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Numerical computation of decentralized fixed modes
Automatica, Vol. 27, No. 2, pp. 375-382, 1991
0005-1098/91 $3.00 + 0.00 Pergamon Press plc © 1991 International Federation of Automatic Control
Printed in Great Britain.
Brief Paper
Numerical Computation of Decentralized Fixed Modes* R. V. PATELt and P. MISRA:~
Abstract--In this paper, we use an algebraic characterization of "fixed modes" of a decentralized linear multivariable system to show that the fixed modes are related to the "blocking zeros" of certain subsystems derived from the given decentralized system. A numerical algorithm is then presented which enables us to compute the fixed modes in a reliable manner. Examples are provided to illustrate the main results of the paper.
terizations in terms of transmission zeros of certain "sub-systems" of the given system. The determination of d.f.ms by these approaches can be computationally expensive for systems having high order and/or a large number of "stations", since many transmission zero computation tests would be required. Anderson (1982) gives a transfer function characterization. However, the characterization does not provide an efficient and numerically reliable method by which d.f.ms may be computed. Anderson and Clements (1981) give an algebraic characterization which provides valuable insight into the properties of d.f.ms and conditions under which they occur. The characterization requires the partitioning of the set of stations into two disjoint subsets and involves a rank test, but as will be discussed later, a direct application of the result to find d.f.ms can be computationally expensive. One of the most straightforward ways of computing d.f.ms is the method suggested by Wang and Davison (1973). This gives the fixed modes as those eigenvalues of the state matrix which are unaltered when random decentralized feedback is applied. However as pointed out in Misra and Patei (1986) and Vaz and Davison (1989), the method based on eigenvalue computation can be numerically unreliable. This is explained further in Section 5. In this paper, we relate the concept of "blocking zeros" (Patel, 1986) of a linear multivariable system to the fixed modes of decentralized systems. It is shown how such a characterization leads to a numerically reliable algorithm for computing d.f.ms.
1. Introduction IN RECENT years there has been considerable interest in the study of decentralized control of large scale linear muitivariable systems such as those which arise in developing control strategies for large flexible space structures, e.g. West-Vukovich et al., 1984 or multi-machine power systems (e.g. Davison and Tripathi, 1978). The decentralized structure of these systems is a consequence of the constraints that are imposed on the information flow within the system, usually because of the locations of various sensors and actuators. By judiciously locating these sensors and actuators, a structure can be chosen for a decentralized controller which makes it considerably simpler to implement than a "centralized" controller. The structure of a decentralized controller is an important issue in the control of large-scale systems. This is because of the existence of "decentralized fixed modes" (d.f.ms) (e.g. Wang and Davison, 1973; Anderson and Clements, 1981; Armentano and Singh, 1982; Corfmat and Morse, 1976; West-Vukovich et al., 1984). D.f.ms are those modes of the system which are invariant under the implementation of all decentralized controllers having a particular structure. Therefore, if a d.f.m, corresponding to a particular decentralized structure is unstable or has other undesirable characteristics, the decentralized controller will not be able to remedy the situation. One aspect of the design problem, therefore, is to develop methods of determining a structure for a decentralized controller such that there are no d.f.ms or no undesirable d.f.ms. Consequently, it is of interest to investigate the conditions under which these modes occur, and develop a numerically efficient and reliable method for computing them. Several characterizations of d.f.ms have been obtained in recent years, e.g. (e.g. Misra and Patel, 1986; Davison and Wan.g., 1985; Tarokh, 1985; Patel and Misra, 1984; Davison and Ozgiiner, 1983; Seraji, 1982; Anderson, 1982; Anderson and Clements, 1981; Corfmat and Morse, 1976; Wang and Davison, 1973). Some of these references provide charac-
2. Preliminaries Definition 1. A linear time-invariant multivariable system described by N
i(t) = Ax(t) + ~ Biui(t )
(2. la)
i=1
yi(t) = Cix(t ),
i = 1. . . . .
N
where x(t) ~ ~n, ui(t ) ~ R,,,i, yi(t) ~ R pi, i = 1. . . . . called an "N station decentralized system".
(2.1b) N, is
Definition 2. Given the system (2.1), if we define a set ~ of block-diagonal matrices K as K = { K I K = block diag (KI . . . . .
KN), K, e R "'×p'}
(2.2)
then the set of d.f.ms of (2.1) with respect to • is defined as A(A, B i, Ci, K ) = I'-'1 o ( A + ~ BiKiCi) IK~
* Received 2 October 1986; revised 1 December 1988; received in final form 18 June 1990. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor T. Ba~ar under the direction of Editor A. P. Sage. t Department of Electrical and Computer Engineering, Concordia University, Montreal, Canada H3G 1M8. Author to whom all correspondence should be addressed. ~Department of Electrical Engineering, Wright State University, Dayton, OH 45435, U.S.A. and Concordia University, Montreal, Canada.
\
i=1
(2.3)
where o(.) denotes the set of eigenvalues of the matrix (-).
Theorem 1. A scalar ~. e o(A) is a d.f.m, of the system described by (2.1) if and only if for some partition of the set Q = {1 . . . . . N) into disjoint subsets if2c = {iI . . . . . it,} and Q0 = (ik+l ..... iN), rank
Cik+l N
375
< n. 0
0
(2.4)
376
Brief Paper the matrix (o). If it ~t a(A), then it is a blocking zero of
Proof. See Anderson and Clements (1981). Remarks. Since, i t ~ a ( A ) , all "possible" candidates for d.f.ms are known a priori. Further, since the partition in (2.4) is disjoint, any mode which is controllable and observable from one or more stations cannot be a d.f.m, and these modes can be eliminated from consideration. Theorem 2. Any multi-input, multi-output, single-station system (A, B, C) can be reduced by means of an orthogonal state coordinate transformation T to a condensed form (F, G, H) called a "block upper Hessenberg form" (BUHF) such that F = T r A T , G = T r B and H = CT, with F~
• .-
F~.k
F,.~+,
F2I
F22
"'"
&.k
F=.~+~
0
&2
• ""
&.~
F3.~+,
" F,,
F=
(A,
B , C ) i f C(M,, - A )
3. Characterization and computation o f d.f.ms In this section, we will investigate the conditions under which the inequality in (2.4) is satisfied. It will be shown later that these conditions can be easily used to derive an efficient and numerically reliable algorithm to compute the d.f.ms of (2.1). Consider a system (F, G, H) with F ~ R ~×~, G ~ R ~×" and H ~ R p×~. Assume that F is a cyclic matrix with an eigenvalue it of multiplicity r. Also assume without loss of generality that the matrix F is in USF (see Fact 1). The system (F, G, H) therefore has the following structure:
F=
G=
0
0
-..
0
0
• • "
Ii'!],
Fk,t, 0
Or
it
0T
0
.
.
"'" A,, A,,+1 .
"'"
.
f3,r
f3,r + 1
,
Fk.k+l Fk+l.k+
Or
0
"'"
-0 r
0
-
1
I]
••
it
f,.,+l
0
it
g~
H = [H 1 H 2 . - - Hk Hi,+ iI
(2.5) c=
J
H = [H~h2'''
h, h,+.l.
(3.1)
d gL,
where F q ¢ R t, ,×6 ,, H i e R v × 6 - t and G l e ~ l°×'. The integers l~, i = 0 . . . . . k are defined as follows: 1o = rank (B), k
li=rank(Fi+Li), i = 1 . . . . .
t B = O.
k and ~ l,=t~ where k* is the i=o
dimension of the controllable subsystem;/~ = n if the system is controllable and/a < n if the system is uncontrollable.
Proof. The proof is by construction and can be found in Patel (1981), Paige (1981) and Van Dooren (1981). Reduction of a system to a block Hessenberg form has also been reported in several other publications (e.g. Nour-Eldin, 1977; Tse et al., 1978; Konstantinov et al., 1981), although non-orthogonal transformations have been used in some cases. Remarks. A similar result can be stated for reducing the triple to a "block lower Hessenberg form" (BLHF). Using Theorem 2, we can easily obtain a minimal order subsystem from the given triple (A, B, C) by first determining the controllable subsystem (A (c), B (el, C (c)) and then computing the observable subsystem (A (o), B (o), C (o)) of (A (c), B (c),
C(~>) Fact 1. The system (A, B, C) can be reduced by means of a unitary_state coordinate transformation U to a condensed from (F, (~,/Q) = (UHAU, UnB, CU) _where U H denotes the conjugate transpose of U, such that F is in an upper Schur form (USF). In this condensed form, all eigenvalues of A appear along the diagonal of /~ as real or complex_scalars, and can be made to appear along the diagonal of F in any desired order by appropriate choice of U (Golub and Van Loan, 1989). Fact 2. The system (A, B, C) can be reduced by means of an orthogonal state coordinate transformation T to a condensed form ([', G, 17t) = ( T r A T , TTB, CT) such that F is in real Schur form (RS_F). The eigenvalues of A appear along the diagonal of F with real eigenvalues as scalars and complex-conJugate pairs of eigenvalues as 2 × 2 blocks. These scalars and 2 x 2 blocks can be arranged in any desired order by appropriate choice of T (Golub and Van Loan, 1989). Definition 3. A scalar it ~ C is a "blocking zero" (Patel, 1986) of the system (A, B, C) with A a cyclic matrix, if C adj ( M ~ - A ) B = 0 where adj(°) denotes the adjoint of
Since it ~ O(Fll ) and F is a cyclic matrix, rank ( i t l - F ) = n - 1, which implies that ~.g+l ~ 0 , i = 2 . . . . . r. We can now state the following result:
Theorem 3. Let the system (F, G, H) defined above have an uncontrollable and unobservable mode at X. Also, assume (without loss of generality) that the eigenvalue 3. corresponding to this mode is in the (n, n)th position of F. Then rank [ M ~
F
G ] < n if and only if it is a blocking
zero of the system (F, ¢~,/4) where
/~=
Fll f13 0T )1" +f23 .
"'" "" "
0"T
''"
0
fl.r+l
]
f2,r+l
:
,
it +?r r + l j
F ;1 LglJ
Proof. See A p p e n d i x .
