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ON SENSOR FAULTS ESTIMATION USING SLIDING MODE

OBSERVERS

Anna Filasova and Dusan Krokavec

Abstract This paper discusses the problem of designingthe sliding-mode-based sensor faults estimation in a generalstructure suitable on the actuator as well as sensor faultsdetection and estimation. The problem addressed is indicatedas an unified algebraic approach giving sufficient conditionsof solution. Lyapunov inequality implying from two linearmatrix inequalities are outlined to posses a stabile solutionfor the modified optimal estimator parameters in the standardestimator structure. An example is presented to explain theprocedures for the execution of a sensor fault estimation andto illustrate the properties of the proposed design method inthe continuous time system.

Index Terms Fault detection, fault estimator, sliding mode,Lyapunov inequality, convex optimization, linear matrix in-equalities.

I. INTRODUCTION

The essential aspect for designing the fault-tolerant control

requires a complete diagnosis procedure capable of solving

the fault detection and isolation problem. This procedure

composes the residual signal generation followed by its

evaluation within the decision-making process. The residuals

are derived from implicit information in the functional or an-

alytical relationships, which exist between the measurements

taken from the process and the data obtained from a process

model. The overall procedure of fault detection using thestate estimation principle covers the residuals generation by

the state observers and their evaluation. Here, the estimation

error, or some function of it, is used as a fault residual. The

standard design approach can be found e.g. in [4], [7], [12].

A special position among observer-based methods is oc-

cupied by the sliding mode observers. Essentially, the sliding

mode observer uses discontinuous control action to drive

the observer error trajectory toward a specific hyper-plane

in the error space, and then the trajectory is maintained to

slide on this until the origin of the state space is reached.

If a sliding-mode is formulated using a Lyapunov approach

then it guarantee that the observer error first reaches the

prescribed sliding mode in finite time from any initial state,

and then remains on it there after by a discontinuous control.

A complete Lyapunov stability standpoint for the design of

sliding observers was presented in [8]

Supported by existence of linear matrix inequality (LMI)

solvers, the intention is to reformulate given problem and

The work presented in this paper was supported by VEGA, Grant Agencyof Ministry of Education and Academy of Science of Slovak Republic under

Grant No. 1/0328/08. This support is very gratefully acknowledged.A. Filasova, as well as D. Krokavec are with the Department of

Cybernetics and Artificial Intelligence, Faculty of Electrical Engineer-ing and Informatics, Technical University of Kosice, Kosice, Slovakia,anna.filasova@tuke.sk, dusan.krokavec@tuke.sk

optimize it over LMI constraints [3], [10], [17]. In that

sense in [18] was proposed the sliding mode observer design

method detection, on which design condition in terms of LMI

were given. Of course, sliding mode observers of another

structure are proposed to explore in fault detection and

reconstruction (e.g. see [11], [15], [16]).

The work presented here is still limited to the assumption

presented in [18], and its modification given in [6]. The main

contribution is to present the design method as a modification

of above mentioned principle for sensor faults estimation in

continuous-time linear MIMO systems. To maintain the same

formalism, Lyapunov inequality is used only as stability

condition, and demonstrating the application suitability based

on the unified algebraic approach [17]. Two standard forms

of the linear matrix inequalities are used to posses the

sufficient condition for the observer gain matrix solution

existence, as well as a new observer gain matrix structure is

introduced to solve the potentially feasible task for a sensor

fault estimation in given sliding mode estimator structure.

The necessary modifications was motivated in the sense

to obtain by the sensor faults un-destroying sliding mode

regime. The method in here presented form enables extension

considering bounded real lemma conditions, as well as to

design the active sensor fault control reconfiguration. To thebest of authors knowledge, the modification presented has

not yet been fully investigated in this field.

