disturbance observers for rigid mechanical systems - equivalence, stability,and design

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Erwin Schrijvere-mail: [email protected]

Johannes van Dijke-mail: [email protected]

Laboratory of Mechanical Automation,Cornelis J. Drebbel Institute

for Systems Engineering,University of Twente, P.O. Box 217,

7500 AE Enschede, The Netherlands

Disturbance Observers for RigidMechanical Systems:Equivalence, Stability,and DesignMechanical (direct-drive) systems designed for high-speed and high-accuracy aptions require control systems that eliminate the influence of disturbances like cogforces and friction. One way to achieve additional disturbance rejection is to extendusual (P(I)D) controller with a disturbance observer. There are two distinct waysdesign, represent, and implement a disturbance observer, but in this paper it is showthe one is a generalization of the other. A general systematic design procedure for dbance observers that incorporates stability requirements is given. Furthermore,shown that a disturbance observer can be transformed into a classical feedback struenabling numerous well-known tools to be used for the design and analysis of disturobservers. Using this feedback interpretation of disturbance observers, it will be shthat a disturbance observer based robot tracking controller can be constructed thequivalent to a passivity based controller. By this equivalence not only stability proothe disturbance observer based controller are obtained, but it also provides more tparent controller parameter selection rules for the passivity based controller.@DOI: 10.1115/1.1513570#

Keywords: Observers, Mechanical System, Design, Unknown Input Observer, PasBased, Tracking, Manipulators

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1 IntroductionAfter the introduction of the disturbance observer as an U

known Input~state space! observer@1#, it has been used and analyzed for linear systems by several researchers@2,3#. The key ideaof a disturbance observer is to augment the plant with a fictitiautonomous dynamical system that generates the disturbanceobserver is then constructed to estimate the states of both theand the disturbance generator. By using the reconstructed dibance state for feedback, disturbance rejection is accomplisAs the feedback signal is only an estimate of the actual disbance acting on the system, the disturbance observer doecontrol the plant, i.e., a normal feedback controller is still needto achieve performance. The benefit of the disturbance observthat it adds disturbance rejection to the nominal feedback conler without effecting the performance. This separation propeenables two independent design stages for the overall contrdesign: one for performance and one for disturbance rejectio

Several years after the introduction of the disturbance obseconcept, the two-degree-of-freedom~2-dof! controller was de-scribed@4–8#. In the 2-dof controller configuration, 1 dof is useto design the command input response, while the other is desigto obtain disturbance rejection. In this paper, it is shown thatdisturbance rejection properties of the 2-dof controller is equilent to the previously known disturbance observer of Johnson@1#,for a linear plant and for some specific design choices. By virof the equivalence, we will denote the disturbance rejectionof the 2-dof structure again a disturbance observer. To distingthe two design methodologies, we denote the original disturbaobserver as introduced by Johnson an Unknown Input DisturbaObserver~UIDO!, and the one resulting from the 2-dof controll

Contributed by the Dynamic Systems and Control Division of THE AMERICANSOCIETY OF MECHANICAL ENGINEERSfor publication in the ASME JOURNAL OFDYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by theASME Dynamic Systems and Control Division, July 2001; final revision, Nov. 20Associate Editor: E. A. Misawa.

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structure as a Disturbance Estimating Filter~DEF!. Note that, as aresult of the equivalence, the terms UIDO and DEF only referthe design methodologyand/or therepresentation, and not to thedisturbance observer itself.

As the 2-dof structure is more general than the original distbance observer as introduced by Johnson, we will mainly focusthe 2-dof structure in this paper. However, because of thequivalence, some properties known for the UIDO can be traformed to the DEF, and vice versa. Furthermore, by closelyamining the equivalence, a new structure for the design oflow-pass filter in the DEF design is derived, which enables theof the disturbance observer for non-minimum-phase plants.

Although the disturbance observer was originally designedobtain additional disturbance rejection for linear systems, soresearchers also included the disturbance observer to makecontroller more robust~e.g.,@9#!. The disturbance observer makethe real plant behave like the nominal plant model for a certfrequency range, thereby simplifying the design of the feedband feedforward controllers. However, it has recently been shothat only a carefully designed disturbance observer enlargesstability margin @3,10–12#. A systematic design procedure thincorporates these stability requirements, is presented in this p~Section 2.4!.

The stability properties of disturbance observer applied to nlinear plants are not shown rigorously yet, except for rigid robmanipulators@8,13#. In these disturbance observer based rocontroller designs, the robot is treated as a set of linear decoumoving mass systems, plus disturbances acting on these seplinear systems representing the nonlinearities, coupling terms,any other~unmodeled, but bounded! effects. In @8#, it is shownthat this type of robot controller results in the same error dynaics as with a specific Passivity-Based controller, thereby estabing stability of the error dynamics for the disturbance obserbased approach to robot tracking control. By comparing thesulting error dynamics, however, no insight is gained in howsimilarity is achieved. Besides, it only shows that the error d

1.

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namics are equivalent. In contrast, this paper will show thatdisturbance observer based controller and the passivity basedtroller themselves are equivalent. To accomplish this equivaleit will be shown~Section 2.5! that the disturbance observer cantransformed into a classical feedback structure. This impliesbasically any feedback controller design tool can be used to cstruct a disturbance observer. Furthermore, the resulting feedrepresentation enables a structural analysis of disturbance obers applied to nonlinear systems. This will be illustrated in Sect4, where a disturbance observer based robot tracking controlldesigned, based on intuitive arguments. Using the feedback ipretation of the disturbance observer, it will be shown thatresulting controller is equivalent to the passivity based controas used in@8# ~Section 4.4!.

The outline of the paper is as follows: First, the DEF distbance observer structure is introduced and analyzed in SectioBased on some general and intuitive arguments together wirobustness analysis, this results in a systematic design proce~Section 2.4!. In Section 2.5, it will be shown that the disturbanobserver can be restructured in a feedback form, which rendeinteresting interpretation of disturbance observers for linearchanical plants.

