 1, 2, . . . ) N. The optimal control law for SD with respect to the cost J is obtained through the standard procedure (4) as
u”=-Kx,
(7)
K = R-‘BT,
(8)
where
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M. E. Sezer and D. D. Siljak
and P is the unique symmetric positive definite solution of the Riccati equation A;P
+ PAD - PBR-‘BTP
+ Q = 0.
(9)
The control u”, when applied to S, yields the optimal cost
P(x*) = x,TPx,.
(10)
Due to the block diagonal structure of the matrices AD, B, Q and R, we have {P,, Pz, . . . , PN} and K = diag {K,, K2, . . . , K,+.,}, so that the components of the control ~8 are
P =diag
UT=-&xi,
i=l,2,
. . . . N,
(11)
i.e., the control is decentralized as required. It is also obvious that each local control ~9 in (11) is optimal for the corresponding decoupled subsystem S$‘, and yields an optimal value Jq(Xio) = XXPixio for the associated cost Ji, which is the share of the i-th subsystem in the overall cost JO(xO) in (10). When we apply the decentralized control u0 to the interconnected system S in (2), the closed-loop system &~=(A,+A,-BK)~
(12)
would result in a cost J@(x,J which is, in general, different from the optimal cost J”(xO) due to the interconnections among the subsystems. This leads to the concept of suboptimality defined as in u’?:
Definition I The control law u” is said to be suboptimal for the system S with degree p if there exists a positive number F such that
P(xo)
5
p-‘JG(xo)
(13)
for all x0.
To derive a sufficient condition we define a matrix M as
for suboptimality
of the control
uc for S,
where tLi are positive numbers, and I,, is the identity matrix of order n;. We
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state the following: Theorem I Suppose there exists a matrix M such that the matrix F(M) = AEM-‘I’ + M-‘PAc + (I - M-‘)(Q + KTRK) is negative semidejinite.
(15)
Then the control u” is suboptimal for S with degree
/L = min {.Li}* We- note that when CL,= yz = . . . = pN = p, i.e., M = PI, negative semidefiniteness of F(M) in (1.5) is equivalent to negative semidefiniteness of the matrix. F(p)
= A:P + PA, - (1 - /_L)(Q+ KrZ?K),
(16)
which is the result of Theorem (4.3) in (8). It can, however, be verified by simple examples that there are cases where F(p) is never negative semidefinite for any p while a matrix M exists such that F(M) is negative semidefinite. This is due to the fact that allowing different pi’s in F(M) corresponds to a scaling of the interconnection matrices A, whereby certain block rows of Ac are increased (decreased) while the corresponding block columns are decreased (increased) by the same amount. We also note that suboptimality as implied by Theorem I is not, in general, sufficient for the stability of the closed-loop interconnected system 5? in (12). However, suboptimality implies stability if the pair (A, + Ac, Q”‘) is observable, for then the pair [A, +A, -BK, (Q + K%K)“*] would be observable [(9), Theorem 3.61 so that stability follows from the fact that f@&) is finite. Definition of suboptimality and Theorem I provide a means of assessing the sensitivity of the interconneced system 9 to changes in the interconnection structure: 9 is suboptimal and stable as long as Ac is such that F(M) is negative semidefinite for some matrix M and the pair (A, + A=, Q”‘) is observable. We now turn our attention to another aspect of sensitivity analysis: Can s, for fixed Act tolerate distortions in the suboptimal control u” caused by insertion of nonlinearities or introduction of the gain changes and phase shifts into the feedback loops? To answer this question we first consider the system
where the nonlinearity
@,(a)= Vol. 312. No. 3/4, pp. 179-197. Printed in Great Britain
@(a) is of the form
[4T(ad, 42’(a2),. . . , ISIS,
September/October
1981
i = 1,2,. . . , N,
(18) 183
M. E. Sezer and D. D. Siljak
with 4i(o;) being continuous the following:
vector-valued
functions
of ai = - K,x,. We state
Theorem II Suppose that F(M) is negative semidefinite for some M, and that the pair (At, + Ac, Q’j2) is observable. Then, the equilibrium x = 0 of & is asymptotically stable in the large for all nonlinearities @ whose components satisfy the conditions KioyR;‘ai 5 oydi(R;‘oi)
5 KioToiy i = 1, 2, . . . . , N
(19)
where Ki = 1 - (pi - A,~)12 with Ap being an arbitrarily small positive number, and Ki < ~0.
Next, we consider the case when the nonlinearities 4i in (18) are replaced by stable linear time-invariant systems Li having states ti, inputs ai and transfer function matrices L,(s). Denoting the resulting system by S,, we state the following: Theorem III Suppose that F(M) is negative semidefinite for some M, and that the pair (A, + Ac, Q’12) is observable. Then, the equilibrium x = 0, zi = 0, i = 1,2, . . ., N, of S, is asymptotically stable (in the large ) provided that Li(jti)RI’+
R;‘L~(jw)-(2-/.~i
+Ap)Ry’?
0, i = 1, 2, . . ., N,
(20)
for all w, where Ap is an arbitrarily small positive number.
Theorems II and III can be considered as generalizations of the wellknown criteria for gain and phase margin to interconnected systems with multivariable subsystems. To relate these results to classical concepts of gain and phase margin, let us consider a special case, where the subsystems Si have single inputs, i.e., mi = 1. In this case, ai and ai in (18) as well as L,(S) and Ri in (4) are scalars, and we obtain the following result from Theorems II and III: Corollary I Suppose that each Si has a single input, F(M) is negative semidefinite for some M, and the pair (A, + Ac, Q’12) is observable, so that S is stable. Then i-th feedback loop of S has (i) infinite gain margin ; (ii) at least 2 cos-‘(1 - cLi/2)phase margin and (iii) at least 50 pi% gain reduction tolerance.
By “i-th feedback loop of S has infinite gain margin” we mean as in (5) that arbitrarily large increases in the gain of the local feedback (11) do not destroy stability of S. Similar interpretation can be given to loop phase margin and loop gain reduction tolerance.
184
Journal of The
Franklin Institute Pergamon Press Ltd.
Sensitivity of Large-scale Control Systems The above results show that suboptimal systems can tolerate a large class of distortions in the control in pretty much the same way as optimal systems (4). Moreover, the similarity between the conditions (19) and (20) and the corresponding conditions of (5) for optimal multivariable systems suggests the possibility that a modification of the suboptimal control (7) might result in an interconnected system which is optimal with respect to a modified performance measure. To investigate this possibility, let us assume that F(M) is negative semidefinite for some matrix M. We define the modified weighting matrices I? and 0 as l? = diag{R,, I&, . . . , I&}
(21)
where
(22) and
(j = M-‘(Q - PBR~‘BrP) + M-‘PB@-‘BTPM-’ - A;M-‘P - M-‘PAc, (23) and state the following: Theorem IV Suppose that F(M) is negative semidefinite for some M. Then, (i) the matrix 0) is positive semidefinite; (ii) the decentralized control fi0
=
_
&,
K
=
R-IBTM-lp
(24)
is optimal for S with respect to the modified cost
&(x0,u)
=
(x’& + u =&)dt,
(25)
and
J’(x,,) = x,TM-‘Px,,; (iii) if, in addition, the pair (A, + Ac, Q”*) is observable, then the control 1’ stabilizes S.
Vol. 312. No. 3/4. pp. 179-197. Printed in Groat Britain
September/October
1981
185
M. E. Sezer and D. D. Siljak From (22) and (24) it follows that the components control are
of the modified optimal
(26) where us are local controls that are optimal for the decoupled subsystems Ss with respect to their local performance measures Ji. Thus, Theorem IV implies that once the suboptimality of the decentralized control u” is established using F(M), then increasing the gains of the feedback loops around the subsystems properly, we can recover the optimality of the overall system. Obviously, this means that the modified closed-loop interconnected system will have less sensitivity to distortions of the local controls. In particular, when the subsystems have single input, each feedback loop will have infinite gain margin, -+60” phase margin and 50% gain reduction tolerance. Theorem IV concludes our discussion on the sensitivity properties of decentrally controlled suboptimal interconnected systems. While the suboptimality criterion of_ Theorem I provides a measure of sensitivity of the closed-loop system S to interconnections among the subsystems, Theorems I, III, IV and Corollary I take care of the sensitivity of the system with a fixed interconnection structure to distortions of the suboptimal control. A natural sequel to this study is to investigate the sensitivity of the system to simultaneous perturbations in the interconnection structure and the control. This we consider next. III. Connective Suboptimality Let us consider the interconnected system in (1) again, where now we assume that the subsystems S, have single input, i.e., Bi = bi, i = 1, 2, . . ., N, and the interconnection
(27)
matrices are of the form A, = eJif, i = 1, 2, . . ., N.
