HYBRID OBSERVERS FOR LOCALLY LIPSCHITZ SYSTEMS WITH HIGH RELATIVE DEGREE

8th IFAC Symposium on Nonlinear Control Systems University of Bologna, Italy, September 1-3, 2010

HYBRID OBSERVERS FOR LOCALLY LIPSCHITZ SYSTEMS WITH HIGH RELATIVE DEGREE Denis Efimov1, Leonid Fridman2 1

2

University of Bordeaux, IMS-lab, Automatic control group 351 cours de la libération, 33405 Talence, France [email protected]

Departamento de Ingeniería de Control y Robótica, Facultad de Ingeniería UNAM, Edificio "Bernardo Quintana", Ciudad Universitaria D.F., México [email protected]

Abstract: State observer design procedure is proposed for nonlinear locally Lipschitz systems with high relative degree (from the available for measurements output to nonlinearity). The solution is based on logic-based control and high order super-twisting observer. The approach is applicable to nonlinear systems with bounded solutions. the nonlinear system (1) were carried out in (Barbot and Floquet, 2007; Fridman, et al., 2008). All these solutions are obtained under assumption on globality of Lipschitz property for unknown inputs and nonlinearities. The observer based on global second-order sliding mode differentiator (Pisano and Usai, 2007) is designed in (Bejarano, et al., 2008) for nonlinear systems with unknown inputs which could be unbounded.

1. INTRODUCTION The state observer design problem for nonlinear systems has been an area of intensive research during the last two decades (see, for example, Besançon (2007) or Nijmeijer and Fossen (1999) and references therein). Application of the slidingmode approach allows one to ensure finite time convergence of a part of estimation error to zero even in the presence of unknown inputs. Additionally, the equivalent control method may help to estimate unknown inputs values affecting on the system that is useful for fault detection and estimation (see, for example, (Edwards, et al., 2002; Spurgeon, 2008; Weitian and Saif, 2008) and references therein). For the first order sliding mode observers the application of low-pass filters is required that leads to deterioration of estimation error. Recently developed sliding mode observers based on step-bystep super-twisting differentiation provide finite time exact observation without additional filters (Levant, 1993; 1998; 2003; Moreno and Osorio, 2008). Such observers ensure finite time exact estimation for considered classes of systems with unknown inputs (Barbot and Floquet, 2007; Bejarano, et al., 2007).

In work (Efimov and Bobtsov, 2009) a solution of the problem is proposed for perturbed system (1) with locally Lipschitz function f . In this case there exist no observer gains providing convergence of estimation error globally. The solution is obtained under assumption that derivative of the output depends on the nonlinear function f (relative degree one). Growing observer gains are updated by eventbased algorithm. On each step the new gains are substituted in the observer using logic-based scheme if the previous observer gains fail to satisfy some performance criteria (fail to ensure convergence of the estimation error). Infinite growth of the gains in the case of disturbances presence is avoided. In the present work the result of (Efimov and Bobtsov, 2009) is extended to the case of high relative degree, that leads to more complex sliding mode techniques application. Step by step super-twisting differentiation algorithms are used in this work (other kinds of differentiators can be adopted in a similar way (see (Stotsky and Kolmanovsky, 2002) for their performance comparison)). Preliminaries are introduced in section 2. Main results are presented in section 3. Results of application for a satellite system are discussed in section 4.

The class of Lipschitz nonlinear systems has seen much attention: x = A x + φ( y ) + B f ( x, d ) , y = C x ,

(1)

where x ∈ R n is state vector; d ∈ R m is disturbing input; y ∈ R p is available for measurements output and functions φ : R p → R n and f : R n + m → R p are Lipschitz continuous (function f globally); constant matrices A , B , C have appropriate dimensions. An advantage of this class of systems consists in fact, that almost all nonlinear systems of the form

2. PRELIMINARIES The symbol | ⋅ | is used for the Euclidean norm designation. The norm of Lebesgue measurable and essentially bounded function d : R+ → R m of time t ≥ 0 will be defined as

x = F ( x, d ) , y = C x , n+m

→ R n is locally Lipschitz continuous, can be where F : R reduced to (1) at least locally.

|| d || [ t0 ,t ) = ess sup { | d( t ) |, t ∈ [ t0 , t ) } .

