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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust Nonlinear Control (in press) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1198

Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems Leonid Fridman1, Yuri Shtessel2,*,y, Christopher Edwards3 and Xing-Gang Yan3 1

Division de Ingenieria Electrica, Facultad de Ingenieria, National Autonomous University of Mexico, UNAM, 04510, Mexico, D.F., Mexico 2 Department of Electrical and Computer Engineering, The University of Alabama in Huntsville, Huntsville, AL 35899, U.S.A. 3 Control & Instrumentation Group, Department of Engineering, University of Leicester, Leicester LE17RH, U.K.

SUMMARY In this paper, a higher-order sliding-mode observer is proposed to estimate exactly the observable states and asymptotically the unobservable ones in multi-input–multi-output nonlinear systems with unknown inputs and stable internal dynamics. In addition the unknown inputs can be identified asymptotically. Numerical examples illustrate the efficacy of the proposed observer. Copyright # 2007 John Wiley & Sons, Ltd. Received 8 May 2006; Revised 14 December 2006; Accepted 9 March 2007 KEY WORDS:

higher-order sliding observers; unknown input observers

1. INTRODUCTION State observation and unknown input reconstruction for multi-input–multi-output (MIMO) nonlinear systems is one of the most important problems in modern control theory [1]. The problem of robust state observation continues to be actively studied using sliding modes, see, for example, [2–5]. The corresponding implementation effects were extensively studied in [6]. Sliding-mode observation strategies possess such attractive features as * *

insensitivity (more than robustness) with respect to unknown inputs; the possibility of using the equivalent output error injection as a further source of information.

*Correspondence to: Yuri Shtessel, Department of Electrical and Computer Engineering, The University of Alabama in Huntsville, Huntsville, AL 35899, U.S.A. y E-mail: [email protected]

Copyright # 2007 John Wiley & Sons, Ltd.

L. FRIDMAN ET AL.

Step-by-step vector-state reconstruction by means of sliding modes has been presented in [2, 7–9]. These observers are based on a transformation to a triangular or the Brunovsky canonical form and successive estimation of the state vector using the equivalent output error injection. The corresponding conditions for linear time-invariant systems with unknown inputs were obtained in [4, 9–11]. Moreover, the above-mentioned observers theoretically ensure finitetime convergence for all system states. Unfortunately, the realization of step-by-step observers is based on conventional sliding modes, leading to filtration at each step due to discretization or non-idealities of the analog devices used to implement the schemes. In order to avoid the necessity for filtration, hierarchical observers were recently developed in [10, 12] iteratively using the continuous super-twisting algorithm, based on second-order sliding-mode ideas [13]. The super-twisting structure is also used in the modified version of the step-by-step observer in [9]. Unfortunately, these observers are also not free of drawbacks: the super-twisting algorithm provides the best possible asymptotic accuracy of the derivative estimation at each single realization step [13]. In particular, the accuracy is proportional to the sampling step d for the discrete realization in the absence of noise, and to the square root of the input noise magnitude if the discretization error is negligible. The step-by-step and hierarchical observers use the output of the super-twisting algorithm as a noisy input at the next step. As a result, the r1 overall observation accuracy is of the order d1=2 ; where r is the observability index of the system. Similarly in the presence of measurement noise with magnitude e; the estimation r accuracy is proportional to e1=2 which requires measurement noises not exceeding 1016 for a fourth-order observer implementation to achieve an accuracy of 101 : The use of higher-order sliding-mode differentiators [14] for exact observer design for linear systems with unknown inputs, initially transformed to the Brunovsky canonical form, is suggested in [9]. This work has shown that the accuracies increase to d and e1=ðrþ1Þ ; respectively. In this paper, an exact observer scheme for the nonlinear systems with unknown inputs is proposed based on two steps: * *

transformation of the system to the Brunovsky canonical form; the application of higher-order sliding-mode differentiators for each component of the output error vector.

The proposed scheme ensures exact finite-time state estimation of the observable variables and asymptotic exact estimation of the unobservable variables for the case when the system has stable internal dynamics. Also the unknown inputs can be identified asymptotically.

