Content uploaded by Juergen Hahn

Author content

All content in this area was uploaded by Juergen Hahn on Nov 17, 2017

Content may be subject to copyright.

1

A Methodology for Fault Detection, Isolation, and Identification for

Nonlinear Processes with Parametric Uncertainties

Srinivasan Rajaramana,b Juergen Hahnb,∗ M. Sam Mannana,b

aMary Kay O’ Connor Process Safety Center

Department of Chemical Engineering

Texas A&M University

College Station, Texas 77843-3122, USA

bDepartment of Chemical Engineering

Texas A&M University

College Station, Texas 77843-3122, USA

Abstract

This paper presents a novel methodology for systematically designing a fault detection, isolation,

and identification algorithm for nonlinear systems with known model structure but uncertainty in

the parameters. The proposed fault diagnosis methodology does not require historical operational

data and/or a priori fault information in order to achieve accurate fault identification. This is

achieved by a two-step procedure consisting of a nonlinear observer, which includes a parameter

estimator and a fault isolation and identification filter. Parameter estimation within the observer

is performed by using the unknown parameters as augmented states of the system and robustness

is ensured by application of a variation of Kharitonov’s theorem to the observer design. The filter

design for fault reconstruction is based upon a linearization, which has to be repeatedly

computed at each step where a fault is to be identified. However, this repeated linearization does

not pose a severe drawback since linearization of a model can be automated and is

computationally not very demanding for models used for fault detection. It is not possible to

simultaneously perform parameter estimation and fault reconstruction since faults and the

parametric uncertainty influence one another. Therefore, these two tasks are performed at

different time scales, where the fault identification takes place at a higher frequency than the

parameter estimation. It is shown that the fault can be reconstructed under some realistic

assumptions and the performance of the proposed methodology is evaluated on a simulated

chemical process exhibiting nonlinear dynamic behavior.

Keywords: Fault detection; Fault isolation; Fault identification; State and parameter estimation;

Hurwitz stability; Kharitonov's theorem

1 Introduction

Early and accurate fault detection and diagnosis is an essential component of operating

modern chemical plants in order to reduce downtime, increase safety and product quality,

∗ Corresponding author. Phone: +1 (979) 845 3568

Fax: +1 (979) 845 6446

E-mail: hahn@tamu.edu

2

minimize impact on the environment, and reduce manufacturing costs1,2. As the level of

instrumentation in chemical plants increases, it is essential to be able to monitor the variables and

interpret their variations. While some of these variations are due to changing operating

conditions, others can be directly linked to faults. Extracting essential information about the state

of a system and processing the data for detecting, isolating, and identifying abnormal readings

are important tasks of a fault diagnosis system3, where the individual goals are defined as:

• Fault detection: a Boolean decision about the existence of faults in a system.

• Fault isolation: determination of the location of a fault, e.g., which sensor or actuator is

not operating within normal limits.

• Fault identification: estimation of the size and type of a fault.

There exist numerous techniques for performing fault diagnosis4. The majority of these

approaches are based upon data from past operations in which statistical measures are used to

compare the current operating data to earlier conditions of the process where the state of the

process was known. While these techniques are often easy to implement, they do have the

drawback that it is not possible to perform fault identification, that a large amount of past data is

required, that the method may not be able to detect a fault if operating conditions have changed

significantly, or that processes exhibiting highly nonlinear behavior may be difficult to diagnose5.

In order to address these points, this paper presents an approach for fault diagnosis based upon

nonlinear first-principles models, which can include parametric uncertainty. Incorporating

fundamental models into the procedure allows for accurate diagnosis even if operating conditions

have changed, while the online estimation of model parameters takes care of plant-model

mismatch. The parameter estimation is performed using an augmented nonlinear observer, where

a concept from Kharitonov’s theory6 about stability under the influence of parametric uncertainty

is utilized in order to ensure a certain level of robustness for the designed observer. The fault

diagnosis itself uses the computation of residuals (i.e., the mismatch between the measured

output and estimated output using the model) for fault detection3 and appropriate filters are

designed to achieve fault isolation and identification as well. Since it is not possible to

simultaneously perform parameter estimation and fault detection, due to the interactions of these

two tasks, an approach where these computations are taking place at different time scales is

implemented. It is shown that fault detection, isolation, and identification for nonlinear systems

containing uncertain parameters can be performed under realistic assumptions with the presented

approach.

