stabilization of differential linear repetitive processes ... · tion of 2d systems in [7, 24, 35,...

Stabilization of differential linear repetitive processes saturatedsystems by state feedback control

Said KRIRIM, Abdelaziz HMAMEDFaculty of Sciences Dhar El Mehraz

Department of PhysicsB.P. 1796 Fes-Atlas

MOROCCOsaidkririm, hmamed [email protected]

Abstract: The stabilization of linear differential linear repetitive processes subject to saturating controls is ad-dressed. Sufficient conditions obtained via a linear matrix inequality (LMI) formulation are stated to guaranteeboth the local stabilization and the satisfaction of some performance requirements. The method of synthesis con-sists in determining simultaneously a state feedback control law and an associated domain of safe admissible statesfor which the stability of the closed-loop system is guaranteed. Two cases are considered: the first one, the controlmay saturate and limits may be attained. The second one, the control does not saturate and limits are avoided.

Key–Words: Differential linear repetitive processes, State-feedback control, Lyapunov functions, Saturation, LMIs.

1 Introduction

As is well known, many practical systemscan be modeled as two-dimensional 2D systems[18, 19], such as those in image data processing andtransmission, thermal processes, gas absorption andwater stream heating. During the last few decades, theinvestigation of 2D systems in the control and signalprocessing fields has attracted considerable attentionand many important results have been reported to theliterature. Among these results, the H∞ filtering prob-lem for two-dimensional 2D linear systems describedby Roesser and FornasiniMarchesini (FM) models in[21, 25, 26, 22, 23, 8, 9, 38, 37, 30, 28, 29, 31, 32, 34],for 2D linear parameter-varying systems, the relatedwork can be found in [36, 29], stability and stabiliza-tion of 2D systems in [7, 24, 35, 27], H∞ control for2-D nonlinear systems with delays and the nonfragileH∞ and l2 − l1 problem for Roesser-type 2D systemsin [33]. However, because there is no systematicand general approach to analyze linear repetitiveprocesses systems, many problems still remain.On the other hand, Many physical systems completethe same finite duration operation over and overagain. Repetitive processes have this characteristicwhere a series of sweeps or passes are made throughdynamics defined over a finite duration known asthe pass length. Once each pass is complete, theprocess resets to the original location and the next onebegins. The output on each pass is termed the passprofile and the notation for scalar or vector valued

variables is yk(t), 0 ≤ t ≤ α < ∞, k ≥ 0, where yis the scalar or vector valued variable, the integer kis the pass number and α is the pass length. Also theprevious pass profile contributes to dynamics of thenext one and the result can be oscillations in the passprofile sequence {y}k that increase in amplitude frompass-to-pass (k) and cannot be controlled by standardsystems theory.This paper studies the stability of differential linearrepetitive processes with input saturation where thedynamics along the pass are governed by a linearmatrix differential equation and the pass-to-passdynamics by a discrete linear matrix equation. Thestability theory ([1]) for linear repetitive processes isof the bounded-input bounded-output (BIBO) typeand is based on an abstract model in a Banach spacesetting that includes a large range of examples asspecial cases.On the other hand, an important problem whichis always inherent to all dynamical systems is thepresence of actuator saturations. The class of systemswith saturations has attracted great interest over thethree last decades. Even for linear systems, thisproblem has been an active area of research for manyyears. In general, nonlinear control has to be used, asonly simple cases may be handled via linear controllaws,([2]) and the references therein. Two mainapproaches have been developed in the literature:The first, the so-called positive invariance approach,is based on the design of controllers which workinside a region of linear behavior where saturations

WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS Said Kririm, Abdelaziz Hmamed

E-ISSN: 2224-266X 141 Volume 14, 2015

do not occur,([3]), ([4] ), ([5]), ([6]) and the refer-ences therein. One can also cite the work of ([7]),([10]), ([11]), where the synthesis of the controlleris presented as a technique of partial eigenstructureassignment. This resolution was also associated tothe constrained regulator problem. In this work,the controller designed with this technique will bereferred as an ”unsaturating controller”.