The above result can be easily applied in characterizing d.f.ms of (2.1). Once the set f~ has been partitioned into disjoint subsets f2c and ff2o, the matrix G will be the partitioned matrix [Gg~ Gi:" • " Gik] and the matrix H will r Hik+2 T "" . BiN] T T" be the partitioned matrix [H ,k+l One way to obtain the partitions f2 c and Q0 is to find all the stations from which it is uncontrollable and all the stations from which it is unobservable. This can be done merely by inspection once A has been reduced to its USF F with it at position ( n , n ) and (1, 1) to check for uncontrollability and unobservability respectively. For complex-conjugate pairs of eigenvalues, we can avoid the use of complex arithmetic by reducing A to an RSF, with the corresponding 2 × 2 blocks at the bottom right and top left corners to check for uncontrollability and unobservability respectively. However, the information obtained from the above inspection may not necessarily give a disjoint partition i.e. ff~cN f2o 4: Q where Q is the null set. If there is a station y such that y E £~,. N 20, then for (2.4) to hold, it must necessarily be uncontrollable and
Brief Paper unobservable from the yth station. Let u) = ff2~f3 f~0 *: O. To verify the rank condition in (2.4), it is necessary to assign each element of ud to Q~ or Q0 such that f2c I') Q0 = O and at the same time, the partition should be such that if A is a d.f.m, of (2.1), then the rank condition in (2.4) is satisfied. The problem of computing the d.f.ms of the system described by (2.1) can_be divided into two smaller problems: (1) Obtaining a set A such that the set of fixed modes A c A c o(A). T h e set /k consists of all possible candidates for d.f.ms; and (2) obtaining a disjoint partition of t2 (if it exists) such that the rank condition (2.4) is satisfied. To find the set /~___o(A) that consists of the possible candidates for the d.f.ms, we remove all the eigenvalues of A that are controllable and observable from the same station. This can be accomplished by reductions to B U H F and B L H F using orthogonal transformations as described in Theorem 2.
Remark. All ).~ • A need not be d.f.ms of the given system. The set .~ contains those eigenvalues of the system which are possible candidates for d.f.ms. Usually the set A is a very small subset of o(A). Having obtained/~, we now need to examine each element of A to determine whether or not it is a d.f.m, of the system. We denote the subsystem obtained at the end of the above procedure (after the controllable and observable subsystems from all stations have been removed) by an hth order system (`4,/ii, Ci), i = 1. . . . . N. The elements of /k are the eigenvalues of `4. In order to obtain the disjoint partitions K2~ and f~o (if they exist) that also satisfy (2.4), it will be necessary to evaluate at some complex value A several "transfer function relations" of the form Sq = C',(AI._, - ` 4 ) - ' / i j
(3.3)
where i, j • ~ and i 4: j. The matrices .4, l i s and Ci above are obtained from the system (A, Bj, Ci) as shown below: Let
-I" .,2 0 T
A
~
).
• . .
a2, r
a2,r+l
• .
J
Or
0
"'"
)~
a~, + 1 !
Or
0
--"
0
~.
1 C i = [ C . ci2 • " • ci,,- c,,,.+,].
Bi =
(3.4)
for which the last rows of the input matrices ~it, l = 1. . . . . N, are zero vectors. Step 3: Let the sets ~c and 2 0 b e ^ g i v e n by ~ c = {i~'. . . . . i~*, t'~+1. . . . . iv}, Qo = {is+l . . . . . iv, i~*+~. . . . . i~v}, where the "asterisked" elements are the ones which correspond to the stations from where Aq is either controllable but unobservable or uncontrollable but observable. Form W = ~c N ~o = {is+ 1. . . . . iv} and set ~ c = {i~ . . . . . i~*), ~2o = {i*+1 . . . . . i~,} and k = 1. Step 4: (a) Corresponding to the k-th (asterisked) station of Q0, form Siv+~t = Civ+,(Zqln 1 - `4)-l /it, l e ~ . (b) For l e ~c, (i) If Siv+d ~ 0 and I e Q~, go to Step 5. (ii) If Sic+kilO and l • qJ, Set ~ = Q¢ 69 {l} (remove l from (2~) Set f~0 = f~o ~ {1} (add l to Qo) Set W = • @ {1}, go to Step 4(c). (iii) If Si,+k t = 0 for all l • ~2~, go to Step 4(c). (c) If all (asterisked) stations of Qc are exhausted, go to Step 6. Else, set k = k + 1 and go to Step 4(a). Step 5: There is no disjoint partition for which (2.4) is satisfied. Therefore, Aq is not a d.f.m. Go to Step 7. Step 6: Set Q~ = ~ ; the disjoint partition satisfying (2.4) is given by ~2o and Q~. Therefore ;tq is a d.f.m. Set A = A ~) ;tq and go to Step 7. Step 7: If q = ri, stop; else set q = q + 1 and go to Step l(b). At the end of Algorithm 1, the set A(c_A) contains all the eigenvalues of A which are d.f.ms of (2.1).
Remarks. When A has some complex-conjugate pairs of eigenvalues, we can use the reduction to RSF in Steps 1 and 2 to avoid complex arithmetic. In Step 4b, the condition in (i) implies that for at least one station that must appear in any disjoint partition of f2, the transfer function relation evaluated at 2~q is non-zero. This, in turn, means that there is no disjoint partition of Q that satisfies (2.4) for ;~q. The condition in (ii) corresponds to the case where the station is in both ~ and Qo. If the condition is satisfied, then the station cannot be included in ~q~. So we include it in (2o and flo and remove it from ~ and qJ. In order to conform with the value of k in subsequent steps, we shall assume that the operation of set addition in Step 4b (ii) corresponds to including the station I at the end of the set t)0. Condition (iii) does not provide any additional information about forming disjoint partitions of f~ except to say that Aq could be a d.f.m. If W is the empty set but ,~q is not a d.f.m., then condition (i) would be satisfied. 4. Discussion o f the results In this section, we will discuss various computational and numerical properties of the proposed algorithms.
Then
.4=
All 0T :
a13 ,~,+a23 :
0T
0
•. . ...
~ll,r+ 1 ] a2,r+ 1 ! ,
' " " &+a,,+l_l
r"-,1
,3.,,
Lb~,jJ Next, we will discuss a systematic procedure which enables us to find the required disjoint partition of fl into ~2~ and flo.
Algorithm 1 (To find disjoint sets f ~ and flo). Step h (a) S e t q = l a n d A = O . (b) Transform (,4,/it, (~t), l = 1. . . . . N, such that ,4 is in USF with Zq in its (1,1) position. (c) Form a set f2o where ~qo contains all the stations for which the first columns of the output matrices (~t, 1 = 1. . . . . N, are zero vectors. Step 2: (9) Transform (,4,/it, (~t), 1 = 1. . . . . N, such that A is in USF with ;tq in its ~a, h) position. (b) Form a set ~q~ where fl~ contains all the stations AUTO 27:2-K
377
4.1. General remarks. 1. Note that the computational procedure developed in this paper assumes that the state matrix is cyclic. This restricts the class of systems for which we can compute d.f.ms. However, it should be pointed out that the eigenvalue problem for non-cyclic matrices is often likely to be poorly conditioned (Stewart, 1973). Therefore, any procedure that requires knowledge of the eigenvalues of such non-cyclic state matrices will give inaccurate results. 2. Decentralization can be considered as one class of constraints that can be imposed on a feedback structure. A more general class can be defined via arbitrary constraints on the elements of feedback matrices. For such systems, the proposed method will not be suitable for computing "fixed modes" whereas the eigenvalue approach of Wang and Davison (1973) can be used. 3. The characterization of d.f.ms in terms of blocking zeros of certain subsystems derived from the given decentralized system is consistent with the characterization given by Davison and Wang (1985) of d.f.ms in terms of transmission zeros. Such characterizations provide "structural" information about the decentralized system. It should be mentioned that the characterization does not impose any restriction on the structure of the decentralized system.
378
Brief Paper
Simpler results can of course be obtained when the system has certain additional properties, e.g., block diagonal structure for A, or interconnected systems with only input and output matrices in block diagonal form. 4. The algorithm for obtaining * , uses only orthogonal state coordinate transformations and is numerically "backward stable" (Van Dooren, 1981). This is a desirable property of the algorithm from the point of view of its application to very high order systems. For greater reliability, singular value decomposition (Patel, 1981) can be used for the reduction of the system to its BUHF. 5. The breakdown of the operations count required for the proposed algorithms is as follows: (a) Obtaining the matrix /i: This step requires several reductions to block lower and upper Hessenberg forms and involves approximately
~=ln2k(3nk+i~=k(mi+pl)) operations, where n, = n, the dimension of the original system; n k, k -> 2, is the dimension of the subsystem that is uncontrollable and/or unobservable from the previous k - 1 stations; r is an integer which is the smaller of N and the minimum number of stations from which the entire system is controllable and observable. (b) Finding the elements of the set A: These elements correspond to the eigenvalues of fi~ and typically r/<< n. Note however that since we need to evaluate the transfer function relations Sq, it would be advantageous to have A in upper Hessenberg form. Reducing the entire system to upper Hessenberg form would require approximately n2( 5 n + ~ (mi+pi)) i=1 operations. (c) Obtaining the partitions ~ and ~o:_For this step we need to rearrange the eigenvalues of A to determine the stations from which a particular element of ,~ is uncontrollable and/or unobservable. In this case, since the value of the shift in the Q R algorithm is known (Golub and Van Loan, 1989), this requires approximately h 2 operations for each Q R step. Including the transformations on B~ and t~, we need approximately
~2 h + ~ (m i + pi)
4.2. Illustrative example. We shall now illustrate Algorithm 1 by an example of a decentralized system with ~ c = {4,1, 2, 5, 6, 7}, f f 2 o = { 1 , 2 , 5 , 6 , 7 , 3 } , f ~ = { 4 } and Qo= {3}. Therefore, using the notation in the algorithm, s = 1 and v = 6. Given below are the results at various steps of the algorithm for checking if a scalar ).q • .,~ is a d.f.m, of the given system: k=l Step 4a:
[$31S32S34S35S36S37] at Aq • .,~. Let $34 = 0 and S31, 536 ~h 0. Step 4b(i): This condition is not satisfied ($34 = 0). Step 4b(ii): Since S31, $36 ~=0, remove stations 1 and 6 from ~c and qJ and include them in fat,. This gives Q ~ = { 2 , 4 , 5 , 7 } , W = { 2 , 5 , 7 } , f2~={4} and fao = {3, 1,6} i~+k = 3. Evaluate
k=2 Step 4a: iv+k = 1. Let $12~0. Step 4b(i): This condition is not satisfied (S14 = 0).^ Step 4b(ii): Since $12 ~a 0, remove station 2 from ~2c and qJ and include it in fa o. This gives ~c = {4, 5, 7}, = {5, 7}, f2c = {4} and f20 = {3, 1,6, 2}. k=3 Step 4a: i~÷ k = 6. Let S6t = 0 for all l • ~2¢. k=4 Step 4a: i~+k = 2. Let S2t = 0 for all l • (2¢. Step 6: ).q is a d.f.m, and the disjoint partition fao= {1,2,3,6} and fac= {4, 5, 7} satisfies the r a n k condition (2.4). In the above example, if either $24 or $64 were non-zero, then ~,q would not be a d.f.m. If $65 and $67 were non-zero, but $54 and $74 were zero, then ~-q would be a d.f.m, with f 2 0 = { 1 , 2 , 3 , 5 , 6 , 7 } and f 2 c = { 4 }. The algorithm ends when a disjoint partition is found or else when a conclusion is reached that no disjoint partition (for the value of ;tq under consideration) satisfying (2.4) exists.