(28)
In (28), ~ij are fixed matrices, and eir E [- 1, l] are the elements of an N x N interconnection matrix E = (eij)NxN and represent the gains of the interconnections among the subsystems. To emphasize the dependence of the interconnections on the parameters, we describe the interconnected system of (2) with a slight difference in notation as SE : 1 = [A, + &(E)]x
+ Bu,
(2%
where
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Sensitivity of Large-scale
Control Systems
Following the suboptimal control strategy of the previous section, we associate with S, the block diagonal cost J(xO, U) of (5) and obtain a decentralized control u0 that is optimal for the decoupled system SD obtained from S, by setting E = 0 in (3.3). We could now proceed to consider suboptimality of the control u ’ for S, for each fixed E using Theorem I. Instead, we required that the control u0 be robust, that is, it should be a stabilizing suboptimal control under arbitrary structural perturbations described by any and all interconnection matrices E. Thus, we need (7): Definition II The control law (7) is said to be connectively p if there exists a positive number p such that
Pyx,)
5
suboptimal for S, with degree
/c’JO(xo),
(30)
for all x0 and all E.
Our immediate interest is to delineate a class of interconnection matrices A, for which there exist a connectively suboptimal and stabilizing decentralized control. For this purpose, let us assume without loss of generality that the pairs (Ai, bi) of the subsystems Si are in controllable canonical form, that is,
,i=l,2
For each interconnection
,...,
N.
(31)
matrix ~ij = (a:J of (28), we define an integer rij as
1, =
zf+: (4 -P}*
I
- 5 ni,
Aijf
O
(32)
Aij = 0
i-l
Each integer lij defines a boundary for nonzero elements of the corresponding matrix A,. Two typical cases are illustrated by Fig. 1.
“I
“1
FIG. 1. Vol. 312, No. 314, pp. 179-197, September/October Printed in Great Britain
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M. E. Sezer and D. D. Siljak
We recall from (10) that if the integers f, are such that for any index set LJ ={i,, iZ,. . . ,im}C{l,. . . ,N} and any permutation $ = {jr, j2, . . . ,j,,,} of (33) then the interconnected system SE of (29) is stabilizable by the decentralized control law u” regardless of what the matrix E is, that is, it is connectively stabilizable. This, however, does not imply automatically that the control u” is also suboptimal. In fact, to establish suboptimality via Theorem I we have to choose the cost J(x,,, U) properly. For this we first need the following: Lemma I Suppose that the numbers numbers vi such that
1, satisfy the inequality
(33).
Then there exist
VI- ~i + 1 - I, > 0, i, j = 1, 2, . . ., N.
(34)
We now choose the blocks Qj of the state weighting matrix Q in J(x,, u) of (5)
Qi = where pi are arbitrary
Q(S)QiQ(S),
(35)
positive definite matrices, and Di(S) = diag {a”~,a”~+‘,. . ., 8”~+‘;-‘}
(36)
with S being a positive number to be determined. Also, keeping in mind that, due to the assumption on the dimension of the subsystem inputs, the input weighting matrix R in J(x,, U) has the form R = diag {p], p2, . . ., pN}, we chose the scalars pi as: oi = 6*(“~+“~), i = 1, 2, . . ., N.
(37)
With this choice of J(x,, u), we have: Theorem V Suppose that the numbers 1, satisfy (33), and that the cost J(xO, u) are defined by (3.5)-(37). Then, number 8 such that whenever S < s the control suboptimaf with degree u = p(S) for S, and is also p(S)+ 1 as S +O.
the matrices Q and R of there exists a positive law u” is connectively stabilizing. Furthermore
We note that the above choice of the cost matrices Q and R leads to high-gain feedback matrices kT = [ki,8-*“1, kizS-““i-“, . . ., ki”,S~2]which, in turn, result in closed-loop decoupled subsystems having eigenvalues of the order of S-‘. It is, in fact, this high gain characteristics of the suboptimal control 188
Journal of The
Franklin lnztitute Pergamon Press Lfd
Sensitivity of Large-scale Control Systems together with the structure closed-loop system
of the interconnection
SE: 1 =
[A, + ii,(E)
matrix 2,
that makes the
- BK]x,
(38)
connectively suboptimal. Clearly, SE also has the sensitivity properties described in Corrollary I, that is, for sufficiently small 6, each feed-back loop of SE has infinite gain margin, approximately 60” phase margin and 50% gain reduction tolerance. Finally, we would like to consider a special class of interconnected systems where interconnections enter additively through the inputs of the subsystems, that is, A, = biK%,i, j = 1, 2, . . . ., N, where & are fixed constant have the structure
vectors.
Clearly, for such systems,
(39) & matrices
(40) when the pairs (A, bi) are in the form of (31), are
SO
that the numbers 1, in (32)
iii = nj - ni, i = 1, 2, . . ., N.
(41)
Thus, (33) is satisfied, and Theorem V ensures the existence of a suboptimal decentralized control law. Furthermore, Theorem IV implies that by increasing the gains of the local controls the closed-loop interconnected system can be made optimal. A similar result has been recently obtained by Yasuda and colleagues (11) in an attempt to construct a performance measure with respect to which optimal control for S is decentralized. IV. Overlapping Subsystems In the previous sections we considered the sensitivity of decentrally controlled large-scale systems. A natural question at this point can be to ask whether a similar sensitivity analysis can be carried over for systems with other types of constraints on the control than decentralization. In this section we shall make an attempt to answer this question for a system with overlapping subsystems (8). Let us consider a system S described as
(42)
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M. E. Sezer and D. D. Siljak
where the ni vectors xi(t) constitute the state x = (~7, XT, xT)~ of S, the mi vectors ui(t) are the components of the input u = (UT, UT)’ of S, and A, and Bi are constant matrices of appropriate dimensions. We assume that the inputs U, and u2 of S are restricted to be of the form UI = -
w,,x,+K,*x*)
u2 =
(J&*x*
-
+
(43)
K23x3).
Our aim is to choose suboptimal controls u, and u2 satisfying the constraints in (43) and investigate the sensitivity of the resulting system to nonlinear distortions of these controls. For this purpose, we define 2, = (XT, xT)r and A?~ = (x,‘, XT)’ to be overlapping components of the state x of S, and form an expanded vector 2 = (iy, 2:)‘. Clearly, x’ and x are related by 4 = TX,
(44)
where
L,
0 0
0
0
0 0
In, 0 0 - I”,
L1
T= The transformation
3:
4l,
(44) defines an expanded
;:
[Ii
which can be considered
system
A,, Al21 0 A 22 II 0 ;21 __u’jk-22_ ____-
=
L
A::
(45)
Ul u2
0
32
to be composed
19
of two interconnected
(46)
subsystems
j’,:i, =A,i,+A,2i2++,U, (47)
3, : k, = A2i2+ A2,i, + B2U2,
with the matrices A,, A2, B,, B2, A,, and A,, being obvious from (46). From (47) and the definition of 2, and Z2 it is clear that any decentralized control for S would satisfy the constraint in (43) provided that the relation (44) holds also for the trajectories of the resulting closed-loop systems. Leaving this latter requirement aside for the moment, we proceed to computation of the suboptimal decentralized control u,, u2 for S. For this purpose, we rewrite the description of S in (46) as
S:k=(A,+A,)n+Bu, 190
(48) Journal
of The Franklin Institute Pergamon Press Ltd.
