計測自動制御学会東北支部 第 218 回研究集会 (2004.10.9)
資料番号 218-16

線形システムとその電力相補システムにおけるグラミアンと2次モード の新しい性質
Novel Properties of the Gramians and Second-Order Modes ofLinear State-Space Systems and Their Power ComplementarySystems○越田俊介，阿部正英，川又政征○ Shunsuke Koshita, Masahide Abe, Masayuki Kawamata
東北大学大学院工学研究科

Graduate School of Engineering, Tohoku University
キーワード : 状態空間シ ステム (state-space system), 有界実シ ステム (bounded-real system), 電力相補シ ステム(power complementary system)，グ ラミアン (Gramian)，2 次モード (second-order mode)

連絡先 : 〒 980-8579

仙台市青葉区荒巻字青葉 6-6-05 東北大学大学院工学研究科 電子工学専攻 川又研究室

越田俊介，Tel.: (022)217-7095，Fax.: (022)263-9169，E-mail: kosita@mk.ecei.tohoku.ac.jp

1.

Introduction

eratures on signal processing address this theory
mainly from the viewpoint of the transfer func-

The Gramians and second-order modes of lintions and thus there has been few researches on theear state-space systems are signiﬁcant factors since
power complementary systems by the state-space
they play important roles in many ﬁelds of anaapproach.
log and digital signal processing.

For example,
In this paper, we ﬁrst derive a novel theorem

the problem of quantization eﬀects of digital ﬁlters,on continuous-time power complementary systems
which can be solved by the Gramians and secondthrough the state-space approach. Then, we reveal
order modes, has attracted many researchers since
an important property of the second-order modes
quantization eﬀects depend highly on ﬁlter strucby applying the new theorem to the second-ordertures and the state-space approach is the most efmodes. Moreover, we extend these results to discretefective and elegant method that can be used to ﬁndtime state-space systems and formulate a new exthe optimal structures.
pression for the evaluation of minimum quantizaThe theory on power complementary systems istion eﬀects.
also of considerably practical signiﬁcance in many
The organization of this paper is as follows. Inﬁelds of digital signal processing such as digital ﬁlSection 2, the continuous-time state-space systems,
ter design, ﬁlter banks, and so on. However, litGramians and second-order modes are introduced.

–1–

In Section 3, the theory on bounded-real systems

dand power complementary systems is addressed.

u(t)This theory is frequently used in our main discus-bx(t)ẋ(t)

1/ssion. Especially, The bounded-real Riccati equa-cy(t)Ations play central roles in derivation of our new theorems. In Section 4, our main result is presented;
Fig. 1

new theorems on the Gramians and second-order

Continuous-time state-space system.

modes of continuous-time and discrete-time state-

For the system (A, b, c, d), the solutions K andspace systems are established. In Section 5, a new

W of the following Lyapunov equations are called

formula for evaluating minimum attainable value

the controllability Gramian and the observabilityof coeﬃcient sensitivity is presented as an applica-Gramian, respectively:

tion of our new theorems. This formula shows theeﬀectiveness and high performance of digital ﬁlterswith minimum coeﬃcient sensitivity structure. In

AK + KAt = −bbt

(4)

At W + W A = −ct c.

(5)Section 6, a numerical example is given in order to

The Gramians K and W are symmetric and pos-verify our theorems.

itive deﬁnite, that is, K = K t > 0 and W =W t > 0, because the system (A, b, c, d) of H(s) is2.

The Gramians and SecondOrder Modes of ContinuousTime State-Space Systems

Consider the following state-space equations forassumed to be stable, controllable and observable.
2Then, the eigenvalues θ12 , θ22 , · · · , θNof the matrix

product KW are all positive. The positive squareroots θ1 , θ2 , · · · , θN of the eigenvalues are called thesecond-order modes of the system1, 2) .

a stable single-input/single-output continuous-timesystem of order N with the transfer function H(s):It should be noted that the Gramians depend on
realization of the system, while the second-orderẋ(t) = Ax(t) + bu(t)(1)

modes depend only on the transfer function. In they(t) = cx(t) + du(t)(2)

literatures on control system theory, the second-

where u(t), y(t) and x(t) are the scalar input, thescalar output and the state vector of size N × 1,
and the matrices A, b, c and d are coeﬃcient ma-order modes are also called Hankel singular values
because the eigenvalues of KW are equal to the
singular values of the Hankel operator of H(s).

trices with appropriate size. The block diagramof this system is given in Figure 1. The system

3.