Then, in Section 3, it will be shown that for some specidesign choices of the DEF design, the DEF and the UIDOequivalent, implying that the UIDO structure is a subset ofDEF structure. Some consequences of the equivalence are g

In the last section, we will focus on disturbance observersmechanical plants specifically. Using the feedback interpretaof the DEF, an intuitively designed disturbance observer barobot tracking controller will be transformed into a Passivbased controller~Section 4.1!. By this equivalence, stability of thedisturbance observer based tracking controller can be guaran

2 Disturbance Estimating FilterThe DEF disturbance observer structure that originated fr

the 2-dof control structure is depicted in Fig. 1@4–6,8#. The dis-turbance observer is based on using the inverse nominal pmodel Pn

21(s) to estimateu0 . By directly feeding back the dif-ference betweenu0 andu, which is thus an estimate of the distubanced, disturbance rejection is accomplished. However, dirfeedback ofd cannot be realized, as in general the inverse plmodel is non-proper. Besides, direct feedback would also resuan algebraic loop. Therefore, the filterQ(s) is introduced. Thedesign of the disturbance observer is now reduced to the desigthe filter Q(s).

This section will continue with a derivation of some geneproperties of the DEF, resulting in specific requirements onQ(s)~Section 2.1!. The sensitivity and complementary sensitivity funtions realized by the DEF are described in Section 2.2. Thespecific structure forQ(s) is selected in Section 2.3. The deriverequirements onQ are summarized in a systematic design produre ~Section 2.4!. Finally, an interesting feedback interpretatioof disturbance observers is presented in Section 2.5.

Fig. 1 Disturbance estimating filter

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2.1 Analysis. The transfer functions realized by the~SISO!disturbance observer of Fig. 1 are@8#:

Huay5y

ua5

PPn

Q~P2Pn!1Pn(1)

Hdy5y

d5

PPn~12Q!

Q~P2Pn!1Pn(2)

Hjy5y

j5

PQ

Q~P2Pn!1Pn(3)

~d is the disturbance input,j is measurement noise, andua is thenew control input!. Assume that the nominal model of the plantcorrect~i.e., P5Pn). Then the transfer-functions simplify to:

Huay5P (4)

Hdy5P~12Q! (5)

Hjy5Q (6)

BecauseHuay is equal toP, the feedback controller that is used iconjunction with the disturbance observer~see Fig. 2! does notnotice the presence of the disturbance observer. So, the dibance observer and the feedback controller can be designedpendently~which is known as the separation property! as long asthe nominal model is correct.

As mentioned, the design of the disturbance observer comdown to the design of the filterQ(s). From an engineering pointof view, some requirements onQ(s) can be derived. The requirements are:

1. Relative degree—In order to enable practical implementation of the disturbance observer,Q(s) should have a relative degreerq larger than or equal to the relative degreerp of the nomi-nal plant model.

2. Global shape—As disturbances should be rejected as muas possible, from the transfer functionHdy ~5!, it can be concludedthatQ(s) should be unity. However, as a result of the requiremeon the relative degree, this will in general only be possible foparticular frequency range. Besides, from the transfer functHjy ~6!, it is clear thatQ(s) should go to zero for high frequencies in order to reject measurement noise. As a result,Q(s) shouldbe a low-pass filter.

3. Peaking—From Hdy ~5!, is can be concluded that the maxmum amplification of 12Q(s) should be small for all frequen-cies. However, fromHjy ~6!, it follows that alsoQ(s) should besmall. This is a conflicting requirement.

2.1.1 Model Misfits. Now suppose the filterQ(s) has beendesigned as a low-pass filter and applies to all the requiremderived before. The properties of the DEF can then be determifor the case that the nominal plant model does not matchactual plant. First, consider high frequencies. For high frequencQ(s)'0, resulting in:

Fig. 2 Disturbance observer combined with a controller

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Hry5PPn

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Hdy5PPn~12Q!

Pn5P (8)

Hjy5PQ

Pn50 (9)

Clearly, for high frequencies the system behaves like the pwithout DEF. For low frequencies,Q(s)'1, so:

Hry5PPn

P5Pn (10)

Hdy5PPn~12Q!

P50 (11)

Hjy52P

P51 (12)

It follows that for low frequencies, the resulting system behalike the nominal plant. Robustness for parameter uncertainttherefore guaranteed@5,6#. However, for this to be true, the DEFshould result in robust stable dynamics. The robustness propeare further analyzed in the next section.

2.2 Sensitivity Analysis. To continue with a more structuraanalysis of the DEF, notice that the disturbance observer cawritten in the form of Fig. 3. Assume the plant is SISO. From thfigure, the sensitivity-functionsT5L/(11L) and S51/(11L)~whereL is the loop gain! of the disturbance observer are easderived as@3,13#:

T5QP

Q~P2Pn!1Pn(13)

S5Pn~12Q!

Q~P2Pn!1Pn(14)

This results inT5Q andS5(12Q) for P5Pn which shows theimportance of the filterQ(s) in the design of the disturbancobserver. Using Eqs.~13! and ~14!, the requirements derived before (i12Q(s)i` and iQ(s)i` small! can simply be restated arequiring limited peaking of the sensitivity and complementasensitivity function~Small Gain Theorem@14#! @15#.

When a controllerC(s) is applied ~see Fig. 2!, the realizedtransfer functions are given by~assumeP5Pn):

Hry5PC

11PC5Tc (15)

Hdy5P~12Q!

