Enlarged finite time exact compensation in discrete disturbed systems

Enlarged Finite Time Exact Compensation in Discrete Disturbed Systems

Larbi AFIFI

Laboratory MACS, Faculty of Sciences Ain Chock B.P.5366-Maˆ arif, Casablanca, Morocco

l.afifi@fsac.ac.ma, larbi

afifi@yahoo.fr Maria HAKAM

Laboratory MACS, Faculty of Sciences Ain Chock B.P.5366-Maˆ arif, Casablanca, Morocco

[email protected] Abdelhakim CHAFIAI

Laboratory MACS, Faculty of Sciences Ain Chock B.P.5366-Maˆ arif, Casablanca, Morocco

a.chafi[email protected] Mehdi BENSSASSI

Laboratory MACS, Faculty of Sciences Ain Chock B.P.5366-Maˆ arif, Casablanca, Morocco

[email protected] Abstract

In this work, we study the notion of enlarged exact compensation for

a class of discrete linear distributed systems. It’s an extension to the dis-

crete and regional cases of previous works on the continuous finite time

remediability problem. It consists to study, with respect to the output

operator (sensors), the existence of an input one ensuring the reduction

of a disturbance effect on the output by bringing the observation in a

given tolerance zone C. We show that under convenient hypothesis and

with a convenient choice of the input operator (actuators), the consid-

ered control problem admits a unique solution and we show how to find

it. Then, we extend this notion as well as the developed approach to

the regional discrete case.

Keywords: Enlarged exact remediability, control, observation, actuators, sensors, region.

1 Considered system and notations

Without loss of generality , we consider a class of linear distributed systems [2, 4, 6, 7, 8] described by the following discrete equation

(S

)

⎧ ⎨

⎩

z

= φz

+ Bu

+ f

; k = 0, · · · , N − 1 z

∈ X

(1)

where φ ∈ L (X), B ∈ L (U, X ), z

, f

and u

∈ U are respectively the state, the disturbance, the control at step k and N is an integer, N ≥ 1. X and U are supposed to be Hilbert spaces. The system (1) is augmented by the output equation

(E

) y = Cz (2)

where C ∈ L (X, Y ), z = (z

, ...., z

)

, y = (y

, ...., y

)

with y

= Cz

for k = 1, · · · , N . Y is the observation space, a Hilbert space.

The state of the system (1) at the final step N , noted z

, is given by

z

= φ

z

+

N

φ

Bu

+

N−1

i=0

φ

f

and the corresponding observation is given by

y

= Cφ

z

+

N−1

i=0

Cφ

Bu

+

N−1

i=0

Cφ

f

Let H

et H

be the operators defined by

H

: U

−→ X

u = (u

, ..., u

)

−→ H

u =

N

φ

Bu

(3)

and

H

: X

−→ X f = (f

, ..., f

)

−→ H

f =

N−1

i=0

φ

f

(4) The observation at the final step N becomes

y

= Cφ

z

+ CH

u + CH

f or

y

= Cφ

z

+ K

u + R

f (5) where K

and R

are the operators defined by

K

u = CH

u =

N

Cφ

Bu

(6)

and

R

f = CH

f =

N−1

i=0

Cφ

f

(7)

Let us note that the systems (1) and (2) can be a discrete version of con- tinuous time equations. Indeed, if we consider the system described by the following linear state equation

⎧ ⎨

⎩

z(t) =

Az(t) + Bu(t) + g(t) ; 0 < t < T

z

∈ X

(8)

augmented by the following output equation

y(t) = Cz(t) ; 0 < t < T (9) where A generates a strongly continuous semi-group (s.g.f.c) (S(t))

on the state space X, B ∈ L (U, X ), C ∈ L (X, Y ), u ∈ L

(0, T ; U) and g ∈ L

(0, T ; X), then, for τ =

N

> 0 with N sufficiently large, the corresponding discrete ver-

sion is as follows

⎧ ⎨

⎩

z

= φz

+ B u

+ f

; 0 ≤ k ≤ N − 1 z

∈ X

(10)

augmented by the output equation

y

= Cz

; 0 ≤ k ≤ N − 1 (11)

where z

= z(kτ ); u

= u(kτ ) and

φ = S(τ ); B u

=

0

S(τ − s)Bu

(s)ds ; f

=

0

S(τ − s)g

(s)ds (12) u

and g

are respectively the restrictions of u and g to the time interval [kτ, (k + 1)τ [. If τ is small, u

and g

can be assumed to be constant on the interval [kτ, (k + 1)τ [.