The

The first attempts to generalize sliding mode observers for 978-3-902661-80-7/10/$20.00 © 2010 IFAC

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all

such

functions

with

property

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NOLCOS 2010 Bologna, Italy, September 1-3, 2010

|| d ||[ 0,+ ∞ ) = || d || < + ∞ we will further denote as MR m . As

and consider new observation vector ξ = [ ξ1T ξT2 ]T , where

usual, continuous function σ : R+ → R+ belongs to class K

ξ1 = K −1x1 , ξ 2 = x 2 , from (2) we have:

if it is strictly increasing and σ ( 0 ) = 0 ; additionally it belongs to class K∞ if it is also radially unbounded.

ξ1 = G ξ1 + R ξ 2 + s f1 ( x, d ) ,

(3a)

ξ 2 = A 3K ξ1 + A 4ξ 2 + f 2 ( K ξ1 ) ;

(3b)

G = K −1A1K , R = K −1b aT2 , s = K −1b . Due to assumption 2 and structure of the matrix K we have ⎡0 1 … 0 ⎤ ⎡0 … 0 ⎤ ⎡0 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥, s=⎢ ⎥, ⎥, R=⎢ G=⎢ ⎢0 0 … 1 ⎥ ⎢0 … 0 ⎥ ⎢0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣s⎦ ⎣⎢ r1 … rn2 ⎦⎥ ⎣⎢ g1 g 2 … g n1 ⎦⎥

If for all initial conditions x0 ∈ R n and inputs d ∈ MR m the solutions x( t , x0 , d ) ( y ( t , x0 , d ) = h( x( t , x0 , d ) ) ) of the system (1) are defined for all t ≥ 0 , then the system is called forward complete. 3. MAIN RESULTS

ξ1,1 = y , g = [ g1 ...g n1 ] , r = [ r1 ...rn2 ] .

In this work we suppose that the state vector x and the disturbing input d in the system (1) are bounded without precise information on their upper bounds.

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

A s s u m p t i o n 1 . Let for the system (1) || x || < +∞ and d ∈ MR m . □

v1 = −λ | v1 − y | sign( v1 − y ) + υ1 ,

In the sequel for simplicity of consideration we impose some structural restrictions on the system (1).

v j = γ j [ − λ | v j − v j −1 | sign( v j − v j −1 ) + υ j ] ,

υ1 = −μ sign( v1 − y )

where λ > 0 , μ > 0 , γ j ∈ {0,1} are coefficients to be defined later. Consider the error σ = v − ξ1 , which dynamics due to (4), (5) takes form: σ1 = −λ | σ1 | sign( σ1 ) + υ1 − ξ1,2 , υ1 = −μ sign( σ1 ) , (6a)

(2)

x 2 = A3 x1 + A 4 x 2 + f 2 ( x1 ) , x = [ x1T xT2 ]T ,

where x1 ∈ R n1 , x 2 ∈ R n2 , n = n1 + n2 , the functions f1 and f 2 are lo ca lly Lipschitz continuous; (ii) relative degree of the system is n1 > 1 , i.e.

σ j = γ j [ − λ | v j − v j −1 | sign( v j − v j −1 ) + υ j ] − ξ1, j +1 , (6b) υ j = γ j [ − μ sign( v j − v j −1 ) ] , 2 ≤ j ≤ n1 − 1 .

To clarify properties of the system (6) note that the equations (6a) and (A1), (A2) in the appendix coincide. Let γ j = 0 ,

n −1

cT A1i b = 0 , 0 ≤ i ≤ n − 2 ; cT A11 b ≠ 0 ;

(iii) the matrix pair ( A1 , c ) is observable, the matrix A 4 is Hurwitz. □ Part (iii) of the assumption deals with observability of the system (2), part (ii) is technical and the only restriction is presented in the equation (2), that x 2 variable affects on n1 th derivative of the output y .