2. SYSTEMS DYNAMICS Consider the following MIMO locally stable system x’ ¼ f ðxÞ þ GðxÞjðtÞ

ð1Þ

y ¼ hðxÞ where f ðxÞ 2 Rn ; hðxÞ ¼ ½h1 ; h2 ; . . . ; hm T 2 Rm ; GðxÞ ¼ ½g1 ; g2 ; . . . ; gm 2 Rnm ; x 2 Rn ; y 2 Rm ; j 2 Rm ; and gi 2 Rn 8i ¼ 1; . . . ; m are smooth vector and matrix functions defined on an open set O Rn : Copyright # 2007 John Wiley & Sons, Ltd.

Int. J. Robust Nonlinear Control (in press) DOI: 10.1002/rnc

HIGHER-ORDER SLIDING-MODE OBSERVER

Assumptions ðIsidori [15]Þ At a neighbourhood of any point x 2 O (i) The system in (1) is assumed to have a vector relative degree r ¼ fr1 ; r2 ; . . . ; rm g; i.e. Lgj Lkf hi ðxÞ ¼ 0

8j ¼ 1; . . . ; m 8k5ri 1

8i ¼ 1; . . . ; m ð2Þ

Lgj Lrf i 1 hi ðxÞ=0 for at least one 14j4m (ii) The m m matrix

2

Lg1 ðLfr1 1 h1 Þ

6 6 L ðLr2 1 h Þ 2 6 g1 f 6 EðxÞ ¼ 6 .. 6 6 . 4

Lg1 ðLfrm 1 hm Þ

Lg2 ðLrf 1 1 h1 Þ

Lg2 ðLrf 2 1 h2 Þ

.. .

.. .

Lg2 ðLfrm 1 hm Þ

Lgm ðLfr1 1 h1 Þ

3

7 Lgm ðLfr2 1 h2 Þ 7 7 7 7 .. 7 7 . 5

ð3Þ

Lgm ðLfrm 1 hm Þ

is nonsingular; (iii) The distribution G ¼ spanfg1 ; g2 ; . . . ; gm g is involutive. A well-known property of systems of the form in (1) which satisfy assumptions (i) and (ii) is summarized in the following lemma [15]. Lemma Suppose that assumptions (i) and (ii) are valid for the system (1). Then the row vectors dh1 ðxÞ; dLf h1 ðxÞ; . . . ; dLfr1 1 h1 ðxÞ dh2 ðxÞ; dLf h2 ðxÞ; . . . ; dLfr2 1 h2 ðxÞ ð4Þ

.. . dhm ðxÞ; dLf hm ðxÞ; . . . ; dLfrm 1 hm ðxÞ are linearly independent.

&

The lemma conditions are also interpreted in [16] as the notion of local weak observability.

3. PROBLEM FORMULATION AND MAIN RESULT The problem considered in this paper is to design an asymptotic observer that generates the # # estimates xðtÞ; jðtÞ for the state xðtÞ; and the disturbance jðtÞ of the system (1)–(3) given the measurements y ¼ hðxÞ; i.e. # xðtÞjj ¼ 0 lim jjxðtÞ

ð5Þ

# jðtÞjj ¼ 0 lim jjjðtÞ

ð6Þ

t!1

t!1



L. FRIDMAN ET AL.

3.1. Coordinate transformation The system given by (1)–(3) with an involutive distribution G ¼ spanfg1 ; g2 ; . . . ; gm g and P total relative degree r ¼ m r i¼1 i 5n can be presented in a new basis that is introduced as follows: 1 0 1 0 i 1 0 i f1 ðxÞ hi ðxÞ x1 C B C B C B B xi C B fi ðxÞ C B Lf hi ðxÞ C C B 2C B 2 C B C B C B C B r fxT ; ZT gT : xi ¼ B . C ¼ B . C ¼ B C 2 R i 8i ¼ 1; . . . ; m .. C B . C B . C B . C B . C B . C B A @ A @ A @ Lrf i 1 hi ðxÞ fir ðxÞ xir i

i

0

x1

B B 2 Bx B x¼B B .. B . @

xm

1

0

C C C C C; C C A

B B B Z¼B B B @

Z1

1

0

frþ1 ðxÞ

1

C B C C B Z2 C C B frþ2 ðxÞ C C¼B C C B .. .. C C C B . . A @ A

Znr

ð7Þ

fn ðxÞ

It is well known (see Proposition 5.1.2 on p. 222 of [15]) that if assumption (i) is satisfied then it is always possible to find n r functions frþ1 ðxÞ; . . . ; fn ðxÞ such that the mapping m n FðxÞ ¼ colff11 ðxÞ; . . . ; f1r1 ðxÞ; . . . ; fm 1 ðxÞ; . . . ; frm ðxÞ; frþ1 ðxÞ; . . . ; fn ðxÞg 2 R

