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A Design Method for Control System to Attenuate Periodic Input Disturbances Using Disturbance Observers for Time-Delay Plants 37

A Design Method for Control System toAttenuate Periodic Input Disturbances UsingDisturbance Observers for Time-Delay Plants

Jie Hu1 , Kou Yamada2 , Tatsuya Sakanushi3 ,

Yuki Nakui4 , and Yusuke Atsuta5 , Non-members

ABSTRACTIn this paper, we examine a design method for con-

trol system to attenuate periodic input disturbancesusing disturbance observers for time-delay plants.The disturbance observers have been used to esti-mate the disturbance in the plant. Several papers ondesign methods of disturbance observers have beenpublished. Recently, parameterizations of all distur-bance observers and all linear functional disturbanceobservers for time-delay plants with any input dis-turbance were clarified. If parameterizations of allsuch observers for time-delay plants with any inputdisturbance are used, there is a possibility that wecan design control systems to attenuate input dis-turbances effectively. However, no paper examinesa design method for control system using parame-terizations of all disturbance observers and all linearfunctional disturbance observers for time-delay plantswith any input disturbance. In this paper, to atten-uate periodic input disturbances effectively, we pro-pose a design method for control system using theseparameterizations.

Keywords: Time-Delay Plant, Periodic Distur-bance, Disturbance Observer, Parameterization

1. INTRODUCTION

In this paper, we examine a design method forcontrol system to attenuate periodic input distur-bances using the parameterization of all disturbanceobservers for time-delay plants with any input distur-bance. A disturbance observer is used in the motion-control field to cancel the disturbance or to make theclosed-loop system robustly stable [1–8]. Generally,the disturbance observer consists of the disturbancesignal generator and observer. And then, the distur-bance, which is usually assumed to be step distur-bance, is estimated using observer. Since the distur-bance observer has simple structure and is easy to un-derstand, the disturbance observer is applied to many

Manuscript received on July 12, 2012 ; revised on September12, 2012.1,2,3,4,5 The authors are with Department of Mechani-

cal System Engineering, Gunma University 1-5-1 Tenjin-cho, Kiryu 376-8515 Japan., E-mail: [email protected], [email protected], [email protected] [email protected]

applications [1–6, 8]. However, Mita et al. point outthat the disturbance observer is nothing more than analternative design of an integral controller [7]. Thatis, the control system with the disturbance observerdoes not guarantee the robust stability. In addition,in [7], an extended H∞ control is proposed as a ro-bust motion control method which achieves the dis-turbance cancellation ability. This implies that usingthe method in [7], a control system with a disturbanceobserver can be designed to guarantee the robust sta-bility. From other viewpoint, Kobayashi et al. con-sider the robust stability of the control system witha disturbance observer and examine an analysis ofparameter variations of disturbance observer [8]. Inthis way, robustness analysis of control system witha disturbance observer has been considered.

On the other hand, another important controlproblem is the parameterization problem, the prob-lem of finding all stabilizing controllers for a plant[9–14]. If the parameterization of all disturbance ob-servers for any disturbance could be obtained, wecould express previous studies of disturbance observerin a uniform manner. In addition, disturbance ob-server for any disturbance could be designed system-atically. From this viewpoint, Yamada et al. examineparameterizations of all disturbance observers and alllinear functional disturbance observers for plants withany input disturbance [15]. Yamada et al. expandthe result in [15] and propose parameterizations of alldisturbance observers and all linear functional distur-bance observers for time-delay plants with any inputdisturbance [16]. If parameterizations of all distur-bance observers and all linear functional disturbanceobservers for time-delay plants with any input dis-turbance in [16] are used, there is a possibility thatwe can design control systems to attenuate input dis-turbances effectively. However, no paper examinesa design method for control system using parame-terizations of all disturbance observers and all linearfunctional disturbance observers for time-delay plantswith any input disturbance.

In this paper, in order to show the effectiveness ofthese parameterizations, we propose a design methodfor control system to attenuate periodic input distur-bances effectively using these parameterizations fortime-delay plants with any input disturbance. First,

38 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.11, NO.1 February 2013

the disturbance observer and the linear functionaldisturbance observer for time-delay plants with anyinput disturbance are introduced. Next, to attenu-ate periodic input disturbances effectively, a designmethod for control system using these parameteri-zations of all disturbance observers and of all linearfunctional disturbance observers for time-delay plantswith any input disturbance is proposed. In addition,control characteristics of control system using theseparameterizations are clarified. Note that the repeti-tive control system [17] is well known as an effectivecontrol system to attenuate periodic disturbances. Itis shown that the proposed method can attenuate pe-riodic disturbances effectively without using repeti-tive controllers. A design procedure is also given. Fi-nally, a numerical example is illustrated to show theeffectiveness of the proposed method.

NotationR the set of real numbers.R(s) the set of real rational functions

with s.RH∞ the set of stable proper real ra-

tional functions.U the unimodular procession in

RH∞. That is, P (s) ∈ Umeans that P (s) ∈ RH∞ andP−1(s) ∈ RH∞.

