Global sliding-mode observer with adjusted gains for locally Lipschitz systems

Automatica 47 (2011) 565–570

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Brief paper

Global sliding-mode observer with adjusted gains for locally Lipschitz systems✩ D. Efimov a,∗ , L. Fridman b a

University of Bordeaux, IMS lab, Automatic control group, 351 cours de la libération, 33405 Talence, France

b

Departamento de Ingeniería de Control y Robótica, Facultad de Ingeniería UNAM, Edificio‘‘T’’, Ciudad Universitaria D.F., Mexico

article

abstract

info

Article history: Available online 15 January 2011

A state observer design procedure is proposed for nonlinear locally Lipschitz systems with high relative degree from the available for measurements output to the nonlinearity. The possible presence of disturbances is taken into account. The solution is based on logic-based control and the high order supertwisting observer. The approach is applicable to nonlinear systems with bounded solutions. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Observers Nonlinear systems Sliding Mode Supervisory control Disturbance signals

1. Introduction The state observer design problem for nonlinear systems has been an area of intensive research during the last two decades. There exist a lot of solutions in the area dealing with diverse forms of system models (see, for example, Besançon, 2007, Nijmeijer & Fossen, 1999 and references therein). Application of sliding-mode observers allows one to ensure finite-time convergence of a part of the estimation error to zero even in the presence of unknown inputs. Additionally, equivalent control methods may help to estimate the values of unknown inputs affecting the system, which is useful for fault detection (see, for example, Edwards, Spurgeon, & Hebden, 2002, Spurgeon, 2008, Weitian & Saif, 2008 and references therein). Such observers based on first order sliding modes must use low-pass filters in each step, thereby leading to the deterioration of the estimation error. Recently developed sliding-mode observers based on step-by-step super-twisting differentiation provide finite-time exact estimation without additional filters (Bejarano, Poznyak, & Fridman, 2007; Floquet & Barbot, 2007; Levant, 1993, 1998, 2003). The following class of Lipschitz nonlinear systems has seen much attention: x˙ = Ax + ϕ(y) + Bf(x, d), n

y = Cx

(1) m

where x ∈ R is the state vector; d ∈ R is a disturbing input; y ∈ Rp is the available for measurements output and ϕ : Rp → Rn , ✩ A shorter version of this paper was presented during the 8th IFAC Symposium on Nonlinear Control Systems, University of Bologna — Italy, September, 01–03 2010. This paper was recommended for publication in revised form by Associate Editor Raul Ordóñez under the direction of Editor Miroslav Krstic. ∗ Corresponding author. Tel.: +33 5 40 00 25 05; fax: +33 5 56 37 15 45. E-mail addresses: [email protected], [email protected] (D. Efimov), [email protected] (L. Fridman).

0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.12.003

f : Rn+m → Rp are Lipschitz continuous (globally in the case of f); matrices A, B, C have appropriate dimensions. An advantage of this class of systems consists in the fact that almost all nonlinear systems of the form x˙ = F(x, d), y = Cx, where F : Rn+m → Rn is locally Lipschitz continuous, can be reduced to (1) at least locally. The first attempts to generalize sliding-mode observers for the nonlinear system (1) are made in Floquet and Barbot (2007), Fridman, Shtessel, Edwards, and Xing-Gang (2008). All these solutions are obtained under the assumption of globality of the Lipschitz property for unknown inputs and nonlinearities. The observer based on the global second order sliding-mode differentiator (Pisano & Usai, 2007) is designed in Bejarano, Fridman, and Pisano (2008) for nonlinear systems with unknown inputs which may be unbounded. In the work of Efimov and Bobtsov (2009), a solution to the problem is proposed for the perturbed system (1) with a locally Lipschitz function f. In this case, there exist no observer gains which can provide global convergence of the estimation error. The solution is obtained under the assumption that the derivative of the output depends on the nonlinear function (relative degree one) and it is based on a conventional sliding-mode observer (Edwards et al., 2002). Growing observer gains are updated by an eventbased algorithm. On each step the new gains are substituted in the observer using a logic-based scheme if the previous observer gains fail to satisfy some performance criteria (e.g., fail to ensure convergence of the estimation error). The infinite growth of the gains in the case where disturbances are present is avoided. In this work the result of Efimov and Bobtsov (2009) is extended to the case of high relative degree; such a consideration leads to the application of more complex sliding-mode techniques, i.e. the step-by-step super-twisting differentiation algorithms (Bejarano et al., 2007; Besançon, 2007; Cannas, Cincotti, & Usai, 2002) are used in this work (other kinds of differentiators can be adopted