(A, b, c, d) is assumed to be a minimal realizationof H(s), that is, the system (A, b, c, d) is controllable and observable. The coeﬃcient matrices andthe transfer function are related as
H(s) = d + c(sI − A)−1 b.

Bounded-Real Systems andPower Complementary Systems

This section introduces the well-known important theory on bounded-real systems and power(3)

complementary systems. This theory is frequently–2–if there exists P = P t > 0 such that
1|H(jΩ)||H(jΩ)|

0.8
Magnitude

At P + P A + c t c+(P b + ct d)r−1 (P b + ct d)t = 00.6(8)where r = γ 2 − d2 > 0.

0.4

Lemma 2 A state-space system (A, b, c, d) is bounded0.2

real if and only if its dual system (At , ct , bt , d) is0 −1
10Fig. 20bounded-real. Moreover, if P = P t > 0 is a solu-110Frequency Ω [rad/s]

10tion to (8), then Q = γ 2 P −1 is a solution to thedual equation

Power complementary systems. (γ = 1)

AQ + QAt + bbt

used in the derivation of our result.

+(Qct + bd)r−1 (Qct + bd)t = 0.

3.1

Deﬁnition of Bounded-Real Systems and Power ComplementarySystems(9)

We call Eqs. (8) and (9) the bounded-real Riccati
equations. Note that solutions of the bounded-realRiccati equations yield a state-space representation

A linear system H(s) is called bounded-real ifof power complementary systems as follows.

and only if
Lemma 3 Given a stable bounded-real state-space|H(jΩ)| <= γ,∀Ω ∈ ,(6)system H(s) = d + c(sI − A)−1 b with minimal

and a pair of bounded-real linear systems H(s) andrealization, its power complementary system H(s)H(s) is said to be power complementary if

is described by|H(jΩ)|2 + |H(jΩ)|2 = γ 2 ,∀Ω ∈ .

H(s) = w + l(sI − A)−1 b(7)For example, if H(s) is lowpass, then H(s) is high-where w and l are given as

pass as illustrated in Figure 2. While if H(s) is1bandstop, then H(s) is bandpass.

3.2

Bounded-Real Riccati Equations

The bounded-real systems satisfy the so-called(10)w = r2(11)l = −(P b + ct d)t /w.

(12)Considering the dual system, we can also see that
H(s) has the following representation

bounded-real Riccati equations. The following lem-H(s) = w + c(sI − A)−1 m(13)m = −(Qct + bd)/w.

(14)

mas are well-known results relating bounded-real
wheresystems to the Riccati equations3, 4) .
Lemma 1 A state-space system (A, b, c, d) of H(s)with minimal realization is bounded-real if and only–3–

4.

Novel Properties of the Grami-Theorem 2 If a stable system H(s) = d + c(sI −−1
ans and Second-Order Modes A) b with minimal realization is bounded-real, thenthe second-order modes are all bounded by γ, thatIn this section, our main result is presented. Theis,

theorems presented in this section reveal novel propθi <=γerties of the Gramians and second-order modes.
Theorem 1 Let K and W be the controllability
and observability Gramians of a stable linear system H(s) = d+c(sI −A)−1 b with minimal realization. Let H(s) be the power complementary systemof H(s). Then, there exist matrices P , Q, W andK such thatfor 1 <=i<= N.

(20)

Proof: The proof is straightforward. From Theorem 1, it is easily seen that K <= Q and W <= P.
2Hence we have KW <= γ I. This result showsθi <= γ.

If |H(jΩ)| = γ for all ω, K and W

are equal to Q and P , respectively, and therefore
θi = γ for 1 <=i<= N.