The second approach allows saturations to takeeffect while guaranteeing asymptotic stability, ([12]),([13]) and the references therein. This approach, al-lowing the control to be saturated, leads to a boundedregion of stability along the pass which is ellipsoidaland symmetric. This region can easily be obtainedby the resolution of a set of LMIs. In ([14]), besidesthe saturated character of the control, additional con-straints on the increment or rate are taken into account.Further, its results combine this technique with theformer one. The output feedback problem is also stud-ied in ([7]), ([15]) using the tools of this approach. Inthis work, the designed controller with this techniquewill be referred as a saturating controller.The main challenge in these two approaches is to ob-tain a large enough domain of initial states which en-sures stability for the closed-loop system, despite thepresence of saturations, ([16]), ([17]), ([12]), ([13]).More precisely, this paper investigates the differen-tial linear repetitive processes systems. The obtainedthe differential linear repetitive processes is then de-scribed as a convex combination of 2l differential lin-ear repetitive processes systems. The aim of thiswork is the design of stabilizing state-feedback con-trollers for this class of systems. To this end, suf-ficient conditions of stabilizability under LMI formare presented. This formulation enables saturatingstate-feedback controllers to be derived. Furthermore,the unsaturating controller case for differential linearrepetitive processes systems is also considered in thispaper. In fact, stabilizability conditions are derivedsuch that the linear behavior is always guaranteed.These conditions are also given under LMI form.

Notation : we use standard notation throughoutthis paper. The notation P > 0(< 0) is used for posi-tive (negative) definite matrices. ∗ stands for the sym-metric term of the diagonal elements of square sym-metric matrix. I denotes the identity matrix with ap-propriate dimension. the superscript ”T” representsthe transpose. sym(A) indicates AT + A, diag(...)stands for a block-diagonal matrix.

2 Problem formulation

Consider the differential linear repetitive processesdescribed by the following state-space model over0 ≤ t ≤ β, k ≥ 0:

xk+1(t) = Axk+1(t) +B0yk(t) +Bsat(uk+1(t))yk+1(t) = Cxk+1(t) +D0yk(t) +Dsat(uk+1(t))

(1)where on pass k, xk(t) ∈ Rn is the state vectorand yk(t) ∈ Rm is the pass profile vector, anduk+1(t) ∈ Rl is the control vector. respectively; A,B0, B, C, D0, D are time-invariant real matrices withappropriate dimensions.the state initial vector on each pass and the initial passprofile (on pass 0). The form of these considered hereis

xk+1(0) = dk+1, k ≥ 0y0(t) = f(t)

(2)

The saturation function used here is the standardsymmetric one defined as follows for i = 1, . . . , l:

sat(u) = (sat(ui)) =

1 if ui > 1ui if −1 ≤ ui ≤ 1−1 if ui < −1

(3)Note that in the general case of saturations, whenmaximums are any positive real numbers (i.e., theith element of the real control vector w saturates atQi > 0), the change of variables ui = wi

Qican be used:

to use directly the definition of the saturation functionin (3), it is then only necessary to replace the original

matrix Bu =

[BD

]with Budiag(Q1, ..., Ql) for the

differential linear repetitive processes system (1).Further, the state-feedback control is used such that:

uk+1(t) =[K1 K2

] [ xk+1(t)yk(t)

](4)

where matrix K =[K1 K2

]is the state-

feedback gain to be designed.Furthermore, define the sets ε(P, ρ) and £(H) asfollows:

ε(P, ρ) = {x ∈ Rn/xTPx ≤ ρ;P = P T > 0} (5)

£(H) = {x ∈ Rn/|(H)ix| ≤ 1} (6)

where ρ is a positive scalar.


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The problem we are addressing thereafter is to findstabilizing state-feedback controllers for the differen-tial linear repetitive processes system (1) with satura-tion on the control (3) by using state-feedback con-trol (4). We address the problem from two points ofview: first, saturating controls are allowed, so nonlin-ear behavior may occur (Thus, a saturating controlleris used). Second, the behavior is limited to be linear,so saturating controls are not allowed (an un saturat-ing controller is designed).

3 Preliminaries

This section is devoted to some preliminaries useful tothe development in the sequel: the first lemma makesit possible to write the saturated closed-loop system asa convex combination of 2l linear systems. Besides,conditions of stability for the differential linear repet-itive processes systems are then presented. Finally, atechnical lemma giving a sufficient stability conditionis provided.