2 Notion of enlarged exact remediability

In the ”normal” case, i.e. without disturbance and without control term, the system (1) becomes

⎧ ⎨

⎩

z

= φz

; k = 0, · · · , N − 1 z

∈ X

The state z

at step k is given by

z

= φ

z

and the corresponding observation y

is given by

y

= Cφ

z

But in the ”abnormal” case where the system is excited by a known or unknown

disturbance f = (f

, · · · , f

)

, the observation at step k becomes

y

= Cφ

z

+

i=0

Cφ

f

; k = 1, · · · , N Generally, we have y

= Cφ

z

; k = 1, · · · , N.

The problem consists to study the existence of an input operator B, with respect to the output one, in order to reduce the effect of this disturbance at the final step N by bringing the observation in a given zone of tolerance C , i.e.

such that:

For any f = (f

, · · · , f

)

∈ X

, there exists u = (u

, · · · , u

)

∈ U

satisfying

y

= Cφ

z

+

N−1

i=0

Cφ

Bu

+

N−1

i=0

Cφ

f

∈ C (13) where C is a convex, closed and nonempty subset of Y . This leads to the following definition

Definition 2.1

i) A disturbance f ∈ X

is said to be C -remediable if there exists u ∈ U

such that

y

= Cφ

z

+ K

u + R

f ∈ C where K

and R

are respectively defined by (6) and (7).

ii) The system (1) augmented by (2) is said to be C -remediable if any distur- bance f ∈ X

is C -remediable.

Remark 2.2 Let us note that:

1) f is C -remediable ⇐⇒ Im(K

) ∩C

= ∅ where C

= C − Cφ

z

− R

f . 2) If C =

Cφ

z

, then we have a problem of exact remediability.

3) If C = B(0, ), then a disturbance f C -remediable is said -remediable, and f is -remediable ⇐⇒ Im(K

) ∩ B (R

f, ) = ∅

⇐⇒ P

(R

f) <

where P

is the orthogonal projection on F, F is a closed subset of Y .

4) If f is -remediable, then for any

> , f is

-remediable. The converse is not true.

5) The cost increases when decreases.

3 Enlarged exact remediability with minimum energy

Let C be a convex, closed and nonempty subset of Y and f ∈ X

. We consider the following problem (P

) of enlarged exact remediability with minimum energy

(P

)

min J(u) with J(u) = u

subject to y

∈ C (14)

We suppose that the disturbance f is C -remediable, the problem (P

) is then well defined and has a unique solution v

in the set of admissible controls defined by

U

=

u ∈ U

/y

∈ C The solution v

of (P

) is characterized by

J (v

)(v − v

) ≥ 0 ; ∀ v ∈ U

i.e.

v

, v − v

≥ 0 ; ∀ v ∈ U

The problem (P

) is a generalization of the following problem (P

) of exact compensation

(P

)

min u

subject to K

u + R

f = 0 (15) It is sufficient to consider C =

Cφ

z

. If u

is the solution of (P

), then

v

≤ u

Hence, the optimal cost corresponding to (P

) is reduced with respect to that corresponding to (P

).

The problem (P

) is also a generalization of the -remediability one, it sufficient to consider C = B (0, ).

If C

and C

are two nonempty, closed and convex subsets of Y such that C

⊂ C

, then the C

-remediability implies the C

-remediability, and the cost is decreasing when C is increasing.

For the resolution of the problem (P

), we use an extension of the Hilbert Uniqueness Method (HUM) combined with an appropriate penalization method.

