Yet Another Tutorial of Disturbance Observer: Robust Stabilization and Recovery of Nominal Performance

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by  Hyungbo Shim

Yet Another Tutorial of Disturbance Observer:
Robust Stabilization and Recovery of Nominal
Performance
HyungboSHIM1†, GyunghoonPARK1, YoungjunJOO2, JuhoonBACK3, NamHoonJO4
1.ASRI,DepartmentofElectricalandComputerEngineering,SeoulNationalUniversity,Korea;
6 2.DepartmentofElectricalEngineeringandComputerScience,UniversityofCentralFlorida,USA;
1
3.SchoolofRobotics,KwangwoonUniversity,Korea;
0
2 4.DepartmentofElectricalEngineering,SoongsilUniversity,Korea

n
u
J Abstract:

0 Thispaperpresentsatutorial-stylereviewontherecentresultsaboutthedisturbanceobserver(DOB)inviewofrobuststabilization
2 andrecoveryofthenominalperformance.TheanalysisisbasedonthecasewhenthebandwidthofQ-filterislarge,anditisexplained

inapedagogicalmannerthat,eveninthepresenceofplantuncertaintiesanddisturbances,thebehaviorofrealuncertainplantcanbe
]
Y madealmostsimilartothatofdisturbance-freenominalsystembothinthetransientandinthesteady-state.TheconventionalDOBis
S interpretedinanewperspective,anditsrestrictionsandextensionsarediscussed.
.
s
Keywords: Disturbanceobserver,robuststabilization,robusttransientresponse,disturbancerejection.
c
[

2
v
5 1 Introduction anddisturbancegrows.Finally,DOBhasthebenefitofde-
7
signsimplicity(whileitstheoreticalanalysisisnotsimple)
0
Robust control via disturbance observer (DOB) has
2 so that it has been employed in many industrial applica-
0 manyadvantagesoverotherrobustcontrolmethods.Inpar- tions.
. ticular, it is an inner-loop controller and its primary role
1 Because of the benefits, a large number of research
0 is just to compensate uncertainty in the plant and exter-
works have been reported in the literature, including
6 nal disturbances into the plant, so that the inner-loop be-
1 survey-style papers [1,3,4], monographs [5,6], and a re-
haves like a nominal plant without disturbances and un-
: lated paper [2,7,57,58] under the name of ‘active dis-
v
certainties.Therefore,anyouter-loopcontrollerthatisde-
i turbance rejection control (ADRC).’ On the other hand,
X signed just for the nominal plant without considering ro-
this paper presents yet another tutorial of DOB as a sum-
r bustness should work, and this enables modular design
a maryofrecentfindingsbytheauthors,inlessformalstyle
of controllers; that is, the outer-loop controller deals with
(for example, we avoid the theorem-proof style of writ-
nominalstabilityandnominalperformance,andtheinner-
ing).WeviewtheDOB1 asanoutputfeedbackrobustcon-
loopDOBcaresforrobustnessagainstuncertaintyanddis-
troller which, under certain conditions such as minimum-
turbances.Inthissense,DOBisincontrasttootherrobust
phasenessoftheplantandlargebandwidthoftheQ-filters,
controlmethodssuchasH control,adaptivecontrol,or
∞ enables robust stabilization against arbitrarily large para-
sliding mode control, and there is much design freedom
metricuncertainty(aslongastheuncertainparametersare
fortheouter-loopcontrollerinDOB-basedrobustcontrol.
bounded and their bounds are known a priori), and re-
When there is no uncertainty and disturbance, the DOB-
coveryofnominalsteady-stateandtransientperformance.
basedrobustcontrolshowsthebestnominalperformance
This perspective will lead us to the underlying principles
withoutinterventionoftheinner-loopDOB,whiletheper-
oftheDOBthathaslargebandwidthofQ-filters.
formancedegradesgraduallyastheamountofuncertainty
†Correspondingauthor.E-mail:[email protected].:[phone].ThisworkwassupportedpartiallybyUniversityofFlorida,USA,andbythe
NationalResearchFoundationofKorea(NRF)grantfundedbytheKoreagovernment(MISP)(2015R1A2A2A01003878).
1 TheoriginofDOB,whichwascalledaloadtorqueestimator,datesbackto[8].Itwasmoreorlessanestimatorratherthanarobustcontroller.
2
2 SystemDescriptionforAnalysis form of x and z is to emphasize their different roles that
willbeseenshortly.Theintegerν iscalledtherelativede-
Thesystemsdealtwithinthispaperarethesingle-input-
gree of the plant. It is emphasized that the eigenvalues of
single-outputlineartime-invariantsystemsgivenby
thematrixSarethezerosofP(s)in(2)(seetheAppendix
fortheproof),andz˙ = Sz iscalledthezerodynamicsof
x˙ =Ax+bu+Ed, x∈Rn, u∈R,
(1) thesystem.Then,wesaythesystemisofminimumphase
y =cx, d∈Rq, y ∈R,
if and only if the matrix S is Hurwitz. For designing the
DOB,itisnotnecessarytoconvertthegivenplantintothe
where u is the input, y is the output, x is the state, and d
normalform.Therepresentation(3)isjustfortheanalysis
is the external disturbance. The disturbance signal d(t) is
inthispaper.