AT transpose of A.σ({·}) largest singular value of {·}.diag (a1, · · · , an) an n × n diagonal matrix with

ai as its i-th diagonal element.[A BC D

]represents the state space de-scription C(sI −A)−1B +D.

L{·} the Laplace transformation of{·}.

2. DISTURBANCE OBSERVER AND LIN-EAR FUNCTIONAL DISTURBANCE OB-SERVER

In this section, we briefly introduce a disturbanceobserver and a linear functional disturbance observerfor time-delay plants with any input disturbance andexplain the problem considered in this paper.

Consider the plant written by{x(t) = Ax(t) +B (u(t− L) + d(t− L))y(t) = Cx(t) +D (u(t− L) + d(t− L))

, (1)

where x ∈ Rn is the state variable, u ∈ Rp is thecontrol input, y ∈ Rm is the output, d ∈ Rp is theperiodic disturbance with period T > 0 (T ∈ R) sat-isfying

d(t+ T ) = d(t) (∀t ≥ 0), (2)

A ∈ Rn×n, B ∈ Rn×p, C ∈ Rm×n and D ∈ Rm×p.It is assumed that (A,B) is stabilizable, (C,A) is de-tectable, u(t) and y(t) are available, but d(t) is un-available. The transfer function y(s) in (1) is denoted

by

y(s) = G(s)e−sLu(s) +G(s)e−sLd(s), (3)

where

G(s) = C (sI −A)−1

B +D ∈ Rm×p(s). (4)

When the disturbance d(t) is not available, inmany cases, the disturbance estimator named thedisturbance observer is used. The disturbance ob-server estimates the disturbances d(t) in (1) usingavailable measurements. Available measurements ofthe time-delay plant in (1) are the control inputu(t− δ)(L ≥ δ ≥ 0) and the output y(t). For simplic-ity, we select δ = L, and then the general form of thedisturbance observer d(s) for (1) is written by

d(s) = F1(s)y(s) + F2(s)e−sLu(s), (5)

where F1(s) ∈ Rp×m(s), F2(s) ∈ Rp×p(s), d(s) =L{d(t)} and d(t) ∈ Rp(t). That is, the general formof the disturbance observer d(s) is shown in Fig. 1. In

++

u(s) y(s)

F1(s)

dà(s)

++

d(s)

G(s)eàsL

F2(s)eàsL

Fig.1: Structure of a disturbance observer and thatof a linear functional disturbance observer

the following, we call the system d(s) in (5) a distur-bance observer for time-delay plants with any inputdisturbance, if

limt→∞

(d(t− L)− d(t)

)= 0 (6)

is satisfied for any initial state x(0), control input u(t)and disturbance d(t).

According to [16], there exists a disturbance ob-server d(s) satisfying (6) if and only if m ≥ p andG(s) is biproper and of minimum phase, that is, D isof full rank and

rank

[A− sI B

C D

]= n+min(m, p)

(∀ℜ{s} ≥ 0). (7)

In addition, when above-mentioned expressions hold,the parameterization of all disturbance observers fortime-delay plants G(s)e−sL with any input distur-bance is written by (5), where

F1(s) = D(s)N∗(s) +Q(s)N⊥(s) ∈ RHp×m∞ (8)


and

F2(s) = −I ∈ RHp×p∞ , (9)

respectively, where N(s) ∈ RHm×p∞ and D(s) ∈

RHp×p∞ are coprime factors of G(s) on RH∞ satis-

fying

G(s) = N(s)D−1(s), (10)

N∗(s) is a pseudo inverse of N(s) satisfying

N∗(s)N(s) = I (11)

and Q(s) ∈ RHp×(m−p)∞ is any function.

When one of following expressions:1. m ≥ p.2. G(s) is of minimum phase.3. G(s) is biproper.does not hold, there exists no disturbance observerfor time-delay plants with any input disturbance sat-isfying (6) [16]. Since many plants in the motion-control field are strictly proper and of non-minimumphase, this is a problem for the disturbance observerfor time-delay plants with any input disturbance to besolved. When a disturbance observer for time-delayplants with any input disturbance is used to attenuatedisturbances such as in [1–6], even if d(s) satisfying(6) cannot be designed, the control system can bedesigned to attenuate disturbance effectively. Thatis, in order to attenuate disturbances, it is enough toestimate (I − F (s))e−sLd(s), where F (s) ∈ RHp×p

∞ .From this point of view, when G(s) is strictly properand of non-minimum phase, Yamada et al. defined alinear functional disturbance observer for time-delayplants with any input disturbance [16].