Sensitivity of Large-scale Control Systems where A, = diag {A,, AZ}, B = diag {B,, &} and A
C
=
0
[ A,,
A2
(49)
01 ’
and we associate with S a block diagonal cost
where $, = x’(0). We now choose the cost associated
with S to be
(x’Qx + Im
J(x,, u) =
0
uTRu)dt,
(51)
where x0 = x(O) and Q = TTQT.
(52)
It has been shown in (8) that if U=-G,
(53)
is any control for S which yields the cost j@(io), then the control u=-Kx,
K=ET,
(54)
for S yields a cost j@(x,) such that J@(x,) = J@( TX,).
(55)
Furthermore, the solutions of the resulting closed-loop systems satisfy (43). Having the relation (55) in hand, we design the control (53) to be the optimal control for the decoupled system
(56) with respect to the cost J in (50), and choose the control for S as in (54). Since k is block diagonal, this control will satisfy the constraint (43). Furthermore, if the matrix
P(M) = AgwP
+ WPA,
+ (I
-
Ax-$6 + RTRk),
(57)
where &f = diag{p ,I “,+“*, pJ,,2+n3} and p is the optimal cost matrix for SD, is vol.312. No.
3/4, pp. 179-197. September/October Printed in Great Britain
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191
M. E. Sezer and D. D. Siljak
negative semidefinite
for some positive numbers pI and I_L~, then P&J
5 ~~‘f~P&, I_L= min {pi, pz}
(58)
so that (55) implies J@(xo) I /.L-*x;TrPTx(). Once the finiteness closed-loop system
of the cost is established
(59) by (59) then the resulting
=(A-BK)x,
$:i
(60)
where A and B are the system matrices defined in (42), is stable provided the pair (A, Q”*) is observable. To investigate the sensitivity of the system 9 to nonlinearities in the control paths, we now consider the system &:~=Ax+B@(-Kx)
where the nonlinearity
(61)
Cpis of the form
a$-Kx)=
with & and #2 being continuous
41(
-
(62)
[
functions.
We state the following:
Theorem VI Supp.ose that P(k) is negative semidejinite for some positive numbers CL, a,nd p2, and that the pair (A, Q’12) is observable. Then, the equilibrium x = 0 of S@ is asymptotically stable in the large for all nonlinearities @ with com-
ponents satisfying
where Ki = 1 - (cci - Ap)/2 with Ap being an arbitrarily small positive number, i?i < QZ, and Ri are the diagonal blocks of the cost matrix R.
Although the result good example of how timality can be used systems which would 192
of Theorem VI is somewhat expected, it provides a the sensitivity analysis through the concept of subopto make clear the sensitivity properties of particular otherwise be very difficult to study. Journal of The
Franklin Institute Per~amon Press Ltd.
Sensitivity of Large-scale
Control Systems
References (1) H. W. Bode, “Network Analysis and Feedback Amplifier Design”, Van Nostrand, New York, 1945. (2) G. J. Thaler and R. G. Brown, “Analysis and Design of Feedback Control System”, McGraw-Hill, New York, 1960. (3) R. E. Kalman, “When is a linear control system optimal?” Trans. Am. Sot. Mech. Engrs Series, Vol. 86, pp. l-10, 1964. (4) B. D. 0. Anderson and J. B. Moore, “Linear Optimal Control”, Prentice-Hall, Englewood Cliffs, NJ, 1971. and phase for multiloop (5) G. Safonov M. Athans, regulators”, IEEE Vol. AC-22, 173-179, 1977. D. D. “Large-Scale Dynamic Stability and NorthHolland, York, 1978. R. Krtolica D. D. “Suboptimality of stochastic control estimation”, IEEE Vol. AC-25, 76-83, 1980. M. Ikeda, D. Siljak D. E. “Decentralized control overlapping information J. Optim. Appl. Vol. 1981, in Control: A Approach”, (9) W. Wonham, “Linear Springer, York, 1979. M. Ikeda D. D. “On decentrally large-scale systems”, Vol. 16, 331-334, 1980. optimizable interconnected K. Yasuda, Hikata and Hirai, “On Proc. 19th Conf. on and Control, New Mexico, 10-12, 1980, 536-537. (12) E. Sezer D. D. “Decentralized stabilization structure of large-scale systems”, 13th Asilomar on Circuits, and Comput., Grove, CA, 5-7, 1979, 176-181. Appendix : Proofs oj Propositions Proof of Theorem I We note that J@(&) = Jim ’ xT(t)(Q+ KTRK)x(t)dt, 7-mI cl where x(t) is the solution of (12) corresponding get
to x(0) = x0. From (8), (9) and (15) we
- x~(~)M-‘Px(T) Thus, negative definiteness
where p = min{pi}, establishing 179-197,
3
+ x;M-‘Pxo.
(A3)
of F(M) implies J@(x,) I x,TM-‘Px, 5 /.-‘x;Pxo,
Vol. 312, No.
(AlI
inequality
(A4)
(13).
193
M. E. Sezer and D. D. Siljak Proof of Theorem II Let v(x) = X%~‘PX, be a candidate for a Liapunov function straightforward computations, we obtain
(AS)
of .$,. Using
(8), (9), (15) and (19), by
d(x) = x’[(A,
+ A,)=M-‘P
+ Mm’P(A, + A,)]x + 2xTM-‘PB@( - Kx)
= xT[F(M)-
Q + (2M-’ - Z)K=RK]x +2 2 p;‘x:P,B&(i=l
R;‘B:Pixi)
= - x’(Q + ApK=RK)x
(A@
so that d(x) 5 0. On the other hand, the observability assumption together with the fact that B(O) = 0, which is implied by (19) and continuity of @(a), ensure that G(x) is not identically zero along any trajectory of 3, except the equilibrium x = 0. This completes the proof. Proof of Theorem ZZI Let the collection of the linear system Li inserted into the feedback loops be denoted by 2, which has the state z = (zT, z:, . . , zj?i)‘, the input (T = COT,a:, . . . , CT:)’ with ai = Kixi, i = 1, 2, . . ., N, and the block diagonal transfer function 1L(s) = diaglMs), L*(s), . . J&(S)}. Obviously, (x = 0, z = 0) is the equilibrium of &. For any x(0) = x0 we have x;M-‘Px,, = x’(T)M-‘Px(7)
- ,-$ I
Using (8), (17) (15) and negative semidefiniteness x;M-‘Px,
-
[x=(t)M-‘Px(t)]dt.
(A7)
of F(M), we obtain from (A7)
T~T(t)(Q+ApM-‘KTRK)x(t)dt I0
2 I 7 [x’(t)(Z - 2M-’ - ApM-‘)K*RKx(t) 0
- 2x=(t)M-‘PBu(t)]dt.
(-48)
Taking the limit as r -+ 2, the right hand side (RHS) of (A8) can be written using Parseval’s relation as RHS =&j_=
a
{x*(jo)(Z - 2M-’ - ApM-‘)K=RKx(jo)
- x*(jo)M-‘PBu(jo)
- u*(jw)B=M-‘Px(jw)}dw where x(jo) and u(jw) are Fourier transforms x(jw) are related by
of x(t) and u(f). With z(O) = 0, u(jw) and
u(jw) = L(jw)u(jw) = - L(jo)Kx(jw).
194
(A9)
(AlO) Journal of The Franklin Institute Pergamon Press Ltd
Sensitivity Substituting
RHS = &
of Large-scale
Control Systems
(AlO) into (A9) we obtain m
I_ x*(jo)M-‘K’R[L(jw)R-
+ R-‘L*(jw)
m
- (21- M + ApI)R-‘lRKx(jo)do
(Al 1)
and (20) implies that RHS 2 0. Thus, (A8) gives m
x’(t)(Q
+ APM-‘KTRK)x(t)dt
5 x;M-‘Pxo I m.