assumedtobesmooth(i.e.,differentiableasmanytimesas
necessarywithrespecttotimet),andweassumethatd(t)
anditsderivativesareuniformlybounded.ThematricesA, 3 RequiredActionforDOB
b,c,andEareofappropriatesizes,andareassumedtobe
uncertain. In particular, we assume that system (1) (with- The behavior of (3) is unexpected because the plant
outthedisturbancetermEd)isaminimalrealizationofthe P(s)isuncertain,andthus,thequantitiesφ,ψ,g,G,and
transferfunction S areconsequentlyuncertain.Hence,onemaywanttode-
signacontrolinputusuchthatthesystem(3)behaveslike
β sm+β sm−1+···+β
P(s)= m m−1 0 =c(sI−A)−1b itsnominalplant:
sn+α sn−1+···+α
n−1 0
(2)
y =x , x˙ =x , i=1,...,ν−1,
inwhich,allparametersα andβ areuncertain,butβ (cid:54)= 1 i i+1
i i m
0andthesignofβm(whichisso-calledthehigh-frequency x˙ν =fn(x,z)+gnu¯, (4)
gain of P(s)) is known. System (1) can always be trans- z˙ =h (x,z)
n
formedto
where u¯ is an external input that is designed by another
y =x
1
(outer-loop) controller. Comparing (3) and (4), it is seen
x˙ =x
1 2 that system (4) is the system (3), in which, there is no
.
.. disturbance and the uncertain f, g, and h are replaced
(3)
with their nominal f , g , and h , respectively, where
x˙ =x n n n
ν−1 ν
f (x,z) = φ x+ψ z andh (x,z) = S z+G x.While
x˙ =φx+ψz+gu+gd=:f(x,z)+gu+gd n n n n n n
ν thereplacementoff andgcanbeachievedifthecontrolu
z˙ =Sz+Gx+d =:h(x,z,d )
z z in(3)becomesthesameas
whereν :=n−m,x=[x ,··· ,x ]T ∈Rν,andz ∈Rm.
1 ν 1 g
Here, the notation f and h are defined for convenience, u =−d+ (−f(x,z)+f (x,z))+ nu¯,
candidate g n g
which will be used frequently later. The disturbance sig-
nals d and d are linear combinations of d and its deriva-
z
the replacement of h is a difficult task because the z-
tives(fordetails,refertotheAppendix,e.g.,equation(a1)).
subsystemisnotdirectlyaffectedbythecontrolu.Hence,
All the matrices φ ∈ R1×ν, ψ ∈ R1×m, g ∈ R1×1,
insteadofreplacinghinthez-subsystemof(3),letusin-
S ∈ Rm×m, and G ∈ Rm×ν are uncertain, but the sign
troduceanewstatez ∈ Rm (whichwillbeimplemented
of g is known and g (cid:54)= 0 (this is because g is in fact β , n
m
in the controller) and construct a new (dynamic) desired
whichisclarifiedintheAppendix).RefertotheAppendix
inputas
for the derivation from (1) to (3). If the plant has the in-
putdisturbanceonly(likeinFig.1),thentheplantcanbe
writtenasin(3)withoutthetermdz inthez-subsystem. z˙n =hn(x,zn),
The representation (3) is called the normal form [9, u =−d+ 1(−f(x,z)+f (x,z ))+ gnu¯. (5)
10].2 The reason for writing the system state in the split desired g n n g
2 Itisthenameofthestructurelikethewell-knowncontrollability/observabilitycanonicalformoftheplant.Therepresentation(3)ofthetransfer
function(2)canalsobedirectlyderived.See[10,p.513–514]forthisprocedure.Bythisprocedure,itisalsoseenthat,inacertaincoordinate,theterm
Gxin(3)candependonlyonx1,sothat,GxcanbewrittenlikeGx=G(cid:48)x1=G(cid:48)y.
3
Now if u ≡ u (for all t ≥ 0), then the system (3) Fig. 1, is performing all the afore-mentioned tasks when
desired
becomes the bandwidth of the Q-filter is sufficiently large (which
will be clarified in the next section). In the figure, the
y =x , x˙ =x , i=1,...,ν−1, (6) dotted-blockistheplantP(s)withtheDOB,andC(s)is
1 i i+1
x˙ =f (x,z )+g u¯, (7) the outer-loop controller that is designed for the nominal
ν n n n
modelP (s)(or(4))oftheactualplantP(s).SinceC(s)is
z˙ =h (x,z ), (8) n
n n n
designed without considering disturbance and plant’s un-
z˙ =h(x,z,d ). (9)
z certainty, it is the responsibility of the DOB to make the
dotted-blockbehavelikeP (s)sothattheclosed-loopsys-
n
Clearly,thesystem(6)–(8)yieldsthesamebehavioras(4).
temwithC(s)operatesasexpected.Then,thedesignfac-
Atthesametime,thez-subsystem(9)becomesstand-alone
toristheso-calledQ-filtersQ (s)andQ (s)inthefigure.
