arxiv:0904.0322v1 [math.oc] 2 apr 2009tro inducing a mo del-free trol con and with restricted mo del...
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MODEL-FREE CONTROL AND INTELLIGENT
PID CONTROLLERS: TOWARDS A POSSIBLE
TRIVIALIZATION OF NONLINEAR
CONTROL?
Mi hel FLIESS
∗,∗∗Cédri JOIN
∗,∗∗∗
∗INRIA-ALIEN
∗∗LIX (CNRS, UMR 7161), É ole polyte hnique
91128 Palaiseau, Fran e
Mi hel.Fliess�polyte hnique.edu
∗∗∗CRAN (CNRS, UMR 7039), Nan y-Université
BP 239, 54506 Vand÷uvre-lès-Nan y, Fran e
Cedri .Join� ran.uhp-nan y.fr
Abstra t. We are introdu ing a model-free ontrol and a ontrol with a restri ted
model for �nite-dimensional omplex systems. This ontrol design may be viewed
as a ontribution to �intelligent� PID ontrollers, the tuning of whi h be omes
quite straightforward, even with highly nonlinear and/or time-varying systems.
Our main tool is a newly developed numeri al di�erentiation. Di�erential algebra
provides the theoreti al framework. Our approa h is validated by several numeri al
experiments.
1. INTRODUCTION
Writing down simple and reliable di�erential
equations for des ribing a on rete plant is al-
most always a daunting task. How to take into
a ount, for instan e, fri tions, heat e�e ts, ageing
pro esses, hara teristi s dispersions due to mass
produ tion, . . . ? Those severe di� ulties explain
to a large extent why the industrial world is
not willing to employ most te hniques stemming
from �modern� ontrol theory, whi h are too often
based on a �pre ise� mathemati al modeling, in
spite of onsiderable advan es during the last �fty
years. We try here
1to over ome this unfortunate
1This ommuni ation is a slightly modi�ed and updated
version of (Fliess & Join [2008a℄), whi h is written in
Fren h. Model-free ontrol and ontrol with a restri ted
model, whi h might be useful for hybrid systems (Bourdais,
Fliess, Join & Perruquetti [2007℄), have already been
applied in several on rete ase-studies in various domains
(Choi, d'Andréa-Novel, Fliess & Mounier [2009℄, Gédouin,
Join, Delaleau, Bourgeot, Chirani & Callo h [2008℄, Join,
situation thanks to re ent fast estimation meth-
ods.
2Two ases are examined:
(1) Model-free ontrol is based on an elementary
ontinuously updated lo al modeling via the
unique knowledge of the input-output behav-
ior. It should not be onfused with the usual
�bla k box� identi� ation (see, e.g., (Ker-
s hen, Worden, Vakakis & Golinval [2006℄,
Sjöberg, Zhang, Ljung, Benveniste, Delyon,
Glorenne , Hjalmarsson & Juditsky [1995℄)),
where one is looking for a model whi h is
valid within an operating range whi h should
be as large as possible.
3Let us summarize
our approa h in the monovariable ase. The
Masse & Fliess [2008℄, Villagra, d'Andréa-Novel, Fliess &
Mounier [2008a,b℄). Other appli ations on an industrial
level are being developed.
2See, e.g., (Fliess, Join & Sira-Ramírez [2008℄, Mboup,
Join & Fliess [2009℄) and the referen es therein.
3This is why we use the terminology �model-free� and not
�bla k box�.
input-output behavior of the system is as-
sumed to �approximatively� governed within
its operating range by an unknown �nite-
dimensional ordinary di�erential equation,
whi h is not ne essarily linear,
E(y, y, . . . , y(a), u, u, . . . , u(b)) = 0 (1)
We repla e Eq. (1) by the following �pheno-
menologi al� model, whi h is only valid dur-
ing a very short time interval,
y(ν) = F + αu (2)
The derivation order ν, whi h is in general
equal to 1 or 2, and the onstant parameter αare hosen by the pra titioner. It implies that
ν is not ne essarily equal to the derivation
order a of y in Eq. (1). The numeri al value
of F at any time instant is dedu ed from
those of u and y(ν), thanks to our numeri al
di�erentiators (Fliess, Join & Sira-Ramírez
[2008℄, Mboup, Join & Fliess [2009℄). The de-
sired behavior is obtained by implementing,
if, for instan e, ν = 2, the intelligent PID
ontroller
4(i-PID)
u = −F
α+
y∗
α+KP e+KI
∫
e+KDe (3)
where
• y∗ is the output referen e traje tory,
whi h is determined via the rules of
�atness-based ontrol (see, e.g., Fliess,
Lévine, Martin & Rou hon [1995℄, Rotella
& Zambettakis [2007℄, Sira-Ramírez &
Agrawal [2004℄);
• e = y − y∗ is the tra king error;
• KP , KI , KD are the usual tuning gains.
(2) Assume now that a restri ted or partial
model of the plant is quite well known and
is de�ned by Eq. (1) for instan e. The plant
is then governed by the restri ted, or in om-
plete, modeling,
5
E(y, y, . . . , y(a), u, u, . . . , u(b)) +G = 0
where G stands for all the unknown parts.
6
If the known system, whi h orresponds to
E = 0, is �at, we also easily derive an
intelligent ontroller, or i- ontroller, whi h
gets rid of the unknown e�e ts.
Di�erential algebra is brie�y reviewed in Se t. 2
in order to
• derive the input-output di�erential equa-
tions,
4This terminology, but with other meanings, is not new in
the literature (see, e.g., Åström, Persson & Hang [1992℄).
5Sin e we are not employing the terminology �bla k box�,
we are also here not employing the terminology �grey box�.
6Any mathemati al modeling in physi s as well as in
engineering is in omplete. Here the term G does not need
to be �small� as in the lassi approa hes.
• de�ne minimum and non-minimum phase
systems.
Numeri al di�erentiation of noisy signals is exam-
ined in Se t. 3. Se t. 4 states the basi prin i-
ples of our model-free ontrol. Several numeri al
experiments
7are reported in Se t. 5. We deal
as well with linear
8and nonlinear systems and
with monovariable and multivariable systems. An
anti-windup strategy is sket hed in Se t. 5.4.2.
Se t. 6 studies the ontrol with a partially known
medeling. One of the two examples deals with
a non-minimum phase system.
9The numeri al
simulations of Se t. 5.1 and Se t. 6.2 show the that
our ontrollers behave mu h better than lassi
PIDs.
10Several on luding remarks are dis ussed
in Se t. 7.