6412)
(A12) together with the observability assumption implies that the solutions of L% corresponding to the initial condition x(0) = x0, z(0) = 0 tend to zero as t + M. Since x0 is arbitrary, this is equivalent to saying that the poles of the transfer function [sl- A, - Ac + BL(s)K] are stable. The proof then follows from the argument in (5) that all poles of $. which do not appear as a pole of this transfer function (if there are any) should be poles of Y, which are assumed to be stable, so that all the poles of & are stable. This completes the proof.
Proof of Corollary I Under the conditions
of the corrollary, K 5 bi(Oj)/oi
(19) reduces to
I l?i 5GO”,i = 1, 2, . . ., N,
(A13)
where ui and di(oi) are scalars, from which (i) and (iii) follow. Similarly, (20) becomes Re fi(jw)2 1 - 9, where Ii(s) is a scalar transfer function, Ii = exp cpi(jo), which gives Ipi I
i E N,
(Af4)
and (ii) follows from (A14) upon choosing -(pi - Ap)/2].
COS-‘[I
Proof of Theorem IV Using (8) and (15) in (23) we obtain
0 = Q - F(M)+
8, P,&[(l - 2&)R;’
+ /L;~R;‘]B:P~,
(Al%
and (i) is established by the definition of Ri in (22) and the fact that Q is positive semidefinite and F(M) is negative semidefinite. Also, it can easily be verified that (A, + A,)=M-‘P
+ M-‘P(AD + A,) - M-‘PBk’B=M-‘P
+ 0 = 0,
6416)
proving (ii). Finally, observability of the pair (A, + Ac, Q”‘) implies observability of the pair {A, +A= - Bl?, (e+ ~~~K)“*} (9), from which stability of the modified closed-loop system follows, proving (iii). This completes the proof. Vol. 312. No. 3/4, pp. 179-197, Printed in Great Britain
September/October
1981
195
M. E. Sezer and D. D. Siljak Proof of Lemma I
a combinatorial
problem which can be solved using graph-theoretic therefore, omitted. A similar result has been proved in (12). Proof of Theorem V
We first show that with Ai, b, as in (31), Qi as in (35) and pi as in (37), the positive definite solution of the Riccati equation ATPi + PiA, - p:‘Pib,bTPi P, =
GDi(G)P,(S)Di(S), i E N,
(‘418)
where pi(a) > 0 for any 6 > 0, and lim p,(S) = pi, > 0. For this, we substitute (A17) and pre- and post-multiply
b;%;‘(s)
(Al@ into
to get 6419)
where
A,(S) = 6Di(S)AiDL’(6)
=
(A20)
_._________________--,iEN.
Since the pair_ [A,(S), b,] is controllable and the pair [Ai( Q”‘] is observable, (A19) implies that Pi(S) > 0. Positive definiteness of the limit Pi, follows from continuity of f&(6). Now, to prove Theorem V, we introduce D = diag{D,, D2,. . . , 0,) choose /_l,=j&=...= pN = CL,define F(p) = pD-‘F(M)D-‘, and use (7) (IS), (35), (37) and (A18) to get F(/.L) = S(D-‘A:Dp where Q = diag {Q,, Q?, . . ., &} consider the matrix GPDArD-’
+ PDA~D-‘)
- (I-
p)(Q + PBBT~)
and p = P(6) = diag{P,(S),
(A21)
P,(6), . . ., p,,,(S)}. Now,
= [eii~Pi(s)Di(s)A,,Di’(6)1,,,
6422)
from the definition of the numbers Iii in Since 6Di(6)AijD7’(6) = [a$?“‘~ “J+‘~(~~~)]~~~~,, (32) and from (34) we conclude that the matrix in (A22) is of the order 6”, where v = min {vi - vi + 1 - iii}. Thus, as 6 +O, the first term in &)
approaches
0 regardless
of the values of e,,. Since, Q + pBBTp > 0 for all 6 < O,jt follows that for sufficiently small 6, there exists a I_L> 0 arbitrarily close to 1 such that F(p) < 0 for any E. This completes the proof.
196
Journal of The Franklin
Institute Pergamon Press Ltd.
Sensitivity
of Large-scale
Control
Systems
Proof of Theorem VI We first note that (2, + &)T
= TA,
(A23
and BT=B
(A24)
as can easily be verified by direct substitution. expressed as @( - Kx) = @(-
Also, the nonlinearity
zzi) =
in (62) can be
&$---p__, 1
(-425)
)
7
[
where &, and i?? are the diagonal blocks of l?. We now choose u(x) = x’(T%?‘PT)x,
(A26)
as a candidate for Liapunov function for 3,. Following similar steps as in the proof of Theorem II, and using (63) and (A23)-(A25), the derivative of t’(x) can be majorized as d(x) I - x’(Q + ApKTMm’RK)x and the result follows from the observability proof.
Vol. 312. No 3/4. pp 179-197. September/October Prmted in Great Britain
1981
I 0,
(AU)
of the pair (A, Q”‘). This completes
the
197
of Santa
Clara, Santa
Clara,
CA 95053,
U.S.A. ABSTRACT-The purpose of this paper is to consider sensitivity of suboptimal decentralized control schemes for large-scale systems composed of interconnected subsystems. It is shown that sensitivity of suboptimal systems with respect to distortions in the local control laws can be expressed in terms of the classical notions of the gain and phase margin measured directly by the degree of suboptimality of the decentralized control. This establishes the degree of suboptimality as an index of both the system performance regarding the optimality criterion and the robustness to uncertainties in the interconnections among the subsystems as well as distortions of the individual control laws, thus raising further the confidence in suboptimal designs of large-scale systems.
I. Introduction Over the years, control
systems evolved from simple servomechanisms to sophisticated control and estimation schemes for multivariable systems. The increase of complexity has been a consequence of the emerging new technologies that involved a large number of components related in nonsimple ways to each other in order to achieve high performance under stringent design constraints. One of the essential roles of feedback has been to achieve a satisfactory control of processes having parameters which are either not known exactly due to modeling errors, or are varying in time during operation. In classical control theory (1, 2), sensitivity of feedback control systems to parameter uncertainty has been considered by frequency domain methods whereby the gain and phase margin have served to measure the ability of the closed-loop system to withstand gain and phase changes in the open-loop dynamics. Once it has been shown (3) that optimal linear-quadratic regulators satisfy a frequency domain condition, the same sensitivity measures have been shown (3) to apply to optimal control systems designed to meet time domain performance criteria. This opened up a real possibility to establish the gain and phase margin, as well as the tolerance to nonlinearities, as performance characteristics of the optimal multivariable feedback systems (5), thus increasing considerably the applicability of the linear-quadratic control design. A further increase of complexity has appeared in large-scale systems comtThe research reported herein was supported Under Grant No. ECS-8011210.
by the National
SOn leave from the Electrical Engineering Department, University,
Science Foundation
Middle East Technical
Ankara, Turkey.
0 The Franklin lnslitute 00160032/81/090179-19$02.00/O
179
M. E. Sezer and D. D. Siijak
posed of interconnected subsystems, which is characterized by multiple controllers associated with individual subsystems. This nonclassical information and control structure constraint leads to decentralized control laws that have to be implemented by local feedback loops. The decentralized control schemes have been proven to be effective in handling the modeling uncertainties in the interconnections among the subsystems, which are invariably present in the control problems of large-scale systems (6). It has also been shown that the concept of suboptimality (7) provides a suitable framework for generating the decentralized control laws that are insensitive to perturbations in the interconnection structure. The purpose af this paper is to consider sensitivity of the suboptimal decentralized schemes and establish the robustness of the schemes in terms of the classical sensitivity measures of the gain and phase margin. In this way, the dual purpose of the sensitivity analysis is established since the tolerable uncertainties are quantitatively characterized in both the plant and the control structure. The plan of the paper is as follows: In Section II, we first summarize briefly the suboptimal design approach for generating decentralized feedback laws for large-scale systems, and then investigate the sensitivity properties of the resulting closed-loop systems. We show that suboptimal systems can tolerate distortions in the local controls, characterized as insertion of memoriless nonlinearities or stable linear systems into the feedback loops, in pretty much the same way as optimal systems. In particular, when the subsystems constituting the overall system have single input, this result has the same interpretation of sensitivity as provided by gain and phase margin. In this section, we also show that once a suboptimal decentralized control is obtained, simple gain adjustments in each local feedback loop restore optimality, thus verifying that suboptimal systems having similar sensitivity properties as optimal systems is not a coincidence. In Section III, we broaden our sensitivity analysis by formulating the concept of connective suboptimality, and characterize a class of large-scale systems for which a connectively suboptimal decentralized control always exists. We show that for that class of systems a choice of performance criterion can be made, which leads to an almost optimal decentralized feedback scheme, and a closed-loop system that has almost the same sensitivity properties as optimal systems. In Section IV, we examine the sensitivity of the suboptimal design of a system composed of overlapping subsystems to demonstrate the use of suboptimality approach in the sensitivity analysis of large-scale systems under other types of structural constraints on control than decentralization. Finally, the Appendix contains the proofs of the propositions stated in the paper. II. Interconnected Systems: We consider a system described as
Gain and Phase Margin
S composed
of N interconnected
Si : ii = AiXi+ 5 Ai$j + Biui, i = 1) 2, . . . 9 NT j=l
180
subsystems
Si
(1)
Journal of The Franklin Institute Pergamon Press Ltd.