A B
and does not affect the output y. In other words, the state
Conventionally, they are taken as a stable low-pass filter
z, which was observable from y in (3), has now become
givenby
unobservable by the desired input u .3 This is the
desired
costtopayforenforcingthenominalinput-outputbehav-
c (τs)µ−1+···+c (τs)+c
iorof(4),or(6)–(8),upontherealplant(3).Sincewedo Q (s)=Q (s)= µ−1 1 0
A B (τs)µ+a (τs)µ−1+···+a
not have any information of z from the output y (when µ−1 0
(10)
this nominal input-output behavior is achieved) and have
where µ ≥ ν and τ is a positive constant that determines
no more freedom left in the input u (= u ) to con-
desired thebandwidth.Asτ getssmaller,thebandwidthofthisfil-
trolz-subsystem,wehavetoaskthatz-subsystemisstable ter becomes larger.4 We take c = a (in order to have
0 0
itself(i.e.,S isHurwitz)sothatthestatez(t)doesnotdi-
thedc-gainone)andc =c =···=c =0
µ−1 µ−2 µ−(ν−1)
vergeunderboundedxandd .
z sothattherelativedegreeoftheQ-filterisatleastν.The
Inordertoimplementthecontrolideadiscussedsofar,
latterpropertyisrequiredinordertoimplementtheblock
therearestilltwomorechallenges.First,toimplement(5), P−1(s)Q (s)inthefigure,whichthenbecomesaproper
n B
thestatexneedstobeestimatedbecausexisnotdirectly
transferfunctiontogether.So,thedesigntaskistochoose
measured but is used to compute the nominal values of τ, a , and the remaining c appropriately for the control
i i
fn(x,zn) and hn(x,zn). This problem may be solved by goal that the DOB with C(s) robustly stabilizes the plant
a state observer, but a robust estimation of x is necessary
(againsttheuncertaintyoftheplant)andrejectstheeffect
sincethesystem(3)isuncertainandisaffectedbydistur- oftheexternaldisturbanceontheoutputy(sothatthenom-
bances. Second, since u contains unknown quanti-
desired inalperformanceisrecovered).
ties such as d, f(x,z), and g, we cannot compute it di-
rectly. Instead, we have to estimate u (t) and drive
desired
4 ACloserLookatConventionalDOB
u(t)totheestimate.
Forsimplicityofpresentation,letusconsider,fromnow
on,ageneralexampleofathirdorderuncertainplantwith
relativedegreeν = 2(i.e.,n = 3andm = 1)in(2),and
itsnominalmodel
β s+β
P (s)= n,1 n,0 , β (cid:54)=0.
n s3+α s2+α s+α n,1
n,2 n,1 n,0
Fig.1 Closed-loopsystemwithDOBstructure;risareference
signal,anddisaninputdisturbance.TheDOBcanalsobecom-
This nominal model has the normal form realization
binedwithC(s)whenimplemented,whichbecomesthenafeed-
backcontrollerhavingtwoinputsrandy. (like (3)) with gn = βn,1, Sn = −βn,0/βn,1, φn,2 =
β /β −α , φ = −(β /β )φ −α , ψ =
n,0 n,1 n,2 n,1 n,0 n,1 n,2 n,1 n
It may be rather surprising and exciting to see that the −(β /β )φ −α ,andG = [G ,G ] = [1,0].
n,0 n,1 n,1 n,0 n n,1 n,2
conventional disturbance observer [3,4,11], depicted in Suppose that the Q-filter has the same relative degree as
3 Thisisnotpossibleinpracticebecauseudesired containsunknownquantitiesandsowecannotletu ≡ udesired.However,sincetheDOBwill
estimateudesiredandletu≈udesired,thedegreeofobservabilityofzatleastgetsweakened.
4 ItistriviallyverifiedthattheQ-filtershaveitspoleatλ/τ withλbeingarootofsµ+aµ−1sµ−1+···+a0.Sothebandwidthisproportionalto
1/τ.
4
P (s)andistakenas 4.1 RobustobserverisembeddedinP−1(s)Q (s).
n n B
Itisseenthatequation(13)isacascadeoftwosubsys-
a /τ2
Q (s)=Q (s)= 0 . (11) tems
A B s2+(a /τ)s+(a /τ2)
1 0
z˙ =S z +G q =h (q,z ) (15)
n n n n n n
ArealizationofthefilterQ (s)(inFig.1)isobtainedby and
A (cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35)
q˙ 0 1 q 0
1 1
= + y. (16)
(cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35)
p˙1 = 0 1 p1 + 0 u q˙2 −τa02 −aτ1 q2 τa02
p˙2 −τa02 −aτ1 p2 τa02 (12) Sinceitisnotyetcleartoseeanobserverinit,letustrans-
form (q ,q ) into (q¯ ,q¯ ) := (q + (a /a )τq ,q ). A
y =p . 1 2 1 2 1 1 0 2 2
p 1
simplecomputationyields
Ontheotherhand,thetransferfunction
(cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35)
q¯˙1 = −aτ1 1 q¯1 + aτ1 y
P−1(s)Q (s)= (s3+αn,2s2+αn,1s+αn,0)(a0/τ2) q¯˙2 −τa02 0 q¯2 τa02
n B (βn,1s+βn,0)(s2+(a1/τ)s+(a0/τ2)) =(cid:34)01(cid:35)(cid:34)q¯1(cid:35)+(cid:34)aτ1(cid:35)(y−q¯ ). (17)
inFig.1canberealizedas5 00 q¯2 τa02 1
       Thisistheveryformofthehigh-gainrobustobserverfor
z˙ S G G z 0
n n n,1 n,2 n x(butnotforz),studiedin[12]andmanyothers.Accord-
      
q˙1=0 0 1 q1+0y, (13) ingto[12],thestateq¯(t)approachesclosetox(t)∈R2of
      
q˙2 0 −τa02 −aτ1 q2 τa02 (3) when τ is sufficiently small (so, the name ‘high-gain’
  follows,asseenin(17)).Thisistrueeventhoughtheob-
z
n
(cid:104) (cid:105)  server has no information about the system (3) (so, it is a
uˆn = −ψgnn, −g1n(φn,1+ τa02), −g1n(φn,2+ aτ1) q1 robust observer), and the residual error between q¯(t) and
q2 x(t) (when t tends to infinity) becomes smaller as τ gets
+ 1 a0y. smaller [12]. Moreover, from the transformation between
gnτ2 q¯andq,weseethat(q(t)−q¯(t)) → 0asτ → 0.Thisis
theunderlyingreasonwhyq(t)canbeusedastheestimate
See [13] for more detailed derivation. It is noted that uˆ
n ofx(t)whenthebandwidthoftheQ-filterislarge(i.e.,τ
canbeequivalentlyrewritteninasimplerwayas
issmall).Onceq(t)isusedinsteadofx(t),werevisit(15)
andseethatitisnothingbut(5)withxreplacedwithitses-
1 (cid:16) (cid:16)a a a (cid:17)(cid:17)
uˆ = −(ψ z +φ q)− 0q + 1q − 0y timate.Verily,thenominalz-subsystem(5)isimplemented
n g n n n τ2 1 τ 2 τ2
n in the part of controller, P−1(s)Q (s). So it can be said
1 n B
= (q˙ −f (q,z )) that the role of P−1(s)Q (s) is to construct the nominal
g 2 n n n B
n z-subsystemandyieldthestateestimateofx.