Remark 1. Only Se t. 2 is written in an abstra t
algebrai language. In order to understand the
sequel and, in parti ular, the basi prin iples of
our ontrol strategy, it is only required to admit
the input-output representations (1) and (4), as
well as the foundations of �atness-based ontrol
(Fliess, Lévine, Martin & Rou hon [1995℄, Rotella
& Zambettakis [2007℄, Sira-Ramírez & Agrawal
[2004℄).
2. NONLINEAR SYSTEMS
2.1 Di�erential �elds
All the �elds onsidered here are ommutative and
have hara teristi 0. A di�erential �eld
11K is a
�eld whi h is equipped with a derivation
ddt, i.e.,
a mapping K → K su h that, ∀ a, b ∈ K,
• ddt(a+ b) = a+ b,
7Those omputer simulations would of ourse be impos-
sible without pre ise mathemati al models, whi h are a
priori known.
8The remark 9 underlines the following ru ial fa t: the
usual mathemati al riteria of robust ontrol are be oming
pointless in this new setting.
9Non-minimum phase systems are today beyond our
rea h in the ase of model-free ontrol. This is ertainly
the most important theoreti al question whi h is left open
here.
10We are perfe tly aware that su h a omparison might be
obje ted. One ould always argue that an existing ontrol
synthesis in the huge literature devoted to PIDs sin e
Ziegler & Ni hols [1942℄ (see, e.g., Åström & Hägglund
[2006℄, Åström & Murray [2008℄, Besançon-Voda & Gentil
[1999℄, Dattaet, Ho & Bhatta haryya [2000℄, Dindeleux
[1981℄, Franklin, Powell & Emami-Naeini [2002℄, John-
son & Moradi [2005℄, Lequesne [2006℄, O'Dwyer [2006℄,
Rotella & Zambettakis [2008℄, Shinskey [1996℄, Visioli
[2006℄, Wang, Ye, Cai & Hang [2008℄, Yu [1999℄) has been
ignored or poorly understood. Only time and the work of
many pra titioners will be able to on�rm our viewpoint.
11See, e.g., (Chambert-Loir [2005℄, Kol hin [1973℄) for
more details and, in parti ular, (Chambert-Loir [2005℄) for
basi properties of usual �elds, i.e., non-di�erential ones.
• ddt(ab) = ab+ ab.
A onstant c ∈ K is an element su h that c = 0.The set of all onstant elements is the sub�eld of
onstants.
A di�erential �eld extension L/K is de�ned by two
di�erential �elds K, L su h that:
• K ⊆ L,
• the derivation of K is the restri tion to K of
the derivation of L.
Write K〈S〉, S ⊂ L, the di�erential sub�eld of L
generated by K and S. Assume that L/K is �nitely
generated, i.e., L = K〈S〉, where S est �nite. An
element ξ ∈ L is said to be di�erentially algebrai
over K if, and only if, it satis�es an algebrai
di�erential equation P (ξ, . . . , ξ(n)) = 0, where Pis a polynomial fun tion over K in n+1 variables.The extension L/K is said to be di�erentially
algebrai if, and only if, any element of L is
di�erentially algebrai over K. The next result is
important:
L/K is di�erentially algebrai if, and only if, its
trans enden e degree is �nite.
An element of L, whi h is non-di�erentially al-
gebrai over K, is said to di�erentially trans en-
dental over K. An extension L/K, whi h is non-
di�erentially algebrai , is said to be di�erentially
trans endental. A set {ξι ∈ L | ι ∈ I} is said to be
di�erentially algebrai ally independent over K if,
and only if, there does not exists any non-trivial
di�erential relation over K:
Q(. . . , ξ(νι)ι , . . . ) = 0, where Q is a polynomial
fun tion over K, implies Q ≡ 0.Two su h sets, whi h are maximal with respe t
to set in lusion, have the same ardinality, i.e.,
they have the same number of elements: this is the
di�erential trans enden e degree of the extension
L/K. Su h a set is a di�erential trans enden e
basis. It should be lear that L/K is di�erentially
algebrai if, and only if, its di�erential trans en-
den e degree is 0.
2.2 Nonlinear systems
2.2.1. General de�nitions Let k be a given dif-
ferential ground �eld. A system
12is a �nitely
generated di�erentially trans endental extension
K/k. Let m be its di�erential trans enden e de-
gree. A set of (independent) ontrol variables u =(u1, . . . , um) is a di�erential trans enden e ba-
sis of K/k. The extension K/k〈u〉 is therefore
di�erentially algebrai . A set of output variables
y = (y1, . . . , yp) is a subset of K.
12See also (Delaleau [2002, 2008℄, Fliess, Join & Sira-
Ramírez [2008℄, Fliess, Lévine, Martin & Rou hon [1995℄)
whi h provide more referen es on the use of di�erential
algebra in ontrol theory.
Let n be the trans enden e degree of K/k〈u〉 andlet x = (x1, . . . , xn) be a trans enden e basis
of this extension. It yields the generalized state
representation:
Aι(xι,x,u, . . . ,u(α)) = 0
Bκ(yκ,x,u, . . . ,u(β)) = 0
where Aι, ι = 1, . . . , n, Bκ, κ = 1, . . . , p, are
polynomial fun tions over k.
The following input-output representation is a
onsequen e from the fa t that y1, . . . , yp are
di�erentially algebrai over k〈u〉:
Φj(y, . . . ,y(Nj),u, . . . ,u(Mj)) = 0 (4)
where Φj , j = 1, . . . , p, is a polynomial fun tion
over k.
2.2.2. Input-output invertibility
• The system is said to be left invertible if,
and only if, the extension
13 k〈u,y〉/k〈y〉is di�erentially algebrai . It means that one
an ompute the input variables from the
output variables via di�erential equations.
Then m ≤ p.• It is said to be right invertible if, and only
if, the di�erential trans enden e degree of
k〈y〉/k is equal to p. This is equivalent sayingthat the output variables are di�erentially
algebrai ally independent over k. Then p ≤m.
The system is said to be square if, and only if,
m = p. Then left and right invertibilities oin ide.
If those properties hold true, the system is said to
be invertible.
2.2.3. Minimum and non-minimum phase sys-
tems The ground �eld k is now the �eld R of real
numbers. Assume that our system is left invert-
ible. The stable or unstable behavior of Eq. (4),
when onsidered as a system of di�erential equa-
tions in the unknowns u (y is given) yields the
de�nition of minimum, or non-minimum, phase
systems ( ompare with Isidori [1999℄).