11PC5PSc~12Q!5P•S (16)

Hjy5PC1Q

11PC5Tc1

Q

11PC5T (17)

where Sc51/(11PC) and Tc5PC/(11PC) are the sensitivityand complementary sensitivity functions of the controlled systwithout DEF. From the transfer functions~16! and~17!, it is clear

Fig. 3 Equivalent form of the disturbance observer

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that the peaking in 12Q(s) and Q(s) should not coincide withthe peaking of the sensitivity (Sc) and complementary sensitivityfunctions (Tc) which would have been realized by the controllwithout the DEF, to limit the peaking of the overall sensitivifunctions. As a result, the DEF and the controllerC(s) cannot bedesigned independently!

2.3 Low-Pass Filter Q. Despite the fact thatQ(s) shouldadhere to some basic requirements as derived before, there isa fair amount of freedom to selectQ(s). This freedom can beused to include a specific disturbance model in the disturbaobserver design. To illustrate this, suppose the disturbances trejected can be described by 1/sni. The Internal Model Principle@16# then prescribes that the controller should also include a fa1/sni, i.e., it should includeni integrating actions. For a specifichoice of Q(s), the disturbance observer indeed implemeni-integrating actions. To achieve this,Q(s) can be selected according to the following structure@5,8,13#:

Q~s!511(m51

nq2rqf msm

11(m51nq f msm

(18)

where nq is the order ofQ(s) and rq is the relative degree oQ(s). As a result of this specific choice forQ(s), 1/(12Q) ~seeFig. 3! can be written as:

1

12Q~s!5

11(m51nq amsm

(m5nq2rq11nq amsm

511(m51

nq amsm

sni~ani1(m51

rq21ani1msm!

(19)

where ni5nq2rq11. It follows that the disturbance observeimplementsnq2rq11 pure integrators. As a result of the selectstructure forQ(s), only two of the three parametersnq , rq andnican be selected freely.

Although this structure~Eq. ~18!! is presented by many researchers@5,8,13#, in here we present a new structure. This nestructure has the benefit that it enables the application of the Dto non-minimum-phase plants. Assume that the nominal, linstable, SISO plant model is proper~relative degreerp>0):

Pn~s!5N~s!

D~s!5

Nu~s!Ns~s!

D~s!(20)

wherenp is the order ofPn(s) andnn the order of the polynomialN(s). The relative order of the plant isrp5np2nn>0. The nu-merator polynomial is divided into two sectionsNu ~of ordernu)andNs . The only requirement on this separation is that the ponomial Ns is stable. The new structure for theQ(s) filter is nowgiven by:

Q~s!511(m51

ni21f msm1(m51nu gmsni1m21

11(m51nq f msm

(21)

In this new structure, the coefficientsg1 , . . . ,gnnare selected

such that the numerator polynomial ofQ can be written asNu(s)•(r ni21sni211 . . . 1r 1s1r 0) for somer 0 , . . . ,r ni21 .

It is straightforward to verify that this new structure compliwith all requirements derived so far, including the requiremethat 1/(12Q) should include a model of the disturbance genetor. By using this new structure, the disturbance observer strucof Fig. 3 is implementable even for non-minimum-phase plantsthe unstable poles ofPn

21 are cancelled by the right-half-planzeros ofQ(s).

Note that the order of the disturbance modelni realized by theDEF is now given byni5nq2rq2nu11. Note furthermore thatfor minimum-phase plants,Nu can be selected equal to 1 whicrenders the new structure~Eq. ~21!! equal to the old structure~Eq. ~18!!.

DECEMBER 2002, Vol. 124 Õ 541

Fig. 4 Feedback interpretation of the disturbance observer

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2.4 Design Procedure. In this section, an overview is givenof choices to be made to design a disturbance observer. Thesequences of every step are discussed.

1. Determine the nominal plant. It is stated in many papersonly Q(s) is a design entity. However, from~10!, it followsthat for low frequencies the plant is transformed into tnominal plant, and it might therefore be interesting to selPn deliberately different fromP. From ~10!, one could con-clude thatPn should be selected as simple as possiblefacilitate the feedback controller design. However, forPÞPn , stability and performance might be in danger. AftselectingPn5N/D, the numerator polynomialN(s) has tobe divided into two partsNu andNs ~see Eq.~20!!. The onlyrequirement on this separation is that the polynomialNs isstable. To minimize the order of the finalQ(s) filter, theorder nu of Nu should be minimal~i.e., if the plant has noright half plane zeros,Nu should be selected as a static garesulting innu50).

2. Select the relative degreerq>rp of the filter Q(s). For rq5rp1q, there isq320 dB/decade high-frequency roll-ofin the transfer function from measurement noisej to thecontrol actionu. More roll-off might be needed for robustness. However, excessive roll-off will result in peakingthe sensitivity and complementary sensitivity functions. Fthermore, the additional roll-off is only effective when thfeedback controller includes the same amount of roll-off~seeEq. ~17!!.

3. Determine the desired number of integrating actionsni . Forexample the desired shape of the sensitivity function~14!might be a selection criterion. More integrating actions iprove the disturbance rejection for low frequencies, butsults also in more peaking of the sensitivity and complemtary sensitivity functions. The order ofQ(s) can now bedetermined:

nq5ni1rq1nu21 (22)

4. Select the cut-off frequency ofQ(s). This will depend on,among others, the amount of measurement noise and dibances present in the system, the amount of model untainty, and the desired bandwidth of the overall system wha controllerC(s) is applied, see Fig. 2. As already statedSection 2.2, the cut-off frequency should be selected sthat the peaks inQ(s) and 12Q(s) do not coincide with thepeaks of the sensitivity (Sc) and complementary sensitivitfunctions (Tc) which would have been realized by the cotroller without the DEF. From this step onwards, the desof the DEF requires knowledge of the controllerC(s).

5. The next step is to determine the actual filterQ(s), using thestructure of Eq.~18!. Several authors@5,6,8# suggest a But-terworth or binomial model to be used to select the plocations~i.e., de coefficientsf m , mP@1, . . . ,nq#). The lo-cation of the zeros then follows from~21!. Afterwards, itshould be checked thatiQ(s)i` and i12Q(s)i` are small.Further research has to be done to derive methods to sQ(s) according to the structure~21! while minimizingiQ(s)i` and i12Q(s)i` .