whereq˙2isjustashorthandof−τa02q1−aτ1q2+τa02y(which 4.2 Estimationofu isperformedbyQ (s).
ismotivatedbythelastrowof(13)).Finally,fromFig.1, desired A
theinputuisgivenby In the previous subsection, it is seen that q(t) → x(t)
approximately.Now,inordertohavethepropertyu(t) →
1
u (t),weexpectfromtheequation(14)that
u=u¯+y −uˆ =u¯+p − (q˙ −f (q,z )). (14) desired
p n 1 g 2 n n
n
1
p (t)→−u¯(t)+ (q˙ (t)−f (q(t),z (t)))+u (t).
Withthis,weclaimtwofindings. 1 g 2 n n desired
n
5 AssumingthatPn(s)isrealizedasin(4)withtheinputuˆnandtheoutputv,theinversePn−1(s)canbewrittenas(becausex1istheoutputandx2
isthederivativeoftheoutput)
1
z˙n=Snzn+Gn,1v, uˆn= (v¨−φn,1v−φn,2v˙−ψnzn).
gn
Now,sinceQB(s)(assumingtheinputisyandtheoutputisq1)canberealizedasq˙1=q2,q˙2=−τa02q1− aτ1q2+ τa02y.Then,Pn−1(s)QB(s)can
bewrittenas(13)sincev=q1,v˙ =q2,andv¨=q˙2.
5
Wsene,ctlhaeimntthhiasti,sifthτeicsassme.allandai andci aresuitablycho- srotaobtlsea.rIets1c/hτartaimcteesritshtiecreoqoutsatoiofns2is+s2a+saτ+1s+gaggnτa=02:wph(oss)e,
1 gn 0 f
To see this, let us analyze the (inner) closed-loop sys- andthus,stabilityof(18)isdeterminedbythepolynomial
tem(12)and(14)(withu¯,q,y,andzn viewedasexternal pf(s). Since g (cid:54)= 0 and the sign of g is known, by letting
inputs). Before going into details, it should be mentioned the nominal value gn have the same sign, the polynomial
that the analysis of the behavior of p1(t), or the system pf(s) is Hurwitz (because a0 > 0 and a1 > 0 from the
(12) and (14) is not very trivial because the p term in- stabilityoftheQ-filter).ThisissimplebecausetheQ-filter
1
sideuof(14)cancelsthe(2,1)-elementofthe2×2sys- (and thus, the polynomial pf(s)) is just of second order.
tem matrix in (12) so that the stability of the system (12) However,ifahigherorderQ-filter,like(10),isemployed,
with(14)isseeminglylost(i.e.,thesystemmatrixof(12) thenpf(s)becomesmorecomplicated(see[15])as
doesnotlooklikebeingstable).Stabilityisinfactnotlost
(cid:18) (cid:19)
because the (negative) feedback effect of p1 is provided p (s)=sµ+ a + g−gnc sµ−1+···
throughthestateq.Toseethis,weemployanothercoordi- f µ−1 gn µ−1
(cid:18) (cid:19) (cid:18) (cid:19)
nanatdepcha:n=gepfro−m(1pq¨1,.pI2n)dienetdo,(ap1t,epd2io)ubsycpa1lc:u=latpi1on−leg1naqd˙2s + a1+ g−gngnc1 s+ a0+ g−gngnc0 . (19)
2 2 gn 2
to6
Then it is not straightforward to ensure that p (s) is Hur-
f
(cid:34)p˙ (cid:35) (cid:34) 0 1 (cid:35)(cid:34)p (cid:35) witz for all variation of g, and for this, the coefficients ai
1 1
= (18) andc shouldbecarefullydesigned.Thisobservationhas
p˙ −g a0 −a1 p i
2 gnτ2 τ 2 been made in [13–15]. Fortunately, a design procedure of
(cid:34) (cid:35)
+ 0 (cid:110)(1− g )(u¯+ fn(q,zn))− 1 (f(x,z)+gd)(cid:111) ai and ci has been developed which makes pf(s) remain
a0 gn gn gn Hurwitz for arbitrarily large variation of g as long as the
τ2
upper and lower bounds of g are known. See [16,17] for
in which, the (negative) (2,1)-element of the system ma- thegeneralcase,but,formakingthispaperself-contained,
trix appears again. Assume, for the time being, that this wenowquotefrom[14,15]howtochoosea whenµ=ν,
i
system is stable and that the input term (the braced c = a , and c = 0 for i = 1,··· ,µ − 1 (this se-
0 0 i
term {···} of (18)) is constant. Then, we can con- lection in fact applies to any plant having relative degree
clude that p2(t) → 0 and p1(t) → (gn/g) × ν). In this case, we have pf(s) = sν + aν−1sν−1 +
(thebracedterm{···}of(18)).Bypluggingthisinto(14) ···+a s+(g/g )a . First, choose a ,··· ,a so that
1 n 0 ν−1 1
(withp1 = p1 + g1nq˙2),onecaneasilyverifythatu(t) → ρ(s):=sν−1+aν−1sν−2+···+a1isHurwitz.Then,find
udesired(t). k¯ >0suchthatsρ(s)+k =sν+aν−1sν−1+···+a1s+k
The issue here is that the input term is not constant isHurwitzforall0<k ≤k¯.Suchk¯alwaysexists.Indeed,
but a time-varying signal. Nevertheless, if p-dynamics is consider the root locus of the transfer function 1/(sρ(s))
