High-order sliding-mode observation for linear systems with unknown inputs

Our reference: NAHS 323 P-authorquery-v8

AUTHOR QUERY FORM

Journal:Nonlinear Analysis: Hybrid Systems

Article Number: 323

Please e-mail or fax your responses and any corrections to:

E-mail: [email protected]

Fax: +44 1392 285879

Dear Author,

Please check your proof carefully and mark all corrections at the appropriate place in the proof (e.g., by using on-screen annotationin the PDF file) or compile them in a separate list.

For correction or revision of any artwork, please consult http://www.elsevier.com/artworkinstructions.

Any queries or remarks that have arisen during the processing of your manuscript are listed below and highlighted by flags in theproof. Click on the ‘Q’ link to go to the location in the proof.

Locationin article

Query / remark click on the Q link to goPlease insert your reply or correction at the corresponding line in the proof

Q1 The abstract is different in the TeX file and the PDF manuscript. We have followed the TeX file. Please check,and correct if necessary.

Q2 An extra opening parenthesis is inserted. Please check, and correct if necessary.

Thank you for your assistance.

of ...1...

http://www.elsevier.com/artworkinstructions





Nonlinear Analysis: Hybrid Systems xx (xxxx) xxx–xxx

Contents lists available at ScienceDirect

Nonlinear Analysis: Hybrid Systems

journal homepage: www.elsevier.com/locate/nahs

High-order sliding-mode observation for linear systems with unknowninputsLeonid Fridman a, Jorge Davila b,∗, Arie Levant ca Engineering Faculty, National Autonomous University of Mexico (UNAM), Ciudad Universitaria, 04510, D.F., Mexicob Section of Graduate Studies and Research, National Polytechnic Institute (ESIME-UPT). Av. Ticoman 600, 07340, D.F., Mexicoc Applied Mathematics Department, School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel

a r t i c l e i n f o

Keywords:Sliding modesObserver designNonlinear observers

a b s t r a c t

A∧high-order sliding-mode observer is designed for linear systems with unknown inputs.

The concepts of strong observability and strong detectability are studied and their relationwith the relative degree of a plant is established.

∧High-order sliding-mode-based observers

for linear time-invariant systems with unknown inputs satisfying the∧condition of strong

observability or strong detectability are developed.© 2010 Elsevier Ltd. All rights reserved.

1. Introduction1

1.1. Preliminaries and motivation2

Observation of system states in the presence of unknown inputs is one of themost important problems in∧modern control3

theory. Usually the observers for such systems are designedunder the assumption that only the outputs are available,∧andnot4

their derivatives. In particular, it is required that the terms containing unknown inputs appear already in the first derivative5

of the system outputs. Q16

Sliding-mode-based robust state observation∧has been successfully developed in

∧variable structure theory within recent7

years (see [1–7]).∧Sliding-mode-based observation has such attractive features as8

• insensitivity (more than robustness) with respect to unknown inputs;9

∧• the possibility of using the equivalent output injection in order to obtain additional information.10

Further analysis has shown that these observers are very useful for fault detection [8–10]. However∧, for those observers11

the fault detection is realized via equivalent output injection, while the estimations of the observable states were made by12

traditional smooth (usually Luenberger) observers without differentiators.∧This leads to their main limitation: the output13

of the system should have a relative degree∧1 with respect to the unknown input. This condition is very restrictive even for14

velocity observers for mechanical systems [11–15].15

Step-by-step vector-state reconstruction by means of sliding modes is studied in [16–18]. These observers are based on16

a system transformation to a triangular form and successive estimation of the state vector using the equivalent injection17

signal. The corresponding sufficient conditions for observation of linear time-invariant systems with unknown inputs were18

obtained in [18]. Moreover, such observers theoretically ensure finite-time convergence for all system states. Unfortunately,19

the realization of step-by-step sliding-mode observers is based on conventional sliding modes requiring filtration at each20

step due to the imperfections of analog devices or discretization effects.21

∗ Corresponding address: Automatic Control Department, CINVESTAV-IPN, Av. IPN 2508, Col. San Pedro Zacatenco, 04510 Gustavo A. Madero, Mexico.Tel.: +52 5556763097.

E-mail addresses: [email protected] (L. Fridman), [email protected], [email protected] (J. Davila), [email protected] (A. Levant).

1751-570X/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.nahs.2010.09.003

http://dx.doi.org/10.1016/j.nahs.2010.09.003

http://www.elsevier.com/locate/nahs

http://www.elsevier.com/locate/nahs

mailto:[email protected]




http://dx.doi.org/10.1016/j.nahs.2010.09.003

2 L. Fridman et al. / Nonlinear Analysis: Hybrid Systems xx (xxxx) xxx–xxx

Hierarchical observerswhichwere recently developed in [19] avoid the filtration. They iteratively use∧a continuous super- 1

twisting controller [20]. Amodified version of the super-twisting controller is also used in the step-by-step observer∧of [18]. 2

Unfortunately,∧those observers are also not free of drawbacks: 3

1. The super-twisting algorithm provides the∧best possible asymptotic accuracy of the derivative estimation at each single 4

realization step [21]. In particular, with discrete measurements the accuracy is proportional to the sampling step τ in 5

the absence of∧noise, and to the square root of the input noise magnitude, if the above discretization error is negligible. 6

The step-by-step and hierarchical observers use the output of the super-twisting algorithm as noisy input at the next 7

step. As a result, the overall observation accuracy is of the order τ1

2r−1 , where r is the observability index of the system. 8

This means, for example, that in order to implement the fourth-order derivative observer with∧0.1 precision, and the 9

unknown fifth derivative being less than 1 in its absolute value, the∧practically impossible discretization step τ = 10−8

10

is needed. 11

2. Similarly, in the presence of the measurement noise with magnitude ε the estimation accuracy is proportional to ε12r , 12

which requires measurement noises∧not exceeding 10−16 for the fourth-order observer implementation under the above 13

conditions. 14

3. The step-by-step observers [18] provide for∧bounded input–bounded state (BIBS) finite-time stability only, restricting 15

the application of these observers to the class of the systems for which the upper bound of the initial conditionsmight be 16

estimated in advance. Moreover, it works only under conditions of full relative degree, i.e.,∧where the sum of the relative 17

degrees of the outputs with respect to the unknown inputs∧is equal to the dimension of the system. 18

At the same time the rth-order robust exact sliding-mode-based differentiator [22] removes the first issue providing for 19

∧r∧th-derivative accuracy proportional to the discretization step τ , and resolves the second one providing for the accuracy 20