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D. Efimov, L. Fridman / Automatica 47 (2011) 565–570

in a similar way, see Stotsky and Kolmanovsky (2002) for their performance comparison). The principal novelty of the proposed solution consists in the combination of different (local) observation approaches in one global estimator under discrete logic-based algorithm supervision. The estimation error convergence and boundedness in the presence of bounded disturbances are proven for the obtained hybrid system. Preliminaries are introduced in Section 2. Main results are presented in Section 3. Results of application to a satellite system are discussed in Section 4.

from (2) we have:

ξ˙ 1 = Gξ1 + Rξ2 + sf1 (x, d), ξ˙ 2 = A3 Kξ1 + A4 ξ2 + f2 (Kξ1 ); −1

G=K

A1 K,

The symbol |d| represents the Euclidean norm of the vector d ∈ Rm , the norm of measurable and essentially bounded functions d : R+ → Rm of time t ≥ 0 is defined as follows

‖d‖[t0 ,t ) = ess sup{|d(t )|, t ∈ [t0 , t )}. We will further denote the set of all such functions which satisfy ‖d‖[0,+∞) = ‖d‖ < +∞ as MRm . As usual, the continuous function σ : R+ → R+ belongs to class K if it is strictly increasing and σ (0) = 0; additionally it belongs to class K∞ if it is also radially unbounded. If for all initial conditions x0 ∈ Rn and inputs d ∈ MRm the solutions x(t , x0 , d)(y(t , x0 , d) = Cx(t , x0 , d)) of system (1) are defined for all t ≥ 0, then the system is called forward complete. 3. Main results In this work we suppose that the state vector x and the disturbing input d in system (1) are bounded without precise information on their upper bounds. Assumption 1. Let for system (1) ‖x‖ < +∞ and d ∈ MRm .



R=K

−1

baT2

(3b)

,

s=K

−1

b.

Due to Assumption 2 and the structure of K we have



0

.

2. Preliminaries

(3a)

.. G=  0 g1

1

.. .

0 g2

··· .. . ··· ···

0



..  . , 

1 gn1



0

 ..

. R=  0 r1

··· .. . ··· ···

0



..  . , 

0 rn2

0

   ..  . s=   0 s

ξ1,1 = y,

g = [g1 · · · gn1 ],

r = [r1 · · · rn2 ].

Consider the step-by-step super-twisting observer (Besançon, 2007):

 v˙ 1 = −λ |v1 − y|sign(v1 − y) + υ1 , υ˙ 1 = −µsign(v1 − y)  v˙ j = γj [−λ |vj − v˙ j−1 |sign(vj − v˙ j−1 ) + υj ], υ˙ j = γj [−µsign(vj − v˙ j−1 )], 2 ≤ j ≤ n1 − 1,

(4)

(5)

vn1 = vn1 −1 ,

where λ > 0, µ > 0, γj ∈ {0, 1} are coefficients to be defined later. Consider the error σ = v − ξ1 , whose dynamics due to (4), (5) take the form:

In what follows, for simplicity of consideration, we impose some structural restrictions on system (1).

 σ˙ 1 = −λ |σ1 |sign(σ1 ) + υ1 − ξ1,2 , υ˙ 1 = −µsign(σ1 ), (6a)  σ˙ j = γj [−λ |vj − v˙ j−1 |sign(vj − v˙ j−1 ) + υj ] − ξ1,j+1 , (6b)

Assumption 2. Let p = 1 and

υ˙ j = γj [−µsign(vj − v˙ j−1 )],

(i) system (1) can be represented in the form: x˙ 1 = A1 x1 + b[f1 (x, d) + aT2 x2 ], x˙ 2 = A3 x1 + A4 x2 + f2 (x1 ),

y = cT x1 ,

x = [xT1 xT2 ]T ,

(2a) (2b)

where x1 ∈ R , x2 ∈ R , n = n1 + n2 and the functions f1 and f2 are locally Lipschitz continuous; (ii) the relative degree of the system is n1 > 1, i.e. n1

cT Ai1 b = 0,

n2

0 ≤ i ≤ n − 2;

n −1

cT A11

b ̸= 0;

(iii) the matrix pair (A1 , c) is observable, matrix A4 is Hurwitz.



