Parametrizing Linear Systems Daniel Robertz (joint work with F. - - PowerPoint PPT Presentation

Parametrizing Linear Systems

Daniel Robertz (joint work with F. Chyzak, A. Quadrat)

Lehrstuhl B f¨ ur Mathematik

29.10.2010

Outline

◮ Module-theoretic approach to linear systems ◮ Parametrizing linear systems ◮ Injective parametrizations

• 1. Module-theoretic approach to linear systems

Linear System

D ring (field, integral domain, Ore algebra, . . . ) R ∈ Dq×p, F left D-module R y = 0, y ∈ Fp. “optimal” answer for us: P ∈ Dp×r s.t. ker(R.) = im(P.) not possible in general Example. D = k a (skew) field.

◮ Gaussian elimination singles out parameters ◮ injective parametrization

Example

de Rham complex: D = R[∂x, ∂y, ∂z], e.g. F = C ∞(Ω), Ω ⊆ R3 convex 0 → R → F

@ ∂x ∂y ∂z 1 A .

− − − − − − → F3×1

@ ∂z −∂y −∂z ∂x ∂y −∂x 1 A .

− − − − − − − − − − − − − − − → F3×1

` ∂x ∂y ∂z ´ .

− − − − − − − − − − → F → 0 0 ← M ← D

. @ ∂x ∂y ∂z 1 A

← − − − − − − D1×3

. @ ∂z −∂y −∂z ∂x ∂y −∂x 1 A

← − − − − − − − − − − − − − − − D1×3

. ` ∂x ∂y ∂z ´

← − − − − − − − − − − D ← 0

Module-theoretic approach to linear systems

Σ : R y = 0, R ∈ Dq×p D = ring of functional operators M = D1×p/D1×q R independent of eq.’s chosen for Σ F = signal space If F is an injective cogenerator for DM then M

homD( · ,F) SolF(M)

is a categorical duality.

Malgrange, Oberst, Pommaret, Quadrat, Willems, Zerz, . . .

Module-theoretic approach to linear systems

Σ : R y = 0, R ∈ Dq×p, M = D1×p/D1×q R F injective cogenerator for DM: M

• D1×p
• D1×q

.R

• D1×r

exact

.S

• if and only if

(apply homD( · , F)) SolF(M) Fp

R.

Fq

S.

Fr exact. Fundamental principle (Ehrenpreis, Malgrange, Palamodov) for D = C[∂1, . . . , ∂n] acting by differentiation on F: e.g. F ∈ { complex-valued C ∞-functions on Rn, complex-valued distributions on Rn, formal / convergent power series }

• 2. Parametrizing linear systems

Parametrization and torsion-freeness

Let M be given by the finite presentation M

• D1×p

ρ

• D1×q.

.R

• Assume P is a parametrization, i.e.

D1×r D1×p

.P

• D1×q

.R

• is an exact sequence of left D-modules.

Then M is torsion-free: Fr

ι∗

• SolF(M)
• D1×r

D1×p

.P

• ρ
• D1×q

.R

• M
• ι

Example

• F. Dubois, N. Petit, and P. Rouchon,

Motion Planning and Nonlinear Simulations for a Tank Containing a Fluid,

• Proc. ECC, Karlsruhe (Germany), 1999.

   φ1(t − 2) + φ2(t) − 2 ˙ φ3(t − 1) = 0, φ1(t) + φ2(t − 2) − 2 ˙ φ3(t − 1) = 0. D := Q [∂, δ] (differential time-delay operators) R :=

• δ2

1 −2 δ ∂ 1 δ2 −2 δ ∂

• ∈ D2×3,

M := D1×3/D1×2 R

Parametrizability test

M = D1×p/D1×q R

• M⊤ = Dq×1/R Dp×1

D1×1

@ 2 δ ∂ 2 δ ∂ δ2 + 1 1 A .