much faster than this input signal, then the assumption withthegainparameterk.Sincetherootlocusincludesall
of constant input holds approximately in the relatively points in the complex plane along the real axis to the left
fast time scale, and p1(t) quickly converges to its de- ofanoddnumberofpolesandzeros(fromtheright)ofthe
sired(time-varying)valueapproximately.Thisapproxima- transferfunction,andsince1/(sρ(s))hasnozerosandhas
tionbecomesmoreandmoreaccurateasp-dynamicsgets allpolesintheleft-halfplaneexceptoneattheorigin,the
faster. (See [10, Sec. 9.6] for rigorous treatment of this rootlocusstartingattheoriginmovestotheleftasthegain
statement.) A way to make p-dynamics faster is to take kincreasesalittlefromzero,whiletheothersremaininthe
smallerτ,whichisseenfromthelocationofeigenvalues. openleft-halfplaneforthesmallvariationofkfromzero.
(One may argue that, by taking smaller τ, the evolution Theclosed-loopofthetransferfunction1/(sρ(s))andthe
ofthehigh-gainobserverstateq(t)getsalsofasterthatis gainkhasitscharacteristicequationsρ(s)+k,andthisim-
containedintheinputterm.Whilethisistrue,thestateq(t) pliestheexistenceof(possiblysmall)k¯ > 0.Withsuchk¯
quicklyconvergestotherelativelyslowx(t),andafterthat, athand,nowchoosea = k¯/max{g/g }wherethemax-
0 n
theinputtermbecomesrelativelyslowlyvarying.7) imum is known while g is uncertain. (So, a often tends
0
Finally,letusinspectwhetherthesystem(18)isactually to be small.) For the general case, this idea is repeatedly
6 Duringthecalculation,onemaynotethatp˙2=−aτ1p2+ τa02(u¯+ gfnn − g1n(f+gu+gd))inwhichu=u¯+p1− g1nq˙2+ gfnn =u¯+p1+ gfnn.
7 Theargumenthereisnotveryrigorous,butjustdeliversunderlyingintuition.See[13,14]formorepreciseproofsusingthesingularperturbation
theory[10].
6
applied. nator D (s) of Q (s) with (τs) = s. Note that the poly-
b b
Atlast,wenotethat,ifthevariationofgissmallsothat nomial(22)isnothingbuttheproductofpf(s)in(19)and
g ≈ g , then the term g −g may be almost zero so that theHurwitzpolynomialD1(s).
n n b
p (s)remainsHurwitzforallg sincetheQ-filterisstable Sincetherootsofthecharacteristicpolynomial(20)ap-
f
sothatsµ+a sµ−1+···+a itselfisHurwitz.There- proachtherootsof(21)and1/τ timestherootsof(22)as
µ−1 0
fore, with small uncertainties, the stability issue of p (s) τ → 0, robust stability of the overall feedback system is
f
doesnotstandoutandp (s)isautomaticallyHurwitz. guaranteedif
f
(A) N(s) is Hurwitz (i.e., the z-subsystem of (3) is sta-
ble),
5 Robust Stability of DOB-based Control
(B) D D +N N isHurwitz(i.e.,C(s)internallystabi-
System n c n c
lizesP (s)(notP(s))),
n
(C) p (s)remainsHurwitzforallvariationsofg(=β )
From the discussions so far, we know that, with Hur- f m
witz p (s) and small τ, the high-gain observer subsys- andifthebandwidthofQ-filterissufficientlylarge(i.e.,τ
f
tem (16) or (17) and the subsystem (18) are stable. It issufficientlysmall).Ifanyrootof(21)or(22)appearsin
is however not enough for the stability of the overall theopenright-halfcomplexplane,thentheoverallsystem
closed-loop system, and let us take the outer-loop con- becomesunstablewithlargebandwidthofQ-filter.There-
troller C(s) into account as well. Inspecting robust sta- fore,theaboveconditions(A),(B),and(C)arenecessary
bility of the overall system with DOB is indeed not an and sufficient for robust stability of DOB-based control
easy task in general. To see the extent of difficulty, let systemsundersufficientlylargebandwidthofQ-filters,ex-
us express P(s) = N(s)/D(s), P (s) = N (s)/D (s), cept the case when any root of (21) or (22) has zero real
n n n
Q (s) = N (s)/D (s), Q (s) = N (s)/D (s), and partbecause,inthiscase,itisnotclearinwhichdirection
A a a B b b
C(s) = N (s)/D (s) where all ‘N’ and ‘D’ stand for therootsof(20)approachtherootsof(21)and1/τ times
c c
numeratoranddenominatorcoprimepolynomials,respec- therootsof(22).
