Page 1

Simultaneous Sensor and Actuator Fault Reconstruction and Diagnosis

using Generalized Sliding Mode Observers

R. Raoufi†∗and H. J. Marquez‡

Abstract—A new filter for state and fault estimation in a class

of nonlinear systems is presented in this paper. The observer

benefits from both sliding mode control and singular systems

theory. The novelty of this approach is based upon dealing with

systems prone to faults at sensors and actuators during the

course of the system’s operation coincidentally. Conditions and

proofs of conversion for the proposed observer are presented.

A noticeable feature of the proposed approach is that the state

trajectories do not leave the sliding manifold even in presence

of sensor/actuator faults. This allows for actuator faults to

be reconstructed based upon information retrieved from the

equivalent output error injection signal. Due to employing a

generalized state space form (singular system theory), the sensor

faults are also estimated.

I. INTRODUCTION

It is often the case when dealing with complex systems

requiring safe operation, that some form of supervisory

function is needed to indicate undesired process states or

“faults”. Faulty signals can exist in actuators, sensors and

process components that can deteriorate normal operation or

even lead to instability. Taking immediate and appropriate

actions in order to preserve safe operation while avoiding

possibly catastrophic damages is crucial. Thus, fault

detection and isolation (FDI) is of significant technical

importance. Model-based FDI schemes employ measured

variables and use mathematical models of the system

to detect abnormal conditions. Once a fault is detected,

diagnosis and isolation is performed, and decisions and

counteractions are then taken. Model-based FDI schemes

have proved effective in many successful implementations;

(see [1] and the references therein). In particular, observer-

based techniques for FDI have drawn significant attention

(see for example [2], [3], [4], [5], [12] and references

therein). This paper deals with observer design for fault

detection and estimation using sliding mode observers

(SMOs) in a generalized state space form inspired by

singular system theory. Due to their ability to cope with

model uncertainties, SMOs offer great potential in fault

detection applications [7].

[8] introduced the use of the sliding mode approach in

observer design and used the Lyapunov theory to prove

stability. [6] proposed an alternative approach to the design

of a sliding mode observer using a discontinuous sliding

∗Corresponding author. Email: Raoufi@ece.ualberta.ca

†Department of Electrical and Computer Engineering, The University of

Alberta, Alberta, Canada.

‡Department of Electrical and Computer Engineering, The University of

Alberta, Alberta, Canada.

term fed back through a suitable gain. SMOs for linear

unknown input systems were studied in [9]. [10] proposed

a canonical sliding observer form design for linear system

where a sufficient condition for stability based on linear

matrix inequalities (LMIs) was derived. Observation of

linear systems with unknown inputs via high-order sliding-

mode was addressed in [16] and [17]. More recently,

development of sliding mode observers for unknown input

systems was proposed in [18]. A robust fault detection

method for nonlinear systems with disturbances was studied

in [2] where strict geometric conditions where exploited.

Applications of SMO for fault tolerant control of linear

systems were addressed in [11] and [19].

The main focus of this paper is to explain the design

of sliding mode observers for the problem of fault

reconstruction and FDI. Recently, [5] and [7] proposed

that using the variable structure control law of the SMO

and the concept of equivalent output injection, a fault

can be reconstructed to any required accuracy for linear

systems. Sensor fault reconstruction using SMO was studied

in [15], further extending the results for linear systems

with disturbance and uncertainty. For nonlinear Lipschitz

systems, [13] addressed SMO based fault reconstruction by

assuming that disturbances are matched and can be lumped

into the so-called matching condition. Inspired by the

theory of singular systems, [21] proposed a new generalized

state-space observer design to estimate unknown signals

for a class of nonlinear systems. In [12] an interesting

method to design descriptor observers for systems with

measurement noise and application to sensor fault diagnosis

was proposed. State/noise observer for descriptor systems

with application to sensor fault diagnosis was also studied in

[22]. The approaches of these references play an important

role in inspiring our observer design.

Fault reconstruction is excellent for directly isolating the

flaws within a system by revealing which sensor or actuator

is faulty and is useful for diagnosing incipient and small

faults. The detailed knowledge of the fault’s shape, obtained

from fault reconstruction, can highly facilitate the fault

tolerant control design. However, in practical systems, it is

often the case where actuators and also sensors suffer from

faults during the course of the system’s operation. Both

Actuators and Sensors can suffer from faults either alone,

at separate times or simultaneously. In this case, detection

and reconstruction of all faults is highly important. The

co-existence of unknown fault at both some sensor(s) and

2010 American Control Conference

Marriott Waterfront, Baltimore, MD, USA

June 30-July 02, 2010

FrC21.5

978-1-4244-7425-7/10/$26.00 ©2010 AACC7016

Page 2

actuator(s) has not been addressed in any earlier design

of the sliding mode observers or other fault reconstruction

schemes. Clearly the sensor fault corrupts measurement,

therefore making it harder to reconstruct the actuator faults.