D3×1

„ δ2 1 −2 δ ∂ 1 δ2 −2 δ ∂ « .

D2×1 M⊤ D1×1 D1×3

. @ 2 δ ∂ 2 δ ∂ δ2 + 1 1 A

• D1×2

. „ δ2 1 −2 δ ∂ 1 δ2 −2 δ ∂ «

• not exact

D1×1 D1×3

. @ 2 δ ∂ 2 δ ∂ δ2 + 1 1 A

• D1×2

. „ δ2 1 −2 δ ∂ 1 δ2 −2 δ ∂ «

• D1×2

R′ := „ 1 −1 −δ2 − 1 2 δ ∂ «

Parametrizability test

D1×1 D1×3

. @ 2 δ ∂ 2 δ ∂ δ2 + 1 1 A

• D1×2

. „ δ2 1 −2 δ ∂ 1 δ2 −2 δ ∂ «

• D1×2

R′ := „ 1 −1 −δ2 − 1 2 δ ∂ «

• We have

t(M) = D1×2 R′/D1×2 R = 0. In particular, (δ2 − 1) (φ1(t) − φ2(t)) = 0, φ1 − φ2 is 2-periodic.   2 δ ∂ 2 δ ∂ δ2 + 1   is a parametrization of the subsystem    φ1(t) − φ2(t) = 0, −φ2(t − 2) − φ2(t) + 2 ˙ φ3(t − 1) = 0.

Parametrizability test

In fact, we compute ext1

D(M⊤, D) ∼

= t(M).

↓ t(M) ↓ ↓ ↓ ↓ 0 ← − M ← − F0 ← − F1 ← − homD(M⊤, D) ← − 0 ↓ ↓ ↓ ↓ 0 ← − K ⊗D M ← − K ⊗D F0 ← − K ⊗D F1 ← − homD(M⊤, K) ← − 0 ↓ ↓ ↓ ↓ 0 ← − (K/D) ⊗D M ← − (K/D) ⊗D F0 ← − (K/D) ⊗D F1 ← − homD(M⊤, K/D) ← − 0 ↓ ↓ ↓ ↓ ext1

D(M⊤, D)

↓

Parametrizability test

We can compute R′ :=

• 1

−1 −δ2 − 1 2 ∂ δ

• ,

R′′ :=

• δ2

−1 1 −1

• which satisfy

R = R′′ R′. Here we have ker(R′) = 0. ⇒ t(M) ∼ = D1×2/(D1×2 R′′), M/t(M) ∼ = D1×3/(D1×2 R′). M/t(M) corresponds to the parametrizable subsystem    φ1(t) − φ2(t) = 0, −φ2(t − 2) − φ2(t) + 2 ˙ φ3(t − 1) = 0.

System Module Homological Algebra autonomous elements t(M) = 0 ext1

D(MT, D) = 0

controllable, parametrizable t(M) = 0 ext1

D(MT, D) = 0

parametrization exti

D(MT, D) = 0,

is parametrizable reflexive i = 1, 2 . . . . . . . . . exti

D(MT, D) = 0,

. . . projective 1 ≤ i ≤ gld(D) flatness free . . .

Contributions to this classification: Pommaret-Quadrat, Oberst, Fliess.

Presentations

M = D1×p/D1×q R t(M) ∼ = D1×q′ R′/D1×q R R = R′′ R′, R′′ ∈ Dq×q′ ker(.R′) = D1×r′ R2 M/t(M) ∼ = D1×p/D1×q′ R′ t(M) ∼ = D1×q′/D1×(q+r′) R′′ R2

Block-triangular presentation

t(M) ∼ = D1×q′/D1×(q+r ′)

• R′′

R′

2

• ,

M/t(M) ∼ = D1×p/D1×q′ R′ Then, 0 → t(M) → M → M/t(M) → 0 and 0 → t(M) → D1×(p+q′)/D1×(q′+q+r ′)   R′ −Iq′ R′′ R′

2

  → M/t(M) → 0 are equivalent extensions. Can integrate R y = 0 in cascade: R y = 0 ⇐ ⇒        R′ ζ = η, R′′ η = 0, R′

2 η

=

• 3. Injective parametrizations

Injective parametrizations

In general, we do not get an injective parametrization for SolF(M). Moreover:    ˙ x2(t) − u2(t) = 0, ˙ x1(t) − sin(t) u1(t) = has an injective parametrization    u1(t) = ˙ x1(t)/ sin(t), u2(t) = ˙ x2(t), but it is singular at t = 0.