2 Previous work

Extensive research on fault detection has been undertaken over the last few decades5,7,8.

The majority of methods are based on statistical techniques9,10, however, a significant body of

literature also exists for fault detection and identification of measurement bias based upon first

principles models5,11.

Work on model-based fault detection has included unknown input observers (UIO)13,

which are based upon the idea that the state estimation error can be decoupled from unknown

inputs3 acting as disturbances and thereby decoupling residuals from uncertainties. This concept

was generalized in subsequent work14 for detecting and isolating both sensor and actuator faults

by considering the case when unknown inputs also appear in the output equation. A different

approach, but using a similar concept of decoupling the disturbances from the residuals for fault

detection, made use of the eigenstructure assignment of the observer14,15. In this technique, the

observer is designed to de-couple the residual from unknown inputs rather than from the state-

3

estimation error. The above two approaches work satisfactorily for LTI systems, however, they

can result in poor performance for nonlinear systems because these systems are not always affine

in the unknown inputs. To cope with nonlinearities of the process, it has been proposed to

develop nonlinear unknown input observers16 for residual generation, which requires a suitable

nonlinear state transformation. However, the conditions under which these transformations exist

are very restrictive in nature. Moreover, it is assumed that the unknown disturbances affect the

system in a piecewise linear affine fashion. The restrictive conditions required to develop UIOs

and attain eigenstructure assignment has led to frequency domain and optimization methods for

robust fault diagnosis3. The main emphasis is on formulation of robust fault diagnosis and

isolation (FDI) problems using frequency performance criteria. Limitations of using an

optimization method for robust fault diagnosis are due to the assumption that no modeling errors

are present or that the modeling errors can be viewed as disturbances17,18.

A major advantage of observer-based fault diagnosis techniques as compared to data-

driven techniques is that the residual’s sensitivity to faults of a specific frequency range can be

tailored. Uncertainties in the model can then be taken into account by considering them to be

slowly time-varying faults. However, this involves the risk that fault signals of low frequency

may not be detected because the enhancement of robustness is associated with an accompanying

decrease of the ability to detect slowly time-varying fault signals19,20. To overcome this difficulty,

it was proposed to use adaptive observers21,22 where certain effects of nonlinearities and model

uncertainties may be handled as unknown parameters that can decoupled from the residuals. The

formulation of the above problem is based on the assumption that the slowly time-varying

unknown parameters appear as an affine unknown input to the system. However, most chemical

processes are nonlinear and exhibit exponential dependence of unknown parameters in the

process model, e.g. The activation energy. Moreover, since industrial processes operate in

closed-loop with appropriate output feedback to attain certain performance objectives, it is

important to not just detect and isolate instrument faults but to reconstruct them at the same time

in order to implement fault tolerant control23.

3 Preliminaries

In section 3.1 observer based fault diagnosis for LTI systems is reviewed. Required

background information about the concept of stability of an interval family of polynomials is

presented in section 3.2. This serves as a foundation for the presentation of the new technique in

section 4.

3.1 Fault diagnosis for LTI systems

Consider a linear time invariant system with no input

s

fCxy

Ax

x

+=

=

(1)

where n

x

R∈is a vector of state variables and m

yR∈is a vector of output variables, n is the

number of states, and m refers to the number of output variables. A and C are matrices of

appropriate dimensions and fs is the sensor fault of unknown nature with the same dimensions as

the output. Assuming the above system is observable, a Luenberger observer for the system can

be designed

ˆˆ ˆ

()

ˆˆ

x

Ax L y y

yCx

=

+−

=

(2)

4

where L is chosen to make the closed loop observer stable and achieve a desired observer

dynamics. Further, define a residual3

0

ˆ

() ( )( ( ) ( ))

t

rt Qt y y d

τ

τττ

=− −

∫ (3)

which represents the difference between the observer output and the actual output passed through

a filter Q(t). Taking a Laplace transform of equations (1)-(3) results in

1

() ()[ ( ( )) ] ()

s

rs Qs I CsI A LC L f s

−

=−−− (4)

where Q(t) is chosen s.t Q(s) is a

R

H

∞

-matrix24. It can be shown that

1) () 0rt = ( ) 0

s

if f t =.

