On the constructive computation of flat outputs over an Ore algebra

MotivationsConstructive computation of flat outputs for nonlinear systems

Numerical exampleConcluding remarks

On the Constructive Computation ofFlat Outputs over an Ore Algebra

V. Morio, F. Cazaurang and A. Zolghadri

University of Bordeaux/IMS labAutomatic Control Department

351, cours de la Libération, 33400, Talence, France

http://extranet.ims-bordeaux.fr/aria

[email protected]

9th IEEE International Symposium on Computer-aidedControl System Design

September 3-5, 2008, San Antonio, Texas

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http://extranet.ims-bordeaux.fr/aria

mailto:[email protected]



Outline

1 MotivationsSome Historical ConsiderationsFlatnes-based DesignFlatness Necessary and Sufficient Conditions

2 Constructive computation of flat outputs for nonlinear systemsModule TheoryProblem statementProposed Methodology

3 Numerical example

4 Concluding remarks

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Some Historical ConsiderationsFlatnes-based DesignFlatness Necessary and Sufficient Conditions

Outline



3 Numerical example


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Some Historical Considerations

Since early 80’s, much attention has been paid to providegeneric, formalized, efficient and practical nonlinear tools forengineering purposes.Objective: extension of mature control methods developed withina linear settingNonlinear Systems may be ranked into two main classes:

Nonlinear

Systems

“True” nonlinear

systems

Specific tools

predictive control

Lyapunov control,nonlinear H∞, ...

“Pseudo”

nonlinear systems

Equivalent to lineartrivial systems

Feedback linearizationtechniques, differentialflatness

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Some Historical Considerations

Differential flatness has been introduced in 1991 by Fliess,Lévine, Martin and RouchonSince, this concept has been applied to a number of problems:

- Robust control: H∞, LPV, ...- Trajectory generation- More recently, fault diagnosis and parameter estimation

Some advantages:- Direct open-loop trajectory generation without integration of

differential algebraic equations- The equivalence with linear trivial systems allows the use of robust

linear trajectory tracking methods- No nonlinear unobservable dynamics (which may be potentially

unstable)

Flatness-based control=Trajectory Planning + Trajectory Tracking

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Outline



3 Numerical example


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Flatness-based Design

Definition

The nonlinear system ˙x(t) = f (x(t),u(t)), with x(t) = (x1(t), . . . ,xn(t)):state and u(t) = (u1(t), . . . ,um(t)): control, m ≤ n, is (differentially) flatif and only if there exists z(t) = (z1(t), . . . ,zm(t)) such that:

z(t) and its successive derivatives ˙z(t), ¨z(t),. . . , are independent,

z(t) = Φ

(x(t),u(t), u(t), . . . ,u(α)(t)

)(linearizing output),

Conversely, x and u can be expressed as: x(t) = Ψx

(z(t), z(t), . . . ,z(β−1)(t)

)u(t) = Ψu

(z(t), z(t), . . . ,z(β )(t)

)The elements of z are called flat outputs. Thus, nonlinear systemtrajectories are equivalent to those of the trivial system z(β ) = v .

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Outline



3 Numerical example


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Flatness Necessary and Sufficient Conditions

General formulations of flatness necessary and sufficientconditions are now well-established for linear multidimensionalsystems and also for nonlinear systems.Unfortunately, most of proposed algorithms are difficult toimplement in some formal computing tools.

Question (open)

How to choose a particular set of simple candidate flat outputs:well adapted to sensor measurementsand/or endowed with a physical meaning

Need for more systematically control and guidance design ofnonlinear systems which have the potential for a flat charaterization

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Module TheoryProblem statementProposed Methodology

Outline



3 Numerical example


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Module theory

Main characteristicsRedefine systems properties in a more intrinsic wayDevelop effective algorithms to check these structural propertiesAllow us to bring back the geometric properties of a differentialmanifold to their functional propertiesUnified framework to deal with several classes of linearmultidimensional systems (differential time-delay systems,multidimensional discrete systems,...)Less conventional mathematical framework

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Outline



3 Numerical example


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Problem statement

Definition

Let Sz(γ) be the subset of flat outputs depending on a finite numberof state derivatives (implicit form) such that:

Sz(γ) ={

z ∈ Sz |z = Φimpl

(x(t), x(t), . . . ,x (γ)(t)

)}Then, the set Sz,r of reduced-order flat outputs is defined by:

Sz,r = minγ

(Sz(γ))

