Lyapunov function design for finite-time convergence analysis: “Twisting” controller for second-order sliding mode realization

Automatica 45 (2009) 444–448

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Lyapunov function design for finite-time convergence analysis: ‘‘Twisting’’ controller for second-order sliding mode realizationI Andrei Polyakov, Alex Poznyak ∗ Automatic Control Department, CINVESTAV-IPN, A.P. 14-740, C.P. 07360 Mexico, D.F, Mexico

article

info

Article history: Received 26 November 2007 Received in revised form 25 March 2008 Accepted 30 July 2008 Available online 10 December 2008 Keywords: Lyapunov function Finite-time convergence Characteristic method High-order sliding mode

a b s t r a c t A generalization of the Zubov method of a Lyapunov function design is presented. It is based on the characteristic method application and is related to resolving the first-order partial differential equation of a special type. A successful resolution of this equation guaranties a finite-time convergence for the corresponding dynamics given by an ordinary differential equation with a discontinuous right-hand side. The suggested method is illustrated by its application to the so-called ‘‘twisting’’ controller stability analysis. The constructed Lyapunov function as well as its level line sections is graphically illustrated. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. On the Lyapunov function approach In the seminal work published by A.M. Lyapunov at the end of the 19th century there were presented some very simple (but, philosophically, very profound) theorems (thereafter referred to as the direct Lyapunov method) for deciding the stability or instability of an equilibrium point of an ODE. The idea of this approach consists in the generalization of the concept of ‘‘energy’’ and its ‘‘power’’, the usefulness of which lies in the fact that the decision on stability can be made by investigating the differential equation itself (in, fact, its right-hand side only) and not by finding its exact solution. The considered class of ODE contained basically the so-called regular differential equations with right-hand sides continuous on both variables: time and state. Later several authors (Filippov, 1988; Utkin, Guldner, & Shi, 1999) considered a more wide class of differential equations where the right-hand side was discontinuous on its variables. Modern approaches usually treat such class of differential equations as differential inclusions (DI).

This class of systems includes as a subclass the so-called Variable Structure Systems (VSS) which has attracted considerable research interest in the last three decades (Levant, 1993; Levant & Fridman, 2004; Pan & Furuta, 2006; Utkin, 1992). Although Lyapunovtype methods have been widely used in practice Edwards and Spergeon (1998) and Shtessel, Shkolnikov, and Levant (2007), the need for non-smooth Lyapunov functions in particular has been recognized for non-smooth system models (Bacciotti & Ceragioli, 1999). So, to analyze VSS with sliding modes, where ‘‘kinks’’ form an essential part of dynamics, a non-smooth Lyapunov-like analysis is required (Orlov, 2006; Ryan, 1998; Sontag & Sussman, 1995). 1.2. Motivation One of the most important features of VSS systems with sliding modes is a finite-time convergence to a sliding surface (or manifold), or in other words, a finite-time reaching phase treach . The corresponding Lyapunov functions are non-smooth in this case. For example (see Utkin (1992)), for the first-order sliding mode systems, the simplest representative of which is x˙ (t ) = −r sign [x (t )] ,

r >0

(1)

if x > 0 if x < 0 if x = 0

(2)

with I This paper was not presented at any IFAC meeting. This paper was

recommended for publication in revised form by Associate Editor Andrew R. Teel under the direction of Editor Hassan K. Khalil. ∗ Corresponding address: Automatic Control Department, CINVESTAV-IPN, Av.IPN-2508, Apartado Postal 14-740Col. San Pedro Zacatenco, 07300 Mexico D.F, Mexico. E-mail addresses: [email protected] (A. Polyakov), [email protected] (A. Poznyak). 0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.07.013

1

( sign[x] :=

−1 ∈ [−1, 1]

evidently, the Lyapunov function is V (x) = |x|, and hence, its timederivative (for x 6= 0) taken on the trajectories of (1) satisfies V˙ (x (t )) = −r

A. Polyakov, A. Poznyak / Automatica 45 (2009) 444–448

that implies 0 ≤ V (x (t )) = V (x (0)) − rt and defines treach = V (x (0)) /r. For the second-order sliding modes (see Levant (1993)), such as x¨ (t ) = −r1 sign[x(t )] − r2 sign[˙x(t )],

r1 > r2 > 0

(3)