2

These new theorems can be extended to discreteP = W +W

(15)time systems. That is, the following two theorems

Q = K +K

(16)

can be established.

where P and Q are positive deﬁnite solutions ofTheorem 3 Let K d and W d be the controllabil-

Eqs. (8) and (9), respectively, W is the observabil-

ity and observability Gramians of an N -th order

ity Gramian of H(s) = w +l(sI −A)−1 b, and K isstable discrete-time system Hd (z) = dd + cd (zI −the controllability Gramian of H(s) = w + c(sI −Ad )−1 bd with minimal realization, and let H d (z)A)−1 m. The coeﬃcients w, l and m in H(s) are

be the power complementary system of Hd (z). K dobtained from Eqs. (11), (12) and (14).

and W d are given as the positive deﬁnite solutions

Proof: Since H(s) is stable, a solution P = P t > 0of the following discrete-time Lyapunov equations:of Eq. (8) is represented as
P =X +Y(17)

where X and Y are given as the solutions of theK d = Ad K d Atd + bd btd(21)

W d = Atd W d Ad + ctd cd .

(22)Then, there exist matrices P d , Qd , W d and K dsuch that

following Lyapunov equations:
At X + XA = −ct c(18)

At Y + Y A = −(P b + ct d)r−1 (P b + ct d)t (. 19)

Pd = Wd + Wd

(23)

Qd = K d + K d .

(24)

In the above equations, P d and Qd are given as

From Eqs. (5) and (18), it is obvious that X = W .

positive deﬁnite solutions of the following discrete-Moreover, from Eqs. (11), (12) and (19), it follows

time bounded-real Riccati equations

that Y = W . These relationships show Eq. (15).
The proof of Eq. (16) is completely dual to theproof of Eq. (15).

P d − Atd P d Ad − ctd cd−(Atd P d bd + ctd dd )r1−1 (Atd P d bd + ctd dd )t = 0(25)2The above theorem yields an important property

Qd − Ad Qd Atd − bd btd−(Ad Qd ctd + bd dd )r2−1 (Ad Qd ctd + bd dd )t = 0(26)of the second-order modes as follows.

–4–where r1 = r2 = γ 2 − d2 − btd P d bd = γ 2 − d2 −modes as follows6) :cd Qd ctd . W d is the observability Gramian of H d (z) =S = (tr[K d ] + 1) (tr[W d ] + 1)
N2
Smin =
θdi + 1 .

wd + ld (zI − Ad )−1 bd and K d is the controllability
Gramian of H d (z) = wd + cd (zI − Ad )−1 md . The

12(32)

i=1

coeﬃcients wd , ld and md are obtained as
12(31)We are now ready to construct a new formula for

wd = r1 = r2(27)the evaluation of the minimum coeﬃcient sensitiv-ld = −(Atd P d bd + ctd dd )t /wd(28)ity Smin . Applying Theorem 4 to Eq. (32) gives

md = −(Ad Qd ctd + bd dd )/wd .

(29)

the upper bound of Smin as follows:
2Smin <= (N γ + 1) .

Theorem 4 Let the second-order modes of Hd (z)(33)

be θd1 ,This new formula shows the eﬀectiveness of mini-θd2 , · · · , θdN . If Hd (z) is bounded-real, then themum coeﬃcient sensitivity structure, especially insecond-order modes of the discrete-time system are

the case of narrowband digital ﬁlters. As is well

all bounded by γ, that is,known, the coeﬃcient sensitivity of digital ﬁltersθdi <=γ

for 1 <
=i<= N.

(30)

with direct form tends to be inﬁnity as ﬁlter bandwidth approaches zero. On the other hand, we can

Proof: It is easy to prove the above two theorems

see from Eq. (33) that the coeﬃcient sensitivity ofwith the help of the bilinear transformation s =digital ﬁlters with minimum coeﬃcient sensitivityα(z − 1)/(z + 1).

structure is at most (N γ + 1)2 , where γ is generally2equal to 1. This fact emphasizes that the minimum

5.

A New Formula for Evaluating Minimum Attainable
Value of Quantization Eﬀects

coeﬃcient sensitivity structure has excellent noise
performance and that it is very suitable to implementation of digital ﬁlters on ﬁnite wordlength signal processors.

In this section, a new formula for evaluating minimum attainavle value of quantization eﬀects is presented as an application of Theorem 4.