ProblemThe computation of Sz,r may be recast as the determination ofconstructive algorithms enabling to compute:

1 A reduced-order basis ω of the free left D-module L .2 An integrating factor M of the basis ω satisfying d(M.ω) = 0

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Outline



3 Numerical example


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Proposed methodology

Process Modeling

Implicit linear model ΣL

Computation of a basis ω of

the free left D-module LL

Implicit nonlinear model ΣNL

Computation of the (linear)

variational system P(ΣNL)STEP 1


the free left D-module LNLSTEP 2

Computation of an integrating

factor M such that d(Mω) = 0STEP 3

Integration of dy = M.ω STEP 4

Flat Outputs

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STEP 1: Variational left D-module

Consider the system x = f (x ,u) with x ∈ X , dim X = n, u ∈ Rm,and f a smooth vector field on the manifold X , satisfyingrank

(∂ f∂u

)= m, and the equivalent underdetermined implicit

system F (x , x) = 0 with x ∈ X , dim X = n, rank(

∂ f∂ x

)= n−m

(invariant by endogenous dynamic feedback).By using the Ore algebra formalism, we define P(F ) as thevariational system associated to F :

P(F ) =∂F∂x

+∂F∂ x

Z

Then, we can define the variational D-module L associated toP(F ) over the ring D = M [Z ;σ ,δ ] such as:

L = D1×n/(D1×(n−m)P(F ))

Return

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STEP 2: Reduced-order basis of a free left D-module

Theorem [1]

Consider an order basis M(Z ) ∈ Dm,s with order ~ω and degree ~µ, andλ ∈ {1, . . . ,s} the set of columns indices of M(Z ).

1 If r1 = . . . = rm = 0, then M(Z ) = M(Z ) is an order basis of degree~ν =~µ and order ~ω.

2 Otherwise, let π be an index such that rπ 6= 0. Then, an orderbasis M(Z ) of degree ~ν =~µ +~eλ and order ~ω with coefficients inM can be obtained via the formulas:

M(Z )l ,k = M(Z )l ,k − rl

rπ

·M(Z )π,k , {l ,k}= {1, . . . ,m}, l 6= π

M(Z )π,k =(

Z − δ (rπ)rπ

)·M(Z )π,k − ∑

l 6=π

σ(pl)rπ

· M(Z )l ,k

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STEP 2: Reduced-order basis of a free left D-module

Theorem

Let L = D1×(n−m)/(D1×nP(F )∗) be the transposed left D-module ofL , where P(F )∗ is the formal adjoint of P(F ) on the adjoint ring D∗.

1 A reduced-order basis Q(Z ) ∈ Dn,m satisfying P(F )Q(Z ) = 0 canbe obtained by computing a reduced-order basis Q∗(Z ) ∈ Dm,n

such that:Q∗(Z )P(F )∗ = 0

2 Then a reduced-order left multiplier T (Z ) ∈GLn(D) can beobtained by computing a weak Popov form W (Z ) ∈ Dn,m ofQ(Z ) ∈ Dn,m such that

T (Z ) ·Q(Z ) = W (Z )3 Finally, a reduced-order basis ω of L is given by:

ω = (Im,0m,n−m)T (Z )dx .

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STEP 2: Reduced-order basis of a free left D-moduleDefinition of a performance metric based on the number of iterationsneeded to converge towards a reduced-order basis.

Jacobson normal forms (total number or orbits):

Q(Z ) matrix T (Z ) matrixReduced-order bases (number of iterations):

Q(Z ) matrix T (Z ) matrixReturn

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STEP 3: Computation of an integrating factor

Given a basis ω of the free left D-module L , it is somewhatdifficult to define a generic structure of the unimodular matricesM ∈GLm(D) satisfying d(M.ω) = 0.We propose to consider instead a particular parameterization ofthe generalized moving frame structure equations:

dω = µω

d(µ) = µ ∧µ

d(M) = −Mµ

A matrix M is sought in triangular form so as to decrease thenumber of candidate solutions of the associated systems ofPDE, thus providing a better computational tractability.

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Definition

For any arbitrary integer m, let GLm(D) be the set of unimodularmatrices of size m×m.The subset GL∆m(D)⊂GLm(D) of unipotent matrices consists of anym×m matrix M having the following structure:

M =

1 M12 . . . M1,m0 1 . . . M2,m...

. . ....

0 0 1 Mm−1,m0 0 . . . 0 1

where each nonzero term Mij is an arbitrary Z -polynomial withmeromorphic coefficients.