ε1

r+1 . Unfortunately, its∧straightforward application requires the boundedness of the unknown (r + 1)th derivative. In 21

practice∧this means that still only semiglobal observation of stable linear systems is allowed. 22

The∧high-order sliding-mode observers recently developed in [23–25] provide for

∧global finite-time convergence to zero 23

of the estimation error and for the best possible accuracy in the strongly observable case. However, the application of that 24

observer is confined to the class of∧systems having a nonsingular special matrix of high-order partial derivatives. It turns 25

that this is just a restriction of the transformation method suggested in the above cited papers. The technique of weakly 26

observable subspaces and corresponding Molinari transformations [26] is proposed in [27,28] to avoid that restriction. 27

1.2. Main contribution 28

The observation algorithms are developed for the general case of strongly observable linear∧time-invariant (LTI) systems 29

with bounded unknown Lebesgue-measurable inputs, providing for∧global finite-time exact observation of the state vector. 30

1.3. Structure of the paper 31

In Section 2 we discuss the problem statement. This algorithm is designed in three steps. First, in Section 3.1, the state of 32

the system is divided into special components in order to avoid contamination by noises in the subspace not affected by the 33

unknown inputs. Second, an algorithm∧for ensuring the boundedness of the estimation error is developed in Section 3.2. 34

Finally the algorithm for the full reconstruction of the state is presented in Section 3.3. Section 4 contains an example 35

illustrating the proposed algorithm. 36

2. System description 37

Consider a∧linear time-invariant system with unknown inputs (LTISUI) 38

x = Ax + Bu(t) + Eζ (t), 39

y = Cx + Du(t) + Fζ (t), (1) 40

where x ∈ X ⊆ Rn are the system states, y ∈ Y ⊆ Rp is the vector of the system outputs, u(t) ∈ U ⊆ Rq0 is a vector control 41

input, ζ (t) ∈ W ⊆ Rm,m ≤ p, are the unknown inputs (disturbances or system nonlinearities), and the known matrices 42

A, B, C, D, E, F have suitable dimensions. The equations are understood in the Filippov sense [29] in order to provide 43

∧the possibility of using discontinuous signals in controls and observers. Note that Filippov solutions coincide with the usual 44

solutions, when the right-hand sides are continuous. It is assumed also that all∧inputs considered allow the existence of 45

solutions and their extension to the whole semi-axis t ≥ 0. 46

Without loss of generality it is assumed that 47

rank[EF

]= m. 48

The aim of this article is to build an observer providing for the exact (preferably finite-time convergent) estimation of 49

the states and of the unknown input. Obviously, it can be assumed without loss of generality that the known input u(t) is 50

equal to zero (i.e., u(t) = 0). 51

L. Fridman et al. / Nonlinear Analysis: Hybrid Systems xx (xxxx) xxx–xxx 3

The following notation is used in the paper. Let G ∈ Rn×m be a matrix. If rankG = n, then define the∧right-hand side1

pseudoinverse of G as the matrix G+= GT (GGT )−1. If rankG = m, then define the

∧left-hand side pseudoinverse of G as the2

matrix G+= (GTG)−1GT . For a matrix J ∈ Rn×m, n ≥ m, with rank J = r , we define one of the matrices J⊥ ∈ Rn−r×n, such3

that rank J⊥ = n − r and J⊥J = 0. The notation J⊥⊥∈ Rr×n corresponds to one of the matrices such that rank J⊥⊥

= r and4

J⊥(J⊥⊥)T = 0. It is clear that the matrices and J⊥ and J⊥⊥ are not unique and5

rank[J⊥

J⊥⊥

]= n.6

Two main definitions are introduced∧for designing the observers.7

Definition 1 ([30]). System (1) is called strongly observable∧if for any initial state x(0) and any unknown input ζ (t), y(t) ≡ 08

with ∀t ≥ 0 implies that also x ≡ 0.9

Definition 2 ([30]). System (1) is called strongly detectable∧if for any initial state x(0) and any unknown input ζ (t), y(t) ≡ 010

with ∀t ≥ 0 implies that x → 0 as t → ∞.11

3. Observer design12

3.1. State separation13

System (1) can be unstable; this characteristic makes∧the application of a

∧high-order sliding-mode algorithm impossible.14

On the other hand, it is possible that not all the outputs of the system contain information affected by the unknown inputs.15

In this subsection the outputs which do not contain information affected by the unknown inputs are separated from the rest16

of the outputs. A suitable transformation is applied to the system to separate the states that could be reconstructed by the17

outputs unaffected∧by the unknown inputs from the rest of the state.18

Let p3 = rank F . Select a matrix F⊥∈ R(p−p3)×n as19

F⊥=

[F 1⊥

F 2⊥

],20

such that F 1⊥

∈ Rp1×p, F 2⊥

∈ Rp2×p, and21

F 1⊥F = 0, and F 1

⊥CAiE = 0, ∀ i = 0, . . . , n − 1,22

F 2⊥F = 0, andF 2

⊥CAj−1E = 0, for some 0 ≤ j < n,23

rank[F 1⊥

F 2⊥

]= p − p3.24

Choose a matrix F⊥⊥∈ Rp3×p, and apply the output transformation25 [

F⊥

F⊥⊥

]y(t).26

The transformed output takes the form27 y1y2y3

=

C1C2C3

x +

D1D2D3

u +

00F3

ζ . (2)28

Note that rank F3 = rank F = p3.29

Definition 3. Consider the system (1). Define the vector of partial relative degrees of the output y(t) with respect to the30

unknown vector input ζ (t) as the vector (r1, . . . , rp) composed of the integers ri, i = 1, . . . , p. Each partial relative degree31

ri satisfies the following requirements:32

• ri = 0∧if fi = 0, where fi is the i

∧th row of the matrix F3

∧.33

• If fi = 0, then ri is the integer such that34

ciAjE = 0, j = 0, . . . , ri − 2, ciAri−1E = 0, ri ≤ n − 1 (3)35

where ci is the i∧th row of the matrix C

∧.36

∧• ri = ∞

∧if ciAjE = 0 for all j = 0, . . . , n − 1.37

In other words F 1⊥y corresponds to the outputs with partial relative degree equal to infinity, and F 2

⊥y corresponds to the38

outputs with∧finite positive partial relative degree. The vector y1(t) ∈ Rp1 is composed of all the outputs with partial relative39

degree equal to ∞, the components of y2(t) ∈ Rp2 correspond to the outputs with finite partial relative degree such that40