2 ≤ j ≤ n1 − 1 . According to theorem A1 if the upper bound

L>0

is

known

such

that

|| ξ1,2 || ≤ L

and

|| ξ1,2 || = || ξ1,3 || ≤ L , then tacking coefficients as in theorem

A1 (Polyakov and Poznyak, 2008; 2009) (see also the recommendation below the theorem) it is possible to ensure σ1 ( t ) = v1 ( t ) − y ( t ) = 0 for all t ≥ T , where T ≥ 0 is an upper estimate on finite time of convergence for the system (A1), (A2) evaluated in theorem A1. Then v1 ( t ) = ξ1,1 ( t ) = ξ1,2 ( t ) for all t ≥ T . Since the signal

The outline of this section is as follows. In the first part following (Besançon, 2007) we introduce step-by-step supertwisting 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, which is used for verification of the given upper estimates for || x || and || d || accuracy. In the third part the equations of the proposed hybrid state observer for the system (2) are presented and global convergence of estimation error is proven.

ξ1,1 is available for measurements, by proper choice of v1 ( 0 ) the quantity σ1 ( 0 ) = 0 always can be guaranteed, then the estimate (A3) on the value of T is given in the appendix. Set γ 2 = 1 at the time instant t = T and v2 ( T ) = ξ1,2 ( T ) = v1 ( T ) , then σ2 ( T ) = 0 and from (6b) for j = 2 we have

σ2 = − λ | σ2 | sign( σ2 ) + υ2 − ξ1,3 , υ2 = −μ sign( σ2 ) ,

A. Step-by-step super-twisting observer Define K −1 = [cT A1T cT

(5)

υ j = γ j [ − μ sign( v j − v j −1 ) ] , 2 ≤ j ≤ n1 − 1 , vn1 = vn1 −1 ,

A s s u m p t i o n 2 . Let p = 1 and (i) system (1) can be presented in the form: x1 = A1x1 + b [ f1 ( x, d ) + aT2 x 2 ] , y = cT x1 ,

(4)

that again has form (A1), (A2). Therefore, v2 ( t ) = ξ1,2 ( t ) = ξ1,3 ( t ) for all t ≥ 2 T

( A1T ) n −1 cT ]T

providing that || ξ1,3 || ≤ L and || ξ1,3 || = || ξ1,4 || ≤ L . Repeat928

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ing this procedure step-by-step for 2 ≤ j ≤ n1 − 1 and setting up γ j = 1 for t ≥ ( j − 1) T and v j ( j T ) = v j −1 ( j T ) we ob-

C. Global robust hybrid observer for locally Lipschitz nonlinear systems Assume that there exists t ′ ≥ jT such, that | σ j ( t ′ ) | > 0 ,

tain v j ( t ) = ξ1, j ( t ) = ξ1, j +1 ( t ) for all t ≥ j T

then it means that the constants X 0 , D0 have been chosen not sufficiently high. Taking for X 0 and D0 new higher values it is necessary to repeat all described above steps, that can be formalized as follows:

under assumption that || ξ1, j || ≤ L , 2 ≤ j ≤ n1 , || ξ1, 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 σ this implies v( t ) = ξ1 ( t ) for t ≥ ( n1 − 1) T . As the last step define γ n1 = 1 for t ≥ ( n1 − 1) T in the equation z = γ n1 [ A3 K v + A 4 z + f 2 ( K v ) ] ,

X i = hx ( i, X i −1 ) , Di = hd ( i, Di −1 ) ,

(8)

X 0 > 0 , D0 > 0 , i = 1, 2, 3,...N ≤ + ∞ ;

μi = κ Li , λi =

(7)

−1 Ti = 2 kmin k

where the variable z estimates the vector x 2 . The dynamics of estimation error e = x 2 − z is governed by the following equation: e = A4 e .

7 ( κ − 1) Li ,

(9)

2 / 7 Li /( κ − 1) , Li = L( X i , Di ) ;

z = γ n1 ( t )[ A 3K v + A 4 z + f 2 ( K v )] ;

(10)

v1 = −λ i | v1 − y | sign( v1 − y ) + υ1 ,

(11a)

υ1 = −μi sign( v1 − y ) ,

According to assumption 2 the matrix A 4 is Hurwitz, then the variable z asymptotically converges to x 2 . Therefore, the whole state x is observed by (4), (5), (7) (components x1 in finite time and x 2 asymptotically).

