Dynamic sliding mode control design using attracting ellipsoid method

Automatica 47 (2011) 1467–1472

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Brief paper

Dynamic sliding mode control design using attracting ellipsoid method✩ Jorge Davila a,∗ , Alexander Poznyak b a

Section of Graduate Studies and Research ESIME-UPT, National Polytechnic Institute (IPN), Mexico D.F., Mexico

b

Automatic Control Department, CINVESTAV-IPN, Mexico D.F., Mexico

article

info

Article history: Received 4 February 2010 Received in revised form 2 September 2010 Accepted 14 January 2011 Available online 2 March 2011 Keywords: Sliding mode Robust control Uncertain linear systems

abstract A methodology for the design of sliding mode controllers for linear systems subjected to matched and unmatched perturbations is proposed. It is considered that the control signal is applied through a firstorder low-pass filter. The technique is based on the existence of an attracting (invariant) ellipsoid such that the convergence to a quasi-minimal region of the origin using the suboptimal control signal is guaranteed. The design procedure is given in terms of the solution of a set of Matrix Inequalities. A benchmark example illustrating the design is given. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction The design of controllers for dynamical systems in the presence of bounded external perturbations is one of the most important problems in the control theory. The Sliding Mode (SM) control is a very efficient methodology to control dynamic plants operating under uncertain conditions (see, Utkin, Guldner, and Shi (1999)). One of the main features of SM controllers is their insensitivity in the presence of matched perturbations (perturbations that lie on the same subspace than the control). The designing of a controller that effectively reduces the effect of the non-matched perturbations is still an open problem. A controller ensuring local practical stability for systems with matched and unmatched uncertainties is designed using a technique of eigenvalues assignment in Hui and Zak (1993). Edwards and Spurgeon (1998) study the design of the SM controller parameters by the minimization of a quadratic functional and the robust eigenvalue assignment. Hsu, Costa, and Cunha (2003) design a model-reference SM unit-vector control to a linear system with relative degree one. Andrade-Da Silva, Edwards, and Spurgeon (2009) present an SM Output Feedback Controller to stabilize an uncertain linear system in the presence of perturbations. The solution is presented as an optimization problem and solved using LMI’s (see, Boyd, Ghaoui, Feron, and Balakrishnan (1994), Poznyak (2008)). Castanos and Fridman (2006)

✩ This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Fabrizio Dabbene under the direction of Editor Roberto Tempo. ∗ Corresponding author. Tel.: +52 55 57296000x46133; fax: +52 55 55863394. E-mail addresses: [email protected], [email protected] (J. Davila), [email protected] (A. Poznyak).

0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.02.023

provide a controller minimizing the effect of unmatched perturbations by means of a combination of integral SM and H∞ techniques. A unit control is proposed to stabilize a perturbed linear system in the presence of unmatched perturbations by Choi (2008). Dynamic SM controllers (see Sira-Ramírez (1993) and references therein), realized for example, by the inclusion of a firstorder smoothing filter for the discontinuous control, allow the design of robust controllers in combination to smoothing the control signals. The effect of the introduction of this dynamics have been studied by Fridman (2001), Fridman (2002). The invariant subspace approach (see Kurzhanski and Valyi (1996), Blanchini (1999) and the references therein) is a technique used for the design of controllers for perturbed systems. A particular technique applied for the suppression of bounded perturbations in linear systems is the invariant ellipsoid method (Nazin, Polyak, & Topunov, 2007). Polyakov and Poznyak (2009) present an innovative approach applying the invariant ellipsoid method to minimize the effect of unmatched perturbations on linear systems. In this article a methodology for the design of the time constant, the surface and gains of a dynamic SM controller is presented. As a result the computed controller and filter parameters guarantee the global convergence of state and control to the suboptimal attracting ellipsoid around the origin. As far as the authors know, it is the first paper providing a methodology for the design of the parameters of a dynamic SM controller, represented by a first-order smoothing filter, to guarantee suboptimal solution to the global stabilization problem. A computational version of the method is given as the solution of a set of LMI’s. 2. Problem statement Consider the following perturbed linear system x˙ (t ) = Ax(t ) + Bu(t ) + Df (t , x)

(1)