For θ ∈ Y

≡ Y , let

θ

= (

N

B

(φ

)

C

θ

U

)

(16)

·

is a semi-norm. If

Ker(C

) = { 0 } , then ·

is a norm on Y if and only if (S

) + (E

) is weakly remediable. Under this condition, we consider the space

F

= Y

F

is a Hilbert space with the inner product

θ, σ

=

N−1

k=0

B

(φ

)

C

θ, B

(φ

)

C

σ

U

and the operator Λ

defined on Y by

Λ

θ =

N−1

k=0

Cφ

BB

(φ

)

C

θ

admits a unique extension as an isomorphism from F

−→ F

such that Λ

θ, σ

= θ, σ

; ∀ θ, σ ∈ F

The following result gives a characterization of the optimal control ensuring the C -compensation of a disturbance f and shows how to find it.

1

This hypothesis is true in the usual case where the observation is given by means of

sensors [2, 5, 7].

Theorem 3.1

If C is a convex, closed and nonempty subset of Y and if C ∩ (Cφ

z

+ R

f + F

) = ∅ then

i- there exists a unique θ

in F

such that

Λ

θ

+ Cφ

z

+ R

f ∈ C (17) and

θ

, y − Λ

θ

− Cφ

z

− R

f

≥ 0 ; ∀ y ∈ C ∩ (Cφ

z

+ R

f + F

) (18) ii- The control u

defined by

u

= K

θ

(19)

is the unique solution of the problem (P

). Moreover, u

is optimal with u

U^N

= θ

Proof:

Unicity: Let θ

and σ

∈ F

be such that for any y ∈ C ∩ (Cφ

z

+ R

f + F

), we have

θ

, y − Λ

θ

− Cφ

z

− R

f

≥ 0

and

σ

, y − Λ

σ

− Cφ

z

− R

f

≥ 0

then for y = Λ

σ

+ Cφ

z

+ R

f in the first inequality and y = Λ

θ

+ Cφ

z

+ R

f in the second one, we obtain

θ

, Λ

σ

− Λ

θ

≥ 0 and

σ

, Λ

θ

− Λ

σ

≥ 0

Hence

θ

, Λ

σ

− Λ

θ

+ σ

, Λ

θ

− Λ

σ

= θ

− σ

, Λ

σ

− Λ

θ

= − θ

− σ

≥ 0 consequently, θ

= σ

.

Existence: It’s more complex and is done in stages.

For α > 0 , we consider the following criterion with two variables

J

(y, v) = 1

α y

− y

+ v

(20) where y ∈ Y ; v ∈ U

and y

is given by (5). We consider the corresponding minimization problem

(P

)

min J

(y, v)

y ∈ C et v ∈ U

(21)

We have the following result concerning the existence of a solution for (P

).

Proposition 3.2

1- The minimization problem (P

) admits a solution (y

, v

).

2- (y

, v

) verifies the inequalities 1

α y

− y

, y − y

≥ 0 ; ∀ y ∈ C (22) and

− 1

α y

− y

, y

− y

+ v

, v − v

≥ 0 ; ∀ v ∈ U

(23) Proof:

1- Let (y

, v

)

be a minimizing sequence.

J

(y

, v

) inf

(y,v)∈C×U^N

J

(y, v) when k −→ + ∞

The sequence k −→ J

(y

, v

) is bounded because it is convergent. Conse-

quently, there exists c

> 0 such that J

(y

, v

) ≤ c

; ∀ k ≥ 0.

Since v

= J

(y

, v

) − 1 α

y

v^(k)α ,f

− y

≤ J

(y

, v

)

then v

≤ c

; ∀ k ≥ 0, i.e. (v

)

is bounded in U

. The map v = (v

, ..., v

)

∈ U

−→ K

v =

N−1

i=0

Cφ

Bv

∈ Y is linear and continuous, then there exists c

> 0 such that

K

v = y

− y

≤ c

v ; ∀ v ∈ U

Consequently, (y

vα^(k),f

)

is bounded in Y . On the other hand y

v^(k)α ,f

− y

= αJ

(y

, v

) − α v

≤ αJ

(y

, v

) ≤ αc

; ∀ k ≥ 0 then (y

)

is bounded in Y .

Finally, the sequence (y

, v

)

is bounded in Y × U

. Hence, one can extract a convergent subsequence which converges to a limit (y

, v

).