II. SYSTEM MODEL

The formulation and study of the concepts of the sliding-

mode fault observer design is based in the next on the system

canonical model which is described as

q(t) = Aq(t) + B(u(t) + fa(t)) (1)

y(t) = Cq(t) + fs(t) (2)

where q(t) IRn, u(t) IRr, and y(t) IRm, are thestate, the input and the output vector variables, respectively,

matrices A IRnn, B IRnr , and C IRmn arereal matrices. Without loss of generality it is assumed, for

simplicity that

A =

A11 A12A21 A22

, B =

0

B2

, C =

0 L

(3)

where A11 IR(nm)(nm), B2 IR

mm is a non-

singular matrix, and L IRmm is an orthogonal matrix,i.e. LTL = Im. (The results of [9] implies the existenceof a nonsingular transform matrix to have this structure, and

one way to obtain it is given in the Appendix). The bounded

2010 Conference on Control and Fault Tolerant SystemsNice, France, October 6-8, 2010

WeA2.2

978-1-4244-8154-5/10/$26.00 2010 IEEE 44

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real signals fa(t) IRr , fs(t) IR

m represent actuator and

sensor faults, respectively.

Throughout the paper it is assumed the pair (A,C) is

observable, and the memory-less output controller is used,

where m = r.

III . SLIDING MOD E OBSERVER

The state observer associated to the system (1), (2) is

considered in the form

qe(t) = Aqe(t) + Bu(t) + J(y(t) ye(t)) + H(t) (4)

ye(t) = Cqe(t)) (5)

where qe(t) IRn, ye(t) IR

m are the state and the output

vector variable estimate, respectively, (t) IRm is the feed-forward output error injection signal, J IRnm is the esti-mator gain matrix, and H IRnm is the design parametermatrix satisfying condition rank(CH) = rank(B) = m.This condition is necessary for the existence of the unique

equivalent control, and the independence of the sliding mode

dynamics from the fault signals.

The problem is to design a parameter H that sliding mode

estimator provides asymptotic faults tracking estimation de-

coupled from sliding mode dynamics.

Assembling (1), (2) with (4), (5) after some algebraic

manipulations it can be obtainedq(t)

qe(t)

=

A 0

J C AJC

q(t)

qe(t)

+

+

B B 0 0

B 0 J H

u(t)

fa(t)fs(t)(t)

(6)

It is straightforward to show using the state transformationq(t)

eq(t)

= Te

q(t)

qe(t)

, Te = T

1e =

I 0

I I

(7)

that with the notation

A=

I 0

I I

A 0

JC AJC

I 0

I I

=

=

A 0

0 AJ C

(8)

B=

I 0

I I

B B 0 0

B 0 J H

=

=

B B0 0

0 B J H (9)

respectively, and as a result (6) can be reformulated as

q(t) = Aq(t) + Bu(t) (10)

where

eq(t) = q(t) qe(t), ey(t) = Cqe(t) (11)

q(t) =

qT(t) eTq (t)

(12)

u(t) =

uT(t) fTa (t) fTs (t)

T(t)

(13)

The equation (10) is formulated in such a way that the sliding

manifold design conditions can now be derived.

A. Sliding Mode Motion

Theorem 3.1: Considering the error dynamics (10), then

there exists a stable sliding motion that is independent of

fTa (t) and fTs (t) if there exists a symmetric positive definite

matrix P > 0, P IRnn such that the inequalities

P = PT > 0 (14)

CT(P A + ATP)CTT < 0 (15)

conditioned by the equalities

CTP B = 0, H = P1CTCB (16)

BTP B = (CB )TCB , CTP J = 0 (17)

are satisfied, where CT is the orthogonal complement to

CT.

Proof: To obtain the state error estimate stability

condition the Lyapunov function can be defined as follows

v(eq(t)) = eTq(t)P eq(t) > 0 (18)

where P = PT > 0. Then evaluating derivative ofv(eq(t))with respect to t it can be obtained

v(eq(t)) = eTq (t)P eq(t) + e

Tq (t)Peq(t) < 0 (19)

Since (10) implies

eq(t) = (AJC)eq(t) + Bfa(t) J fs(t) H(t) (20)

then inserting (20) into (19) gives

eTq (t)Peq(t) < 0 (21)

where

eTq (t) =

eTq (t) fTa (t) f

Ts (t)

T(t)

(22)

0 > P =

=

P(AJC)+ (AJC)TP P B PJ PH 0 0 0

0 0

0

(23)

(Hereafter, denotes the symmetric item in a symmetricmatrix).