Sensitivity of Large-scale where xi(t) is Si, and Ai, A, that the pair compact form
Control Systems
an ni vector-the state of Si, ui(t) is an mi vector-the input of and Bi are constant matrices of appropriate dimensions such (A, Bi) is controllable. For convenience, we describe S in a as S:i = (A, + A,)x + Bu
(2)
AD = diag{A,, AZ,. . . , u = (UT, UT,. . . , u;)~, where x = (XT,XT,. . . ,xi)‘, AN}, B = diag {B,, I??, . . . , BN}, and Ac = (Aij)NxN To obtain a decentralized control for the system S, we consider the decoupled subsystems Sp:~;=Aixi+Biui,
i= 1, 2, . . . . N,
which are obtained from Sj by setting A, = 0 in (1). The collection be described as
(3) of Sq can
S,:l=A~+Bu, where x, u, A, and B are as defined for S of (2). With the decoupled SD we associate a quadratic cost J(x,, u) =
I0
(4) system
s (x’Qx + uTRu)dt,
where x0=x(O), Q = diag {Q,, Qz, . . . , QN} is a symmetric positive semidefinite matrix with blocks Qi as constant ni X ni matrices such that the pair (A,, Q”‘) is observable, and R = diag {R,, I?*, . . . , RN} is a symmetric positive definite matrix with blocks Ri as constant mi X mi matrices. The cost J in (5) can be considered to be the sum of N costs
JiCxiO,
ui)
=
I
r (XyQiXi + u;rRiui)dt, i = 1, 2, . . . , N,
0
(6)
associated with the decoupled subsystems ST of (3) where x,~ = Xi(O), i = 1, 2, . . . ) N. The optimal control law for SD with respect to the cost J is obtained through the standard procedure (4) as
u”=-Kx,
(7)
K = R-‘BT,
(8)
where
Vol. 312. No. 3/4. pp 179-197. September/October Printed in Great Britain
1981
181
M. E. Sezer and D. D. Siljak
and P is the unique symmetric positive definite solution of the Riccati equation A;P
+ PAD - PBR-‘BTP
+ Q = 0.
(9)
The control u”, when applied to S, yields the optimal cost
P(x*) = x,TPx,.
(10)
Due to the block diagonal structure of the matrices AD, B, Q and R, we have {P,, Pz, . . . , PN} and K = diag {K,, K2, . . . , K,+.,}, so that the components of the control ~8 are
P =diag
UT=-&xi,
i=l,2,
. . . . N,
(11)
i.e., the control is decentralized as required. It is also obvious that each local control ~9 in (11) is optimal for the corresponding decoupled subsystem S$‘, and yields an optimal value Jq(Xio) = XXPixio for the associated cost Ji, which is the share of the i-th subsystem in the overall cost JO(xO) in (10). When we apply the decentralized control u0 to the interconnected system S in (2), the closed-loop system &~=(A,+A,-BK)~
(12)
would result in a cost J@(x,J which is, in general, different from the optimal cost J”(xO) due to the interconnections among the subsystems. This leads to the concept of suboptimality defined as in u’?:
Definition I The control law u” is said to be suboptimal for the system S with degree p if there exists a positive number F such that
P(xo)
5
p-‘JG(xo)
(13)
for all x0.
To derive a sufficient condition we define a matrix M as
for suboptimality
of the control
uc for S,
where tLi are positive numbers, and I,, is the identity matrix of order n;. We
182
Journal of The
Franklin Institute Pergamon Presc Ltd.
Sensitivity of Large-scale
Control Systems
state the following: Theorem I Suppose there exists a matrix M such that the matrix F(M) = AEM-‘I’ + M-‘PAc + (I - M-‘)(Q + KTRK) is negative semidejinite.
(15)
Then the control u” is suboptimal for S with degree
/L = min {.Li}* We- note that when CL,= yz = . . . = pN = p, i.e., M = PI, negative semidefiniteness of F(M) in (1.5) is equivalent to negative semidefiniteness of the matrix. F(p)
= A:P + PA, - (1 - /_L)(Q+ KrZ?K),
(16)
which is the result of Theorem (4.3) in (8). It can, however, be verified by simple examples that there are cases where F(p) is never negative semidefinite for any p while a matrix M exists such that F(M) is negative semidefinite. This is due to the fact that allowing different pi’s in F(M) corresponds to a scaling of the interconnection matrices A, whereby certain block rows of Ac are increased (decreased) while the corresponding block columns are decreased (increased) by the same amount. We also note that suboptimality as implied by Theorem I is not, in general, sufficient for the stability of the closed-loop interconnected system 5? in (12). However, suboptimality implies stability if the pair (A, + Ac, Q”‘) is observable, for then the pair [A, +A, -BK, (Q + K%K)“*] would be observable [(9), Theorem 3.61 so that stability follows from the fact that f@&) is finite. Definition of suboptimality and Theorem I provide a means of assessing the sensitivity of the interconneced system 9 to changes in the interconnection structure: 9 is suboptimal and stable as long as Ac is such that F(M) is negative semidefinite for some matrix M and the pair (A, + A=, Q”‘) is observable. We now turn our attention to another aspect of sensitivity analysis: Can s, for fixed Act tolerate distortions in the suboptimal control u” caused by insertion of nonlinearities or introduction of the gain changes and phase shifts into the feedback loops? To answer this question we first consider the system
where the nonlinearity
@,(a)= Vol. 312. No. 3/4, pp. 179-197. Printed in Great Britain
@(a) is of the form
[4T(ad, 42’(a2),. . . , ISIS,
September/October
1981
i = 1,2,. . . , N,
(18) 183
M. E. Sezer and D. D. Siljak
with 4i(o;) being continuous the following:
vector-valued
functions
of ai = - K,x,. We state
Theorem II Suppose that F(M) is negative semidefinite for some M, and that the pair (At, + Ac, Q’j2) is observable. Then, the equilibrium x = 0 of & is asymptotically stable in the large for all nonlinearities @ whose components satisfy the conditions KioyR;‘ai 5 oydi(R;‘oi)
5 KioToiy i = 1, 2, . . . . , N
(19)
where Ki = 1 - (pi - A,~)12 with Ap being an arbitrarily small positive number, and Ki < ~0.
Next, we consider the case when the nonlinearities 4i in (18) are replaced by stable linear time-invariant systems Li having states ti, inputs ai and transfer function matrices L,(s). Denoting the resulting system by S,, we state the following: Theorem III Suppose that F(M) is negative semidefinite for some M, and that the pair (A, + Ac, Q’12) is observable. Then, the equilibrium x = 0, zi = 0, i = 1,2, . . ., N, of S, is asymptotically stable (in the large ) provided that Li(jti)RI’+
R;‘L~(jw)-(2-/.~i
+Ap)Ry’?