The similar discussion can be made for the evaluation of minimum roundoﬀ noise. Details are omitted here.

As we stated in Section 1, the Gramians andsecond-order modes play crucial roles in analysis
and minimization of quantization eﬀects such as

6.

A Numerical Example

roundoﬀ noise and coeﬃcient sensitivity of digitalThis section gives a numerical example to ver-ﬁlters1, 5, 6) . It should be particularly noted that

ify our theorems. Consider the following 2nd-orderthe second-order modes determine the minimum

continuous-time transfer function H(s):attainable value of quantization e. For example,H(s) =the coeﬃcient sensitivity S of an N -th order dig-

0.0032s2 + 0.0063.
s2 + 0.1123s + 0.0063(34)

ital ﬁlter H(z) and its minimum value Smin canThis transfer function has an lowpass magnitude

be obtained from the Gramians and second-order

response with the gain γ = 1 which is indicated in–5–

can be conﬁrmed. Hence Theorem 1 is veriﬁed to10.9

be valid.

0.8

From Eqs. (36) and (37), the second-order modes

Magnitude

0.7of H(s) are obtained as follows:

0.60.5

θ1 = 0.6822 < 1,0.40.3

θ2 = 0.1838 < 1.

(41)

Therefore, the validity of Theorem 2 is conﬁrmed.

0.2

Theorems 3 and 4 for discrete-time systems can0.1
0 −2
10Fig. 3−1010

11010Frequency Ω [rad/s]

be veriﬁed in a similar way.

7.

Magnitude response of H(s). (γ = 1)

Figure 3. A state-space realization (A, b, c, d) ofConclusion

This paper has established novel theory on theGramians and second-order modes of linear state-

this system is given as follows:A bspace systems. It has been shown that a positivec d⎛−0.1123⎜=⎝
1−0.0063
010−0.0004

0.00630.0032deﬁnite solution of the bounded-real Riccati equa⎞⎟⎠ . (35)tion consists of the Gramians of linear systems andtheir power complementary systems. Using this result, the upper bound of the second-order modes

The Gramians K, W and a solution P of the bounded- of linear systems has been described. This upper
real Riccati equation (8) are computed asbound of the second-order modes has given a new
4.4527
0formula for evaluating minimum attainable value of
K=(36)
0704.0407
coeﬃcient sensitivity and shown the excellent per
0.0280 0.0031
formance of digital ﬁlters with minimum coeﬃcientW =(37)
0.0031 0.0005
sensitivity structure.
0.1123 0.0063
P =.
(38)
0.0063 0.0007
From Eqs. (11), (12) and (38), a power comple-References

mentary system of the above system is obtainedasH(s) =

1.0000s2
s2 + 0.1123s + 0.0063(39)

and the observability Gramian of this system iscomputed asW =0.08430.00320.00320.0002

.

(40)

Consequently, from (37), (38) and (40) it is veriﬁedthat W + W = P . In a similar way, K + K = Q–6–1) C. T. Mullis and R. A. Roberts: Roundoﬀ noise
in digital ﬁlters: Frequency transformations andinvariants, IEEE Trans. Acoust., Speech, Signal
Processing, ASSP-24, 538/550 (1976)
2) B. C. Moore: Principal component analysis inlinear systems: Controllability, observability, andmodel reduction, IEEE Trans. Automatic Control,
AC-26, 17/32 (1981)
3) P. C. Opdenacker and E. A. Jonckheere: A contraction mapping preserving balanced reduction
scheme and its inﬁnity norm error bounds, IEEETrans. Circuits and Systems, CAS-35, 184/189(1988)
4) R. Ober: Balanced parametrization of classes oflinear systems, SIAM J. Control and Optimization, 29-6, 1251/1287 (1991)

5) S. Y. Hwang: Minimum uncorrelated unit noise instate-space digital ﬁltering, IEEE Trans. Acoust.,Speech, Signal Processing, ASSP-25, 273/281(1977)
6) M. Kawamata, M. Iwatsuki, and T. Higuchi: Balanced realizations as minimum sensitivity structures in linear systems, Transactions of the Society of Instrument and Control Engineers, 21-9,900/906 (1985)–7–