High Order Sliding Mode Observers and Differentiators–Application to Fault Diagnosis Problem

Somefault diagnosis schemes using unknown input observer UIO and conventionalfirst order sliding mode observer SMO adopt this strategy.. One limitation of the existing fault diagnosis sch

Differentiators–Application to Fault Diagnosis Problem

Mehrdad Saif, Weitian Chen, and Qing Wu

School of Engineering Science

Simon Fraser University

require-these capabilities in other complex systems Generally speaking, the term fault

is referred to any disturbances, errors, malfunctions or failures in the functionalunits that can lead to undesirable or intolerable behavior of a system Somefactors that have contributed to automatic fault detection, isolation, and ac-commodation (FDIA) problem to become an active area for research are: 1) theincreasingly sophisticated industrial and consumer goods as a result of advances

in electronics and computer technologies; 2) more interests in FDIA in turing and process industries mainly due to economics and safety reasons; and3) greater concerns over the air pollution and the environment in general

manufac-To ensure the normal operation, and increase the safety and reliability of thecontrol systems in many applications, the problem of fault detection, isolation,identification, and accommodation has received considerable attention over thepast two decades Fault detection signals the occurrence of a fault Fault isola-tion determines the locations and/or type of the fault, and fault identificationspecifies the magnitude of the fault The information provided by the diagnosticsystem could assist in the development of fault accommodation strategies whichwould guarantee fail-safe operation of the control system In recent years thedesign and analysis of fault diagnosis schemes using model-based analytical re-

dundancy approaches has been a subject of many research studies Patton et al [1], Gertler [2], Chen and Patton [3], Patton et al [4] survey some of the works

in this area

Because of the existence of system complexities such as nonlinearities, turbances, and uncertainties in a typical complex control system, fault diagnosis

dis-G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 321–344, 2008.

for such dynamical systems still pose a number of challenging problems Amongstvarious uncertainties, unknown inputs are one type of uncertainty that hasreceived considerable attention To deal with the unknown inputs, robust ap-proaches are often employed Two robust strategies have been developed fordealing with the unknown inputs One is to completely remove their effect Somefault diagnosis schemes using unknown input observer (UIO) and conventionalfirst order sliding mode observer (SMO) adopt this strategy For example, UIO-based schemes can be found in [5, 6, 7, 8], and the SMO-based ones were proposed

by [9, 10, 11, 12, 13, 14, 15] The other strategy is to attenuate the unknown

input effect to some minimum level in certain sense, such as minimizing the H ∞

gain of the unknown inputs Fault diagnosis schemes using this strategy can befound in [16, 17, 18] and the references listed therein Generally speaking, fol-lowing this strategy leads to losing the invariant property to matched unknowninputs

One limitation of the existing fault diagnosis schemes using UIOs or tional first order SMOs is that the relative degrees from the inputs and/or theunknown inputs to the outputs must be one Because many physical systemssuch as satellite control systems, and mechanical systems can not satisfy thiscondition, new fault diagnosis strategies beyond using UIOs or conventional firstorder SMOs are needed One promising strategy is to use the recently developedhigh order sliding mode techniques such as high order sliding mode observersand differentiators

conven-The well known problems with using the conventional first order sliding modesare the relative degree one requirement and the chattering effect In order to dealwith these limitations while preserving the main properties of the conventionalfirst order sliding modes such as finite-time convergence, and robustness withrespect to disturbances, high order sliding modes have been designed for bothcontrol [19, 20], and system state observation [21, 22, 23, 24, 25, 26, 27, 28, 29].Sliding mode observers, which could be used to remove the relative degree onerestriction, were designed in [32, 33] based on the so called equivalent controlconcept with a need for using low-pass filters In order to avoid the use of low-pass filters, high order sliding mode observers based on twisting algorithms wereproposed [21, 22, 23, 24, 25, 26, 27, 28, 29] These high order sliding modeobservers do not require the relative degree from the disturbances to the slidingmanifold to be one, and can totally remove the chattering effect if properlydesigned Because of these two advantages, high order sliding mode observerscan be used for fault diagnosis in systems with relative degrees from the inputsand/or the unknown inputs to the outputs that are greater than one

Based on high order sliding modes, arbitrary-order exact robust tors have also been studied in the literature–see [20] and references cited Theproposed differentiators can provide exact estimation for the derivatives of asignal of any order if there is no measurement noise When noise is present, theestimation errors of the derivatives will be small if the magnitude of the noise issmall These properties make high order sliding mode differentiators appealing

differentia-in fault diagnosis

Although high order sliding mode observers and differentiators have appealingproperties that could be used in fault diagnosis, their great potential has notbeen well recognized in the fault diagnosis community and there are very fewresults in this direction A fault diagnosis scheme using second order slidingmode observers was proposed in [30], while a fault diagnosis using high ordersliding mode differentiators was designed in [31].