Defining the congruence transform matrices

Tq =

I

I I

I

0

, Tq =

I I

I I

0

0

(24)

then pre-multiplying left-hand side of (23) by Tq , and right-

hand side of (23) by TTq gives

0 > P = TqPT

Tq =

=

P(AJ C)+ (AJ C)TP P(BH) P J 0 0

0

(25)

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Since (25) can be rewritten as P A + A

TP P(B H) 0 0 0

0

P

0

0

JC 0 I

CT

0

I

JT

P 0 0

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where 0 < Q = QT IR(nm)(nm) is a positive definitematrix. Then the derivative of v(1(t)) with respect to t isas follows

v(1(t)) = T1 (t)Q

11(t) + T1 (t)Q

11(t) < 0 (49)

Since

1(t) = A111(t) (50)

inserting (50), (42) into (49) the following holds

Q1CTP AC

TT(CTP CTT)1+

+(CTP CTT)1CTATP CTTQ1 < 0(51)

CTP ACTT(CTP CTT)1Q+

+Q(CTP CTT)1CTATP CTT < 0(52)

respectively. Setting

Q = (CTP CTT)1 (53)

(52) implies (15). That is if the sliding motion is stable the

sliding mode observer is stable.

Corollary 3.2: Since L in (3) is a square matrix of full

rank, the orthogonal complement to CT takes form

CT =

E 0

(54)

where E IR(nm)(nm) is a non-zero matrix. Then using(3), (54) the condition (16) takes form

CTP B =

=

E 0 P11 P12

P21 P22

0

B2

= EP12B2 = 0

(55)

where P11 IR(nm)(nm), and a trivial solution can be

obtained if the weighting matrix P of Lyapunov function is

block-diagonal.

Considering (55) let P > 0 is a block diagonal matrix ofthe form

P = diag

X W

(56)

0 < X= XT IR(nm)(nm), 0 < W= WT IRmm.Substituting, (3), (56) into (17) yields

BTP B =

0 BT2 X 0

0 W

0

B2

=

= BT2 W B2 = (CB )TCB = BT2 L

TLB2

(57)

It is obvious that (57) implies

P = diag

X LTL

= P = diag

X Im

(58)

since L is orthogonal.

C. Design Condition

As noted before, sliding mode estimator parameter design

condition can be described in the form of matrix inequalities

but the conditions (14), (15) cannot be directly used to com-

pute the sliding-mode observer parameter J. Summarizing,

it can be proved the following results.

Theorem 3.3: Considering that P with the diagonal block

structure as in (58) is a solution of (14), (15), then anysolution of J exists if there exist a symmetric positive

definite matrix R > 0, R IRmm and a matrix Y IRmm such that

F RFTM F R+ GTYT

R

< 0 (59)

where

FT =

BTP 0 0, G =

C 0 I

(60)

M =

P A + ATP 0 0

0 0

0

< 0 (61)

The observer gain matrix is then given by (35).

Proof: It can be seen that (26), (35) has an open form

0 >

P A + A

TP 0 0

0 0

0

P B0

0

YC 0 I

C

T

0

I

YTBTP 0 0

(62)

To solve the problem, using (60), (61) inequality (62) can be

written as

F Y G + GT

Y

T

F

T

M < 0 (63)Now there exists a positive definite symmetric matrix R> 0such that

F Y G + GTYTFT M + GTYTR1Y G < 0 (64)

In this case, after completing the square it can be obtained

(F R+GTYT)R1(F R+GTYT)TF RFTM

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Remark 3.2: A conservative solution can be obtained us-

ing the congruence transform matrix Tq as defined in (24),

where the obtained conditions are closest to those presented

in [6]. It is evident that problem of regularization of (23)

can be solved using bounded real lemma principle as it is

presented in [9].

The feed-forward output error injection signal (t) can be

chosen to be discontinuous regulation policy as

(t) = (t)sign(W ey(t)) = (t)ey

ey(t)(68)

where (t) is a gain high enough to enforce the slidingmotion.