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TheoremLet ω be a reduced-order basis of the free left D-module L . Then, asufficient condition for L to admit an integrable basis is that the twofollowing propositions are satisfied:

1 The basis ω matches the recursive expression:

ωm =Card(c J)

∑i=1

bi ,m

(xc Ji

)dxc Ji

+Card(J)

∑j=1

cj ,m

x(ρ

Jj)

Jj

dx(ρ

Jj)

Jj

ωm−i =n∑

k=1fk ,m−i

x(0,...,∑

m−1p=m−i+1 δp−1,p )

,x(ρ

J,...,ρ

J+∑

mp=m−i+1 δp−1,p )

J,x

(0,...,∑mp=m−i+1 δp−1,p )

c J

dxk

+n∑

k=1

∑m−1p=m−i+1 δp−1,p

∑s=0

gk ,s,m−i

(x(s)k ,x

)dx(s)

k +Card(J)

∑k=1

ρJk

+∑mp=m−i+1 δp−1,p

∑s=ρ

Jk

hk ,s,m−i

(x(s)Jk

,x

)dx(s)

Jk

+Card(c J)

∑k=1

∑mp=m−i+1 δp−1,p

∑s=0

ek ,s,m−i

(x(s)c Jk

,x)

dx(s)c Jk

2 The system of partial differential equations associated to theconstraint d(µ) = µ ∧µ admits a solution. Return

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STEP 4: Integration of dy = M.ω

If ω = T dx defines a basis of the free left D-module L , we havefound another basis v = Mω which is integrable for a certainintegrating factor M ∈GLm(D).Hence, some reduced-order flat outputs can be computed byintegrating:

dy = M.ω

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Example: 3-tank process

Nonlinear model of an academic hydraulic process:Sc

dh1dt = −az10Sn

√2gh1−az13Sn

√2g (h1−h3)+Q1

Scdh2dt = −az20Sn

√2gh2 +az32Sn

√2g (h3−h2)+Q2

Scdh3dt = −az30Sn

√2gh3−az32Sn

√2g (h3−h2)

+az13Sn√

2g (h1−h3)

where (h1,h2,h3): states, and (Q1,Q2): inputs.24 / 33




Process Modeling

Implicit linear model ΣL


the free left D-module LL

Implicit nonlinear model ΣNL

Computation of the (linear)

variational system P(ΣNL)STEP 1


the free left D-module LNLSTEP 2

Computation of an integrating

factor M such that d(Mω) = 0STEP 3

Integration of dy = M.ω STEP 4

Flat Outputs

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We consider the ring of differential operators D = M [Z ;σ ,δ ],where σ = idM , δ = d

dt = LτXand M represents the field of

meromorphic functions from X to R.The variational system associated to the nonlinear model isobtained by computing its Frechet derivative:

P(F ) =∂F∂x

+∂F∂ x

Z

=(−1

2K1√

h1−h3−1

2K2√

h3−h2

12

K1√h1−h3

+ 12

K2√h3−h2

+Z)

where K1 = az32SnSc

√2g and K2 = az13Sn

Sc

√2g.

Let L = D1×3/(D1×1P(F )) be the variational module associatedto P(F ).

Return

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We compute the adjoint Ore matrix P(F )∗ on the adjoint ringD∗ = M [Z ;σ∗,δ ∗], where σ∗ = idM and δ ∗ =− d

dt =−LτX

P(F )∗ =

−1

2K1√

h1−h3

−12

K2√h3−h2

12

K1√h1−h3

+ 12

K2√h3−h2

+Z

Then, we compute a left nullspace basis Q(Z ) of P(F )∗ by usingthe formal reduced-order basis algorithm:

Q(Z ) =

−K2K1

√h1−h3h3−h2

1+ K2K1

√h1−h3h3−h2

+ 2√

h1−h3K1

Z1 00 1

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Computation of a weak Popov form W (Z ) of Q(Z ), satisfyingT (Z )Q(Z ) = W (Z ):

T (Z ) =

1 K2K1

√h1−h3h3−h2

−1− K2K1

√h1−h3h3−h2

− 2√

h1−h3K1

Z0 1 00 0 1

where

W (Z ) =

0 01 00 1

Since the weak Popov form W (Z ) comprises min(m,n) = 2nonzero rows, then the variational left D-module L is free.

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A reduced-order basis ω is given by ω = (Im,0m,n−m)T (Z )dx , i.e.