ð8Þ

% O Rn ; which means is a local diffeomorphism in a neighbourhood of any point x 2 O x ¼ F1 ðx; ZÞ

ð9Þ

Furthermore, for a system given by (1)–(3) with an involutive distribution G ¼ spanfg1 ; g2 ; . . . ; gm g i.e assumption (iii) it is always possible to identify the functions frþ1 ðxÞ; . . . ; fn ðxÞ in such a way that Lgj fi ðxÞ ¼ 0

8i ¼ r þ 1; . . . ; n 8j ¼ 1; . . . ; m

ð10Þ

% O Rn : in a neighbourhood of any point x 2 O Taking into account Equations (7) and (8), the system given byP(1)–(3) with an involutive distribution G ¼ spanfg1 ; g2 ; . . . ; gm g and a total relative degree r ¼ m i¼1 ri 5n can be written in the form x’ i ¼ Li xi þ ci ðx; ZÞ þ li ðx; Z; jðtÞÞ 8i ¼ 1; . . . ; m ð11Þ Z’ ¼ qðx; ZÞ Copyright # 2007 John Wiley & Sons, Ltd.

ð12Þ Int. J. Robust Nonlinear Control (in press) DOI: 10.1002/rnc


where 2

0

6 60 6 Li ¼ 6 6. 6 .. 4 0

1 0 0 1 .. .

.. .

0 0

0 B B B B i l ðx; Z; jðtÞÞ ¼ B B B @

0

3

7 07 7 7 2 Rri ri ; .. 7 . 7 5 0

0 .. . j¼1

B B B i c ðx; ZÞ ¼ B B B @

0

0

Pm

0

Lgj Lrf i 1 hi ðxÞjj ðtÞ

1

0

C B C B C B C B C¼B C B C B A @

1

0 0 .. . Lrf i hi ðxÞ

0

C B C B C B C¼B C B C B A B @

0 .. . Lrf i hi ðF1 ðx; ZÞÞ

0 0 .. . Pm

j¼1

1

0

Lgj Lrf i 1 hi ðF1 ðx; ZÞÞjj ðtÞ

C C C C C C C A

1 C C C C C C C A

8i ¼ 1; . . . ; m

Remark P In this paper, it has been assumed that the total relative degree r ¼ m i¼1 ri 5n: The developments, however, are also applicable to the case when r ¼ n with minor modifications. In this situation, there will be no internal dynamics and all the results will be finite time in nature. 3.2. Internal dynamics It is assumed that for some norm-bounded x ¼ x% ðtÞ : jjx% ðtÞjj4Lx ; there exists an unique and norm-bounded solution of the equations of the internal dynamics (12) Z ¼ Z% ðtÞ : jjZ% ðtÞjj4LZ : This norm-bounded solution of the internal dynamics (12) is assumed to be locally asymptotically stable: this means that, first of all, it is stable in a Lyapunov sense and, second, there exists an e > 0 such that 8Z% ðt0 Þ satisfying jjZðt0 Þ Z% ðt0 Þjj5e ) limt!1 jjZðtÞ Z% ðtÞjj ¼ 0: Such an assumption guarantees that there exists a domain Y : jjZðt0 Þjj 2 LZðt0 Þ so that a solution Z ¼ Zðt; t0 Þ; Zðt0 Þ 2 Y; asymptotically converges to a solution Z ¼ Z% ðt; t0 Þ with some unknown initial condition Z% ðt0 Þ 2 Y and forced by x ¼ x% ðtÞ; i.e. limt!1 jjZðt; t0 Þ Z% ðtÞjj ¼ 0: Remark Of course, not all systems satisfy this assumption (in the same way not all systems have stable zero dynamics for instance). In addition, for general nonlinear systems, this requirement may be difficult to check. 3.3. Higher-order sliding-mode observer Definition 1 System (1)–(3) is said to be locally detectable, if Pm * total relative degree is r ¼ i¼1 ri 5n; Copyright # 2007 John Wiley & Sons, Ltd.


L. FRIDMAN ET AL.

* *

the distribution G ¼ spanfg1 ; g2 ; . . . ; gm g is involutive; the internal dynamics (12) are locally asymptotically stable.