0 < ri < n − 1 for i = 1, . . . , p2, and the output y3(t) ∈ Rp3 is composed by all the outputs with partial relative degree41

equal to 0 with respect to the unknown inputs.42


Consider the system output (2) and the first equation of (1). It is necessary to separate the state dynamics contaminated 1

by the unknown inputs from the ‘‘clean’’ state dynamics. 2

Define ny1 as the rank of the observability matrix of the pair (C1, A) (see [31]). Let the matrix Uy1 ∈ Rny1×n be composed 3

∧of the first ny1 linearly independent rows of the observability matrix. The matrix Uy1 is further called the reduced order 4

observability matrix of the pair (C1, A). 5

Choose one of the matrices Uy1 ∈ R(n−ny1 )×n∧such that 6

Uy1 UTy1 = 0, rank

[Uy1Uy1

]= n 7

and define the transformation matrix 8

Ty =

[Uy1Uy1

]. (4) 9

∧Apply the state transformation ξ = Tyxwith Ty defined by (4). The system (1) is transformed into the form 10[

ξ1ξ2

]=

[A11 0A21 A22

] [ξ1ξ2

]+

[B1B2

]u +

[0E2

]ζ , (5) 11y1

y2y3

=

C11 00 C22C31 C32

[ξ1ξ2

]+

D1D2D3

u +

00F3

ζ , (6) 12

where ξ1 ∈ Rny1 , ξ2 ∈ R(n−ny1 ). 13

It is important to∧note that the subsystem 14

ξ1 = A11ξ1 + B1u,y1 = C11ξ1 + D1u,

(7) 15

is observable. 16

Remark 1. Recall that this transformation is necessary only in the case when not all∧outputs are affected by the unknown 17

inputs, i.e., p1 = 0. 18

In the following two assumptions are introduced: 19

Assumption 1. The quadruple {A, C, E, F} is strongly detectable. 20

Assumption 2. The unknown input ζ (t) is a Lebesgue-measurable function and is bounded, i.e. ‖ζ (t)‖ ≤ ζ+. 21

The observer is designed in two steps. First, the convergence of the estimation error to a bounded vicinity of the origin 22

is ensured. Second, the bounded estimation error is forced to vanish using a∧technique based on the application of the high- 23

order sliding-mode differentiator. 24

3.2. Error stabilization 25

It is clear that the eigenvalues of the matrix A from (1) are the union of the∧sets of eigenvalues of the matrices A11, A22 26

from (5). 27

Consider the system (5), (6).Select a gain matrix 28

L =

[L11 0 00 L22 L23

]∈ Rn×p, 29

where L11 ∈ Rny1×p1 , L22 ∈ R(n−ny1 )×p2 , L23 ∈ R(n−ny1 )×p3 ,∧such that A11 − L11C11, A22 − L22C22 − L23C32

∧are Hurwitz. The 30

gain matrix L exists due∧Assumption 1. 31

The Luenberger part of the observer takes on the form 32[z1z2

]=

[A11 0A21 A22

] [z1z2

]+

[B1B2

]u +

[L11 0 00 L22 L23

] y1 − y1y2 − y2y3 − y3

, (8) 33

where y1, y2, y3 are the output estimations 34y1y2y3

=

C11 00 C22C31 C32

[z1z2

]+

D1D2D3

u. (9) 35


The corresponding error system is1 [˙e1˙e2

]=

[A11 0A21 A22

] [e1e2

]+

[0E2

]ζ (t), (10)2 y1

y2y3

=

C11 00 C22C31 C32

[e1e2

]+

00F3

ζ (t), (11)3

where e = ξ − z, y = y − y, and the matrices A11, A21, A22, E2 are defined as4

A11 = A11 − L11C11, A21 = A21 − L23C31,5

A22 = A22 − L22C22 − L23C32, E2 = E2 − L23F3.6

It is possible to∧define7

C =

C11 00 C22C31 C32

=

C1

C2

C3

.8

∧Eqs. (10) and (11) can be rewritten in a compact form as9

˙e = Ae + Eζ (t), (12)10

y = C e + Fζ (t). (13)11

3.3. Convergence to zero enforcement12

Obtain the matrices13

UT1i =

cT1i (c1iA)T · · · (c1iAn1i−1)T

14

where c1i, i = 1, . . . , p1, is the i∧th row of the matrix C1, and n1i is the integer defined as15

n1i = rankcT1i (c1iA)T · · · (c1iAn−1)T

T.16

Recall that the matrix U1i is the observability matrix for the pair (c1i, A).17

It is clear∧that the selection of the matrices U1i allows us to write the

∧following equalities:18

c1i(c1iA)

...

(c1iAn1i−1)

e =

y1i˙y1i...

y(n1i−1)1i

19

where y1i denotes the ith row of the vector y1. Surely, the output-estimation error y1 is measurable. Apply the differentia-20

tor [22] to each component of y1:21

vi1 = wi1 = −αn1iN1/n1ii |vi1 − y1i|(n1i−1)/n1isign(vi1 − y1i) + vi2,22

vi2 = wi2 = −α(n1i−1)N1/(n1i−1)i |vi2 − wi1|

(n1i−2)/(n1i−1)sign(vi2 − wi1) + vi3,23

...24

vi,n1i−1 = wi,n1i−1 = −α2N1/2i |vi,n1i−1 − wi,n1i−2|

1/2sign(vi,n1i−1 − wi,n1i−2) + vin1i ,25

vin1i = −α1Nisign(vin1i − wi,n1i−1), (14)26

where Ni > 0 and the constants αi are recursively chosen sufficiently large for all the components as in [22]. In particular,27

one of the possible choices is α1 = 1.1, α2 = 1.5, α3 = 2, α4 = 3, α5 = 5, α6 = 8, which is sufficient for ri ≤ 6.28

The∧components obtained, vij , can be arranged in the vector29

vTi =

vTi1 vT

i2 · · · vTin1i

.30

For all vi and U1i, i = 1, . . . , p1, the equality vi = U1ie holds after finite time.31


It is possible to find the extended matrices U1extended and vextended as∧

1

U1extended =

U11U12...

