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Model reduction of systems with zeros interlacing the poles
fiYSTt[MS h CONTROL U |RS
ELSEVIER
Systems & Control Letters 30 (1997) 19-24
Model reduction of systems with zeros interlacing the poles B. Srinivasan, P. Myszkorowski* Institut d'Automatique, t~eole Polytechnique Fkdkrale de Lausanne, CH-IOI5 Lausanne, Switzerland Received 21 June 1996; revised 17 October 1996
Abstract The problem of reducing a system with zeros interlacing the poles (ZIP) on the real axis is considered. It is proved that many model reduction methods, such as the balanced truncation, balanced residualization, suboptimal and optimal Hankel approximations, inheril: the ZIP property. Properties of the Hankel singular values of ZIP systems are also listed. (~) 1997 Elsevier Science B.V. Keywords." Model reduction; inheritance; Hankel singular values
I. Introduction Model reduction is a crucial step in the process of designing a controller (see e.g., [8] for a r6sum6 of major model reduction methods). However, little has been said so far abou! the type of system properties that are inherited by the reduced model from its original. Needless to say, it is important to know what system is or can be obtained once a model reduction is performed. Without pretending to be a breakthrough in this domain, the present note analyses model reduction of a rather specific class of systems corresponding to RC circuits in the network domain. The driving-point impedance function~; of these circuits share a nice property of interlacing zeros and poles on the negative real axis [4]. It turns out that this feature is preserved by a number of model reduction procedures, including balanced truncation [7], singular perturbation approximation of balanced systems or balanced residualization [6], and the optimal Hankel-norm reduction [2]. An entry point to the results presented here can be found in [3] or [1]. It has been shown there that for a very special case of ZIP systems, the bounds on the
~ norm and model reduction error, while performing balanced truncation, are tight. Although the latter issue is interesting, it cannot be extended to a generic ZIP system or to other reduction techniques. Instead, Jg~ bounds on generic ZIP systems are investigated.
2. Equivalence Definition I. n--I
.....
1-[j:~ (s + z/)
t ~ s ) = ~ ~-ff7-~ " 7 - ~ l li=ll.S --t- ai )
is ZIP iff0 < a i < z i < a i + l holds for i = 1 . . . . . n - 1. Without loss of generality K is assumed to be positive in the sequel.
Theorem 2. The following statements are equivalent: (i) G(s) is a strictly proper Z I P system. (ii) G(s) can be written as
G(s)= ~ * Corresponding author. Tel.: +41 21 693 3838; fax: +41 21 693 2574; e-mail: [email protected]. 0167-6911/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PH SO 1 6 7 - 6 9 1 1 ( 9 6 ) 0 0 0 7 2 - 2
i=1
bi sq-ai
with ai > 0, bi > 0, ai • aj
Vi c j .
(1)
B. Srinivasan, P. Myszkorowski/ Systems & Control Letters 30 (1997) 19-24
20
P being symmetric leads to the eigendecomposition P = TTNT with T T = T 1. Then some algebra leads to
(iii) G(s) has a diagonal realization
with [V
A = d i a g ( - a l , -a2 . . . . . -a~ ) l ,/g2
and
BT = c =
. . .
TATT Z q- ZTAT T T q- TBBT T T = O,
(6)
TATTTS q- ZTAT T q- TCTCT T = O.
(7)
(iv) G(s) has a principle axis balanced realization
Comparing (4), (5) with (6), (7), the similarity transformation for obtaining a principal axis balanced realization can be inferred to be
d[0/
,4=TAT T,
where A is symmetric and B x = C.
Proof. Equivalence (i) ¢=> (ii) ¢:~ (iii) is well known. The proof of these statements are provided for completeness. The proof then proceeds in the following manner: (i) => (ii), (ii) =~ (i), (iii) => (iv), and (iv) (iii). (ii) ¢~ (iii) is omitted as trivial. (i) ~ (ii): It suffices to show that b i = Res(G(s))l~=-a~ > 0. Elementary algebra obtains n--1
Res(G(s))[s=_a, = K
I~/=l (zj - ai) ylnk:l,k+i(ak -- ai )"
(2)
Since sign(H~-il(zj - a i ) ) = ( - 1 ) i-1 : sign(l-[k~i(ak - ai)), the result follows. (ii) ~ (i): Using (2) and the fact that bi > 0 results in
(Hn-I) sign \7_s(zj -
(
rI
= sign
(ak - ai)
\k:l.k4i : ( - 1 ) i-1
gi.
(3)
n--I Consider the function 0 ( x ) = l--[j=1 (zj - x ) . Then from (3), sign(0(ai )) = ( - 1 )i-I and sign(0(ai+l )) = ( - 1 )i = _ sign(0(ai)). Consequently, the function 0(x) has an odd number of zeros in each of intervals ]ai, ai+l [. Since degree(0)--n - 1, the inequalities ai < z i < ai+l hold for i = 1 . . . . . n - 1. (iii) :=~ (iv): Since A is a diagonal matrix and B T = C, there exists a symmetric matrix, P, satisfying both controllability and observability Lyapunov equations, viz., AP + PA T + BB T = 0 ,
(4)
ATp + PA +
(5)
cTc
=
O.