The derivatives xij ðtÞ 8i ¼ 1; . . . ; m 8j ¼ 1; . . . ; ri of the measured outputs yi ¼ hi ðxÞ can be estimated in finite time by the higher-order sliding-mode differentiator [14]. This can be written in the form z’ i0 ¼ vi0 vi0 ¼ li0 jzi0 yi ðtÞjðri =ðri þ1ÞÞ sign ðzi0 yi ðtÞÞ þ zi1 z’ i1 ¼ vi1 vi1 ¼ li1 jzi1 vi0 jððri 1Þ=ri Þ sign ðzi1 vi0 Þ þ zi2 ð13Þ

.. . z’ iri 1 ¼ viri 1 viri 1 ¼ liri 1 jziri 1 viri 2 jð1=2Þ sign ðziri 1 viri 2 Þ þ ziri z’ iri ¼ liri sign ðziri viri 1 Þ for i ¼ 1; . . . ; m: By construction, # 1 ðxÞ ¼ z1 ; x# 11 ¼ f# 11 ðxÞ ¼ z10 ; . . . ; x# 1r1 ¼ f r1 r1 1

#’ 1 ðxÞ ¼ z1 x#’ 1r1 ¼ f r1 r1

.. .

ð14Þ

m m #m #m #m x# m 1 ¼ f1 ðxÞ ¼ z0 ; . . . ; xrm ¼ frm ðxÞ ¼ zrm 1 ;

#’ m 1 x#’ m r1 ¼ frm ðxÞ ¼ zrm

Therefore, the following exact estimates are available in finite time: 0

x# i1

1

0

# i ðxÞ f 1

1

0

C B C B C B x# i C B f B 2 C B # i2 ðxÞ C C C B B r x# i ¼ B C ¼ B C2Ri B .. C B .. C B . C B . C A @ A @ #fi ðxÞ #xi ri ri

x# 1

1

B C B x# 2 C B C B C 8i ¼ 1; . . . ; m x# ¼ B C 2 Rr B .. C B . C @ A x# m

ð15Þ

Next, integrating Equation (12), with x# replacing x # Z# Þ Z’# ¼ qðx;

ð16Þ

and with some initial condition Z# ðt0 Þ 2 Y from the stability domain Y of the internal dynamics (12), a solution Z# ðtÞ is obtained. This solution Z# ðtÞ converges asymptotically to an unknown Copyright # 2007 John Wiley & Sons, Ltd.



(unobservable) solution ZðtÞ that passes through an unknown initial condition Zðt0 Þ: In other words, the asymptotic estimate Z# ðtÞ of ZðtÞ can be obtained locally: 1 1 0# 0 frþ1 ðxÞ Z# 1 C C B B # rþ2 ðxÞ C B Z# 2 C B f C B C B B C C¼B ð17Þ Z# ¼ B C B . C B . C . . B . C B C . A @ @ A Z# nr f# n ðxÞ Finally, the asymptotic estimate for the mapping (8) is identified as # 1 ðxÞ; #1 # ...;f # m ðxÞ; # m # f# rþ1 ðxÞ; # n ðxÞg # ¼ colff # ...;f # 2 Rn FðxÞ 1 # . . . ; fr1 ðxÞ; 1 # . . . ; frm ðxÞ;

ð18Þ

The asymptotic estimate x# of the state vector x can be easily identified via Equations (9) and (18) as # Z# Þ x# ¼ F1 ðx;

ð19Þ

It is worth noting that the operation (19) is also local and can be performed, for example, by inverting a Jacobian of the map (18) that is nonsingular in a vicinity of some point x: Combining the last equations in the ith subsystem in (11) in a new system, we obtained: m X x’ iri ¼ Lrf i hi ðF1 ðx; ZÞÞ þ Lgj Lrf i 1 hi ðF1 ðx; ZÞÞjj ðtÞ 8i ¼ 1; . . . ; m ð20Þ j¼1