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J. Davila, A. Poznyak / Automatica 47 (2011) 1467–1472

where x(t ) ∈ Rn is the state variable, u(t ) ∈ Rm is the control inputs vector, and f (t , x) ∈ Rq is the vector of perturbations. The short notation x, u, f will be introduced for simplicity. Differential equations are understood in the Filippov sense (Filippov, 1988) in order to provide the possibility to use discontinuous signals in controls. Filippov solutions coincide with the usual solutions, when the right-hand sides are Lipschitzian. It is assumed also that all considered inputs allow the existence of solutions and their extension to the whole semi-axis t ≥ 0. Let the sliding variable σ ∈ Rm be given by

σ = Cx + u,

C ∈ Rm×n .

(2)

Suppose that the sliding mode controller is applied through the following dynamic u˙ = −H (u + M Sign σ ) + ξ

W44 = [Im + (Kξ − Im )−1 ]−1 0 < αmin ≤ αi ≤ αmax ,

 0 ≤ τ1 ≤

m −

(12)

 h2k αk

−1

(13)

k=1

βu ≥ αui > 0,

βx ≥ αxi > 0,

(14)

hmin ≤ hi ≤ hmax .

(15)

Then, the controller (2)–(3), designed with the computed parameters, ensures the convergence of σ to zero in finite time. Proof. Consider Vσ = 12 σ T σ . Which derivative is V˙ σ = σ T (CAx + ∑ (CB − H )u + CDf − HMsign(σ )+ξ ). Since |σ T signσ | = ni=1 |σi | ≥

‖σ ‖, σ T V˜ ≤ ‖σ ‖·‖V˜ ‖, ∀ V˜ ∈ Rm the inequality V˙ σ < 0 is satisfied

(3)

if

M = diag {M1 , M2 , . . . , Mm }

(4)

H = diag {h1 , h2 , . . . , hm }

(5)

¯‖ ‖CAx + (CB − H )u + CDf + ξ ‖ < ‖H M (16) T ¯ where M = (M1 , . . . , Mm ) . Define Y = ξ , X = CAx + (CB − H )u +

Sign σ := sign σ1 ,

sign σ2 ,



...,

sign σm

T

(6)

where ξ is a ‘‘dynamic noise’’ and satisfies

ξ T Kξ ξ ≤ 1 ,

Kξ = KξT > Im .

(7)

Mi =

αi + αxi ‖x‖2 + αui ‖u‖2 ,

i = 1, . . . , m

(8)

where αi , hi , αxi and αui are parameters to be designed 0 < hmin ≤ hi ≤ hmax , 0 < αmin ≤ αi ≤ αmax , 0 ≤ αxi ≤ βx , 0 ≤ αui ≤ βu .

(9)

f Qf f < f0 + x Qx x, Qf > 0, f0 > 0, Qx > 0. T

N (1 + Λ)N

+ h2m (αm + αxm xT x + αum uT u)   where N = CA CB − H CD . Taking Λ−1 = Kξ − I > 0. Last inequality can be rewritten as x u f

(10)

 2 T µ N (I + (Kξ − I )−1 )N     −  

m −

 h2i

Theorem 1. Let (C 0 , αi0 , H 0 , αx0i , αu0i , τ10 ) be a solution of the following set of inequalities

T W14

0 W22 0 T W24

 W11 =

m −

0 0 W33 T W34

In −

k=1

 W22 =

m −

W14 W24  ≥0 W34  W44

τ1 f0

Qx , W14 = (AT C T )

 h2k αuk

Im , W24 = (BT C T − H T )

k=1

W33 =

τ1 f0

Qf ,

 <

0

 0

m −

 h2i αui

W34 = (DT C T )

(11)

Im

i=1

0

0    x   u  0  f 0



m

−

h2i αi

− 1.

(17)

i=1

On the other hand, the perturbations satisfy the inequality (10). Using S-theorem (Poznyak, 2008), the Eq. (10) implies (17) if there exist τ1 such that the following inequalities are satisfied:

 0 ≤ τ1 ≤

m −

 h2i

αi − 1,

i=1





 h2k αxk

αxi In

0

The controller is designed in three steps. First the gains of the controller and actuator are chosen such that the sliding variable converges to zero. Second, we show that the dynamics of the closed loop system over the sliding surface converges to a neighborhood of the origin which belongs to an interior region of an attracting ellipsoid. Finally, the state trajectories are enforced to converge to the smallest invariant ellipsoid.

