An algebraic analysis approach to
mathematical system theory
NETCA Workshop:
Verification and Theorem Proving for Continuous Systems
Oxford, 26/08/05
Alban Quadrat
INRIA Sophia Antipolis,
CAFE Project,
2004 route des lucioles, BP 93,
06902 Sophia Antipolis cedex,
France.
Alban.Quadrat@sophia.inria.fr
www-sop.inria.fr/cafe/Alban.Quadrat/index.html
Synthesis problems
Transfer functions
• Finite-dimensional system:
1
ẋ(t) = x(t)+u(t), x(0) = 0 ⇒ x̂(s) = û(s).
(s − 1)
• Differential time-delay system:
ẋ(t) = x(t) + u(t), x(0) = 0,
(
0, 0 ≤ t ≤ 1,
y(t) =
x(t − 1), t ≥ 1,
e−s
⇒ ŷ(s) = û(s).
(s − 1)
• System of partial differential equations:
2
∂ z (x, t) − ∂ 2 z (x, t) = 0,
∂t2 ∂x2
∂z (0, t) = 0, ∂z (1, t) = u(t),
∂x ∂x
y(t) = ∂z (1, t),
∂t
(1 + e−2 s)
⇒ ŷ(s) = −2 s
û(s).
(1 − e )
• The poles of the transfer functions (1, 1, k π i, k ∈ Z)
belong to C+ = {s ∈ C | Re(s) ≥ 0} ⇒ unstability.
A module approach of synthesis problems
1. An integral domain A of SISO stable systems is
chosen (e.g., A = RH∞, H∞(C+), Â. . . ).
2. The plant is defined by a transfer matrix:
P ∈ K q×r , K = Q(A) = {n/d | 0 6= d, n ∈ A} .
3. We write P as:
− N ) ∈ Aq×p,
(
(D
P = D−1 N = Ñ D̃−1,
(Ñ T D̃T )T ∈ Ap×r .
(e.g., D = d Iq , N = d P , D̃ = d Ir , Ñ = d P ).
!
y
(D − N) = 0,
u
3. y = P u ⇔ ! ! (?)
y Ñ
= z.
u D̃
4. Synthesis problems are reformulated in terms
of the properties of (?).
• Linear algebra over rings is module theory
⇒ a module approach to synthesis problems.
Stable algebras A of SISO systems
1. RH∞ = n(s)
n
d(s)
∈ R(s) | deg n(s) ≤ deg d(s),
d(s) = 0 ⇒ Re(s) < 0 }
1
1 =
h1 = s−1 s+1 , 1 , s−1 ∈ RH
∞
s−1 s+1 s+1
s+1
⇒ h1 ∈ Q(RH∞) = R(s).
2. A = {f (t) + +∞
i=0 ai δt−ti | f ∈ L1 (R+ ),
P
(ai)i≥0 ∈ l1(Z+), 0 = t0 ≤ t1 ≤ t2...}
and  = {ĝ | g ∈ A} the Wiener algebras.
e−s
−s e−s , s−1 ∈ Â ⇒ h ∈ Q(Â).
e
h2 = s−1 = s+1 , 2
s−1 s+1 s+1
s+1
3. C+ = {s ∈ C | Re s > 0}. The Hardy algebra
H∞(C+) = { holomorphic functions f in C+ |
k f k∞= sups∈C+ | f (s) |< +∞}.
(1+e−2 s )
h3 = (1−e−2 s) , 1 + e−2 s, 1 − e−2 s ∈ H∞(C+)
⇒ h3 ∈ Q(H∞(C+)).
Example
• We consider the transfer matrix:
e−s
s−1
P = .
e−s
(s−1)2
• Let us consider A = H∞(C+) and K = Q(A).
• We have:
−s (s−1) e−s u = 0,
y1 =
e u,
y
(s+1) 1
−
(s−1) (s+1)
−s ⇒ 2
y2 = e
u
s−1 e−s u = 0,
y2 − (s+1)
(s−1)2 2
s+1
!
y
⇒R = 0,
u
e−s
s−1 0 − s+1
s+1
with R = ∈ A2×3.
2
e−s
0 s−1 − (s+1)2
s+1
| {z } | {z }
D −N
• We have:
P = D−1 N ∈ K 2.