445

2.2. The generalization of the Zubov method Let V (x, y) be a function differentiable almost everywhere. Then its derivative on the trajectories of (5) is estimated as follows V˙ = h∇x V , g i + ∇y V , a + bu¯ + f





the finite-time convergence analysis and the reaching time estimation have been done by a geometrical setting (Levant, 2007). Such proof seems to be difficult to extend to multidimensional case. Besides, in Orlov (2006) to analyze the convergence of x(t ) governed by (3) there has been suggested the Lyapunov function approach with the Lyapunov function

where h·, ·i means the scalar product in an Euclidean space, ∇y V = (| ∂ V |, | ∂ V |, . . . , | ∂ V |)T and C = (C1 , C2 , . . . , Cn )T is the ∂ y1 ∂ y2 ∂ yn vector with non-negative components. Denoting

V (x, x˙ ) = r1 |x| + x˙ 2 /2

hj (x, y, γj ) := aj (x, y) + bj (x, y)¯u(x, y) + γj ,

(4)

which guarantees only asymptotic (non-finite-time) convergence since in this case it turns out that V˙ (x(t ), x˙ (t )) ≤ 0. The modern approaches to the finite-time convergence analysis of high-order sliding mode systems are based on homogeneity principle (Bacciotti & Rosier, 2001; Hong, Huang, & Xu, 2001; Orlov, 2005; Shtessel et al., 2007). However, it does not allow to estimate the reaching time. Thus, the unified approach to the problems of the finite-time convergence analysis and the reaching time estimation of the high-order sliding mode systems have never been presented before. So, the motivation of this study is based on the following remarks: – the geometrical approach for the convergence analysis and reaching time estimation cannot be definitely extended to multidimensional case; – the Lyapunov functions providing finite-time convergence of the high-order sliding systems have never been designed; – the finite-time converged Lyapunov function permits to estimate the reaching time even in the presence of a bounded deterministic noise. 1.3. Structure of the paper In the next section the main ideas of the Lyapunov function design providing a finite-time convergence are discussed. Then, using the suggested approach, the process of the Lyapunov function designing for the so-called ‘‘twisting’’ controller is given. Finally, it is illustrated by a numerical example. 2. The method of the Lyapunov function design with finitetime convergence 2.1. System description Consider an affine control system given by x˙ = g (x, y) y˙ = a(x, y) + b(x, y)u + f (t , x, y)



 ≤ h∇x V , g i + ∇y V , a + bu¯ + ∇y V , C

(8)

j = 1, n

the inequality (8) might be rewritten (in the component-wise form) as dV dt

≤

k X ∂V i =1

∂ xi

g i ( x, y ) +

   k X ∂V ∂V hj x, y, Cj sign . ∂ y ∂ yj j j =1

Let us try to find the Lyapunov function V (x, y) as an absolute continuous positive definite solution of the following partial differential equation k ∂V X ∂V + hj (x, y, γj ) = −qV ρ (9) ∂ xi ∂ yj i =1 j =1 h i where γj = Cj sign ∂∂yV and q > 0, ρ > 0 are some positive j k X

gi (x, y)

parameters. Remark 1. (a) In the case ρ < 1 we obtain the Lyapunov function with finite time convergence treach ≤ q(11−ρ) [V (x(0), y(0))]1−ρ . (b) For ρ ≥ 1 the positive definite solution of (9) implies only an asymptotic convergence. To solve the partial differential equation (9) we need to define precisely h i the piecewise constant function γj ,which depends on sign ∂∂yV . Fortunately, since γj takes only two values Cj and j

−Cj , we may consider (9) supposing that γj is one of these two constants. Then, obtaining V (x, y) as a solution of (9), one can try define the parameters γj in an appropriate way. The idea of the Lyapunov function design as the solution of the partial differential equation was presented by Zubov (1964). The well-known variable gradient method (see, for example, Khalil (2002)) is also based on this idea. The presented generalization of the Zubov method permits the right-hand side of (9) to be depended on the function V too. Only this assumption allows us to find the Lyapunov function with finite-time convergence.



(5)

where x ∈ R , y ∈ R are components of the state-space vector, g : Rk × Rn → Rk , a : Rk × Rn → Rn are smooth system vector functions, u ∈ Rm is a vector of control inputs, b : Rk × Rn → Rn×m are control-gain matrix, and f : R+ × Rk × Rn → Rn is an uncertain measurable but bounded function (in fact, a deterministic noise), i.e., k

fj (t , x, y) ≤ Cj ,

∀x ∈ Rk , ∀y ∈ Rn , ∀t ≥ 0, j = 1, n.