2) () 0rt ≠ ( ) 0

s

if f t ≠.

indicating that the value of r(t) predicts the existence of a fault in the system24.

Figure 1. Schematic of Dedicated Observer Scheme (DOS) for a system with 2 measurements

In addition, if one uses the dedicated observer scheme as shown for a system with two

outputs in Figure 1, then the fault detection system can also determine the location of the fault:

3) ( ) 0

i

rt = ,

() 0

si

if f t =, 1,2,3,......., .im=

4) ( ) 0

i

rt ≠ ,

() 0

si

if f t ≠, 1, 2, 3,......., .im=

where i represents the th

imeasurement. A fault detection system that satisfies all of the above

conditions is called as a Fault detection and isolation filter (FDIF). A fault detection and isolation

filter becomes a fault identification filter (FIDF) if additionally the following condition is

satisfied25:

5) ,

lim( ( ) ( )) 0, 1, 2,3......, .

isi

trt f t i m

→∞ −==

In order to meet the above conditions, the following restrictions on the choice of Q(s) are

imposed:

(a) ( ) 0, Qs s≠∀∈.

5

(b) 11

() [ ( ( )) ]Qs I CsI A LC L

−−

=− − − 1

[( ) ]CsI A L I

−

=

−+.

Linear, observer-based fault detection, isolation, and identification schemes work well if

an accurate model exists for the process over the whole operating region and if appropriate

choices are made for L and Q.

3.2 Robust stability of an interval polynomial family

Consider a set ( )s

δ

of real polynomials of degree n of the form

234

12 3 4

() n

on

ssssss

δ

δδ δ δ δ δ

=++++++

where the coefficients lie within given ranges:

[

]

+−

∈000 ,

δδδ

,

[

]

+−

∈111 ,

δδδ

, …,

[

]

+−

∈nnn

δδδ

,

Denote that

[

]

n

δ

δ

δ

δ

,,, 10 …=

and define a polynomial ()s

δ

by its coefficient vector δ. Furthermore, define the hyperrectangle

of coefficients.

{

}

ni

iii

n,,2,1,0,,:: 1…=≤≤ℜ∈=Ω +−+

δδδδδ

Assuming that the degree remains invariant over the family, so that

[

]

+−

∉nn

δδ

,0. A set of

polynomials with above properties is called an interval polynomial family. Kharitonov’s theorem

provides a necessary and sufficient condition for the Hurwitz stability of all members contained

in this family.

Theorem 1 (Kharitonov’s Theorem)

Every polynomial in the family ()s

δ

is Hurwitz if and only if the following four extreme

polynomials are Hurwitz6.

.)(

,)(

,)(

,)(

6

6

5

5

4

4

3

3

2

210

6

6

5

5

4

4

3

3

2

210

6

6

5

5

4

4

3

3

2

210

6

6

5

5

4

4

3

3

2

210

…

…

…

…

+++++++=

+++++++=

+++++++=

+++++++=

−++−−++++

−−++−−+−+

++−−++−+−

+−−++−−−−

sssssss

sssssss

sssssss

sssssss

δδδδδδδδ

δδδδδδδδ

δδδδδδδδ

δδδδδδδδ

(5)

While this theorem has been used extensively in parametric approaches to robust control,

it will be utilized for developing observers that handle parametric uncertainties and nonlinearities

in the model.