W11



i =1

3. Controller design

 0  0

+ ξ T (1 + Λ−1 )ξ

< h21 (α1 + αx1 xT x + αu1 uT u) + · · ·

The aim of this article is to propose a design methodology for αi , H , Rxi , Rui and C , to ensure global convergence of the state trajectories to a suboptimal ellipsoid of a minimum possible volume around the origin.



x u f

  T

 T

The perturbations are suppose to be from the ‘‘strip-sector’’: T

 T x u f

The components of the matrix of gains M have the form:



CDf . Raising to the square both sides of (16) and introducing the Λ-Inequality (Poznyak, 2008) for the left hand side ‖X + Y ‖2 ≤ X T (I + Λ)X + Y T (I + Λ−1 )Y valid for the defined X , Y ∈ Rm , and any 0 < Λ = ΛT ∈ Rm×m . Inequality (16) is implied by:

       

m −

 h2i

αxi In −

i =1

τ1 f0

 Qx

0

 0

m −

h2i αui

i =1

0

0 



0

Im

   0    τ1  f0

Qf

− N T (I + (Kξ − I )−1 )N ≥ 0. Applying the Schur complements (see, for example Poznyak (2008)) we obtain inequality (11). Notice that while (11) ensures the convergence of σ to zero, the remaining inequalities (12)–(15) correspond to the controller restrictions (9). 

J. Davila, A. Poznyak / Automatica 47 (2011) 1467–1472

Once the systems begins to slide on the surface, the equality

σ = 0 is satisfied, it implies that the equivalent control  becomes  u = −Cx. Define the extended variable z T = xT uT and introduce the following Lyapunov like function

1469

variable w , Eq. (23) can be written as

 V˙ z + γ Vz = w T (Ξ1 + γ Ξ2 )w + τ2

m −

 αi

≤β

(24)

i=1

Vz = z T P z z ,

Pz = diag{Px Pu }.

(18)

Theorem 2. Let (C 0 , αi0 , H 0 , αu0i , αx0i , τ10 , τ20 , τ30 , τ40 , τ50 , γ 0 , β 0 , Px0 , Pu0 ) be any solution of the set of inequalities (11)–(15),

 β ≥ τ2

m −

‖u + Cx‖2 = wT Ξ4 w = 0

 αi + τ4 + τ5

(19)

i=1



−τ3 C T

F1

−τ3 C  T  D Px  0

Px D 0 −τ4 Qf 0 0

F2 0 −HPu Pu

0

where Ξ2 = diag {Px , Pu , 0(q+2m)×(q+2m) }. After a finite time transient the system trajectories reach the sliding surface, since that moment, the equality σ = u + Cx = 0 is satisfied. So, it is considered that:

0 −P u H 0 −τ2 Im 0

where

Ξ4 = diag



0 Pu   0 ≤0  0 −τ5 Kξ

(20)

(25)



CT C C

CT Im



 , 0(q+2m)×(q+2m) .

As a result of the convergence of σ to zero, and applying the Finsler Lemma (Poznyak, 2008) to Eqs. (24) and (25) we obtain that there exist τ3 ≥ 0 such that:

F1 = (A − BC )T Px + Px (A − BC ) + γ Px

 + τ2



m

−

αxi In +

τ4 f0

i =1

 F2 = γ Pu + τ2

m −

 w T (Ξ1 + γ Ξ2 − τ3 Ξ4 )w + τ2

Qx − τ3 C T C

m −

 αi

≤ β.

 αui Im − Pu H − HPu − τ3 Im

i =1

the controller (3), (8) and (2) designed with this parameter selection guarantees the existence of an attracting invariant ellipsoid of z γ (ε(Θ ) = z T Θ z, with Θ = β Pz ).