(6)

Supposing that a stabilizing control u is already designed as u = u¯ (x, y)

2.3. Method of characteristics

n

The solution of (9) can be found using the so-called Method of Characteristics (El’sgol’ts, 1961). According to this method we can formulate the following lemma. Lemma 1. If an absolute continuous positive definite function V (x, y) satisfies the following system of ODE (in the symbolic form) dx1 g1 (x, y)

= ··· =

(7)

(which, in fact, is admitted to be discontinuous) we have to prove the stability (asymptotic or finite-time) of zero solution (0, 0) of the system (5) with the control (7) using the Lyapunov function technique.

= ··· =

dxk gk (x, y)

dyn hn (x, y, γn )

=

=

dy1 h1 (x, y, γ1 )

dV

−qV ρ

(10)

for kxk2 + kyk2 > 0, then the same function V (x, y) is a solution of (9).

446

A. Polyakov, A. Poznyak / Automatica 45 (2009) 444–448

Proof. For kxk2 + kyk2 > 0 we have V (x, y) > 0, dxi = dV dV −gi (x, y) qV ρ , dyj = −hj (x, y, γi ) qV ρ and

Xn ∂ V ∂V dV = dx + dy i=1 ∂ x j =1 ∂ y i j  X Xn ∂ V dV ∂V k gi (x, y) + hj (x, y, γi ) =− ρ j=1 ∂ y i =1 ∂ x qV i j Xk

implies (9).

Suppose that the system of ODE (10) can be solved and its first integrals are i = 1, n + k.

(11)

Since any function of constants is a constant too (in the partial case, zero), the function V (x, y, γ , q, ρ) can be found as a solution of nonlinear algebraic equation

Φ (ϕ1 (V , x, y, γ , q, ρ), . . . , ϕn+k (V , x, y, γ , q, ρ)) = 0

For an arbitrary absolute continuous function V (x, y) we have

∂V ∂V ∂V ∂V x˙ + y˙ = y + (u + f (t , x, y)) dt ∂x ∂ x   ∂y ∂ y ∂V ∂V ∂V ∂V ∂V ≤ y + u + C sign =y − sign[x]γ ∂x ∂y ∂y ∂x ∂y   ∂V γ = r1 + r2 sign[xy] − γ0 , γ0 = C sign x . (16) ∂y dV



ϕi (V , x, y, γ , q, ρ) = const := ci ,

3.2. Partial differential equation

(12)

where Φ (ϕ1 , . . . , ϕn+k ) is an arbitrary function. An explicit analytical solution of the last equation (if it exists) defines the Lyapunov function candidate. However, the function Φ and the parameters γ , q, ρ should be chosen in such a way that the function V (x, y) should be an absolute continuous and positive define. Only such functions V (x, y) may be considered as a Lyapunov function candidate. Remark 2. Obviously, this approach does not provide a formal algorithm for a Lyapunov function design, but only helps to derive a specific form of a Lyapunov function candidate that reduces the problem of a Lyapunov function design to the problem of the correct parameters definition which guarantee the absolute continuity and positive definiteness of the obtained ‘‘energetic’’ function. Below we will illustrate the proposed method designing the Lyapunov function with finite-time convergence (0 < ρ < 1) for the second-order sliding system with ‘‘twisting’’ controller. 3. Lyapunov function design for twisting control algorithm

=

According to the proposed method Eq. (9) becomes

∂V ∂V − sign[x]γ = −kV 1/2 (17) ∂x ∂y where k > 0 is a positive parameter. Supposing that γ0 is a constant y

let us find the solution of (17). The corresponding system (10) of the characteristic ODE is dx y

=

dy

dV

√ . −k V

=

− sign[x]γ

(18)

Its first integrals are as follows

ϕ1 (x, y) = |x| +

y2 2γ

√ ,

ϕ2 (V , y) =

y sign[x]

−

γ

2 V k

.

(19)

Select Φ as

√ Φ (ϕ1 , φ2 ) = k0 ϕ1 + ϕ2 = 0

(20)

where k0 is a real parameter. Substituting (19) in to (20) leads to the equation

√ 2 V k

y sign[x]

=

γ

s + k0 |x| +

y2 2γ

.