tively. Then, the overall system in Fig. 1 is stable if and It is again emphasized that one group of roots have
only if the characteristic polynomial (we omit ‘(s)’ for more and more negative real parts as τ → 0 (when (22)
convenience) is Hurwitz), and this confirms that some part inside the
overall system operates faster than other parts. This ob-
N(N N D +D D N )D +N DD D (D −N ) (20) servation goes along with the previous discussions in the
n c b n c b a n c b a a
state-space (Section 4). Another way to appreciate (21)
is Hurwitz (if there is no unstable pole-zero cancellation and (22) is the following. The polynomial (22) corre-
between P−1(s) and Q (s), or P (s) is of minimum sponds to the dynamics which governs the behavior that
n B n
phase)[15].ItisnotedthatthepolynomialsN(s)andD(s) u(t) → udesired(t) while (21) determines the behavior
are uncertain and so can vary. Hence, QA and QB need whenu(t)=udesired(t).Indeed,whenu(t)=udesired(t),
to be designed such that (20) remains Hurwitz under all the z-subsystem becomes stand-alone and the uncertain
variations of N(s) and D(s). This may sound very chal- terms f and g are replaced with fn and gn so that stabil-
lenging,butwithlargebandwidthofQ-filters,ithasbeen ity of z-subsystem (or N(s)) and stability of the nominal
provedin[15,18]thattherootsof(20)formtwoseparate closed-loop(orDn(s)Dc(s)+Nn(s)Nc(s))arerequired.
groups,whichhelpsdealingwiththischallenge.Onegroup
ofrootsapproaches,asτ →0,therootsof
6 RobustTransientResponse
N(s)(Dn(s)Dc(s)+Nn(s)Nc(s)), (21) For some industrial applications, robust transient re-
sponse(inadditiontorobuststeady-stateresponse)isvery
andtheothergroupofrootsapproaches1/τ timestheroots important. For example, if a controller has been designed
of to satisfy some time-domain specifications such as ris-
D1(s)(D1(s)+γN1(s)) (22) ing time, overshoot, and settling time for a nominal plant
b a a
model, then it is desired that the same transient perfor-
where γ represents (g − g )/g , which can be simply
n n
written as γ = lim P(s)P−1(s) − 1 (recalling that mance is maintained for the real plant under disturbances
s→∞ n
anduncertainties.By‘robusttransientresponse,’wemean
g = β and g = β ). The superscript ‘1’ in (22) im-
m n n,m
pliesthatτ issetto1;forexample,D1(s)isthedenomi- that the output trajectory y(t) of the real plant remains
b
7
closetotheoutputy (t)forallt ≥ 0(i.e.,fromthe See[14,24]formoredetails.
nominal
initial time) under disturbances and uncertainties, where
y (t) is supposed to be the output of the nominal
nominal
closed-loopsystem(withthesameinitialcondition).How
canweachieverobusttransientresponse?Inorderfory(t)
of(3)tobethesameasy (t)forallt ≥ 0,wehave
nominal
tohavey(i)(t)=y(i) (t)fori=0,1,··· ,ν.Thistask
nominal
isachievedifu(t)=u (t)forallt≥0,8 So,thesig-
desired
naly (t)shouldbeunderstoodasthenominaloutput Fig.2 DOBstructurewithsaturationforrobusttransientresponse
nominal
resultedbytheinteractionbetweenC(s)andthenominal
plant(6)–(8).whichisactuallytheactionrequiredforthe
DOB.9 still lacks though. In fact, this is another reason 7 Extensions
why u(t) should converge quickly to u (t). It is in-
desired
deedbecause,ifu(t)convergestou (t)ratherslowly, Theanalysisofthispaperallowsmoreextensionsasfol-
desired
then the time interval for y(ν)(t) (cid:54)≈ y(ν) (t) becomes lows.
nominal
largeandso,evenafteru(t)convergestou (t),two • Nonlinear plant: The analysis of the previous sec-
desired
signals y(t) and y (t) are already different and so tionsalsoappliestosingle-input-single-outputnonlinear
nominal
afterward. plantsaslongastheyhavewell-definedrelativedegree
sothattheplantcanberepresentedasinthenormalform
A remaining problem is that, while making the con-
vergence u(t) → u (t) faster, u(t) may incur very like(3).10 See[14]formoredetails.
desired
large over/undershoot before it gets close to u (t), • MIMOplant:Multi-input-multi-outputplantsalsoadmit
desired
and this large excursion makes robust transient of y(t) theDOB.See[24]fordetails.
much difficult because large difference between y(ν)(t) • Reduced-orderDOB:SincethesameQ-filtersarefound
and y(ν) (t) for short time period may be enough inFig.1,theycouldbemergedintooneinordertore-
nominal
to hamper the property y(t) ≈ y (t) during and ducethedimensionoftheDOB.See[25,26].
nominal
• Exact(notapproximate)rejectionofdisturbances:Nom-
even after the short time period. (The situation is related
inal performance recovery (or disturbance rejection)
to the peaking phenomenon, which has been studied in
studied in this paper is based on the convergence u →
[22], of the high-gain observer that is embedded in the
P−1(s)Q (s) block.) A well-known remedy is to insert udesired. This convergence, however, is inherently ap-
n B
proximate because estimation of u (t) is approx-
a saturation element in the feedback loop [23] in order to desired
preventthelargeexcursionofthecontrolsignalu(t)from imate and the convergence of u(t) to the estimate of
u (t)isalsoapproximate,althoughtheapproxima-
entering the plant. This technique has been taken for the desired
tionbecomesmoreandmoreaccurateasτ getssmaller.