5. Numerical examples In this section, we consider two numerical examples to illustrate the proposed algorithms.
Example 1. Consider a 3-station decentralized system. The
operations for this step. If ,% contains a complexconjugate pair of eigenvalues, then an implicit double shift can be used to get an RSF. (d) The number of transfer function relations S¢ that need to be evaluated to find the disjoint partition depends on the given system data and therefore cannot be specified a priori. However, a reasonable operations count for each /~q • A is approximately N-s (n-l) z n-l+~mi+ i=1
the count will be considerably higher because it requires several rank tests on systems of order greater than n.
)
2Pi i=1
where s and N - s are respectively the number of elements in f~0 and (2c. Note that in Step (a), for most practical systems, hi, k >- 2, would generally be much smaller than n, i.e. most of the modes of the system would typically be controllable and observable from many of the stations. Therefore, taking into account all the operations mentioned above and dropping the less significant terms from each step, we require approximately (2 + y)n 2 n(~ + 4) + ~'~ (m i + Pi) i=1 operations for computing all the decentralized fixed modes, where y is the number of transfer function relations S~j that we need to compute in order to find the disjoint partition of Q. If the aproach in Anderson and Clements (1981) is used,
matrices describing the system are given in Table 1. The state matrix has eigenvalues at { - 2 , - l . 5 , - 1 . 0 , 3 . 0 , 2 . 5 , 2 . 0 , 1.5, 1.0}. It is found that A = {2.0} i.e. only A = 2.0 is a possible d.f.m. Next, it is found that ~ = 2.0 is unobservable from stations 1 and 2 and uncontrollable from stations 1 and 3. Therefore ~o = {il, i~} and ~c = {i~, i~}. Following the steps of Algorithm 1, it is found that Szt =C21(~ql.-l-All)-lBll and 523 = are both zero matrices for Aq = 2.0. The elements of the matrices Set and $23 are given in Table 2. They are of the order of 10 -16 and can be safely assumed to be zero. Therefore, we have the partition Q , = {il, is} and fac = i/e} which are disjoint and satisfy the condition in (2.4) i.e. = 2.0 is a d.f.m, of the given system.
C21(~Lqln_l --An)-l/~13
Example 2. In this example, we will illustrate a difficulty that may be encountered in deciding whether or not a particular eigenvalue is a d.f.m, using the test proposed in Wang and Davison (1973). For instance, depending on how a " r a n d o m " decentralized feedback matrix affects the closed-loop eigenvalue problem, it may not always be possible to say conclusively by inspection of the open-loop and closed-loop eigenvalues whether or not there are any d.f.ms. The difficulty is increased further when the eigenvalue problem for the open-loop state matrix is ill-conditioned. The characterization and computational algorithms presented here enable us to conclude with greater certainty if a given mode is a d.f.m, or not. The data for Example 2 is given in Table 3. For several randomly generated values of feedback matrices with their elements of the order 102 , it was found that certain eigenvalues of the system do not "move"
B
1
=
-.6824007787 1.253223829 -.0725179078
-.0687101141 .0992129486 .1207746598
.9728813679 •6338502954 .07251791178
.0952676686 -.2252498152 -.2314675213
C2 =
.2216458770 .6386855981 3.578384236
=
.0066780601 •1715238903 .2092921292
,8321248154 1.874876846 -.4003742177
-
2
-
.0905768308 •1417124980 .0112031403
.0737554812 -.7779391327 .6048139599
- . 2 6 6 0 5 2 7 3 8 0
- . 1072959550 -.0477814471 -.0280553044
.0515436172 -.2226901107 -.1581541911
.3060021972 .0698972579 -.0994345890
-
-
•1535121891 .0948070092 .0858390289
.8059427381 - 1.471703558 .5700763364
1.103299168 .6507314098 -3.687097343 -
B3 =
3.833588112 1.692145555 -9.401666250 - 10.21076863 - 1.277505940 - 1.136995967 -2.544374131 3.580616126 -
7.808913485 8.940485304 31.47867615 17.03165000 4•700713320 31.63366095 17.08815424 14.47725433
- 1.339597088 -1.031929416 -1.250335699 - 1.102425449 -1.463336353 - 1.659477876 -2.282092104 - 1.798583916 8.869414499 19.28474045 18.08277192 19.94822168 9.939200440 26.58611759 25.04543540 17.02037591
.8816532905 .0878917469
24.06096115 16.40558839 25.20750633 23.84960895 10.15737156 19.48795695 29.74830670 17.73934984
1.091710896 -6.163631092 .3658471056 - 1.249825322 -6.527468755 -4.228199859 -3.664980257 -5.785447163 -
.1560969942 •1258808196 .3357822914
-.1697021187
- . 7 4 4 6 1 8 0 0 0 0
- 1.726710642
1.223098002 .0120458116 -3.700738679 -
-2.620038311 2.912205174 -5.644899094 -2.165972056 -3.186760175 -2.458138918 2.620038311 .5667218645
.1354525359 -11.85191042 5.593010537 .9773381096 10.98858465 -5.916479189 -5.126096156 - 11.47372649
.5276480743 ] .0120458116 l 2.940754836 _]
- . 0 5 9 4 5 0 1 7 7 0
- 1.110223024625157d - 16 -4.440892098500626d - 16 0 . ~ +00 -5.551115123125783d - 17 - 1.110223024625157d - 16 1.110223024625157d - 16
F - 1.110223024625157d - 16 |-2.220446049250313d - 16 L 2.220446049250313d - 16 F-8.326672684688674d - 17 | -1.110223024625157d - 16 [ 1.110223024625157d - 16
$21 =
Sza =
TABLE 2. TRANSFER RELATIONS S21 AND $23
4.440892098500626d - 16J
-2.220446049250313d - 1 6 ]
- 3. 330669073875470d - 16 ] -4.440892098500626d - 16 [ 4.440892098500626d - 16 J
J
.o636o8o2ool
1.440410961 ] 1.113991924 1 - . 1697021187 [ =1 .0472149362 |
-
- 3.144045973 3.494646208 -6.773878912 -2.599166467 -3.824112210 -2.949766702 -3.144045973 .6800662374
.1354525359 .5280631913 - . 1838540200 .9773381096 1.391388959 .6866298675 .4770129007 1.906247120
t Note that the above matrices have been rounded off to 10 decimal places• A copy of the exact matrices can be obtained from the authors.
C3
-.8816532905 -.6965370444 .7413375337
B
3.698135576 4.471485536 6.808655713 9.233430521 6.577811264 4.365195826 4.982190956 2.204831037
-10.48015324 11.64882069 -22.57959637 -8.663888222 -12.74704070 -9.832555673 -10.48015324 2.266887458
4.902280128 8.663631092 5.242845433 9.358517860 10.33803799 6.026323164 7.475549489 5.785447163
--2.632399367 .6965370444 .1.251321667
-8.384122594 9.319056555 -18.06367710 -6.931110578 -10.19763256 -7.866044538 -8.384122594 1.813509966
-2.902280128 14.57598539 -13.01717766 -9.358517860 10.90157850 7.438961098 4.590977560 16.85292654
C 1=
-6.288091946 6.989292416 - 13.54775782 -5.198332933 -7.648224421 -5.899533404 -6.288091946 1.360132475
A=
-
TABLE 1. VARIOUS MATRICESt IN EXAMPLE 1
-13.10019155 14.56102587 -28.22449547 -10.82986028 -15.93380088 -12.29069459 -13.10019155 2.833609322
] ] [ | | / | _1
L~ ',,,I
t~
5.9 -4.6 -6.0 -3.0 8.0 2.5
e.v.s
o/l
Some
- 1 . 8 0 5 9 d + 01 - 7 . 4 8 5 0 d + 01
00 01 00 00 01 01 01 01 01 00
1 . 7 5 9 3 d + 01 7 . 3 7 1 2 d + 01
+ + + + + + + + +
C2=
-2.1500d -3.4050d 6.5000d -6.4000d -3.3000d -3.8600d -3.2100d 2.0000d -2.8550d 2.1500d
2 . 3 5 2 6 d + 01 - 6 . 2 4 2 3 d + 01
00 01 00 01 01 01 01 00 01 00
- 4 . 7 3 2 3 d + 01 4 . 2 8 8 0 d + 01
+ + + + + + + + + +
CI=
A=
6.0000d 3.0500d -7.5000d 1.2000d 2.9500d 3.1200d 3.0300d -6.2000d 1.9800d 3.8000d + + + + + + + + + +
+ + + + + + + + +
- 2 . 6 3 7 4 d + O1 - 1 . 1 0 6 7 d + 02
01 02 00 01 02 02 02 O0 02 O0
+ + + + + + + +
8 . 9 4 1 0 d + O0 3 . 7 8 3 1 d + 01
+ + + + + + + + + +
00 01 00 00 01 01 01 00 01 00
8 . 7 7 1 0 d + 00 3 . 6 9 4 6 d + 01
- 2 . 3 5 7 7 d + 01 2 . 1 3 8 5 d + 01
1 . 0 1 0 5 d + 01 3 . 8 9 5 0 d + 01
- 5 . 0 6 8 7 d + 01 4 . 1 2 7 0 d + 01
+
-5.359871211893602d +3.066671782119345d 4.909457853960178d 3.095864118165678d 5.833571910641285d -4.595795866440305d - 5.995408391604500d - 2.999789534354356d 8.012032299929436d 2.499967133489732d
c/l e.v.s for k 0 "~ 10 - 2
+ + + + + + + + + +
00 00i 00 00 00 00 00 00 00 00
1.981705713622234d +5.349405101541089d 4.794988863438237d - 2.272479821115828d 5.806693718096567d -4.594925581006383d - 6.003566998696624d -3.000821692878285d 7.999957971270422d 2.500010816079302d
kq ~ 102
c/l e.v.s for
+ + + + + + + + + +
02 04i 00 00 00 00 00 0(3 00 00
14 16i 00 01i 01 00 04 01 01 01
- 2 . 2 0 0 0 d - 02 - 5 . 1 0 0 0 d - 02
2.007855000236994d + :t:5.349407732726706d + 8.994344754265372d + +2.495595121512005d + -6.761211902258197d -4.466914156228984d + -4.752607649352673d + 4.101839349739446d + -4.457406647868712d + 1.240053279603918d +
k o -~ 10 l°
+ 00 + 01 +00 + 00 + 01 + 01 + 01 + 00 + 01 + O0
0.0000d + 00 - 4 . 0 9 9 6 d + 01
c/l e.v.s for
02 01 00 00 01 01 01 01 01 00
0.0000d - I. 5 5 0 0 d 0.0000d 0.0000d - 1.5500d -1.5500d -1.5500d -7.5000d - 1.550(O O.O000d
- 1.7100d
01 01 00 01 01 01 01 01 01 01 + + + + + + + +
+ + + + + + + + + +
-4.0000d -4.0900d 0.0000d - 1.8300d -4.2770d -4.1020d -4.2690d -4.0990d -4.1020d