In the rest of this chapter, the design of a second order sliding mode observerand a high order sliding mode differentiator is first presented Then, based onthe second order sliding mode observer and the high order sliding mode differen-tiator, two fault diagnosis schemes are proposed Illustrative examples are given

to show the effectiveness of the proposed fault diagnosis schemes Finally, someconclusions are drawn

2 High Order Sliding Mode Observers and Differentiators

In this section, we shall propose a second order sliding mode observer for a class

of nonlinear systems The development here follows that of [23] Then, a highorder sliding mode differentiator developed in the literature will be presented

The nonlinear dynamic systems under study is described in the state spaceform as

in any compact region of x.

Only the scalar case x1, x2∈  is considered because the design method can

be extended to the vector case easily Moreover, it is easy to see that the relative

degree from u to y is two.

For the sake of designing and analyzing a diagnostic observer, the followingassumptions are introduced

Assumption 1 There exist two positive constants k1 and k2 such that

Assumption 2 The uncertainty function ξ(t, x, u) satisfies



dξ(t, x, u) dt  < δξ+

(5)

where ξ+ and δξ+ are two positive numbers

Based on the system dynamics (1), a second order sliding mode observer isproposed as follows

˙ˆx1= ˆx2+ z1 xˆ1(0) = x1

˙ˆx2= f (t, ˆ x1, ˆ x2, u) + z2 ˆx2(0) = 0ˆ

struc-The system dynamics before ˜x1 reaches the sliding manifold is

˙˜x1= ˜x2− λ1|˜x1| 1/2 sign(˜ x1)− v1

˙v1= α1sign(˜ x1)

˙˜x2= F (t, x1, ˆ x1, x2, ˆ x2, u) (9)

where F (t, x1, ˆ x1, x2, ˆ x2, u) = f (t, x1, x2, u) −f(t, ˆx1, ˆ x2, u)+ξ(t, x1, x2, u) Based

on Assumption 1 and 2, we have

for any possible t, x1, ˆx1, x2, ˆx2, and u.

The convergence of a super-twisting second-order sliding mode observer has

been studied by Davila et al in [23], where F (t, x1, ˆ x1, x2, ˆ x2, u) is bounded by

a constant f+ Here, we are ready to use the similar approach to investigate theconvergence of the proposed second-order sliding mode observer (6)-(8), where

F ( ·) only satisfies the condition (10) and (11).

Theorem 1 The first variable pairs (ˆ x1, ˙ˆ x1) converge to (x1, ˙ x1) in finite time,

if the condition (10) holds for system (1), and the parameters of the observer (7) are selected according to the following criteria:

α1> k1x˜1M + ξ+ or α1>

k1˙˜x1 0+ ξ+ (12)

λ1>  4α1

where ˜ x1M and ˙˜ x10 will be defined later.

Proof From (9) and (10), the state estimation errors ˜ x1 and ˜x2 satisfy thedifferential inclusion

Here and in the following part, all differential inclusions are defined in the

Fil-ippov sense Using the identity d |x|/dt = ˙x sign(x), we obtain the derivative of

in Figure 1 Since the initial values of the observer are set to (ˆx1, ˆ x2) = (x1, 0),

the trajectory enters the half-plane ˜x1> 0 with a positive initial value ˙˜ x10= ˜x2

and the half-plane ˜x1< 0 with a negative value of ˜ x2

In quadrant 1 (˜x1 > 0, ˙˜ x1 > 0), the trajectory is confined between the axis

1

1M k

) ,

~ (

1 1 1 1

1

k k M

Then, let’s consider the boundedness curve in quadrant 4 (˜x1 > 0, ˙˜ x1 < 0),

where based on (18), ˙˜x1continues to decrease until ¨x˜1returns back to zero from

a negative value Therefore, the boundedness curve consists of two parts Thefirst part drops down from (˜x1M , 0) to (˜ x1M , ˙˜ x1M in), where ¨x˜1M in = 0 implies

˙˜x1M in reaches the smallest value of ˙˜x1 [see Fig 1, line (b)]

Let the right-hand side of (15) be zero in the worst case, we have ˙˜x1M in =

−2

λ1(k1x˜1M + α 

1)˜x 1/21

M , where α 

1 = α1+ ξ+ Since in quadrant 4, ˙˜x < 0, the

trajectory approaches ˜x1 = 0 Thus, the second part of the boundedness curve

in the fourth quadrant is the horizontal trajectory from (˜x1M , ˙˜ x1M in ) to (0, ˙˜ x1M in)[see Figure 1, line (c)]

Based on (12), (13) and (16), we can derive

˙˜x1M in<˙x˜10 (19)

If we define ˙˜x1M in= ˙˜x11, ˙˜ x12, , ˙˜ x1i , as the intersection points of the system

