On the Constructive Computation of Flat Outputs over an Ore Algebra

MotivationsConstructive computation of flat outputs for nonlinear systems

Example

On the Constructive Computation ofFlat Outputs over an Ore Algebra

V. Morio, F. Cazaurang and A. Zolghadri

University Bordeaux 1IMS lab

Automatic Control Department351, cours de la Libération, 33400, Talence, France

[email protected]://www.laps.u-bordeaux1.fr/aria

17th IFAC World CongressJuly 6-11, 2008, Seoul, Korea

V. Morio, F. Cazaurang and A. Zolghadri On the Constructive Computation of Flat Outputs over an Ore Algebra

mailto:[email protected]

http://www.laps.u-bordeaux1.fr/aria


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Outline

1 MotivationsSome Historical ConsiderationsFlatnes-based DesignFlatness Necessary and Sufficient Conditions

2 Constructive computation of flat outputs for nonlinear systemsRecalls on Module TheoryStatement of the problemProposed Methodology

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Some Historical ConsiderationsFlatnes-based DesignFlatness Necessary and Sufficient Conditions

Outline



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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



Example


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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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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Flatness Necessary and Sufficient Conditions

General formulations of flatness necessary and sufficientconditions are now well-established for linear multidimensionalsystems and also for nonlinear systems:

- In a linear setting, some constructive algorithms have been derivedand implemented in some software packages (e.g. OREMODULES,STAFFORD and JANET MapleTMlibraries)

- As regards nonlinear systems, the derivation of formal and effectivealgorithms is still an open topic.

Various formalisms and mathematical tools:- Deformation theory of pseudogroup structures [4],- module theory [1], Jets of infinite order [2],- “very” formal series [4], Generalized Cartan moving frames [4],- Jacobson normal forms [4], Characteristic sets,- Gröbner bases, Janet bases [3], Order bases [1],- ...



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Flatness Necessary and Sufficient Conditions

Some algorithms have been proposed to check if a nonlinearsystem is flat and, if so, to compute some candidate flat outputs.Unfortunately, most of them seems to be not very effective anddifficult to implement 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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Recalls on Module TheoryStatement of the problemProposed Methodology

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Recalls on Module theory

Algebraic analysis of functional systemsClassification of linear multidimensionnal systems according toalgebraic properties of associated D-modulesRedefine systems properties in a more intrinsic wayDevelop effective algorithms to check these structural properties

Advantages

Allows to bring back the geometric properties of a differentialmanifold to their functionnal propertiesUnified framework to deal with several classes of linearmultidimensional systems (differential time-delay systems,multidimensional discrete systems,...)

DrawbacksLess conventional mathematical framework



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Statement of the problem

Definition

Let Sz(γ) be the subset of flat outputs which involves a reducednumber of state derivatives 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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Proposed methodology

Process Modeling

Implicit linear model ΣL

Computation of a basis ω of

the free left D-module LL

Implicit nonlinear model ΣNL INIT

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



Example


Implicit nonlinear model

Consider the systemx = f (x ,u)

with x ∈ X , dim X = n, u ∈ Rm and f a smooth vector field on themanifold X , satisfying rank

(∂ f∂u

)= m. It is locally equivalent to the

underdetermined implicit system:

F (x , x) = 0

with x ∈ X , dim X = n, rank(

∂ f∂ x

)= n−m.

RemarkThis implicit representation is invariant by endogenous dynamicfeedback.

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Example


STEP 1: Variational left D-module

Let F be a nonlinear model in implicit form such that:

F (x , x) = 0

The associated variational model is defined by:(∂F∂x

(x(t), x(t)))

ξ (t)+(

∂F∂ x

(x(t), x(t)))

ξ (t) = 0

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 ))

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

Based on effective algorithms recently obtained in the symboliccomputation community:

- Computation of matrix left nullspaces bases over Ore algebras,- Computation of matrix normal forms, e.g. Popov forms.