1.0200d 3.5300d 7.5000d 1.0200d 1.5500d 3.3200d 3.2300d 1.2000d 2.5000d 1.9800d
+ 01 3 . 5 0 0 0 d - 01 - 1.0300d + 00 - 4 . 9 0 1 0 d + 01 9 . 0 0 0 0 d - 02 - 5 . 0 1 8 0 d + 01 5 . 0 0 0 0 d - 02 - 1.3300d + 00 - 1.2400d + 00 3 . 1 0 0 0 d - 01
"-4.8920d
1.7500d 1.3550d -6.5000d 6.0000d 1.7500d 1.8100d 1.6200d -3.1000d 1.2650d -2.5500d
B2 =
01 00 00 01 00 O0 00 00 00 00
1 . 6 8 2 0 d + O0 - 6 . 7 0 0 0 d - 02
+ + + + + + + +
4.0000d 4.1000d -7.5000d 4.0000d 8.0000d 5.5000d 2.5000d -9.1000d 3.9000d -8.5000d
-7.0000d 1.2639d 2.1700d -7.0000d 1.2993d 1.3202d 1.2577d 1. 5 4 0 0 d 1.3148d -4.1600d
00 01 01 01 01 O1 01 00 01 00
TABLE 4. COMPARISON OF SOME OPEN-LOOP AND CLOSED-LOOP EIOENVALUES
1.2830d + 00 2.1835d + 00
00 02 O0 O0 02 02 02 O0 02 O0
+ + + + + + + + + +
2 . 1 8 4 4 d + O1 - 6 . 4 2 6 5 d + O1
+
1.2401d 2.0800d 1.4100d 1.2593d 1.2942d 1.2401d 2.0700d 1.2950d - 1.9200d
-2.5500d - 4.4050d 1.4000d - 1.2800d -4.7000d -4.4100d -4.6500d 9.3000d - 3.2450d - 1.2500d 1.4100d
00 01 01 00 00 01 01 01 01 01
- 2 . 6 8 5 4 d + 01 1 . 9 7 6 8 d + 01
BI=
8.4500d 1.2850d 1.1000d 4.2000d -2.0000d 1.2100d 1.0200d 1.5100d 1.2350d 1.3450d
TABLE 3. VARIOUS MATRICES IN EXAMPLE 2 + 01
+ 01
- 1.0200d
-9.7580d + 00 - 3 . 8 8 7 7 d + 01
4 . 9 0 5 6 d + 01 - 1 . 5 8 0 0 d - 01
0.0000d +00 - 1 . 0 2 0 0 d + 01 -1.0900d + 01 O.O000d + O0 - 5 . 0 0 0 0 d + O0 -1.0900d + 01
+
-1.0900d
01 0.0000d + 00
- 1.0200d
+ + + + + + + + +
00 01 00 01 00 00 01 00 01
+ 00
8.0050d + 00 3 . 5 9 1 7 d + 01
2 . 7 1 6 1 d + 01 2 . 1 2 6 9 d + 01
4.6500d - 1.4000d -3.8000d 1.7500d 5.4000d 5.7000d -1.2200d 3.5500d -1.6050d
- 8.0500d
2.
Brief P a p e r appreciably. However, as the magnitude of the elements of the feedback matrices were gradually increased to 101°, the eigenvalues changed completely. Table 4 compares typical values of some of the open-loop eigenvalues with those of the closed-loop eigenvalues for 2 representative sets of values of feedback gains: k 0 - 102 and kij - 101° where the k0s are the elements of the randomly generated decentralized feedback matrices. It should be noted that the eigenvalues at - 3 . 0 and 2.5 do not change appreciably for k0~102 but are altered completely for k 0 - 1 0 TM. In contrast, corresponding to the eigenvalue at -4.6, we have eigenvalues at -4.59492 and -4.46691 for k O- 1 0 2 and 10TM respectively. Such observations with this and several other examples suggest that some difficulties could arise in computing the set of d.f.ms using the characterization given in (2.3). Applying the algorithms proposed above, it was found conclusively that the system does not have any d.f.ms. This was further confirmed by performing the rank test (using the singular value decomposition) on the system matrix in (2.4) A question that arises from Example 2 is what numerical or "threshold" value should be used for "zero" in the algorithms proposed in this paper. Among the numerical techniques used in the algorithms, the maximum error is accumulated in the reduction of the state matrix to an RSF and hence an error bound on this reduction (Stewart, 1973; Wilkinson, 1965; Golub and Van Loan, 1989) may be used to define a value for "zero".
6. Conclusions In this paper, we have used a characterization of d.f.ms given by Anderson and Ciements (1981) to define d.f.ms in terms of blocking zeros of certain subsystems of the given decentralized system. Based on this characterization, an efficient and reliable method has been proposed for computing d.f.ms. The computational method uses only orthogonal transformations and can be easily implemented with software available in scientific programming packages such as IMSL, EISPACK, LINPACK, etc. Extensive numerical tests carried out so far suggest that the proposed approach is numerically more reliable than existing methods for computing d.f.ms. Acknowledgment--The authors are grateful to the reviewers for their helpful comments and suggestions. This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant A1345. References Anderson, B. D. O. and D. J. Clements (1981). Algebraic characterization of fixed modes in decentralized control. Automatica, 17, 703-712. Anderson, B. D. O. (1982). Transfer function description of decentralized fixed modes. I E E E Trans. Aut. Control, AC-27, 1176-1182. Armentano, V. A. and M. G. Singh (1982). A procedure to eliminate decentralized fixed modes. IEEE Trans. Aut. Control, AC-27, 258-260. Corfmat, J. P. and A. S. Morse (1976). Decentralized control of linear multivariable systems. Automatica, 12, 479-496. Davison, E. J. and N. Tripathi (1978). The optimal decentralized control of a large power system: load and frequency control. 1EEE Trans. Aut. Control, AC-23, 312-325. Davison, E. J. and I). Ozgiiner (1983). Characterization of decentralized fixed modes for interconnected systems. Automatica, 19, 169-182. Davison, E. J. and S. H. Wang (1985). A characterization of decentralized fixed modes in terms of transmission zeros. IEEE Trans. Aut. Control, AC-30, 81-82. Golub, G. H. and C. F. Van Loan (1989). Matrix Computations, 2nd edn. Johns Hopkins University Press, Baltimore. Konstantinov, M., P. Petkov and N. Christov (1981). Invariants and canonical forms for linear multivariable
381
systems under the action of orthogonal transformation groups. Kybernetika, 17, 413-424. Misra, P. and R. V. Patel (1986). Characterization and computation of decentralized fixed modes of multivariable systems, Proc, 1986 Amer. Control Conf., Seattle, 427-432. Nour-Eldin, H. (1977). Minimalrealisierung der MatrixUbertragungsfunktion. Regelungstechnik, 25, 82-87. Paige, C. C. (1981). Properties of numerical algorithms related to computing controllability. IEEE Trans. Aut. Control, AC-26, 130-138. Patel, R. V. (1981). Computation of minimal order state-space realizations and observability indices using orthogonal transformations. Int. J. Control, 33, 227-246. Patel, R. V. and P. Misra (1984). A numerical test for transmission zeros with applications in characterizing decentralized fixed modes. Proc. 23rd IEEE Conf. on Decision and Control, Las Vegas, 1746-1751. Patel, R. V. (1986). On blocking zeros in linear multivariable systems. 1EEE Trans. Aut. Control, AC-31, 239-241. Seraji, H. (1982). On fixed modes in decentralized control systems. Int. J. Control, 35, 775-784. Stewart, G. W. (1973). Introduction to Matrix Computations. Academic Press, New York. Tarokh, M. (1985). Fixed modes in multivariable systems using constrained controllers. Automatica, 21, 495-497. Tse, E. C. Y., J. V. Medanic and W. R. Perkins (1978). Generalized Hessenberg transformations for reduced order modelling of large-scale systems. Int. J. Control, 27, 493-512. Van Dooren, P. (1981). The generalized eigenstructure problem in linear systems theory. IEEE Trans. Aut. Control, AC-26, 111-129. Vaz, A. and E. J. Davison (1989). On the quantitative characterization of approximate decentralized fixed modes uing transmission zeros, Math. Control Signals Syst., 2, 287-302. Wang, S. H. and E. J. Davison (1973). On stabilization of decentralized control systems. IEEE Trans. Aut. Control, AC-18, 473-478. West-Vukovich, G. S., E. J. Davison and P. C. Hughes (1984). The decentralized control of large flexible space structures. IEEE Trans. Aut. Control, AC-29, 866-879. Wilkinson, J. H. (1965). The Algebraic Eigenvalue Problem. Oxford University Press, London.
Appendix Proof of Theorem 3. Since ). is an uncontrollable mode of (F, G, H), it follows from the structure of F and G that, gr+l = 0. Also, since ). is an unobservable mode of (F, G, H) and F is a cyclic matrix, we have rank[M~F]=n-1. Using the fact that 3.~o(Fu) , we can perform some elementary row operations on [)'/~H- F ] to get
rank
LrAInH-
F] = rank
x
*
*
• ..
~
*
Or
0
f2,3
"'"
f2,r
f2,r+ 1
Or
0
0
"'"
f3,r
f3,r+ 1
: Or
0
0
---
0
f~.~+l
Or
0
0
""
0
0
0
=n-1
,~(~) *3(x)
-..
,~(;~) ~r+l(X)
where ¢pj(~.)= hj + Hl(;tln_r - Fll)-lfl/, j = 2 . . . . . r + 1 and , denotes possible non-zero vectors. Since f~.i+l:~0, i = 2 . . . . . r, it follows that ¢~2(2~)= 0. Next, performing the
382
Brief Paper
elementary row operations mentioned above on the matrix G], we obtain
row operations to the matrix
[)'I~IF
G]=n-r+rank f2,, f2,,+1
rank [ ) ' 1 ~ F
~
•
O
Q
Therefore,
gr
l
~3(x) ,,(x)
0G].
Since f/,i+l ~0, i = 2. . . . . r, it follows from the structure of F that A , Therefore,
.
o(F).
×
0
rank[3"l~ F G]=rank[3"l"H-F
0
fr.r+l
0,(~.) O,+,(A)
gT
I
H,(M,_,- F,1)-'G,A
The right-hand side of the above equation is exactly the result that we would get if we were to apply the elementary
if and only if Z is a blocking zero of (F, G,/~). The result of the theorem then follows, completing the proof.