Lemma 3.1 ([12]) Let u ∈ ℜl and v ∈ ℜl, supposethat |vi| ≤ 1, i = 1, . . . , l then

sat(u) ∈ co{Dsu+D−s v}, s ∈ [1, N ],

where the Ds are all the different diagonal matriceswith elements either 1 or 0,D−

s = Il − Ds, N = 2l

and co stands for the convex hull: in such a case,there exist δ1 ≥ 0, . . . , δN ≥ 0, with

∑Ns=1 δs = 1,

such that

sat(u) =

N∑s=1

δs(Dsu+D−s v) (7)

This lemma is used to rewrite the saturated controlproblem using an auxiliary control v that fulfills|vi| ≤ 1. Hence, in state-feedback control, for twogiven feedback matrices K and H with u = Kx andv = Hx, such that |(Hx)i| ≤ 1, by Lemma 1, thereexist δ1 ≥ 0, . . . , δN ≥ 0, with

∑Ns=1 δs = 1, such

that

sat(Kx) =N∑s=1

δs(DsKx+D−s Hx) (8)

The result of Lemma 1 can be extended to differentiallinear repetitive processes systems since the reasoningdepends on the saturation function and not on thenumber of dimensions (independent variables on

which the control depends). Thus, (7) can be writtenas follows:

sat(uk+1(t)) =

N∑s=1

δs(k, t)(Dsuk+1(t)+D−s vk+1(t))

(9)

Hence, using the state-feedback control (4) and

the fact that vk+1(t) = H

[xk+1(t)yk(t)

]with H =[

H1 H2

]and ξk(t) =

[xk+1(t)yk(t)

]∈ £(H) the

closed-loop saturated differential linear repetitiveprocesses system can be rewritten as

[xk+1(t)yk+1(t)

]= Aξk(t)+B

N∑s=1

δs(k, t)(DsK+D−s H)ξk(t)

where matrices A and B are given by

A =

[A B0

C D0

];B =

[BD

]That is[

xk+1(t)yk+1(t)

]=

N∑s=1

δs(k, t)Asξk(t) = A(δ)ξk(t)

(10)where matrices A(δ) and As, are given by

A(δ) =∑N

s=1 δs(k, t)As; As =

[As Bs

0

Cs Ds0

]=

[A+B(DsK1 +D−

s H1) B0 +B(DsK2 +D−s H2)

C +D(DsK1 +D−s H1) D0 +D(DsK2 +D−

s H2)

](11)

Consider now the following differential linear repeti-tive processes autonomous system:[

xk+1(t)yk+1(t)

]= Aξk(t) (12)

Theorem 1 [20] A differential linear repetitive pro-cesses described by (12) is stable along the pass ifthere exist matrices 0 < P1 ∈ Rn×n and 0 < P2 ∈Rm×m such that the following LMI holds P1A+AP1 P1B0 CP2

⋆ −P2 D0P2

⋆ ⋆ −P2

< 0, (13)

In this case, the following is a Lyapunov function ofsystem (12):

V (k, t) = V1(k, t) + V2(k, t)= xTk+1(t)P1xk+1(t) + yTk (t)P2yk(t)

(14)


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Definition 1 :The increment of function V (k, t) givenby:

∆V (k, t) = V1(k, t) + ∆V2(k, t) (15)

Lemma 3.2 [39] A differential linear repetitive pro-cesses described by (12) and is stable along the passif

∆V (k, t) < 0, (16)

4 Main results

With the background of the previous section, suffi-cient conditions are now given for the stabilization ofdifferential linear repetitive processes saturated sys-tems. The conditions presented so far are only usefulfor analysis, but not for synthesis. In order to allowthe synthesis of stabilizing controllers some transfor-mations to LMI form are then worked out. Thus, thissection presents sufficient conditions of stabilizabilityof the differential linear repetitive processes saturatedsystems expressed as LMIs. The two cases are consid-ered separately: saturating controller and unsaturatingcontroller.