4 Robust fault detection, isolation, and identification

4.1 Problem formulation

Consider a nonlinear system with possibly multiple outputs of the following form

(

)

()

s

fxhy

xfx

+=

=

θ

θ

,

,

(6)

where n

x

R∈is a vector of state variables and m

yR∈is a vector of output variables. It is

assumed that

()

θ

,xf is a smooth analytic vector field in n

R

and

(

)

,hx

θ

is a smooth analytic

vector fieled in m

R

. Let k

R

θ

∈ be a parameter vector assumed to be constant with time but a

priori uncertain and fs is the sensor fault of unknown nature with the same dimensions as the

6

output. The goal of this paper is to estimate the state vector with limited information about the

parameters describing the process model and under the influence of output disturbances

s.t. ˆ

lim( ) 0

txx

→∞ −=, where ˆ

x

is the estimate of the state vector, x, and to design a set of filters

()Qt so that the residuals, given by the expression

0

ˆ

() ( )( ( ) ( ))

t

rt Qt y y d

τ

τττ

=− −

∫ have all the

five properties discussed in section 3.1.

One of the main challenges in this research is that both faults and plant-model mismatch

will have an effect on the fault identification. In order to perform accurate state and parameter

estimation, it is desired to have reliable measurements, while at the same time an accurate model

of the process is required to identify the fault. This will be taken into account by performing the

parameter estimation and the fault detection at different time scales. Each time the parameters

are estimated, it is assumed that the fault is not changing at that instance, while the values of the

parameters are not adjusted during each individual fault detection. A variety of different

techniques exist for designing nonlinear closed-loop observers26-30. However, since the class of

problems under investigation includes parametric uncertainty it would be natural to address these

issues through a parametric approach instead of the often used extended Kalman filter or

extended Luenberger observer. The procedure for designing nonlinear observers under the

influence of parametric uncertainty is outlined in the next subsections, which is followed by a

description of the fault detection, isolation, and identification algorithm.

4.2 Estimator design – a parametric approach

A nonlinear system of the form given by equation (6) can be rewritten by viewing the parameters

as augmented states of the system

(

)

()

s

fxhy

xfx

+=

=

θ

θ

θ

,

0

,

(7)

and with a change of notation

()

(

)

=

=0

,

,,

θ

θ

θ

xf

xf

x

x (8)

this results in the following system

(

)

()

s

fxhy

xfx

+=

=

(9)

For the state and parameter estimation step it is assumed that the sensor faults are known, since

they are identified at certain “sampling times” and the assumption is made that they remain

constant over the time interval between two “sampling points”. Furthermore, assume that each

component θi of the parameter vector

[

]

110 ,,,: −

=k

θ

θ

θ

θ

… (10)

can vary independently of the other components and each θi lies within an interval where the

upper and lower bounds are known

{

}

1,...,2,1,0,:: −=≤≤=Π +− ki

iii

θθθθ

(11)

Also, let Π∈= ss

θ

θ

be a vector of constant, a priori unknown parameters and (xss, θss) be an

equilibrium point of equation (7). The augmented system needs to be observable in order to

7

design an observer, which can also estimate the values of the parameters. A sufficient condition

for local observability of a nonlinear system is if the observability matrix

()

(

)

()

()

∂

∂

∂

∂∂

∂

=

−+ xhL

x

xhL

x

x

xh

xW

kn

f

f

o

1

(12)

has rank n+k for (x, θ) = (xss, θss)31. Since the equilibrium points of the system depend upon the

values of the parameters which are not known a priori, it is required that the rank of

(

)

o

Wx is

checked for all Π∈= ss

θ

θ

and the resulting equilibrium points (xss, θss).

In order to proceed it is assumed that the augmented system is observable over the entire

hyperrectangle-like set Π and the equilibrium points corresponding to these parameters values. It

is then possible to design an observer for the augmented system

() ()

()

()

,ˆ

,

0

ˆ,s

xfx Lx y y

yhx f

θθ

θ

θ

=+−

=+

(13)

Where x

~ is the estimate of

x

and

θ

is the estimate of

θ

and

(

)

,Lx

θ

is a suitably chosen

nonlinear observer gain. Also, note that the observer makes use of the assumption that the

measurement fault is known from an earlier identification of the fault. When the observer is

computed for the first time, it has no knowledge about possible sensors faults and assumes that

no sensor fault was initially present.

4.2.1 Determining the family of polynomials for observer design

In this section, the result about Hurwitz stability of an interval family of polynomials from

section 3.2 is utilized to determine a methodology for computing the gain (, )Lx

θ

of the

nonlinear observer given by equation (13). Consider the linearized model of the augmented

process model around an equilibrium point (xss, θss).