For any initial state x(0), control input u(t) anddisturbance d(s), we call d(s) the linear functionaldisturbance observer for time-delay plants with anyinput disturbance if

e−sLd(s)− d(s) = F (s)e−sLd(s) (12)

is satisfied, where F (s) ∈ RHp×p∞ [16]. Available

measurements of the time-delay plant in (1) are thecontrol input u(t − δ)(L ≥ δ ≥ 0) and the outputy(t). For simplicity, we select δ = L, and then thegeneral form of the linear functional disturbance ob-server for time-delay plants with any input distur-bance is written by (5), where F1(s) ∈ Rp×m(s) andF2(s) ∈ Rp×p(s). That is, the general form of the lin-ear functional disturbance observer d(s) is shown inFig. 1. When m = p holds and the time-delay plantG(s)e−sL is stable, the system d(s) in (5) is a linearfunctional disturbance observer for stable time-delayplants with any input disturbance if and only if F1(s),F2(s) and F (s) are written by

F1(s) = I +Q(s), (13)

F2(s) = −G(s)−Q(s)G(s) (14)

and

F (s) = I − F1(s)G(s), (15)

respectively, where Q(s) ∈ RHp×p∞ is any function.

On the other hand, when m = p holds and the time-delay plant G(s)e−sL is unstable, the system d(s) in(5) is a linear functional disturbance observer for un-stable time-delay plants with any input disturbanceif and only if F1(s), F2(s) and F (s) are written by

F1(s) = D(s) +Q(s)D(s), (16)

F2(s) = −N(s)−Q(s)N(s) (17)

and

F (s) = I − F1(s)G(s), (18)

respectively, where N(s) ∈ RHm×p∞ and D(s) ∈

RHm×m∞ are coprime factors of G(s) on RH∞ sat-

isfying

G(s) = D−1(s)N(s) (19)

and

D(s)N(s)− N(s)D(s) = 0 (20)

and Q(s) ∈ RHp×p∞ is any function.

The problem considered in this paper is to pro-pose a design method for control system to attenuateperiodic input disturbances effectively using the pa-rameterizations of all disturbance observers and alllinear functional disturbance observers for time-delayplants with any input disturbance.

3. DESIGN METHOD FOR CONTROLSYSTEM FOR MINIMUM-PHASE ANDBIPROPER PLANTS

In this section, we propose a design method forcontrol system to attenuate periodic input distur-bances effectively using the parameterization of alldisturbance observers for time-delay plants with anyinput disturbance.

When G(s) is of minimum-phase and biproper, wepropose a control system using the parameterizationof all disturbance observers for time-delay plants withany input disturbance as shown in Fig. 2. Here,C(s) ∈ Rp×m(s) and C(s) ∈ Rp×p(s) are controllers,L ≥ 0 is chosen as

L = nT − L, (21)

n is the smallest positive integer that makes L in (21)nonnegative, and F1(s) and F2(s) are given by (8) and(9), respectively.

Next, we clarify control characteristics of the con-trol system in Fig. 2. First, the input-output char-acteristic of control system in Fig. 2 is shown. The


+

+ +

+

àà

+r(s) u(s)

d(s)

y(s)

F1(s)

C(s)

Cê(s)

+F2(s)eàsL

G(s)eàsL

eàsLö

Fig.2: Control system using the parameterization ofall disturbance observers

transfer function from the reference input r(s) to theoutput y(s) and that from the reference input r(s) tothe error e(s) = r(s)− y(s) are written by

y(s) =(I +G(s)e−sLC(s)

)−1G(s)e−sLC(s)r(s) (22)

and

e(s) = r(s)− y(s) =(I +G(s)e−sLC(s)

)−1r(s), (23)

respectively. Therefore, the input-output character-istic in Fig. 2 is specified using C(s).

Next, the disturbance attenuation characteristic ofcontrol system in Fig. 2 is shown. The transfer func-tion from the disturbance d(s) to the output y(s) isgiven by


)−1G(s)e−sL(

I − C(s)e−snT)d(s). (24)

From (24), if C(s) is set satisfying

σ{I − C(jωi)

}≃ 0 (i = 1, . . . , nd), (25)

then the periodic disturbance d(s) with frequencycomponent ωi (i = 1, . . . , nd) written by

ωi =2π

Ti (i = 1, . . . , nd), (26)

is attenuated effectively, where ωndis the maximum

frequency component of the periodic disturbance d(s)with period T . To satisfy (25), C(s) is set accordingto

C(s)

= diag{

1(1 + sτ1)

α1 · · · 1(1 + sτp)

αp

},

(27)

where τi ∈ R (i = 1, . . . , p), and αi (i = 1, . . . , p) arearbitrary positive integers satisfying

σ

[I − diag

{1

(1 + jωiτ1)α1

· · · 1

(1 + jωiτp)αp

}]≃ 0 (i = 1, . . . , nd). (28)

When C(s) is settled as (27), we have

y(jωi)

=(I +G(jωi)e

−jωiLC(jωi))−1

G(jωi)e−jωiL[

I − diag

{1

(1 + jωiτ1)α1

· · · 1

(1 + jωiτp)αp

}]d(jωi) (i = 1, . . . , nd). (29)

Under the condition in (28),

σ {y(jωi)} ≃ 0 (i = 1, . . . , nd) (30)

hold true. This implies that when C(s) is chosenas (27), the control system in Fig. 2 can attenuateperiodic input disturbances effectively without usingrepetitive controllers in [17].