2 ≤ j ≤ n1 − 1.

To clarify properties of system (6) note that Eq. (6a) and (A.1), (A.2) in the Appendix coincide. Let γj = 0, 2 ≤ j ≤ n1 − 1. According to Theorem A.1 if the upper bound L > 0 is known such that ‖ξ1,2 ‖ ≤ L and ‖ξ˙1,2 ‖ = ‖ξ1,3 ‖ ≤ L, then taking coefficients as in Theorem A.1 (Polyakov & Poznyak, 2008, 2009) (see also recommendation below the theorem) it is possible to ensure

σ1 (t ) = v1 (t ) − y(t ) = 0,

σ˙ 1 (t ) = v˙ 1 (t ) − y˙ (t ) = 0

for all t ≥ T , where T ≥ 0 is an upper estimate on finite time of convergence for system (A.1), (A.2) evaluated in Theorem A.1. Then v˙ 1 (t ) = ξ˙1,1 (t ) = ξ1,2 (t ) for all t ≥ T . Since the signal ξ1,1 is available for measurements, through a proper choice of v1 (0), the quantity σ1 (0) = 0 can always be guaranteed, then the estimate (A.3) on the value of T is given in the Appendix. Set γ2 = 1 at the time instant t = T and v2 (T ) = ξ1,2 (T ) = v˙ 1 (T ), then σ2 (T ) = 0 and from (6b) for j = 2 we have

The case of the scalar output y is considered for brevity of presentation (an extension to the case p > 1 is straightforward). Part (iii) of the assumption deals with observability of system (2), and the only restriction is presented in parts (i) and (ii): that x2 and the nonlinear function f1 affect the n1 -th derivative of the output y. The outline of this section is as follows. In the first part, following Besançon (2007), we introduce the step-by-step super-twisting observer equations and substantiate the observer properties for the case when the exact upper estimates for ‖x‖ and ‖d‖ are given. In the second part, a procedure is proposed that is used to verify the accuracy of the given upper estimates for ‖x‖ and ‖d‖. In the third part, the equations of the proposed hybrid state observer for system (2) are presented and global convergence of estimation error is proven.

 σ˙ 2 = −λ |σ2 |sign(σ2 ) + υ2 − ξ1,3 ,

3.1. Step-by-step super-twisting observer

v˙ j (t ) = ξ˙1,j (t ) = ξ1,j+1 (t ) for all t ≥ jT

Define K−1 = cT



AT1 cT · · · (AT1 )n−1 cT

T

and consider the new

observation vector ξ = [ξT1 ξT2 ]T , where ξ1 = K−1 x1 , ξ2 = x2 ;

υ˙ 2 = −µsign(σ2 ),

which once again has the form (A.1), (A.2). Therefore,

v˙ 2 (t ) = ξ˙1,2 (t ) = ξ1,3 (t ) for all t ≥ 2T providing that ‖ξ1,3 ‖ ≤ L and ‖ξ˙1,3 ‖ = ‖ξ1,4 ‖ ≤ L. Repeating this procedure step-by-step for 2 ≤ j ≤ n1 − 1 and setting γj = 1 for t ≥ (j − 1)T and vj (jT ) = v˙ j−1 (jT ) we obtain

under the assumption that ‖ξ1,j ‖ ≤ L, 2 ≤ j ≤ n1 , ‖ξ˙1,n1 ‖ ≤ L. Thus the vector ξ1 (t ) is estimated in finite time (n1 − 1)T or equivalently σ(t ) = 0 for t ≥ (n1 − 1)T . By definition of the error

D. Efimov, L. Fridman / Automatica 47 (2011) 565–570

σ this implies v(t ) = ξ1 (t ) for t ≥ (n1 − 1)T . As the last step define γn1 = 1 for t ≥ (n1 − 1)T in the equation z˙ = γn1 [A3 Kv + A4 z + f2 (Kv)],