ω =(

0 1 00 0 1

) dh1dh2dh3

Return

Since ω1 = dh2 and ω2 = dh3, the integrability conditions of thefree left D-module L are trivially satisfied here.

Return

Finally, by choosing M as the identity matrix, we get dy1 = dh2and dy2 = dh3, and some candidate flat outputs are simply givenby: {

y1 = h2 +C1y2 = h3 +C2

where C1 and C2 are arbitrary constants.29 / 33



Concluding remarks

Contributions

Presentation of a new and constructive algorithm forsystematically computation of flat outputs for nonlinear systems.Computation of a basis of a free left D-module usingreduced-order bases and the weak Popov form.Computation of an integrating factor by choosing a particularparameterization of the generalized moving frame structureequations.

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Concluding remarks

Outlook

By using the adjoint Ore ring, the meromorphic entries of thetransformation matrix may be more complex than expectedduring the first basis computation.The choice of a unique set of pivots may be delicate duringcomputation of reduced-order bases: it is quite difficult to definea metric that could provide a notion of complexity over the field ofmeromorphic functions.

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Concluding remarksCurrent work

Fault-tolerant onboard path planning for atmospheric reentryvehicles using flatness approach

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THANK YOU!

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Appendix For Further Reading

References I

A. Isidori.Nonlinear Control Systems.Springer Verlag, 1989.

M. Spivak.A Comprehensive Introduction to Differential Geometry.Publish or Perish, Inc., 1979.

M. Aschbacher.Finite Group Theory.Cambridge University Press, 2000.

S. S. Chern, W. H. Chen and K. S. Lam.Lectures on differential geometry.World Scientific, 2000.

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References II

M. Fliess, J. Lévine, Ph. Martin and P. Rouchon.Flatness and defect of non-linear systems: introduction, theoryand examples.Int. Journal of Control, 61(6):1327–1361, 1995.

M. Fliess, J. Lévine, Ph. Martin and P. Rouchon.A Lie-Bäcklund approach to equivalence and flatness ofnonlinear systems.IEEE Trans. Automatic Control, 44(5):922–937, 1999.

B. Jakubvzyk and W. Respondek.On linearization of control systems.Bull. Acad. Pol. Sci. Ser. Sci. Math., 26:517–522, 1980.

J. Lévine.On Necessary and Sufficient Conditions for Differential Flatness.arXiv, no. 0605405, 2006.

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References III

B. Charlet, J. Lévine and R. Marino.Sufficient conditions for dynamic state feedback linearization.SIAM J. Control and Optimization, 29(1):38–57, 1991.

J. Lévine and D. V. Nguyen.Flat output characterization for linear systems using polynomialmatrices.Systems & Control Letters, 48(1):69–75, 2003.

E. Aranda-Bricaire, C. H. Moog and J. B. Pomet.A linear algebraic framework for dynamic feedback linearization.IEEE Trans. Automat. Contr., 40(1):127–132, 1995.

V.N. Chetverikov.New flatness conditions for control systems.Proc. of the 5th IFAC Symposium on Nonlinear Control Systems,168–173, St. Petersburg, Russia, 2001.

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References IV

F. Chyzak, A. Quadrat and D. Robertz.Effective algorithms for parametrizing linear control systems overOre algebras.Appl. Algebra Eng., Commun. Comput., 16(5):319–376, 2005.

F. Chyzak, A. Quadrat and D. Robertz.OREMODULES: A symbolic package for the study ofmultidimensional linear systems.Lecture Notes in Control and Information Sciences,352:233–264, 2007.

A. Quadrat and and D. Robertz.Computation of bases of free modules over the Weyl algebras.Journal of Symbolic Computation, 42:1113–1141, 2007.

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References V

B. Beckermann, H. Cheng and G. Labahn.Fraction-free Row Reduction of Matrices of Ore Polynomials.Journal of Symbolic Computation, 41(5):513–543, 2006.

H. Cheng and G. Labahn.Output-sensitive Modular Algorithms for Polynomial MatrixNormal Forms.Journal of Symbolic Computation, 42(7):733–750, 2007.

J.C. McConnell and J.C. Robson.Noncommutative Noetherian Rings.Bull. Amer. Math. Soc., 23(2):579–582, 1990.

D. Avanessoff.Dynamic linearization of non linear systems andparameterization of all solutions.PhD thesis, University of Nice, 2005.

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Technology

On the constructive computation of flat outputs over an Ore algebra