U1p1

, vextended =

v1v2...

vp1

2

it is clear that rankU1extended = ny1 . Construct the matrix U1 ∈ Rny1×n selecting the first ny1 linearly independent rows of 3

U1extended and the vector v composed of the corresponding rows of the matrix vextended, so that the equality v = U1e holds 4

after finite time. 5

The matrix U1∧satisfies ker(U1) = N1. The ny1 columns of the matrix UT

1 form a basis for the subspace N1. 6

Now it is necessary to find the corresponding basis∧for the subspaces R2 and M. Let ri be the vector of partial relative 7

degrees of the output y2i with respect to the unknown inputs for the system in the form (12), (11) where y2i is the i∧th 8

component of the output y2. 9

For every row of C2 obtain 10

UT2i =

cT2i (c2iA)T · · · (c2iAri−1)T

, 11

where c2i, i = 1, . . . , p2, is the ith row of the matrix C2, and ri is the corresponding vector of partial relative degrees of the 12

ith component of the output-estimation error y2 with respect to the unknown inputs. 13

Define the extended matrix U2extended and compute the integer ny2 as 14

U2extended =

UT21 UT

22 · · · UT2p2

,

ny2 = rank(U2extended).(15) 15

Take the full row rank matrix U2 ∈ Rny2×n composed by the first ny2 linearly independent rows of the matrix U2extended . 16

The matrix U2∧satisfies ker(U2) = R2. The columns of the matrix UT

2 form a basis of the subspace R2. 17

Define the matrix 18

Ur =

c21 A

r1−1

c22 Ar2−1

...

c2p2 Arp2−1

. 19

LetM0 = 01×n. Compute the matrix Mi+1 as∧

20

Mi+1 =

MiEUr EF3

⊥ MiAUr AC3

. (16) 21

This computation is performed until 22

rank

U1U2

Mi+1

= rank

U1U2

Mi

23

is satisfied. Let nV be defined as nV = n − rankUT1 UT

2 MTi

T. The integer nV is called the dimension of the weakly 24

unobservable subspace. Select the first n− ny1 − ny2 − nV linearly independent rows of Mi to form the matrix Mreduced such 25

that 26

rank

U1U2

Mreduced

= n − nV . 27

Let∧us name as µ

∧the number of iterations, i.e. the number of matricesMi computed. 28

The matrix Mreduced∧satisfies ker(Mreduced) = M. The columns of the matrix MT

reduced form a basis of the subspace M. 29

Compute the matrixMobs as 30

Mobs =

[U2

Mreduced

]. 31

Let us define the set of all points that cannot be reconstructed by the observer. 32


Definition 4 ([32]). A point x0 ∈ X is called weakly unobservable if there exists an input function ζ (t)∧such that the1

corresponding output y(t) equals zero for all t ≥ 0. The set of all weakly unobservable points of (1) is denoted by V∗2

and is called the weakly unobservable subspace of (1).3

The above mentioned set∧can be represented in terms of the computed matrices as follows:4

Theorem 1 ([30]). The weakly unobservable subspace is given by5

ker[

U1Mobs

]= V∗.6

Define ny as the rank of the observability matrix of the pair (C, A), and nV as the dimension of the set V∗. Let N be the7

unobservable subspace of the pair (C, A).8

Chose n − ny linearly independent vectors Ni ∈ N∧such that {N1, . . . ,N(n−ny)} is a basis of N . Obtain the matrix9

∧N = [N1 N2 · · · Nn−ny ] .10

Define the subspace VN as the complement of N in V∗, such that11

VN ⊕ N = V∗.12

Note that VN belongs to the observable subspace of the pair (A, C).13

Chose n − ny − nV linearly independent vectors VNi ∈ VN , i = 1, . . . , n − ny − nV such that {VN1 , . . . , VNn−ny−nV} is a14

basis of VN . Compute the matrix VN = [VN1 VN2 · · · VNn−ny−nV]and define the matrix V∗ ∈ Rn×nV as V∗ = [VN N].15

Select a matrix Q ∈ Rm×n such that16

(A − EQ )VN = V∗Ξ , (C − FQ )VN = 0, Q

Uy1Mobs

V TN

= 0 (17)17

for some matrix Ξ = 0, Ξ ∈ RnV×nV .18

The system (12), (13)∧can be written in the equivalent form19

˙e = (A − EQ )e + Bu + E(Q e + ζ ) (18)20 y1(t)y2(t)y3(t)

=

C1

C2

(C3 − F3Q )

e +

D1D2D3

u +

00F3

(Q e + ζ ).21

Obtain the vector of partial relative degrees of the output y2i with respect to the unknown inputs in the new22

representation of the system (18).23

For every row of C2 obtain24

UT2i =

cT2i (c2i(A − EQ ))T · · · (c2i(A − EQ )ri−1)T

,25

where c2i, i = 1, . . . , p2, is the ith row of the matrix C2, and ri is the corresponding partial relative degree of the i∧th26

component of the output-estimation error y2 with respect to the unknown inputs.27

Notice that the equalities28 c2i

c2i(A − EQ )...

c2i(A − EQ )ri−1

e =

y2i˙y2i...

y(ri−1)2i

29

are satisfied. Apply the∧high-order sliding-mode differentiator [22]) to each component of y2 as follows:30

˙vi1 = wi1 = −αri N1/rii |vi1 − y2i|(ri−1)/risign(vi1 − y2i) + vi2,31

˙vi2 = wi2 = −αri−1N1/(ri−1)i |vi2 − wi1|

(ri−2)/(ri−1)sign(vi2 − wi1) + vi3,32

...33

˙vi,ri−1 = wi,ri−1 = −α2N1/2i |vi,ri−1 − wi,ri−2|

1/2sign(vi,ri−1 − wi,ri−2) + viri ,34

˙viri = −α1Nisign(viri − wi,ri−1), (19)35

where vij and wij are the components of the vectors vi ∈ Rri and wi ∈ Rri−1 respectively. The parameter Ni is chosen36

sufficiently large for each∧output-estimation error; in particular, Ni > |di|ζ+ is required, where di = ¯c2i

¯Ari−1 ¯E. The constants37

αi are chosen recursively sufficiently large for all the components as in [22]. In particular, one of the possible choices is


α1 = 1.1, α2 = 1.5, α3 = 2, α4 = 3, α5 = 5, α6 = 8, which is sufficient for ri ≤ 6. Note that (19) has a recursive form, 1

useful for the parameter adjustment. 2

For each component of y2 form the vector 3

vTi =

vTi1 vT

i2 · · · vT1,ri

. 4

It is clear that the equalities vi = U2ie, i = 1, . . . , p2, hold after finite time. 5

Define the extended matrices U2extended, vextended and compute the integer ny2 as 6

vextended =vT1 vT

2 · · · vTp2

TU2extended =

UT21 UT

22 · · · UT2p2

T,

ny2 = rank(U2extended).