Since the finite-time exact estimates x#’ ir1 of x’ ir1 8i ¼ 1; . . . ; m are available via the higher-order # Z# for x; Z in (20), the asymptotic sliding-mode differentiator (13), (14), and using the estimates x; # estimate jðtÞ of the disturbance jðtÞ in (1) can be identified 20 1 0 13 # Z# ÞÞ Lrf 1 h1 ðF1 ðx; x#’ 1r1 6B C B C7 7 6B # C B r2 # Z# ÞÞ C 6B x’ 2 C B Lf h2 ðF1 ðx; C7 r2 C 7 6 C B B # Z# ÞÞ6B C B C7 # ¼ E 1 ðF1 ðx; ð21Þ jðtÞ 6B . C B C7 . 6B .. C B C7 .. 6B C B C7 4@ A @ A5 r 1 m # Z# ÞÞ #x’ m Lf hm ðF ðx; rm Based on the developments in this section, the following theorems are true. Theorem 1 If system (1)–(3) is locally detectable in the sense of Definition 1, the higher-order sliding-mode observer (13), (14), (19), (21) asymptotically estimates the state x and the disturbance jðtÞ in the system, and hence the goals of the observer design (5) and (6) are met. When the total relative degree of the system is r ¼ n; all the states are estimated in finite time. Copyright # 2007 John Wiley & Sons, Ltd.


L. FRIDMAN ET AL.

Theorem 2 Suppose that system (1)–(3) is locally detectable in the sense of Definition 1 and the measured outputs are corrupted with noise which is a Lebesgue-measurable function of time with maximal magnitude e: Then the higher-order sliding-mode observer (13), (14), (19), (21) ensures a state observation error accuracy of the order of e2=ð%rþ1Þ ; r% ¼ max ri ; i ¼ 1; . . . ; m: Theorem 3 Suppose that the outputs of system (1)–(3) are measured at discrete sampling times with a sufficiently small sampling interval d: Then the higher-order sliding-mode observer (13), (14), (19), (21), after some transient, ensures a state observation error accuracy of the order of d2 : Remark When the total the relative degree of the system r ¼ n; all the states are estimated in finite time.

4. EXAMPLES Example 1 Consider a satellite system which is modelled as in [17] as r’ ¼ v v’ ¼ ro2

kg M þd r2

2vo yo r m where r is the distance between the satellite and the Earth centre, v is the radial speed of the satellite with respect to the Earth, o is the angular velocity of the satellite around the Earth, m and M are the mass of the satellite and the Earth, respectively, kg represents the universal gravity coefficient, and y is the damping coefficient. The quantity d which affects the radial velocity equation is assumed to be a disturbance which is to be reconstructed/estimated. Let x :¼ colðx1 ; x2 ; x3 Þ :¼ ðr; v; oÞ: The satellite system can be rewritten as follows: 1 0 x2 0 1 C B 0 C B k B x1 x2 1 C B C 3 B C ð22Þ x’ ¼ B x21 C C þ @ 1 AdðtÞ B C B A @ 2x2 x3 0 k2 x3 x1 o ’ ¼

y ¼ x1

ð23Þ

where y is the system output, k1 ¼ kg M and k2 ¼ y=m: Copyright # 2007 John Wiley & Sons, Ltd.



By direct computation, it follows that Lg hðxÞ ¼ 0;

Lg Lf hðxÞ ¼ 1

and thus the system (22)–(23) has global relative degree 2. Choose the coordinate transformation as T : x1 ¼ x1 ; x2 ¼ x2 ; Z ¼ x21 x3 : Note that for x1 =0; this transformation is invertible and an analytic expression for the inverse can be obtained as x1 ¼ x1 ; x2 ¼ x2 ; x3 ¼ Z=x21 ; since x1 ¼ r is the distance of the satellite from the centre of the Earth x1 =0: It follows that in the new coordinate system colðx1 ; x2 ; ZÞ; system (22)–(23) can be described by x’ 1 ¼ x2 2

Z k1 x’ 2 ¼ 3 2 þ d x1 x1 Z’ ¼ k2 Z Clearly, the system internal dynamics are asymptotically stable since k2 > 0: Therefore, system (21)–(22) is locally asymptotically observable. From (13) and (14), the higher-order slidingmode differentiator is described by z’ 10 ¼ n10 n10 ¼ l10 jz10 yj2=3 sign ðz10 yÞ þ z11 z’ 11 ¼ n11 n11 ¼ l11 jz11 n10 j1=2 sign ðz11 n10 Þ þ z12 z’ 12 ¼ l12 sign ðz12 n11 Þ