J

is continuous on Y × U

, then J

(y

, v

) ≤ lim inf

k

J

(y

, v

) = inf

(y,v)∈C×U^N

J

(y, v) Since C is closed, (y

, v

) ∈ C × U

and

(y,v)∈C×U

inf

J

(y, v) ≤ J

(y

, v

) Then J

(y

, v

) = inf

(y,v)∈C×U^N

J

(y, v), i.e. (y

, v

) is a solution of (P

).

2- (y

, v

) satisfies the following necessary condition

J

(y

, v

)(y − y

, v − v

) ≥ 0 ; ∀ (y, v) ∈ C × U

We have

J

(y

, v

)(y, v) = 2 α

y

− y

, y

− Cφ

z

− R

f

− 2

α y

− y

, y

+2 v

, v

then

J

(y

, v

)(y − y

, v − v

) = 2 α

y

− y

, y

− y

− 2

α y

− y

, y − y

+ 2 v

, v − v

The necessary condition becomes

1

α y

− y

, y

− y

− 1

α y

− y

, y − y

+ v

, v − v

≥ 0 (24) By replacing respectively in (24), v by v

and y by y

, we obtain the inequalities (22) and (23). We deduce the following characterization result.

Proposition 3.3 (y

, v

) is characterized by

b

, y − y

≥ 0 ; ∀ y ∈ C (25) where b

is given by

b

= 1

α (y

− y

) (26)

and v

= (v

, · · · , v

) with

v

= B

(φ

)

C

b

; k = 0, ..., N − 1 (27) Proof: The element b

given by (26) appears in inequality (22) of propo- sition 3.2. We have

− 1

α y

− y

, y

− y

+ v

, v − v

= −

b

,

N−1

i=0

Cφ

B (v

− v

)

+

N

v

, v

− v

= −

i=0

B

(φ

)

C

b

, v

− v

+

N−1

i=0

v

, v

− v

=

N

v

− B

(φ

)

C

b

, v

− v

By reporting this relation in (23), we obtain

N−1

i=0

v

− B

(φ

)

C

b

, v

− v

≥ 0; ∀ v ∈ U

(28)

which implies

v

= B

(φ

)

C

b

; i = 0, ..., N − 1 (29)

Before proceeding to the limit, we show the following preliminary convergence results.

Proposition 3.4

1- The sequence (y

− y

) converges strongly in Y to 0.

2- The sequence J

(y

, v

) is increasing and bounded, then convergent.

3- The sequence (y

, v

, b

) is bounded in Y × U

× F

. Proof:

1- Under the C -remediability hypothesis, there exists a control ˆ u = (ˆ u

, ..., u ˆ

) in U

such that y

∈ C . By noting ˆ y = y

, we have (ˆ y, u) ˆ ∈ C × U

, and hence

J

(y

, v

) ≤ J

(ˆ y, u) = ˆ u ˆ

(30) then for every α > 0, we have

y

− y

= αJ

(y

, v

) − α v

≤ αJ

(y

, v

) ≤ α u ˆ

consequently lim

α−→0

y

− y

= 0.

2- i- Using (30), we deduce that the sequence (J

(y

, v

))

is bounded.

ii- For δ ≤ α, we have

J

(y, v) ≤ J

(y, v) ; ∀ (y, v) ∈ C × U

consequently

J

(y

, v

) ≤ J

(y

, v

) ≤ J

(y

, v

) then J

(y

, v

) increases when α −→ 0.

3- i- (v

)

is bounded because

v

≤ J

(y

, v

) ≤ J

(ˆ y, u) = ˆ u ˆ

ii- The map v ∈ U

−→ y

∈ Y is affine and continuous, then there exists c

> 0 such that

y

− Cφ

z

− R

f ≤ c

v ; ∀ v ∈ U

The sequence (y

)

is then bounded because (v

)

is.

This shows that y

= (y

− y

) + y

is bounded in Y .

iii- Generally, (b

)

is not bounded in Y but it is bounded in F

, because v

=

N

v

U

=

N

B

(φ

)

C

b

U

=

N

B

(φ

)

C

b

U

= b

To conclude, we show the following result where we give the solution of the initial problem (P

) and its characterization.