(20) 7

Take the full row rank matrix U2 ∈ Rny2×n composed by the first ny2 linearly independent rows of the matrix U2extended and 8

the vector v ∈ Rny2 by selecting the corresponding rows of the vector vextended such that the equality U2e = v holds. 9

The matrix U2∧satisfies ker(U2) = R2. The columns of the matrix UT

2 form a basis of the subspace R2. 10

Consider the derivatives of order ri − 1 of each component of y2. The following equality holds: 11yr1−121

yr2−122...

yrp2−12p2

=

c21(A − EQ )r1−1

c22(A − EQ )r2−1

...

c2p2 (A − EQ )rp2−1

e. 12

Define the matrices 13

Ur =

¯c21(A − EQ )r1−1

¯c22(A − EQ )r2−1

...¯c2p2 (A − EQ )rp2−1

, vr =

v1r1v2r2...

vp2 rp2

. 14

∧Set M0 = 01×n and ρ0 = 01×1. Define the matrix ρi as 15

ρi =

Mi¯E

Ur¯E

F3

⊥

ρivr∫ t

0y3dt

. 16

For each component of the vector ρi compute 17

βij = α2N1/2i |βij − ρij|

1/2sign(βij − ρij) + γij

γij = α1Nisign(γij − βij)(21) 18

where ρij is the jth component of the vector ρi; Ni > ‖Di‖ζ+, where Di is the i

∧th row of the matrix 19

D =

MiEUr EF3

⊥

(A − EQ )E. 20

Notice that the matrix D is computed for each matrix Mi∧appearing below. 21

Compute the matrix Mi+1 and the vector ρi+1: 22

Mi+1 =

MiEUr EF3

⊥ Mi(A − EQ )

Ur(A − EQ )

C3

, ρi+1 =

γi1γi2...

γiκ

. (22) 23

Here κ is the number of rows of the matrix

(MiE)T (Ur E)T F T3

T⊥

. 24


This computation is performed until1

rank

U1U2

Mi+1

= n − nV2

is satisfied. Let µy be the number of computed matrices Mi. Select the first n − ny1 − ny2 − nV linearly independent rows of3

Mi+1 and the corresponding rows of ρi+1 to form the matrix Mreduced and the vector ρreduced such that4

rank

U1U2

Mreduced

= n − nV, Mreducede = ρreduced.5

It is clear that, by construction, the following equality is satisfied:6 U1U2

Mreduced

e =

vv

ρreduced

.7

Define the matrix V∗ ∈ Rn×nV such that the nV columns of the matrix form a basis for the subspace V∗. It can be∧obtained8

by selecting V T∗as the orthogonal complement of the matrix

UT1 UT

2 MTreduced

T . Name as Te∧the matrix9

Te

U1U2

Mreduced

V T∗

.10

Compute the matrices11

TeAT−1e =

A1

A2

A3

, TeE =

0E2E3

,12

TeB =

B1

B2

B3

,

C1C2C3

T−1e =

C1

C2

C3

.13

The observer takes the form∧

14

x = T−1y

z + (Te)−1

vv

ρreduced

−z3 + χ + T2ξ2

(23)15

where the matrices T2 and T3 are chosen such that16

T2E2 + T3F3 = E3, (24)17

and the variables χ, z and ξ are given by18

χ = [A3 − T2A2 − T3C3]

ξ1

ξ2

ξ3

+ T3y3 + [B3 − T3D3 − T2B2]u (25)19

Tez =

z1z2z3

, TeTyx =

ξ1

ξ2

ξ3

. (26)20

Theorem 2. Let Assumptions 2 and 1 be satisfied. The state of the system (1) is asymptotically estimated by the observer (8),21

(23)–(26).22

Proof. If the matrices A11, A22, A33 are Hurwitz, then also A is Hurwitz.23

∧Choose the Lyapunov function of the system24

V = eTHe,25


where H is a symmetric positive-definite matrix. The matrix H is chosen as the solution of the Lyapunov equation 1

HA + ATH = −I. 2

It is used here that A is an Hurwitz matrix. Calculate the derivative 3

V = eT (HA + ATH)e + (eTHEζ (t) + ζ T (t)ETHe) 4

V ≤ −I‖e‖2+ 2‖e‖ ‖H‖ ‖E‖ζ+. 5

V is negative definite with ζ (t) = 0. Thus, if ζ satisfies∧Assumption 2, we obtain that the estimation error e converges to a 6

bounded vicinity of the origin e = 0.∧From that moment, ˙e also remains uniformly bounded. 7

The application of the observer (8), (9) produces the estimation error subsystem given by 8

˙e1 = A11e1y1 = C11e1.

(27) 9

Since A11 is Hurwitz, the estimation error e1 = ξ1 − z1 asymptotically converges to zero. 10

Consider the transformation Tee. The transformed matrices of system (18) are 11

¯A =

¯A11 0 0¯A21

¯A22 0¯A31

¯A32¯A33

,¯B =

¯B1¯B2¯B3

,¯E =

0¯E2¯E3

, (28) 12

¯C =

¯C11 0 00 ¯C22 0¯C31

¯C32 0

. (29) 13

Let prove that the form given in (28) is valid for this transformation∧. 14

The inverse of the transformation matrix Te can be represented as 15

T−1e = [Te1 Te2 Te3 ] 16

where Te1 ∈ Rn×ny1 , Te2 ∈ Rn×(n−ny1−nV ), Te3 ∈ Rn×nV . By definition the matrices Te1 , Te2 and Te3 satisfy 17

Ue1Ty1 = I, Te1 ∈ ker[Mn

V T∗

](30) 18

MnTe2 = I, Te2 ∈ ker[U1

V T∗

]⊆ N1 (31) 19

V T∗Te3 = I, Te3 ∈ ker

[U1Mn

]= V∗ (32) 20

The matrix Te3 can be represented as Te3 = (V T∗)+ = V∗(V T

∗V∗)

−1. 21

Apply the transformation Ty to each matrix of (18): 22

Te1(A − EQ )T−1e1 =

U1(A − EQ )Te1 U1(A − EQ )Te2 U1(A − EQ )V∗(V T∗V∗)

−1

Mn(A − EQ )Te1 Mn(A − EQ )Te2 Mn(A − EQ )V∗(V T∗V∗)

−1

V T∗(A − EQ )Te1 V T

∗(A − EQ )Te2 V T

∗(A − EQ )V∗(V T

∗V∗)

−1

. 23

Let us prove that the terms U1(A − EQ )Te2 , U1(A − EQ )V∗(V T∗V∗)

−1, Mn(A − EQ )V∗(V T∗V∗)

−1 are equal to zero: 24

• U1(A − EQ )Te2 = U1ATe2 − U1EQTe2 . Recall that the matrix U1 is the observability matrix of the pair (C1, A). It follows 25

from Definition 3 that U1E = 0. It is clear from (31) that Te2 ∈ N1, and by definition ATe2 ∈ N1. It follows that 26

U1(A − EQ )Te2 = 0. 27

• Recall that V∗ is (A − EQ )-invariant. It is clear that (A − EQ )V∗ = V∗Ξ for some Ξ ∈ RnV×nV . The term U1(A − 28