Roughly speaking, particular kind of Gaussian elimination:- Adapted to non-commutative rings,- Including additional degree constraints to control coefficient growth.



Example



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



Example



Let L = D1×n/(D1×(n−m)P(F )) be a finitely presented leftD-module. Then, L is a free left D-module iff there exists twomatrices Q ∈ Dn×m and T ∈ Dm×n satisfying:{

kerD(.Q) = D1×(n−m)P(F )TQ = Im

where{

kerD(.Q) = {λ ∈ D1×n|λQ = 0}D1×(n−m)P(F ) = {λ ∈ D1×n|∃µ ∈ D1×(n−m),λ = µP}

If Q and T exist, the residue classes of the rows of T define abasis of the free left D-module L .This condition is equivalent to solving the Bézout identities:(

P(F )T

)(S Q) = In, (S Q)

(P(F )

T

)= In

where S ∈ Dn×(n−m) corresponds to a right-inverse of P(F ).



Example



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 .



Example


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):

Reduced-order bases (number of iterations):



Example



Reduction of the number of iterations to obtain a reduced-orderbasis of the free left D-module L

In comparison to the approach based on Jacobsondecompositions, the size of the solution space is reduced due tothe order constraints.If two or more rows have the same degree, there exists a degreeof freedom in the pivot sequence associated to the choice ofparticular meromorphic coefficients.

RemarkThe definition of a particular metric, allowing to classify arbitrarymeromorphic functions, could help improving the decision-makingprocess.

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Example


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.



Example



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.



Example



The constraint d(µ) = µ ∧µ must also be met: dµij =j−1

∑k=i+1

µik ∧µkj if j > i +1

dµij = 0 otherwise

where i ∈ {1, . . . ,m−1} and j ∈ {1, . . . ,m}.Consider that the general expression of the m-th term of thebasis ω is given by:

ωm =n

∑i=1

K

∑k=0

fi ,k ,m(x)dx (k)i

where K is an arbitrary integer.



Example



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



Example


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



Example

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.V. Morio, F. Cazaurang and A. Zolghadri On the Constructive Computation of Flat Outputs over an Ore Algebra


Example


Process Modeling

Implicit linear model ΣL


the free left D-module LL

Implicit nonlinear model ΣNL INIT

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



Example


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 system x = f (x ,u) with rank(

∂ f∂u

)= m is locally equivalent

to the underdetermined implicit system F (x , x) = 0 withrank

(∂ f∂ x

)= n−m.

The associated variational system is obtained by computing itsFrechet 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. Return



Example


Let L = D1×3/(D1×1P(F )) be the variational module associatedto P(F ).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 of P(F )∗ by using theformal reduced-order basis algorithm.

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Example


Initialization: Residual R0 = P(F )∗; Mahler system M0 = I3.

1st iteration: pivot p1 =−12

K1√h1−h3

M1(Z ) =

Z 0 0

−K2K1

√h1−h3h3−h2

1 0

1+ K2K1

√h1−h3h3−h2

0 1

, R1(Z ) =

− 12

K1√(h1−h301

2nd iteration: pivot p2 =−1

2K1√

h1−h3

M2(Z ) =

Z 2 0 0

−K2K1

√h1−h3h3−h2

1 0

1+ K2K1

√h1−h3h3−h2

+ 2√

h1−h3K1

Z 0 1

,R2(Z ) =

− 12

K1√(h1−h300

Only 2 iterations are needed here to obtain the transformationmatrix while the formula gives OQ = 4 iterations. In comparison,there exists OQ = 9 candidate Jacobson factorizations of P(F ).