(,)

(,)

ss ss

s

sss s

xAx x

yCx x f

θ

θ

=

=+

(14)

Where ),( ssss

xA

θ

is the Jacobian of

(

)

θ

,xf at the point ),( ssss

x

θ

and

(,)

s

sss

Cx

θ

=

(,) (,)

(, ) (, )

ss ss ss ss

xx

hx hx

x

θθ

θθ

θ

∂∂

∂∂

. The characteristic polynomial of the system, which

determines its stability, is given by

2

12

()det (,) (,) (,) (,) .

n

ssss ossss ssss ssss

ssIAx x xsxs s

δθδθδθδθ

=− = + + ++

(15)

It can be seen that the coefficients of the characteristic polynomial are nonlinear functions of the

parameter vector ss

θ

and ss

x. Assuming that ),( ssss

xf

θ

satisfies the conditions of the implicit

8

function theorem i.e. 0

),(

,

≠

∂

∂

ssss

x

x

xf

θ

θ

then ss

x can be solved for a given ss

θ

, i.e.

ss

x=)(ss

θ

φ

,:kn

R

R

φ

→. The characteristic polynomial in s given by equation (15) can then be

rewritten as

2

12

2

12

( ) ( ( ), ) ( ( ), ) ( ( ), )

( ) ( ) ( )

n

o ssss ssss ssss

n

oss ss ss

ssss

ss s

δ δ φθ θ δ φθ θ δ φθ θ

δθ δθ δθ

=++ ++

=++ ++

(16)

where, ( ( ), ) ( ), 0,1, 2,3, , 1.

issssissin

δφθ θ δθ

== −While it is generally not possible to derive

an analytic expression of the coefficients

(

)

11

,, ,

on

δδ δ

−

…… as a function of ss

θ

, ss

xcan be

evaluated by numerically solving the equation 0),(

=

ssss

xf

θ

for ∏

∈

ss

θ

. Since ( , )

f

x

θ

is

assumed to be a smooth vector function, the coefficients of the characteristic polynomial are

continuous functions of ),( ssss

x

θ

. Therefore, by discretizing the set

∏

and evaluating the

maximum and minimum values for each coefficient )( ssi

θδ

over all the points in the set

∏

, the

hyperrectangle of coefficients Ω as described in section 3.2 can be obtained. In case of a multi-

dimensional parameter

θ

, discretizing the set

∏

can be computationally expensive, however,

advanced NLP algorithms exist that facilitate the calculation of the required bounds on the

coefficients. For the case where

θ

is a scalar, the range within which the coefficients vary can be

determined by plotting )( ssi

θδ

against ss

θ

0,1, 2, 3, , 1.in

∀

=− Figure 2 shows a typical plot

of the coefficient versus the one-dimensional parameter

θ

.

12345678

4

6

8

10

12

14

16

Plot of coefficient as a function of the parameter

θ

min

ss

δ

min

i

(θ

ss

)

δ

max

i

(θ

ss

)

θ

max

ss

Figure 2: Sample plot of the coefficient as a function of a one-dimensional real parameter

To enforce that the estimation error decays asymptotically for the linearized system, the observer

gains are chosen to satisfy the following condition

9

((,) (,)(,)) ,

ss ss ss ss ss ss ss

Ax Lx Cx

λθ θ θ θ

−

−∈∀∈∏ (17)

Where (.)

λ

refers to the eigenvalues of the matrix. The following section focuses on computing

appropriate gains L by making use of Kharitonov’s theorem.

4.2.2 Observer gain computation

Since it is assumed that the augmented system given by equation (7) is observable over the entire

hyperrectangle-like set ∏and the equilibrium points corresponding to these parameters values, it

is possible to find an invertible transformation ),( ssss

xT

θ

∏

∈

∀

ss

ts

θ

.. the LTI system given by

equation (14) can be transformed into an observer canonical form (Appendix A), considering one

output at a time as

zCy

zxAz ssss

=

=),(

θ

(18)

where, 11

(,), (,) (, )(, ) (, ), (,) (, )

s

sss ssss ssss ssss ssss ssss ssss

zTx xAx Tx Ax T x CCx T x

θ

θθθθ θθ

−−

== =

,

and

1

2

2

1

0

(,)10 00

(,)01 00

(,) (, ) 0 010

(,) 0 001

(,) 0 000

nssss

nssss

ss ss

ss ss

ss ss

ss ss

x

x

Ax x

x

x

δθ

δθ

θδθ

δθ

δθ

−

−

=

[]