(7)

where the variable z estimates the vector x2 . The dynamics of the estimation error e = x2 − z are governed by the following equation: e˙ = A4 e. According to Assumption 2, matrix A4 is Hurwitz and thus z asymptotically converges to x2 . Therefore, the whole state x is observed by (4), (5), (7) (components x1 in finite time and x2 asymptotically). As the next step, it is necessary to evaluate L. For the locally Lipschitz function f1 for all x ∈ Rn , d ∈ MRm :

|f1 (x, d)| ≤ α(1 + |(x, d)|) ≤ α(1 + |x| + ‖d‖) for some function α ∈ K . Take some guess values X0 > 0, D0 > 0 such that ‖x‖ ≤ X0 and ‖d‖ ≤ D0 , then we obtain: |f1 (x, d)| ≤ α(1 + X0 + D0 ), |ξ1 | ≤ ψ X0 , |ξ˙1,n1 | ≤ (ψ|g| + |r|)X0 + |s|α(1 + X0 + D0 ), ψ = max{1, λmin (KT K)−0.5 }. Thus, L = L(X0 , D0 ) where L(X , D) = max{ψ X , (ψ|g| + |r|)X + |s|α(1 + X + D)}. 3.2. Observer gains verification Unfortunately the guessed values X0 , D0 are not known and in the general case for some particular 1 ≤ j ≤ n1 the inequality ‖ξ1,j ‖ ≤ L(X0 , D0 ) can be violated for a given X0 , D0 . Therefore, it is necessary to propose a procedure to validate the values of X0 , D0 . To do so one can use property of finite-time convergence to zero of the available for measurements signal σ1 (t ) = v1 (t ) − y(t ) on the first step, i.e. if the choice of L is correct then σ1 (t ) = 0 for t ≥ T . Next, the constrain σj (t ) = 0 has to be satisfied for t ≥ jT . Therefore, failure of one of these conditions can be used for detect consistency of the values X0 and D0 . 3.3. Global hybrid observer for locally Lipschitz nonlinear systems Assume that there exists t ′ ≥ jT such that |σj (t ′ )| > 0, then this means that the constants X0 , D0 have been chosen not sufficiently high. Taking for X0 and D0 new higher values it is necessary to repeat all the steps described above, which can be formalized as follows: Xi = hx (i, Xi−1 ), X0 > 0,

D0 > 0,

Di = hd (i, Di−1 ),

 λi = 7(κ − 1)Li ,   1 ¯ Ti = 2k− Li = L(Xi , Di ); min k 2/7 Li /(κ − 1), µi = κ L i ,

z˙ = γn1 (t )[A3 Kv + A4 z + f2 (Kv)];

v˙ 1 = −λi |v1 − y|sign(v1 − y) + υ1 , υ˙ 1 = −µi sign(v1 − y),  v˙ j = γj (t )[−λi |vj − v˙ j−1 |sign(vj − v˙ j−1 ) + υj ], υ˙ j = γj (t )[−µi sign(vj − v˙ j−1 )], 2 ≤ j ≤ n1 − 1, vn1 = v˙ n1 −1 ;  0 if ti ≤ t < ti + (j − 1)Ti ; γj (t ) = 2 ≤ j ≤ n1 ; 1 if t ≥ ti + (j − 1)Ti 

ti+1 =

min {tj′ },

1≤j≤n1 −1

tj′ = arg inf{σj (t ) ̸= 0}, t ≥ti +jTi

(8)

i = 1, 2, 3, . . . , N ≤ +∞;

t0 = 0 1 ≤ j ≤ n1 − 1,

(9)

(10) (11a) (11b)

(12) (13)

567

where the discrete systems (8) have well defined strictly increasing solutions for any X0 > 0, D0 > 0 for all i ≤ 1; κ > 5 is a fixed parameter. Formulas for kmin and k¯ are presented in the Appendix. The following result describes stability properties of this hybrid observation algorithm. Theorem 1. Let Assumptions 1 and 2 hold and discrete systems (8) have well defined strictly increasing to infinity solutions for any X0 > 0, D0 > 0 for all i ≥ 1. Then for any κ > 5 for system (1) along with algorithm (8)–(13) it holds that - ‖z‖ < +∞, ‖v‖ < +∞, ‖υ‖ < +∞; - there exists the last iteration N < +∞ of the algorithm and 0 ≤ TN0 < +∞ such, that |x1 (t ) − Kv(t )| = 0 for all t ≥ TN0 ; - for any ε > 0 there exists TN0 ≤ Tε < +∞ such that