From (22) and (24), the role of C(s) is differentfrom that of C(s). C(s) specifies the input-outputcharacteristic, while C(s) specifies the disturbance at-tenuation characteristic. That is, the control systemin Fig. 2 has one of two-degree-of-freedom structures.

Finally, the condition that the control system inFig. 2 is stable is clarified. From (22) and (24), it isobvious that the control system in Fig. 2 is stable ifand only if following expressions hold.1. C(s) makes the unity feedback control system in{

y(s) = G(s)e−sLu(s)u(s) = −C(s)

(31)

stable.2. Q(s) ∈ RH

p×(m−p)∞ .

3. C(s) ∈ RHp×p∞ .

4. DESIGN METHOD FOR CONTROLSYSTEM FOR NON-MINIMUM-PHASEAND/OR STRICTLY PROPER PLANTS

In this section, we propose a design methodfor control system to attenuate periodic input dis-turbances effectively using the parameterization ofall linear functional disturbance observers for non-minimum-phase and/or strictly proper time-delayplants with any input disturbance.

Even if G(s) is of non-minimum-phase and/orstrictly proper, we can design a control system usingthe parameterization of all linear functional distur-bance observers for time-delay plants with any inputdisturbance in Fig. 2. Here, C(s) ∈ Rp×m(s) andC(s) ∈ Rp×p(s) are controllers, L ≥ 0 is chosen as(21), n is the smallest positive integer that makes Lin (21) nonnegative. When G(s) is stable, F1(s) andF2(s) are given by (13) and (14), respectively. WhenG(s) is unstable, F1(s) and F2(s) are given by (16)and (17), respectively.

Next, we clarify control characteristics of the con-trol system in Fig. 2. First, the input-output char-acteristic of control system in Fig. 2 is shown. Thetransfer function from the reference input r(s) to the


output y(s) and that from the reference input r(s) tothe error e(s) = r(s)− y(s) are written by


)−1G(s)e−sLC(s)r(s) (32)

and

e(s) = r(s)− y(s) =(I +G(s)e−sLC(s)

)−1r(s), (33)

respectively. Therefore, the input-output character-istic in Fig. 2 is specified using C(s). For example,the output y(t) follows the step reference input r(t)without steady state error if

(I +G(0)C(0))−1

= 0. (34)

Next, the disturbance attenuation characteristic ofcontrol system in Fig. 2 is shown. The transfer func-tion from the disturbance d(s) to the output y(s) isgiven by

y(s)

=(I +G(s)e−sLC(s)

)−1G(s)e−sL{

I − C(s) (I +Q(s)) N(s)e−snT}d(s)

=(I +G(s)e−sLC(s)

)−1D−1(s)e−sL{

I − N(s)C(s) (I +Q(s)) e−snT}N(s)d(s).

(35)

From (35), if C(s) is set satisfying

σ{I − N(jωi)C(jωi) (I +Q(jωi))

}≃ 0

(i = 1, . . . , nd), (36)

then the periodic disturbance d(s) with frequencycomponent ωi (i = 1, . . . , nd) written by

ωi =2π

Ti (i = 1, . . . , nd), (37)

is attenuated effectively. When G(s) is of minimumphase, that is N(s) is of minimum phase, there existsNr(s) ∈ RHm×m

∞ satisfying

N(s)Nr(s)

= Q(s)

= diag{

1(1 + τ1s)

α1 · · · 1(1 + τms)

αm

},

(38)

where αi (i = 1, . . . ,m) are arbitrary positive integersto make Nr(s) proper and τi ∈ R (i = 1, . . . ,m) areany positive real numbers satisfying

σ{I − Q(jωi)

}= σ

[I − diag

{1

(1 + jωiτ1)α1

· · · 1

(1 + jωiτm)αm

}]≃ 0 (i = 1, . . . , nd). (39)

Using Nr(s), if C(s) is selected as

C(s) = Nr(s) (I +Q(s))−1

, (40)

where Q(s) is selected so that (I +Q(s)) ∈ U , thenwe have

y(jωi)

=(I +G(jωi)e


D−1(jωi)e−jωiL{

I − Q(jωi)}N(jωi)d(jωi) (i = 1, . . . , nd).

(41)


σ {y(jωi)} ≃ 0 (i = 1, . . . , nd) (42)

hold true. That is, the periodic input disturbanceswith frequency components ωi (i = 1, . . . , nd) isattenuate effectively without using repetitive con-trollers in [17]. On the other hand, when G(s) is ofnon-minimum phase, that is N(s) is of non-minimumphase, there exists ˆNr(s) ∈ RHm×m

∞ satisfying

N(s) ˆNr(s)

= Q(s)Ni(s)

= diag{

1(1 + τ1s)

α1 · · · 1(1 + τms)

αm

}Ni(s), (43)

where Ni(s) ∈ RH∞ is an inner function satisfyingNi(0) = I, αi (i = 1, . . . ,m) are arbitrary posi-tive integers to make ˆNr(s) proper and τi ∈ R (i =1, . . . ,m) are any positive real numbers satisfying