υ j = γ j ( t )[ − μi sign( v j − v j −1 ) ] , 2 ≤ j ≤ n1 − 1 , vn1 = vn1 −1 ;

As the next step it is necessary to evaluate L . For locally Lipschitz and continuous function f1 for all x ∈ R n , d ∈ MR m :

ti +1 = min {t ′j } , t0 = 0 ,

v j = γ j ( t )[ − λi | v j − v j −1 | sign( v j − v j −1 ) + υ j ] , (11b)

⎧0 if ti ≤ t < ti + ( j − 1) Ti ; γ j(t ) = ⎨ 2 ≤ j ≤ n1 ; (12) ⎩1 if t ≥ ti + ( j − 1) Ti , 1≤ j ≤n1 −1

t ′j = arg inf{ σ j ( t ) ≠ 0} , 1 ≤ j ≤ n1 − 1 , t ≥ ti + jTi

| f1 ( x, d ) | ≤ α(1+ | ( x, d ) |) ≤ α(1+ | x | + || d ||)

where the discrete systems (8) have well defined strictly increasing solutions for any X 0 > 0 , D0 > 0 for all i ≥ 1 ;

for some function α ∈ K . If we assume that || x || ≤ X 0 and || d || ≤ D0 for some X 0 > 0 , D0 > 0 , then we may write:

κ > 5 is a fixed parameter, formulas for kmin and k are presented in the appendix.

| f1 ( x, d ) | ≤ α(1 + X 0 + D0 ) , | ξ1 | ≤ ψ X 0 ,

| ξ1, n1 | ≤ ( ψ | g | + | r |) X 0 + | s | α(1 + X 0 + D0 ) , T

ψ = max{1, λ min ( K K )

−0.5

(13)

The following result describes stability properties of this hybrid observation algorithm.

}.

T h e o r e m 1 . Let assumptions 1−2 hold and discrete systems (8) have well defined strictly increasing to infinity solutions for any X 0 > 0 , D0 > 0 for all i ≥ 1 . Then for any κ > 5 for the system (1) with the algorithm (8)–(13) it holds that

Thus, L = L( X 0 , D0 ) where L( X , D ) = max{ ψ X , ( ψ | g | + | r |) X 0 + + | s | α(1 + X 0 + D0 )}.

B. Observer gains verification Unfortunately the values X 0 , D0 are not known and in general case for a particular 1 ≤ j ≤ n1 the inequality || ξ1, j || ≤ L( X 0 , D0 ) can be violated for given X 0 , D0 .

− || z ||< + ∞ , || v ||< + ∞ , || υ ||< + ∞ ; − there exists the last iteration N < + ∞ of the algorithm and 0 ≤ TN0 < + ∞ such, that | x1 ( t ) − K v( t ) | = 0 for all t ≥ TN0 ;

Therefore, it is necessary to propose a procedure for chosen values X 0 , D0 validation.

− for any ε > 0 there exists TN0 ≤ Tε < + ∞ such that | x 2 ( t ) − z ( t ) | ≤ ε for all t ≥ Tε .

To do so one can use the property of convergence in finite time 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 constraint σ j ( t ) = 0 has to be satisfied for t ≥ j T . Therefore, failure

P r o o f . The properties of the algorithm (8)–(13) follow by the consideration above. For any i ≥ 1 at time instant ti the values X i , Di are derived from the equation (8) (for i = 0 the initial conditions X 0 , D0 are used). Further, the values of twisting observer gains λ i , μi and the step time Ti are calculated in accordance with (9). If constants X i , Di have

of one of these conditions can be used for detection of the values X 0 and D0 incorrectness.