Applying the S-Theorem (see Poznyak (2008)) to combine Eqs. (7), (10) and (26), we obtain that (23) is satisfied if there exist τ4 , τ5 ≥ 0 such that ∑mthe inequality (20) is satisfied simultaneously with β − τ2 ( i=1 αi ) ≥ τ4 + τ5 . This inequality is implied by (19).  Theorem 3. Let (C 0 , αi0 , H 0 , αu0i , αx0i , τ10 , τ20 , τ30 , τ40 , τ50 , γ 0 , β 0 , Px0 ,

Proof. The derivative of Vz along z is given by the following equality

Pu0 ) be the solution of the optimization problem:

V˙ z = xT Px [(A − BC )x + Df ] + [(A − BC )x + Df ]T Px x + uT Pu [ξ − H (u + Msign σ )]

maximize :

+[ξ − H (u + Msign σ )]T Pu u. (21)  T  T T T T T Define the variable w = x u f (Msign σ ) ξ . Adding and subtracting the term τ2 (Msignσ )T (Msignσ ) to Eq. (21), that can be written as:

 V˙ z = w T Ξ1 w + τ2

m −

(26)

i=1

γ tr (Pz ). β

subject to (11)–(15), (19) and (20). The controller (3) designed with this selection of parameters ensures the existence of a quasi-minimal γ attracting invariant ellipsoid ε( β Pz ). γ

Proof. The attracting invariant ellipsoid takes the form ε( β Pz ) = γ

z T β Pz z the rest of the proof is a consequence of the ellipsoid definition. 

 αi

(22)

i=1

4. Computational implementation

where



F3  0  Ξ1 = DT Px  0 0

0 F4 0 −HPu Pu

Px D 0 0 0 0

0 −Pu H 0 −τ2 Im 0

 F3 = (A − BC ) Px + Px (A − BC ) + τ2 T

The above mentioned results provide the quasi-optimal solution to the stabilization problem. However, the resulting inequalities contain at least bilinear terms which increases the difficulty to find a computational solution. In this section the inequalities are modified to obtain an implementable form of the design process. Let introduce = Px−1 , Y = CPx−1 , Ru = ∑m the following ∑m variables X∑ Pu H , α¯ = ¯ x = i=1 αxi , α¯ u = m i=1 αi , α i=1 αui .



0 Pu   0 0 0 m −

 αxi In

i =1

 F4 = τ2

m −

Theorem 4. Let (X 0 , Y 0 , R0u , α¯ 0 , α¯ x0 , α¯ u0 , τ10 , τ20 , τ30 , τ40 , τ50 , τ60 , γ 0 , β 0 , Pu0 , G01 , G02 ) be any solution of the set of inequalities:

 αui Im − HPu − Pu H .

i=1

The variable z will be asymptotically ultimately bounded if the following inequality holds V˙ z ≤ −γ Vz + β . Equivalently V˙ z + γ Vz ≤ β.

˜ 11 W  0  

(23)

The ultimately bounded region can be represented as an attracting γ invariant ellipsoid defined by lim z T β Pz z ≤ 1. In terms of the new

0 0

AT Pu BT

Qf f0 D 0

DT

0 mh2min α¯ u G2

  0   A

0 BPu

0

−RTu

˜ 11 = mh2min α¯ x In − W

τ1

τ1 f0

Qx ,

τ6 X 0



0 −Ru  

0 >0

  0  L

(27)

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J. Davila, A. Poznyak / Automatica 47 (2011) 1467–1472

 ¯ F1 −τ3 Y  T  D  0 0

−τ3 Y T F¯2

D 0

0

−τ4 Qf

0

−R u

0 0

−τ2 Im

Pu



0

−R u 0

0 Pu   0 ≤0  0 −τ5 Kξ

(28)

1

F¯1 = XAT − Y T BT + AX − BY + τ2 α¯ x G1 + γ X + τ4 Q˜ x F¯2 = γ Pu + (τ2 α¯ u − τ3 )Im − RTu − Ru

β ≥ τ2 α¯ + τ4 + τ5 1 X

 τ6  0

YT

L L

L

G1 X

  ≥0

(30)

Pu

0 ≤ α¯ x ≤ mβx ,

(31)

Im

0 ≤ α¯ u ≤ mβu

mαmin ≤ α¯ ≤ mαmax ,

τ1 , τ3 , τ4 , τ5 ≥ 0,

(32)

X > 0, L > 0,

(33)

τ2 , τ6 > 0.