• Properties of P can be studied by means of the
matrix R with entries in the Banach algebra A.
Doubly coprime factorizations
P = D −1 N = Ñ D̃ −1 ∈ K q×r ,
R = (D − N ) ∈ Aq×p,
T T T p×r
R̃ = (Ñ D̃ ) ∈ A .
• Definition: P admits a doubly coprime factor-
ization if there exist
0 = (D 0 0 ) ∈ Aq×p,
R − N
T T
0
R̃ = (Ñ 0 D̃0 )T ∈ Ap×r ,
such that:
S = (X T Y T )T ∈ Ap×q ,
S̃ = (−Ỹ X̃) ∈ Ar×p,
R0
! !
Iq 0
S R̃0 = = Ip.
S̃ 0 Ir
• Theorem: Let us define the A-modules:
M = A1×p/(A1×q R), N =A1×p / A1×r T
R̃ .
Then, we have the following equivalences:
1. P admits a doubly coprime factorization.
2. The A-modules M/t(M ) and N/t(N ) are free
of rank respectively q and r.
3. The A-modules A1×p RT and A1×p R̃ are free
of rank respectively q and r.
Internal stabilizability
• Let A be an integral domain of SISO stable plants.
• K = {n/d | 0 6= d, n ∈ A} field of fractions of A.
• P ∈ K q×r a plant.
• C ∈ K r×q a controller.
• The closed-loop system is defined by:
u1 + e1 y1
C
+
y2 e2 + u2
P
+
u1 , u2 : external inputs, e1 , e2 : internal inputs, y1 , y2 : outputs.
! ! ! (
u1 Iq −P e1 y1 = e2 − u2,
= ,
u2 −C Ir e2 y2 = e1 − u1.
• Definition: C internally stabilizes P if the trans-
!−1
Ir −P
fer matrix T = satisfies:
−C Ir
(Iq − P C)−1 (I − Pq C)−1 P
T = C(Iq − P C)−1 Ir + C (Iq − P C)−1 P
∈ A(q+r)×(q+r).
L2 − L2 stability if A = H∞(C+),
• Internal stability ⇔
L − L stability if A = Â.
∞ ∞
Internal stabilizability
−1 N = Ñ D̃ −1 ∈ K q×r ,
P =D
R = (D − N ) ∈ Aq×p,
T T T r×p
R̃ = (Ñ D̃ ) ∈ A .
• Theorem: Let us define the A-modules:
M = A1×p/(A1×q R), N =A1×p / A1×r T
R̃ .
Then, we have the following equivalences:
1. P is internally stabilizable.
2. The A-modules M/t(M ) and N/t(N ) are pro-
jective of rank respectively q and r.
3. The A-modules A1×p RT and A1×p R̃ are pro-
jective of rank respectively q and r.
• Corollary: P = D−1 N is internally stabilizable
iff ∃ S = (X T Y T )T ∈ K p×q such that:
!
X D −X N
1. S R = ∈ Ap×p,
Y D −Y N
2. R S = D X − N Y = Iq .
The controller C = Y X −1 internally stabilizes P .
Example
• We consider the transfer matrix (A = H∞(C+)):
e−s
(s−1) 2
P = −s ∈K , K = Q(A).
e
(s−1)2
• We have P = D−1 N where R is defined by:
e−s
s−1 0 − s+1
s+1 2×3 .
R= ∈A
2
e−s
0 s−1 − (s+1)2
s+1
• The matrix S = (X T Y T )T ∈ K 3×2 defined by
2
(s−1)
s−1
b s+1 + s−1 2 2 (b − 1) (s+1)
(s−1) 1 2 b + s+1
S= 2 − s−1
b (s+1) ,
s+1 s−1
(s−1)
−a (s+1) 2a
− s+1
2
4 e (5 s − 3) (s + 1)3 − 4 (5 s − 3) e−(s−1)
a= , b= ∈ A,
(s + 1) (s + 1) (s − 1)2
satisfies S R ∈ A3×3 and R S = D X −N Y = I2.
⇒ P is internally stabilized by C = Y X −1, i.e.:
−4 (5 s − 3) e (s − 1)2
C= (1 2).
3
(s + 1) ((s + 1) − 4 (5 s − 3) e −(s−1) )