0, i = 1, 2, . . ., N,
(20)
for all w, where Ap is an arbitrarily small positive number.
Theorems II and III can be considered as generalizations of the wellknown criteria for gain and phase margin to interconnected systems with multivariable subsystems. To relate these results to classical concepts of gain and phase margin, let us consider a special case, where the subsystems Si have single inputs, i.e., mi = 1. In this case, ai and ai in (18) as well as L,(S) and Ri in (4) are scalars, and we obtain the following result from Theorems II and III: Corollary I Suppose that each Si has a single input, F(M) is negative semidefinite for some M, and the pair (A, + Ac, Q’12) is observable, so that S is stable. Then i-th feedback loop of S has (i) infinite gain margin ; (ii) at least 2 cos-‘(1 - cLi/2)phase margin and (iii) at least 50 pi% gain reduction tolerance.
By “i-th feedback loop of S has infinite gain margin” we mean as in (5) that arbitrarily large increases in the gain of the local feedback (11) do not destroy stability of S. Similar interpretation can be given to loop phase margin and loop gain reduction tolerance.
184
Journal of The
Franklin Institute Pergamon Press Ltd.
Sensitivity of Large-scale Control Systems The above results show that suboptimal systems can tolerate a large class of distortions in the control in pretty much the same way as optimal systems (4). Moreover, the similarity between the conditions (19) and (20) and the corresponding conditions of (5) for optimal multivariable systems suggests the possibility that a modification of the suboptimal control (7) might result in an interconnected system which is optimal with respect to a modified performance measure. To investigate this possibility, let us assume that F(M) is negative semidefinite for some matrix M. We define the modified weighting matrices I? and 0 as l? = diag{R,, I&, . . . , I&}
(21)
where
(22) and
(j = M-‘(Q - PBR~‘BrP) + M-‘PB@-‘BTPM-’ - A;M-‘P - M-‘PAc, (23) and state the following: Theorem IV Suppose that F(M) is negative semidefinite for some M. Then, (i) the matrix 0) is positive semidefinite; (ii) the decentralized control fi0
=
_
&,
K
=
R-IBTM-lp
(24)
is optimal for S with respect to the modified cost
&(x0,u)
=
(x’& + u =&)dt,
(25)
and
J’(x,,) = x,TM-‘Px,,; (iii) if, in addition, the pair (A, + Ac, Q”*) is observable, then the control 1’ stabilizes S.
Vol. 312. No. 3/4. pp. 179-197. Printed in Groat Britain
September/October
1981
185
M. E. Sezer and D. D. Siljak From (22) and (24) it follows that the components control are
of the modified optimal
(26) where us are local controls that are optimal for the decoupled subsystems Ss with respect to their local performance measures Ji. Thus, Theorem IV implies that once the suboptimality of the decentralized control u” is established using F(M), then increasing the gains of the feedback loops around the subsystems properly, we can recover the optimality of the overall system. Obviously, this means that the modified closed-loop interconnected system will have less sensitivity to distortions of the local controls. In particular, when the subsystems have single input, each feedback loop will have infinite gain margin, -+60” phase margin and 50% gain reduction tolerance. Theorem IV concludes our discussion on the sensitivity properties of decentrally controlled suboptimal interconnected systems. While the suboptimality criterion of_ Theorem I provides a measure of sensitivity of the closed-loop system S to interconnections among the subsystems, Theorems I, III, IV and Corollary I take care of the sensitivity of the system with a fixed interconnection structure to distortions of the suboptimal control. A natural sequel to this study is to investigate the sensitivity of the system to simultaneous perturbations in the interconnection structure and the control. This we consider next. III. Connective Suboptimality Let us consider the interconnected system in (1) again, where now we assume that the subsystems S, have single input, i.e., Bi = bi, i = 1, 2, . . ., N, and the interconnection
(27)
matrices are of the form A, = eJif, i = 1, 2, . . ., N.
(28)
In (28), ~ij are fixed matrices, and eir E [- 1, l] are the elements of an N x N interconnection matrix E = (eij)NxN and represent the gains of the interconnections among the subsystems. To emphasize the dependence of the interconnections on the parameters, we describe the interconnected system of (2) with a slight difference in notation as SE : 1 = [A, + &(E)]x
+ Bu,
(2%
where
186
Journal of The
Franklin Institute Pergamon Press Ltd.
Sensitivity of Large-scale
Control Systems
Following the suboptimal control strategy of the previous section, we associate with S, the block diagonal cost J(xO, U) of (5) and obtain a decentralized control u0 that is optimal for the decoupled system SD obtained from S, by setting E = 0 in (3.3). We could now proceed to consider suboptimality of the control u ’ for S, for each fixed E using Theorem I. Instead, we required that the control u0 be robust, that is, it should be a stabilizing suboptimal control under arbitrary structural perturbations described by any and all interconnection matrices E. Thus, we need (7): Definition II The control law (7) is said to be connectively p if there exists a positive number p such that
Pyx,)
5
suboptimal for S, with degree
/c’JO(xo),
(30)
for all x0 and all E.
Our immediate interest is to delineate a class of interconnection matrices A, for which there exist a connectively suboptimal and stabilizing decentralized control. For this purpose, let us assume without loss of generality that the pairs (Ai, bi) of the subsystems Si are in controllable canonical form, that is,
,i=l,2
For each interconnection
,...,
N.
(31)
matrix ~ij = (a:J of (28), we define an integer rij as
1, =
zf+: (4 -P}*
I
- 5 ni,
Aijf
O
(32)
Aij = 0
i-l
Each integer lij defines a boundary for nonzero elements of the corresponding matrix A,. Two typical cases are illustrated by Fig. 1.
“I
“1
FIG. 1. Vol. 312, No. 314, pp. 179-197, September/October Printed in Great Britain
1981
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M. E. Sezer and D. D. Siljak
We recall from (10) that if the integers f, are such that for any index set LJ ={i,, iZ,. . . ,im}C{l,. . . ,N} and any permutation $ = {jr, j2, . . . ,j,,,} of (33) then the interconnected system SE of (29) is stabilizable by the decentralized control law u” regardless of what the matrix E is, that is, it is connectively stabilizable. This, however, does not imply automatically that the control u” is also suboptimal. In fact, to establish suboptimality via Theorem I we have to choose the cost J(x,,, U) properly. For this we first need the following: Lemma I Suppose that the numbers numbers vi such that
1, satisfy the inequality
(33).
Then there exist
VI- ~i + 1 - I, > 0, i, j = 1, 2, . . ., N.
(34)
We now choose the blocks Qj of the state weighting matrix Q in J(x,, u) of (5)
Qi = where pi are arbitrary
Q(S)QiQ(S),
(35)
positive definite matrices, and Di(S) = diag {a”~,a”~+‘,. . ., 8”~+‘;-‘}
(36)
with S being a positive number to be determined. Also, keeping in mind that, due to the assumption on the dimension of the subsystem inputs, the input weighting matrix R in J(x,, U) has the form R = diag {p], p2, . . ., pN}, we chose the scalars pi as: oi = 6*(“~+“~), i = 1, 2, . . ., N.
(37)
With this choice of J(x,, u), we have: Theorem V Suppose that the numbers 1, satisfy (33), and that the cost J(xO, u) are defined by (3.5)-(37). Then, number 8 such that whenever S < s the control suboptimaf with degree u = p(S) for S, and is also p(S)+ 1 as S +O.
the matrices Q and R of there exists a positive law u” is connectively stabilizing. Furthermore
We note that the above choice of the cost matrices Q and R leads to high-gain feedback matrices kT = [ki,8-*“1, kizS-““i-“, . . ., ki”,S~2]which, in turn, result in closed-loop decoupled subsystems having eigenvalues of the order of S-‘. It is, in fact, this high gain characteristics of the suboptimal control 188
Journal of The
Franklin lnztitute Pergamon Press Lfd
Sensitivity of Large-scale Control Systems together with the structure closed-loop system
of the interconnection
SE: 1 =
[A, + ii,(E)
matrix 2,
that makes the
- BK]x,
(38)
connectively suboptimal. Clearly, SE also has the sensitivity properties described in Corrollary I, that is, for sufficiently small 6, each feed-back loop of SE has infinite gain margin, approximately 60” phase margin and 50% gain reduction tolerance. Finally, we would like to consider a special class of interconnected systems where interconnections enter additively through the inputs of the subsystems, that is, A, = biK%,i, j = 1, 2, . . . ., N, where & are fixed constant have the structure
vectors.