|x2 (t ) − z(t )| ≤ ε for all t ≥ Tε . Proof. If at some step N ≥ 0 the constant LN is correctly assigned in (9) (the guess values XN , DN have been chosen to satisfy ‖x‖ ≤ XN , ‖d‖ ≤ DN ), then according to Besançon (2007) the step-by-step super-twisting differentiator (11) ensures finite-time exact estimation of the state x1 , next the observer (10) ensures asymptotical estimation of x2 due to Assumption 2. To prove that (8), (9), (13) work properly and such a N ≥ 0 indeed exists let us consider the algorithm (8)–(13) operation. For any i ≥ 1 at time instant ti the guess values Xi , Di are derived from Eqs. (8) (for i = 0 the initial conditions X0 , D0 are used). Further, the values of the twisting observer gains λi , µi and the step time Ti are calculated in accordance with (9). If the constants Xi , Di have been chosen correctly, v(t ) = K−1 x1 (t ) for all t ≥ ti + (n1 − 1)Ti . The value ti + Ti in this case indicates the time instant when available for measurements signal σ1 (t ) = y(t ) − v1 (t ) should reach zero, at time instants ti + jTi the signals σj have to approach zero, 2 ≤ j ≤ n1 − 1. At this point the ‘‘offline’’ part of the calculations in algorithm (8)–(13) is finished (it is assumed that all these computations are done at the time instant ti ) and ‘‘on-line’’ operations are initiated. During the ‘‘on-line’’ part, observers (10), (11) attempt to estimate the system (2) state vector values x(t ). Algorithm (11) defines the step-by-step expansion of the super-twisting observer. Due to (13) each next step 2 ≤ j ≤ n1 is activated if the corresponding errors σk on the previous steps 1 ≤ k ≤ j − 1 are identically zero. This ensures boundedness of the vectors v and υ (vector v is not continuous due to initial conditions choice for different iterations, i.e. vj (ti + jTi ) = v˙ j−1 (ti + jTi ), 2 ≤ j ≤ n1 − 1). Observer (10) is activated for t ≥ ti + (n1 − 1)Ti , when e˙ = A4 e, where e = x2 − z, therefore z always stays bounded. If for some tj′ ≥ ti + jTi the condition |σj (tj′ )| > 0 is satisfied, 1 ≤ j ≤ n1 − 1, which implies that the values Xi , Di have not been taken sufficiently high. Then ti+1 = tj′ and it is necessary to repeat all steps of the algorithm. By assumptions, systems (8) have strictly increasing solutions Xi , Di for all i ≥ 0. Since ‖x‖ < +∞ and ‖d‖ < +∞, for strictly increasing sequences Xi , Di there exists an index i = N < +∞ such that ‖x‖ < XN , ‖d‖ < DN . In this case it holds that |σj (t )| = 0 for t ≥ tN + jTN , 1 ≤ j ≤ n1 − 1 and TN0 = tN + (n1 − 1)TN . Finally note that it may be the case that |σ(t )| = 0 for all t ≥ TN0 , but ‖x‖ ≥ XN and/or ‖d‖ ≥ XN . In this case due to Assumption 2 the equality x1 (t )−Kv(t ) = 0 holds, which ensures convergence of the estimation error e(t ) to the desired neighborhood of the origin for some finite time Tε ≥ TN0 .  According to the result of Theorem 1, the observer (8)–(13) provides global finite-time convergence of the observation error to an ε -neighborhood of zero for any ε > 0 for all initial conditions x0 ∈ Rn and d ∈ MRm when the corresponding solutions are bounded (Assumption 1 is satisfied). The main restrictions on the class of admissible systems for the proposed approach are formulated in Assumption 2.

568

D. Efimov, L. Fridman / Automatica 47 (2011) 565–570

Remark 1. Let us stress that application of an adaptive control approach for the continuous tuning of the gains µ, λ in the sense of works (Lei, Wei, & Lin, 2005; Zhou, Lu, & Lü, 2006) is not possible here due to the properties of the super-twisting differentiator. To resolve this problem the discrete algorithm (8), (9) is applied in this work.  Remark 2. The only restriction imposed on the functions hx , hd is that the solutions of systems (8) are strictly increasing for all i ≥ 0 (a designer is free with their choice, for instance, hx (i, X ) = X + 1 or hd (i, D) = 2D are admissible). Optimization of the functions hx , hd can guarantee convergence of the algorithm with the minimum number of steps or at least provide a desired upper estimate on the number of algorithm steps. If hx (i, X ) = exp(γ i) for some γ > 0, then N = round{γ −1 ln(‖x‖)}, where round{·} is the rounding-off operator to the closest largest integer number.  Remark 3. In practice, the condition tj′ = arg inf{|σj (t )| > εσ ,i },

1 ≥ j ≥ n1 − 1

Fig. 1. Norms of the observation error σ.