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upper estimate on the number of the algorithm steps. For example, let hx ( i, X i ) = exp( γ i ) for some γ > 0 , then

been chosen correctly, v ( t ) = K −1x1 ( t ) for all t ≥ ti + ( n1 − 1) Ti . The value ti + Ti in this case indicates the time instant when the available for measurements signal σ1 ( t ) = y ( t ) − v1 ( t ) should reach for zero, at the time instants ti + j T the signals σ j have to approach zero,

N = round { γ −1 ln( || x || )} , where round{} ⋅ is rounding-off operator to closest bigger integer number. □

R e m a r k 3 . In practice the condition

2 ≤ j ≤ n1 − 1 . The observers (10), (11) attempt to estimate the system (2) state vector values x( t ) . Algorithm (11) defines 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 identical zeros. That ensures boundedness of the vectors v and υ (the vector v is not continuous due to initial conditions choice for different iterations, i.e. v j ( ti + j Ti ) = v j −1 ( ti + j Ti ) , 2 ≤ j ≤ n1 − 1 ). The observer

t ′j = arginf{| σ j ( t ) | > εσ } , 1 ≤ j ≤ n1 − 1 t ≥ti + jTi

should be used in (13) for some small constant εσ > 0 ( σ ( t ) never converges to zero due to accuracy of the algorithm (10)−(12) numerical realization). □ 4. APPLICATION Following (Fridman, et al., 2008) consider a satellite system with additional control of angular velocity:

(10) is activated for t ≥ ti + ( n1 − 1) Ti , when e = A 4 e , where e = x 2 − z , therefore the variable z always stay bounded.

ρ = v ; v = ρ ω2 − k g M ρ−2 + d ;

ω = −2vω / ρ − θω / m + α v u , where ρ > 0 is the distance between the satellite and the Earth centre, 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, k g is the universal gravity coef-

If for some t ′j ≥ ti + jTi the condition | σ j ( t ′j ) | > 0 is satisfied, 1 ≤ j ≤ n1 − 1 , it implies that the values X i , Di have been taken not sufficiently high. Then ti +1 = t ′j and it is necessary to repeat all steps of the algorithm. Since || x ||< + ∞ and || d ||< + ∞ , for strictly increasing sequences X i , Di there exists an index i = N < + ∞ such that || x ||< X N , || d ||< DN . In this case it holds that | σ j ( t ) | = 0 for

ficient, 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.

t ≥ t N + jTN , 1 ≤ j ≤ n1 − 1 and TN0 = t N + ( n1 − 1) TN .

Finally note, that it may be the case that | σ ( t ) | = 0 for all

Introducing change of variables ξ1 = [ ρ v ]T , ξ2 = ρ2ω and new parameters k1 = k g M , k2 = θ / m and k3 = α we can

≥ TN0 ,

t but || x || ≥ X N and/or || d ||≥ DN . In this case due to assumption 2 the equality x1 ( t ) − K v ( t ) = 0 holds, that ensures convergence of the estimation error e( t ) to the desired

neighborhood of the origin for some finite time Tε ≥ TN0 .

(14)

transform (14) to the form (3): 3 2 ξ1,1 = ξ1,2 , ξ1,2 = ξ2 / ξ1,1 − k1 / ξ1,1 +d ,

■

(15)

2 ξ 2 = −k2 ξ 2 + k3 ξ1,1 ξ1,2u , y = ξ1,1 .

According to the result of Theorem 1 the observer (8)−(13) provides global finite time convergence of the estimation error to ε -neighborhood of zero for any ε > 0 for all initial conditions x0 ∈ R n and d ∈ MR m when the corresponding

The system (15) has relative degree 2, then n1 = 2 and n2 = 1 (assumption 2 is satisfied). The control is taken as follows 2 2 u = u ( ξ1 ) = ξ1,2 / (1 + ξ1,1 ξ1,2 ) ,

solutions are bounded (assumption 1 is satisfied). The main restrictions on class of admissible for the approach systems (1) are formulated in assumption 2.

ensuring desired altitude dynamics for (14). Then, 3 2 f1 ( ξ1, ξ2 , d ) = ξ2 / ξ1,1 − k1 / ξ1,1 +d ,

R e m a r k 1 . Let us stress that application of the adaptive control approach for continuous tuning of the gains μ , λ in the sense (Lei, et al., 2005; Zhou, et al. 2006) is difficult here due to properties of the super-twisting differentiator. To resolve this problem the discrete algorithms (8) are applied in □ this work.