4.1 The saturating controller

Theorem 2 For a given scalar ρ > 0, if there existmatrices H1 ∈ Rl×n, H2 ∈ Rl×m, K1 ∈ Rl×n,K2 ∈ Rl×m, and symmetric positive definite matrices0 < P1 ∈ Rn×n and 0 < P2 ∈ Rm×m such that thefollowing LMI conditions hold true, for s = 1, . . . , N :

Υ(s) =

P1As + AsTP1 P1B

s0 CsTP2

⋆ −P2 DsT0 P2

⋆ ⋆ −P2

< 0,

(17)where matrices As, Bs

0, Cs and Ds0 are given by (11),

and

ε(P, ρ) ⊂ £(H) (18)

with P = diag(P1, P2), then the differential linearrepetitive processes system (12) is stable along the

pass ∀ξ0 =[xk+1(0)y0(t)

]∈ ε(P, ρ).

Prof: Assume that condition (18) holds true; usingthe condition of stability (13) for the closed loopsystem given by (10), one obtains Υ(s) < 0, fors = 1, . . . , N . �The previous result states the stabilizability conditionalong the pass for the closed-loop system. In the next,the LMI formulation of these conditions is derived.

The state-feedback saturating controller can then besynthesized.Corollary 1:For a given scalar ρ > 0, if there existmatrices Z1, Z2, U1, U2, W1 = W T

1 > 0, andW2 = W T

2 > 0 such that the following LMIs holdtrue:

Ψ(s) =

Ψs11 +ΨsT

11 Ψs12 Ψs

13

⋆ −W2 Ψs23

⋆ ⋆ −W2

< 0, s = 1, . . . , N

(19) µ (U1)i (U2)i⋆ W1 0⋆ ⋆ W2

> 0, i = 1, . . . , l (20)

where (U1)i and (U2)i hold for the ith row of matricesU1 and U2 respectively; µ = 1/ρ while matrices Ψs

11,Ψs

12, Ψs13, Ψs

23 are given by

Ψs11 = AW1 +B(DsZ1 +D−

s U1)

Ψs12 = B0W2 +B(DsZ2 +D−

s U2)

Ψs13 = W1C

T + (ZT1 D

Ts + UT

1 D−Ts ))DT

Ψs23 = W2D

T0 + (ZT

2 DTs + UT

2 D−Ts ))DT

then the differential linear repetitive processes system

(12) is stable along the pass ∀ξ0 =

[xk+1(0)y0(t)

]∈

ε(P, ρ) with P = diag(P1, P2), when the controllergain is given by

K =[Z1W

−11 Z2W

−12

](21)

Moreover, the set £(H) is given by (8) with

H =[U1W

−11 U2W

−12

](22)

Prof:Post- and pre-multiply Υ(s) by the followingmatrix:

Θ = diag(P−11 , P−1

2 , P−12 )

Replacing matrices As, Bs0, Cs and Ds

0 by their ex-pressions given by (11) ∀s ∈ [1;N ], one obtains (19)with W = diag(W1,W2), Wi = P−1

i , Zi = KiWi

and Ui = HiWi, for i = 1, 2.On the other hand, the inclusion (18) is equivalentto ρ(H)iP

−1(H)Ti ≤ 1, i = 1, . . . , l. Developingequivalently as follows:ρ(HW )iW

−1(HW )Ti ≤1,that is ρ(U)iW

−1(U)Ti ≤ 1, Using Schur comple-ment, one obtains:[

µ (U)i⋆ W

]> 0, i = 1, . . . , l (23)


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Finally, using µ = 1/ρ, W = diag(W1,W2) andU = [U1, U2], the LMIs (20) follows.�Example 1:As an example, the metal rolling process is con-sidered . This process is an extremely commonindustrial process where, in essence, deformation ofthe workpiece takes place between tow rolls withparallel axes revolving in opposite directions.Thus the following linear differential equation repre-sent Metal Rolling dynamics: ([40]).