t ≤ti +jTi

may be used in (13) for some small constant εσ ,i > 0 (σ(t ) never converges to zero due to accuracy of the algorithm (10)–(12), numerical realization or some other noise presence). This constant may depend on i ≥ 0 and the parameters λi , µi (the accuracy of the sliding mode depends both on the sampling time and on the differentiator parameters).  Remark 4. The step-by-step super-twisting algorithm (11) realizes the differentiation of the signal y for a correctly chosen bound L (see the Appendix for the constant L introduction reasons). The discrete logic-based gain update algorithm (8), (9) coupled with the algorithm (11) ensures its independence on L.  4. Application Following Fridman et al. (2008) consider a satellite system with additional control of angular velocity:

ρ˙ = v; v˙ = ρω2 − kg Mρ−2 + d; ω˙ = −2vω/ρ − θ ω/m + αv u,

(14)

where ρ > 0 is the distance between the satellite and the Earth center, v ∈ R is the radial speed of the satellite with respect to the Earth, ω ∈ R is the angular velocity of the satellite around the Earth; m and M are masses of the satellite and the Earth, respectively, kg is the universal gravity coefficient, and θ is the damping coefficient; d ∈ R is the disturbing input, u ∈ R is the control, α is a control gain. Only distance to the Earth ρ is assumed available for measurements. Introducing the change of variables ξ1 = [ρ v]T , ξ2 = ρ 2 ω and new parameters k1 = kg M , k2 = θ /m and k3 = α we can transform (14) to the form (3):

ξ˙1,1 = ξ1,2 , ξ˙1,2 = ξ2 /ξ13,1 − k1 /ξ12,1 + d, ξ˙2 = −k2 ξ2 + k3 ξ12,1 ξ1,2 u, y = ξ1,1 .

(15)

Fig. 2. Observer gains λi .

then −2 ρmin (k1 + X ) + D ≤ α(1 + X + D), −2 α(s) = max{1, ρmin k1 }s.

According to Theorem 1, the observer (8)–(13) has to ensure robust state estimation for any initial conditions and bounded disturbances. We will consider two cases: (i) ρ(0) = 107 and d(t ) = 104 sin(0.1t ); (ii) ρ(0) = 106 and d(t ) = 103 sin(0.1t ), thus, the peculiarity of this system consists in large possible deviations of initial conditions and disturbances. Let

System (15) has relative degree 2, then n1 = 2 and n2 = 1 (Assumption 2 is satisfied). The control is taken as follows

v(0) = 0, ω(0) = 6.3156 × 10−4 , m = 10, −11 24 M = 5.98 × 10 , kg = 6.67 × 10 ,

u = u(ξ1 ) = ξ1,2 /(1 + ξ12,1 ξ12,2 ),

θ = 2.5 × 10−5 ,

ensuring the desired altitude dynamics for (14). Then,

and X0 = 105 , D0 = 0, then the graphic of the norm of the error σ is plotted in Fig. 1 and the observer gain λi increasing is shown in Fig. 2 (in logarithmic scale) for cases (i) and (ii), 0 ≤ t ≤ 10 s. According to this simulation the observer gain quickly increases ensuring the required quality of observation. Asymptotical error fluctuations are proportional to the accuracy of the simulation performed in MATLAB 7.0.1.

f1 (ξ1 , ξ2 , d) = ξ2 /ξ13,1 − k1 /ξ12,1 + d, f2 (ξ1 ) = k3 ξ12,1 ξ1,2 u. The physical constraint ξ1,1 > ρmin ≫ 1 implies that −2 |f1 (ξ1 , ξ2 , d)| ≤ ρmin (|ξ2 | + k1 ) + |d|,

α = 1.56 × 10−9 ,

ρmin = 105

D. Efimov, L. Fridman / Automatica 47 (2011) 565–570

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

5. Conclusion The procedure for a hybrid state observer design for nonlinear locally Lipschitz systems with high relative degree is proposed. Possible presence of signal uncertainties is taken into account. The solution is based on the super-twisting observer with adjusted gains applicable to nonlinear systems with bounded solutions. Finite-time convergence of the state estimation error to any neighborhood of the origin is guaranteed. Computer simulations confirm applicability and performance of the proposed observer. Appendix. Lyapunov function for super-twisting algorithm Consider the controlled system (Polyakov & Poznyak, 2008, 2009)