DOBstructurein[14,24]asinFig.2.Bythesaturationel-
However, if the disturbance is generated by a known
ement, the closed-loop system loses ‘global’ stability and
generating model (which is called an exosystem), then
theregionofattractioninthestate-spaceisrestricted.But,
exact rejection of the disturbance, without relying on
bytakingtheinactiverangeofthesaturationelementsuf-
smallnessofτ,ispossible.(Onetypicalexampleisthe
ficientlylarge,onecansecurearbitrarilylargeregionofat-
sinusoidaldisturbancewithknownfrequency.)Thetool
traction(whichisso-called‘semi-global’stabilizationthat
used for this purpose is the well-known internal model
is often enough in practice). In fact, the saturation should
principle[27],andthecontrollerdesignhasbeenstudied
notbecomeactiveduringthesteady-stateoperationofthe
under the name of output regulation in, e.g., [28]. The
control system. In the DOB based control system, it be-
comesactivejustforshorttransientperiodwhenu(t)ex- DOB(morespecifically,theQ-filters)canbemodifiedto
periences unnecessarily large excursion due to small τ. includethegeneratingmodelofthedisturbance,sothat
8 Ifu(t)=udesired(t)fromtheinitialtimet=0,thentheouter-loopcontrollerC(s)feelsasiftheinitialconditionoftheplantis(x(0),zn(0))(not
(x(0),z(0))).Toseethis,referto(6)–(9).
9 Therearetwomoreapproachesintheliteraturetoachieverobusttransientresponse(whiletheunderlyingprinciplethatu(t)→udesired(t)quickly
isthesame).Oneistheuniversalcontrollerof[19]withhigh-gainobserver.ThisismuchsimilartoDOB,butthereisnoinnerloopofQA(s)in
Fig.1(whichyields1/(1−QA(s))).Instead,itsroleisplayedbyalargestaticgain.TheotheroneistheL1adaptivecontrolof[20,21],forwhicha
constructivedesignmethodofthecontroller
10See[9]abouthowtotransformanonlinearsystemwithwell-definedrelativedegreeintothenormalform.
8
the internal model principle holds for the closed-loop 8 Restrictions
system. The initial result in this direction is found in
We have so far looked at the conventional DOB and
[29].Forembeddingthegeneratingmodelofsinusoidal
its extensions, and found that the DOB is a powerful tool
disturbancewithrobuststabilization,referto[16,17,30].
for robust control and disturbance rejection. These bene-
• Different Q-filters: The conventional DOB consists of
fitscameundertherequirementthatthebandwidthofthe
two same Q-filters, while their roles are inherently dif-
Q-filterissufficientlylarge,andthisinturnimposedafew
ferentasdiscoveredinSection4.Suchobservationhas
restrictionsasfollows.
triggeredsubsequentworks[18,30,54]inwhichtheQ-
• Unmodelleddynamics:Ifthereisunmodelleddynamics
filtersarerefinedseparatelyforcertainpurposes.Forin-
in the real plant, then the relative degree ν of the plant
stance,sincetheestimationofu (t)ismainlyper-
desired
P(s) may be different from that of the nominal model
formed by Q (s), it is sufficient to embed the gener-
A
P (s).Inthiscase,ithasbeenactuallyreportedin[34]
atingmodelofdisturbancejustintoQ (s)(butnotinto n
A
thatlargebandwidthmayleadtoinstabilityoftheover-
Q (s))fortheexactdisturbancerejection[30].Another
B
allsystem,eventhougharemedyforafewcasesisalso
example is to use higher order of Q (s) (while the or-
B
suggestedin[34].Seealso[35].
derofQ (s)beingkeptthesame)inordertohavemore
A
• Non-minimum phase plant: If the real plant is of non-
reductionoftheeffectofmeasurementnoise[54].
minimum phase (i.e., the zero dynamics is unstable),
• Use of state feedback controller C: Since the estimate
thenlargebandwidthofQ-filtersmakestheoverallsys-
oftheplant’sstatexisprovidedbytheDOB,theouter-
temunstable,asdiscussedbefore.Toovercomethisre-
loopcontrollerCcanbeofstatefeedbacktypeifthefull
striction, another structure with different Q-filters has
state is considered as (x,z ) where z is also provided
n n
been proposed in [36], but the problem is still open in
bytheDOB.See[55]forthiscombination.
general.
• Input saturation: In practice, it is natural that the con-
• Signofhighfrequencygain:Wehaveseenthatthesign
trol input u is limited. A preliminary result on DOB-
ofthe(uncertain)high-frequencygaing(orβ )should
basedcontrollerunderinputsaturationhasbeenreported m
be known. If it is not the case, then, as a workaround,
in[56],wheretheauthorspresentedanLMItofindthe
the Nussbaum gain technique [37] has been employed
control gain for a state feedback controller and the pa-
in[38],whilemorestudyiscalledforinthisdirection.
rametersofDOBatthesametime.