0005-1098/91 $3.00 + 0.00 Pergamon Press plc © 1991 International Federation of Automatic Control
Printed in Great Britain.
Brief Paper
Numerical Computation of Decentralized Fixed Modes* R. V. PATELt and P. MISRA:~
Abstract--In this paper, we use an algebraic characterization of "fixed modes" of a decentralized linear multivariable system to show that the fixed modes are related to the "blocking zeros" of certain subsystems derived from the given decentralized system. A numerical algorithm is then presented which enables us to compute the fixed modes in a reliable manner. Examples are provided to illustrate the main results of the paper.
terizations in terms of transmission zeros of certain "sub-systems" of the given system. The determination of d.f.ms by these approaches can be computationally expensive for systems having high order and/or a large number of "stations", since many transmission zero computation tests would be required. Anderson (1982) gives a transfer function characterization. However, the characterization does not provide an efficient and numerically reliable method by which d.f.ms may be computed. Anderson and Clements (1981) give an algebraic characterization which provides valuable insight into the properties of d.f.ms and conditions under which they occur. The characterization requires the partitioning of the set of stations into two disjoint subsets and involves a rank test, but as will be discussed later, a direct application of the result to find d.f.ms can be computationally expensive. One of the most straightforward ways of computing d.f.ms is the method suggested by Wang and Davison (1973). This gives the fixed modes as those eigenvalues of the state matrix which are unaltered when random decentralized feedback is applied. However as pointed out in Misra and Patei (1986) and Vaz and Davison (1989), the method based on eigenvalue computation can be numerically unreliable. This is explained further in Section 5. In this paper, we relate the concept of "blocking zeros" (Patel, 1986) of a linear multivariable system to the fixed modes of decentralized systems. It is shown how such a characterization leads to a numerically reliable algorithm for computing d.f.ms.
1. Introduction IN RECENT years there has been considerable interest in the study of decentralized control of large scale linear muitivariable systems such as those which arise in developing control strategies for large flexible space structures, e.g. West-Vukovich et al., 1984 or multi-machine power systems (e.g. Davison and Tripathi, 1978). The decentralized structure of these systems is a consequence of the constraints that are imposed on the information flow within the system, usually because of the locations of various sensors and actuators. By judiciously locating these sensors and actuators, a structure can be chosen for a decentralized controller which makes it considerably simpler to implement than a "centralized" controller. The structure of a decentralized controller is an important issue in the control of large-scale systems. This is because of the existence of "decentralized fixed modes" (d.f.ms) (e.g. Wang and Davison, 1973; Anderson and Clements, 1981; Armentano and Singh, 1982; Corfmat and Morse, 1976; West-Vukovich et al., 1984). D.f.ms are those modes of the system which are invariant under the implementation of all decentralized controllers having a particular structure. Therefore, if a d.f.m, corresponding to a particular decentralized structure is unstable or has other undesirable characteristics, the decentralized controller will not be able to remedy the situation. One aspect of the design problem, therefore, is to develop methods of determining a structure for a decentralized controller such that there are no d.f.ms or no undesirable d.f.ms. Consequently, it is of interest to investigate the conditions under which these modes occur, and develop a numerically efficient and reliable method for computing them. Several characterizations of d.f.ms have been obtained in recent years, e.g. (e.g. Misra and Patel, 1986; Davison and Wan.g., 1985; Tarokh, 1985; Patel and Misra, 1984; Davison and Ozgiiner, 1983; Seraji, 1982; Anderson, 1982; Anderson and Clements, 1981; Corfmat and Morse, 1976; Wang and Davison, 1973). Some of these references provide charac-
2. Preliminaries Definition 1. A linear time-invariant multivariable system described by N
i(t) = Ax(t) + ~ Biui(t )
(2. la)
i=1
yi(t) = Cix(t ),
i = 1. . . . .
N
where x(t) ~ ~n, ui(t ) ~ R,,,i, yi(t) ~ R pi, i = 1. . . . . called an "N station decentralized system".
(2.1b) N, is
Definition 2. Given the system (2.1), if we define a set ~ of block-diagonal matrices K as K = { K I K = block diag (KI . . . . .
KN), K, e R "'×p'}
(2.2)
then the set of d.f.ms of (2.1) with respect to • is defined as A(A, B i, Ci, K ) = I'-'1 o ( A + ~ BiKiCi) IK~
* Received 2 October 1986; revised 1 December 1988; received in final form 18 June 1990. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor T. Ba~ar under the direction of Editor A. P. Sage. t Department of Electrical and Computer Engineering, Concordia University, Montreal, Canada H3G 1M8. Author to whom all correspondence should be addressed. ~Department of Electrical Engineering, Wright State University, Dayton, OH 45435, U.S.A. and Concordia University, Montreal, Canada.
\
i=1
(2.3)
where o(.) denotes the set of eigenvalues of the matrix (-).
Theorem 1. A scalar ~. e o(A) is a d.f.m, of the system described by (2.1) if and only if for some partition of the set Q = {1 . . . . . N) into disjoint subsets if2c = {iI . . . . . it,} and Q0 = (ik+l ..... iN), rank
Cik+l N
375
< n. 0
0
(2.4)
376
Brief Paper the matrix (o). If it ~t a(A), then it is a blocking zero of
Proof. See Anderson and Clements (1981). Remarks. Since, i t ~ a ( A ) , all "possible" candidates for d.f.ms are known a priori. Further, since the partition in (2.4) is disjoint, any mode which is controllable and observable from one or more stations cannot be a d.f.m, and these modes can be eliminated from consideration. Theorem 2. Any multi-input, multi-output, single-station system (A, B, C) can be reduced by means of an orthogonal state coordinate transformation T to a condensed form (F, G, H) called a "block upper Hessenberg form" (BUHF) such that F = T r A T , G = T r B and H = CT, with F~
• .-
F~.k
F,.~+,
F2I
F22
"'"
&.k
F=.~+~
0
&2
• ""
&.~
F3.~+,
" F,,
F=
(A,
B , C ) i f C(M,, - A )
3. Characterization and computation o f d.f.ms In this section, we will investigate the conditions under which the inequality in (2.4) is satisfied. It will be shown later that these conditions can be easily used to derive an efficient and numerically reliable algorithm to compute the d.f.ms of (2.1). Consider a system (F, G, H) with F ~ R ~×~, G ~ R ~×" and H ~ R p×~. Assume that F is a cyclic matrix with an eigenvalue it of multiplicity r. Also assume without loss of generality that the matrix F is in USF (see Fact 1). The system (F, G, H) therefore has the following structure:
F=
G=
0
0
-..
0
0
• • "
Ii'!],
Fk,t, 0
Or
it
0T
0
.
.
"'" A,, A,,+1 .
"'"
.
f3,r
f3,r + 1
,
Fk.k+l Fk+l.k+
Or
0
"'"
-0 r
0
-
1
I]
••
it
f,.,+l
0
it
g~
H = [H 1 H 2 . - - Hk Hi,+ iI
(2.5) c=
J
H = [H~h2'''
h, h,+.l.
(3.1)
d gL,
where F q ¢ R t, ,×6 ,, H i e R v × 6 - t and G l e ~ l°×'. The integers l~, i = 0 . . . . . k are defined as follows: 1o = rank (B), k
li=rank(Fi+Li), i = 1 . . . . .
t B = O.
k and ~ l,=t~ where k* is the i=o
dimension of the controllable subsystem;/~ = n if the system is controllable and/a < n if the system is uncontrollable.
Proof. The proof is by construction and can be found in Patel (1981), Paige (1981) and Van Dooren (1981). Reduction of a system to a block Hessenberg form has also been reported in several other publications (e.g. Nour-Eldin, 1977; Tse et al., 1978; Konstantinov et al., 1981), although non-orthogonal transformations have been used in some cases. Remarks. A similar result can be stated for reducing the triple to a "block lower Hessenberg form" (BLHF). Using Theorem 2, we can easily obtain a minimal order subsystem from the given triple (A, B, C) by first determining the controllable subsystem (A (c), B (el, C (c)) and then computing the observable subsystem (A (o), B (o), C (o)) of (A (c), B (c),
C(~>) Fact 1. The system (A, B, C) can be reduced by means of a unitary_state coordinate transformation U to a condensed from (F, (~,/Q) = (UHAU, UnB, CU) _where U H denotes the conjugate transpose of U, such that F is in an upper Schur form (USF). In this condensed form, all eigenvalues of A appear along the diagonal of /~ as real or complex_scalars, and can be made to appear along the diagonal of F in any desired order by appropriate choice of U (Golub and Van Loan, 1989). Fact 2. The system (A, B, C) can be reduced by means of an orthogonal state coordinate transformation T to a condensed form ([', G, 17t) = ( T r A T , TTB, CT) such that F is in real Schur form (RS_F). The eigenvalues of A appear along the diagonal of F with real eigenvalues as scalars and complex-conJugate pairs of eigenvalues as 2 × 2 blocks. These scalars and 2 x 2 blocks can be arranged in any desired order by appropriate choice of T (Golub and Van Loan, 1989). Definition 3. A scalar it ~ C is a "blocking zero" (Patel, 1986) of the system (A, B, C) with A a cyclic matrix, if C adj ( M ~ - A ) B = 0 where adj(°) denotes the adjoint of
Since it ~ O(Fll ) and F is a cyclic matrix, rank ( i t l - F ) = n - 1, which implies that ~.g+l ~ 0 , i = 2 . . . . . r. We can now state the following result:
Theorem 3. Let the system (F, G, H) defined above have an uncontrollable and unobservable mode at X. Also, assume (without loss of generality) that the eigenvalue 3. corresponding to this mode is in the (n, n)th position of F. Then rank [ M ~
F
G ] < n if and only if it is a blocking
zero of the system (F, ¢~,/4) where
/~=
Fll f13 0T )1" +f23 .
"'" "" "
0"T
''"
0
fl.r+l
]
f2,r+l
:
,
it +?r r + l j
F ;1 LglJ
Proof. See A p p e n d i x .