6. The final step is checking the robustness of the systemexample by employing the small-gain-theorem. If the syst


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is not robust, the robustness can be increased by reconsing the design choices made. Once robust stability ofDEF is achieved, robust stability of the DEF including thfeedback controllerC(s) should be checked~see Eq.~17!!for the definition of the complementary sensitivity functioof the closed-loop system!.

2.5 Feedback Interpretation. In the previous sections, thedisturbance observer was described for a general linear planP.We now confine ourselves to linearmechanicalplants, to simplifythe analysis and to illustrate the disturbance observer designmore practical level. Finally, we will show that for specific plaand disturbance model combinations~in this case rigid mechanicaplants and 1/sk disturbance models!, the disturbance observer reduces to a simple feedback controller.

Consider a rigid moving mass system~input is force, output isthe velocity!, for which the linear nominal plant model is given a

Pn~s!51

ms(23)

For most mechanical applications, at least one integrating actiorequired to reduce the influence of friction. This integrating actwill be implemented by a disturbance observer. Using the analof the first part of this section, it results that the 1/(12Q) block ofthe disturbance observer in Fig. 3 should contain a factor 1/s. Themost simpleQ filter that accomplishes this, is@5,8,13#:

Q~s!51

st11(24)

wheret is the only design parameter. When we now calculateblock QPn

21 in the feedback path of Fig. 3:

QPn215

ms

st11(25)

we notice that this block contains the factors in the numerator,which will be cancelled by the integrator of the 1/(12Q) blockwhen the loop-gain is computed. As these two factors will noteffective in the loop~as they cancel each other!, it is perhaps moreconvenient to restructure Fig. 3 as is Fig. 4~this is only possible ifQ was designed with the same relative degree asPn , which isactually the case in this development!. For the system defined sfar, the resulting feedback controller block of Fig. 4 evaluates

~12Q~s!!21Q~s!Pn21~s!5

ts11

ts

ms

ts115

m

t(26)

Notice that this is simply a static gain. In conclusion, in this spcific case, we could have designed the disturbance observesimply designing a proportional controller. Only the pre-filtPnQ21 remains to be designed. The function of this pre-filtbecomes clear after computing the transfer function of the innal loop of the disturbance observer~see Fig. 4!, assuming thatP5Pn :


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PQ

Pn~12Q!

11PQ

Pn~12Q!

5Q

12Q1Q5Q (27)

Note from this equation that the pre-filter blockPnQ21 cancelsthe closed loop dynamics of Eq.~27!, and replaces it by the nominal plant model. It is tempting to omit this pre-filter, and considthe closed loop dynamics of Eq.~27! as the new plant for which acontroller ~see Fig. 2! should be designed to achieve the desiroverall performance~e.g., tracking of a reference signal!. How-ever, that would also cancel the desired disturbance rejectionthe loop no longer contains the disturbance generator dynam~Internal Model Principle@16#!.

Remark:A similar result can be obtained if not the velocity othe mass but its position was taken as output, combined witsecond order disturbance observer, i.e.:

Pn51

ms2 (28)

Q52ts11

~ts!312~ts!212ts11(29)

~12Q!21QPn215m

2ts11

t3s12t2 (30)

The resulting feedback controller is in that case a lead-compensator.

3 EquivalenceThis section will show that the DEF disturbance observer str

ture as described in the previous section, is a generalization oUnknown Input Disturbance Observer~UIDO!. The section is or-ganized as follows. First, the UIDO structure as introducedJohnson@1#, is described~Section 3.1!. Then the UIDO structurefor a linear SISO plant is transformed from a state-space desction to a set of transfer-functions~Section 3.2!. This step is per-formed to enable comparison of the UIDO with the DEF~Section3.3!. Finally, in the last section~Section 3.4!, some propertiesknown for the UIDO will be applied on the DEF design and viversa.

3.1 Unknown Input Disturbance Observer. The distur-bance observer structure of Fig. 5, referred to as an UnknoInput Disturbance Observer~UIDO! @2,3,17#, originated from anUnknown Input Observer@18#. For the construction of the distur

Fig. 5 Unknown input disturbance observer „UIDO…


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bance observer, it is assumed that the disturbance signald(t) canbe thought of as being generated by a fictitious autonomousnamic system of order k:

z5Adz (31)

d5Cdz (32)

As an example, assume the disturbances are piece-wise conin time. The Laplace transform ofd(t) is then given byd(s)51/s, resulting inAd50 andCd51. This fictitious disturbancegenerator can be included in the plant, resulting in theaugmentedplant. Where the original plantP(s) has two inputs~one knowninput u(t), and one unknown disturbance inputd(t)), the aug-mented plant has only one inputu(t). And under some mild con-ditions @3#, an observer can be constructed which estimatesonly the states of the plant, but also the statesz of the fictitiousdisturbance generator. Using this estimatedz, an estimate of thedisturbance can be constructed and used for feedback.

3.1.1 Construction of the UIDO.As indicated, the UIDO de-sign method starts with the construction of the augmented plAssume that the nominal, linear, stable, SISO plant modestrictly proper~relative degreerp5np2nn.0) and given by:

Pn~s!5N~s!