(34)

The controller (3) designed with the computed parameters ensures the convergence of the sliding variable (i.e., σ ) to zero and the existence γ of an attracting invariant ellipsoid ε( β Pz ) of the variable z. Proof. For the sliding surface, it is possible to introduce the following estimation



1

 τ6



X −1 0

0  ≥

L



−1



CT Im

 (Im + (Kξ − Im )−1 ) C

Im



τ1

mh2min α¯ x In −

  

f0

0

X

−Φ T  τ6

−1

τ1 f0

Qf

˜ 11 W  0    0   A 0

˜ 11 W  0 

−τ3 XC T F2 0 −HPu Pu

D 0

0 −P u H 0 −τ2 Im 0

−τ4 Qf 0 0



0 Pu   0  ≤ 0.  0 −τ5 Kξ

Consider the linear system x˙ = Ax + Bu + Df with the matrixes A, B, C be defined as.

1 1 0

0  Φ≥0

D=

−1

L

2

−200 3

− 5 

−40  20  ,  −100

3  10   B=  0 , 6 5

0 −1 . 1

The perturbation f is given by A 0

B −H





D . 0

f =

0 mh2min α¯ u Im 0 B −H

0 0

AT BT

Qf f0 D 0

DT

τ1

0 −H T  

f

0  > 0.

  0 

τ6 X 0

(36)

L

0 mh2min α¯ u Pu Pu

  0   A

0 BPu

0

−RTu

0 0

AT Pu BT

Qf f0 D 0

DT

τ1

τ6 X 0



0 −R u  

0  > 0.

  0  L

0.0028 cos(0.4t ) − 0.0879x1 sin(0.4t ) 0.0499 cos(0.4t ) + 0.0049x2 sin(0.4t )



and satisfies the following inequality



Define the nonsingular matrix T1 = diag{In , Pu , I2n+m }. Due T1 is nonsingular, multiplying (36) to the right and to the left by T1 we obtain





For each set of scalar parameters α¯ i0 , α¯ x0 , α¯ u0 , τ10 , τ20 , τ30 , τ40 , τ , τ60 , γ 0 , β 0 the conditions presented in Theorem 4 can be solved as a set of Linear Matrix Inequalities.



Using the Schur’s complements (see Poznyak (2008)) we obtain



L ≥ 0.

0 5

− 3 A=  2 50



where

Φ=

 (I + (Kξ − I )−1 ) Y

The first and second inequalities of (31) are obtained from the Schur’s complements application to the inequality XX ≤ G1 . As a consequence of the above defined matrix G1 and using the inequality 0 ≤ τ3 C T C , the inequality XF1 X ≤ F¯1 is satisfied, hence, inequality (20) is implied by (28). Eqs. (32)–(34) are obtained from the restrictions of the controller gains and the corresponding definitions of the auxiliary variables. 



0



XF1 X −τ3 CX  T  D  0 0

 10

0  

0 1



Using again the Schur’s complements we obtain (30). Let define an other nonsingular matrix T3 = diag{X , I4m }. Multiplying on the left and on the right of Eq. (20) we obtain:



0

mh2min α¯ u Im

0



L

YT LT

5. EXAMPLE

Qx

0

 −

(35)

for some integer τ6 > 0 and some matrix L > 0, L ∈ Rm×m . It follows that Eq. (17) is implied for the inequality



0



−1 −1

In



X

0

(Im + (Kξ − Im ) )    X G2 Pu ≥0 ≥ 0,

Y



τ6 (29)

0

Third equation of (31) is obtained using Schur’s complements to the inequality Pu Pu ≤ G2 . Using the last inequality, Eq. (27) implies (36). Let define a new nonsingular matrix T2 = diag{X , L}. Multiplying on the left and on the right of (35) by T2 we obtain

T

[

130 15

]