Injective parametrizations

SolF(M) has an injective parametrization ⇐ ⇒ M is free Flatness (Fliess, L´ evine, Martin, Rouchon et al., Pomet, . . . ) Hence, we study how to compute bases of free left D-modules.

Remark

M is free of rank p − q ⇐ ⇒ ∃ R ∈ Dq×p, U ∈ GL(p, D) s.t. R U = (Iq 0) and M ∼ = D1×p/D1×q R.

Theorem (Quillen-Suslin; Stafford)

◮ D = K[∂1, . . . , ∂n]:

projective ⇒ free

◮ D = Weyl algebra (char. 0):

stably free of rank ≥ 2 ⇒ free

Computation of bases of free left An-modules

0 ← − M ← − D1×p

.R

← − D1×q ← − 0 split exact, p − q ≥ 2 0 − → ker(.˜ R) − → D1×p

.˜ R

− → D1×q − → 0 (apply involution of D) find E ∈ GL(p, D) such that E ˜ R =               1 ⋆ . . . ⋆ 0 1 ⋆ . . . 0 0 ... ⋆ 0 0 ... 1 0 0 . . . 0 . . . . . . . . . . . . 0 0 . . . 0               ⇒ ker(.(E ˜ R)) = D1×(p−q) (0 Ip−q) ⇒ ker(.˜ R) = D1×(p−q) (0 Ip−q) E define Q := E (0 Ip−q)T, 0 ← − D1×(p−q)

.Q

← − D1×p

.R

← − D1×q ← − 0 basis of M: the rows of (0 Ip−q) E −1 mod R

Stafford’s Theorem

D = An(k) = k[x1, . . . , xn][∂1, . . . , ∂n], where Q ⊆ k

Theorem (Stafford, 1978)

For any a, b, c ∈ D there exist λ, µ ∈ D such that D a + D b + D c = D (a + λ c) + D (b + µ c). Example. n = 3, D = k[x1, x2, x3][∂1, ∂2, ∂3] For a = ∂1, b = ∂2, c = ∂3 may choose λ = 0, µ = x1: (−x1 ∂3 − ∂2) (a + 0 · c) + ∂1 (b + x1 · c) = c

Stafford’s Theorem – algorithmic!

D = An(k) = k[x1, . . . , xn][∂1, . . . , ∂n], where Q ⊆ k

Theorem (Stafford)

For any a, b, c ∈ D there exist λ, µ ∈ D such that D a + D b + D c = D (a + λ c) + D (b + µ c).

◮ Hillebrand & Schmale (2001) ◮ Leykin (2004)

Maple package Stafford developed for OreModules

Gaussian elimination using Stafford’s result

? E ∈ GL(m, D): E ·      v1 v2 . . . vm      =      1 . . .     , v ∈ Dm unimodular, m > 2 ∃ λ, µ ∈ D : D(v1 + λ vm) + D(v2 + µ vm) + Dv3 + . . . + Dvm−1 = D ∃ b1, . . . , bm−1 ∈ D : b1 v ′

1 + b2 v ′ 2 + b3 v ′ 3 + . . . + bm−1 v ′ m−1 = 1

         v1 + λ vm v2 + µ vm v3 . . . vm−1 vm          =:          v ′

1

v ′

2

v ′

3

. . . v ′

m−1

v ′

m

        

• 

        v ′

1

v ′

2

v ′

3

. . . v ′

m−1

v ′

1 − 1

        

• 

        1 v ′

2

v ′

3

. . . v ′

m−1

v ′

1 − 1

        

• 

        1 . . .         