D~s!5

bnnsnn1bnn21snn211 . . . 1b0

snp1anp21snp211 . . . 1a0(33)

As the plant model is strictly proper, the plant model can be wten in observer canonical form@19#:

x5Ax1Bu (34)

y5Cx (35)

with

A5F 0 0 ¯ 0 2a0

1 0 ¯ 0 2a1

] ] ] ]

0 0 ¯ 1 2anp21

G@np3np#

(36)

B5F b0

]

bnp21G

@np31#

(37)

C5@0 ¯ 1#@13np# (38)

Write the fictitious disturbance generator model~of orderk! alsoin observer canonical form~see~31! and ~32!!:

Ad5F 0 0 ¯ 0 2q0

1 0 ¯ 0 2q1

] ] ] ]

0 0 ¯ 1 2qk21

G@k3k#

(39)

Cd5@0 ¯ 0 1#@13k# (40)

Combining the plant with the disturbance generator, the amented plant~Fig. 5! is constructed, resulting in the state-spamatrices:

Auido5FA BCd

0 AdG (41)

Buido5FB0 G (42)

Cuido5@C 0# (43)


Fig. 6 Analogy of DEF and UIDO

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Hostetter@18# and Benschop@20# proved that, if the pair$A,C%and the pair$Ad ,Cd% are both observable, and no eigenvaluesAd coincide with a zero of the plant~no pole-zero cancellation!,then also the augmented plant is observable. An observer westimates both the statex of the original plant and the statez of thedisturbance generator can then be constructed. From the estimstatez, an estimate of the disturbanced is available and can beused to compensate for the input disturbance~see Fig. 5!.

Comparing the UIDO with the DEF, the UIDO has apparensome advantages:

1. The UIDO method can incorporate disturbances other t1/sk transparently.

2. Apart from estimating the disturbance states, the UIDO aestimates the system states. Instead of using a feedbacktroller to stabilize and control the plant, state-feedbacknow applicable without an auxiliary observer.

3.2 Transfer functions of the UIDO. The UIDO structureis normally written in a state-space representation. This stspace representation will be transformed to a set of transfer ftions to enable comparison of the UIDO method with the Dmethod. Only SISO plants are considered. The transfer functof interest, realized by the UIDO, are illustrated in Fig. 6b and aregiven by:


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Hyd5Kuido~sI2Auido1LCuido!21Ly (44)

Hud5Kuido~sI2Auido1LCuido!21Buidoy (45)

where

Kuido5@0 Cd# (46)

L5FLp

LkG (47)

Lp5@ l 1 ¯ l np#T (48)

Lk5@ l np11 ¯ l np1k#T (49)

The poles of both transfer functions~44! and ~45! are given by:

det~sI2Auido1LCuido!50 (50)

which, by construction ofL, are all stable and specified for example by a binomial or Butterworth model, or resulted fromLQG design. Determining the poles, which is straightforward blaborious, gives:

(51)


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whereJ(s)5 l np1k det(sI2F) and:

F530 ¯ 0 2

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l np1k

1 ¯ 0 2l np12

l np1k

] ] ]

0 ¯ 1 2l np1k21

l np1k

4@~k21!3~k21!#

(52)

Notice thatJ(s) is a function ofLk only, and has a degree o(k21). It also shows that althoughLp places the roots of det(sI2A1LpC), the combination of bothLp and Lk determines theoverall closed loop poles of the disturbance observer.

In order to retrieve the transfer-functions realized by the UIDthe zeros of~44! and~45! have to be determined also. The zeros~45! are given by@19#:

detFsI2Auido1LCuido 2Buido

Kuido 0 G50 (53)

Writing out the left-hand side of this equation results in:

detFsI2Auido1LCuido 2Buido

Kuido 0 G52detF det~sI2A1LpC! 2N~s!

l np1ksk211 . . . 1 l np11 0 G

52N~s!•J~s! (54)

Equivalently, the zeros of~44! are given by:

detFsI2Auido1LCuido 2L

Kuido 0 G5$snp1anp21snp211 . . . 1a0%• l np1k det~sI2F!

5det~sI2A!•J~s! (55)

Using the derived formulas for the poles and zeros, the tranfunctions~44! and ~45! can be rewritten as:

Hyd5det~sI2A!•J~s!

det~sI2Auido1LCuido!(56)

l

o


f

O,of

fer

Hud52N~s!•J~s!

det~sI2Auido1LCuido!(57)

Now comparison of the UIDO with the DEF is possible, as bostructures are represented in terms of transfer-functions.

3.3 Equivalence. To show the equivalence between thDEF and the UIDO, the derived transfer functions for the UIDof the previous subsection are used. Suppose a UIDO hasdesigned. First it is shown that this disturbance observer canrepresented~using the derived transfer functions! in a structureequivalent to a DEF. This equivalent structure has a filterQ(s)that will be referred to asQUIDO(s), to indicate that this filterresulted from the UIDO design.

As a DEF design is normally implicitly based on a disturbanmodel 1/sni, assume in the following that the UIDO is also dsigned forni-th order disturbances such that det(sI2Ad)5sni ~i.e.,k5ni).

Comparing Fig. 6a andb, it follows that for equivalence,Hud

should be2Q andHyd should equalQPn21. It is easily derived,

that if Hud is equal to2Q, that then alsoHyd equalsQPn21, viz.:

QPn2152HudPn

215N~s!•J~s!

det~sI2Auido1LCuido!•

D~s!

N~s!

5det~sI2A!•J~s!

det~sI2Auido1LCuido!5Hyd (58)

So, all that remains to show equivalence of the UIDO andDEF, is proving that2Hud complies to the structure ofQ used ina DEF design~Eq. ~21!!. The transfer function2Hud ~57!, whichwe will indicate byQUIDO , can be rewritten using~51!:

2Hud5N~s!•J~s!

det~sI2A1LpC!sni1N~s!•J~s!=QUIDO (59)

We used the fact that the UIDO was designed for the disturbamodel 1/sn1. Note that the order ofQUIDO is np1ni .

We now design aQ filter for a DEF ~indicated byQDEF),according to the structure of Eq.~21! and show that this filter isequivalent toQUIDO ~Eq. ~59!!. We use the design parametenq5np1ni which is obviously needed to obtain equivalencWith this design choice,QDEF is given by:

QDEF~s!511(m51

ni21f msm1(m51nu gmsk1m21

11(m51ni21f msm1~(m51

nu gmsk1m212(m51nu gmsk1m21!1(m5k

np1ni f msm

5Nu~s!•~r ni21sni211 . . . 1r 1s1r 0!