The above result can be easily applied in characterizing d.f.ms of (2.1). Once the set f~ has been partitioned into disjoint subsets f2c and ff2o, the matrix G will be the partitioned matrix [Gg~ Gi:" • " Gik] and the matrix H will r Hik+2 T "" . BiN] T T" be the partitioned matrix [H ,k+l One way to obtain the partitions f2 c and Q0 is to find all the stations from which it is uncontrollable and all the stations from which it is unobservable. This can be done merely by inspection once A has been reduced to its USF F with it at position ( n , n ) and (1, 1) to check for uncontrollability and unobservability respectively. For complex-conjugate pairs of eigenvalues, we can avoid the use of complex arithmetic by reducing A to an RSF, with the corresponding 2 × 2 blocks at the bottom right and top left corners to check for uncontrollability and unobservability respectively. However, the information obtained from the above inspection may not necessarily give a disjoint partition i.e. ff~cN f2o 4: Q where Q is the null set. If there is a station y such that y E £~,. N 20, then for (2.4) to hold, it must necessarily be uncontrollable and
Brief Paper unobservable from the yth station. Let u) = ff2~f3 f~0 *: O. To verify the rank condition in (2.4), it is necessary to assign each element of ud to Q~ or Q0 such that f2c I') Q0 = O and at the same time, the partition should be such that if A is a d.f.m, of (2.1), then the rank condition in (2.4) is satisfied. The problem of computing the d.f.ms of the system described by (2.1) can_be divided into two smaller problems: (1) Obtaining a set A such that the set of fixed modes A c A c o(A). T h e set /k consists of all possible candidates for d.f.ms; and (2) obtaining a disjoint partition of t2 (if it exists) such that the rank condition (2.4) is satisfied. To find the set /~___o(A) that consists of the possible candidates for the d.f.ms, we remove all the eigenvalues of A that are controllable and observable from the same station. This can be accomplished by reductions to B U H F and B L H F using orthogonal transformations as described in Theorem 2.
Remark. All ).~ • A need not be d.f.ms of the given system. The set .~ contains those eigenvalues of the system which are possible candidates for d.f.ms. Usually the set A is a very small subset of o(A). Having obtained/~, we now need to examine each element of A to determine whether or not it is a d.f.m, of the system. We denote the subsystem obtained at the end of the above procedure (after the controllable and observable subsystems from all stations have been removed) by an hth order system (`4,/ii, Ci), i = 1. . . . . N. The elements of /k are the eigenvalues of `4. In order to obtain the disjoint partitions K2~ and f~o (if they exist) that also satisfy (2.4), it will be necessary to evaluate at some complex value A several "transfer function relations" of the form Sq = C',(AI._, - ` 4 ) - ' / i j
(3.3)
where i, j • ~ and i 4: j. The matrices .4, l i s and Ci above are obtained from the system (A, Bj, Ci) as shown below: Let
-I" .,2 0 T
A
~
).
• . .
a2, r
a2,r+l
• .
J
Or
0
"'"
)~
a~, + 1 !
Or
0
--"
0
~.
1 C i = [ C . ci2 • " • ci,,- c,,,.+,].
Bi =
(3.4)
for which the last rows of the input matrices ~it, l = 1. . . . . N, are zero vectors. Step 3: Let the sets ~c and 2 0 b e ^ g i v e n by ~ c = {i~'. . . . . i~*, t'~+1. . . . . iv}, Qo = {is+l . . . . . iv, i~*+~. . . . . i~v}, where the "asterisked" elements are the ones which correspond to the stations from where Aq is either controllable but unobservable or uncontrollable but observable. Form W = ~c N ~o = {is+ 1. . . . . iv} and set ~ c = {i~ . . . . . i~*), ~2o = {i*+1 . . . . . i~,} and k = 1. Step 4: (a) Corresponding to the k-th (asterisked) station of Q0, form Siv+~t = Civ+,(Zqln 1 - `4)-l /it, l e ~ . (b) For l e ~c, (i) If Siv+d ~ 0 and I e Q~, go to Step 5. (ii) If Sic+kilO and l • qJ, Set ~ = Q¢ 69 {l} (remove l from (2~) Set f~0 = f~o ~ {1} (add l to Qo) Set W = • @ {1}, go to Step 4(c). (iii) If Si,+k t = 0 for all l • ~2~, go to Step 4(c). (c) If all (asterisked) stations of Qc are exhausted, go to Step 6. Else, set k = k + 1 and go to Step 4(a). Step 5: There is no disjoint partition for which (2.4) is satisfied. Therefore, Aq is not a d.f.m. Go to Step 7. Step 6: Set Q~ = ~ ; the disjoint partition satisfying (2.4) is given by ~2o and Q~. Therefore ;tq is a d.f.m. Set A = A ~) ;tq and go to Step 7. Step 7: If q = ri, stop; else set q = q + 1 and go to Step l(b). At the end of Algorithm 1, the set A(c_A) contains all the eigenvalues of A which are d.f.ms of (2.1).
Remarks. When A has some complex-conjugate pairs of eigenvalues, we can use the reduction to RSF in Steps 1 and 2 to avoid complex arithmetic. In Step 4b, the condition in (i) implies that for at least one station that must appear in any disjoint partition of f2, the transfer function relation evaluated at 2~q is non-zero. This, in turn, means that there is no disjoint partition of Q that satisfies (2.4) for ;~q. The condition in (ii) corresponds to the case where the station is in both ~ and Qo. If the condition is satisfied, then the station cannot be included in ~q~. So we include it in (2o and flo and remove it from ~ and qJ. In order to conform with the value of k in subsequent steps, we shall assume that the operation of set addition in Step 4b (ii) corresponds to including the station I at the end of the set t)0. Condition (iii) does not provide any additional information about forming disjoint partitions of f~ except to say that Aq could be a d.f.m. If W is the empty set but ,~q is not a d.f.m., then condition (i) would be satisfied. 4. Discussion o f the results In this section, we will discuss various computational and numerical properties of the proposed algorithms.
Then
.4=
All 0T :
a13 ,~,+a23 :
0T
0
•. . ...
~ll,r+ 1 ] a2,r+ 1 ! ,
' " " &+a,,+l_l
r"-,1
,3.,,
Lb~,jJ Next, we will discuss a systematic procedure which enables us to find the required disjoint partition of fl into ~2~ and flo.
Algorithm 1 (To find disjoint sets f ~ and flo). Step h (a) S e t q = l a n d A = O . (b) Transform (,4,/it, (~t), l = 1. . . . . N, such that ,4 is in USF with Zq in its (1,1) position. (c) Form a set f2o where ~qo contains all the stations for which the first columns of the output matrices (~t, 1 = 1. . . . . N, are zero vectors. Step 2: (9) Transform (,4,/it, (~t), 1 = 1. . . . . N, such that A is in USF with ;tq in its ~a, h) position. (b) Form a set ~q~ where fl~ contains all the stations AUTO 27:2-K
377
4.1. General remarks. 1. Note that the computational procedure developed in this paper assumes that the state matrix is cyclic. This restricts the class of systems for which we can compute d.f.ms. However, it should be pointed out that the eigenvalue problem for non-cyclic matrices is often likely to be poorly conditioned (Stewart, 1973). Therefore, any procedure that requires knowledge of the eigenvalues of such non-cyclic state matrices will give inaccurate results. 2. Decentralization can be considered as one class of constraints that can be imposed on a feedback structure. A more general class can be defined via arbitrary constraints on the elements of feedback matrices. For such systems, the proposed method will not be suitable for computing "fixed modes" whereas the eigenvalue approach of Wang and Davison (1973) can be used. 3. The characterization of d.f.ms in terms of blocking zeros of certain subsystems derived from the given decentralized system is consistent with the characterization given by Davison and Wang (1985) of d.f.ms in terms of transmission zeros. Such characterizations provide "structural" information about the decentralized system. It should be mentioned that the characterization does not impose any restriction on the structure of the decentralized system.
378
Brief Paper
Simpler results can of course be obtained when the system has certain additional properties, e.g., block diagonal structure for A, or interconnected systems with only input and output matrices in block diagonal form. 4. The algorithm for obtaining * , uses only orthogonal state coordinate transformations and is numerically "backward stable" (Van Dooren, 1981). This is a desirable property of the algorithm from the point of view of its application to very high order systems. For greater reliability, singular value decomposition (Patel, 1981) can be used for the reduction of the system to its BUHF. 5. The breakdown of the operations count required for the proposed algorithms is as follows: (a) Obtaining the matrix /i: This step requires several reductions to block lower and upper Hessenberg forms and involves approximately
~=ln2k(3nk+i~=k(mi+pl)) operations, where n, = n, the dimension of the original system; n k, k -> 2, is the dimension of the subsystem that is uncontrollable and/or unobservable from the previous k - 1 stations; r is an integer which is the smaller of N and the minimum number of stations from which the entire system is controllable and observable. (b) Finding the elements of the set A: These elements correspond to the eigenvalues of fi~ and typically r/<< n. Note however that since we need to evaluate the transfer function relations Sq, it would be advantageous to have A in upper Hessenberg form. Reducing the entire system to upper Hessenberg form would require approximately n2( 5 n + ~ (mi+pi)) i=1 operations. (c) Obtaining the partitions ~ and ~o:_For this step we need to rearrange the eigenvalues of A to determine the stations from which a particular element of ,~ is uncontrollable and/or unobservable. In this case, since the value of the shift in the Q R algorithm is known (Golub and Van Loan, 1989), this requires approximately h 2 operations for each Q R step. Including the transformations on B~ and t~, we need approximately
~2 h + ~ (m i + pi)
4.2. Illustrative example. We shall now illustrate Algorithm 1 by an example of a decentralized system with ~ c = {4,1, 2, 5, 6, 7}, f f 2 o = { 1 , 2 , 5 , 6 , 7 , 3 } , f ~ = { 4 } and Qo= {3}. Therefore, using the notation in the algorithm, s = 1 and v = 6. Given below are the results at various steps of the algorithm for checking if a scalar ).q • .,~ is a d.f.m, of the given system: k=l Step 4a:
[$31S32S34S35S36S37] at Aq • .,~. Let $34 = 0 and S31, 536 ~h 0. Step 4b(i): This condition is not satisfied ($34 = 0). Step 4b(ii): Since S31, $36 ~=0, remove stations 1 and 6 from ~c and qJ and include them in fat,. This gives Q ~ = { 2 , 4 , 5 , 7 } , W = { 2 , 5 , 7 } , f2~={4} and fao = {3, 1,6} i~+k = 3. Evaluate
k=2 Step 4a: iv+k = 1. Let $12~0. Step 4b(i): This condition is not satisfied (S14 = 0).^ Step 4b(ii): Since $12 ~a 0, remove station 2 from ~2c and qJ and include it in fa o. This gives ~c = {4, 5, 7}, = {5, 7}, f2c = {4} and f20 = {3, 1,6, 2}. k=3 Step 4a: i~÷ k = 6. Let S6t = 0 for all l • ~2¢. k=4 Step 4a: i~+k = 2. Let S2t = 0 for all l • (2¢. Step 6: ).q is a d.f.m, and the disjoint partition fao= {1,2,3,6} and fac= {4, 5, 7} satisfies the r a n k condition (2.4). In the above example, if either $24 or $64 were non-zero, then ~,q would not be a d.f.m. If $65 and $67 were non-zero, but $54 and $74 were zero, then ~-q would be a d.f.m, with f 2 0 = { 1 , 2 , 3 , 5 , 6 , 7 } and f 2 c = { 4 }. The algorithm ends when a disjoint partition is found or else when a conclusion is reached that no disjoint partition (for the value of ;tq under consideration) satisfying (2.4) exists.