B=TB,
C'T = TcT.
(8)
Since B T = C , / ~ v = ~ holds. Also, A T = T A T T T = TAT T = ~ . (iv) ~ (iii): The proof is complete if it is verified that in a diagonal representation given by
c 10 )' Bvd=Cd holds. Since A-is symmetric there exists an orthogonal matrix, R, which can diagonalizeA~, i.e.A=RTAdR. The canonical form will then have A d : R A R T, B d = R B and C d = C R T. If / } T = ~ , it is clear that in the diagonal representation BS = Cd. []
3. Properties Proposition 3. The Hankel singular values o f a Z I P system are distinct. Conversely, a system satisfying ~T = ~ with distinct Hankel singular values is a Z I P one. Proof. (=~) The Hankel singular values of a ZIP system, ai, are the eigenvalues of the matrix P defined in (4). The fact that P has distinct eigenvalues is proved by contradiction. Suppose Vl and v2 are two distinct normalized eigenvectors of P corresponding to the same eigenvalue ai. Multiply Eq. (4) on the left by v~ and on the right by vl to get vTAPv, + VVlPAvl + vTBBTvl=O,
(BTvl)2 = 2 a i v ~ ( _ A ) v , =2ailb(_A)l/2vj U2,
(9) (10)
where (--A) 1/2 is obtained by taking the square root of the diagonal elements. Note that all elements of the diagonal matrix A are negative. Similarly, (BTvz)2=2~TiH(--A)I/2v2H 2. Multiplying Eq. (4) on the left by vlx and on the right by v2 one
B. Srinivasan. P. Myszkorowski/Systems & Control Letters 30 (1997) 19 24 obtains v~APv2 + v~PAv2 + vT BBTv2 = 0 ,
(11)
(BTvl)(BTv2 ) = 2o-iv[(-A )v2
=2tri((-A)l/Zvl )T((--A)I/2
V2).
(12)
Substituting (10) in (12) one gets
21
Theorem 2(iii). Recall that the Hankel singular values are the eigenvalues of P satisfying (4). Noticing that the sum of eigenvalues of a matrix is its trace one gets, E O'i =trace(P). With Pii being the diagonal terms of P, the diagonal elements of (4) give the relationships bi Pii = - - , 2ai
--2aiPii q- bi=O,
(16)
((-A)l/2vj)T((_A)I/Zv 2) = + II(-A)~/2vlld II(-A)'/2v2ll •
(13)
From the Cauchy-Schwarz inequality, (-A)l/Zvl and (-A)l/2v2 are parallel and hence are vl and v2. Since both these vectors are considered normalized vl = ± v2 follows. So no two distinct eigenvectors exist for the same eigenvalue of P and hence the eigenvalues are distinct. ( ~ ) It is shown first that if in a principal axis balanced realization (i)/}r = ~ and (ii) (7i are distinct, then A~ is symmetric. The proof then follows from Theorem 2(iv). Subtracting (6) if, am (7) one gets
(A~- A'T) Z = Z (A" - A"T)
(14)
which means that ( ~ - A~T) commutes with Z. Since S is a simple matrix it commutes with (A~ - A~T) if and only if these matrices share the same set of eigenvectors [5]. S is a diagonal matrix with distinct diagonal entries, and hence only the unit vectors are eigenvectors. Thus, a matrix which commutes with Z must be diagonal and so is ( L I - A~T). Since (A~ - A ~T) is skewsymmetric by construction it must be the null matrix, which implies A~= A~T. [] It is well known [7:] that the g/g~ norm of the system is bounded from below by twice the sum of its Hankel singular values. Also for a very special case of the ZIP systems it has been shown in [1] that the abovementioned bound is tight. In the following proposition, the result is extended to all ZIP systems. Proposition 4. The bound on the g ~ norm provided by the Hankel sin~rular values & tight f o r a Z I P system, i.e., n
Ila(s)llo~ = 2 ~-~ai,
(15)
2 Z . O'i = 2 trace(P) =
Z.
2
i=1
Pii =
i= I
+ bi Z - , --
(17)
i=l ai
The proof is complete if it is shown that this bound is actually attained at some frequency, i.e., [G(io9)[ = ~ bi/ai for some o9. Clearly from (1),
Ia(0)l = £
_bi
(18)
i:1 a i '
and hence the proof.
[]
The preceding development applies also to proper but not strictly proper ZIP systems (with a slight abuse of the ZIP definition). For such systems, an additional system zero gets introduced either before the first or after the last pole. Corollary 5. Consider the proper but not strictly proper Z I P system G (. .s.).- - h - ~N-(~s ,)
deg(N)=deg(D)=n,
K=G(~).