EQ )V∗(V T∗V∗)

−1 can be written as U1V∗Ξ(V T∗V∗)

−1. By definition, U1V∗ = 0. It follows that 29

U1(A − EQ )V∗(V T∗V∗)

−1= 0. 30

• By definitionMnV∗ = 0. It follows that 31

Mn(A − EQ )V∗(V T∗V∗)

−1= MnV∗Ξ = 0. 32


∧Define1

¯A11 = U1(A − EQ )Te1 ,¯A21 = Mn(A − EQ )Te1 ,

¯A22 = Mn(A − EQ )Te2 ,2

¯A31 = V T∗(A − EQ )Te1 ,

¯A32 = V T∗(A − EQ )Te2 ,

¯A33 = V T∗(A − EQ )V∗(V T

∗V∗)

−1.3

The transformed matrix TeB consists of the matrices ¯B1 = U1B,¯B2 = MnB and ¯B3 = V T

∗B.4

The transformed matrix E takes the form5

TeE =

U1Mn

V T∗

E =

U1E¯E2¯E3

.6

It follows from the definition of U1 and the Definition 3 that the term U1E = 0.7

The transformed matrix ¯C takes the form∧

8 C1

C2

(C3 − F3Q )

T−1e =

C1Te1 C1Te2 C1V∗(V T∗V∗)

−1

C2Te1 C2Te2 C2V∗(V T∗V∗)

−1

(C3 − F3Q )Te1 (C3 − F3Q )Te2 (C3 − F3Q )V∗(V T∗V∗)

−1

.9

By definition∧, V∗

⊂ ker C; then10

(C3 − F3Q )V∗(V T∗V∗)

−1= 0, C2V∗(V T

∗V∗)

−1= 0, C1V∗(V T

∗V∗)

−1= 0.11

It is clear from the definitions of Te1 and Te2 that12

Te1 ∈ ker C2,13

Te2 ∈ ker C1.14

It follows that C1Te2 = 0 and C2Te1 = 0.15

The remaining submatrices are given by16

¯C11 = C1Ty1 ,¯C22 = C2V∗(V T

∗V∗)

−1,17

¯C31 = (C3 − F3Q )Te1 ,¯C32 = (C3 − F3Q )Te2 .18

The matrices D1, D2, D3 and F3 have the same form as is given in (18). The form (28) is proved.19

If Assumption 1 is satisfied then20

rank[−

¯E2F3

]= m.21

Let the Rosenbrock matrix for the subsystem {¯A11,

¯A21,¯A22,

¯E2, C, F3} be given by22

Rs(s) =

sI −

¯A11 0 0−

¯A21 sI −¯A22 −

¯E2¯C11 0 00 ¯C22 0¯C31

¯C32 F3

.23

Notice that rank Rs(s0) = n − nV + m for all s0 ∈ C.24

The corresponding Rosenbrock matrix is given by25

R(s) =

sI −¯A11 0 0 0

−¯A21 sI −

¯A22 0 −¯E2

−¯A31 −

¯A32 sI −¯A33 −

¯E3¯C11 0 0 00 ¯C22 0 0¯C31

¯C32 0 F3

.26


The matrix R(s) has the same rank∧as the following matrix: 1

R(s) =

sI −¯A11 0 0 0

−¯A21 sI −

¯A22 −¯E2 0

¯C11 0 0 00 ¯C22 0 0¯C31

¯C32 F3 0−

¯A31 −¯A32 −

¯E3 sI −¯A33

. 2

Consider the case when rank−

¯E2F3

< m. There

∧exists a matrix G such that 3

G[−

¯E2F3

]=

[H1 0H2 0

]4

for some matrices H1 and H2 and 5

−G ¯E3 = [¯E31¯E32

]. 6

Then rank R(s) is given by 7

rank

sI −¯A11 0 0 0 0

−¯A21 sI −

¯A22 H1 0 0¯C11 0 0 0 00 ¯C22 0 0 0¯C31

¯C32 H2 0 0−

¯A31 −¯A32

¯E31¯E32 sI −

¯A33

. 8

It is clear that rank(R(s0)) < n + m for any value of s0. The set of invariant∧zeros of this system is infinite. 9

∧Assuming that 10

rank[−

¯E2F3

]= m, (33) 11

it follows that the Rosenbrock matrix Rs(s) has rank(R(s0)) = n + m for any s0 ∈ C. 12

The Rosenbrock matrix is reduced to 13

R(s) =

Rs(s) 00

−¯A31 −

¯A32 −¯E3 sI −

¯A33

. (34) 14

The submatrix Rs(s0) is full rank for any s0 ∈ C. It is clear that rank R(s0) < n+m for all the values of s0 for which s0(I − A33)Q2 15

loses rank. It is known that the invariant∧zeros of the quadruple {A, C, E, F} are invariant under nonsingular 16

transformations. It is∧clear that the eigenvalues of the matrix A33 coincides with the invariant

∧zeros of the quadruple 17

{A, C, E, F}. 18

Define the estimation error e = x − x∧; then from Eq. (23) obtain 19

e = x − T−1y

z + (Te)−1

vρn

−z3 + χ + T2ξ2

. 20

Multiply the last equation by the matrix Ty and obtain 21

Tye = e + (Te)−1

vρn

−z3 + χ + T2ξ2

. 22

Nowmultiply the last equation by the matrix (TeTe): 23

(Te)Tye = (Te)e −

vρn

−z3 + χ + T2ξ2

. 24


Define1

(Te)Tyx = ξ =

ξ1ξ2ξ3

, (Te)Tyx =ˆξ =

ˆξ 1ˆξ 2ˆξ 3

, (Te)Tyz = z =

z1z2z3

,2

(Te)Tye = ξ =

ξ1ξ2ξ3

, Tee =

e1e2e3

.3

Note that by definition U1e2 = 0, U1(ξ3 − χ − T2ξ2) = 0, Mn(ξ3 − χ − T2ξ2) = 0 and consequently4