Sk(t) +α

MSk(t) =

α

α1Sk−1(t) +

α

MSk−1(t)−

α

Mα2FM (t)

(24)

Where:Sk(t):current passes through the rolls.Sk−1(t):previous passes through the rolls.M: is the lumped mass of the roll-gap adjustingmechanismα1: the stiffness of the adjustment mechanism springα2: the hardness of the metal stripα = α1α2

α1+α2:the composite stiffness of the metal strip

and the roll mechanismFM (t):the force developed by the motor. is the inputfunction which is assumed here to be constrainedas |FM (t)| ≤ FMmax (with FMmax represent themaximum driving force)The linear differential equation (17) can be modeledas a system (1) by imposing :

xk+1(t) =

(sk(t)sk(t)

), yk(t) =

(sk−1(t)sk−1(t)

)Thus The linear differential equation (23) can bewrithed:

xk+1(t) =

(0 1

− αM

0

)xk+1(t)+

(0 0αM

αα1

)yk(t)+

(0

− αMα2

)FM (t)

yk+1(t) =

(1 0

− αM

0

)xk+1(t)+

(0 0αM

αα1

)yk(t)+

(0

− αMα2

)FM (t)

In these design studies, the data used are α1 = 600,α2 = 2000 and M = 100 This yields α = 461.54,resulting in the state-space model matrices in (1):

A =

(0 1

−4.61538 0

), B0 =

(0 0

4.61538 0.76923

)

C =

(1 0

−4.61538 0

), D0 =

(0 0

4.61538 0.76923

)

B =

(0

−0.00231

), D =

(0

−0.00231

)

For this data and ρ = 100, This model is both unstablealong the pass (A has eigenvalues (0 + 2.1483i, 0- 2.1483i)) Corollary 1 can be successfully appliedhere since the LMIs (19) and (20) are feasible, with asolution give

K1 = [542.6908 434.9946]K2 = 103[1.6313 0.2719]H1 = [189.7717 143.6449]H2 = [810.1543 135.0257]

020

4060

80100

0

50

100−4

−2

0

2

4

ik

stat

e ve

ctor

xk+

11

Figure 1. The evolution of the first component ofxk+1(t) using the saturating controller

020

4060

80100

0

50

100−4

−2

0

2

4

ik

stat

e ve

ctor

yk1

Figure 2. The evolution of the first component ofyk(t) using the saturating controller

020

4060

80100

0

50

100−10

−5

0

5

10

ik

stat

e ve

ctor

xk+

12

Figure 3. The evolution of the second component ofxk+1(t) using the saturating controller


E-ISSN: 2224-266X 145 Volume 14, 2015

020

4060

80100

0

50

100−15

−10

−5

0

5

10

15

ik

stat

e ve

ctor

yk2

Figure 4. The evolution of the second component ofyk(t) using the saturating controller

020

4060

80100

0

50

100−1

−0.5

0

0.5

1

ik

cont

rol u

k+11

Figure 5. The evolution of the saturating controlleruk+1(t)

020

4060

80100

0

50

100−1

−0.5

0

0.5

1

ik

cont

rol u

k+12


4.2 The unsaturating controller

Consider now the differential linear repetitive pro-cesses system (1) with constrained control (3). Inthe previous section, the design of a saturating con-troller was studied (saturation of control was allowed),whereas in this section, saturation is not allowed andthe synthesis will guarantee that the state evolves in-side a region of linear behavior given by £(F ) (F be-ing the controller gain). Thus, this case can be seen asa particular case of the saturating one.

Theorem 3 For a given scalar ρ > 0, if there exist

matrices F1 ∈ Rl×n, F2 ∈ Rl×m, and symmetricpositive definite matrices 0 < P1 ∈ Rn×n and0 < P2 ∈ Rm×m such that the following LMIconditions hold true:

Υ(s) =

P1A+ ATP1 P1B0 CTP2

⋆ −P2 DT0 P2

⋆ ⋆ −P2

< 0,

(25)

ε(P, ρ) ⊂ £(H) (26)

where[A B0

C D0

]=

[A+BF1 B0 +BF2

C +DF1 D0 +DF2

]and P = diag(P1, P2), then the differential linearrepetitive processes system (12) is stable along the

pass ∀ξ0 =[xk+1(0)y0(t)

]∈ ε(P, ρ).