σ{I − Q(jωi)Ni(jωi)

}= σ

[I − diag

{1

(1 + jωiτ1)α1

· · · 1

(1 + jωiτm)αm

}Ni(jωi)

]≃ 0 (i = 1, . . . , nd). (44)

Using ˆNr(s), if C(s) is selected as

C(s) = ˆNr(s) (I +Q(s))−1

, (45)

where Q(s) is selected so that (I +Q(s)) ∈ U , thenwe have

y(jωi)

=(I +G(jωi)e


D−1(jωi)e−jωiL{

I − Q(jωi)Ni(jωi)}N(jωi)d(jωi)

(i = 1, . . . , nd). (46)


σ {y(jωi)} ≃ 0 (i = 1, . . . , nd) (47)


hold true. That is, the periodic input disturbanceswith frequency components ωi (i = 1, . . . , nd) isattenuate effectively without using repetitive con-trollers in [17].

Note that Nr(s) ∈ RHm×m∞ satisfying (38) can be

designed using the method in [18]. ˆNr(s) ∈ RHm×m∞

satisfying (43) can be designed using the method in[19–22]. The method in [19] makes Ni(s) in (43) benot necessarily a diagonal functional matrix, but themethod in [21, 22] makes Ni(s) in (43) by a diagonalfunctional matrix. Therefore, in order to attenuatethe periodic disturbance d(s) effectively, we had bet-ter design ˆNr(s) using the method in [21, 22].

From (32) and (35), the role of C(s) is differentfrom that of C(s). C(s) specifies the input-outputcharacteristic, while C(s) specifies the disturbance at-tenuation characteristic. That is, the control systemin Fig. 2 has one of two-degree-of-freedom structures.

Finally, the condition that the control system inFig. 2 is stable is clarified. From (32) and (35), it isobvious that the control system in Fig. 2 is stable ifand only if following expressions hold.1. C(s) makes the unity feedback control system in{

y(s) = G(s)e−sLu(s)u(s) = −C(s)

(48)

stable.2. Q(s) ∈ RHp×p

∞ .3. C(s) ∈ RHp×p

∞ .

5. DESIGN PROCEDURE

In this section, a design procedure of control sys-tem using the linear functional disturbance observerin Fig. 2 is presented.

A simple design procedure of control system usingthe linear functional disturbance observer in Fig. 2 issummarized as follows:1. Obtain coprime factors N(s), D(s), N(s) and D(s)of the plant G(s) on RH∞ satisfying (10), (19) and(20). A state space description of N(s), D(s), N(s)and D(s) of the plant G(s) satisfying (10), (19) and(20) are obtained using the method in [23].2. Design C(s) to attenuate periodic input distur-bance d(s) effectively using the method described inSection 4..3. Design C(s) to stabilize the unity feedback controlsystem in (48). Such controller C(s) can be designedusing the parameterization of all stabilizing modifiedSmith predictors for multiple-input/multiple-outputplants in [24].According to [24], when G(s)e−sL is stable, the pa-rameterization of all stabilizing modified Smith pre-dictors C(s) for stable plants G(s)e−sL is written as

C(s) = Qc(s)(I −G(s)Qc(s)e

−sT)−1

, (49)

where Qc(s) ∈ RHm×m∞ is any function and settled to

specify the input-output characteristic. For example,

in order for the output y(t) to follow the step referenceinput r(t) without steady state error, Qc(s) must besettled to satisfy

G(0)Qc(0) = I. (50)

On the other hand, when G(s)e−sL is unstable andunstable poles si(i = 1, . . . , n) of G(s) satisfies si =sj(i = j; i = 1, . . . , n; j = 1, . . . , n), the parameteriza-tion of all stabilizing modified Smith predictors C(s)for unstable plants G(s)e−sL is written as

C(s) = C1(s)(I −G(s)C1(s)e

−sT)−1

, (51)

where C1(s) is given by

C1(s) = D(s)(Gu(s) + G−1

u (s)Qc(s)), (52)

Gu(s) ∈ RHm×m∞ is a function satisfying

N(si)Gu(si)e−siL = I (∀i = 1, . . . , n) , (53)

Gu(s) =f(s)

n∏i=1

(s− si)

I ∈ Rm×m(s), (54)

f(s) is a Hurwitz polynomial with n-th degree andQc(s) ∈ RHm×m

∞ is any function and settled to spec-ify the input-output characteristic. For example,when G(s) has a pole at the origin, the output y(s)follows the step reference input r(s) without steadystate error, independent from Qc(s) ∈ RHm×m

∞ in(52). On the other hand, when G(s) has no pole atthe origin, for the output y(t) to follow the step ref-erence input r(t) without steady state error, Qc(s)must be settled to satisfy

N(0)(Gu(0) + G−1

u (0)Qc(0))= I. (55)

6. NUMERICAL EXAMPLE

In this section, we show a numerical example toillustrate the effectiveness of the proposed method.