• Measurement noise: Large bandwidth of Q-filters may
• Discrete-time implementation of DOB: While all the
alsoyieldinsufficientreductionofnoiseathighfrequen-
above results are discussed in the continuous-time do-
cieswhilewereferto[39,40,54]forapossiblemodifica-
main, for implementing the DOB in the digital de-
tionoftheDOBstructuretoenhancethenoisereduction
vices, DOB is constructed in the discrete-time domain.
capability.
At first glance, stability seems to remain guaranteed if
the continuous-time DOB is discretized by fast sam- We emphasize that, if the bandwidth of Q-filter is
pling. However, fast sampling (with zero-order hold) severely limited by some reason, then the desired steady-
ofacontinuous-timesystemintroducesadditionalzeros state/transient performance and the robust stabilization
(which are called sampling zeros [31,32]), and worse may not be obtained simply because the analysis so far
yet, at least one of them is always unstable when the is no longer valid. Fortunately, an appropriate bandwidth
relative degree ν ≥ 3. This causes another trouble that of Q-filter for robust stabilization and disturbance rejec-
the sampled-data model of the plant becomes a non- tion,variesfromsystemtosystem,anditturnsoutthatthe
minimumphasesystem,whichseemsviolatingthenec- bandwidthofQ-filterneednotbetoolargeinmanyprac-
essary condition for stability as discussed earlier. In ticalcases.Forinstance,areasonablechoiceofbandwidth
[33],ithasbeenfoundthattheQ-filtersinthediscrete- workedintheexperimentsof[41,42].Theanalysisinthe
time domain can be specially designed to take care of related papers yields an upper bound for τ that works, in
unstable sampling zeros as well while maintaining all theory.But,sinceitisoftentooconservative,findingsuit-
goodpropertiesinthecontinuous-timedomain. ableτ isdoneusuallybyrepeatedsimulationsorbytuning
itinexperiments.
On the other hand, there are other approaches that do
not explicitly rely on large bandwidth of Q-filter for ro-
bust stability [3,43–50]. Among them, the most popular
9
one is to employ the tool ‘small-gain theorem’ [43–45]. [7] Z.Gao,“Activedisturbancerejectioncontrol:Fromanenduringidea
However, it gives a sufficient condition for stability and toanemergingtechnology,”inProc.ofthe10thInt.Workshopon
so may yield conservatism. More specifically, for an un- RobotMotionandControl,2015.
certain plant with multiplicative uncertainty (i.e., P(s) = [8] K.Ohishi,K.Ohnishi,andK.Miyachi,“Torque-speedregulationof
DCmotorbasedonloadtorqueestimationmethod,”inProc.ofJIEE
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(cid:107)∆(cid:107) <(cid:107)(1+P C)/(Q+P C)(cid:107) forallω[45].
s=jω n n s=jω [10]H.K.Khalil,NonlinearSystems.PrenticeHall,3rdEd.,2002.
Then,thesizeofuncertaintyisseverelylimited(consider-
[11]K. Kong and M. Tomizuka, “Nominal model manipulation for
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Anotheroneisworkingwiththestate(notoutput)mea- controlsystems,”Int.J.ofControl,AutomationandSystems,vol.11,
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thatthedisturbanceisanoutputofageneratingmodel(the [12]H. K. Khalil and L. Praly, “High-gain observers in nonlinear
feedbackcontrol,”Int.J.ofRobustandNonlinearControl,vol.24,
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no.6,pp.993–1015,2014.
disturbancebyestimatingtheexosystem’sstate,areoften
[13]H.ShimandY.Joo,“Statespaceanalysisofdisturbanceobserver
proposed and combined with many well-established con-
andarobuststabilitycondition,”inProc.ofConf.onDecisionand
trollers such as sliding mode control or model predictive
Control,pp.2193–2198,2007.
control.See,forexample,[5,6]formoredetails.
[14]J. Back and H. Shim, “Adding robustness to nominal output-
Inspiteoftheserestrictions,studyofDOBunderlarge feedbackcontrollersforuncertainnonlinearsystems:Anonlinear
bandwidthofQ-filtersisworthwhilesinceitillustratesthe version of disturbance observer,” Automatica, vol. 44, no. 10,
pp.2528–2537,2008.
roleofeachcomponentofDOB,andyieldsusefulinsights
[15]H.ShimandN.H.Jo,“Analmostnecessaryandsufficientcondition
forfurtherstudyofDOB.Italsoshowsidealperformance
for robust stability of closed-loop systems with disturbance
thatcanbeachievedunderarbitrarilylargeparametricun-
observer,”Automatica,vol.45,no.1,pp.296–299,2009.
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Finally,weclosethistutorialwithadisclaimerthatthe stability,”inProc.ofConf.onDecisionandControl,pp.949–954,
purposeofthistutorialisnottosurveyexhaustivelistofre- 2012.
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in disturbance observer with robust stability.” to appear in IEEE
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Trans.onAutomaticControl.
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rial is written in less formal manner, and more rigorous,
based on generalized Q-filter design scheme.” in Proc. of
theorem-proofstyle,argumentscanbefoundintherefer- Int. Conf. of Control, Automation and Systems, pp. 1211–1216,
encescitedinthetext. 2014.
[19]L. B. Freidovich and H. K. Khalil, “Performance recovery of
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Appendix
IndustrialElectronics,vol.57,no.9,pp.3197–3206,2010.
The conversion from (1) to (3) is done by the follow-
[43]C. J. Kempf and S. Kobayashi, “Disturbance observer and
feedforwarddesignforahigh-speeddirect-drivepositioningtable,” ing procedure. First, it is noted that cAi−1b = 0 for