Row operations

B B B B B B B @ 1 . . . −v′

2

1 . . . −v′

3

1 . . . . . . . . . . . . ... . . . . . . −v′

m−1

. . . 1 −v′

1 + 1

. . . 1 1 C C C C C C C A · B B B B B B B @ 1 . . . −1 1 . . . 1 . . . . . . . . . . . . ... . . . . . . . . . 1 . . . 1 1 C C C C C C C A · B B B B B B B @ 1 . . . 1 . . . 1 . . . . . . . . . . . . ... . . . . . . . . . 1 c1 c2 c3 . . . cm−1 1 1 C C C C C C C A · B B B B B B B @ 1 . . . λ 1 . . . µ 1 . . . . . . . . . . . . ... . . . . . . . . . 1 . . . 1 1 C C C C C C C A

ci := (v ′

1 − v ′ m − 1) bi,

i = 1, . . . , m − 1

Power series case

Let k be a field, char(k) = 0 resp. k ∈ {R, C}.

Definition

A noetherian prime ring D is very simple if ∀ d1, . . . , d4 ∈ D, d4 regular, ∃ λ, µ ∈ D: D d1 + D d2 + D d3 = D (d1 + d4 λ d3) + D (d2 + d4 µ d3).

Remark

(a) dim k[[t]] = dim k{t} = 1. (b) k[[t]]∂ and k{t}∂ are simple.

Theorem (cf. McConnell, Robson)

A commutative ring, dim(A) < ∞, not artinian. If D := A∂ is simple, then dim D = dim A.

Theorem (Coutinho-Holland, 1988)

D simple noetherian ring of Krull dim. 1 ⇒ D is very simple.

Example

   ˙ x2(t) − u2(t) = 0, ˙ x1(t) − sin(t) u1(t) = has an injective parametrization    u1(t) = ˙ x1(t)/ sin(t), u2(t) = ˙ x2(t), but it is singular at t = 0.

Example

R =

∂ −1 ∂ − sin(t)

• ,
• R =

 

−∂ −∂ − sin(t) −1

 

B B @ 1 ∂ 1 1 2 1 1 C C A · B B @ 1 −1 1 1 1 1 C C A · B B @ 1 1 1 1 1 1 C C A · B B @ 1 1 1 1 1 1 C C A · B B @ −∂ −∂ − sin(t) −1 1 C C A = B B @ 1 − sin(t) −∂ 1 C C A @ 1 − sin(t) 1 −∂ + 1 1 1 A · @ 1 −1 1 1 1 A · @ 1 1 cos(t) sin(t) − cos(t) ∂ − sin(t) 1 1 A · @ 1 1 1 1 1 A · @ sin(t) ∂ 1 A = @ 1 1 A

Example

V := e E = B B B B @ cos(t) sin(t) −1 + cos(t) sin(t) cos(t) ∂ − 2 sin(t) −1 (cos(t) sin(t) − 1) ∂ + 2 cos(t)2 − 1 − cos(t) sin(t)2 cos(t) sin(t) ∂ − 1 − sin(t) (cos(t) sin(t) − 1) (cos(t) sin(t) − 1) ∂ − 1 − cos(t) sin(t) ∂ − 3 cos(t)2 + 1 (cos(t) ∂ − 2 sin(t)) ∂ (sin(t) − cos(t) + cos(t)3) ∂ − 3 cos(t)2 sin(t) + sin(t) + cos(t) (cos(t) sin(t) − 1) ∂2 − 2 sin(t)2 ∂ 1 C C C C A .

V −1 =   

∂ −1 ∂ − sin(t) cos(t) ∂ − 2 sin(t) − cos(t) ∂ + 2 sin(t) −1 −1 + cos(t) sin(t) − cos(t) sin(t)

   ∈ D4×4.