ξ1 = U1e1 − v5

ξ2 = Mn

[e1e2

]− ρn.6

The convergence of∧these coordinates is a consequence of the application of the

∧high-order sliding-mode differentiator.7

The error∧dynamics for ξ3 = ξ3 −

ˆξ 3 is given by8

˙ξ 3 =

˙ξ 3 − χ − T2˙ξ 2. (35)9

Notice that10

T3y3 = T3C3ξ1 + T3D3u + T3F3ζ (36)11

T2ξ2 = T2A2ξ + T2B2u + T2E2ζ . (37)12

Let us substitute (36), (37) into (35):13

˙ξ 3 = A33ξ . (38)14

After∧a finite time, ξ1 and ξ2 converge to zero

∧; from that moment it follows that the

∧dynamics of ξ3 is given by15

˙ξ 3 =

¯A33ξ3. (39)16

The matrix ¯A33 is Hurwitz by Assumption 1. Now the asymptotic convergence of ξ3 → 0 is a direct consequence.∧With the17

mixed coordinates, only asymptotic convergence can be guaranteed. �18

The asymptotic convergence of the observer is produced for the detectable part of the system. In this case, the mix of19

coordinates produces an asymptotic reconstruction of the states.20

Let us define a particular case of the∧systems studied.21

Definition 5 ([30]). System (1) is called strongly observable, if for any initial state x(0) and anyunknown input ζ (t), y(t) ≡ 022

with ∀t ≥ 0 implies that also x ≡ 0.23

A possible alternative to Assumption 1 appears:24

Assumption 3. The quadruple {A, C, E, F} is strongly observable.25

Corollary 3. Let Assumptions 2 and 3 be satisfied. The state of the system (1) is estimated after a finite-time transient by the26

observer (8), (23)–(26).27

Proof. The proof is a consequence of the equivalence of the following statements [32]:28

(i) The quadruple {A, E, C, F} is strongly observable.29

(ii) The quadruple {A, E, C, F} has no invariant zeros.30

(iii) The weakly unobservable subspace contains only the origin, V∗= {0}.31

From the above mentioned equivalence it follows that the matrix Te will be composed uniquely by the matrices U1 andMn,32

and the correction factor will be composed by v and ρn. As a consequence∧, finite-time observation is achieved. �33

4. Example34

The effectiveness of the observation and fault detection algorithm is tested on an example. Consider a linear∧time-35

invariant system36

x = Ax + Eζ37

y = Cx + Fζ .38


The values of the matrices A, E, C, F are as follows: 1

A =

0 1 0 0 0 0 0 01 −3 0 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 −1 1 1 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 1 4 −6 −1 2 0

, E =

0 0 00 0 00 0 01 0 00 1 00 0 02 0 00 0 0

2

C =

1 0 0 0 0 0 0 00 0 1 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 0 1 0 0

, F =

0 0 00 0 00 0 10 0 0

. 3

The eigenvalues of A are 0.3028, −3.3028, −1.618, −1, 0.618, 1, 1, 1. Note that the system is unstable. The system 4

has an invariant zero∧at s0 = −3. 5

Let the unknown inputs be given by ζ = [ζ1 ζ2 ζ3], where ζ1(t) = 0.5 sin(2t)+0.43, ζ2(t) = 0.5 sin(4/5t+π/5)+ 6

0.4, and ζ3(t) is a square signal. 7

The matrices F⊥ and F⊥⊥ are obtained as 8

F⊥=

1 0 0 00 1 0 00 0 0 1

, F⊥⊥

=0 0 1 0

. 9

The transformation matrix Ty is obtained as Ty = I8×8. 10

Since the matrix A is unstable, the Luenberger gain L is chosen as 11

L =

12 0 0 021 0 0 00 16 0 00 91 0 00 226 0 00 0 6 00 0 13 00 0 17 0

. 12

The eigenvalues of the matrix A are −1, − 2, − 3, − 4, − 5, − 6, − 7, − 8. 13

The vector of partial relative degrees of the system in the form (12), (13) is (∞, 2, 2, 0). 14

The matrices U1, U2, Mreduced are given by 15

U1 =

[1 0 0 0 0 0 0 0

−12 1 0 0 0 0 0 0

], U2 =

0 0 1 0 0 0 0 00 0 −16 1 0 0 0 00 0 0 0 0 1 0 00 0 0 0 0 −6 1 0

, 16

Mreduced =0 0 0 32 −2 0 −6 1

, V T

∗=

0 0 0 0 1 0 0 2

. 17

The matrices Q , U1, U2, Mreduced are 18

Q =

0 0 0 0 1 0 0 00 0 0 0 4 0 0 00 0 0 0 0 0 0 0

, U1 =

[1 0 0 0 0 0 0 00 1 0 0 0 0 0 0

], 19

U2 =

0 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 0

, 20

Mreduced =0 0 −330 0 23 0 1 0

, V T

∗=

0 0 0 0 0 0 0 1

. 21

The∧second-order differentiator is applied to the components of the output y1 and y2: 22

˙vi1 = w1 = −α3N1/3i |v1 − y2i |

2/3sign(v1 − y2i) + v2, 23

˙vi2 = w2 = −α2N1/2i |v2 − w1|

1/2sign(v2 − w1) + v3, 24

˙vi3 = −α1Nisign(v3 − w2). 25


0 1 2 3 4 5 6 7 8 9 10

t [s]

–25

–20

–15

–10

–5

0

5

10

15

20

25

Fig. 1. Estimation error x − x.

t [s]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.2

0

0.2

0.4

0.6

0.8

1

Fig. 2. Convergence of the estimation error ξ1 .

Here Ni, i = 1, 2, take the values N1 = 1, N2 = 8.5 and the gains of the differentiator are α1 = 1.1, α2 = 1.5, α3 = 2.1

The vector v is obtained as2

v =

v11v12v21v22

.3

The matrixM1 and the vector ρ1 are given by4

M1 =0 0 −330 0 23 0 1 0

, ρ11 =

−2 1 0

v12v22∫ t

0y3dt

.5

The∧first-order differentiator (21) is applied to ρ11 :6

β11 = α2N1/21 |β11 − ρ11 |

1/2sign(β11 − ρ11) + γ117

γ11 = α1N1sign(γ11 − β11).8

The matrices T2, T3, A3, A2, C3 are given by9

T2 =0 10 0 0 −0.5

, T3 = 0,10

A3 =0 0 −655.8 −131.8 79.2 30.4 4.4 −2.2

,11

A2 =

0 0 16 1 0 0 0 00 0 −183.2 −3.2 −5.2 −2.4 −0.4 0.20 0 0 0 6 1 0 00 0 −36.4 −6.4 −33.4 −4.8 0.2 0.40 0 −3279.4 17.6 8.6 −39.8 −10.8 2.4

,12

C3 =0 0 17 1 0 0 0 0

.13


t [s]

–5

–4

–3

–2

–1

0

1

2

3

4

5

0 1 2 3 4 5 6


t [s]0 1 2 3 4 5 6 7 8 9 10

–30

–20

–10

0

10

20

30

40


x 106

t [s]0 1 2 3 4 5 6 7 8 9 10

–7

–6

–5

–4

–3

–2

–1

0

1

Fig. 5. System states.