On the other hand, reconstruction of the sensor faults also

remain challenging and unsolved due to faulty actuators.

Thus, fault reconstruction/identification brings important

benefits to the system and thus in this paper, we aim at

reconstructing the faults when they coexist at sensors and

actuators during the system operation.

What is unique about our approach is that it involves a new

design of a robust sliding mode observer in a generalized

state space form when faults occur at both sensors and

actuators coincidentally. To cope with the sensor faults,

inspired in [21], a generalized state space form is employed

such that the augmentation results in a descriptor system

form. This singular (descriptor) formulation, provides the

possibility for sensor fault estimation. In this singular form,

we define a sliding surface and the proposed filter forces the

trajectories of the estimation error to approach the sliding

surface and remain there afterwards. The actuator fault

satisfies the so-called matching condition and is targeted

by the sliding mode controller for reconstruction. These

features allow for actuator faults to be reconstructed based

on information retrieved from the equivalent output error

injection signal. Thus, not only the actuator faults but

also sensor faults are reconstructed. In addition, the states

of the system are also estimated by the proposed robust

observer which is very important due to the corruption

of the measurements by sensor faults. As a result, the

proposed observer will be called the generalized sliding

mode observer (GSMO) here on after.

II. PRELIMINARIES AND ASSUMPTIONS

Consider a dynamical system affected by sensor and actuator

faults of the form:

?

where x ∈ Rnrepresents the system state, u ∈ Rmthe

control input, y ∈ Rpthe measured system output and t ∈

R+. Throughout this paper we assume that p > m. The

set (A,B,C,D,BΦ) is of real constant known matrices of

appropriate dimensions where D ∈ Rp×(p−m)and (A,C)

is an observable pair. fa(t) : R+→ Rmdenotes the fault

(unknown input) that is bounded in the Euclidean norm as

˙ x(t) = Ax(t) + B(u(t) + fa(t)) + BΦΦ(x,t)

y(t) = Cx(t) + Dfs(t)

(1)

?fa(t)? ≤ ρ.

(2)

The function fs(t) : R+→ Rp−mis the sensor fault where

D is the corresponding distribution matrix with full columns

rank. Therefore, without loss of generality, we can assume

there exist a nonsingular change of coordinate S0(yet to be

designed) which provides the following geometric condition

associated with D:

?

S0D =

0

D2

?

(3)

where D2?= 0, D2∈ R(p−m)×(p−m)and is invertible. This

assumption implies that certain sensors are prone to fault

and not all of them. The known nonlinear function Φ(x,u,t)

satisfies a Lipschitz condition locally on a set D ⊂ Rnin

which

?Φ(x1,u,t) − Φ(x2,u,t)? ≤ LΦ?(x1− x2)?

where x1,x2 ∈ D and LΦ ∈ R+is a known positive

constant called Lipschitz gain or Lipschitz constant [23]. If

D = Rn, the function Φ is said to be globally Lipschitz.

Throughout this paper, we assume that the system to be

addressed is at least locally Lipschitz on a set D. We make

the following two well-known assumptions:

(4)

Minimum Phase Condition: For every complex number s

with nonnegative real part

?

Matching Condition: Assume that there exists an arbitrary

matrix F ∈ Rm×pand P = PT> 0 ∈ Rn×nsatisfying

BTP = FC

rank

sIn− A

C

B

0

?

= n + rank(B)

(5)

(6)

III. SOME PRELIMINARY LEMMAS

Lemma 1. [20] Consider system (1). There exists a solution

P = PT> 0 such that BTP = FC if and only if

rank(CB) = rank (B)

(7)

Lemma 2. Given the system (1) with rank(CB) = rank (B)

and associated assumption (5), there exist nonsingular trans-

formation matrices T and S such that

?

SCT−1=

0C4

TAT−1=

A1

A3

C1

A2

A4

0

?

?

, TB =

?

?

B1

0

0

D2

?

?

, TBΦ =

?

BΦ1

BΦ2

?

,

?