3. NUMERICAL DIFFERENTIATION
The interested reader will �nd more details and
referen es in (Fliess, Join & Sira-Ramírez [2008℄).
We refer to (Mboup, Join & Fliess [2009℄) for
ru ial developments whi h play an important r�le
in pra ti al implementions.
13This extension k〈u,y〉/k〈y〉 is alled the residual dy-
nami s, or the zero dynami s ( ompare with Isidori [1999℄).
3.1 General prin iples
3.1.1. Polynomial signals Consider the polyno-
mial time fun tion of degree N
xN (t) =
N∑
ν=0
x(ν)(0)tν
ν!
where t ≥ 0. Its operational, or Lapla e, transform(see, e.g., Yosida [1984℄) is
XN (s) =
N∑
ν=0
x(ν)(0)
sν+1(5)
Introdu e
dds, whi h is sometimes alled the alge-
brai derivation. Multiply both sides of Eq. (5) by
dα
dsαsN+1
, α = 0, 1, . . . , N . The quantities x(ν)(0),ν = 0, 1, . . . , N , whi h satisfy a triangular sys-
tem of linear equations, with non-zero diagonal
elements,
dαsN+1XN
dsα=
dα
dsα
(
N∑
ν=0
x(ν)(0)sN−ν
)
(6)
are said to be linearly identi�able (Fliess & Sira-
Ramírez [2003℄). One gets rid of the time deriva-
tives sµ dιXN
dsι, µ = 1, . . . , N , 0 ≤ ι ≤ N , by
multiplying both sides of Eq. (6) by s−N, N > N .
Remark 2. The orresponden e between
dα
dsαand
the produ t by (−t)α (see, e.g., Yosida [1984℄)
permits to go ba k to the time domain.
3.1.2. Analyti signals A time signal is said to
be analyti if, and only if, its Taylor expansion is
onvergent. Trun ating this expansion permits to
apply the previous al ulations.
3.2 Noises
Noises are viewed here as qui k �u tuations
around 0. They are therefore attenuated by low-
pass �lters, like iterated integrals with respe t to
time.
14
4. MODEL-FREE CONTROL: GENERAL
PRINCIPLES
It is impossible of ourse to give here a omplete
des ription whi h would trivialize for pra titioners
the implementation of our ontrol design. We hope
that numerous on rete appli ations will make
su h an endeavor feasible in a near future.
14See (Fliess [2006℄) for a pre ise mathemati al theory,
whi h is based on nonstandard analysis.
4.1 Lo al modeling
(1) Assume that the system is left invertible. If
there are more output variables than input
variable, i.e., p m, pi k up m output
variables, say the �rst m ones, in order to
get an invertible square system. Eq. (2) may
be extended by writing
y(n1)1 = F1 + α1,1u1 + · · ·+ α1,mum
. . .
y(np)p = Fm + αm,1u1 + · · ·+ αp,mum
(7)
where
• nj ≥ 1, j = 1, . . . , p, and, most often,
nj = 1 or 2;• αj,i ∈ R, i = 1, . . . ,m, j = 1, . . . , p, arenon-physi al onstant parameters, whi h
are hosen by the pratitioner su h that
αj,iui and Fj are of the same magnitude.
(2) In order to avoid any algebrai loop, the
numeri al value of
Fj = y(nj)j − αj,1u1 − · · · − αj,mum
is given thanks to the time sampling
Fj(κ) = [y(nj)j (κ)]e −
m∑
i=1
αj,iui(κ− 1)
where [•(κ)]e stands for the estimate at the
time instant κ.(3) The determination of the referen e traje to-
ries for the output variables yj is a hieved in
the same way as in �atness-based ontrol.
Remark 3. It should also be pointed out that, in
order to avoid algebrai loops, it is ne essary that
in Eq. (4)
∂Φj
∂y(nj)j
6≡ 0, j = 1, . . . , p
It yields
nj ≤ Nj
Numeri al instabilities might appear when
∂Φj
∂y(nj)
j
is losed to 0. 15
Remark 4. Our ontrol design lead with non-
minimum phase systems to divergent numeri al
values for the ontrol variables uj and therefore
to the inappli ability of our te hniques.
Remark 5. Remember that the Equations (1) and
(4) are unknown. Verifying therefore the proper-
ties dis ussed in the two previous remarks may
only be a hieved experimentally within the plant
operating range.
15This kind of di� ulties has not yet been en ountered
whether in the numeri al simulations presented below nor
in the quite on rete appli ations whi h were studied until
now.
4.2 Controllers
Let us restri t ourselves for the sake of notations
simpli ity to monovariable systems.
16If ν = 2 in
Eq. (2), the intelligent PID ontroller has already
been de�ned by Eq. (3). If ν = 1 in Eq. (2), repla eEq. (3) by the intelligent PI ontroller, or i-PI,
u = −F
α+
y∗
α+KP e+KI
∫
e (8)
Remark 6. Until now we were never obliged to
hose ν � 2 in Eq. (2). The previous ontrollers
(3) and (8) might then be easily extended to to
the generalized proportional integral ontrollers,
or GPIs, of (Fliess, Marquez, Delaleau & Sira-
Ramírez [2002℄).
Remark 7. In order to improve the performan es
it might be judi ious to repla e in Eq. (3) or in
Eq. (8) the unique integral term KI
∫
e by a �nite
sum of iterated integrals
KI1
∫
e+KI2
∫ ∫
e+ · · ·+KIΛ
∫
. . .
∫
e
where
•∫
. . .∫
e stands for the iterated integral of
order Λ,• the KIλ , λ = 1, . . . ,Λ, are gains.
We get if KIΛ 6= 0 an intelligent PI
ΛD or PI
Λ
ontroller. Note also that setting Λ = 0 is a
mathemati al possibility. The la k of any integral
term is nevertheless not re ommended from a
pra ti al viewpoint.
Let us brie�y ompare our intelligent PID on-
trollers to lassi PID ontrollers:
• We do not need any identi� ation pro edure
sin e the whole stru tural information is on-
tained in the term F of Eq. (2), whi h is
eliminated thanks to Eq. (3).
• The referen e traje tories, whi h are hosen
thanks to �atness-based methods, is mu h
more �exible than the traje tories whi h are
usually utilized in the industry. Overshoots
and undershoots are therefore avoided to a
large extent.
5. SOME EXAMPLES OF MODEL-FREE
CONTROL
A zero-mean Gaussian white noise of varian e 0.01is added to all the omputer simulations in order
to test the robustness property of our ontrol
design. We utilize a standard low-pass �lter with
16The extension to the multivariable ase is immediate.