(9) trajectory starting from (0, ˙˜ x10) with the axis ˜x1 = 0, the inequality (19)

ensures the finite-time convergence of the state (0, ˙˜ x1i) to ˜x1= ˙˜x1= 0

Remark 1 The boundedness curve consists of segments (a), (b), and (c) is the

“worst” case of the trajectory Actually, (˜x1, ˙˜ x1) moves along the direction of(a), (b), (c) within the boundedness curve

Remark 2 The choice of α1and λ1depends on the bound of uncertainty and theinitial state estimation error in the worst case The theoretical result is consistent

with that only when the bound of F ( ·) is known In applications, a sufficiently

large α1 is preferred in order to satisfy (12) and (13)

Now, we consider the finite time convergence of ˜x2 Obviously, when ˜x1reachesthe sliding manifold, i.e., ˙˜x1= 0, z1= ˜x2, the dynamics of the estimation error

be proved in a similar manner as that in Theorem 1

In this subsection, the design of the high order sliding mode differentiator(HOSMD) developed in [20] will be presented

Let f (t) = f0(t) + n(t) be a function on [0, ∞), where f0(t) is an unknown base function with the n −th derivatives having a Lipschitz constant L, and n(t)

is a bounded Lebague-measurable noise with unknown features The problem ofhigh-order sliding-mode robust differentiator design is to find real-time robustestimations of ˙f0(t), ¨ f0(t), · · · , f (n)

0 (t) being exact when n(t) = 0 The proposed

HOSMD in [20] takes on the following form

where λ0, λ1, · · · , λ n are positive design parameters

With respect to the HOSMD given by (22), the following three results areproved in [20]

Theorem 2 If n(t) = 0 and all the parameters are chosen properly, then after

a finite transient, the following equalities are true

z0= f0(t); zi = vi −1 = f0(i) (t), i = 1, 2, · · · , n (23)

Theorem 3 If |n(t)| = |f(t) − f0(t) | ≤  and all the parameters are chosen properly, then after a finite transient, the following inequalities are obtained

Theorem 4 Let τ be the constant sampling time If n(t) = 0 and all the

param-eters are chosen properly, then after a finite transient, the following inequalities are obtained

where f a (t, x1, x2, u) represents a process fault The time profile function β(t) is

a step function described as

β(τ ) =



0, if τ < 0,

and T f is the time instant at which the fault occurs

A diagnostic observer is proposed in [30] as

˙ˆx2= f (t, ˆ x1, ˆ x2, u) + z2+ β(t − T m) ˆ M (t) xˆ2(0) = 0

ˆ

where ˆM (t) is the fault estimator, and other terms are the same as those in

previous section Tmis the time for activating the fault estimator

Assumption 3 Assume all the states are observed via sliding mode before the

activation of the wavelet network, and Tm < T f

Output Layer

ij

i

m m

m

− +

)

⋅

Fig 2 Structure of the three-layer wavelet network

Fault detection is achieved using the following logic:



No fault has occured, and ˆM (t) is set to zero if|˜y(t)| < δ f

Fault has occured, and ˆM (t) estimate is needed if|˜y(t)| ≥ δ f

(29)

where δf is a threshold for robust fault detection With the help of sliding mode,

δ f can be set very small without losing robustness

Moreover, the online estimator for ˆM (t) can be used not only to detect the

occurrence of the fault, but also to determine the location and magnitude of thefault

Based on the wavelet transform theory, wavelet network can achieve universalnonlinear function approximation [34] In this section, we construct a three-layerwavelet network to estimate the fault

The proposed three-layer wavelet network is comprised of an input layer (the

i layer), a wavelet layer (the ij layer), and an output layer (the o layer) The

schematic of this wavelet network is shown in Fig 2

The relationship between the input and output of each node i in the input

layer is represented as follows,

net1i = nii , no1i = f i1(net1i) = net1i , i = 1, · · · , p (30)

where ni i is the input of the wavelet network in which ni1= z1, and ni2= z1− ˆ M

Moreover, in the wavelet layer, a family of wavelets is established by preformingtranslations and dilations on a single fixed function called the mother wavelet

In this study, the first derivative of a Gaussian function, φ(x) = −x exp(−x2/2),

is selected as the mother wavelet This mother wavelet function has a universalapproximation property, since it can be regarded as a differentiable version ofthe Haar mother wavelet, just as the sigmoid is a differentiable version of a step

function [35] For the ijth node in the wavelet layer, we have

i to the node of mother wavelet layer, and q is the total number

of the wavelets with respect to the corresponding input node

In the output layer, the single node o is labelled as

, which adds all inputsignals together

˙

x = Ax + Bu + Dd

where x ∈ R n is the state vector, y = (y1 y2· · · y p)T ∈ R p is the output vector,

and u ∈ R m is the input vector (the output of actuators), and d ∈ R q is abounded unknown input vector which may consist of system uncertainties and/ordisturbances

We shall make the following assumptions

Assumption 4 Matrices A, B, C, D are known.