Proposition 3.5 We consider a convergent subsequence of (y

, v

, b

)

, with the same notation. Its limit (y

, v

, b) is characterized by

1- y

= y

2- v

is the solution of the problem (P

) given by (14).

3- The sequence J

(y

, v

) is increasing with a limit v

and the sequence (v

)

converges strongly to v

in U

.

4- The control v

=

v

, ..., v

is given by

v

= B

(φ

)

C

b ; k = 0, ..., N − 1 (31)

5- The element b is characterized by

b, y − y

≥ 0 ; ∀ y ∈ C ∩ (Cφ

z

+ R

f + F

) (32) Proof:

1- The sequence (v

)

converges weakly to v

and the map v ∈ U

−→

y

− Cφ

z

− R

f ∈ Y is linear and continuous, then (y

)

converges weakly to y

and hence

y

= lim

α−→0

y

= lim

α−→0

(y

− y

) + lim

α−→0

y

= y

2- The set C is closed, then y

= lim

α−→0

y

∈ C . Moreover, if v ∈ U

is such that y

∈ C , then

v

= J

(y

, v

) − 1

α y

− y

≤ J

(y

, v

) ≤ J

(y

, v) = v

(33)

i.e. v

≤ v

, consequently

v

≤ lim inf

α

v

≤ v

Then (v

) is a solution of (P

).

3.i- We show in proposition 3.4 that the sequence J

(y

, v

) is increasing when α is decreasing to 0.

ii- Using (33), for any v ∈ U

such that y

∈ C , we have v

≤ J

(y

, v

) ≤ v

Particulary for v = v

, we obtain

v

≤ J

(y

, v

) ≤ v

(34)

The weak convergence of (v

)

to v

implies v

≤ lim inf

α

v

≤ lim

α

J

(y

, v

) ≤ v

then

α−→0

lim J

(y

, v

) = v

iii- Using the inequality (34), we have

v

≤ lim inf

α

v

≤ lim sup

α

v

≤ v

and then v

converges to v

. Using the weak convergence of (v

)

to v

, we deduce that (v

)

converges strongly to v

in U

.

4- Since v

=

v

, ..., v

with v

= B

(φ

)

C

b

; k = 0, ..., N − 1 then for w = (w

, ..., w

) ∈ U

, we have

v

, w =

b

,

N−1

k=0

Cφ

Bw

=

b

, y

− R

f − Cφ

z

On the other hand y

− R

f − Cφ

z

∈ F

, then from the weak convergence of (v

)

to v

and that of (b

)

to b, we deduce that

v

, w =

b, y

− R

f − Cφ

z

=

b,

N−1

k=0

Cφ

Bw

=

N−1

k=0

B

(φ

)

C

b, w

and hence v

= B

(φ

)

C

b ; k = 0, ..., N − 1.

5- The inequality (25) can be written

b

, y

≤ b

, y ; ∀ y ∈ C

i.e.

b

, y

− R

f − Cφ

z

≤

b

, y − R

f − Cφ

z

; ∀ y ∈ C (35) We have

J

(y

, v

) = 1

α y

− y

+ v

= b

, y

− y

+ v

=

b

, y

− R

f − Cφ

z

−

b

, y

− R

f − Cφ

z

+ v

=

b

, y

− R

f − Cφ

z

−

b

,

N−1

k=0

Cφ

Bv

+ v

=

b

, y

− R

f − Cφ

z

−

N−1

k=0

B

(φ

)

C

b

, v

+ v

=

b

, y

− R

f − Cφ

z

then

b

, y

− R

f − Cφ

z

= J

(y

, v

) Consequently

α−→0

lim

b

, y

− R

f − Cφ

z

= lim

α−→0

J

(y

, v

) = v

=

N

B

(φ

)

C

b, v

=

b,

N−1

k=0

Cφ

Bv

=

b, y

− R

f − Cφ

z

For y − R

f − Cφ

z

∈ F

and when α −→ 0 in (35) , we obtain b, y

− R

f − Cφ

z

≤

b, y − R

f − Cφ

z

; ∀ y ∈ C ∩ (Cφ

z

+ R

f + F

)

i.e.

b, y − y

≥ 0 ; ∀ y ∈ C ∩ (Cφ

z

+ R

f + F

)

Remark 3.6 From the proof, we deduce that θ

= b, u

= v

and y

= Λ

θ

+ Cφ

z

+ R

f.