Then the following properties hoM:
IIG(s)ll
=
IKI,
IKI + 2~-~.ai,
zt < al,
z I > a,,
(19)
n
IlG(s)- G(oc)llo¢ = 2 Z a i .
(20)
i=1
Proof. Note that
Since s i g n ( P ( - a i ) ) = sign(zl - a l ) ( - 1)i-l, zeros of P, and D interlace. Thus, in view of Theorem 2, G(s) can be written as
i--I
s +bi ai
(22)
where tri denote the Hankel singular values o f G(s).
G(s) = K + sign(K(zl - al ))
Proof. First the bound is calculated in terms of ai and b~. Consider the realization of the system as in
Next, observe that ](io9 +zj)/(io~ + aj)] is a monotonously decreasing (increasing) function of o9 for
i=1
B. Srinivasan, P. Myszkorowski/ Systems & Control Letters 30 (1997) 19~4
22
zj > a~ (zy < aj). Since [G(ico)l= 11717(ico + Z/)/ (ico+aj)[, taking the limits at ~o=0 or co=oc as appropriate obtains (19). Finally, (20) follows from (22) and (15). [] Finally, it is interesting to note that if G(s) is ZIP, so is KI + G(s) for any K 1 C ~. To see this, set G(s)=K(N(s)/D(s)) which in turn gives Kl + G(s) =B(s)/A(s)=(KID(s) + KN(s))/D(s). Since sign( B(-ai ) )=sign(K)sign(zl - al ) ( - 1 ) / - I holds independently of K1, zeros of B and A interlace for all K~ C ~.
(v~)Vx-' t,~ = O. However, (v~)TX-'vik ¢; 0 since X is definite. So, any two distinct eigenvectors of the matrix XY obey ( v~ )Z Yv~ = O, (v[)Tx-1 r} = 0, Vi -¢ j or k ¢ l and hence the proposition. [] It has been shown in [2] that the optimal or suboptimal Hankel approximate of a system is the stable part of its all-pass dilation. The following lemmas assess the properties of the all-pass dilation, (~(s), and its stable part.
Lemma 7.
Given a
ZIP system G(s), let
4. Preparatory results Proposition 6. Let X be a definite matrix, Y a symmetric matrix and V the matrix of eigenvectors of XY. Then v T y v and v T x -1V are diagonal. Proof. First, the fact that XY is diagonalizable is proved. Without loss of generality let X be positive definite. Then there exists a square root X 1/2 which is symmetric. Note that ( X I / 2 y x 1/2) is symmetric and admits a decomposition of the form TTDT, where T is orthogonal, and D is diagonal. Then, = xl/2x
- 1 / 2 ( ~ ) X 1 / 2 X - I/2
= x l / 2 ( X 1/2 Y X I/2 ) X - 1 / 2 = X1/2TTDTX-U
= V-1DV
(23)
2
where V = T X -U2.
(24)
Second, it is shown that the eigenvectors vi and vj, corresponding to two distinct eigenvalues 2i and 2j, obey the property vT Yvj = O, vYiX - 1Vj = O, V i ¢ j. Since X is non-singular, and X, Y symmetric, vTyvj can be developed as follows: vTyvi ~--- vTx-I(xY)ui = 2i(vfx-1Ui ) = 2i(vTi X - 10j), vTyvi = v T y v j = v T X - ' ( X Y ) v / = 2 j ( v T X - ' v / ) .
be an all-pass dilation as explained in [2], such that (G - G) is all-pass with IIG - d l l ~ = ~ k + l . Then, (i) .4S= (AS) T is negative-definite, and (ii)/~T= (~S, where
Proof. For the purpose of construction, rows and columns corresponding to the singular value ak+l need to be removed. Since the Hankel singular values are distinct, only the (k + 1)th row and/or column should be deleted from the matrices A,/t, 0 , / ) and -r giving .4,/1, C,/3 and 2~. The all-pass dilation denoted by 2
h
can be performed using the following matrices:
~=
-
F2
'
(27)
(25) (26)
Subtracting (26) from (25) gives (2i - 2 j ) v T X - I v j =0. Since 2i ¢ 22, vTx - l v j = O , V i ¢ j. Also Yvj = 2j = v T x - ' = 0. Third, consider two eigenvectors v~ and v[ corresponding to a non-distinct eigenvalue of XY. Two vectors spanning the necessary space can always be chosen such that ( u ? ) T x - I u [ = O , except when
I
F 1
,
F .= ~2 -- Gk+IL 2
From the properties of all-pass functions, D=ak+l. Also, from the property that (/~T /}r ).~e+/~T (~ _ ~ ) = 0, noting that/~r = ~ obtains
/}T = _/}T(2~ + ak+lI), (28) = - d ( 2 + ~k+ll) r -1.
23
B. Srinivasan, P. Myszkorowski/ Systems & Control Letters 30 (1997) 19-24
The Lyapunov equation for the augmented system provides A-"= --,~T + ~X ~. Noting that - ~T ~,~ = jT £2 + ^
5. Model reduction Theorem 9. Balanced truncation of ZIP systems are ZIP.
SAZ, ,4 can be written as ^
A-=(62+1/i T + ~/i~ - 6k+ 1~T~) F - i .
