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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)