15 f ≤ 1 + xT 400

0.02 0 0



0 0.0001 0

0 0 x. 0.0001



Let us consider the sliding mode controller (3). The dynamic perturbation in the actuator is ξ = 0.05 sin (cos(0.9t ) + sin(0.4t )) − 0.0479 sin(0.4t ), with constant Kξ = 100. The controller restrictions are given by hmin = 0.01. The controller gains H = 0.0135, α = 5515.2, αx = 5541345.8296, αu = 6.5535 and sliding surface σ = −0.0015x1 + 0.0024x2 + 0.0027x3 + u designed according to Theorem 4, were obtained with the selection of parameters τ1 = 0.000054557, τ2 = 0.00000084229, τ3 = 28920.4085, τ4 = 0.00123121, τ5 = 0.198337968, τ6 = 981009.4666, γ = 0.1238. The problem was solved with the quasi-optimal solution tr Pz = 218.53. Let the initial conditions for the system be x(0) = [3 − 2 1]T . The convergence of the state trajectories and the control

J. Davila, A. Poznyak / Automatica 47 (2011) 1467–1472

1471

Fig. 1. State trajectories (above). Zoom of the state trajectory (middle). Control signal (below).

Fig. 2. Convergence of the state trajectories (solid line) to the Attracting Invariant Ellipsoid (dashed line).

signal are shown in Fig. 1, a zoom in on the convergence region is also presented in that Figure. Notice that even in the presence of perturbations, the trajectories converge to a small bounded region of the origin. In Fig. 2 the convergence of the trajectories to the γ quasi-optimal invariant ellipsoid ε( β Px ) is presented. 6. Conclusions The attracting invariant ellipsoid method is applied to the solution of the global stabilization problem of perturbed linear systems. The control signal is injected to the system through a first-order dynamic. The constants, surface and gains of the dynamic sliding mode controller are designed to reduce the effect of unmatched perturbations. The suboptimal solution to the stabilization problem is given as the computational solution of a set of Linear Matrix Inequalities. A benchmark example is presented to illustrate the workability of the method. References Andrade-Da Silva, J., Edwards, C., & Spurgeon, S. (2009). Sliding-mode outputfeedback control based on LMIs for plants with mismatched uncertainties. IEEE Transactions on Industrial Electronics, 56(9), 3675–3683.

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Jorge Davila was born in Mexico City, Mexico, in 1977. He received the B.Sc. and M.Sc. degrees from the National Autonomous University of Mexico (UNAM) in 2000 and 2004, respectively. In 2008, he received the Ph.D. degree and obtained the Alfonso Caso medal for the best Graduate student from the UNAM. He was awarded a Postdoctoral Research position in the Centro de Investigacion y Estudios Avanzados (CINVESTAV) during 2009. He currently holds a Research Professor position at the Section of Graduate Studies and Research of the National Polytechnic Institute (ESIME-UPT). His professional activities are concentrated in the fields of observation of linear systems with unknown inputs, nonlinear observation theory, robust control design, high-order sliding-mode control and their application to the aeronautic and aerospace technologies.

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Alexander S. Poznyak was graduated from Moscow Physical Technical Institute (MPhTI) in 1970. He earned Ph.D. and Doctor Degrees from the Institute of Control Sciences of Russian Academy of Sciences in 1978 and 1989, respectively. From 1973 up to 1993 he served this institute as researcher and leading researcher, and in 1993 he accepted a post of full professor (3E) at CINVESTAV of IPN in Mexico. Actually, he is the head of the Automatic Control Department. He is the director of 33 Ph.D. theses (25 in Mexico). He has published more than 140 papers in different international journals and 9 books including Adaptive Choice of Variants (Nauka, Moscow, 1986), Learning Automata: Theory

and Applications (Elsevier-Pergamon, 1994), Learning Automata and Stochastic Programming (Springer-Verlag, 1997), Self-learning Control of Finite Markov Chains (Marcel Dekker, 2000), Differential Neural Networks: Identification, State Estimation and Trajectory Tracking (World Scientific, 2001) and Advance Mathematical Tools for Automatic Control Engineers. Vol. 1: Deterministic Technique (Elsevier, 2008) and Vol. 2: Stochastic Technique (Elsevier, 2009). He is Regular Member of Mexican Academy of Sciences and System of National Investigators (SNI 3). He is Associated Editor of Ibeamerican Int. Journal on Computations and Systems. He was also Associated Editor of CDC, ACC and Member of Editorial Board of IEEE CSS. He is a member of the Evaluation Committee of SNI (Ministry of Science and Technology) responsible for Engineering Science and Technology Foundation in Mexico.