00001

=C

(19)

The characteristic polynomial of ),( ssss

xA

θ

takes the following form:

21

12 1

() ( , ) ( , ) ( , ) ( , ) .

nn

ossss ssss ssss n ssss

s x xsxs xss

δδθδθδθ δ θ

−

−

=++ + +

Since, the coefficients of the above characteristic polynomial in s are continuous and nonlinear

functions of ),( ssss

x

θ

, the hyperrectangle within which these coefficients can vary independently

from one another can be evaluated from the method discussed in section 4.2.1. The following

analysis provides a method to compute a constant gain vector l for the case of a single output

system such that:

(( , ) ) ,

ss ss ss

Ax lC

λθ θ

−

−

∈∀∈∏

Consider the set of n nominal parameters

{

}

0

1

0

2

0

1

0

0,...,,, −n

δδδδ

together with a set of a priori

uncertainty ranges 11

, ,........, ,

on

δ

δδ

−

∆∆ ∆ which is given by , 0,1, , 1

ii i

in

δδ δ

+−

∆

=− = −…… .

Furthermore, consider the family ( )s

δ

of polynomials,

nn

no ssssss ++++++= −

−

1

1

3

3

2

21 ....)(

δδδδδδ

where the coefficients of the polynomial can vary independently from one another and lie within

the given ranges

10

−=

∆

+≤≤

∆

−1,....,2,1,0,

22

:00 ni

i

ii

i

i

δ

δδ

δ

δδ

Further let there be n free parameters

(

)

121 ,,,, −

=

no lllll to transform the family ()s

δ

into

the family described by

21

00 11 22 1 1

() ( ) ( ) ( ) ( )nn

nn

sl lsls lss

γδ δ δ δ

−

−−

=+++++ + + + +………

The above problem arises, when it is desired to suitably place the closed loop observer poles for

a single output system where the system matrices ,Ac are in observable canonical form and the

coefficients of the characteristic polynomial of

A

are subject to bounded perturbations. The

following theorem guarantees that there exists a free parameter vector l such that the pair ,Ac

can always be stabilized:

Theorem 2 For any set of nominal parameters,

{

}

0

1

0

2

0

1

0

0,...,,, −n

δδδδ

, and for any set of positive

numbers 11

, ,........, ,

on

δ

δδ

−

∆∆ ∆ it is possible to find a vector l such that the entire family ()s

γ

is

stable6.

By Theorem 2 there always exists a .. ( ( , ) ) ,

ss ss ss

lst Ax lC

λθ θ

−

−

∈∀∈∏

, which can

be systematically computed (Appendix B). The result is an observer given by equation (13),

which locally estimates the states and parameters of the system given by equation (6) and with an

observer gain

() ()

1

,,Lx T x l

θ

θ

−

=

. The proposed approach is considerably less computationally

demanding than alternative state and parameter estimation techniques such as extended

Luenberger observers33. Additionally, the presented method yields an analytic expression for

observer gains irrespective of the dimension of the systems but guarantees convergence of the

error dynamics locally around the operating point.

4.3 Fault detection

The purpose of fault detection is to determine whether a fault has occurred in the system. It can

be seen that lim( ) 0

txx

→∞ −≠

in the presence of sensor faults. In order to extract the information

about faults from the system a residual needs to be defined as

0

ˆ

() ( )( ( ) ( ))

t

rt Qt y y d

τ

τττ

=− −

∫, where ( )Qt is any stable filter. It can be verified that

• lim ( ) 0

trt

→∞ = ( ) 0

s

if f t =.

• lim ( ) 0

trt

→∞ ≠ ( ) 0

s

if f t ≠.

Additional restrictions on the class of stable filters ( )Qt will be imposed in the following

sections in order to satisfy desired objectives.