(29) Proof. Let the system matrices be partitioned into
Let/= be the absolute value of F, F =/~S. The balancing similarity transformation, which leads to Z = SS, in this case is given by/=1/2. Performing such a transformation obtains
A-,2) X= \L, L2 '
A=f-~/2(a~+fir ~- 2212 -
(30)
7') D= ( /D2
(31)
Since the original system is ZIP,/}T _ ~ and A~ symmetric. These properties are not destroyed by partitioning and hence /}T = Ci for i = 1, 2 and Aii are symmetric matrices. From Theorem l(iv),
ffk+lCTC)y-ll2s,
/}T = _/}T(2 + ak4 II)/~-U2, d = -- C ( 2 Jr- (Tk4_lJr)F-112S.
The lemma is easily verified from the above equations by noting that (/~ 1/2~t'S/~1,/2) is a sum of three negativedefinite matrices. [-1 Lemma 8. The diagonal representation of the stable and unstable parts of G(s) given by f ~ + ID +
)
(32)
Cil 0 ) is a ZIP system. []
Proof. Let the system matrices be partitioned as in the previous theorem. Balanced residualization gives the reduced model,
\Cdl 0 ) - Cd-.
Proof. Let V be the eigenvectors of A. Note that A = (AS)S, with AS being negative-definite and S symmetric. Then from Proposition 6, Asd = V- 1(.4S) V-T and Sd = v T s v are diagonal matrices. Also, the diagonal representation o f / ] , / ] d = V-1A'V=Asd Sd. The diagonalizing transformation leaves /}d = V-I/} and C'd= C V. Noting that/}v = CS, after some algebra one obtains,/}] = Cd Sd I. Since (/IS) and hence Asd are negative-definite, the signs of the elements in the diagonal of So partition the stable and unstable parts of Ad. If (Sd)ii > 0 the ith mode of-4d is stable and the signs of the ith elements of Bd and t~d agree. So the stable parts of/}d and Cd will have the same ,;ign while their unstable parts will differ in sign. Finally, construct the partitions such that (/7~-)i-(C+)i = ~ t ~ d )
C = ( C 1 C2).
Theorem 10. Sinoular perturbation approximation of balanced systems or balanced residualization of ZIP systems are ZIP.
and
satisfies (g~C)T=ed - and (l~d ) T :
and
i for the stable part and (/Td)i=
--(Cd)i=d--(Bd)i((~d)i for the unstable part. The lemma then--,follow~.. []
Cl - C2A221A12
- d 2 A~l/}2
"
By inspection, it can be seen that the input and the output matrices are equal and the state matrix is symmetric. The theorem follows. [] Theorem 11. Optimal Hankel approximation ZIP systems are ZIP.
of
Proof. Note that (~-(s) of Lemma 8 is the optimal Hankel approximate. Since (/~-)T =~d+ in a stable diagonal representation, (~-(s) is ZIP. [] Theorem 12. Suboptimal Hankel approximations of ZIP systems are ZIP. Proof. When the suboptimal Hankel approximation is performed, there is no removal of rows or columns and hence A-----A~, /7=/?, C--C, D = D and 2~=S in Lemma 7. The all-pass dilation will still result in AS
24
B. Srinivasan, P. Myszkorowski/Systems & Control Letters 30 (1997) 19~4
being negative-definite a n d / ~ a - = ~'S. L e m m a 8 holds and 6:+(s), which is the suboptimal H a n k e l approximate, is ZIP. []
6. Conclusions Properties o f the various representations o f ZIP systems were first considered. It was shown that the H a n k e l singular values o f ZIP systems are distinct and provide a tight b o u n d on the ~ n o r m o f these systems. The inheritance issue was then addressed and it was proved that m a n y model reduction methods preserve the ZIP property.
Acknowledgements The authors thank Prof. Athanasios Antoulas for introducing them to the subject o f model reduction.
References [1] D.F. Enns, Model reduction with balanced realizations: an error bound and a frequency weighted generalization, Proc. 23rd Con['. Decision and Control, Las Vegas (1984) 127-132. [2] K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their 5J~-error bounds, Int. J. Control 39 (1984) 1115 t 193. [3] M. Green and D.J.N. Limebeer, Linear Robust Control (Prentice-Hall, Englewood Cliffs, NJ, 1995). [4] F. Kuo, Network Analysis and Synthesis (Wiley, New York, 1966). [5] P. Lancaster and M. Tismenetsky, The Theory o[' Matrices. Second Edition with Applications (Academic Press, San Diego, 1985). [6] Y. Li and B.D.O. Anderson, Singular perturbation approximation of balanced systems, Int. J. Control 50 (1989) 1379 1405. [7] B.C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction, 1EEE Trans. Automat. Control 26 (1981) 17-32. [8] K. Zhou, J.C. Doyle and K. G/over, Robust and Optimal Control (Prentice-Hall, Upper Saddle River, 1996).