$(m51np f ni1msm2(m51

nu gmsm21%sni1Nu~s!•~r ni21sni211 . . . 1r 1s1r 0!(60)

-

e

To prove thatQUIDO andQDEF are indeed equivalent, we shoukeep the following in mind:

1. In the DEF design we can always selectNu5N and Ns51. For this selection, the poles of the transfer functiQPn

21 are completely determined by the poles ofQ. Further-more,nu5nn .

d

n

2. J(s) is of orderni21, and is a function of the last component ofL ~i.e., Lk) only.

3. By proper selection ofL, we can place the poles of thtransfer functions~Eqs. ~56! and ~57!!, i.e., the roots ofdet(sI2A1LpC)sni1N(s)•J(s) practically anywhere in theleft-half plane.

Comparing Eq.~60! with Eq. ~59!, it follows that:


oce

n

gy

e

lo

te

h

e

au

rfe

eDEF

llerAllw-inlexwitz

illthedy-s

ceof

, theant

k-of

a

s

e

-the

rd

de

n the

neartedan-the

b-

each

~r ni21sni211 . . . 1r 1s1r 0!5J~s!

(m51

np

f ni1msm2 (m51

nn

gmsm215det~sI2A1LpC! (61)

ruido5~np1ni !2~nn1ni21!5rp11

This proves thatQUIDO complies with the proposed structureEq. ~21!. In conclusion, given any UIDO design for a disturbanmodel 1/sk, a DEF can always be constructed that results inactly the same disturbance observer.

3.4 Consequences of the Equivalence.In this subsection,the equivalence of the UIDO and the DEF will be used to lisome well known properties and characterizations of the DEFthe UIDO and vice versa.

3.4.1 Relative Degree.Contrary to the DEF design methodwhere the relative degreerq of the filter Q(s) can be chosenfreely ~that is, under the constraintrq>rp), the UIDO alwaysimplements a filterQ(s) with relative degreerq5rp11. So theUIDO always implements at least 20 dB/decade additional hifrequency roll-off, which might be beneficial for robust stabilitHowever, it might as well be superfluous, resulting in a highorder controller than needed. The fixed relative degree also mthat a DEF cannot~in general! be rewritten in a UIDO structure.

3.4.2 Order of the Disturbance Observer.As indicated inthe design procedure of Section 2.4, the order ofNu should beselected as small as possible to minimize the order of the resuQ filter. However, in the UIDO design, there is no design freedto selectNu , andNu is implicitly always selected asNu5N. Thisagain shows the more general nature of the DEF. Furthermoreplants with stable zeros, the UIDO will result in a higher orddisturbance observer than the DEF would.

3.4.3 Sensitivity Functions.An important aspect involved inDEF designs are the~complementary! sensitivity functions~seeSection 2.2!. The shape of these function can be used to assesrobustness of the resulting controller. However, little literatureavailable that discusses robustness for the UIDO design me~see@3# for one of the exceptions!. But because the UIDO can bwritten in terms of a DEF structure, it follows that the~comple-mentary! sensitivity functions for the UIDO are given by (12QUIDO) andQUIDO . This gives a systematic way to analyze trobustness of the UIDO, similar as in the DEF design.

3.4.4 Pole-zero Cancellation.One of the requirements of thUIDO method to succeed, is that the augmented plant model~i.e.,plant and disturbance generator! is observable. To the best of ouknowledge, similar requirements on pole-zero cancellation hnever been reported for a DEF design. But because of the eqlence, this requirement can directly be translated to the Ddesign.

3.4.5 Alternative Disturbance Models.When the distur-bance to be rejected is not described by 1/sk, but more generallydescribed by 1/(sk1qk21sk211 . . . q0), the UIDO designmethod can simply be used to arrive at a disturbance observethe other hand, the DEF design method is less apparent. Inthe filter Q(s) has to be selected according to the more gen~and, therefore, harder to determine! structure:

Q~s!511(m51

nq2rqgmsm

11(m51nq f msm

(62)

such that:

1. the numerator ofQ(s) contains the unstable transmissiozerosN(s) of the plant,

2. 1/(12Q) includes the factor 1/(sk1qk21sk211 . . . q0).


fex-

kto

,

h-.erans

tingm

, forer

theishod

e

rveiva-EF

. Onact,ral

n

It is not trivial to selectQ(s) such that both requirement arsatisfied. Though, as a result of the equivalence between theand the UIDO, it is always possible to construct aQ(s) that ful-fills both requirements.

4 Disturbance Observer Based Robot ControlIn this section, we consider the design of a tracking contro

for rigid robot manipulators, based on a disturbance observer.individual steps in this design are quite basic and intuitive. Hoever, although the approach is intuitive and basic, it will resultthe same controller as the one resulting from the more compand abstract passivity based approach of Sadegh and Horo@21# and Slotine and Li@22#.

The intuitive design procedure will go as follows: First, we wdesign a feedforward control signal to compensate most ofnonlinear robot dynamics, based on estimates of the robotnamic properties~Section 4.1!. Any discrepancies between thiestimate and the real dynamics, including any~external and/orunmodeled! disturbances, will be cancelled by a linear disturbanobserver~Section 4.2!. The final step in the design is the designthe tracking controller~Section 4.3!. As the plant is linearized bythe disturbance observer and the nonlinear feedforward signaltracking controller can be designed for a linear nominal plmodel.

Finally, in Section 4.4, it will be shown that the designed tracing controller is indeed the passivity based tracking controller@21,22#.