According to Theorem 2 the estimation error x − x is presented in Fig. 1. 1

Consider the transformed estimation error (TeTe)Ty(x − x) = ξ . The convergence of the estimation error ξ1 is demon- 2

strated in Fig. 2. The convergence of the estimation error ξ2 is demonstrated in Fig. 3. Finally the convergence of the estima- 3

tion error ξ3 is demonstrated in Fig. 4. 4

The∧instabilities of the states are presented in Fig. 5. 5


References1

[1] V. Utkin, J. Guldner, J. Shi, Sliding Mode Control in Electromechanical Systems, Taylor and Francis, London, 1999.2

[2] B. Walcott, S. Zak, State observation of nonlinear uncertain dynamical systems, IEEE Trans. Automat. Control 32 (2) (1987) 166–170.3

[3] C. Edwards, S. Spurgeon, Sliding Mode Control: Theory and Applications, Taylor and Francis, London, 1998.4

[4] J. Barbot, M. Djemai, T. Boukhobza, Sliding mode observers, in: W. Perruquetti, J. Barbot (Eds.), Sliding Mode Control in Engineering, Control5

Engineering, Marcel Dekker, New York, 2002, pp. 103–130.6

[5] Y. Shtessel, I. Shkolnikov, M. Brown, An asymptotic second-order smooth sliding mode control, Asian J. Control 5 (4) (2003) 498–504.7

[6] A. Poznyak, Stochastic output noise effects in sliding mode estimations, Internat. J. Control 76 (2003) 986–999.8

[7] C. Edwards, S. Spurgeon, C.P. Tan, On development and applications of sliding mode observers, in: X. Yu, J. Xu (Eds.), Variable Structure Systems:9

Towards 21st Century, in: Lecture Notes in Control and Information Science, Springer-Verlag, Berlin, Germany, 2002, pp. 253–282.10

[8] C. Edwards, S. Spurgeon, R. Patton, Sliding mode observers for fault detection and isolation, Automatica 36 (4) (2000) 541–553.11

[9] Y. Xiong, M. Saif, Sliding mode observer for nonlinear uncertain systems, IEEE Trans. Automat. Control 46 (12) (2001) 2012–2017.12

[10] C. Tan, C. Edwards, Sliding mode observers for robust detection and reconstruction of actuator and sensor faults, Internat. J. Robust Nonlinear Control13

13 (5) (2003) 443–463.14

[11] J. Alvarez, Y. Orlov, L. Acho, An invariance principle for discontinuous dynamic systems with application to a Coulomb friction oscillator, J. Dyn. Syst.15

Meas. Control 122 (2000) 687–690.16

[12] B. Xian, M. Queiroz, D. Dawson, M. McIntyre, A discontinuous output feedback controller and velocity observer for nonlinear mechanical systems,17

Automatica 40 (4) (2004) 695–700.18

[13] J. Davila, L. Fridman, A. Levant, Second-order sliding-mode observer for mechanical systems, IEEE Trans. Automat. Control 50 (11) (2005) 1785–1789.19

[14] J. Davila, L. Fridman, A. Poznyak, Observation and identification of mechanical systems via second order sliding modes, Internat. J. Control 79 (10)20

(2006) 1251–1262.21

[15] Y. Su, P. Müller, C. Zheng, A simple nonlinear observer for a class of uncertain mechanical systems, IEEE Trans. Automat. Control 52 (7) (2007)22

1340–1345.23

[16] H. Hashimoto, V. Utkin, J.X. Xu, H. Suzuki, F. Harashima, VSS observer for linear time varying system, in: Proceedings of IECON’90, Pacific Grove CA,24

1990, pp. 34–39.25

[17] T. Ahmed-Ali, F. Lamnabhi-Lagarrigue, Sliding observer–controller design for uncertain triangular nonlinear systems, IEEE Trans. Automat. Control 4426

(6) (1999) 1244–1249.27

[18] T. Floquet, J. Barbot, A canonical form for the design of unknown input sliding mode observers, in: C. Edwards, E. Fossas, L. Fridman (Eds.), Advances28

in Variable Structure and Sliding Mode Control, in: Lecture Notes in Control and Information Science, Springer-Verlag, Berlin, 2006, pp. 271–292.29

[19] F. Bejarano, A. Poznyak, L. Fridman, Observer for linear time invariant systemswith unknown inputs based on the hierarchical super-twisting concept,30

in: 2006 International Workshop on Variable Structure Systems—VSS’06, Sardinia, Italy, 2006, pp. 208–213.31

[20] A. Levant, Sliding order and sliding accuracy in sliding mode control, Internat. J. Control 58 (1993) 1247–1263.32

[21] A. Levant, Robust exact differentiation via sliding mode technique, Automatica 34 (3) (1998) 379–384.33

[22] A. Levant, High-order sliding modes: differentiation and output-feedback control, Internat. J. Control 76 (9–10) (2003) 924–941.34

[23] L. Fridman, A. Levant, J. Davila, High-order sliding-mode observer for linear systems with unknown inputs, in: 2006 International Workshop on35

Variable Structure Systems—VSS’06, Sardinia, Italy, 2006, pp. 202–207.36

[24] L. Fridman, Y. Shtessel, C. Edwards, X. Yan, Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems,37

Internat. J. Robust Nonlinear Control 18 (4–5) (2008) 399–413.38

[25] L. Fridman, A. Levant, J. Davila, Observation of linear systems with unknown inputs via high-order sliding-modes, Int. J. Syst. Sci. 38 (10) (2007)39

773–791.40

[26] B.P. Molinari, A strong controllability and observability in linear multivariable control, IEEE Trans. Automat. Control 21 (5) (1976) 761–764.41

[27] F. Bejarano, L. Fridman, A. Poznyak, Exact state estimation for linear systems with unknown inputs based on hierarchical super-twisting algorithm,42

Internat. J. Robust Nonlinear Control 17 (18) (2007) 1734–1753.43

[28] L. Fridman, A. Levant, J. Davila, High-order sliding-mode observation and fault detection via weakly unobservable subspace reconstruction, in:44

European Control Conference 2007, Kos, Greece, 2007, pp. 5139–5146.45

[29] A. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988.46

[30] M. Hautus, Strong detectability and observers, Linear Algebra Appl. 50 (1983) 353–368.47

[31] C. Chen, Linear Systems: Theory and Design, Oxford University Press, New York, 1999.48

[32] H.L. Trentelman, A.A. Stoorvogel, M. Hautus, Control Theory for Linear Systems, Springer-Verlag, London, Great Britain, 2001.49

High-order sliding-mode observation for linear systems with unknown inputs

Documents

Transcript of High-order sliding-mode observation for linear systems with unknown inputs