5. Numerical examples In this section, we consider two numerical examples to illustrate the proposed algorithms.
Example 1. Consider a 3-station decentralized system. The
operations for this step. If ,% contains a complexconjugate pair of eigenvalues, then an implicit double shift can be used to get an RSF. (d) The number of transfer function relations S¢ that need to be evaluated to find the disjoint partition depends on the given system data and therefore cannot be specified a priori. However, a reasonable operations count for each /~q • A is approximately N-s (n-l) z n-l+~mi+ i=1
the count will be considerably higher because it requires several rank tests on systems of order greater than n.
)
2Pi i=1
where s and N - s are respectively the number of elements in f~0 and (2c. Note that in Step (a), for most practical systems, hi, k >- 2, would generally be much smaller than n, i.e. most of the modes of the system would typically be controllable and observable from many of the stations. Therefore, taking into account all the operations mentioned above and dropping the less significant terms from each step, we require approximately (2 + y)n 2 n(~ + 4) + ~'~ (m i + Pi) i=1 operations for computing all the decentralized fixed modes, where y is the number of transfer function relations S~j that we need to compute in order to find the disjoint partition of Q. If the aproach in Anderson and Clements (1981) is used,
matrices describing the system are given in Table 1. The state matrix has eigenvalues at { - 2 , - l . 5 , - 1 . 0 , 3 . 0 , 2 . 5 , 2 . 0 , 1.5, 1.0}. It is found that A = {2.0} i.e. only A = 2.0 is a possible d.f.m. Next, it is found that ~ = 2.0 is unobservable from stations 1 and 2 and uncontrollable from stations 1 and 3. Therefore ~o = {il, i~} and ~c = {i~, i~}. Following the steps of Algorithm 1, it is found that Szt =C21(~ql.-l-All)-lBll and 523 = are both zero matrices for Aq = 2.0. The elements of the matrices Set and $23 are given in Table 2. They are of the order of 10 -16 and can be safely assumed to be zero. Therefore, we have the partition Q , = {il, is} and fac = i/e} which are disjoint and satisfy the condition in (2.4) i.e. = 2.0 is a d.f.m, of the given system.
C21(~Lqln_l --An)-l/~13
Example 2. In this example, we will illustrate a difficulty that may be encountered in deciding whether or not a particular eigenvalue is a d.f.m, using the test proposed in Wang and Davison (1973). For instance, depending on how a " r a n d o m " decentralized feedback matrix affects the closed-loop eigenvalue problem, it may not always be possible to say conclusively by inspection of the open-loop and closed-loop eigenvalues whether or not there are any d.f.ms. The difficulty is increased further when the eigenvalue problem for the open-loop state matrix is ill-conditioned. The characterization and computational algorithms presented here enable us to conclude with greater certainty if a given mode is a d.f.m, or not. The data for Example 2 is given in Table 3. For several randomly generated values of feedback matrices with their elements of the order 102 , it was found that certain eigenvalues of the system do not "move"
B
1
=
-.6824007787 1.253223829 -.0725179078
-.0687101141 .0992129486 .1207746598
.9728813679 •6338502954 .07251791178
.0952676686 -.2252498152 -.2314675213
C2 =
.2216458770 .6386855981 3.578384236
=
.0066780601 •1715238903 .2092921292
,8321248154 1.874876846 -.4003742177
-
2
-
.0905768308 •1417124980 .0112031403
.0737554812 -.7779391327 .6048139599
- . 2 6 6 0 5 2 7 3 8 0
- . 1072959550 -.0477814471 -.0280553044
.0515436172 -.2226901107 -.1581541911
.3060021972 .0698972579 -.0994345890
-
-
•1535121891 .0948070092 .0858390289
.8059427381 - 1.471703558 .5700763364
1.103299168 .6507314098 -3.687097343 -
B3 =
3.833588112 1.692145555 -9.401666250 - 10.21076863 - 1.277505940 - 1.136995967 -2.544374131 3.580616126 -
7.808913485 8.940485304 31.47867615 17.03165000 4•700713320 31.63366095 17.08815424 14.47725433
- 1.339597088 -1.031929416 -1.250335699 - 1.102425449 -1.463336353 - 1.659477876 -2.282092104 - 1.798583916 8.869414499 19.28474045 18.08277192 19.94822168 9.939200440 26.58611759 25.04543540 17.02037591
.8816532905 .0878917469
24.06096115 16.40558839 25.20750633 23.84960895 10.15737156 19.48795695 29.74830670 17.73934984
1.091710896 -6.163631092 .3658471056 - 1.249825322 -6.527468755 -4.228199859 -3.664980257 -5.785447163 -
.1560969942 •1258808196 .3357822914
-.1697021187
- . 7 4 4 6 1 8 0 0 0 0
- 1.726710642
1.223098002 .0120458116 -3.700738679 -
-2.620038311 2.912205174 -5.644899094 -2.165972056 -3.186760175 -2.458138918 2.620038311 .5667218645
.1354525359 -11.85191042 5.593010537 .9773381096 10.98858465 -5.916479189 -5.126096156 - 11.47372649
.5276480743 ] .0120458116 l 2.940754836 _]
- . 0 5 9 4 5 0 1 7 7 0
- 1.110223024625157d - 16 -4.440892098500626d - 16 0 . ~ +00 -5.551115123125783d - 17 - 1.110223024625157d - 16 1.110223024625157d - 16
F - 1.110223024625157d - 16 |-2.220446049250313d - 16 L 2.220446049250313d - 16 F-8.326672684688674d - 17 | -1.110223024625157d - 16 [ 1.110223024625157d - 16
$21 =
Sza =
TABLE 2. TRANSFER RELATIONS S21 AND $23
4.440892098500626d - 16J
-2.220446049250313d - 1 6 ]
- 3. 330669073875470d - 16 ] -4.440892098500626d - 16 [ 4.440892098500626d - 16 J
J
.o636o8o2ool
1.440410961 ] 1.113991924 1 - . 1697021187 [ =1 .0472149362 |
-
- 3.144045973 3.494646208 -6.773878912 -2.599166467 -3.824112210 -2.949766702 -3.144045973 .6800662374
.1354525359 .5280631913 - . 1838540200 .9773381096 1.391388959 .6866298675 .4770129007 1.906247120
t Note that the above matrices have been rounded off to 10 decimal places• A copy of the exact matrices can be obtained from the authors.
C3
-.8816532905 -.6965370444 .7413375337
B
3.698135576 4.471485536 6.808655713 9.233430521 6.577811264 4.365195826 4.982190956 2.204831037
-10.48015324 11.64882069 -22.57959637 -8.663888222 -12.74704070 -9.832555673 -10.48015324 2.266887458
4.902280128 8.663631092 5.242845433 9.358517860 10.33803799 6.026323164 7.475549489 5.785447163
--2.632399367 .6965370444 .1.251321667
-8.384122594 9.319056555 -18.06367710 -6.931110578 -10.19763256 -7.866044538 -8.384122594 1.813509966
-2.902280128 14.57598539 -13.01717766 -9.358517860 10.90157850 7.438961098 4.590977560 16.85292654
C 1=
-6.288091946 6.989292416 - 13.54775782 -5.198332933 -7.648224421 -5.899533404 -6.288091946 1.360132475
A=
-
TABLE 1. VARIOUS MATRICESt IN EXAMPLE 1
-13.10019155 14.56102587 -28.22449547 -10.82986028 -15.93380088 -12.29069459 -13.10019155 2.833609322
] ] [ | | / | _1
L~ ',,,I
t~
5.9 -4.6 -6.0 -3.0 8.0 2.5
e.v.s
o/l
Some
- 1 . 8 0 5 9 d + 01 - 7 . 4 8 5 0 d + 01
00 01 00 00 01 01 01 01 01 00
1 . 7 5 9 3 d + 01 7 . 3 7 1 2 d + 01
+ + + + + + + + +
C2=
-2.1500d -3.4050d 6.5000d -6.4000d -3.3000d -3.8600d -3.2100d 2.0000d -2.8550d 2.1500d
2 . 3 5 2 6 d + 01 - 6 . 2 4 2 3 d + 01
00 01 00 01 01 01 01 00 01 00
- 4 . 7 3 2 3 d + 01 4 . 2 8 8 0 d + 01
+ + + + + + + + + +
CI=
A=
6.0000d 3.0500d -7.5000d 1.2000d 2.9500d 3.1200d 3.0300d -6.2000d 1.9800d 3.8000d + + + + + + + + + +
+ + + + + + + + +
- 2 . 6 3 7 4 d + O1 - 1 . 1 0 6 7 d + 02
01 02 00 01 02 02 02 O0 02 O0
+ + + + + + + +
8 . 9 4 1 0 d + O0 3 . 7 8 3 1 d + 01
+ + + + + + + + + +
00 01 00 00 01 01 01 00 01 00
8 . 7 7 1 0 d + 00 3 . 6 9 4 6 d + 01
- 2 . 3 5 7 7 d + 01 2 . 1 3 8 5 d + 01
1 . 0 1 0 5 d + 01 3 . 8 9 5 0 d + 01
- 5 . 0 6 8 7 d + 01 4 . 1 2 7 0 d + 01
+
-5.359871211893602d +3.066671782119345d 4.909457853960178d 3.095864118165678d 5.833571910641285d -4.595795866440305d - 5.995408391604500d - 2.999789534354356d 8.012032299929436d 2.499967133489732d
c/l e.v.s for k 0 "~ 10 - 2
+ + + + + + + + + +
00 00i 00 00 00 00 00 00 00 00
1.981705713622234d +5.349405101541089d 4.794988863438237d - 2.272479821115828d 5.806693718096567d -4.594925581006383d - 6.003566998696624d -3.000821692878285d 7.999957971270422d 2.500010816079302d
kq ~ 102
c/l e.v.s for
+ + + + + + + + + +
02 04i 00 00 00 00 00 0(3 00 00
14 16i 00 01i 01 00 04 01 01 01
- 2 . 2 0 0 0 d - 02 - 5 . 1 0 0 0 d - 02
2.007855000236994d + :t:5.349407732726706d + 8.994344754265372d + +2.495595121512005d + -6.761211902258197d -4.466914156228984d + -4.752607649352673d + 4.101839349739446d + -4.457406647868712d + 1.240053279603918d +
k o -~ 10 l°
+ 00 + 01 +00 + 00 + 01 + 01 + 01 + 00 + 01 + O0
0.0000d + 00 - 4 . 0 9 9 6 d + 01
c/l e.v.s for
02 01 00 00 01 01 01 01 01 00
0.0000d - I. 5 5 0 0 d 0.0000d 0.0000d - 1.5500d -1.5500d -1.5500d -7.5000d - 1.550(O O.O000d
- 1.7100d
01 01 00 01 01 01 01 01 01 01 + + + + + + + +
+ + + + + + + + + +
-4.0000d -4.0900d 0.0000d - 1.8300d -4.2770d -4.1020d -4.2690d -4.0990d -4.1020d