Example

M = D1×p/D1×q R is free with basis

(cos(t) ∂ − 2 sin(t), − cos(t) ∂ + 2 sin(t), −1, 0), (−1 + cos(t) sin(t), − cos(t) sin(t), 0, 0) mod D1×q R.

Equivalently,    ˙ x2(t) − u2(t) = 0, ˙ x1(t) − sin(t) u1(t) = has non-singular injective parametrization (x1, x2, u1, u2)T = P (z1, z2)T P =  

− cos(t) sin(t)2 cos(t) sin(t) ∂ − 1 − sin(t) (cos(t) sin(t) − 1) (cos(t) sin(t) − 1) ∂ − 1 − cos(t) sin(t) ∂ − 3 cos(t)2 + 1 (cos(t) ∂ − 2 sin(t)) ∂ (sin(t) − cos(t) + cos(t)3) ∂ − 3 cos(t)2 sin(t) + sin(t) + cos(t) (cos(t) sin(t) − 1) ∂2 − 2 sin(t)2 ∂

 

Maple package OreModules

for multidimensional linear systems over Ore algebras

◮ decide controllability and parametrizability, ◮ construct (minimal) parametrizations, ◮ decide flatness (also π-freeness),

etc.

◮ tools for motion planning and linear quadratic optimal control

problems, etc.

◮ compute flat outputs for time-varying systems of

• rdinary/partial differential equations of rank at least 2

related packages: Stafford, OreMorphisms, QuillenSuslin, PurityFiltration, . . . http://wwwb.math.rwth-aachen.de/OreModules

homalg (for Maple and for GAP)

◮ ring-independent methods from homological algebra ◮ needs other packages which provide the ring arithmetics ◮ given a functor “on objects”, its part “on morphisms” is

provided by homalg

◮ composition and (left / right) derivation of functors is

automatic

◮ easy to define new functors

http://homalg.math.rwth-aachen.de

Barakat – R., homalg: A meta-package for homologial algebra, Journal of Algebra and Its Applications 7 (3):299–317, 2008.

References

Malgrange, B., Syst` emes ` a coefficients constants, S´ eminaire Bourbaki 246:79–89, 1962–63. Oberst, U., Multidimensional constant linear systems, Acta Appl. Math. 20:1–175, 1990. Pommaret, J.-F. and Quadrat, A., Algebraic analysis of linear multidimensional control systems, IMA Journal of Control and Information 16 (3):275–297, 1999. Pommaret, J.-F. and Quadrat, A., A functorial approach to the behavior of multidimensional control systems, Applied Mathematics and Computer Science, 13:7–13, 2003. J.-F. Pommaret Partial Differential Control Theory Kluwer, 2001

References

Baer, R., Erweiterungen von Gruppen und ihren Isomorphismen,

• Math. Z. 38 (1):375–416, 1934.

Cartan, H. and Eilenberg, S., Homological Algebra, Princeton University Press, 1956. MacLane, S., Homology, Springer, 1995. Rotman, J. J., An Introduction to Homological Algebra, Academic Press, 1979.

References

Chyzak, F. and Quadrat, A. and Robertz, D., Effective algorithms for parametrizing linear control systems over Ore algebras, Applicable Algebra in Engineering, Communications and Computing 16 (5):319–376, 2005. Quadrat, A. and Robertz, D., Parametrizing all solutions of uncontrollable multidimensional linear systems, in: 16th IFAC World Congress, Prague, 2005. Quadrat, A. and Robertz, D. On the blowing-up of stably free behaviours, 44th IEEE CDC and ECC, Seville, 2005.