ELSEVIER
Systems & Control Letters 30 (1997) 19-24
Model reduction of systems with zeros interlacing the poles B. Srinivasan, P. Myszkorowski* Institut d'Automatique, t~eole Polytechnique Fkdkrale de Lausanne, CH-IOI5 Lausanne, Switzerland Received 21 June 1996; revised 17 October 1996
Abstract The problem of reducing a system with zeros interlacing the poles (ZIP) on the real axis is considered. It is proved that many model reduction methods, such as the balanced truncation, balanced residualization, suboptimal and optimal Hankel approximations, inheril: the ZIP property. Properties of the Hankel singular values of ZIP systems are also listed. (~) 1997 Elsevier Science B.V. Keywords." Model reduction; inheritance; Hankel singular values
I. Introduction Model reduction is a crucial step in the process of designing a controller (see e.g., [8] for a r6sum6 of major model reduction methods). However, little has been said so far abou! the type of system properties that are inherited by the reduced model from its original. Needless to say, it is important to know what system is or can be obtained once a model reduction is performed. Without pretending to be a breakthrough in this domain, the present note analyses model reduction of a rather specific class of systems corresponding to RC circuits in the network domain. The driving-point impedance function~; of these circuits share a nice property of interlacing zeros and poles on the negative real axis [4]. It turns out that this feature is preserved by a number of model reduction procedures, including balanced truncation [7], singular perturbation approximation of balanced systems or balanced residualization [6], and the optimal Hankel-norm reduction [2]. An entry point to the results presented here can be found in [3] or [1]. It has been shown there that for a very special case of ZIP systems, the bounds on the
~ norm and model reduction error, while performing balanced truncation, are tight. Although the latter issue is interesting, it cannot be extended to a generic ZIP system or to other reduction techniques. Instead, Jg~ bounds on generic ZIP systems are investigated.
2. Equivalence Definition I. n--I
.....
1-[j:~ (s + z/)
t ~ s ) = ~ ~-ff7-~ " 7 - ~ l li=ll.S --t- ai )
is ZIP iff0 < a i < z i < a i + l holds for i = 1 . . . . . n - 1. Without loss of generality K is assumed to be positive in the sequel.
Theorem 2. The following statements are equivalent: (i) G(s) is a strictly proper Z I P system. (ii) G(s) can be written as
G(s)= ~ * Corresponding author. Tel.: +41 21 693 3838; fax: +41 21 693 2574; e-mail: [email protected]. 0167-6911/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PH SO 1 6 7 - 6 9 1 1 ( 9 6 ) 0 0 0 7 2 - 2
i=1
bi sq-ai
with ai > 0, bi > 0, ai • aj
Vi c j .
(1)
B. Srinivasan, P. Myszkorowski/ Systems & Control Letters 30 (1997) 19-24
20
P being symmetric leads to the eigendecomposition P = TTNT with T T = T 1. Then some algebra leads to
(iii) G(s) has a diagonal realization
with [V
A = d i a g ( - a l , -a2 . . . . . -a~ ) l ,/g2
and
BT = c =
. . .
TATT Z q- ZTAT T T q- TBBT T T = O,
(6)
TATTTS q- ZTAT T q- TCTCT T = O.
(7)
(iv) G(s) has a principle axis balanced realization
Comparing (4), (5) with (6), (7), the similarity transformation for obtaining a principal axis balanced realization can be inferred to be
d[0/
,4=TAT T,
where A is symmetric and B x = C.
Proof. Equivalence (i) ¢=> (ii) ¢:~ (iii) is well known. The proof of these statements are provided for completeness. The proof then proceeds in the following manner: (i) => (ii), (ii) =~ (i), (iii) => (iv), and (iv) (iii). (ii) ¢~ (iii) is omitted as trivial. (i) ~ (ii): It suffices to show that b i = Res(G(s))l~=-a~ > 0. Elementary algebra obtains n--1
Res(G(s))[s=_a, = K
I~/=l (zj - ai) ylnk:l,k+i(ak -- ai )"
(2)
Since sign(H~-il(zj - a i ) ) = ( - 1 ) i-1 : sign(l-[k~i(ak - ai)), the result follows. (ii) ~ (i): Using (2) and the fact that bi > 0 results in
(Hn-I) sign \7_s(zj -
(
rI
= sign
(ak - ai)
\k:l.k4i : ( - 1 ) i-1
gi.
(3)
n--I Consider the function 0 ( x ) = l--[j=1 (zj - x ) . Then from (3), sign(0(ai )) = ( - 1 )i-I and sign(0(ai+l )) = ( - 1 )i = _ sign(0(ai)). Consequently, the function 0(x) has an odd number of zeros in each of intervals ]ai, ai+l [. Since degree(0)--n - 1, the inequalities ai < z i < ai+l hold for i = 1 . . . . . n - 1. (iii) :=~ (iv): Since A is a diagonal matrix and B T = C, there exists a symmetric matrix, P, satisfying both controllability and observability Lyapunov equations, viz., AP + PA T + BB T = 0 ,
(4)
ATp + PA +
(5)
cTc
=
O.
