計測自動制御学会東北支部 第 284 回研究集会 (2013.11.8)
資料番号 284-7

２段階安定化法のパラメータ変化に関する入出力の可視化
Visualization Input-Output Relation with Parametrization of
Two Stage Compensator Design◦ 橋本 佳祐 ∗ ，森 和好 ∗◦Keisuke Hashimoto∗ , Kazuyoshi Mori∗*会津大学 コンピュータ理工学部*The University of Aizu.
キーワード : ２段階安定化 (Two-Stage Compensator Design)，既約分解法 (Factorization Approach)，パラメトリゼーション (Parametrization)，可視化 (Visualization)，離散時間 LTI システム (Discrete-Time

LTI System)

連絡先 : 〒 965-8580 福島県会津若松市一箕町鶴賀字上居合９０ 会津大学コンピュータ理工学部
橋本佳祐，Tel.: (0242)37-2615，Fax.: (0242)37-2747，E-mail: s1180030@u-aizu.ac.jp

Abstract

1.

Introduction

In this paper, we consider the two-stage com-

The factorization approach to control sys-pensator designs for single-input single-output

tems has the advantage that it embraces, within

plants in the framework of the factorization ap-

a single framework, numerous linear systems

proach. Recently one of the authors has givensuch as continuous-time as well as discrete-

the parametrization of stabilizing controllers

time systems, lumped as well as distributed

given by the two-stage compensator design. As
an investigation of the role of the two-stagesystems, one-dimensional as well as multidimensional systems, etc.1, 2, 3) . Hence the re-compensator designs, we implement a visual-sult given in this paper will be able to a number

ization system of the parametrization of stabi-of models in addition to the multidimensional

lizing controllers based on the two-stage com-systems. In factorization approach, when prob-

pensator design, in which the discrete-time LTIlems such as feedback stabilization are studied,systems are considered as a model of the fac-

one can focus on the key aspects of the problem

torization approach.

under study rather than be distracted by thespecial features of a particular class of linear
systems. This approach leads to conceptuallysimple and computationally tractable solutions
to many important and interesting problems4) .

–1–

A transfer matrix of this approach is consid-

So far, the results of the two-stage compen-ered as the ratio of two stable causal trans-sator design use the norm algebras as well as

fer matrices. For a long time, the theory ofthe factorization approach. Because the anal-

the factorization approach had been foundedysis by the norm algebra is based on a concreteon the coprime factorizability of transfer ma-

speciﬁed model, this reduces the attractive-trices, which is satisﬁed by transfer matrices

ness of the factorization approach. Recently,

over the principal ideal domains or the Bézout

one of the authors has given a parametrization

domains.

of stabilizing controllers of the two-stage com-

In some design problems, one uses a so-called

pensator design based on the factorization ap-

two-stage compensator design for selecting anappropriate stabilizing compensator4) . Given

proach only, which is in the form of the YoulaKučera-parametrization5) .

a plant, the ﬁrst stage consists of selecting a

As an investigation of the role of the two-stabilizing compensator for the plant. The sec-stage compensator designs, we implement a vi-

ond stage consists of selecting a stabilizing con-sualization system of the parametrization oftroller for the new closed-loop system that alsostabilizing controllers based on the two-stage

achieves some other design objectives such ascompensator design. In the system, we con-decoupling, sensitivity minimization, etc. Thesider the discrete-time LTI systems as a model

rationale behind this procedure is that the de-of the factorization approach. To implement asign problems are often easier to solve when

visualization system easily, we have employedMathematica5) .

the plant is stable.
Another example is to design buildings newly
attached earthquake-resistant dumpers (Figure 1).

2.

Preliminaries of Factorization Approach

The stabilization problem considered in thispaper follows the papers 6, 7) , in which thefeedback system Σ 4) is as in Figure 2. For further details the reader is referred to the literature4) ,6) , 7) , and 8) .

u1e1

cu2
y1

e2

py2

Figure 2: Feedback system Σ.
We consider that the set of stable causal
transfer functions is an integral domain, denoted by A. The total ring of fractions of A
Figure 1: Earthquake-Resistant Dumpers.

is denoted by F; that is, F = {n/d | n, d ∈A, d = 0}. This F is considered as the set of–2–

all possible transfer functions. Matrices over F3.