IV. ILLUSTRATIVE EXAMPLE

As a more specific the next system model is used for

demonstration, where the system matrices A0, B0, and C0are given below

A0 =

0 1 0

0 0 15 9 5,B0 =

1 3

2 11 5,C

T

0 =

1 1

2 11 1

Defining the transform matrix T1 such that [13]

T11 =

Inm 0

C0

=

1 0 01 2 1

1 1 1

B1 = T11 B0 =

1 37 10

5 9

=

B11B12

T

1

2 = Inm B11B112

0 Im

=

1 0.4615 0.8462

0 1 00 0 1

then with T1c = T12 T

11

A = T1c A0Tc, B = T1c B0, C = C0Tc

the canonical standard matrix parameters are as follows

A =

0.0769 2.1124 0.14202.0000 4.0769 0.3077

1.0000 3.5385 0.8462

, B =

0 07 10

5 9

C =

0 1 00 0 1

and the null space of C gives

CT =

1 0 0

Solving with Self-Dual-Minimization (SeDuMi) package for

Matlab [9] the inequalities (14), (15) for the LMI variable

P, and subsequently the inequality (59) for Y conditioned

by design parameter R= 103I2 the problem was feasiblegiving the solution

P=

1.57541

1

, Y=

0.0070 0.00500.0100 0.0090

Thus, the estimator gain matrix is

J= BY =

0.0000 0.00000.1497 0.1256

0.1256 0.1065

and gives the stable estimator system matrix

Ae = A JC = 0.0769 2.1124 0.14202.0000 4.2266 0.1821

1.0000 3.6640 0.9526

(Ae) =

1.2527 2.0018 0.3910 i

where (Ae) is the eigenvalue spectrum of the matrix Ae.To compare, using (14), (15) together (67) the next stable

eigenvalue spectrum was obtained

(Ae) =

0.7867 6.1762 8.1470

However, the purpose of this example is only to illustrate this

method and does not address issues of numerical stability.

The simulations show the fault at the first sensor, as well astheir respective reconstruction, which visually are identical to

the fault. This shows that the method presented is successful.

V. CONCLUDING REMARKS

Based on Lyapunov function and the standard sliding

mode observer structure the modified principle of the sensor

fault estimation is presented in the paper. The design condi-

tions are derived in the terms of numerical optimization over

LMI constraints using the new structured LMI variables. The

obtained formulation is a convex LMI problem for a full

order estimator design and manipulates the estimator state-

space description matrix parameters in the system canonical

form to apply in continuous-time linear systems.

In particular, with the use of Lyapunov method, it was

shown how to adapt the standard approach to design the

matrix parameters of the sliding-mode-based sensor fault

estimator. The procedure was derived in such way as the

design problem boils down to solving a special structured

problem under LMI constraints. The presented illustrative

example confirms the effectiveness of used techniques.

In the application context, an advantage of the proposed

method is that allows to work directly with triple (A,B,C) in

the optimization over LMI constraints thus allowing the ex-

ploitation of the structural properties which occur frequently

in many practical applications.Note, the inclusion of an actuator fault model into the

state-space equation (1) of the system was motivated by the

structure of the sliding mode switching surface ey(t) = 0,when an actuator fault doesnt destroy the sliding mode, as

(41) implies.

APPENDIX

Let state description of the system (1), (2) with r = m is

q0(t) = A0q0(t) + B0u(t) (A.1)

y(t) = C0q0(t) (A.2)

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.5

1

0.5

0

0.5

1

time [s]

y1(t)

y1e

(t)

Fig. 1. Estimation of the first sensor fault

Defining the transform matrix T11 such that

C1 = C0T1 = 0

Im, T

1

1 = Inm 0

C0

(A.3)

then

B1 = T11 B0 = T

11

B01B02

=

B01

C0B0

=

B11B12

(A.4)

If C0B0 = B12 is a regular matrix (in opposite case thepseudoinverse of B12 is possible to use), then the second

transform matrix T12 can be defined as follows

T12 =

Inm B11B

112

0 Im

, T2 =

Inm B11B

112

0 Im

(A.5)

This results in

B = T12 B1 =

Inm B11B

112

0 Im

B11B12

=

0

B2

(A.6)

where

B11 = B01, B2 = B12 = C0B0 (A.7)

and

C= C1T2 =

0 Im Inm B11B112

0 Im

=

0 Im

(A.8)

Finally, with T1c = T12 T

11 it yields

A = T1c A0Tc = T12 T

11 A0T1T2 (A.9)

Thus, (A.6), (A.8), and (A.9) represent the system canonical

model.

Note, the structure of T11 is not unique and others can

be obtained by permutations of the first nm rows in thestructure defined in (A.3) (e.g. see [13]).

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