4.1 Robot Model. The dynamic equations of motion forrigid robot with n degrees of freedom can be expressed as:

M ~q!q1C~q,q!q1G~q!1d5u (63)

whereq is then31 vector of joint displacements,M (q) the in-ertial matrix, C(q,q)q the vector of centripetal and Corioliforces, G(q) the vector of gravitation forces,d external ~un-modelled! disturbances, andu the control input. Separating thlinear and non-linear parts of Eq.~63! leads to:

M q1ud1d5u (64)

ud5~M ~q!2M !q1C~q,q!q1G~q! (65)

The matrix M5diag(m1 . . . mn) is some constant, positivedefinite, diagonal matrix representing the constant portion ofinertia matrixM (q). The signalud is the collection of all non-linear terms, and will be compensated for with a feedforwasignal ud :

ud5~M ~q!2M !qr1C~q,q!qr1G~q! (66)

In this equation,qr is some reference signal which will be definelater. A symbol with a hat~•! indicates an estimate. The differencbetweenud and ud , and all other remaining disturbancesd, willbe compensated for by a disturbance observer, as described inext subsection.

4.2 Disturbance Observer. As the feedforward signalud isbased on estimates of the robot dynamic properties, the nonliand coupled robot dynamics will not be completely compensafor. We will therefore design a disturbance observer, which ccels not only the errors due to this model uncertainty, but alsodisturbance signald ~see Fig. 7!.

The nominal~linear! plant model used for the disturbance oserver design is given asPn(s)51/Ms. The Q filter for the dis-turbance observer is selected as proposed in section II-E, forindividual joint separately, i.e.:

Q~s!5diagS 1

st111, . . . ,

1

stn11D (67)

Now if the disturbance observer performs optimally, that isd5d1(ud2ud), the dynamics from the new inputua ~see Fig. 7!


Journal of Dynamic Sy

Fig. 7 Disturbance observer for a robot manipulator

Fig. 8 Disturbance observer represented in feedback form

e

hs

4

f

b

ror

ary.

-een

arde is

ow-con--d byis

eter

nceious; it

ent,

arfor

to the output of the robot manipulator is given byq5ua . The newcontrol inputua can therefore be regarded the reference acceltion signal for the robot. With this interpretation ofua , the signalqr can be defined asqr5ua . Note that, in an intuitive way, somesort of feedback linearization has been performed. Note furtmore that all matrices involved in the disturbance observer deare diagonal~i.e., M andQ are designed diagonal!, so that in thelinearized system each individual joint can be treated separat

To analyze the controller defined so far, the disturbanceserver is represented in the feedback form~as derived in Section2.5!, resulting in the structure of Fig. 8. The transfer matricesKaandKv are given by:

Ka=M•PnQ215diagS st111

s. . .

stn11

s D (68)

Kv=~12Q~s!!21Q~s!Pn21~s!5diagS m1

t1. . .

mn

tnD (69)

From these equations it is straightforward to compute the ctrol action of the disturbance observer and the feedforward sigud , using the fact thatKaKv5M1Kv /s ~from Eqs. ~68! and~69!!, andqr5ua :

u5ud1Kv•~Kaqr2q!

5~M ~q!2M !qr1C~q,q!qr1G~q!1M qr1Kvqr2Kvq

5M ~q!qr1C~q,q!qr1G~q!2Kvev (70)

whereev5q2qr . This result will be used later on in Section 4.

4.3 Tracking Controller. All that remains in the design othe disturbance observer based tracking controller, is the desigthe tracking controller. As remarked before, the dynamicstween the new control inputua=qr and the outputq is given byq5ua . A basic tracking controller for this system is given by:

ua5qd22ZLe2L2e (71)

stems, Measurement, and Control

ra-

er-ign

ely.ob-

on-nal

.

n ofe-

where e5q2qd , Z5diag(z1 . . . zn).0, L5diag(l1 . . . ln).0andqd is the reference trajectory. This controller leads to the erdynamics:

~s212ZLs1L2!e50 (72)

By proper selection of the bandwidthL and dampingZ, theseerror-dynamics can be designed stable and as fast as necess

4.4 Overall Controller Dynamics. In the previous subsections, a disturbance observer based tracking controller has bdesigned, using fairly basic~linear! techniques. The only designparameters aret i for the disturbance observer~Eq. ~67!!, and thedamping and bandwidth parametersz i , l i of the tracking control-ler ~Eq. ~71!!. The selection of these parameters is straightforwas the influence of each parameter on the overall performanctransparent. The overall stability of the resulting system has hever not been addressed yet. For that purpose, checking thetroller equations~Eqs. ~70! and ~71!!, reveals that we have designed a passivity based controller equal to the one propose@8,22,23#. It is well proven that this passivity based controllerLp input/output stable with respect to the pair (d,ev) for all pP@1, #, and that the controller can be extended with a paramupdate law to estimate the parameters inM (q), C(q,q)q andG(q) without changing passivity properties@8#. As a result of theequivalence, this stability result also applies for the disturbaobserver based tracking controller as designed in the prevsubsections. Note that this stability result by itself is not newhas already been reported by Bickel and Tomizuka@8#. However,we show that not only the resulting error dynamics are equivalbut also that the controllers themselves are equivalent.

Remark:The stability result even holds~although performancewill probably degrade! if the nonlinear feedforward signalud isselected zero, i.e.,M5M , C(q,q)50 andG(q)50. This selec-tion of the robot property estimates implies that all nonlineand/or coupling terms in the robot dynamics are compensated


so

tki

vo

i

et

n

o

n

on

or

ceyn.

rdigh-

ver

ori,i-

nrol

ofch

ardn-

rol

n-

s to

wne-

f as.,

a-

ts:

by the disturbance observer. Remark furthermore that the spestructure of Eq.~71! is also not important for the stability resultsThe more general controller:

ua5F121~s!e1F2

21~s!qd (73)

also yields stability, as long asF1(s) andF2(s) are strictly properand stable@23#.