References

Quadrat, A. and Robertz, D., On the Monge problem and multidimensional optimal control, in: 17th MTNS, Kyoto, Japan, 2006. Quadrat, A. and Robertz, D., Baer’s extension problem for multidimensional linear systems, in: 18th MTNS, Virginia Tech, USA, 2008. Quadrat, A. and Robertz, D., Computation of bases of free modules over the Weyl algebras,

• J. Symbolic Computation, 42:1113–1141, 2007

Quadrat, A. and Robertz, D., Controllability and differential flatness of linear analytic ordinary differential systems, in: 19th MTNS, Budapest, 2010.

References

Quadrat, A., Syst` emes et Structures: Une approche de la th´ eorie math´ ematique des syst` emes par l’analyse alg´ ebrique constructive, Habilitation thesis, Univ. de Nice Sophia Antipolis, 2010. Cluzeau, T. and Quadrat, A., Factoring and decomposing a class of linear functional systems, Linear Algebra Appl. 428(1):324–381, 2008. Fabia´ nska, A. and Quadrat, A., Applications of the Quillen-Suslin theorem to multidimensional systems theory, in: Gr¨

• bner bases in control theory and signal processing, pp.

23–106, Radon Ser. Comput. Appl. Math. 3, de Gruyter, 2007.

References

Stafford, S. T. Module structure of Weyl algebras,

• J. London Math. Soc. 18:429–442, 1978.

Coutinho, S. C. and Holland, M. P. Module structure of rings of differential operators,

• Proc. London Math. Soc. 57:417–432, 1988.

Hillebrand, A. and Schmale, W. Towards an Effective Version of a Theorem of Stafford,

• J. Symbolic Computation, 32:699–716, 2001

Leykin, A. Algorithmic proofs of two theorems of Stafford,

• J. Symbolic Computation, 38:1535–1550, 2004

Dmodules in Macaulay2 (with Tsai, H.)

References

Barakat, M. and Robertz, D., homalg: A meta-package for homologial algebra, Journal of Algebra and Its Applications 7 (3):299–317, 2008. Chyzak, F. and Quadrat, A. and Robertz, D., OreModules: A symbolic package for the study of multidimensional linear systems, in: Chiasson, J. and Loiseau, J.-J. (eds.), Applications

• f Time-Delay Systems,

LNCIS 352, 233–264, Springer, 2007. Cluzeau, T. and Quadrat, A., OreMorphisms: A homological algebra package for factoring and decomposing linear functional systems, in: Loiseau, J.-J., Michiels, W., Niculescu, S.-I., Sipahi, R. (eds.), Topics in Time-Delay Systems: Analysis, Algorithms and Control, LNCIS, Springer, 2008. Robertz, D. Formal Computational Methods for Control Theory, PhD thesis, RWTH Aachen University, 2006

References

Fliess, M., Some basic structural properties of generalized linear systems, Systems & Control Letters 15:391–396, 1990. Fliess, M., L´ evine, J., Martin, P. and Rouchon, P., Flatness and defect of nonlinear systems: introductory theory and examples,

• Int. J. Control 61:1327–1361, 1995.

Polderman, J. W. and Willems, J. C., Introduction to Mathematical Systems Theory: A Behavioral Approach, TAM 26, Springer, 1998. Pillai, H. and Shankar, S., A behavioral approach to control of distributed systems, SIAM J. Contr. Opt. 37:388–408, 1999.

References

Lomadze, V. and Zerz, E., Control and interconnection revisited: the linear multidimensional case, in: 2nd Int. Workshop on Multidimensional (ND) Systems, Czocha Castle, 2000. Zerz, E., Topics in Multidimensional Linear Systems Theory, LNCIS 256, Springer, 2000. Dubois, F. and Petit, N. and Rouchon, P., Motion Planning and Nonlinear Simulations for a Tank Containing a Fluid, in: Proc. ECC, Karlsruhe (Germany), 1999. Petit, N. and Rouchon, P., Dynamics and solutions to some control problems for water-tank systems, IEEE Trans. Autom. Contr. 47 (4): 594–609, 2002.