Prrof: The proof follows readily if one replaces Kby F in the proof of Theorem 4.1 and removes thesaturated convex writing of the control. This can bedone, as in this case, the state is restricted to evolveinside the linear region of behavior given by condition(26).�In the next result, the LMI formulation of these con-ditions, that enables the unsaturating state-feedbackcontrol to be derived, is given:Corollary 1:For a given scalar ρ > 0, if there existmatrices Z1, Z2, W1 = W T

1 > 0, and W2 = W T2 > 0

such that the following LMIs hold true: Ψ11 +ΨT11 Ψ12 Ψ13

⋆ −W2 Ψ23

⋆ ⋆ −W2

< 0, (27)

µ (Z1)i (Z2)i⋆ W1 0⋆ ⋆ W2

> 0, i = 1, . . . , l (28)

where (Z1)i and (Z2)i hold for the ith row of matricesZ1 and Z2 respectively; µ = 1/ρ while matrices Ψ11,Ψ12, Ψ13, Ψ23 are given by

Ψ11 = AW1 +BZ1

Ψ12 = B0W2 +BZ2

Ψ13 = W1CT + ZT

1 DT

Ψ23 = W2DT0 + ZT

2 DT


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then the differential linear repetitive processes system

(12) is stable along the pass ∀ξ0 =

[xk+1(0)y0(t)

]∈

ε(P, ρ) with P = diag(W−11 ,W−1

2 ), when thecontroller gain is given by

F =[Z1W

−11 Z2W

−12

](29)

Prof: The proof follows readily from Corollary 1.Example 2:Consider the same system studied in the saturatingcontroller case of Example 1. If no saturation isallowed in the control signal, Corollary 2 can be usedto synthesize the controller. In this case, the LMIs(27) and (28) are feasible, with a solution given by:F1 = [231.9679 166.5527]F2 = [627.1912 104.5319]

020

4060

80100

0

50

100−0.2

−0.1

0

0.1

0.2

0.3

ik

stat

e ve

ctor

xk+

11

Figure 7. The evolution of the first component ofxk+1(t) using the saturating controller

020

4060

80100

0

50

100−0.2

−0.1

0

0.1

0.2

0.3

ik

stat

e ve

ctor

yk+

11

Figure 8. The evolution of the first component ofyk(t) using the saturating controller

020

4060

80100

0

50

100−1

−0.5

0

0.5

1

ik

stat

e ve

ctor

xk+

12

Figure 9. The evolution of the second component ofxk+1(t) using the saturating controller

020

4060

80100

0

50

100−1.5

−1

−0.5

0

0.5

1

ik

stat

e ve

ctor

yk+

12

Figure 10. The evolution of the second component ofyk(t) using the saturating controller

020

4060

80100

0

50

100−100

−50

0

50

100

150

ik

cont

role

Uk+

11


020

4060

80100

0

50

100−40

−20

0

20

40

60

ik

cont

role

Uk+

12



E-ISSN: 2224-266X 147 Volume 14, 2015

5 Conclusions

In this paper, the problem of the stabilizability ofthe differential linear repetitive processes saturatedsystems has been studied using state-feedback con-trol. Two different cases are considered guaranteeingstability along the pass: the design of saturating andunsaturating controllers. The first one allows satura-tion to take effect, while the second limits the systemsevolution to the region of linear behavior. Sufficientconditions of stability along the pass are derived foreach case. The synthesis of the required controllersare also given under LMI form. Numerical examplesare provided to illustrate the results.

References:

[1] E. Rogers, K. Galkowski, D. H. Owens, ”Con-trol Systems Theory and Applications for Lin-ear Repetitive Processes”, in Control and Infor-mation Sciences, Springer-Verlag, Berlin, Ger-many, 2007.

[2] Z. Lin,A. Saberi,A. Stoorvogel, ”Semi-globalstabilization of linear discrete-time systems sub-ject to input saturation via linear feedback. AnARE-based approach”, In Proceedings of the33rd CDC. Lake Buena FL. Vista, 1994

[3] A. Benzaouia, C. Burgat, ”Regulator prob-lem for linear discrete-time systems with nonsymmetrical constrained control”, InternationalJournal of Control, vol. 48, no. 6, pp. 2441-2451,1988.

[4] F. Blanchini, ”Set invariance in control”, Auto-matica, vol. 35, pp. 1747-1767, 1999.