µ = κ L, κ > 5 ; λ = 7(κ − 1)L; γ = [κ − 1 κ κ + 1]L; g = 8/7[κ − 1 κ κ + 1](κ − 1)−1 ; rg ∈ R3 ; k = 0.5[min{kmax } + max{kmin }],

√



√



kmin = 1/ g exp(rg − 0.5π / g − 1) ∈ R3 , kmax = 1/ g exp(rg + 0.5π / g − 1) ∈ R3 ;

σi = [−1 0 1],      √ π σi √ , ki = γi k gi − exp rgi − √ 2 gi − 1    σi exp − 2√πgσ−i 1  , i = 1, 2, 3,  i k0i =  √ γi k gi − exp rgi − 2√πgσ−i 1 kmin = min{k},

i

where x ∈ R is the state, u ∈ MR , ϕ ∈ MR are the control and the unknown disturbing input. The super-twisting control algorithm has the form:

s and m are calculated for these parameters for current values of variables x and y (the choice k = 1 is admissible). From the equations above, increasing kmin follows from increasing κ > 5. Fortunately, this leads to an improvement in the value of T , for instance if x(0) = 0, then

u = u2 − λ |x|sign[x],

1 T ≤ 2k− min k 2/7 L/(κ − 1).

x˙ = u(t ) + ϕ(t ),

(A.1)

λ > 0; (A.2) u˙ 2 = −µsign[x], µ > 0. t Define y(t ) = ϕ(t )−µ 0 sign[x(τ )]dτ . It is assumed that ‖ϕ‖ ≤ L and ‖ϕ‖ ˙ ≤ L with some known L ∈ R+ . 

Theorem A.1 (Polyakov & Poznyak, 2008). Let µ > 5L and 32L < λ2 < 8(µ − L) then system (A.1), (A.2) has a Lyapunov function  √ 0.25k2 (yγ −1 sign[x] + k0 sem )2 if xy ̸= 0; 2 2 −2 V (x, y) = 2k y λ if x = 0;  0.5|x| if y = 0, γ = µ − Lsign[xy],    π sign[xy]  √  √ k = γ k g − exp rg − √ , 2 g −1    π sign[xy] sign[xy] exp − 2√g −1   , g = 8γ λ−2 , k0 =  √ π sign [ xy] γ k g − exp rg − 2√g −1 e

rg − √π

2 g −1

√ rg + √π < k g < e 2 g −1 ,

rg = −(g − 1)−0.5 arctan((g − 1)−0.5 ), s = 2γ |x| − λ |x|sign[x]y + y2 ,



m = (g − 1)−0.5 arctan



√  λg |x|sign[x] − 2y , √ 2 g − 1y

that is positive definite and absolutely continuous in R2 (continuously differentiable for xy ̸= 0) and V˙ ≤ −kmin

√

V,

kmin = min{k},

t ≥ 0.

The finite time of convergence to zero, T > 0, of V (t ) admits the estimate: 1 T ≤ 2k− min



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

The following worst case estimates can be used in practice: 1 T ≤ 2k− min



max{V (x(0), L), V (x(0), −L)}.

To compute kmin and V the following simplified set of parameters values can be used:





(A.3)