, SD =

(8)

A1 ∈ Rm×m, A4 ∈ R(n−m)×(n−m), C1 ∈ Rm×m,

C4∈ R(p−m)×(n−m); B1, C1and D2are invertible.

Proof. See Appendix.

Lemma 3. [9] Consider system (1) and assume that

rank (CB) = rank (B). Then the pair (A4,C4) is detectable

if and only if

?

for all s such that Re(s) ≥ 0.

Lemma4.

Considersystem

rank (CB) = rank (B),then

(6) always holds in the new coordinates of Lemma 2.

rank

sIn− A

C

B

0

?

= n + m

(9)

(1)and

matching

assumethat

thecondition

Proof. See Appendix.

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IV. SMO DESIGN: A DESCRIPTOR APPROACH

Due to Lemma 2, system (1) in the new coordinates ˜ x :=

(xT

1,xT

2)T= Tx and ˜ y := (yT

1,yT

2)T= Sy is

?

˙ x1= A1x1+ A2x2+ B1(u + fa(t)) + BΦ1Φ(T−1˜ x,t)

y1= C1x1

(x1∈ Rm)

?

We partition S as

?

So that the variable x1 can be obtain from the measured

output y by

x1(t) = C−1

(10)

(11)

˙ x2= A3x1+ A4x2+ BΦ2Φ(T−1˜ x,t)

y2= C4x2+ D2fs(t)(x2∈ Rn−m)

S =

S1

S2

?

,S1∈ Rm×p,S2∈ R(p−m)×p

1S1y(t).

(12)

First, we employ the following state augmentation

?

which leads to the following descriptor plant with singular

E ∈ R(n+p−2m)×(n+p−2m)with rank(E) = n − m

?

where

?

?

Subsystem (10) is rewritten as

x3=

x2

fs

?

, x3∈ Rn+p−2m

E ˙ x3= A4x3+ A3x1+ BΦ2Φ(T−1˜ x,t) + Fsfs(t)

y2= C4x3

(13)

E =

In−m

0

0

0

?

, A4 =

?

A4

0

0

−Ip−m

?

?

, A3 =

?

A3

0

?

BΦ2 =

BΦ2

0

?

,Fs =

?

0

Ip−m

, C4 = [C4 D2].

˙ x1= A1x1+ A2x3+ B1(u(t) + fa(t)) + BΦ1Φ(T−1˜ x,t)

where A2=?A2 0m×(n−m)

We now put forward the following important Lemma.

(14)

?.

Lemma 6. The inverse (E + V C4)−1exists for some gain

V ∈ R(n+p−2m)×(p−m).

Proof. See Appendix.

Due to the structure of V , the following useful properties are

followed

(E + V C4)−1=

?

In−m

−D−1

0

2C4

D−1

2V−1

2

?

(15)

(E + V C4)−1Fs=

?

0

D−1

2V−1

2

?

(16)

C4(E + V C4)−1V =

?

[C4 D2]

In−m

−D−1

0

2C4

D−1

2V−1

2

??

0

V2

?

= In−m.

(17)

We will employ the system structures (10) and (13) in the

new observer design. Consider the following generalized

sliding mode observer structure

(E + V C4)˙ z = A3ˆ x1+ (A4− L4C4)z + L3(y1− C1ˆ x1)

+A4(E + V C4)−1V y2+ BΦ2Φ(T−1ˆ˜ x,t)

(18)

(19)

ˆ x3= z + (E + V C4)−1V y2

˙ˆ x1= A1ˆ x1+A2ˆ x3+L1(y1−C1ˆ x1)+B1ν(t)+BΦ1Φ(T−1ˆ˜ x,t)

where the novel reduced-order sliding gain structure ν ∈ Rm

and the observer gain˜L are respectively

?

0

?

where ρ0 is some positive scalar. P1, P3 and K will be

determined through the stability proof and A1= A1− As

where As

(20)

ν(t) =

(ρ + ρ0)

BT

?BT

1P1(C−1

1P1(C−1

1

S1y−ˆ x1)

S1y−ˆ x1)?

1

: C−1

: otherwise

1S1y − ˆ x1 ?= 0

(21)

?

(22)

˜L :=

L1

L3

L2

L4

?

:=

?

A1C−1

A3C−1

1

0

1

(E + V C4)P−1

3 K

,

1

1represents a stable design matrix.

Remark. In [13] and [14] and references there in, the sliding

mode controller ν has the following general form:

ν(t) = (ρ + ρ0)

P0(y − Cˆ x)

||P0(y − Cˆ x)||, P0 = PT

0,P0 > 0,ρ,ρ0 > 0

with order p. The novelty of the proposed sliding mode

controller ν(t) in (21) is that under the same assumptions,

it requires just some components of the output y(t).