[5] F. Mesquine,F. Tadeo and A. Benzaouia, ”Regu-lator problem for linear systems with constraintson the control and its increments or rate.”, Auto-matica, vol. 40, no. 8, pp. 1378-1395, 2004.

[6] A. Baddou, F. Tadeo and A. Benzaouia, ”Onimproving the convergence rate of linear con-strained control continuous-time systems with astate observer.”, IEEE Transactions on Circuitsand Systems-I, vol. 55, no. 9, pp. 2785-2794,2008.

[7] A. Hmamed, F. Mesquine,F. Tadeo,M. Benhay-oun and A. Benzaouia, ”Stabilization of 2D sat-urated systems by state feedback control.”, Mul-tidim Syst Sign Process, vol. 21, pp. 277-292,2010.

[8] A. Hmamed, C. El-Kasri, E.H. Tissir, T. Alvarez,F. Tadeo, Robust H∞ filtering for uncertain 2-D continuous systems with delays. Int. J. Innov.Comput. Inf. Control 9(5), 2167.2183 (2013)

[9] B. Boukili, A. Hmamed, A. Benzaouia , A. ElHajjaji, H∞ filtering of Tow-Dimensional T-S Fuzzy Systems. Circuits. Syst. Signal Pro-cess.doi: 10.1007/s00034-013-9720-2(2013)

[10] A. Benzaouia, ”The resolution of equationXA+XBX=HX and the pole assignment prob-lem.”, IEEE Transactions on Automatic Control,vol. 39, no. 10, pp. 2091-2095, 1994.

[11] A. Benzaouia,M. Ait Rami and S. El Faiz, ”Sta-bilization of linear systems with saturation: ASylvester equation approach.”, IMA Journal ofMathematical Control and Information, vol. 21,no. 3, pp. 247259, 2004.

[12] T. Hu, Z. Lin and B. M. Chen, ”Analysis and de-sign for discrete-time linear systems subject toactuator saturation.”, Systems and Control Let-ters, vol. 45, pp. 97-112, 2002.

[13] T. Hu, Z. Lin and B. M. Chen, ”An analysis anddesign method for linear systems subject to ac-tuator saturation and disturbance.”, Automatica,vol. 38, pp. 351-359, 2002.

[14] A. Benzaouia,F. Mesquine,A. Hmamed and H.Aoufoussi, ”Stability and control synthesis fordiscrete-time linear systems subject to actuatorsaturation by output feedback.”, MathematicalProblems in Engineering, pp. 1-11, 2006.

[15] A. Benzaouia,F. Tadeo and F. Mesquine, ”Theregulator problem for linear systems with satu-rations on the control and its increments or rate:An LMI approach.”, IEEE Transactions on Cir-cuits and Systems I, vol. 53, no. 9, pp. 2681-2691, 2006.

[16] E. G. Gilbert and K. T.Tan, ”Linear systems withstate and control constraints: The theory and ap-plication of maximal output admissible sets.”,IEEE Transactions on Circuits and Systems I,vol. 36, no. 11, pp. 10081020, 1991.

[17] A. Benzaouia and A. Baddou, ”Piecewise lin-ear constrained control for continuous-time sys-tems.”, IEEE Transactions on Automatic Con-trol, vol. 44, no. 7, pp. 14771481, 1999.

[18] C.W. Chen, J.S.H. Tsai, L.S. Shieh, Two-dimensional discrete-continuous model conver-sion. Circuits Syst. Signal Process. 18, 565585(1999)


E-ISSN: 2224-266X 148 Volume 14, 2015

[19] T. Kaczorek, Tow-dimensionel Linear Systems.Berlin, Springer-Verlag, 1985.

[20] K. Galkowski, W. Paszke , E. Rogers ,S. Xu ,J. Lam , DH. Owens, ”The stability and controlof differential linear repetitive processes usingan LMI setting.”, IEEE Transactions on Circuitsand Systems II: Analog and Digital Signal Pro-cessing, vol. 50, no. 9, pp. 662666, 2003.