5 ConclusionsIn this paper, it is shown that the two commonly known distin

representations of disturbance observers are equivalent in thaone is a generalization of the other. A general systematic deprocedure for disturbance observers is given. The design prdure incorporates robustness criteria for both the disturbanceserver alone, and for the disturbance observer combined wifeedback controller. Finally, a disturbance observer based traccontroller for rigid robot manipulator is designed. By representthe disturbance observer as a feedback controller, the overalltroller configuration appears to be equivalent to the passibased controller as proposed by Sadegh and Horowitz, and Sland Li. As a result of that equivalence, we can state that, althothe presented disturbance observer based controller is desintuitively, input/ouput stability is guaranteed.

References@1# Johnson, C., 1971, ‘‘Accommodation of External Disturbances in Lin

Regulator and Servomechanism Problems,’’ IEEE Trans. Autom. Con16~6!, pp. 635–644.

@2# Profeta, J., Vogt, W., and Mickle, M., 1990, ‘‘Disturbance Estimation aCompensation in Linear Systems,’’ IEEE Trans. Aerosp. Electron. Syst.,26~2!,pp. 225–231.

@3# Coelingh, H., Schrijver, E., De Vries, T., and Van Dijk, J., 2000, ‘‘DesignDisturbance Observers for the Compensation of Low-Frequency Disbances,’’Proc. of Int. Conf. Motion and Vibration Control (MOVIC) 2000,Sydney, Australia, B. Samali~ed.!, pp. 75–80.

@4# Murakami, T., and Ohnishi, K., 1990, ‘‘Advanced Motion Control iMechatronics—A Tutorial,’’Special Lecture—Proc. of IEEE Int. Workshop oIntelligent Control, Istanbul, Turkey,1, pp. SL9–SL17.

@5# Umeno, T., and Hori, Y., 1991, ‘‘Robust Speed Control of DC ServomotUsing Modern Two Degrees-of-Freedom Controller Design,’’ IEEE Trans. IElectron.,38, pp. 363–368.


cific.

ctt theignce-ob-h aing

ngcon-itytine

ughgned

arrol,

d

ftur-

n

rsd.

@6# Lee, H., and Tomizuka, M., 1996, ‘‘Robust Motion Controller Design fHigh-Accuracy Positioning Systems,’’ IEEE Trans. Ind. Electron.,43, pp. 48–55.

@7# Mita, T., Hirata, M., Murata, K., and Zhang, H., 1998, ‘‘h` Control VersusDisturbance-Observer-Based Control,’’ IEEE Trans. Ind. Electron.,45~3!, pp.488–495.

@8# Bickel, R., and Tomizuka, M., 1999, ‘‘Passivity-Based Versus DisturbanObserver Based Robot Control: Equivalence and Stability,’’ ASME J. DSyst., Meas., Control,121, pp. 41–47.

@9# Ko, R., Halgamuge, S., and Good, M., 1998, ‘‘An Improved FeedforwaExtension and Disturbance Observer Design in High-Speed and HPrecision Tracking Control,’’Proc. of Int. Conf. Motion and Vibration Control(MOVIC), Zurich, Switzerland,2, pp. 745–750.

@10# Kempf, C., and Kobayashi, S., 1996, ‘‘Discrete-Time Disturbance ObserDesign for Systems With Time Delay,’’Proc. of Int. Workshop on AdvancedMotion Control, 1, pp. 332–337.

@11# Endo, S., Kobayashi, H., Kempf, C., Kobayashi, S., Tomizuka, M., and HY., 1996, ‘‘Robust Digital Tracking Controller Design for High-Speed Postioning Systems,’’ Control Eng. Pract.,4~4!, pp. 527–536.

@12# Tesfaye, A., Lee, H., and Tomizuka, M., 2000, ‘‘A Sensitivity OptimizatioApproach to Design of a Disturbance Observer in Digital Motion ContSystems,’’ IEEE/ASME Trans. Mechatronics,5~1!, pp. 32–38.

@13# Hori, Y., Shimura, K., and Tomizuka, M., 1992, ‘‘Position/Force ControlMulti-Axis Manipulator Based on the TDOF Robust Servo Controller for EaJoint,’’ Proc. of American Control Conf., 1, pp. 753–757.

@14# Doyle, J., Francis, B., and Tannenbaum, A., 1992,Feedback Control Theory,McMillam, New York.

@15# Kempf, C., and Kobayashi, S., 1999, ‘‘Disturbance Observer and FeedforwDesign for a High-Speed Direct-Drive Positioning Table,’’ IEEE Trans. Cotrol Syst. Technol.,7~5!, pp. 513–526.

@16# Francis, B., and Wonham, W., 1976, ‘‘The Internal Model Principle of ContTheory,’’ Automatica,12, pp. 457–465.

@17# Johnson, C., 1976,Theory of Disturbance-Accommodating Controllers, Cotrol and Dynamic Systems, Vol. 12, pp. 387–489.

@18# Hostetter, G., and Meditch, J., 1973, ‘‘On the Generalization of ObserverSystems With Unmeasurable, Unknown Inputs,’’ Automatica,9, pp. 721–724.

@19# Franklin, G., Powell, J., and Emani-Naeini, A., 1994,Feedback Control ofDynamic Systems, Addison Wesley.

@20# Benschop, R., Steinbuch, M., and Bosgra, O., 1995, ‘‘Observer for UnknoInputs and Disturbances: A Survey of Literature,’’ Nat. Lab. Unclassified Rport ~RWR-501-RB-95015-rb!.

@21# Sadegh, N., and Horowitz, R., 1990, ‘‘Stability and Robustness Analysis oClass of Adaptive Controllers for Robotic Manipulators,’’ Int. J. Robot. Re9~3!, pp. 74–92.

@22# Slotine, J., and Li, W., 1987, ‘‘On the Adaptive Control of Robot Manipultors,’’ Int. J. Robot. Res.,6~3!, pp. 44–59.

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disturbance observers for rigid mechanical systems - equivalence, stability,and design

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