Define x# 1 ¼ z10 ; x# 2 ¼ z11 ; x’# 2 ¼ z12 : Then,

2 x# :¼ 4

x# 1 x# 2

3 5 2 R2

is an estimate of x and the estimate for Z can be obtained from the equation Z’# ¼ k2 Z# : Therefore, the estimate of the disturbance dðtÞ is available online, and from (21), 2 # ¼ x#’ 2 Z# þ k1 dðtÞ x# 31 x# 21

is a reconstruction for the disturbance dðtÞ: As in [17], the parameters have been chosen as follows: m ¼ 10; M ¼ 5:98 1024 ; kg ¼ 6:67 1011 and y ¼ 2:5 105 : For simulation purposes, choose dðtÞ ¼ e0:002t sinð0:02tÞ: The differentiator gains lji are chosen as l10 ¼ 2 and l11 ¼ l12 ¼ 1: In the following simulation, the initial values x0 ¼ ð107 ; 0; 6:3156 104 Þ for the plant states in the original coordinates whilst for the observer z0 ¼ ð1:001 107 ; 0; 1Þ and Z# 0 ¼ 6:3156 104 (in the transformed coordinate system). Figures 1 and 2 show that the states and the disturbance signal dðtÞ can be reconstructed faithfully. Copyright # 2007 John Wiley & Sons, Ltd.


L. FRIDMAN ET AL.

20 d(t) reconstruction of d(t)

10 0 − 10 − 20 0

200

400

600

800

1000

time [sec] 1 d(t) reconstruction of d(t)

0.5 0 −0.5 −1

0

200

400

600

800

1000

time [sec]

# Figure 1. The disturbance signal dðtÞ and its reconstruction signal dðtÞ:

Example 2 Consider the fifth-order nonlinear system 3 2 3 2 2x1 x2 1 0 7 6 7 6 7 6 x1 7 60 7" 6 0 # 7 7 6 6 j1 ðtÞ 7 6 x3 2x3 x4 7 6 3 7 þ 6 0 1 þ ð2x5 þ sin ðx5 ÞÞ2 7 x’ ¼ 6 7 6 7 6 7 6 7 j2 ðtÞ 6 x3 7 60 7 |fflfflfflfflffl{zfflfflfflfflffl} 6 0 7 4 5 jðtÞ 6 5 4 2x5 þ sin x5 ðx2 4Þ 0 0 2 þ cos x5 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} f ðxÞ

ð24Þ

GðxÞ:¼½g1 ðxÞ;g2 ðxÞ

y1 ¼ h1 ðxÞ ¼ x2

ð25Þ

y2 ¼ h2 ðxÞ ¼ x4 in the domain O ¼ fðx1 ; x2 ; x3 ; x4 ; x5 Þj jx2 j53:5; x1 ; x3 ; x4 ; x5 2 Rg; where x 2 R5 is the system state, y ¼ colðy1 ; y2 Þ is the system output, and jðtÞ ¼ ½j1 ðtÞj2 ðtÞT is the system input which will be reconstructed. By direct computation, it follows that Lg1 h1 ¼ Lg2 h2 ¼ 0 Copyright # 2007 John Wiley & Sons, Ltd.



1.005

x 107 x1

1

Estimate of x1 0.995

0

200

400

600

800

1000

time [sec] 500 x2 Estimate of x2

0 −500

0

200

400

600

800

1000

time [sec] 6.4

x 10−4

x3

6.2

Estimate of x3 6

0

200

400

600

800

1000

time [sec]

Figure 2. The response of system states and their estimates.

and

"

Lg1 Lf h1

Lg2 Lf h1

Lg1 Lf h2

Lg2 Lf h2

#

"

1

0

0

1 þ ð2x5 þ sin x5 Þ2

¼

#

which is nonsingular in R5 : Therefore, system (24)–(25) has relative degree f2; 2g: Further, F ¼ spanfg1 ; g2 g is an involutive distribution and thus Assumptions (i)–(iii) are satisfied. This implies that system (24)–(25) is weakly observable in the domain O: Then, under the coordinate transformation: x11 ¼ x2 ; x12 ¼ x1 ; x21 ¼ x4 ; x22 ¼ x3 ; Z ¼ 2x5 þ sin x5 ; the system (22)–(23) in the new coordinate system can be described by x’ 11 ¼ x12 x’ 12 ¼ 2x12 x11 þ j1 ðtÞ x’ 21 ¼ x22 x’ 22 ¼ ðx22 Þ3 x21 2x22 þ ð1 þ Z2 Þj2 ðtÞ Z’ ¼ ðx11 4ÞZ Copyright # 2007 John Wiley & Sons, Ltd.