Assumption 5 (B D) is of full column rank, and C is of full row rank.

Remark 3 The conventional first order SMOs, which were designed to remove

the effect of the unknown inputs require two conditions to ensure their existence

[9] One is that the invariant zeros of (A, D, C) must have negative real parts The other is that rankCD =rankD = q, which implies that the relative degrees from d to the outputs are one These two conditions, which are called matching

conditions in this article, are also required by UIOs in [5, 6, 7] The latter tion is recently removed in [22], where relaxed matching conditions are allowed.Here, no such conditions are assumed Moreover, the system is not necessarilyrequired to be detectable.

condi-For system (35), the following problems are formulated:

P1– Under what conditions can actuator faults be detected?

P2– Is actuator fault isolation possible, and if so, how many actuator faults can

be isolated simultaneously?

P3– Is it possible to estimate the shape of the actuator faults?

P4– What is the design approach for accomplishing these objectives?

An input-output relation that does not involve the derivatives of either the

known inputs in u or the unknown inputs in d is need To achieve this, we define

a generalized input vector as ud = (u T d T)T and also a new input distribution

matrix Bd = (B D) The reason for having the matrices B and D rather than

B d in the beginning of our problem statement is that it is desirable to treatthe known inputs and unknown inputs separately for the sake of fault diagnosis.Additionally, we introduce the concept of relative degree from the generalized

input vector ud to the ith output yi , 1 ≤ i ≤ p.

Definition 1 For the system in (35) and any 1 ≤ i ≤ p, r i is said to be the relative degree from the input vector u d to the ith output y i if C i A j B d = 0 for

1≤ j ≤ r i − 2 and C i A r i −1 B d = 0, where C i is the ith row of C.

We make another assumption below

Assumption 6 For any 1≤ i ≤ p, r i is finite

Remark 4 If there is any i such that r i is infinite, the ith output yi will not

be affected by either the known inputs or unknown inputs In other words,

the ith output yi does not contain any information about either the knowninputs or unknown inputs, which means this output has no use in fault diagnosisand thus can be removed Therefore, in order to solve actuator fault diagnosisproblem, this assumption is necessary and is generally satisfied Furthermore,this assumption is considered to be the least conservative one

Under Assumption 5, it is easy to derive

Defining O = (C1T · · · (C1A r1−1)T C p T · · · (C p A r p −1)T)T Now select all the

independent rows from O in the following manner: first, pick C1, · · · , C pbecause

C is of full row rank, then find all rows from C A, · · · , C A, which together with

C1, · · · , C p form another set of independent rows of O, and continue until no

dependent rows can be found Then use all the independent rows obtained to

form a new matrix as TO = (C T

1 · · · (C1A l1)T C T

p · · · (C p A l p)T)T, which is of

full row rank and has the same rank as O.

Note that since TO is of full row rank, Tc can be chosen such that T = (T T

O T T

c )T is nonsingular Now let w = (w T

1 w T

2)T = T x with w1 = TO x It is

easy to see that w1 consists of the outputs and their derivatives

Let’s define matrices Y d , M , N u , and N d as

For convenience, several notations are introduced as follows Let φ denote

the empty set, 2M be the set consisting of all the subsets of the set M =

{1, 2, · · · , m}, and N io = (Nio,1 · · · N io,m) For any φ = s = {i1, · · · , i l } ∈ 2 M

with 1≤ l ≤ m, define N io,s = (Nio,i1, · · · , N io,i l ), where ij ∈ {1, 2, · · · , m} for

any 1≤ j ≤ l and N io,i j is the ij-th column of Nio If one takes away all columns

of Nio,s from Nio, the remaining columns of Nioconstitute a new matrix denoted

by ¯N io,s Denote also us as a vector consisting of the i1th,· · · ,i lth component of

u and ¯ u s as a vector consisting the remaining components Let u H be the desired

input vector, that is, u H = u when all actuators are healthy Notations u H s and

¯

u H s are defined the same way as usand ¯u s.

To solve the fault detection and isolation problems, we shall introduce two

concepts One is called generalized actuator fault isolation index (GAF IX), and

the other is called actuator fault detectability

Definition 2 System (35) is said to have a Generalized Actuator Fault Isolation

Index (GAFIX) equal to l if and only if for all sets of the form s = {i1, · · · , i l }, rank(N io,s) = l, where l is the largest number for which this rank condition

holds.

To solve the fault detection and isolation problems, we shall introduce two

concepts One is called generalized actuator fault isolation index (GAF IX), and< /i>

the... is called actuator fault detectability

Definition System (35) is said to have a Generalized Actuator Fault Isolation

Index (GAFIX) equal to l if and only if for