See Se t. 5.3.
lassi ontrollers and the prin iples of Se t. 3.2
with our intelligent ontrollers.
5.1 A stable monovariable linear system
The transfer fun tion
(s+ 2)2
(s+ 1)3(9)
de�nes a stable monovariable linear system.
5.1.1. A lassi PID ontroller We apply the
well known method due to Broïda (see, e.g.,
Dindeleux [1981℄) by approximating the system
(9) via the following delay system
Ke−τs
(Ts+ 1)
K = 4, T = 2.018, τ = 0.2424 are obtained thanksto graphi al te hniques. The gain of the PID
ontroller are then dedu ed (Dindeleux [1981℄):
KP = 100(0.4τ+T )120Kτ
= 1.8181, KI = 11.33Kτ
=
0.7754, KD = 0.35TK
= 0.1766.
5.1.2. i-PI. We are employing y = F + u and
the i-PI ontroller
u = −[F ]e + y⋆ + PI(e)
where
• [F ]e = [y]e − u,• y⋆ is a referen e traje tory,
• e = y − y⋆,• PI(e) is an usual PI ontroller.
5.1.3. Numeri al simulations Fig. 1 shows that
the i-PI ontroller behaves only slightly better
than the lassi PID ontroller. When taking into
a ount on the other hand the ageing pro ess and
some fault a ommodation there is a dramati
hange of situation:
• Fig. 2 indi ates a lear ut superiority of our
i-PI ontroller if the ageing pro ess orre-
sponds to a shift of the pole from 1 to 1.5,and if the previous graphi al identi� ation is
not repeated.
• The same on lusion holds, as seen Fig. 3, if
there is a 50% power loss of the ontrol.
Remark 8. This example shows that it might use-
less to introdu e delay systems of the type
T (s)e−Ls, T ∈ R(s), L ≥ 0 (10)
for tuning lassi PID ontrollers, as often done
today in spite of the quite involved identi� ation
pro edure. It might be reminded that
• the stru ture and the ontrol of systems
of type (10) have been studied in (Fliess,
Marquez & Mounier [2002℄),
• their identi� ation with te hniques stemming
also from (Fliess & Sira-Ramírez [2003℄) has
been studied in (Belkoura, Ri hard & Fliess
[2009℄, Ollivier, Moutaouakil & Sadik [2007℄,
Rudolph & Woittennek [2007℄).
Remark 9. This example demonstrates also that
the usual mathemati al riteria for robust ontrol
be ome to a large irrelevant. Let us however
point out that our ontrol leads always to a pure
integrator of order 1 or 2, for whi h the lassi
frequen y te hniques (see, e.g., Åström & Murray
[2008℄, Franklin, Powell & Emami-Naeini [2002℄,
Rotella & Zambettakis [2008℄) might still be of
some interest.
Remark 10. As also shown by this example some
fault a ommodation may also be a hived without
having re ourse to a general theory of diagnosis.
5.2 A monovariable linear system with a large
spe trum
With the system de�ned by the transfer fun tion
s5
(s+ 1)(s+ 0.1)(s+ 0.01)(s− 0.05)(s− 0.5)(s− 5)
We utilize y = F+u. A i-PI ontroller provides the
stabilization around a referen e traje tory. Fig. 4
exhibits an ex ellent tra king.
5.3 A multivariable linear system
Introdu e the transfer matrix
(
s3
(s+0.01)(s+0.1)(s−1)s0
s+1(s+0.003)(s−0.03)(s+0.3)(s+3)
s2
(s+0.004)(s+0.04)(s−0.4)(s+4)
)
For the orresponding system we utilize after a
few attempts Eq. (7) with the following de oupled
form
y1 = F1 + 10u1 y2 = F2 + 10u2
The stabilization around a referen e traje tory
(y∗1 , y∗
2) is ensured by the multivariable i-PID
ontroller
u1 = 110
(
y∗1 − F1 +KP1e1 +KI1
∫
e1 +KD1e1)
u2 = 110
(
y∗2 − F2 +KP2e2 +KI2
∫
e2 +KD2e2)
where
• e1 = y∗1 − y1, e2 = y∗2 − y2;• KP1 = 1, KI1 = KD1 = 0, KP2 = KI2 = 50,KD2 = 10.
The performan es displayed on Fig. 5 and 6 are
ex ellent. Fig. 6-(b) shows the result if we would
set F1 = F2 = 0: it should be ompared with Fig.
6-(a).
Remark 11. Model redu tion is often utilized for
the kind of systems studied in Se t. 5.2 and
Se t. 5.3 (see, e.g., Antoulas [2005℄, Obinata &
Anderson [2001℄).
5.4 An unstable monovariable nonlinear system
5.4.1. i-PID For y − y = u3we utilize for Eq.
(2) the lo al model y = F + u. The stabilization
around a referen e traje tory y∗ is provided by
the i-PI ontroller
u = −F + y⋆ +KP e+KI
∫
e (11)
where KP = −2, KI = −1. The simulations
displayed in Fig. 7 are ex ellent.
5.4.2. Anti-windup We now assume that ushould satisfy the following onstraints −2 ≤ u ≤0.4. The performan es displayed by Fig. 8 are
medio re if an anti-windup is not added to the
lassi part of the i-PI ontroller. Our solution
is elementary
17: as soon as the ontrol variable
gets saturated, the integral
∫
e in Eq. (11) is
maintained onstant.
18
5.5 Ball and beam
Fig. 10 displays the famous ball and beam exam-
ple, whi h obeys to the equation
19 y = Byu2 −BG sinu, where u = θ is the ontrol variable. Thismonovariable system, whi h is not linearizable by
a stati state feedba k, is therefore not �at. It is
thus di� ult to handle.
20
We have hosen for Eq. (2) y = F + 100u. Inorder to satisfy as well as possible the experimen-
tal onditions, the ontrol variable is saturated:
−π/3 < u < π/3 and −π < u < π. Fig. 11 and
Fig. 12 display two types of traje tories: a Bézier
polynomial and a sine fun tion. We obtain in both
ases ex ellent tra kings thanks to an i-PID on-
troller. In the Figures 11-(b), 12-(b), 11-( ), 12-
( ) the ontrol variable and the estimations of F
17This is a well overed subje t in the literature (see, e.g.,
Bohn & Atherton [1995℄, Hippe [2006℄, Peng, Vran i &
Hanus [1996℄.