4.4 Fault isolation

Fault isolation is synonymous with determining the location of a fault and its computation

imposes additional restrictions on the choice of the filter Q(t). In order to perform fault isolation,

the augmented system given by equation (7) is assumed to be separately locally observable

through each of the outputs y

∏

∈

∀ss

θ

. It should be noted that this requirement is mandatory

11

for the existence of a fault isolation filter25 and hence does not pose a stringent condition for

using the presented approach.

To achieve fault detection as well as isolation, the proposed approach uses a series of

dedicated nonlinear observers as shown in Figure 1. In this method as many residuals are

generated as the number of measurable outputs. It can be verified that

• lim ( ) 0

i

trt

→∞ = ,

() 0

si

if f t

=

• lim ( ) 0

i

trt

→∞ ≠ ,

() 0

si

if f t

≠

, 1,2,3,......., .im

=

for an appropriately chosen filter Q(t).

4.5 Fault identification

In order to estimate the shape and size of the fault, the residuals have to meet the following

objective:

,

lim( () ()) 0

isi

trt f t

→∞

−

= 1, 2 , ,im

=

…

Since a dedicated nonlinear observer scheme is utilized in the proposed approach, it remains to

choose a suitable stable filter ( )

Qt to meet all the conditions for fault detection, isolation, and

identification. It was shown in section 3.1 that an appropriate choice of ( )

Qt for a linear time-

invariant system described by equation (1) is given by

])([)( 1ILAsICsQ +−= −

where ( )

Qs is the Laplace transform of the filter ( )Qt . Similarly, for the nonlinear system given

by equation (9) a linear filter

1

() [ ( , )( ( , )) ( , ) ]

ss ss ss ss ss ss

Qs Cx sI Ax Lx I

θθθ

−

=− +

is locally applicable. Since the equilibrium point ),( ssss

x

θ

is a priori unknown, the fault

identification filter is modified:

1

() [(,)( (,)) (,) ]Qs Cx sI Ax L x I

θθθ

−

=− +

where ( )

Qs is the Laplace transform of the filter at any point ( , )

x

θ

in the state space. However,

since at least as many eigenvalues of (, )

A

x

θ

are identically to zero as there are parameters of

the original system, the above ()Qt is not stable. To overcome the problem of choosing a stable

filter for fault reconstruction, a lower dimensional observer that does not perform the parameter

estimation but only estimates the states, needs to be considered

ˆˆ ˆ ˆ

(, ) (, )( )

ˆˆ

(, ) s

x

fx Lx y y

yhx f

θθ

θ

=+ −

=+

(20)

where )(

ˆtx is the estimate of )(tx and

(,) .. ((, ) (, )(,)) ,

ss ss ss ss ss ss ss ss ss

Lx st Ax Lx Cx

θλθ θ θ θ

−

−

∈∀∈∏ (21)

where ),( ssss

xA

θ

is the Jacobian of ),(

θ

xf at the point ),( ssss

x

θ

and

,

(, )

(,)

s

sss

ss ss

x

hx

Cx x

θ

θ

θ

∂

=∂

Lemma 1. The nonlinear system described by equation (20) in conjunction with the observer of

the augmented system (13) is a locally asymptotic observer to the system given by equation (6) if

fs is known.

12

Proof: Since ),( ssss

xL

θ

is chosen . .st the condition in equation (17) is met, lim( )

t

θ

θ

→∞

−

=0.

Linearizing the system given by equation (20) around the equilibrium point ),( ssss

x

θ

:

ˆˆ ˆ

(,) (, )( )

ˆˆ

(,)

ss ss ss ss

ss ss s

x

Ax x Lx y y

yCx x f

θθ

θ

=+ −

=+

(22)

Similarly, linearizing the system given by the equation (6) around the equilibrium point

),( ssss

x

θ

results in:

(,)

(,)

ss ss

s

sss s

xAx x

yCx x f

θ

θ

=

=+

(23)

The error of the state estimates, xxe ˆ

−

=, is then given by the following equations:

((,) (,)(,))

ss ss ss ss ss ss

eAx Lx Cx e

θ

θθ

=−

(24)

Since ),( ssss

xL

θ

is chosen to satisfy the condition in equation (21), the estimation error in

equation (24) converges asymptotically to zero.