Example


Next, we take the adjoint M2(Z )∗ of the resulting transformationmatrix M2(Z ) on the adjoint Ore ring D∗.By taking the nonzero columns of M2(Z )∗ corresponding to thezero columns of R2(Z )∗, we obtain a reduced order basis Q(Z )of the right nullspace of P(F ) such as:

Q(Z ) =

−K2K1

√h1−h3h3−h2

1+ K2K1

√h1−h3h3−h2

+ 2√

h1−h3K1

Z1 00 1

In order to compute a weak Popov form W (Z ) of Q(Z ), weconsider the augmented matrix:

Qaug(Z ) =(

Q(Z )−I2

)Then, a left nullspace basis of the augmented matrix Qaug(Z )satisfying Maug(Z ).Qaug(Z ) = Raug(Z ) is computed by using theformal reduced-order basis algorithm.



Example


Initialization: Residual R0 = Qaug(Z ); Mahler system M0 = I5.1st iteration: pivot p1 = 1

M1(Z ) =

1

K2K1

√h1−h3h3−h2

0 0 0

0 Z 0 0 00 0 1 0 00 1 0 1 00 0 0 0 1

, R1(Z ) =

0 1+

K2K1

√h1−h3h3−h2

+2√

h1−h3K1

Z

1 00 10 00 −1

2nd iteration: pivot p2 = 1

M2(Z ) =

1

K2K1

√h1−h3h3−h2

−1− K2K1

√h1−h3h3−h2

0 0

0 Z 0 0 00 0 Z 0 00 1 0 1 00 0 1 0 1

, R2(Z ) =

0

2√

h1−h3K1

1 00 10 00 0

3rd iteration: pivot p3 = 1

M3(Z ) =

1

K2K1

√h1−h3h3−h2

−1− K2K1

√h1−h3h3−h2

0 0

0 Z2 0 0 00 0 Z 0 00 1 0 1 00 0 1 0 1

, R3(Z ) =

0

2√

h1−h3K1

1 00 10 00 0



Example


4th iteration: pivot p4 = 1

Maug (Z ) =

1

K2K1

√h1−h3h3−h2

−1− K2K1

√h1−h3h3−h2

−2√

h1−h3K1

Z 0 0

0 Z2 0 0 00 0 Z2 0 00 1 0 1 00 0 1 0 1

, Raug (Z ) =

0 01 00 10 00 0

Once again, only 4 iterations suffices to obtain the requiredtransformation matrix while the formula gives OT = 12 iterations.In comparison, there exists OT = 126 possible Jacobsonfactorizations of Qaug(Z ).By taking the rows of Maug(Z ) corresponding to the n zero rowsof Raug(Z ), we obtain the equality:

1K2K1

√h1−h3h3−h2

−1− K2K1

√h1−h3h3−h2

−2√

h1−h3K1

Z 0 0

0 1 0 1 00 0 1 0 1

·Qaug (Z ) =

0 00 00 0



Example


The left handside matrix may be partitionned such that:

(T (Z ) W (Z )) ·Qaug(Z ) = 0

Then, the requested unimodular matrix satisfyingT (Z ) ·Q(Z ) = W (Z ) is given by:

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.



Example


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

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



Example

Summary

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.⇒ May replace advantageously the method based on Jacobson

factorizations, by controlling the order of the transformation matrixduring the reduction process.

Computation of an integrating factor by choosing a particularparameterization of the generalized moving frame structureequations.⇒ Size reduction of the solution space so as to improve the

computational tractability.



Example

Summary

Outlook

During the first basis computation, the meromorphic entries ofthe transformation matrix may be more complex than expected.⇒ This issue is mainly due to the use of the adjoint Ore ring. A

straightforward computation of a basis of the right nullspace of thevariational module could discard this problem.

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.⇒ A hybrid heuristic metric could be considered, but this question

remains open from the authors point of view.



Example

THANK YOU!

Please send any technical comments or questions to:

[email protected]


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.



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.



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.



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.

S. A. Abramov, H. Q. Le and Z. Li.Univariate Ore polynomial rings in computer algebra.Journal of Mathematical Sciences, 131(5):5885–5903, 2005.



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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On the Constructive Computation of Flat Outputs over an Ore Algebra