18Better performan es would be easily rea hed, as �atness-
based ontrol is tea hing us, with a modi�ed referen e
traje tory.
19The sine fun tion whi h appears in that equation takes
us outside of the theory sket hed in Se t. 2. This di� ulty
may be easily ir umvented by utilizing tg
u
2(see Fliess,
Lévine, Martin & Rou hon [1995℄).
20A quite large literature has been devoted to this example
(see, e.g., (Fantoni & Lozano [2002℄, Hauser, Sastry &
Kokotovi [1992℄, Sastry [1999℄) for advan ed nonlinear
te hniques, and (Zhang, Jiang & Wang [2002℄) for neural
networks). Let us add that all numeri al simulations in
those referen es are given without any orrupting noise.
are presented in the noiseless ase. Compare with
the Figures 11-(d) and 12-(d)) where a orrupting
noise is added.
5.6 The three tank example
The three tank example in Fig. 13 is quite popular
in diagnosis.
21It obeys to the equations:
x1 = −C1sign(x1 − x3)√
|x1 − x3|+ u1/S
x2 = C3sign(x3 − x2)√
|x3 − x2|−C2sign(x2)
√
|x2|+ u2/S
x3 = C1sign(x1 − x3)√
|x1 − x3|−C3sign(x3 − x2)
√
|x3 − x2|y1 = x1
y2 = x2
y3 = x3
where
Cn = (1/S).µn.Sp
√2g, n = 1, 2, 3;
S = 0.0154 m (tank se tion);Sp = 5.10−5 m (pipe se tion between the tanks);g = 9.81 m.s−2
(gravity);µ1 = µ3 = 0.5, µ2 = 0.675 (vis osity oe� ients).
As often in industry we utilize a zero-hold ontrol
(see Fig. 14- ( )). A de oupled Eq. (7) is employed
here as we already did in Se t. 5.3: yi = Fi +200ui, i = 1, 2. Fig. 14-(a) displays the traje toriestra king. The derivatives estimation in Fig. 14-
(b) is ex ellent in spite of the additive orrupting
noise. The nominal ontrols (Fig. 14-( )) are not
very far from those we would have omputed with
a �atness-based viewpoint (see Fliess, Join & Sira-
Ramírez [2005℄). We also utilize the following i-PI
ontrollers
ui =1
200
(
y∗i− Fi + 10ei + 2.10−2
∫
ei
)
i = 1, 2
where y∗i is the referen e traje tory, ei = y∗i −yi. Inorder to get a good estimate of ei we are denoisingyi (see Fig. 14-(d)) a ording to the te hniques of
Se t. 3.
6. CONTROL WITH A RESTRICTED
MODEL
6.1 General prin iples
6.1.1. Flatness The system (4) is assumed to
be square, i.e., m = p, and �at. Moreover y =
21See (Fliess, Join & Sira-Ramírez [2005℄) for details and
referen es. This paper was presenting apparently for the
�rst time the diagnosis, the ontrol and the fault a om-
modation of a nonlinear system with un ertain parameters.
See also Fliess, Join & Sira-Ramírez [2008℄.
(y1, . . . , ym) is assumed to be a �at output, �.e.,
Mj = 0, j = 1, . . . ,m. It yields lo ally
uj = Ψj(y, . . . ,y(Nj)), j = 1, . . . ,m (12)
Flatness-based ontrol permits to sele t easily an
e� ient referen e traje tory y⋆to whi h orre-
sponds via Eq. (4) and Eq. (12) the open loop
ontrol u⋆. Let e = y − y
⋆be the tra king error.
We assume the existen e of a feedba k ontroller
ufeedba k
(e) su h that
u = u⋆ + u
feedba k
(e) (13)
ensures a stable tra king around the referen e
traje tory.
6.1.2. Intelligent ontrollers Repla e Eq. (4) by
Φj(y, . . . ,y(Nj),u, . . . ,u(Mj)) +Gj = 0 (14)
where the Gj , j = 1, . . . ,m, stand for the unmod-
eled parts. Eq. (12) be omes then
uj = Ψj(y, . . . ,y(Nj))+Hj, j = 1, . . . ,m (15)
whereHj 6= Gj in general. Thanks to Eq. (15), Hj
is estimated in the same way as the fault variables
and the unknown perturbations are in (Fliess,
Join & Sira-Ramírez [2008℄). Consider again y⋆
and u⋆as they are de�ned above. The intelligent
ontroller, or i- ontroller, follows from Eq. (13)
u = u⋆ +
H1
.
.
.
Hm
+ u
feedba k
(e)
It ensures tra king stabilization around the refer-
en e traje tory.
6.2 Fri tions and nonlinearities
A point mass m at the end of a spring of length yobeys to the equation
my = −K(y) + F(y)− dy + Fext
(16)
where
• Fext
= u is the ontrol variable;
• d and F(y) are due to omplex fri tion phe-
nomena;
• K(y) = k1y + k3y3exhibits a ubi nonlin-
earity of Du�ng type;
The mass m = 0.5 is known; there is a possible
error of 33% for k1 = 3 and we utilize k1 = 2; dand k3, whi h are unknown, are equal to 5 and 10in the numeri al simulations. For the fri tions,
22
22There is a huge literature in tribology where various
possible fri tion models are suggested (see, e.g., in ontrol
(Olsson, Åström, Canudas de Wit, Gäfvert & Lis hinsky
[1998℄, Nuninger, Perruquetti & Ri hard [2006℄). Those
modelings are bypassed here.
we have hosen for the sake of omputer simula-
tions the well known model due to Tustin [1947℄.
Fig. 15-(a) exhibits its quite wild behavior when
the sign of the speed is hanging.
6.2.1. A lassi PID ontroller The PID on-
troller is tuned only thanks to the restri ted model
my = −k1y + u. Its gains are determined in su h
a way that all the poles of the losed-loop system
are equal to −3: KP = −k1 + 27m, KI = −27m,
KD = 9m.
6.2.2. The orresponding i-PID ontroller Pi k
up a referen e traje tory y⋆. Set
u⋆ = my⋆ + k1y⋆
Our i-PID ontroller is given by
Fext
= u = u⋆ − [G]e + PID(e) (17)
where
• G = F(y) − (k1 − k1)y − k3y3 − dy, whi h
stands for the whole set of unknown e�e ts,
is estimated via
[G]e = m[y]e + k[y]e − Fext
whi h follows from Eq. (16) ([y]e and [y]e arethe denoised output variable and its denoised
2nd-order derivative � see Fig. 15-(d,f));
• PID(e), e = y − y⋆, is the above lassi PID ontroller.