(21)

Obviously, the right-hand side of (21) makes sense for all |x| > 0 if and only if γ > 0, or equivalently, when r1 > r2 + C . Since the left-hand side of (21) is always non-negative we have

√

sign[xy] 2/γ

k0 > − p

1 + 2γ |x|/y2

.

(22)

So, the function V (x, y) can be found from (21) as 3.1. System description and problem formulation V (x, y) =

Consider the controlled system given by



x˙ = y y˙ = f (t , x, y) + u(t )

(14)

where r1 , r2 > 0 are control parameters, and the operator sign[·] is defined by (2). The solution of (13) is understood in the Filippov sense (see Filippov (1988)). It is supposed that

|f (t , x, y)| ≤ C ,

∀x, y ∈ R and ∀t ≥ 0

4

 y sign[x]



γ

s + k0 |x| +

y2 2γ

2  .

(23)

(13)

where x, y ∈ R are the scalar state variables, f (t , x, y) is a measurable but unknown function treated hereafter as an external noise or perturbation, u ∈ R is the so-called ‘‘twisting’’ control (see Levant (1993)) u(t ) = −r1 sign[x(t )] − r2 sign[y(t )]

k2

(15)

with a known constant C . Now the main problem is to design the Lyapunov function which provides the finite-time convergence of the solutions of (13) to the origin (0, 0) under the control (14) and the assumptions above.

3.3. Removal of discontinuities The next step consists in the removal of the discontinuities of the function (23) by special selection of the parameters k and k0 . Considering the partial limits of the function V (x, y) when x tends to zero for any fixed y and when y tends to zero for any fixed x, one can derive if x → 0

then V (x, y) →



4 k2 k20

sign[xy]

γ

k0

+√

2γ

2

y2

| x| . 4 Hence, to eliminate the discontinuities on the lines x = 0 and y = 0 it is sufficient to resolve the following system if y → 0

  

k2



then V (x, y) →

k2

sign[xy]

γ

k2 k20 = 1

k0

+√

2γ

2

= k¯ 2

A. Polyakov, A. Poznyak / Automatica 45 (2009) 444–448

¯ This gives for some positive value k. r k=

γ p ¯ 2γ k − 1 > 0, 2

(3) the corresponding guaranteed reaching time is

s k0 =

2

γ

sign[xy]

√

2γ k¯ − 1

.

k¯ 2 4

y2

and

V (x, 0) =

|x| 4

.

(25)

Combining (24) with (22) we can formulate the following simple result. Lemma 2. If r1 > r2 + C and r2 > C and the parameter k¯ in (24) is selected in such a way that the following inequalities hold 1

√

2(r1 + r2 − C )

< k¯ < √

1

(26)

2(r1 − r2 + C )

then k0 > 0 and condition (22) holds for all xy 6= 0. Proof. (1) Consider the case xy

>

0. Then we need to

check the validity of the following inequality 2 −1/2

− 1 + 2γ+ |x|/y




√

with γ+ = r1 + r1 + C


−1

2γ+ k¯ − 1

>

sign[x ∂∂Vy ]

which

directly results from the inequalities k¯ > (2(r1 + r2 − C ))−0.5 ≥ (2γ+ )−0.5 .


−1 √ > < 0 we have to prove 1 − 2γ− k¯
∂V 2 −0.5 1 + 2γ− |x|/y with γ− = r1 − r1 + C sign[x ∂ y ] that results from 0 < k¯ <√(2(r1 − r2 + C ))−0.5 ≤ √ (2γ− )−0.5 . ¯ (3) Since 2γ√ 2γ− k¯ − 1 < 0 we have + k − 1 > 0 and sign[xy] = sign[ 2γ k¯ − 1] and k0 > 0 for all xy 6= 0.  (2) When xy

(27) if x = 0 if y = 0

3.4. Main theorem The rigorous consideration of the partial derivative ∂∂Vy shows h i

√

that sign ∂∂Vy switches on the line L |x| sign[x] + y = 0 where L is a constant depending on the parameters√ of the system (13)–(15). However, defining γ as function of sign[L |x| sign[x] + y] one has to definitely remove new discontinuities of the function V (x, y). So, the more preferable definition of γ is (28)

that leads to the following claim.

(1) V (x, y) is absolute continuous in all space R2 and continuously differentiable when xy 6= 0; (2) the time derivative of V (x, y) on the trajectories of (13)–(15) satisfies r1 − r2 − C √ ≤ −kmin V dt r1 − r2 + C .

almost always with kmin := min(k);

V (x(0), y(0)).