Consequently its order is equal to m < p. Due to this

reduced-order structure of ν(t), it is possible to tackle

a class of sensor faults at the output for the case where

(p − m) sensors are prone to fault.

We now present Theorem 1 which is the main result of this

section.

Theorem 1. Given the nonlinear uncertain system (1) with

assumptions (2)-(6), consider the GSMO structure (18)-(22).

The observer error dynamics is ultimately bounded with an

arbitrary small upper bound if there exist matrices K, PT

P1 > 0 and PT

feasibility problem has a solution:

P1> 0, P3> 0 and

where

1=

3 = P3 > 0 such that the following LMI

AsT

1 P1+ P1As

A

BT

1+ LIm

P1A2

P1BΦ1

P3ΛBΦ2

−I

T

2P1

Φ1P1

M22+ LIn+p−2m

B

T

Φ2ΛTP3

< 0

(23)

L = LΦ2??T−1??2

(24)

Λ = (E + V C4)−1

(25)

M22= P3ΛA4− KC4− A

T

4ΛTP3− C

T

4KT

(26)

7018

Page 4

Proof. See Appendix.

From the proof of Theorem 1, it can be easily verified that

Thus the upper bound of the error can be significantly

dropped by arbitrarily choosing a high-gain V2.

Ωε=

˜ e : ?˜ e? < 2ω0

??????

P3

0

D−1

λmin(Q)

2V−1

2

??????

+ ε0,ε0> 0

(27)

.

V. IDEAL SLIDING MOTION AND FAULT

RECONSTRUCTION

It is well-known that, in order to confine a stable motion of a

dynamical system onto a sliding surface S, it is necessary to

use a switching gain which is discontinuous about the surface

S [7]. Therefore, due to the structure of the switching gain

(21) and the fact that N(B1) = {∅}, it follows that

S = {t ∈ R+: s(t) = 0 | s(t) = C−1

The error system with respect to the new coordinates can be

written as in (62) and (59). If the feasibility LMI problem in

Theorem 1 is solvable, then it implies that the error dynamics

is ultimately bounded by an arbitrary small upper bound

subject to (27). For simplicity, define

????P3

Consider the Lyapunov function Vs=1

1S1y − ˆ x1}.

(28)

ε := 2ω0

?

0

D−1

λmin(Q)

2V−1

2

?????

2sTP1s. We obtain

+ ε0.

(29)

˙Vs= sTP1˙ s

= eT

≤ ?P1e1?(?As

≤ ?BT

Choose the gain ρ0to satisfy

1P1(As

1e1+ A2e3+ BΦ1eΦ+ B1(fa(t) − ν))

1e1+ A2e3+ BΦ1eΦ?) − ρ0?BT

1P1e1?(?B−T

1P1e1?

1

?(?As

1? + ?A2? + ?BΦ1?)ε − ρ0).

(30)

ρ0≥ ?B−T

Therefore it follows that˙Vs < −η0?B1??λmin(P1)√Vs.

satisfied and an ideal sliding motion will take place on the

surface S and after some finite time ts

e1= ˙ e1= 0, ∀t > ts.

The subsystem (62) in sliding mode is given by

1

?(?As

1?+?A2?+?BΦ1?)ε+η0, η0> 0. (31)

This proves that the well-known reachability condition [7] is

(32)

0 = A2e3+ B1(f − νeq) + BΦ1eΦ

where in the sliding mode the discontinuous signal ν in (21)

must takes on the average νeq(referred to as the equivalent

output error injection [6]) to preserve the sliding motion.

Thus

?νeq− fa(t)? ≤ κ

where

κ = ?B−1

(33)

(34)

1A2+ B−1

1BΦ1?ε

(35)

Therefore, approximately, for some small κ

νeq≈ fa(t).

(36)

And based on the concept of equivalent output error injection

[6], the signal νeq can be approximated to any degree of

accuracy by

νeq≈ (ρ + ρ0)

BT

1P1(C−1

1P1(C−1

1S1y − ˆ x1)

1S1y − ˆ x1)? + δ

?BT

(37)

where δ is a small positive scalar to smooth out the signal

ν [7]. Therefore

ˆfa(t) = (ρ + ρ0)

BT

1P1(C−1

1P1(C−1

1S1y − ˆ x1)

1S1y − ˆ x1)? + δ.