[21] C. El-Kasri, A. Hmamed, E.H. Tissir, F.Tadeo, Robust H∞ filtering for uncertain two-dimensional continuous systems with time-varying delays. Multidimens. Syst. Signal Pro-cess. doi: 10.1007/s11045-013-0242-7. (2013)

[22] C. Du, L. Xie, H∞ Control and Filteringof Two-dimensional Systems(Springer, Berlin,2002)

[23] C. Du, L. Xie, Y. Soh, H∞ Fltering of 2Ddiscrete systems. IEEE Trans. Signal Process.48(6), 1760 1768 (2000)

[24] M. Benhayoun, F. Mesquine, A. Benza-ouia, Delay-dependent stabilizability of 2D de-layed continuous systems with saturating con-trol. Circuits Syst. Signal Process. (2013).doi:10.1007/s00034- 013-9585-4

[25] S. Kririm, A. Hmamed, F. Tadeo,Robust H∞Filtering for Uncertain 2D Singular RoesserModels. Circuits. Syst. Signal Process. doi:10.1007/s00034-015-9967-x (2015)

[26] S. Kririm, A. Hmamed, Robust H∞ filtering foruncertain differential linear repetitive processesvia LMIs and polynomial matrices. Advances inElectrical and Computer Engineering. Tenerife,Canary Islands, Spain January 10-12. (2015)

[27] S. Kririm, A. Hmamed, New Approach to Stabil-ity Analysis for Uncertain 2D Continuous Sys-tems. International Journal of Ecology & Devel-opment. 30(2). (2015)

[28] L. Wu, P. Shi, H. Gao, C. Wang, H∞ filter-ing for 2D Markovian jump systems. Automat-ica 44(7), 1849.1858 (2008)

[29] L. Wu, Z. Wang, H. Gao, C. Wang, H∞ andl2-l1 filtering for two-dimensional linear param-etervarying systems. Int. J. Robust NonlinearControl 17(12), 1129.1154 (2007)

[30] L. Wu, Z. Wang, H. Gao, C. Wang, Filtering foruncertain 2D discrete systems with state delays.Signal Process. 87(9), 2213.2230 (2007)

[31] S. Xu, J. Lam, Y. Zou, Z. Lin, W. Paszke, Ro-bust H∞ filtering for uncertain 2D continu-ous systems. IEEE Trans. Signal Process. 53(5),1731.1738 (2005)

[32] S. Xu, J. Lam, Exponential H∞ filter designfor uncertain Takagi.Sugeno fuzzy systems withtime delay. J. Eng. Appl. Artif. Intell. 17(6),645.659 (2004)

[33] H. Xu, Z. Lin, A. Makur, Non-fragile H2and H∞ filter designs for polytopic two-dimensional systems in Roesser model. Mul-tidimens. Syst. Signal Process. 21(3), 255.275(2010)

[34] R. Yang, L. Xie, C. Zhang, H2 and mixedH2/H∞ control of two-dimensional systems inRoesser model. Automatica 42(9), 1507.1514(2006)

[35] D. Napp,F. Tadeo, A. Hmamed, Stabilizationwith Positivity of nD Systems. Int. J. Innov.Comput. Inf. Control 9(12), 1349-4198 (2013)

[36] C. De Souza, L. Xie, D. Coutinho, Robust filter-ing for 2D discrete-time linear systems with con-vex bounded parameter uncertainty. Automatica46(4), 673681 (2010)

[37] H. Tuan, P. Apkarian, T. Nguyen, T. Narikiyo,Robust mixed H2/H∞ filtering of 2D systems.IEEE Trans. Signal Process. 50(7), 1759.1771(2002)

[38] D. Peng, X. Guan, H∞ filtering of 2D discretestate-delayed systems. Multidimens. Syst. Sig-nal Process. 20(3), 265.284 (2009)

[39] W. Paszke, K. Galkowski, E. Rogers and DH.Owens, ”H∞ control of differential linear repet-itive processes.”, IEEE Transactions on Circuitsand Systems II: Analog and Digital Signal Pro-cessing, vol. 53, no. 1, pp. 3944, 2006.

[40] K. Galkowski, E. Rogers, W. Paszke and DH.Owens, ”Linear Repetitive Process Control The-ory Applied to a Physical Example.”, Inter-national Journal of Applied Mathematics andComputational Science, vol. 13, pp. 8799, 2003.


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