(30)

∂V k2 = ∂x 2

1

k0

1 + 2γ |x| /y2

p

∂V k2 = ∂y 2

y

γ2

+

k0 y2 2γ

√

2γ

k20 sign[x]

+

2

k0 sign[x](y2 + γ |x|)

+

!

! .

p γ 2 |x| + y2 /(2γ )

Since γ , k and k0 switch only on the lines x = 0 and y = 0, it follows that ∂∂Vx and ∂∂Vy are continuous for all x 6= 0 and y 6= 0. Moreover, the left and right limits on these lines are finite everywhere in any bounded region. (3) Consider then the time derivative of the function V (x, y) on the trajectories of system (13):

γ¯ =k V − + dt γ √

dV

r

γ 2



k0 sign[xy] 1 + 2γ |x|/y2

p

γ¯ 1− γ

!

where γ¯ = r1 + r2 sign[xy] − f (t , x, y) sign[x], γ = r1 + γ¯ r2 sign[xy] − C sign[xy] (see (28)). Since, 1 ≤ γ for xy > 0 and γ¯

1≥ γ ≥

r1 −r2 −C r1 −r2 +C .

r

γ 2

for xy < 0 the inequality

p

γ¯ 1− γ



k0 sign[xy] 1 + 2γ |x|/

y2

 ≤−

r1 − r2 − C r1 − r2 + C

(31)

√

holds. So, we have dV ≤ −kmin r11−r22+C . V for x, y 6= 0. dt (4) To complete the proof it is enough to notice that the system (13)–(15) does not have nontrivial solutions with x(t ) ≡ 0 or y(t ) ≡ 0, or in other words, the trajectories cross these switching lines avoiding the movements along them (see Orlov (2006)). Thus, the inequality (29) holds almost always along the trajectory of the system (13)–(15).  Remark 4. Notice that the Main Theorem conditions r1 > r2 + C , r2 > C coincide with the ones in Levant and Fridman (2004). Remark 5. It is not difficult to show that using the designed Lyapunov function (27) the local stability (with a finite-time convergence) of (13)–(14) takes place if the condition (15) holds only in a local neighborhood of the origin D0 provided that the system starts from within domain D(c max ), where D(c ) = {(x, y)T ∈ R2 : V (x, y) ≤ c } and c max = max∀c :D(c )⊆D0 c . 4. Numerical example

Theorem 3. If r1 > r2 + C , r2 > C then the function V (x, y) (27) with γ given by (28) has the properties:

dV

kmin (r1 − r2 − C )

r −r −C

if xy 6= 0

with k, k0 are as in (24) and k¯ satisfying (26).

γ := r1 + (r2 − C ) sign[xy]

2 (r1 − r2 + C ) p

Proof. (1) Obviously the function V (x, y) may have discontinuities only on the lines x = 0 and y = 0. Since the parameters k and k0 (24) are selected in such a way that discontinuities have to be removed (see (25)), the function V (x, y) is continuous by the accepted construction. (2) The partial derivatives of the function V (x, y) are

γ¯ − + γ

So, the Lyapunov function (23) becomes

  2 s  2  2 y sign[x] y k      + k0 |x| + 4 γ 2 γ V ( x, y ) =   k¯ 2 y2 /4   |x|/4

treach ≤

(24)

Therefore, the function V (x, y) (23) in view of the continuity property, discussed above, should satisfy V (0, y) =

447

(29)

Consider the system (13)–(15) with r1 =√1.3, r2 = 0.7, C = 0.4. The Lyapunov function (27) for k¯ = 0.9/ 2 is

 √  (0.9 γ − 1)2    8γ V (x, y) =   0 . 101 25y2   0.25|x|

p |y| +

2γ |x| + y2

√

0.9 γ − 1

!2 if xy 6= 0 if x = 0 if y = 0

where γ = 1.3 + 0.3 sign[xy] and it is presented in Fig. 1. Fig. 2 shows the level lines of the Lyapunov function.

448

A. Polyakov, A. Poznyak / Automatica 45 (2009) 444–448

Fig. 1. The Lyapunov function.

Fig. 2. The level lines of the Lyapunov function.