?BT

(38)

Next, From (27) and (29), it follows that ?x3− ˆ x3? ≤ ε.

Thus

?fs(t) −ˆfs(t)? ≤ ε

(39)

and for some small ε, we can directly conclude that

ˆfs(t) ≈ fs(t).

(40)

It should be pointed out that upper bound of the estimation

error ε is arbitrarily reduced by the choice of a high-gain V2.

VI. CONCLUSION

This paper presents a generalized sliding mode observer

(observer) based fault estimation approach for nonlinear

Lipschitz systems when both sensor and actuator faults exist

coincidentally during the course of the system’s operation.

The approach utilizes a sliding mode observer in a new

generalized state space form. Reconstruction of the actuator

and sensor faults and estimation of system states were

achieved simultaneously.

APPENDIX

Proof of Lemma 2. We have rank (B) = m, therefore with-

out loss of generality, by using a nonsingular transformation

T0, we partition the matrix B as

?

where B1∈ Rm×mwith rank (B1) = m. Now, introduce

a nonsingular coordinate transformation T1as

?

then

?

T0B =

B1

B2

?

(41)

T1=

Im

0

−B2B−1

1

In−m

?

(42)

T1T0B =

Im

0

−B2B−1

1

In−m

?

.

?

B1

B2

?

=

?

B1

0

?

(43)

7019

Page 5

where B1 ∈ Rm×mis nonsingular. Now we partition

CT−1

1

as CT−1

1

= (C1 C4). Therefore

0 T−1

0 T−1

CB = CT−1

0 T−1

1 T1T0B = (C1 C4)

?

B1

0

?

= C1B1

(44)

and consequently, using rank(CB) = rank (B) we have

rank(C1B1) = rank(CB) = rank(B1),

then from the nonsingularity of matrix B1, we directly

conclude that

rank(C1) = m.

(45)

Without loss of any generality, using the proper nonsingular

change of coordinate S0, we partition C1as follows

S0C1=

?

C1

C21

?

(46)

where C1∈ Rm×mand

rank(C1) = m.

(47)

Consequently det (C1) ?= 0. Let

S1=

?

Im

0

−C21C−1

1

Ip−m

?

?

(48)

which yields

S1S0C1=

?

C1

0

.

(49)

Letting S = S1S0, we obtain

SCT−1

0 T−1

1

=

?

C1

0

C12

C4

?

?

.

(50)

Let

T−1

2

=

?

Im

0

−C−1

In−m

1C12

(51)

and T = T2T1T0, then it is easy to obtain

TB =

?

B1

0

?

, SCT−1

0 T−1

1 T−1

?

2

?

=

?

?

C1

0

0

C4

?

?

(52)

SD =

?

Im

0

−C21C−1

11

Ip−m

.

0

D2

=

?

0

D2

. (53)

In the new coordinate, the matrices A and BΦ are

transformed as in (8). This completes the proof.

Proof of Lemma 4. Since rank(CB) = rank (B), using

Lemma 2 it follows that there exist nonsingular similarity

transformation matrices T and S providing the decomposed

structure in (8). Therefore, B = T−1˜B and C = S−1˜CT.

Letting P = TT ˜PT and F =˜FS where˜P is symmetric,

substituting P,F,B,C into the matching condition (6) yields

˜BT ˜P =˜F˜C. We select

?

Then, by straight substitution, it follows that the matrix

equality˜BT ˜P =˜F˜C holds, hence (6) holds.

˜P =

P1

0

0

P3

?

˜F =?

B1TP1C−1

1

0

?

(54)

Proof of Lemma 5. We have

?

= n + p − 2m

then there exists gain V of appropriate dimension such that

rank(E+V C4) = n+p−2m, hence (E+V C4) is invertible.

By design, we adopt the following structure for the gain V

?

where V2∈ R(p−m)×(p−m)and is full rank.

Proof of Theorem 1. We define e1= x1− ˆ x1,e3= x3− ˆ x3

and eΦ= Φ(T−1˜ x,t)−Φ(T−1ˆ˜ x,t). From (19), substituting

z = ˆ x3− (E + V C4)−1V y2into (18) yields

Λ−1(˙ˆ x3− ΛV ˙ y2) = A3ˆ x1+ (A4− L4C4)(ˆ x3− ΛKy2)

+L3C1e1+ A4(E + V C4)−1V y2+ BΦ2Φ(T−1ˆ˜ x,t).

rank

E

C4

?

= rank

In−m

0

C4

0

0

D2

= n − m + rank(D2)

(55)

V =

0(n−m)×(p−m)

V2

?