We consider the problem to design a control sys-tem in Fig. 2 for the output y(t) to follow the stepreference input r(t) written by

r(t) =

[r1(t)r2(t)

]=

[12

](56)

and to attenuate periodic input disturbance d(t) withperiod T = 2[sec] written by

d(t) =

[d1(t)d2(t)

]=

[sin(πt)2 sin(πt)

](57)

effectively for the time-delay plant G(s)e−sL written


by

G(s)e−sL

=

s− 1s(s+ 25)

−1s(s+ 25)

1s(s+ 25)

s− 3s(s+ 25)

e−1.2s

=

0 0 0 0 −0.04 −0.040 0 0 0 0.04 −0.120 0 −25 0 1.04 0.040 0 0 −25 −0.04 1.121 0 1 0 0 00 1 0 1 0 0

e−1.2s,

(58)

where

G(s)

=

s− 1s(s+ 25)

−1s(s+ 25)

1s(s+ 25)

s− 3s(s+ 25)

=

0 0 0 0 −0.04 −0.040 0 0 0 0.04 −0.120 0 −25 0 1.04 0.040 0 0 −25 −0.04 1.121 0 1 0 0 00 1 0 1 0 0

(59)

and

L = 1.2[sec]. (60)

G(s) in (59) is unstable and of non-minimum phase,since poles of G(s) in (59) are in (0, 0) and (−25, 0)and invariant zeroes of G(s) are in (2, 0) and (2, 0).The smallest positive integer n that makes L in (21)nonnegative is n = 1, and then L in (21) is obtainedas

L = 0.8[sec]. (61)

C(s), C(s), F1(s) and F2(s) in Fig. 2 are designedusing the method described in Section 5.. Coprimefactors N(s) ∈ RH∞, D(s) ∈ RH∞, N(s) ∈ RH∞and D(s) ∈ RH∞ of the plant G(s) in (59) on RH∞satisfying (10), (19) and (20) are given by

N(s)

= N(s)

=

s− 1(s+ 2)(s+ 25)

−1(s+ 2)(s+ 25)

1(s+ 2)(s+ 25)

s− 3(s+ 2)(s+ 25)

(62)

and

D(s) = D(s) =

[s

s+ 2 0

0 ss+ 2

]. (63)

Q(s) is selected as Q(s) = 0. C(s) in Fig. 2 is givenby (45), where ˆNr(s) is designed satisfying (43), τ1 =

0.001, τ2 = 0.001, α1 = 1, α2 = 1 and

Ni(s) =

s2 − 4s+ 4s2 + 4s+ 4

0

0 s2 − 4s+ 4s2 + 4s+ 4

. (64)

Next, we design C(s) as (51) in order for the out-put y(t) to follow the step reference input r(t) in (56)without steady state error. One of Gu(s) ∈ RH∞ in(52) satisfying (53) is given by

Gu(s) =

−3750s+ 100

1250s+ 100

−1250s+ 100

−1250s+ 100

. (65)

Gu(s) ∈ R(s) in (54) is set as

Gu(s) =

[s+ 2s 0

0 s+ 2s

]. (66)

In order for the output y(t) to follow the step refer-ence input r(t) in (56), Qc(s) ∈ RH∞ is set as

Qc(s) =

[100

s+ 100 0

0 100s+ 100

]. (67)

Note that G(s) in (59) has a pole at the origin, theoutput y(t) follows the step reference input r(t) in(56) without steady state error, independent fromQc(s) in (67). Substitution of above mentioned pa-rameters for (51) gives C(s).

Using the designed control system in Fig. 2, whenthe initial state x(0) and the disturbance d(t) aregiven by

x(0) =[0.1, 0.1, −0.1, −0.1

]T (68)

and d(t) = [ 0, 0 ]T , respectively, the response ofthe output y(t) = [ y1(t), y2(t) ]T for the step ref-erence input r(t) in (56) is shown in Fig. 3. Here, thebroken line shows the response of the reference inputr1(t), the dotted line shows that of the reference in-put r2(t), the solid line shows that of the output y1(t),and the dotted and broken line shows that of the out-put y2(t). Figure 3 shows that the output y(t) followsthe reference input r(t) without steady state error.

Next, the disturbance attenuation characteristic isshown. When the initial state x(0) and the referenceinput r(t) are given by

x(0) =[0.1 0.1 −0.1 −0.1

]T (69)

and r(t) = [ 0, 0 ]T , respectively, the response ofthe output y(t) = [ y1(t), y2(t) ]T for the periodicinput disturbance in (57) is shown in Fig. 4. Here,the broken line shows the response of the periodicinput disturbance d1(t), the dotted line shows thatof the periodic input disturbance d2(t), the solid line


0 1 2 3 4 5 6 7 8 9 10−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

t[sec]

r 1(t),

r 2(t),

y1(t

), y

2(t)

r1(t)

r2(t)

y1(t)

y2(t)

Fig.3: Response of the output y(t) = [y1(t), y2(t)]T

for the step reference input r(t) in (56)

0 2 4 6 8 10 12 14 16 18 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t[sec]

d 1(t),

d2(t

), y

1(t),

y2(t

)

d1(t)

d2(t)

y1(t)

y2(t)

Fig.4: Response of the output y(t) = [y1(t), y2(t)]T

for the periodic input disturbance d(t) in (57)

shows that of the output y1(t) and the dotted andbroken line shows that of the output y2(t). Figure4 shows that the periodic input disturbance d(t) isattenuated effectively.