L. FRIDMAN ET AL.

It is clear that it is not possible to obtain an analytic inverse for the inverse transformation because the fifth coordinate x5 cannot be expressed analytically as a function of Z since Z ¼ 2x5 þ sin x5 : Clearly, the system internal dynamics are asymptotically stable in the domain O: Therefore, system (24)–(25) is locally asymptotically observable. From (13) and (14), the high-order slidingmode differentiator (13) is described by z’ 10 ¼ n10 n10 ¼ l10 jz10 y1 j2=3 sign ðz10 y1 Þ þ z11 z’ 11 ¼ n11 n11 ¼ l11 jz11 n10 j1=2 sign ðz11 n10 Þ þ z12 z’ 12 ¼ l12 sign ðz12 n11 Þ z’ 20 ¼ n20 n20 ¼ l20 jz20 y2 j2=3 sign ðz20 y2 Þ þ z21 z’ 21 ¼ n21 n21 ¼ l21 jz21 n20 j1=2 sign ðz21 n20 Þ þ z22 z’ 22 ¼ l22 sign ðz22 n21 Þ Define x# 11 ¼ z10 ; x# 12 ¼ z11 ; x’# 12 ¼ z12 ; x# 21 ¼ z20 ; x# 22 ¼ z21 ; x#’ 22 ¼ z22 : Then, 2 13 x# 1 7 6 2 3 6 x# 1 7 x# 1 6 27 4 7 x# :¼ 4 5 :¼ 6 6 #2 7 2 R #x2 6 x1 7 4 5 x# 22 is an estimate of x and the estimate for Z can be obtained from equation Z’# ¼ ðx# 1 4Þ#Z 1

Therefore,

2 # ¼4 jðtÞ

3

x’# 12 þ x# 11 þ 2x# 12 ðx’# 22 þ ðx# 22 Þ2 þ x# 21 þ 2x# 22 Þ=ð1 þ Z# 2 Þ

5

is available online and from (19) it is a reconstruction for the input jðtÞ: For simulation purposes, choose j1 ðtÞ ¼ sinð0:5tÞ and j2 ðtÞ ¼ 0:5 sinð0:5tÞ þ 0:5 cos t: From [16], lji can be chosen as l10 ¼ l20 ¼ 3; l11 ¼ l21 ¼ 1:5; and l12 ¼ l22 ¼ 1:1: The simulation with the initial values x0 ¼ ð0; 0:1; 0; 0:2; 0:2Þ; z0 ¼ ð0; 0; 0; 0:2; 0Þ; and Z# 0 ¼ 0:5 are shown in the following figures. Figure 3 shows the states and the estimates in the original coordinate system. Here only asymptotic convergence is achieved. To obtain the values of x# in terms of x# and Z# ; it has been necessary to embed in the simulation an iteration scheme to extract x# 5 from Copyright # 2007 John Wiley & Sons, Ltd.



0.1 estimate error of x estimate error of x estimate error of x estimate error of x estimate error of x

0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25

0

1

2

3

4

5 6 time [sec]

7

8

9

10

# Figure 3. The original state variables xðtÞ and their estimates xðtÞ:

2

input φ1

1

reconstruction of φ1

0 -1 -2

0

1

2

3

4

5 time [sec]

6

2

7

8

9

10

input φ2 reconstruction of φ2

1

0

-1

0

1

2

3

4

5 time [sec]

6

7

8

9

10

Figure 4. The inputs of system (22)–(23) and their reconstruction signal.

Z# ¼ 2x# 5 þ sin x# 5 given Z# : Figure 4 shows the estimate of the unknown inputs. From the simulation, it is observed that the proposed strategy can reconstruct the input faithfully after approximately 1:1 s: Copyright # 2007 John Wiley & Sons, Ltd.


L. FRIDMAN ET AL.

5. CONCLUSIONS In this paper, an exact observer scheme for nonlinear locally detectable systems with unknown inputs has been proposed based on higher-order sliding-mode concepts. The approach is applicable for a class of nonlinear systems with unknown inputs, which enter affinely. The systematic design approach consists of two steps: first the transformation of the system to the Brunovsky canonical form; and second the application of higher-order sliding-mode differentiators for each coordinate of the output vector error. The proposed scheme ensures exact finite state estimation for the observable variables and asymptotic exact estimation of the unobservable variables for the case when the system has stable internal dynamics. When the total the relative degree of the system r ¼ n; all the states are estimated in finite time. In addition to estimating the states, the unknown inputs can also be identified asymptotically.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

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