are transfer matrices. Let Z be a prime ideal

Two-Stage Compensator Designof A with Z = A. Deﬁne the subsets P and Psof F as follows: P = {a/b ∈ F | a ∈ A, b ∈In some design problems, one uses a so-called

A\Z}, Ps = {a/b ∈ F | a ∈ Z, b ∈ A\Z}.

two-state procedure for selecting an appropriate stabilizing compensator4) . Given a plant

Then, every transfer function in P (Ps ) is calledcausal (strictly causal). Analogously, if every entry of a transfer matrix is in P (Ps ),p, the ﬁrst stage consists of selecting a stabilizing compensator for p. Let c0 ∈ S(p) de-

the transfer matrix is called causal (strictlynote this compensator (that is, an arbitrarycausal).

but ﬁxed compensator of p) and deﬁne p1 =Throughout the paper, the plant we con-p(1 + c0 p)−1 . The second stage consists of se-sider has single-input and single-output, andlecting a stabilizing controller for p1 that also

its transfer function, which is also called a plant
itself simply, is denoted by p and belongs to P.
We can always represent p in the form of a fraction p = nd−1 , where n ∈ A and d ∈ Am×machieves some other design objectives such as
decoupling, sensitivity minimization, etc. The
resulting conﬁguration with its inner and outerloops is shown in Figure 3.

with nonzero d.
For p ∈ P and c, a matrix H(p, c) ∈ F 2×2 isdeﬁned as(1 + pc)−1H(p, c) :=c(1 + pc)−1−p(1 + pc)−1(1 + pc)−1pc0c1

(1)

provided that 1 + pc is a nonzero of A. ThisH(p, c) is the transfer matrix from [ ut1to [ et1ut2 ]tet2 ]t of the feedback system Σ. If 1+pc

is a nonzero of A and H(p, c) ∈ A2×2 , then we
say that the plant p is stabilizable, p is sta-

p1

Figure 3: Two-Stage Compensator Design (y2to u2 ).

bilized by c, and c is a stabilizing controllerof p. In the deﬁnition above, we do not men-

Here we give that the two-stage compensatortion the causality of the stabilizing controller.

design based on Figure 3 cannot give all stabi-

However, it is known that if a causal plant islizing controllers. The following two theorems
have been given by one of the authors5) . The

stabilizable, there always exists a causal stabilizing controller of the plant 7) .
We will denote by S(p) the set of stabilizing

The second is a detailed version of Theorem 1with coprime factorizability.

controllers of P .
It is known that W (p, c) deﬁned below isover A if and only if H(p, c) is over A:c(1 + pc)−1 −pc(1 + cp)−1.
W (p, c) :=pc(1 + pc)−1p(1 + cp)−1ﬁrst Theorem 1 is same as Theorem 5.3.10 of 4) .

Theorem 1 Let p denote a causal plant of P
and c0 a causal stabilizing controller of p (c0 ∈(2)This W (p, c) is the transfer matrix from u1 andP). Further let p1 be p(1 + c0 p)−1 . Denote byc0 + S(p1 ) the following set:
{c0 + c1 | c1 ∈ S(p1 )}.

u2 to y1 and y2 .

–3–

Then
c0 + S(p1 ) ⊂ S(p),c0pc1(3)with equality holding if and only if c0 ∈ Am×n .
Theorem 2 Let p, c0 , p1 be as in Theorem 1.
Let n, d, y, x be in A such thatp = nd−1 , c0 = yx−1 ,ny + dx = 1.

(4)

Figure 5: Feedback from y1 to u1 .
The conﬁguration is as in Figure 6.

Then we havepc0
c1

c0 + S(p1 )= {(x−rn)−1 (y + rd) | r = r1 x2 , r1 ∈ A}.
By Theorem 1, we see that the sum of c0 anda stabilizing controller of p1 , say c1 , is again astabilizing controller of p. This sum, a stabilizing controller of p, is the parallel allocationof c0 and c1 , as shown in Figure 4.