2 f 2 ( ξ1 ) = k3 ξ1,1 ξ1,2u .

The physical constraint ξ1,1 > ρmin >> 1 implies that 2 | f1 ( ξ1 , ξ 2 , d ) | ≤ ρ−min (| ξ2 | + k1 )+ | d | , 2 ( k1 + X ) + D . then α ( X , D ) = ρ−min

R e m a r k 2 . The theorem 1 poses only one restriction on the form of the functions hx , hd , they have to ensure strict increase of the systems (8) solutions for any positive initial conditions. Thus, optimization of these functions shape is possible, that can guarantee convergence of the algorithm with minimum number of steps or at least provide a desired

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) 930

ρ( 0 ) = 107 and d ( t ) = 104 sin(0.1t ) ;

NOLCOS 2010 Bologna, Italy, September 1-3, 2010

APPENDIX

(ii) ρ(0) = 106 and d ( t ) = 103 sin(0.1t ) , thus, the peculiarity of this system consists in large possible deviations of the system (14) initial conditions and disturbances. Let

Consider the controlled system x = u ( t ) + ϕ( t ) ,

where x ∈ R is the state, u ∈ MR , ϕ ∈ MR are the control and unknown disturbing input. The super twisting control algorithm has form: u = u2 − λ x sign[ x ] , λ > 0 ; (A2)

v( 0 ) = 0 , ω( 0 ) = 6.3156 × 10−4 , m = 10 , M = 5.98 × 1024 , k g = 6.67 × 10−11 ,

θ = 2.5 × 10−5 , α = 1.56 × 10−9 , ρmin = 105 .

u2 = −μ sign[ x ] , μ > 0 .

and X 0 = 105 , D0 = 0 , then the corresponding graphic of the norm of observer error σ is plotted in Fig. 1 and increasing gain of the observer λ i is shown in Fig 2 (in logarithmic scale), 0 ≤ t ≤ 10 sec.

t

Define y ( t ) = ϕ( t ) − μ ∫ sign[ x( τ ) ] d τ . 0

It is supposed that || ϕ || ≤ L and || ϕ || ≤ L with some known L ∈ R+ . T h e o r e m A 1 (Polyakov and Poznyak, 2008). Let

6

15

x 10

μ > 5 L and 32 L < λ 2 < 8( μ − L ) then the system (A1), (A2) has Lyapunov function ⎧0.25 k 2 y γ −1sign[ x ] + k s em 2 if xy ≠ 0; 0 ⎪ ⎪ 2 2 −2 V ( x, y ) = ⎨ 2 k y λ if x = 0; ⎪0.5 | x | if y = 0 , ⎪ ⎩

10

(

5

0 -1 10

0

10

x 10

1

γ = μ − L sign[ xy ] , k =

γ k

⎛ π sign[ xy ] g − exp ⎜ rg − ⎜ 2 g-1 ⎝

⎞ ⎟⎟ , ⎠

⎛ π sign[ xy ] ⎞ sign[ xy ]exp ⎜ − ⎟ ⎜ 2 g-1 ⎟⎠ ⎝ k0 = , g = 8 γ λ −2 , ⎡ ⎛ π sign[ xy ] ⎞ ⎤ γ ⎢ k g − exp ⎜ rg − ⎟⎥ ⎜ 2 g-1 ⎟⎠ ⎥⎦ ⎢⎣ ⎝

7

3.5 3 2.5 2 1.5 1

e

0.5 0 -1 10

)

10

Fig. 1. Norms of estimation error σ . 4

(A1)

0

10

rg −

π 2

g-1


g
rg +

π 2

g-1

,

rg = −( g − 1) −0.5 arctan( ( g − 1) −0.5 ) ,

1

10

s = 2 γ | x | −λ | x | sign[ x ] y + y 2 ,

Fig. 2. Observer gains λ i .

⎛ λ g | x | sign[ x ] − 2 y m = ( g − 1) −0.5 arctan ⎜ ⎜ 2 g −1 y ⎝

According to this simulation the observer gain quickly increases ensuring required quality of estimation. Asymptotical error fluctuations are proportional to accuracy of the simulation performed in MATLAB 7.0.1.