6.2.3. Numeri al simulations The performan es
of our i-PID ontroller (17), whi h are displayed
in Fig. 15-( ,d), are ex ellent. When ompared to
the Figures
• 15-(e,f), where
· �atness-based ontrol is employed for
determined the open-loop output and
input variables,
· the loop is losed via a lassi PID on-
troller, whi h does not take into a ount
the unknown e�e ts;
• 15-(g,h), where only a lassi PID ontroller
is used, without any �atness-based ontrol;
the superiority of our ontrol design is obvious.
This superiority is in reasing with the fri tion.
6.3 Non-minimum phase systems
Consider the transfer fun tion
s− a
s2 − (b+ c)s+ bc
a, b, c ∈ R are respe tively its zero and its two
poles. The orresponding input-output system is
non-minimum phase if a > 0. The ontrollable
and observable state-variable representation
x1 = x2
x2 = (b+ c)x2 − bcx1 + u
y = x2 − ax1
(18)
shows that z = x1 is a �at output. The �at
output is therefore not the measured output, as
we assumed in Se t. 6.1.1. Our ontrol design has
therefore to be modi�ed.
6.3.1. Control of the exa t model To a nominal
�at output
23 z⋆ orresponds a nominal ontrol
variable
u⋆ = z⋆ − (b + c)z⋆ + bcz⋆
and a nominal output variable
y⋆ = z⋆ − az⋆ (19)
Introdu e the GPI ontroller (Fliess, Marquez,
Delaleau & Sira-Ramírez [2002℄)
u = u⋆ + γ∫
(u− u⋆) +KP (y − y⋆)+KI
∫
(y − y⋆) +KII
∫∫
(y − y⋆)(20)
where the oe� ients γ,KP ,KI ,KII ∈ R are
hosen in order to stabilize the error dynami s
e = z − z⋆. Ex ellent performan es are displayed
in Figures 16-(a) and (b), where a = 1, b = −1,c = −0.5, even with an additive orrupting noise.
See Figures 16-( ) and (d) for y⋆ and z⋆ whi h is
al ulated by integrating Eq. (19) ba k in time.
6.3.2. Unmodeled e�e ts The se ond line of Eq.
(18) may be written again as
x2 = (b+ c)x2 − bcx1 + u+
where stands for the unmodeled e�e ts, like
fri tions or an a tuator's fault. Repla e the nom-
inal ontrol variable u⋆of Se t. 6.3.1 by
u⋆pert
= u⋆ − []e
where []e is the estimated value of , whi h is
given by
[]e = −(
[y]e − (b+ c)[y]e + bc[y]e − [u]ea
+ u
)
Moreover,
[u]e = u⋆ + γ(u− u⋆) +KP ([y]e − y⋆)+KI([y]e − y⋆) +KII
∫
([y]e − y⋆)
follows from Eq. (20).
Start with the Figures 17 and 18, where = −0.5,a = 1, b = −1, c = −0.5. Fig. 17-(b), where the
nominal ontrol is left unmodi�ed, and Fig. 17-
(e), where it is modi�ed, demonstrate a lear- ut
23This Se tion, whi h is based on previous studies (Fliess
& Marquez [2000℄, Fliess, Marquez, Delaleau & Sira-
Ramírez [2002℄), should make the reading of Se t. 6.3.2
easier.
superiority of our approa h, even with an additive
orrupting noise.
In Fig. 19, is no more assumed to be onstant,
but equal to −0.1y. In the numeri al simulations,
a = 2, b = −1 c = 1. If is not estimated,
it in�uen es the tra king quite a lot even if its
amplitude is weak. When is estimated on the
other hand, the results are ex ellent, even with an
additive orrupting noise.
7. CONCLUSION
The results whi h were already obtained with
our intelligent PID ontrollers lead us to the
hope that they will greatly improve the pra ti al
appli ability and the performan es of the lassi
PIDs, at least for all �nite-dimensional systems
whi h are known to be non-minimum phase within
their operating range:
• the tuning of the gains of i-PIDs is straight-
forward sin e
· the unknown part is eliminated,
· the ontrol design boils down to a pure
integrator of order 1 or 2;• the identi� ation te hniques for implement-
ing lassi PID regulators, whi h are often
impre ise and di� ult to handle, are be om-
ing obsolete.
Model-free ontrol and the ontrol with a re-
stri ted model seem to question the very prin i-
ples of modeling in applied s ien es, at least when
one wishes to ontrol some on rete plant. This
might be a fundamental �epistemologi al� hange,
whi h needs of ourse to be further dis ussed and
analyzed. A natural extension to un ontrolled sys-
tems is being developed via various questions in
�nan ial engineering: see already the preliminary
studies in (Fliess & Join [2008b, 2009a,b℄).
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0 2 4 6 8 10 12 14 16−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
Time (s)
(a) i-PI ontrol
0 2 4 6 8 10 12 14 16−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Time (s)
(b) Output (�); referen e (- -); denoised
output (. .)
0 5 10 15−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
Time (s)
( ) PID ontrol
0 5 10 15−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Time (s)
(d) Output (�); referen e (- -); de-
noised output (. .)
Figure 1. Stable linear monovariable system
0 2 4 6 8 10 12 14 16−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
Time (s)
(a) i-PI ontrol
0 2 4 6 8 10 12 14 16−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Time (s)
(b) Output (�); referen e (- -); denoised
output (. .)
0 5 10 15−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Time (s)
( ) PID ontrol
0 5 10 15−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Time (s)
(d) Output (�); referen e (- -); de-
noised output (. .)
Figure 2. Modi�ed stable linear monovariable system
0 2 4 6 8 10 12 14 16−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Time (s)
(a) i-PI ontrol
0 2 4 6 8 10 12 14 16−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Time (s)
(b) Output (�); referen e (- -); denoised
output (. .)
0 5 10 15−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
Time (s)
( ) PID ontrol
0 5 10 15−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Time (s)
(d) Output (�); referen e (- -); de-
noised output (. .)