5. Conclusion In this paper we present a method of Lyapunov function design providing a finite-time convergence of system dynamics to the origin for a class of ODE with discontinuous right-hand side. It is based on Characteristic Method application and related with resolution of the first-order partial differential equation of a special type. Even the suggested approach does not provide a formal algorithm for a Lyapunov function design, it helps to derive a specific form of a Lyapunov function candidate that reduces the problem of a Lyapunov function design to the problem of the correct parameter definition guaranteeing the absolute continuity and positive definiteness of the obtained ‘‘energetic’’ function. The suggested method is successfully illustrated by its application to the ‘‘twisting’’ controller stability analysis. This approach can be extended to the case when instead of   u (t ) in (13) one uses k (t , x, y) u (t ) where k (t , x, y) ∈ k− , k+ , k− > 0. References Bacciotti, A., & Ceragioli, F. (1999). Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions. ESAIM: Control, Optimisation and Calculus of Variations, 4, 361–376. Bacciotti, A., & Rosier, L. (2001). Lyapunov functions and stability in control theory. In Lecture notes in control and information sciences: Vol. 267. New York: Springer.

Edwards, C., & Spergeon, S. (1998). Sliding mode control: Theory and applications. Taylor & Francis. El’sgol’ts, L. E. (1961). International monographs on advanced mathematics and physics, Differential equations. India, Delhi: Hindustan Publisher Company. Filippov, A. F. (1988). Differential equations with discontinuous right-hand sides. Dordrecht, Boston, London: Kluwer Academic Publishers. Hong, Y., Huang, J., & Xu, Y. (2001). On an output feedback finite-time stabilization problem. IEEE Transactions on Automatic Control, 46(2), 305–309. Khalil, H. K. (2002). Nonlinear systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall. Levant, A. (1993). Sliding order and sliding accuracy in sliding mode control. International Journal of Control, 58(6), 1247–1263. Levant, A., & Fridman, L. (2004). Robustness issues of 2-sliding mode control. In A. Sabanovic, L. Fridman, & S. Spurgeon (Eds.), Variable structure systems: From principles to implementations (pp. 131–156). London: The IEE. Levant, A. (2007). Principles of 2-sliding mode design. Automatica, 43(4), 576–586. Orlov, Y. (2005). Finite time stability and robust control synthesis of uncertain switched systems. SIAM Journal on Control and Optimization, 43(4), 1253–1271. Orlov, Y. (2006). In C. Edwards, E. F. Colet, & L. Fridman (Eds.), Advances in variable structure and sliding mode control, Extended invariance principle and other analysis tools for variable structure systems (pp. 3–22). Berlin, Heidelberg, New York: Springer. Pan, Y., & Furuta, K. (2006). In C. Edwards, E. F. Colet, & L. Fridman (Eds.), Advances in variable structure and sliding mode control, Variable structure control systems using sectors for switching rule design (pp. 89–106). Berlin, Heidelberg, New York: Springer. Ryan, E. P. (1998). An integral invariance principle for differential inclusions with applications in adaptive control. SIAM Journal on Control and Optimization, 36(3), 960–980. Sontag, E. D., & Sussman, H. J. (1995). Nonsmooth control-Lyapunov functions. In Proc. 34th IEEE conf. decision control (pp. 2799–2805). Shtessel, Y. B., Shkolnikov, I. A., & Levant, A. (2007). Smooth second-order sliding modes: Missile guidance application. Automatica, 43(8), 1470–1476. Utkin, V. I. (1992). Sliding modes in control optimization. Berlin: Springer Verlag. Utkin, V. I., Guldner, J., & Shi, J. (1999). Sliding modes in electromechanical systems. London: Taylor & Francis. Zubov, V. I. (1964). Methods of A.M. Lyapunov and their applications. Groningen: P. Noordho: Limited. Andrei Polyakov received the M.Sc. degree in Applied Mathematics in 2003 and the Ph.D. degree in 2005 from Voronezh State University (Russia). From 2004 up to 2007 he was Lecturer with the Applied and System Software Department of this university. He is currently Postdoctoral student at CINVESTAV (The Research and Advance Education Center) in Mexico. His research interests include relay and sliding mode control and timedelay systems stability.

Alex 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 (3-E) at CINVESTAV of IPN in Mexico. Actually, he is the head of the Automatic Control Department. He is the director of 30 Ph.D theses (22 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). He is the Regular Member of Mexican Academy of Sciences and System of National Investigators (SNI -3). He is the Associated Editor of Ibeamerican Int. Journal on ‘‘Computations and Systems’’. He was also the 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.