B=TB,
C'T = TcT.
(8)
Since B T = C , / ~ v = ~ holds. Also, A T = T A T T T = TAT T = ~ . (iv) ~ (iii): The proof is complete if it is verified that in a diagonal representation given by
c 10 )' Bvd=Cd holds. Since A-is symmetric there exists an orthogonal matrix, R, which can diagonalizeA~, i.e.A=RTAdR. The canonical form will then have A d : R A R T, B d = R B and C d = C R T. If / } T = ~ , it is clear that in the diagonal representation BS = Cd. []
3. Properties Proposition 3. The Hankel singular values o f a Z I P system are distinct. Conversely, a system satisfying ~T = ~ with distinct Hankel singular values is a Z I P one. Proof. (=~) The Hankel singular values of a ZIP system, ai, are the eigenvalues of the matrix P defined in (4). The fact that P has distinct eigenvalues is proved by contradiction. Suppose Vl and v2 are two distinct normalized eigenvectors of P corresponding to the same eigenvalue ai. Multiply Eq. (4) on the left by v~ and on the right by vl to get vTAPv, + VVlPAvl + vTBBTvl=O,
(BTvl)2 = 2 a i v ~ ( _ A ) v , =2ailb(_A)l/2vj U2,
(9) (10)
where (--A) 1/2 is obtained by taking the square root of the diagonal elements. Note that all elements of the diagonal matrix A are negative. Similarly, (BTvz)2=2~TiH(--A)I/2v2H 2. Multiplying Eq. (4) on the left by vlx and on the right by v2 one
B. Srinivasan. P. Myszkorowski/Systems & Control Letters 30 (1997) 19 24 obtains v~APv2 + v~PAv2 + vT BBTv2 = 0 ,
(11)
(BTvl)(BTv2 ) = 2o-iv[(-A )v2
=2tri((-A)l/Zvl )T((--A)I/2
V2).
(12)
Substituting (10) in (12) one gets
21
Theorem 2(iii). Recall that the Hankel singular values are the eigenvalues of P satisfying (4). Noticing that the sum of eigenvalues of a matrix is its trace one gets, E O'i =trace(P). With Pii being the diagonal terms of P, the diagonal elements of (4) give the relationships bi Pii = - - , 2ai
--2aiPii q- bi=O,
(16)
((-A)l/2vj)T((_A)I/Zv 2) = + II(-A)~/2vlld II(-A)'/2v2ll •
(13)
From the Cauchy-Schwarz inequality, (-A)l/Zvl and (-A)l/2v2 are parallel and hence are vl and v2. Since both these vectors are considered normalized vl = ± v2 follows. So no two distinct eigenvectors exist for the same eigenvalue of P and hence the eigenvalues are distinct. ( ~ ) It is shown first that if in a principal axis balanced realization (i)/}r = ~ and (ii) (7i are distinct, then A~ is symmetric. The proof then follows from Theorem 2(iv). Subtracting (6) if, am (7) one gets
(A~- A'T) Z = Z (A" - A"T)
(14)
which means that ( ~ - A~T) commutes with Z. Since S is a simple matrix it commutes with (A~ - A~T) if and only if these matrices share the same set of eigenvectors [5]. S is a diagonal matrix with distinct diagonal entries, and hence only the unit vectors are eigenvectors. Thus, a matrix which commutes with Z must be diagonal and so is ( L I - A~T). Since (A~ - A ~T) is skewsymmetric by construction it must be the null matrix, which implies A~= A~T. [] It is well known [7:] that the g/g~ norm of the system is bounded from below by twice the sum of its Hankel singular values. Also for a very special case of the ZIP systems it has been shown in [1] that the abovementioned bound is tight. In the following proposition, the result is extended to all ZIP systems. Proposition 4. The bound on the g ~ norm provided by the Hankel sin~rular values & tight f o r a Z I P system, i.e., n
Ila(s)llo~ = 2 ~-~ai,
(15)
2 Z . O'i = 2 trace(P) =
Z.
2
i=1
Pii =
i= I
+ bi Z - , --
(17)
i=l ai
The proof is complete if it is shown that this bound is actually attained at some frequency, i.e., [G(io9)[ = ~ bi/ai for some o9. Clearly from (1),
Ia(0)l = £
_bi
(18)
i:1 a i '
and hence the proof.
[]
The preceding development applies also to proper but not strictly proper ZIP systems (with a slight abuse of the ZIP definition). For such systems, an additional system zero gets introduced either before the first or after the last pole. Corollary 5. Consider the proper but not strictly proper Z I P system G (. .s.).- - h - ~N-(~s ,)
deg(N)=deg(D)=n,
K=G(~).
Then the following properties hoM:
IIG(s)ll
=
IKI,
IKI + 2~-~.ai,
zt < al,
z I > a,,
(19)
n
IlG(s)- G(oc)llo¢ = 2 Z a i .