Figure 3. Stable linear monovariable system, with an a tuator's fault
0 2 4 6 8 10−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
(a) Referen e (- -) and output
0 2 4 6 8 10−10
−5
0
5
10
15
20
25
Time (s)
(b) Control
0 2 4 6 8 10−25
−20
−15
−10
−5
0
5
10
Time (s)
( ) Estimation of F
0 2 4 6 8 10−4
−3
−2
−1
0
1
2
3
4
5
Time (s)
(d) Estimation of y
Figure 4. Linear monovariable system with a large spe trum
0 1 2 3 4 5 6 7 8 9 10-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
Time (s)
(a) Output's noises
0 1 2 3 4 5 6 7 8 9 10-10
-8
-6
-4
-2
0
2
4
6
8
10
Time (s)
(b) Estimation of y1
0 1 2 3 4 5 6 7 8 9 10-10
-8
-6
-4
-2
0
2
4
6
8
10
Time (s)
( ) Estimation of y2
0 1 2 3 4 5 6 7 8 9 10-100
-80
-60
-40
-20
0
20
40
60
80
100
Time (s)
(d) Estimation of y2
0 1 2 3 4 5 6 7 8 9 10-11
-7
-3
1
5
9
13
17
21
25
Time (s)
(e) Estimation of F1
0 1 2 3 4 5 6 7 8 9 10-200
-100
0
100
200
300
Time (s)
(f) Estimation of F2
0 1 2 3 4 5 6 7 8 9 10-2.5
-2.1
-1.7
-1.3
-0.9
-0.5
-0.1
0.3
0.7
1.1
1.5
Time (s)
(g) Control u1
0 1 2 3 4 5 6 7 8 9 10-20
-10
0
10
20
30
40
50
Time (s)
(h) Control u2
Figure 5. Linear multivariable system
0 1 2 3 4 5 6 7 8 9 10-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Time (s)
(a) i-PID: referen es (- -) and outputs
0 1 2 3 4 5 6 7 8 9 10-5
-3
-1
1
3
5
7
9
11
13
15
Time (s)
(b) PID: referen es (- -) and outputs
Figure 6. Linear multivariable system
0 5 10 15−3
−2
−1
0
1
2
3
Time (s)
(a) Estimation of y
0 5 10 15−1.2
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
Time (s)
(b) Commande
0 5 10 15−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Time (s)
( ) Output (�); referen e (- -); de-
noised output (. .)
Figure 7. Instable nonlinear monovariable system
0 5 10 15−1.4
−1.2
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Time (s)
(a) Commande
0 5 10 15−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Time (s)
(b) Output (�); referen e (- -); de-
noised output (. .)
Figure 8. Instable nonlinear system: saturated ontrol without anti-windup
0 5 10 15−1.2
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Time (s)
(a) Commande
0 5 10 15−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Time (s)
(b) Output (�); referen e (- -); de-
noised output (. .)
Figure 9. Instable nonlinear system: saturated ontrol with anti-windup
y
uθ
Figure 10. The ball and beam example
0 2 4 6 8 10−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (s)
(a) Referen e (- -) and output
0 2 4 6 8 10−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
(b) Control
0 2 4 6 8 10−120
−100
−80
−60
−40
−20
0
20
40
Time (s)
( ) Estimation of F
0 2 4 6 8 10−1.5
−1
−0.5
0
0.5
1
1.5
Time (s)
(d) Referen e (- -); output in the noisy ase
Figure 11. Polynomial traje tory for the ball and beam
0 2 4 6 8 10−1.5
−1
−0.5
0
0.5
1
1.5
Time (s)
(a) Referen e (- -); output
0 2 4 6 8 10−1.5
−1
−0.5
0
0.5
1
1.5
Time (s)
(b) Control
0 2 4 6 8 10−150
−100
−50
0
50
100
150
Time (s)
( ) Estimation of F
0 2 4 6 8 10−2
−1.5
−1
−0.5
0
0.5
1
1.5
Time (s)
(d) Referen e (- -); output in the noisy ase
Figure 12. Sinusoidal traje tory for the ball and beam
q1 q2
x1 x2x3
Tank 1
q20
Pomp 1 Pomp 2
S
SS
( )20, µpS( )32, µpS
q13 q32 q20
( )13, µpS
Tank 3 Tank 2
Figure 13. The 3 tank system
0 100 200 300 400 500-0.01
0.03
0.07
0.11
0.15
0.19
0.23
0.27
0.31
Time (s)
(a) Referen es (- -); outputs
0 100 200 300 400 500-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
Time (s)
(b) Estimation of the derivatives y1 (-)
and y2 (- -), whi h is shifted ba k
0.02
0 100 200 300 400 5000.0e+00
2.0e-05
4.0e-05
6.0e-05
8.0e-05
1.0e-04
1.2e-04
1.4e-04
1.6e-04
1.8e-04
2.0e-04
Time (s)
( ) Control u1 (-) and u2 (- -)
120 130 140 150 160 170 180 190 2000.18
0.19
0.20
0.21
0.22
0.23
0.24
0.25
Time (s)
(d) Denoising
Figure 14. Simulations for the 3 tank system
−0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5−6
−4
−2
0
2
4
6
Speed (m/s)
(a) The fri tion law
0 5 10 15−8
−6
−4
−2
0
2
4
6
Time (s)
(b) Fri tion F(y) (�); Estimation of
the whole unknown e�e ts [G(y)]e
(- -)
0 5 10 15−6
−4
−2
0
2
4
6
8
Time (s)
( ) i-PID ontrol
0 5 10 15−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
(d) i-PID: output (�), referen e (- -);
denoised output (. .)
0 5 10 15−6
−4
−2
0
2
4
6
8
Time (s)
(e) Flatness-based ontrol + lassi
PID
0 5 10 15−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
(f) Flatness-based ontrol + lassi
PID: output (�), referen e (- -), de-
noised output (. .)
0 5 10 15−6
−4
−2
0
2
4
6
8
Time (s)
(g) Non-�atness-based ontrol + PID
0 5 10 15−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
(h) Non-�atness-based ontrol: output
(�), referen e (- -), denoised output
(. .)
Figure 15. The spring with unknown with nonlinearity, fri tion and damping
0 5 10 15−0.40
−0.35
−0.30
−0.25
−0.20
−0.15
−0.10
−0.05
0.00
0.05
Time (s)
(a) u (�); u⋆(- -)
0 5 10 15−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
(b) Output (�); referen e (- -); denoised
output (. .)
0 5 10 15−0.35
−0.30
−0.25
−0.20
−0.15
−0.10
−0.05
−0.00
Time (s)
( ) u (�); u⋆(- -)
0 5 10 15−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
(d) Output (�); referen e (- -); de-
noised output (. .)
Figure 16. Non-minimum phase system
0 5 10 15−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
Time (s)
(a) u (�); u⋆(- -)
0 5 10 15−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
(b) Output (�); referen e (- -); denoised
output (. .)