4 Extension to the regional case

4.1 Problem statement

In this part, we study the possibility to reduce the effect of a disturbance in a region ω ⊂ Ω at the final step N . The regional aspect of the considered problem is motivated by the fact that a system can be C -remediable in a region ω ⊂ Ω but not on the whole domain Ω, and even if it’s C -remediable in Ω, the cost is reduced if we are interested only by a subregion ω ⊂ Ω [3, 2]. Moreover, the C -remediability in a region ω ⊂ Ω is a notion more general than the C - remediability in the whole domain Ω which corresponds to ω = Ω [1].

We consider the disturbed system described by the discrete state equation (S

), defined by (1), with X = L

(Ω). The system (S

) is augmented by the following regional output equation

(E

) y

= Ci

p

z

; k = 1, · · · , N (36) where p

is the restriction operator defined by:

p

: L

(Ω) −→ L

(ω)

z −→ p

z = z

(37)

and i

= p

is the adjoint operator of p

. It is defined by i

: L

(ω) −→ L

(Ω)

ϕ −→ i

ϕ =

ϕ in ω 0 otherwise

(38) Let z

be the state of (S

) at the final step N . The corresponding obser- vation is given by

y

= Ci

p

φ

z

+

N−1

i=0

Ci

p

φ

Bu

+

N−1

i=0

Ci

p

φ

f

(39)

Obviously, if f = 0 and u = 0, y

= Ci

p

φ

z

, the observation is normal.

But, if f = 0, generally y

= Ci

p

φ

z

.

Then we introduce a control term Bu in order to reduce regionally the effect of this disturbance at the final step N by bringing the regional observation in a given zone of tolerance C , i.e. such that

For any f = (f

, · · · , f

)

∈ (L

(Ω))

, there exists u = (u

, · · · , u

)

∈ U

satisfying

y

∈ C

where C is a convex, closed and nonempty subset of Y and y

is given by (39). This leads to the following definition.

Definition 4.1

i) A disturbance f ∈ (L

(Ω))

is said C -ω − remediable if there exists u ∈ U

such that

y

= Ci

p

φ

z

+ K

u + R

f ∈ C where K

and R

are the operators defined respectively by

K

u = Ci

p

H

u =

N−1

i=0

Ci

p

φ

Bu

(40) and

R

f = Ci

p

H

f =

N−1

i=0

Ci

p

φ

f

(41) ii) The system (1) augmented by (36) is said to be C -ω − remediable if any disturbance f ∈ X

is C -ω − remediable.

Let us note that if C =

Ci

p

φ

z

, then we have a problem of regional exact remediability.

4.2 Cheap regional enlarged exact remediability

Let f ∈ (L

(Ω))

, we consider the following problem (P

) of enlarged exact

remediability with minimum energy

where

J(u) = u

and U

=

u ∈ U

/y

∈ C

We suppose that the disturbance f is C -ω − remediable, then U

= ∅ and the problem (P

) is well defined and admits a unique solution v

characterized by

v

, v − v

≥ 0 ; ∀ v ∈ U

As in the previous case, the problem (P

) is resolved using an extension of the HUM approach and a penalization method. For θ ∈ Y

≡ Y , let

θ

N

= (

N−1

k=0

B

(φ

)

i

p

C

θ

U

)

(42)

·

N

is a semi-norm. If ker(p

C

) = 0, then ·

N

is a norm on Y if and only if (S

) + (E

) is ω − weakly remediable. Under this condition, we consider the space

F

= Y

F

is a Hilbert space with the inner product

θ, σ

N

=

N−1

k=0

B

(φ

)

i

p

C

θ, B

(φ

)

i

p

C

σ

U

and the operator Λ

defined on Y by

Λ

θ =

N−1

k=0

Ci

p

φ

BB

(φ

)

i

p

C

θ

has a unique extension as an isomorphism from F

−→ (F

)

such that Λ

θ, σ

= θ, σ

N

; ∀ θ, σ ∈ F

We have the analogous following main result.