on the constructive computation of flat outputs over an ore algebra
DESCRIPTION
9th IEEE Multi-conference on Systems and Control (MSC) / IEEE International Symposium on Computer-aided Control System Design (CACSD), September 3-5, 2008, San Antonio, Texas, USATRANSCRIPT
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
9th IEEE International Symposium on Computer-aidedControl System Design
September 3-5, 2008, San Antonio, Texas
1 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
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
2 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Some Historical ConsiderationsFlatnes-based DesignFlatness Necessary and Sufficient Conditions
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
3 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Some Historical ConsiderationsFlatnes-based DesignFlatness Necessary and Sufficient Conditions
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
4 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Some Historical ConsiderationsFlatnes-based DesignFlatness Necessary and Sufficient Conditions
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
5 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Some Historical ConsiderationsFlatnes-based DesignFlatness Necessary and Sufficient Conditions
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
6 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Some Historical ConsiderationsFlatnes-based DesignFlatness Necessary and Sufficient Conditions
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 .
7 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Some Historical ConsiderationsFlatnes-based DesignFlatness Necessary and Sufficient Conditions
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
8 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Some Historical ConsiderationsFlatnes-based DesignFlatness Necessary and Sufficient Conditions
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
9 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
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
10 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
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
11 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
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
12 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
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
13 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
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
14 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
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
Computation of a basis ω of
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
15 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
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
16 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
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
17 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
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 .
18 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
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
19 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
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.
20 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
STEP 3: Computation of an integrating factor
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.
21 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
STEP 3: Computation of an integrating factor
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
22 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Module TheoryProblem statementProposed Methodology
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
23 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
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
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Example: 3-tank process
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
Computation of a basis ω of
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
25 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Example: 3-tank process
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
26 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Example: 3-tank process
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
27 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Example: 3-tank process
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.
28 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Example: 3-tank process
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
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
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.
30 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
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.
31 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
Concluding remarksCurrent work
Fault-tolerant onboard path planning for atmospheric reentryvehicles using flatness approach
32 / 33
MotivationsConstructive computation of flat outputs for nonlinear systems
Numerical exampleConcluding remarks
THANK YOU!
33 / 33
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.
34 / 33
Appendix For Further Reading
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.
35 / 33
Appendix For Further Reading
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.
36 / 33
Appendix For Further Reading
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.
37 / 33
Appendix For Further Reading
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.
38 / 33