Clearly, for such systems,
(39) & matrices
(40) when the pairs (A, bi) are in the form of (31), are
SO
that the numbers 1, in (32)
iii = nj - ni, i = 1, 2, . . ., N.
(41)
Thus, (33) is satisfied, and Theorem V ensures the existence of a suboptimal decentralized control law. Furthermore, Theorem IV implies that by increasing the gains of the local controls the closed-loop interconnected system can be made optimal. A similar result has been recently obtained by Yasuda and colleagues (11) in an attempt to construct a performance measure with respect to which optimal control for S is decentralized. IV. Overlapping Subsystems In the previous sections we considered the sensitivity of decentrally controlled large-scale systems. A natural question at this point can be to ask whether a similar sensitivity analysis can be carried over for systems with other types of constraints on the control than decentralization. In this section we shall make an attempt to answer this question for a system with overlapping subsystems (8). Let us consider a system S described as
(42)
Vol. 312. No. 3/4, pp. 179-197. Printed in Great Britain
September/October
1981
189
M. E. Sezer and D. D. Siljak
where the ni vectors xi(t) constitute the state x = (~7, XT, xT)~ of S, the mi vectors ui(t) are the components of the input u = (UT, UT)’ of S, and A, and Bi are constant matrices of appropriate dimensions. We assume that the inputs U, and u2 of S are restricted to be of the form UI = -
w,,x,+K,*x*)
u2 =
(J&*x*
-
+
(43)
K23x3).
Our aim is to choose suboptimal controls u, and u2 satisfying the constraints in (43) and investigate the sensitivity of the resulting system to nonlinear distortions of these controls. For this purpose, we define 2, = (XT, xT)r and A?~ = (x,‘, XT)’ to be overlapping components of the state x of S, and form an expanded vector 2 = (iy, 2:)‘. Clearly, x’ and x are related by 4 = TX,
(44)
where
L,
0 0
0
0
0 0
In, 0 0 - I”,
L1
T= The transformation
3:
4l,
(44) defines an expanded
;:
[Ii
which can be considered
system
A,, Al21 0 A 22 II 0 ;21 __u’jk-22_ ____-
=
L
A::
(45)
Ul u2
0
32
to be composed
19
of two interconnected
(46)
subsystems
j’,:i, =A,i,+A,2i2++,U, (47)
3, : k, = A2i2+ A2,i, + B2U2,
with the matrices A,, A2, B,, B2, A,, and A,, being obvious from (46). From (47) and the definition of 2, and Z2 it is clear that any decentralized control for S would satisfy the constraint in (43) provided that the relation (44) holds also for the trajectories of the resulting closed-loop systems. Leaving this latter requirement aside for the moment, we proceed to computation of the suboptimal decentralized control u,, u2 for S. For this purpose, we rewrite the description of S in (46) as
S:k=(A,+A,)n+Bu, 190
(48) Journal
of The Franklin Institute Pergamon Press Ltd.
Sensitivity of Large-scale Control Systems where A, = diag {A,, AZ}, B = diag {B,, &} and A
C
=
0
[ A,,
A2
(49)
01 ’
and we associate with S a block diagonal cost
where $, = x’(0). We now choose the cost associated
with S to be
(x’Qx + Im
J(x,, u) =
0
uTRu)dt,
(51)
where x0 = x(O) and Q = TTQT.
(52)
It has been shown in (8) that if U=-G,
(53)
is any control for S which yields the cost j@(io), then the control u=-Kx,
K=ET,
(54)
for S yields a cost j@(x,) such that J@(x,) = J@( TX,).
(55)
Furthermore, the solutions of the resulting closed-loop systems satisfy (43). Having the relation (55) in hand, we design the control (53) to be the optimal control for the decoupled system
(56) with respect to the cost J in (50), and choose the control for S as in (54). Since k is block diagonal, this control will satisfy the constraint (43). Furthermore, if the matrix
P(M) = AgwP
+ WPA,
+ (I
-
Ax-$6 + RTRk),
(57)
where &f = diag{p ,I “,+“*, pJ,,2+n3} and p is the optimal cost matrix for SD, is vol.312. No.
3/4, pp. 179-197. September/October Printed in Great Britain
1981
191
M. E. Sezer and D. D. Siljak
negative semidefinite
for some positive numbers pI and I_L~, then P&J
5 ~~‘f~P&, I_L= min {pi, pz}
(58)
so that (55) implies J@(xo) I /.L-*x;TrPTx(). Once the finiteness closed-loop system
of the cost is established
(59) by (59) then the resulting
=(A-BK)x,
$:i
(60)
where A and B are the system matrices defined in (42), is stable provided the pair (A, Q”*) is observable. To investigate the sensitivity of the system 9 to nonlinearities in the control paths, we now consider the system &:~=Ax+B@(-Kx)
where the nonlinearity
(61)
Cpis of the form
a$-Kx)=
with & and #2 being continuous
41(
-
(62)
[
functions.
We state the following:
Theorem VI Supp.ose that P(k) is negative semidejinite for some positive numbers CL, a,nd p2, and that the pair (A, Q’12) is observable. Then, the equilibrium x = 0 of S@ is asymptotically stable in the large for all nonlinearities @ with com-
ponents satisfying
where Ki = 1 - (cci - Ap)/2 with Ap being an arbitrarily small positive number, i?i < QZ, and Ri are the diagonal blocks of the cost matrix R.
Although the result good example of how timality can be used systems which would 192
of Theorem VI is somewhat expected, it provides a the sensitivity analysis through the concept of subopto make clear the sensitivity properties of particular otherwise be very difficult to study. Journal of The
Franklin Institute Per~amon Press Ltd.
Sensitivity of Large-scale
Control Systems
References (1) H. W. Bode, “Network Analysis and Feedback Amplifier Design”, Van Nostrand, New York, 1945. (2) G. J. Thaler and R. G. Brown, “Analysis and Design of Feedback Control System”, McGraw-Hill, New York, 1960. (3) R. E. Kalman, “When is a linear control system optimal?” Trans. Am. Sot. Mech. Engrs Series, Vol. 86, pp. l-10, 1964. (4) B. D. 0. Anderson and J. B. Moore, “Linear Optimal Control”, Prentice-Hall, Englewood Cliffs, NJ, 1971. and phase for multiloop (5) G. Safonov M. Athans, regulators”, IEEE Vol. AC-22, 173-179, 1977. D. D. “Large-Scale Dynamic Stability and NorthHolland, York, 1978. R. Krtolica D. D. “Suboptimality of stochastic control estimation”, IEEE Vol. AC-25, 76-83, 1980. M. Ikeda, D. Siljak D. E. “Decentralized control overlapping information J. Optim. Appl. Vol. 1981, in Control: A Approach”, (9) W. Wonham, “Linear Springer, York, 1979. M. Ikeda D. D. “On decentrally large-scale systems”, Vol. 16, 331-334, 1980. optimizable interconnected K. Yasuda, Hikata and Hirai, “On Proc. 19th Conf. on and Control, New Mexico, 10-12, 1980, 536-537. (12) E. Sezer D. D. “Decentralized stabilization structure of large-scale systems”, 13th Asilomar on Circuits, and Comput., Grove, CA, 5-7, 1979, 176-181. Appendix : Proofs oj Propositions Proof of Theorem I We note that J@(&) = Jim ’ xT(t)(Q+ KTRK)x(t)dt, 7-mI cl where x(t) is the solution of (12) corresponding get
to x(0) = x0. From (8), (9) and (15) we
- x~(~)M-‘Px(T) Thus, negative definiteness
where p = min{pi}, establishing 179-197,
3
+ x;M-‘Pxo.