(20)
i=1
Proof. Note that
Since s i g n ( P ( - a i ) ) = sign(zl - a l ) ( - 1)i-l, zeros of P, and D interlace. Thus, in view of Theorem 2, G(s) can be written as
i--I
s +bi ai
(22)
where tri denote the Hankel singular values o f G(s).
G(s) = K + sign(K(zl - al ))
Proof. First the bound is calculated in terms of ai and b~. Consider the realization of the system as in
Next, observe that ](io9 +zj)/(io~ + aj)] is a monotonously decreasing (increasing) function of o9 for
i=1
B. Srinivasan, P. Myszkorowski/ Systems & Control Letters 30 (1997) 19~4
22
zj > a~ (zy < aj). Since [G(ico)l= 11717(ico + Z/)/ (ico+aj)[, taking the limits at ~o=0 or co=oc as appropriate obtains (19). Finally, (20) follows from (22) and (15). [] Finally, it is interesting to note that if G(s) is ZIP, so is KI + G(s) for any K 1 C ~. To see this, set G(s)=K(N(s)/D(s)) which in turn gives Kl + G(s) =B(s)/A(s)=(KID(s) + KN(s))/D(s). Since sign( B(-ai ) )=sign(K)sign(zl - al ) ( - 1 ) / - I holds independently of K1, zeros of B and A interlace for all K~ C ~.
(v~)Vx-' t,~ = O. However, (v~)TX-'vik ¢; 0 since X is definite. So, any two distinct eigenvectors of the matrix XY obey ( v~ )Z Yv~ = O, (v[)Tx-1 r} = 0, Vi -¢ j or k ¢ l and hence the proposition. [] It has been shown in [2] that the optimal or suboptimal Hankel approximate of a system is the stable part of its all-pass dilation. The following lemmas assess the properties of the all-pass dilation, (~(s), and its stable part.
Lemma 7.
Given a
ZIP system G(s), let
4. Preparatory results Proposition 6. Let X be a definite matrix, Y a symmetric matrix and V the matrix of eigenvectors of XY. Then v T y v and v T x -1V are diagonal. Proof. First, the fact that XY is diagonalizable is proved. Without loss of generality let X be positive definite. Then there exists a square root X 1/2 which is symmetric. Note that ( X I / 2 y x 1/2) is symmetric and admits a decomposition of the form TTDT, where T is orthogonal, and D is diagonal. Then, = xl/2x
- 1 / 2 ( ~ ) X 1 / 2 X - I/2
= x l / 2 ( X 1/2 Y X I/2 ) X - 1 / 2 = X1/2TTDTX-U
= V-1DV
(23)
2
where V = T X -U2.
(24)
Second, it is shown that the eigenvectors vi and vj, corresponding to two distinct eigenvalues 2i and 2j, obey the property vT Yvj = O, vYiX - 1Vj = O, V i ¢ j. Since X is non-singular, and X, Y symmetric, vTyvj can be developed as follows: vTyvi ~--- vTx-I(xY)ui = 2i(vfx-1Ui ) = 2i(vTi X - 10j), vTyvi = v T y v j = v T X - ' ( X Y ) v / = 2 j ( v T X - ' v / ) .
be an all-pass dilation as explained in [2], such that (G - G) is all-pass with IIG - d l l ~ = ~ k + l . Then, (i) .4S= (AS) T is negative-definite, and (ii)/~T= (~S, where
Proof. For the purpose of construction, rows and columns corresponding to the singular value ak+l need to be removed. Since the Hankel singular values are distinct, only the (k + 1)th row and/or column should be deleted from the matrices A,/t, 0 , / ) and -r giving .4,/1, C,/3 and 2~. The all-pass dilation denoted by 2
h
can be performed using the following matrices:
~=
-
F2
'
(27)
(25) (26)
Subtracting (26) from (25) gives (2i - 2 j ) v T X - I v j =0. Since 2i ¢ 22, vTx - l v j = O , V i ¢ j. Also Yvj = 2j = v T x - ' = 0. Third, consider two eigenvectors v~ and v[ corresponding to a non-distinct eigenvalue of XY. Two vectors spanning the necessary space can always be chosen such that ( u ? ) T x - I u [ = O , except when
I
F 1
,
F .= ~2 -- Gk+IL 2
From the properties of all-pass functions, D=ak+l. Also, from the property that (/~T /}r ).~e+/~T (~ _ ~ ) = 0, noting that/~r = ~ obtains
/}T = _/}T(2~ + ak+lI), (28) = - d ( 2 + ~k+ll) r -1.
23
B. Srinivasan, P. Myszkorowski/ Systems & Control Letters 30 (1997) 19-24
The Lyapunov equation for the augmented system provides A-"= --,~T + ~X ~. Noting that - ~T ~,~ = jT £2 + ^
5. Model reduction Theorem 9. Balanced truncation of ZIP systems are ZIP.
SAZ, ,4 can be written as ^
A-=(62+1/i T + ~/i~ - 6k+ 1~T~) F - i .