1.0200d 3.5300d 7.5000d 1.0200d 1.5500d 3.3200d 3.2300d 1.2000d 2.5000d 1.9800d
+ 01 3 . 5 0 0 0 d - 01 - 1.0300d + 00 - 4 . 9 0 1 0 d + 01 9 . 0 0 0 0 d - 02 - 5 . 0 1 8 0 d + 01 5 . 0 0 0 0 d - 02 - 1.3300d + 00 - 1.2400d + 00 3 . 1 0 0 0 d - 01
"-4.8920d
1.7500d 1.3550d -6.5000d 6.0000d 1.7500d 1.8100d 1.6200d -3.1000d 1.2650d -2.5500d
B2 =
01 00 00 01 00 O0 00 00 00 00
1 . 6 8 2 0 d + O0 - 6 . 7 0 0 0 d - 02
+ + + + + + + +
4.0000d 4.1000d -7.5000d 4.0000d 8.0000d 5.5000d 2.5000d -9.1000d 3.9000d -8.5000d
-7.0000d 1.2639d 2.1700d -7.0000d 1.2993d 1.3202d 1.2577d 1. 5 4 0 0 d 1.3148d -4.1600d
00 01 01 01 01 O1 01 00 01 00
TABLE 4. COMPARISON OF SOME OPEN-LOOP AND CLOSED-LOOP EIOENVALUES
1.2830d + 00 2.1835d + 00
00 02 O0 O0 02 02 02 O0 02 O0
+ + + + + + + + + +
2 . 1 8 4 4 d + O1 - 6 . 4 2 6 5 d + O1
+
1.2401d 2.0800d 1.4100d 1.2593d 1.2942d 1.2401d 2.0700d 1.2950d - 1.9200d
-2.5500d - 4.4050d 1.4000d - 1.2800d -4.7000d -4.4100d -4.6500d 9.3000d - 3.2450d - 1.2500d 1.4100d
00 01 01 00 00 01 01 01 01 01
- 2 . 6 8 5 4 d + 01 1 . 9 7 6 8 d + 01
BI=
8.4500d 1.2850d 1.1000d 4.2000d -2.0000d 1.2100d 1.0200d 1.5100d 1.2350d 1.3450d
TABLE 3. VARIOUS MATRICES IN EXAMPLE 2 + 01
+ 01
- 1.0200d
-9.7580d + 00 - 3 . 8 8 7 7 d + 01
4 . 9 0 5 6 d + 01 - 1 . 5 8 0 0 d - 01
0.0000d +00 - 1 . 0 2 0 0 d + 01 -1.0900d + 01 O.O000d + O0 - 5 . 0 0 0 0 d + O0 -1.0900d + 01
+
-1.0900d
01 0.0000d + 00
- 1.0200d
+ + + + + + + + +
00 01 00 01 00 00 01 00 01
+ 00
8.0050d + 00 3 . 5 9 1 7 d + 01
2 . 7 1 6 1 d + 01 2 . 1 2 6 9 d + 01
4.6500d - 1.4000d -3.8000d 1.7500d 5.4000d 5.7000d -1.2200d 3.5500d -1.6050d
- 8.0500d
2.
Brief P a p e r appreciably. However, as the magnitude of the elements of the feedback matrices were gradually increased to 101°, the eigenvalues changed completely. Table 4 compares typical values of some of the open-loop eigenvalues with those of the closed-loop eigenvalues for 2 representative sets of values of feedback gains: k 0 - 102 and kij - 101° where the k0s are the elements of the randomly generated decentralized feedback matrices. It should be noted that the eigenvalues at - 3 . 0 and 2.5 do not change appreciably for k0~102 but are altered completely for k 0 - 1 0 TM. In contrast, corresponding to the eigenvalue at -4.6, we have eigenvalues at -4.59492 and -4.46691 for k O- 1 0 2 and 10TM respectively. Such observations with this and several other examples suggest that some difficulties could arise in computing the set of d.f.ms using the characterization given in (2.3). Applying the algorithms proposed above, it was found conclusively that the system does not have any d.f.ms. This was further confirmed by performing the rank test (using the singular value decomposition) on the system matrix in (2.4) A question that arises from Example 2 is what numerical or "threshold" value should be used for "zero" in the algorithms proposed in this paper. Among the numerical techniques used in the algorithms, the maximum error is accumulated in the reduction of the state matrix to an RSF and hence an error bound on this reduction (Stewart, 1973; Wilkinson, 1965; Golub and Van Loan, 1989) may be used to define a value for "zero".
6. Conclusions In this paper, we have used a characterization of d.f.ms given by Anderson and Ciements (1981) to define d.f.ms in terms of blocking zeros of certain subsystems of the given decentralized system. Based on this characterization, an efficient and reliable method has been proposed for computing d.f.ms. The computational method uses only orthogonal transformations and can be easily implemented with software available in scientific programming packages such as IMSL, EISPACK, LINPACK, etc. Extensive numerical tests carried out so far suggest that the proposed approach is numerically more reliable than existing methods for computing d.f.ms. Acknowledgment--The authors are grateful to the reviewers for their helpful comments and suggestions. This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant A1345. References Anderson, B. D. O. and D. J. Clements (1981). Algebraic characterization of fixed modes in decentralized control. Automatica, 17, 703-712. Anderson, B. D. O. (1982). Transfer function description of decentralized fixed modes. I E E E Trans. Aut. Control, AC-27, 1176-1182. Armentano, V. A. and M. G. Singh (1982). A procedure to eliminate decentralized fixed modes. IEEE Trans. Aut. Control, AC-27, 258-260. Corfmat, J. P. and A. S. Morse (1976). Decentralized control of linear multivariable systems. Automatica, 12, 479-496. Davison, E. J. and N. Tripathi (1978). The optimal decentralized control of a large power system: load and frequency control. 1EEE Trans. Aut. Control, AC-23, 312-325. Davison, E. J. and I). Ozgiiner (1983). Characterization of decentralized fixed modes for interconnected systems. Automatica, 19, 169-182. Davison, E. J. and S. H. Wang (1985). A characterization of decentralized fixed modes in terms of transmission zeros. IEEE Trans. Aut. Control, AC-30, 81-82. Golub, G. H. and C. F. Van Loan (1989). Matrix Computations, 2nd edn. Johns Hopkins University Press, Baltimore. Konstantinov, M., P. Petkov and N. Christov (1981). Invariants and canonical forms for linear multivariable
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systems under the action of orthogonal transformation groups. Kybernetika, 17, 413-424. Misra, P. and R. V. Patel (1986). Characterization and computation of decentralized fixed modes of multivariable systems, Proc, 1986 Amer. Control Conf., Seattle, 427-432. Nour-Eldin, H. (1977). Minimalrealisierung der MatrixUbertragungsfunktion. Regelungstechnik, 25, 82-87. Paige, C. C. (1981). Properties of numerical algorithms related to computing controllability. IEEE Trans. Aut. Control, AC-26, 130-138. Patel, R. V. (1981). Computation of minimal order state-space realizations and observability indices using orthogonal transformations. Int. J. Control, 33, 227-246. Patel, R. V. and P. Misra (1984). A numerical test for transmission zeros with applications in characterizing decentralized fixed modes. Proc. 23rd IEEE Conf. on Decision and Control, Las Vegas, 1746-1751. Patel, R. V. (1986). On blocking zeros in linear multivariable systems. 1EEE Trans. Aut. Control, AC-31, 239-241. Seraji, H. (1982). On fixed modes in decentralized control systems. Int. J. Control, 35, 775-784. Stewart, G. W. (1973). Introduction to Matrix Computations. Academic Press, New York. Tarokh, M. (1985). Fixed modes in multivariable systems using constrained controllers. Automatica, 21, 495-497. Tse, E. C. Y., J. V. Medanic and W. R. Perkins (1978). Generalized Hessenberg transformations for reduced order modelling of large-scale systems. Int. J. Control, 27, 493-512. Van Dooren, P. (1981). The generalized eigenstructure problem in linear systems theory. IEEE Trans. Aut. Control, AC-26, 111-129. Vaz, A. and E. J. Davison (1989). On the quantitative characterization of approximate decentralized fixed modes uing transmission zeros, Math. Control Signals Syst., 2, 287-302. Wang, S. H. and E. J. Davison (1973). On stabilization of decentralized control systems. IEEE Trans. Aut. Control, AC-18, 473-478. West-Vukovich, G. S., E. J. Davison and P. C. Hughes (1984). The decentralized control of large flexible space structures. IEEE Trans. Aut. Control, AC-29, 866-879. Wilkinson, J. H. (1965). The Algebraic Eigenvalue Problem. Oxford University Press, London.
Appendix Proof of Theorem 3. Since ). is an uncontrollable mode of (F, G, H), it follows from the structure of F and G that, gr+l = 0. Also, since ). is an unobservable mode of (F, G, H) and F is a cyclic matrix, we have rank[M~F]=n-1. Using the fact that 3.~o(Fu) , we can perform some elementary row operations on [)'/~H- F ] to get
rank
LrAInH-
F] = rank
x
*
*
• ..
~
*
Or
0
f2,3
"'"
f2,r
f2,r+ 1
Or
0
0
"'"
f3,r
f3,r+ 1
: Or
0
0
---
0
f~.~+l
Or
0
0
""
0
0
0
=n-1
,~(~) *3(x)
-..
,~(;~) ~r+l(X)
where ¢pj(~.)= hj + Hl(;tln_r - Fll)-lfl/, j = 2 . . . . . r + 1 and , denotes possible non-zero vectors. Since f~.i+l:~0, i = 2 . . . . . r, it follows that ¢~2(2~)= 0. Next, performing the
382
Brief Paper
elementary row operations mentioned above on the matrix G], we obtain
row operations to the matrix
[)'I~IF
G]=n-r+rank f2,, f2,,+1
rank [ ) ' 1 ~ F
~
•
O
Q
Therefore,
gr
l
~3(x) ,,(x)
0G].
Since f/,i+l ~0, i = 2. . . . . r, it follows from the structure of F that A , Therefore,
.
o(F).
×
0
rank[3"l~ F G]=rank[3"l"H-F
0
fr.r+l
0,(~.) O,+,(A)
gT
I
H,(M,_,- F,1)-'G,A
The right-hand side of the above equation is exactly the result that we would get if we were to apply the elementary
if and only if Z is a blocking zero of (F, G,/~). The result of the theorem then follows, completing the proof.