(A3)
of F(M) implies J@(x,) I x,TM-‘Px, 5 /.-‘x;Pxo,
Vol. 312, No.
(AlI
inequality
(A4)
(13).
193
M. E. Sezer and D. D. Siljak Proof of Theorem II Let v(x) = X%~‘PX, be a candidate for a Liapunov function straightforward computations, we obtain
(AS)
of .$,. Using
(8), (9), (15) and (19), by
d(x) = x’[(A,
+ A,)=M-‘P
+ Mm’P(A, + A,)]x + 2xTM-‘PB@( - Kx)
= xT[F(M)-
Q + (2M-’ - Z)K=RK]x +2 2 p;‘x:P,B&(i=l
R;‘B:Pixi)
= - x’(Q + ApK=RK)x
(A@
so that d(x) 5 0. On the other hand, the observability assumption together with the fact that B(O) = 0, which is implied by (19) and continuity of @(a), ensure that G(x) is not identically zero along any trajectory of 3, except the equilibrium x = 0. This completes the proof. Proof of Theorem ZZI Let the collection of the linear system Li inserted into the feedback loops be denoted by 2, which has the state z = (zT, z:, . . , zj?i)‘, the input (T = COT,a:, . . . , CT:)’ with ai = Kixi, i = 1, 2, . . ., N, and the block diagonal transfer function 1L(s) = diaglMs), L*(s), . . J&(S)}. Obviously, (x = 0, z = 0) is the equilibrium of &. For any x(0) = x0 we have x;M-‘Px,, = x’(T)M-‘Px(7)
- ,-$ I
Using (8), (17) (15) and negative semidefiniteness x;M-‘Px,
-
[x=(t)M-‘Px(t)]dt.
(A7)
of F(M), we obtain from (A7)
T~T(t)(Q+ApM-‘KTRK)x(t)dt I0
2 I 7 [x’(t)(Z - 2M-’ - ApM-‘)K*RKx(t) 0
- 2x=(t)M-‘PBu(t)]dt.
(-48)
Taking the limit as r -+ 2, the right hand side (RHS) of (A8) can be written using Parseval’s relation as RHS =&j_=
a
{x*(jo)(Z - 2M-’ - ApM-‘)K=RKx(jo)
- x*(jo)M-‘PBu(jo)
- u*(jw)B=M-‘Px(jw)}dw where x(jo) and u(jw) are Fourier transforms x(jw) are related by
of x(t) and u(f). With z(O) = 0, u(jw) and
u(jw) = L(jw)u(jw) = - L(jo)Kx(jw).
194
(A9)
(AlO) Journal of The Franklin Institute Pergamon Press Ltd
Sensitivity Substituting
RHS = &
of Large-scale
Control Systems
(AlO) into (A9) we obtain m
I_ x*(jo)M-‘K’R[L(jw)R-
+ R-‘L*(jw)
m
- (21- M + ApI)R-‘lRKx(jo)do
(Al 1)
and (20) implies that RHS 2 0. Thus, (A8) gives m
x’(t)(Q
+ APM-‘KTRK)x(t)dt
5 x;M-‘Pxo I m.
6412)
(A12) together with the observability assumption implies that the solutions of L% corresponding to the initial condition x(0) = x0, z(0) = 0 tend to zero as t + M. Since x0 is arbitrary, this is equivalent to saying that the poles of the transfer function [sl- A, - Ac + BL(s)K] are stable. The proof then follows from the argument in (5) that all poles of $. which do not appear as a pole of this transfer function (if there are any) should be poles of Y, which are assumed to be stable, so that all the poles of & are stable. This completes the proof.
Proof of Corollary I Under the conditions
of the corrollary, K 5 bi(Oj)/oi
(19) reduces to
I l?i 5GO”,i = 1, 2, . . ., N,
(A13)
where ui and di(oi) are scalars, from which (i) and (iii) follow. Similarly, (20) becomes Re fi(jw)2 1 - 9, where Ii(s) is a scalar transfer function, Ii = exp cpi(jo), which gives Ipi I
i E N,
(Af4)
and (ii) follows from (A14) upon choosing -(pi - Ap)/2].
COS-‘[I
Proof of Theorem IV Using (8) and (15) in (23) we obtain
0 = Q - F(M)+
8, P,&[(l - 2&)R;’
+ /L;~R;‘]B:P~,
(Al%
and (i) is established by the definition of Ri in (22) and the fact that Q is positive semidefinite and F(M) is negative semidefinite. Also, it can easily be verified that (A, + A,)=M-‘P
+ M-‘P(AD + A,) - M-‘PBk’B=M-‘P
+ 0 = 0,
6416)
proving (ii). Finally, observability of the pair (A, + Ac, Q”‘) implies observability of the pair {A, +A= - Bl?, (e+ ~~~K)“*} (9), from which stability of the modified closed-loop system follows, proving (iii). This completes the proof. Vol. 312. No. 3/4, pp. 179-197, Printed in Great Britain
September/October
1981
195
M. E. Sezer and D. D. Siljak Proof of Lemma I
a combinatorial
problem which can be solved using graph-theoretic therefore, omitted. A similar result has been proved in (12). Proof of Theorem V
We first show that with Ai, b, as in (31), Qi as in (35) and pi as in (37), the positive definite solution of the Riccati equation ATPi + PiA, - p:‘Pib,bTPi P, =
GDi(G)P,(S)Di(S), i E N,
(‘418)
where pi(a) > 0 for any 6 > 0, and lim p,(S) = pi, > 0. For this, we substitute (A17) and pre- and post-multiply
b;%;‘(s)
(Al@ into
to get 6419)
where
A,(S) = 6Di(S)AiDL’(6)
=
(A20)
_._________________--,iEN.
Since the pair_ [A,(S), b,] is controllable and the pair [Ai( Q”‘] is observable, (A19) implies that Pi(S) > 0. Positive definiteness of the limit Pi, follows from continuity of f&(6). Now, to prove Theorem V, we introduce D = diag{D,, D2,. . . , 0,) choose /_l,=j&=...= pN = CL,define F(p) = pD-‘F(M)D-‘, and use (7) (IS), (35), (37) and (A18) to get F(/.L) = S(D-‘A:Dp where Q = diag {Q,, Q?, . . ., &} consider the matrix GPDArD-’
+ PDA~D-‘)
- (I-
p)(Q + PBBT~)
and p = P(6) = diag{P,(S),
(A21)
P,(6), . . ., p,,,(S)}. Now,
= [eii~Pi(s)Di(s)A,,Di’(6)1,,,
6422)
from the definition of the numbers Iii in Since 6Di(6)AijD7’(6) = [a$?“‘~ “J+‘~(~~~)]~~~~,, (32) and from (34) we conclude that the matrix in (A22) is of the order 6”, where v = min {vi - vi + 1 - iii}. Thus, as 6 +O, the first term in &)
approaches
0 regardless
of the values of e,,. Since, Q + pBBTp > 0 for all 6 < O,jt follows that for sufficiently small 6, there exists a I_L> 0 arbitrarily close to 1 such that F(p) < 0 for any E. This completes the proof.
196
Journal of The Franklin
Institute Pergamon Press Ltd.
Sensitivity
of Large-scale
Control
Systems
Proof of Theorem VI We first note that (2, + &)T
= TA,
(A23
and BT=B
(A24)
as can easily be verified by direct substitution. expressed as @( - Kx) = @(-
Also, the nonlinearity
zzi) =
in (62) can be
&$---p__, 1
(-425)
)
7
[
where &, and i?? are the diagonal blocks of l?. We now choose u(x) = x’(T%?‘PT)x,
(A26)
as a candidate for Liapunov function for 3,. Following similar steps as in the proof of Theorem II, and using (63) and (A23)-(A25), the derivative of t’(x) can be majorized as d(x) I - x’(Q + ApKTMm’RK)x and the result follows from the observability proof.
Vol. 312. No 3/4. pp 179-197. September/October Prmted in Great Britain
1981
I 0,
(AU)
of the pair (A, Q”‘). This completes
the
197