⎞ ⎟⎟ , ⎠

that is positive definite and absolute continuous in R 2 (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 ≤ 2 kmin V ( x( 0 ), y ( 0 ) ) . □

5. CONCLUSION The procedure for 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 logic-based control approach and super-twisting observer. 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.

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

max{V ( x( 0 ), L ), V ( x( 0 ), − L )} .

To compute kmin and V the following simplified set of parameters values can be used: μ = κ L , κ > 5 ; λ = 7( κ − 1) L ; γ = [ κ − 1 κ κ + 1] L ;

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Control, December 12-15, Sevilla, Spain, pp. 1911 – 1916. Levant A. (1993). Sliding order and sliding accuracy in sliding-mode control. Int. J. Contr., 58(6), pp. 1247–1263. Levant A. (1998). Robust exact differentiation via slidingmode technique. Automatica, 34(3), pp. 379–384. Levant A. (2003). Higher-order sliding modes, differentiation and output-feedback control, Int. J. of Control, 76 (9/10), pp. 924-941. Moreno J.A., Osorio M. (2008). A Lyapunov approach to second-order sliding mode controllers and observers. Proc. 47th IEEE Conf. on Decision and Control, Dec. 2008, Cancun, Mexico, pp. 2856−2861. Nijmeijer H., Fossen T.I. (1999). New Directions in Nonlinear Observer Design. London, U.K.: Springer-Verlag. Pisano A., Usai E. (2007). Globally convergent real-time differentiation via second order sliding modes. Int. J. Systems Science, 38(10), pp. 833−844. Polyakov A., Poznyak A. (2008). Lyapunov Function Design for Finite-Time Convergence Analysis of "Twisting" and "Super-Twisting" Sliding Mode Controllers. Proc. 10th IEEE Workshop on Varaible Structure Systems, pp. 153158. Polyakov A., Poznyak A. (2009). Lyapunov function design for finite-time convergence analysis: "Twisting" controller for second-order sliding mode realization. Automatica, 45(2), pp. 444-448. Spurgeon S.K. (2008). Sliding mode observers: a survey. Int. J. Systems Science, 39(8), pp. 751–764. Stotsky A., Kolmanovsky I. (2002). Application of input estimation techniques to charge estimation and control in automotive engines. Control Engineering Practice, 10, pp. 1371–1383. Weitian C., Saif M. (2008). Actuator fault diagnosis for uncertain linear systems using a high-order sliding-mode robust differentiator (HOSMRD). Int. Robust and Nonlinear Control, 18(4-5), pp. 413−426. Zhou J., Lu J., Lü J. (2006). Adaptive Synchronization of an Uncertain Complex Dynamical Network. IEEE Trans. Autom. Control, 51(4), pp. 652–656.

g = 8 / 7[ κ − 1 κ κ + 1]( κ − 1) −1 ; rg ∈ R3 ;

k = 0.5[ min{ kmax } + max{ kmin }] ,

( g exp ( r

) g −1 )∈ R ;

kmin = 1/ g exp rg − 0.5 π / g − 1 ∈ R 3 , kmax = 1/

g

+ 0.5 π /

3

kmin = min{ k } , σi = [ − 1 0 1] ,

ki =

γi k

⎛ πσi gi − exp ⎜ rgi − ⎜ 2 gi -1 ⎝

⎞ ⎟ , ⎟ ⎠

⎛ πσi ⎞ σi exp ⎜ − ⎟ ⎜ 2 g -1 ⎟ i ⎝ ⎠ , i = 1, 2, 3 , k0 i = ⎡ ⎛ πσi ⎞ ⎤ γ i ⎢ k gi − exp ⎜ rg i − ⎟⎥ ⎜ 2 g i -1 ⎟⎠ ⎥⎦ ⎢⎣ ⎝ 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 of kmin follows by increasing of κ > 5 . Fortunately this leads to the value T improvement, for instance, if x( 0 ) = 0 , then −1 T ≤ 2 kmin k

2 / 7 L /( κ − 1) .

(A3)

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