(29) Proof. Let the system matrices be partitioned into
Let/= be the absolute value of F, F =/~S. The balancing similarity transformation, which leads to Z = SS, in this case is given by/=1/2. Performing such a transformation obtains
A-,2) X= \L, L2 '
A=f-~/2(a~+fir ~- 2212 -
(30)
7') D= ( /D2
(31)
Since the original system is ZIP,/}T _ ~ and A~ symmetric. These properties are not destroyed by partitioning and hence /}T = Ci for i = 1, 2 and Aii are symmetric matrices. From Theorem l(iv),
ffk+lCTC)y-ll2s,
/}T = _/}T(2 + ak4 II)/~-U2, d = -- C ( 2 Jr- (Tk4_lJr)F-112S.
The lemma is easily verified from the above equations by noting that (/~ 1/2~t'S/~1,/2) is a sum of three negativedefinite matrices. [-1 Lemma 8. The diagonal representation of the stable and unstable parts of G(s) given by f ~ + ID +
)
(32)
Cil 0 ) is a ZIP system. []
Proof. Let the system matrices be partitioned as in the previous theorem. Balanced residualization gives the reduced model,
\Cdl 0 ) - Cd-.
Proof. Let V be the eigenvectors of A. Note that A = (AS)S, with AS being negative-definite and S symmetric. Then from Proposition 6, Asd = V- 1(.4S) V-T and Sd = v T s v are diagonal matrices. Also, the diagonal representation o f / ] , / ] d = V-1A'V=Asd Sd. The diagonalizing transformation leaves /}d = V-I/} and C'd= C V. Noting that/}v = CS, after some algebra one obtains,/}] = Cd Sd I. Since (/IS) and hence Asd are negative-definite, the signs of the elements in the diagonal of So partition the stable and unstable parts of Ad. If (Sd)ii > 0 the ith mode of-4d is stable and the signs of the ith elements of Bd and t~d agree. So the stable parts of/}d and Cd will have the same ,;ign while their unstable parts will differ in sign. Finally, construct the partitions such that (/7~-)i-(C+)i = ~ t ~ d )
C = ( C 1 C2).
Theorem 10. Sinoular perturbation approximation of balanced systems or balanced residualization of ZIP systems are ZIP.
and
satisfies (g~C)T=ed - and (l~d ) T :
and
i for the stable part and (/Td)i=
--(Cd)i=d--(Bd)i((~d)i for the unstable part. The lemma then--,follow~.. []
Cl - C2A221A12
- d 2 A~l/}2
"
By inspection, it can be seen that the input and the output matrices are equal and the state matrix is symmetric. The theorem follows. [] Theorem 11. Optimal Hankel approximation ZIP systems are ZIP.
of
Proof. Note that (~-(s) of Lemma 8 is the optimal Hankel approximate. Since (/~-)T =~d+ in a stable diagonal representation, (~-(s) is ZIP. [] Theorem 12. Suboptimal Hankel approximations of ZIP systems are ZIP. Proof. When the suboptimal Hankel approximation is performed, there is no removal of rows or columns and hence A-----A~, /7=/?, C--C, D = D and 2~=S in Lemma 7. The all-pass dilation will still result in AS
24
B. Srinivasan, P. Myszkorowski/Systems & Control Letters 30 (1997) 19~4
being negative-definite a n d / ~ a - = ~'S. L e m m a 8 holds and 6:+(s), which is the suboptimal H a n k e l approximate, is ZIP. []
6. Conclusions Properties o f the various representations o f ZIP systems were first considered. It was shown that the H a n k e l singular values o f ZIP systems are distinct and provide a tight b o u n d on the ~ n o r m o f these systems. The inheritance issue was then addressed and it was proved that m a n y model reduction methods preserve the ZIP property.
Acknowledgements The authors thank Prof. Athanasios Antoulas for introducing them to the subject o f model reduction.
References [1] D.F. Enns, Model reduction with balanced realizations: an error bound and a frequency weighted generalization, Proc. 23rd Con['. Decision and Control, Las Vegas (1984) 127-132. [2] K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their 5J~-error bounds, Int. J. Control 39 (1984) 1115 t 193. [3] M. Green and D.J.N. Limebeer, Linear Robust Control (Prentice-Hall, Englewood Cliffs, NJ, 1995). [4] F. Kuo, Network Analysis and Synthesis (Wiley, New York, 1966). [5] P. Lancaster and M. Tismenetsky, The Theory o[' Matrices. Second Edition with Applications (Academic Press, San Diego, 1985). [6] Y. Li and B.D.O. Anderson, Singular perturbation approximation of balanced systems, Int. J. Control 50 (1989) 1379 1405. [7] B.C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction, 1EEE Trans. Automat. Control 26 (1981) 17-32. [8] K. Zhou, J.C. Doyle and K. G/over, Robust and Optimal Control (Prentice-Hall, Upper Saddle River, 1996).