Figure 6: Composite Stabilized Feedback withc0 and c1 based on Feedback from y1 to u1 .

pc0c1

Based on this feedback, the following result
has also been given in 5) .
Corollary 1 Let Py1 u1 denote c0 (1 + pc0 )−1 .
Then we have{(1 + c0 c1 )−1 c0 | c1 ∈ S(py1 u1 ), 1 + c0 c1 = 0}= {(x−rn)−1 (y + rd) | r = −r1 y 2 , r1 ∈ A,

Figure 4: Composite Stabilized Feedback withx−rn = 0}.

c0 and c1 .
The theorems were based on the feedback

4.

from y2 to u2 (cf. Figures 2 and 3). Even so, we(5)Visualization of Two-Stage
Compensator Designnote that, from Figure 2, we have two inputs
u1 and u2 and two outputs y1 and y2 . Thus

Now that we have a parametrization of two-

we can consider alternative two-stage compen-

stage compensator design, we are able to vi-

sator design based on other input(s) and othersualize the parameterization of the two-stage

output(s). Let us consider the two-stage com-compensator design. To implement this visualization, we employ Mathematica9) .

pensator design based on the feedback from y1

In this implementation, we consider the discrete-

to u1 . In this case, the feedback system is as
in Figure 5.

time LTI systems as a model of the factorization approach.

–4–

compensator design. By usiging them, we con-

Let the plant p besider to design further robust stabilizning con-

−3d + d2
2 + 2dtroller.

9and d2+dand the inputs u1 and u2 be d−7
2 +3 ,

respectively. In this case, the coprime factor-

参考文献

ization is given as1) C. A. Desoer, R. W. Liu, J. Murray, andy = 1, x = 2−d/2, n = 1/(4(d−3)d, x = (1+d)/2.

R. Saeks, “Feedback system design: The
fractional representation approach to anal-4.1

ysis and synthesis,” IEEE Trans. Automat.
Visualization of Youla-Kučera-parametrization
Contr., vol. AC-25, pp. 399–412, 1980.

Figure 7 is the visualization based on Youla2) M. Vidyasagar, H. Schneider, and B. A.

Kučera-parametrization:

Francis, “Algebraic and topological aspects

y + rd,x − rnof feedback stabilization,”

IEEE Trans.

Automat. Contr., vol. AC-27, pp. 880–894,

where 1 <=r<= 2.
1
Figure 7 is the visualization with r = d−aand 1 < a <= 2.

1982.
3) K. Mori, “Parametrization of all strictly
causal stabilizing controllers,”

4.2

Visualization of Two-Stage Compensator Design

Figure 9 is the visualization based on Theorem 1, where 1 <=r<= 2.

Trans. Automat. Contr., vol. AC-54, pp.
2211–2215, 2009.
4) M. Vidyasagar,

Control System Synthe-

sis: A Factorization Approach, Cambridge,

Figure 10 is the visualization based on The1
and 1 < a <orem 1, where with r = d−a= 2.

4.3

IEEEMA: MIT Press, 1985.
5) K. Mori, “Multi-stage compensator design

Visualization of Alternative TwoStage Compensator Design— a factorization approach —,” in Proceedings of The 8th Asian Control Conference(ASCC 2011), 2011, 1012-1017.

Figure 9 is the visualization based on Theo6) V. R. Sule,rem 1, where 1 <=r<= 2.
Figure 10 is the visualization based on The-commutative rings:

The matrix case,”

SIAM J. Control and Optim., vol. 32, no.

1
and 1 < a <orem 1, where with r = d−a= 2.

5.

“Feedback stabilization over6, pp. 1675–1695, 1994.

Conclusion and Future Works 7) K. Mori and K. Abe,

“Feedback stabi-lization over commutative rings: FurtherThis paper reports a visualization of two-study of coordinate-free approach,” SIAM

stage compensator design. We have obtained

J. Control and Optim., vol. 39, no. 6, pp.

visual input-output relations with varying pa-1952–1973, 2001.

rameter. We have used two types of two-stage–5–8) S. Shankar and V. R. Sule, “Algebraic geometric aspects of feedback stabilization,”
SIAM J. Control and Optim., vol. 30, no.
1, pp. 11–30, 1992.
9) P. Wellin, Programming with Mathematica: An Introduction, Cambridge University Press, 2013.

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Figure 7: Visualization based on
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= 2.

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Figure 8: Visualization based on
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1<=r<
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68

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Figure 9: Visualization with 1 <=r<= 2.

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Figure 10: Visualization with r = d−aand

1<a<= 2.

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Figure 11: Visualization of Alternative
Two-Stage Compensator Design with
1<=r<= 2.

y1y2
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Figure 12: Visualization of Alternative
1
Two-Stage Compensator Design with r = d−aand 1 < a <= 2.

–9–6810