In this way, it is shown that using the proposedmethod, we can easily design control system to fol-low the reference input r(t) and to attenuate periodicinput disturbance d(t) effectively.

7. CONCLUSIONS

In this paper, we proposed a design method forcontrol system to attenuate periodic input distur-bances effectively using the parameterizations of alldisturbance observers and all linear functional distur-bance observers for time-delay plants with any inputdisturbance. Results obtained in this paper are asfollows:1. For minimum-phase and biproper time-delayplants with periodic input disturbance, a design

method for control system to attenuate periodic inputdisturbance effectively using the parameterization ofall disturbance observers is proposed. It is shown thatthe disturbance observer works in only the case thatdisturbances exist.2. Control characteristics of proposed control systemfor minimum-phase and biproper time-delay plantsare clarified. We have shown that the proposed con-trol system can attenuate periodic disturbances effec-tively without using repetitive controllers. A designmethod of controller to attenuate periodic input dis-turbance effectively is given.3. For non-minimum-phase and/or strictly propertime-delay plants with periodic input disturbance, adesign method for control system to attenuate peri-odic input disturbance effectively using the param-eterization of all linear functional disturbance ob-servers is proposed.4. Control characteristics of proposed control systemfor non-minimum-phase and/or strictly proper time-delay plants are clarified. A design method of con-troller to attenuate periodic input disturbance effec-tively is given.5. Proposed control system includes three parametersto be designed, C(s), C(s) and Q(s). It is shown thatthe role of C(s) is different from that of C(s) andQ(s). The role of C(s) is to specify the input-outputcharacteristic. The role of C(s) and Q(s) is to specifythe disturbance attenuation characteristic.6. A design procedure of control system using linearfunctional disturbance observer is presented.7. A numerical example is shown to illustrate the ef-fectiveness of the proposed method.

References

[1] K. Ohishi, K. Ohnishi and K. Miyachi, Torque-speed regulation of DC motor based on loadtorque estimation, Proc. IEEJ IPEC-TOKYO,Vol.2, pp.1209-1216, 1983.

[2] S. Komada and K. Ohnishi, Force feedback con-trol of robot manipulator by the accelerationtracing orientation method, IEEE Transactionson Industrial Electronics, Vol.37, No.1, pp.6-12,1990.

[3] T. Umeno and Y. Hori, Robust speed controlof DC servomotors using modern two degrees-of-freedom controller design, IEEE Transactionson Industrial Electronics, Vol.38, No.5, pp.363-368, 1991.

[4] M. Tomizuka, On the design of digital trackingcontrollers, Transactions of the ASME Journalof Dynamic Systems, Measurement, and Con-trol, Vol.115, pp.412-418, 1993.

[5] K. Ohnishi, M. Shibata and T. Murakami,Motion control for advanced mechatronics,IEEE/ASME Transaction on Mechatronics,Vol.1, No.1, pp.56-67, 1996.

[6] H.S. Lee and M. Tomizuka, Robust motion con-


troller design for high-accuracy positioning sys-tems, IEEE Transactions on Industrial Electron-ics, Vol.43, No.1, pp.48-55, 1996.

[7] T. Mita, M. Hirata, K. Murata and H. Zhang,H∞ control versus disturbance-observer-basedcontrol, IEEE Transactions on Industrial Elec-tronics, Vol.45, No.3, pp.488-495, 1998.

[8] H. Kobayashi, S. Katsura and K. Ohnishi, Ananalysis of parameter variations of disturbanceobserver for motion control, IEEE Transactionson Industrial Electronics, Vol.54, No.6, pp.3413-3421, 2007.

[9] G. Zames, Feedback and optimal sensitiv-ity: model reference transformations, multiplica-tive seminorms and approximate inverse, IEEETransactions on Automatic Control, Vol.26,pp.301-320, 1981.

[10] D. C. Youla, J. J. Bongiorno and H. Jabr,Modern Wiener–Hopf design of optimal con-trollers. Part I: The single-input-output case,IEEE Trans. Automatic Control, Vol.AC-21,No.3, pp.3-13, 1976.

[11] C. A. Desoer, R. W. Liu, J. Murray and R.Saeks, Feedback system design: The fractionalrepresentation approach to analysis and synthe-sis, IEEE Trans. Automatic Control, Vol.AC-25,No.3, pp.399-412, 1980.

[12] M. Vidyasagar, Control System Synthesis - Afactorization approach -, MIT Press, 1985.

[13] M. Morari and E. Zafiriou, Robust Process Con-trol, Prentice-Hall, 1989.

[14] J.J. Glaria and G.C. Goodwin, A parameteriza-tion for the class of all stabilizing controllers forlinear minimum phase systems, IEEE Transac-tions on Automatic Control, Vol.39, pp.433-434,1994.