0 5 10 15−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
−0.0
Time (s)
( ) Perturbation (- -); estimated per-
turbation (�)
0 5 10 15−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Time (s)
(d) u (�); u⋆(- -); u⋆
pert
(. .)
0 5 10 15−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
(e) Output (�); referen e (- -); denoised
output (. .)
Figure 17. The non-minimum phase system where the �rst e�e t is not modeled
0 5 10 15−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
Time (s)
(a) u (�); u⋆(- -)
0 5 10 15−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
(b) Output (�); referen e (- -); denoised
output (. .)
0 5 10 15−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
−0.0
Time (s)
( ) Perturbation (- -); estimated per-
turbation (�)
0 5 10 15−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Time (s)
(d) u (�); u⋆(- -); u⋆
pert
(. .)
0 5 10 15−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
(e) Output (�); referen e (- -); denoised
output (. .)
Figure 18. The non-minimum phase system where the �rst e�e t is not modeled
0 2 4 6 8 10 12 14 16 18 20−0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Time (s)
(a) u (�); u⋆(- -)
0 2 4 6 8 10 12 14 16 18 20−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (s)
(b) Output (�); referen e (- -); denoised
output (. .)
0 2 4 6 8 10 12 14 16 18 20−0.025
−0.020
−0.015
−0.010
−0.005
0.000
0.005
0.010
Time (s)
( ) Perturbation (- -); estimated per-
turbation (�)
0 2 4 6 8 10 12 14 16 18 20−0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Time (s)
(d) u (�); u⋆(- -); u⋆
pert
(. .)
0 2 4 6 8 10 12 14 16 18 20−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
(e) Output (�); referen e (- -); denoised
output (. .)
Figure 19. The non-minimum phase system where the se ond e�e t is not modeled
0 5 10 15−5
0
5
10
15
20
Time (s)
0 5 10 15−4
−2
0
2
4
6
8
10
12
14
16
Time (s)
0 5 10 15−8
−6
−4
−2
0
2
4
6
8
10
12
Time (s)
0 5 10 15−6
−4
−2
0
2
4
6
8
10
12
Time (s)
0 5 10 15−5
0
5
10
Time (s)
0 5 10 15−6
−4
−2
0
2
4
6
8
10
Time (s)
0 1 2 3 4 5 6 7 8 9 108
9
10
11
12
13
14
15
16
17
18
Time (s)
0 5 10 15−6
−4
−2
0
2
4
6
Time (s)
0 5 10 15−8
−6
−4
−2
0
2
4
6
Time (s)
0 5 10 15−8
−6
−4
−2
0
2
4
6
Time (s)
0 1 2 3 4 5 6 7 8 9 10-0.1
0.3
0.7
1.1
1.5
1.9
2.3
2.7
Time (s)
0 1 2 3 4 5 6 7 8 9 10-80
-60
-40
-20
0
20
40
60
80
Time (s)
0 1 2 3 4 5 6 7 8 9 10-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Time (s)
0 1 2 3 4 5 6 7 8 9 10-1
0
1
2
3
4
5
Time (s)
0 1 2 3 4 5 6 7 8 9 10-2.5
-2.1
-1.7
-1.3
-0.9
-0.5
-0.1
0.3
0.7
1.1
1.5
Time (s)
0 1 2 3 4 5 6 7 8 9 10-30
-20
-10
0
10
20
30
40
Time (s)
0 1 2 3 4 5 6 7 8 9 10-0.01
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Time (s)
0 5 10 15−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Time (s)
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
1.2
Time (s)
0 1 2 3 4 5 6 7 8 9 10-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0 5 10 15−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0 5 10 15−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0 5 10 15−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0 5 10 15−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
q1 q2
l1 l2l3
c1 c3 c2
p1 p2
s
ss
q13 q32 q20
m1 m3 m2
q20
0 5 10 15−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
Time (s)
0 1 2 3 4 5 6 7 8 9 10-10
-8
-6
-4
-2
0
2
4
6
8
10
Time (s)
0 1 2 3 4 5 6 7 8 9 10-10
-8
-6
-4
-2
0
2
4
6
8
10
Time (s)
0 1 2 3 4 5 6 7 8 9 10-100
-80
-60
-40
-20
0
20
40
60
80
100
Time (s)
0 5 10 15−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Time (s)
0 5 10 15−1.5
−1.0
−0.5
0.0
0.5
1.0
Time (s)
0 5 10 150.0
0.5
1.0
1.5
Time (s)
0 5 10 15−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
Time (s)
0 5 10 15−0.5
0.0
0.5
1.0
1.5
2.0
2.5
Time (s)
0 5 10 15−1.5
−1.0
−0.5
0.0
0.5
1.0
Time (s)
0 1 2 3 4 5 6 7 8 9 10-4e-04
-3e-04
-2e-04
-1e-04
0e+00
1e-04
2e-04
3e-04
4e-04
Time (s)
0 5 10 150.0
0.5
1.0
1.5
2.0
2.5
Time (s)
0 5 10 15−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
Time (s)
0 5 10 15−15
−10
−5
0
5
10
15
Time (s)
0 5 10 150.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0 5 10 15−15
−10
−5
0
5
10
15
Time (s)
0 5 10 150.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0 5 10 15−15
−10
−5
0
5
10
15
Time (s)
0 5 10 15−8
−6
−4
−2
0
2
4
6
8
Time (s)
0 5 10 15−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0 5 10 150.0
0.5
1.0
1.5
Time (s)
0 5 10 15−1.5
−1.0
−0.5
0.0
0.5
1.0
Time (s)
0 5 10 150.0
0.5
1.0
1.5
2.0
2.5
Time (s)
0 5 10 15−1.5
−1.0
−0.5
0.0
0.5
1.0
Time (s)
i
y0
y
0 1 2 3 4 5 6 7 8 9 10-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
Time (s)
0 1 2 3 4 5 6 7 8 9 108
9
10
11
12
13
14
15
16
17
18
Time (s)
0 5 10 15−4
−2
0
2
4
6
8
10
Time (s)
0 1 2 3 4 5 6 7 8 9 10-0.20
-0.16
-0.12
-0.08
-0.04
0.00
0.04
0.08
0.12
0.16
0.20
Time (s)
0 5 10 15−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
0 1 2 3 4 5 6 7 8 9 10-2.0
-1.6
-1.2
-0.8
-0.4
0.0
0.4
0.8
1.2
1.6
2.0
Time (s)
top related