References Bejarano, F. J., Fridman, L., & Pisano, A. (2008). Global hierarchical observer for linear systems with unknown inputs. In Proc. 47th conf. on decision and control (pp. 1883–1888), Cancun, Mexico. Bejarano, F. J., Poznyak, A., & Fridman, L. (2007). Hierarchical second-order sliding-mode observer for linear time invariant systems with unknown inputs. International Journal of Systems Science, 38(10), 793–802. Besançon, G. (Ed.) (2007). Lecture notes in control and inforamtion science: vol. 363. Nonlinear observers and applications. Berlin: Springer Verlag. Cannas, B., Cincotti, S., & Usai, E. (2002). An algebraic observability approach to chaos synchronisation by sliding differentiators. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 49(7), 1000–1006. Edwards, C., Spurgeon, S. K., & Hebden, R. G. (2002). On development and applications of sliding-mode observers. In J. Xu, & Y. Xu (Eds.), Lecture notes in control and information science, Variable structure systems: towards XXIst century (pp. 253–282). Berlin (Germany): Springer Verlag. Efimov, D.V., & Bobtsov, A.A. (2009). Hybrid robust observer for locally Lipschitz nonlinear systems. In Proc. ECC 2009. (pp. 478–483), Budapest. Floquet, T., & Barbot, J. P. (2007). Super twisting algorithm-based step-bystep sliding mode observers for nonlinear systems with unknown inputs. International Journal of Systems Science, 38(10), 803–815. Fridman, L., Shtessel, Y., Edwards, C., & Xing-Gang, Yan. (2008). Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems. International Journal of Robust and Nonlinear Control, 18(4–5), 399–413. Lei, H., Wei, J., & Lin, W. (2005). A global observer for observable autonomous systems with bounded solution trajectories. In Proc. 44rd IEEE conf. on decision and control (pp. 1911–1916), Sevilla, Spain. December 12–15. 2005. Levant, A. (1993). Sliding order and sliding accuracy in sliding-mode control. International Journal of Control, 58(6), 1247–1263. Levant, A. (1998). Robust exact differentiation via sliding-mode technique. Automatica, 34(3), 379–384. Levant, A. (2003). Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control, 76(9/10), 924–941. Nijmeijer, H., & Fossen, T. I. (1999). New directions in nonlinear observer design. London (UK): Springer-Verlag. Pisano, A., & Usai, E. (2007). Globally convergent real-time differentiation via second order sliding modes. International Journal of Systems Science, 38(10), 833–844. Polyakov, A., & Poznyak, A. (2008). Lyapunov function design for finite-time convergence analysis of ‘‘Twisting’’ and ‘‘Super-Twisting’’ sliding mode controllers. In Proc. 10th IEEE workshop on variable structure systems (pp. 153–158). Polyakov, A., & Poznyak, A. (2009). Lyapunov function design for finite-time convergence analysis: ‘‘Twisting’’ controller for second-order sliding mode realization. Automatica, 45(2), 444–448. Spurgeon, S. K. (2008). Sliding mode observers: a survey. International Journal of Systems Science, 39(8), 751–764. Stotsky, A., & Kolmanovsky, I. (2002). Application of input estimation techniques to charge estimation and control in automotive engines. Control Engineering Practice, 10, 1371–1383. Weitian, C., & Saif, M. (2008). Actuator fault diagnosis for uncertain linear systems using a high-order sliding-mode robust differentiator (HOSMRD). International Journal of Robust and Nonlinear Control, 18(4–5), 413–426.

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D. Efimov, L. Fridman / Automatica 47 (2011) 565–570

Zhou, J., Lu, J., & Lü, J. (2006). Adaptive synchronization of an uncertain complex dynamical network. IEEE Transactions on Automatic Control, 51(4), 652–656.

D. Efimov received the M.S. degree in Control Systems from the Saint-Petersburg State Electrical Engineering University, Russia, in 1998, the Ph.D. degree in Automatic Control from the same university in 2001, and the Dr.Sc. degree in Automatic control in 2006 from Institute for Problems of Mechanical Engineering RAS, SaintPetersburg, Russia. From 2000 to 2009 he was research assistant of the Institute for Problems of Mechanical Engineering RAS, Control of Complex Systems Laboratory. From 2006 to 2007 he was with the LSS, Supelec, France. From 2007 to 2009 he was working in the Montefiore Institute, University of Liege, Belgium. In 2009 he is joining the Automatic control group, IMS lab., University of Bordeaux I, France. His main research interests are nonlinear oscillations analysis, observation and control, switched and hybrid systems stability.

L. Fridman received the M.S. degree in mathematics from Kuibyshev State University, Samara, Russia, in 1976, the Ph.D. degree in applied mathematics from the Institute of Control Science, Moscow, Russia, in 1988, and the Dr.Sc. degree in control science from Moscow State University of Mathematics and Electronics, Moscow, Russia, in 1998. From 1976 to 1999, he was with the Department of Mathematics, Samara State Architecture and Civil Engineering Academy. From 2000 to 2002, he was with the Department of Postgraduate Study and Investigations at the Chihuahua Institute of Technology, Chihuahua, Mexico. In 2002, he joined the Department of Control, Division of Electrical Engineering of Engineering Faculty, National Autonomous University of Mexico (UNAM), México. He has published over 250 technical papers. His main research interests are variable structure systems. Dr. Fridman is an Associate Editor of the International Journal of System Science, Journal of Francklin Institute and Conference Editorial Board of IEEE Control Systems Society, Member of TC on Variable Structure Systems and Sliding mode control of IEEE Control Systems Society. He was working as an invited professor in 14 universities and scientific centers of France, Germany, Italy and Israel.