[15] K. Yamada, I. Murakami, Y. Ando, T. Hagi-wara, Y. Imai, D.Z. Gong and M. Kobayashi,The parametrization of all disturbance observersfor plants with input disturbance, The 4th IEEEConference on Industrial Electronics and Appli-cations, pp.41-46, Xi’an, China, 2009.

[16] K. Yamada, D.Z. Gong, T. Hagiwara, I. Mu-rakami, Y. Ando, Y. Imai and M. Kobayashi,The parametrization of all disturbance observersfor time-delay plants with input disturbance,Fourth International Conference on InnovativeComputing, Information and Control, pp.1343-1346, 2009.

[17] T. Nakano, T. Inoue, Y. Yamamoto and S. Hara,Repetitive Control, SICE Publications, 1989.

[18] K. Yamada and K. Watanabe, State space designmethod of filtered inverse system, Transactionsof the Society of Instrument and Control Engi-neers, Vol.28, pp.923-930, 1992.

[19] K. Yamada and K. Watanabe, A state spacedesign method of stable filtered inverse system,

Transactions of the Society of Instrument andControl Engineers, Vol.32, pp.862-870, 1996.

[20] K. Yamada, K. Watanabe and Z.B. Shu, A StateSpace Design Method of Stable Filtered InverseSystems and Its Application to H2 SuboptimalInternal Model Control, Proceedings of Interna-tional Federation of Automatic Control WorldCongress’96, pp.379-382, 1996.

[21] K. Yamada and W. Kinoshita, New designmethod of stable filtered inverse systems, Pro-ceedings of 2002 American Control Conference,pp.4738-4743, 2002.

[22] K. Yamada and W. Kinoshita, New state spacedesign method of stable filtered inverse systemsand their application, Transactions of the Insti-tute of Systems, Control and Information Engi-neers, Vol.16, pp.85-93, 2003.

[23] C.N. Nett, C.A. Jacobson and M.J. Balas, Aconnection between state-space and doubly co-prime fractional representation, IEEE Transac-tions on Automatic Control, Vol.29, pp.831-832,1984.

[24] K. Yamada, N. T. Mai, Y. Ando, T. Hagi-wara, I. Murakami and T. Hoshikawa, A designmethod for stabilizing modified Smith predictorsfor multiple-input/multiple-output time-delayplants, Key Engineering Materials, Vol.459,pp.221-233, 2011.

Jie Hu was born in Beijing, China, in1986. She received a B.S. degree in CivilEngineering from North China Univer-sity of Technology, Beijing, China, in2008. She is currently M.S. candidatein Mechanical System Engineering atGunma University. Her research inter-ests include observer and repetitive con-trol.

Kou Yamada was born in Akita,Japan, in 1964. He received B.S. andM.S. degrees from Yamagata University,Yamagata, Japan, in 1987 and 1989, re-spectively, and the Dr. Eng. degreefrom Osaka University, Osaka, Japan in1997. From 1991 to 2000, he was withthe Department of Electrical and Infor-mation Engineering, Yamagata Univer-sity, Yamagata, Japan, as a research as-sociate. From 2000 to 2008, he was

an associate professor in the Department of Mechanical Sys-tem Engineering, Gunma University, Gunma, Japan. Since2008, he has been a professor in the Department of Mechan-ical System Engineering, Gunma University, Gunma, Japan.His research interests include robust control, repetitive con-trol, process control and control theory for inverse systems andinfinite-dimensional systems. Dr. Yamada received the 2005Yokoyama Award in Science and Technology, the 2005 Electri-cal Engineering/Electronics, Computer, Telecommunication,and Information Technology International Conference (ECTI-CON2005) Best Paper Award, the Japanese Ergonomics So-ciety Encouragement Award for Academic Paper in 2007, the


2008 Electrical Engineering/Electronics, Computer, Telecom-munication, and Information Technology International Confer-ence (ECTI-CON2008) Best Paper Award and Fourth Interna-tional Conference on Innovative Computing, Information andControl Best Paper Award in 2009.

Tatsuya Sakanushi was born inHokkaido, Japan, in 1987. He receiveda B.S. and M.S. degrees in Mechan-ical System Engineering from GunmaUniversity, Gunma, Japan, in 2009 and2011, respectively. He is currently adoctoral student in Mechanical SystemEngineering at Gunma University. Hisresearch interests include PID control,observer and repetitive control. He re-ceived Fourth International Conference

on Innovative Computing, Information and Control Best Pa-per Award in 2009.

Yuki Nakui was born in Saitama,Japan, in 1988. He received a B.S. de-gree in Mechanical System Engineeringfrom Gunma University, Gunma, Japan,in 2010. He is currently M.S. candi-date in Mechanical System Engineeringat Gunma University. His research in-terests include observer and control ap-plication.

Yusuke Atsuta was born in Chiba,Japan, in 1988. He is currently B.S. can-didate in Mechanical System Engineer-ing at Gunma University. His researchinterests include observer and controlapplication.

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