,

(56)

(57)

Adding V ˙ y to both sides of (13) and then subtracting it from

the above equation, one obtains

Λ−1˙ e3= A3e1−L3C1e1+(A4−L4C4)e3+Fsfs(t)+BΦ2eΦ

By choosing L3= A3C−1

o

4e3+ ΛFsfs(t) + ΛBΦ2eΦ

where A

is expressed as

?

consequently, by design, we can choose a high-gain V2 to

reduce the amplification of the sensor fault fs(t) to any

arbitrary low magnitude. Furthermore, notice that ∀s ∈ C+,

?

= rank

0Ip−m

= rank

C4

??

?

?

From Lemma 3, we know (A4,C4) is a detectable pair. Thus

?

which means that the pair (ΛA4,C4) is detectable and we

can choose a matrix L4such that (ΛA4−L4C4) is a stable

7020

(58)

1, it follows that

˙ e3= ΛA

(59)

o

4:= (A4−L4C4). From (16) the sensor fault term

ΛFsfs(t) =

0

D−1

2V−1

2

?

fs(t)

rank

sIn+p−2m− ΛA4

C4

??

?

In+p−2m

0

?

Λ−1

0

??

sIn+p−2m− ΛA4

C4

??

sΛ−1− A4

?

= rank

sK

Ip−m

??

sE − A4

C4

sIn−m− A4

0

C4

?

??

= rank

sE − A4

C4

?

= rank

0

Ip−m

D2

= p − m + rank

sIn−m− A4

C4

.

rank

sIn+p−2m− ΛA4

C4

?

= n+p−2m,

∀s ∈ C+, (60)

Page 6

matrix. Let L4= Λ−1L4, we obtain the stable matrix ΛA

From (10) and (20), we obtain

o

4.

˙ e1= (A1−L1C1)e1+A2e3+B1f−B1ν(t)+BΦ1eΦ (61)

Let L1= (A1− As

˙ e1= As

1)C−1

1, then

1e1+ A2e3+ B1(f − ν(t)) + BΦ1eΦ

Consider V1= eT

(62)

1P1e1and V3= eT

3P3e3. We define

V = V1+ V3.

Then the derivative of V1and V3are given by:

˙V1= eT

+eT

+eT

1(AST

1P1+ P1As

1)e1+ eT

1B

1P1A2e3+ eT

3AT

2P1e1

1P1BΦ1eΦ1+ eΦT

1P1B1(f − ν) + (f − ν)TBT

T

Φ1P1e1

1PT

1e1

(63)

˙V3= eT

+fs(t)TFT

3(A

0T

4ΛTP3+ P3ΛA

sΛTP3e3+ eT

0

4)e3+ eT

3P3ΛBΦ2eΦ+ eT

3P3ΛFsfs(t)

ΦB

T

Φ2ΛTP3e3

(64)

From (12) it follows that e1= C−1

the switching gain (21) and (2) we obtain

1S1y − ˆ x1. Then using

eT

1P1B1(fa(t) − ν(t)) =

1P1B1fa(t) − (ρ + ρ0)?eT

1P1B1? − (ρ + ρ0)?eT

Consequently, by choosing L4= Λ−1P−1

to (63), (64) and (65), the stability criteria ˙V

equivalent to the following inequality

eT

1P1B1?2

?eT

1P1B1?≤

ρ?eT

1P1B1? = −ρ0?eT

1P1B1? < 0

(65)

3 V and with regard

< 0 is

˜ eT

?

AsT

1 P1+ P1As

A

BT

1+ LIm

P1A2

P1BΦ1

P3ΛBΦ2

−I

T

2P1

Φ1P1

M22+ LIn+p−2m

B

??

?T. Thus if −Q < 0,

T

Φ2ΛTP3

?

−Q

˜ e

+eT

3P3ΛFsfs(t) + fs(t)TFT

sΛTP3e3 < 0

where ˜ eT=?

e1

e3

eΦ

˙V ≤ −?˜ e?(λmin(Q)?˜ e? − 2?P3ΛFs?ω0),

and for

?˜ e? ≥ 2λmin(Q)−1?P3ΛFs?ω0

it follows that˙V < 0 which guarantees that the magnitude

of the error is ultimately bounded with respect to the set:

Ωε=

?

˜ e : ?˜ e? <2?P3ΛFs?ω0

λmin(Q)

+ ε0,ε0> 0

?

(66)

where ε0 is an arbitrarily small positive scalar. This

completes the proof.

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