Note that the gains for the observers given by equation (20) can be computed using the

technique presented in section 4.2.2. Similar observability conditions as in section 4.2. can be

derived for the existence of gains that guarantee stability of the closed-loop observers in the

neighborhood of the operating point.

For practical purposes the original system given by equation (6) in the absence of faults is

considered locally stable around the operating point as the parameters vary in the hyperrectangle

as defined by equation (11). In other words it is assumed that the Jacobian ),( ssss

xA

θ

∏

∈

∀

ss

θ

is Hurwitz stable.

Using the above assumption, a stable linear fault identification filter ( )Qt ..st the residual,

0

ˆ

() ( )( ( ) ( )),

t

rt Qt y y

τ

ττ

=− −

∫ having the property that lim ( )

s

trt f

→∞

=

has the following state space

representation:

)

ˆ

(

)

ˆ

)(

~

,

ˆ

()

~

,

ˆ

(

yyICr

yyxLxA

−+=

−+=

ξ

θξθξ

(25)

where y

ˆ and ˆ

x

are the output and state estimates obtained via the observer given by equation (20)

and n

R∈

ξ

is a state with initial condition 0)0(

=

ξ

.

Putting all these pieces together, the fault detection, isolation, and identification filter

consists of the observers given by equation (13) and (20), and is computed in parallel with

equation (25) in order to generate residuals. The filter is recomputed at each time step by

linearizing the model at the estimate of the location in state space of the augmented system.

In the presence of unknown sensor faults, the estimate

θ

for some ∏∈

ss

θ

may diverge

from the actual value and therefore the stability of the overall fault diagnosis system cannot be

guaranteed. To overcome this problem, parameter estimation and fault reconstruction are

performed at different time scales and it is assumed that the algorithm is initialized when no

sensor fault occurs until a time .

o

tst for some 2

ˆ

0, 0

o

yy t

εε

>−≤∀≥

. The sensor fault is of

the following form:

13

1:

() () , 0:

o

soo

o

tt

f t f t S(t - t ) S(t - t ) tt

≥

==

<

The above assumption ensures that the parameter estimate from equation (17) converges to its

actual value with a desired accuracy

2, ( ) 0

ss

θθ ηηε

−≤ >

(26)

before onset of faults in the original process. Additionally, the parameters are adapted

periodically by the augmented observer (13) in order to take process drifts into account.

In summary, the presented fault diagnosis system performs parameter estimation and

fault reconstruction at different time scales, where the fault identification takes place at a higher

frequency than parameter estimation. The values of the parameters are assumed to stay constant

during the fault identification, while the faults are assumed constant during parameter estimation.

Figure 3 illustrates this two-time scale behavior, where stages 2 and 3 are repeated alternatively

throughout the operation and the time between the start of each stage is decided by the nature of

the process. However, in general the parameter estimation is only performed sporadically and

requires only short periods of time, so that the fault can be identified for the vast majority of the

time.

Figure 3: Schematic fault identification for systems with time-varying parameters

5 Case study

5.1 Fault diagnosis of a reactor with uncertain parameter

To illustrate the main aspects of the investigated observer-based fault diagnosis scheme, a non-

isothermal CSTR is considered with coolant jacket dynamics, where the following exothermic

irreversible reaction between sodium thiosulfate and hydrogen peroxide is taking place34.

OHOSNaOSNaOHOSNa 242263222322 442

+

+

→+ (27)

1 2 3

1) No fault present

2) Parameter estimation

3) Short time period

1) No parameter updating

2) Fault identification assuming

the knowledge of the parameter

3) Long time period

1) Fault assumed constant

(value from previous

identification)

2) Parameter estimation

3) Short time period

Start of operation Time

14

process

parameters values process

parameters values

s

F 120 /minL cp 4.2 (/ )JgK

Ain

C 1 ()/mol L ws

F 30 (/min)L

V 100 L UA 20000 (/ )JsK

o

k 4.11E+13 /(min )Lmol w

V 10 ()L

E

76534.704 (/ )Jmol

w

ρ

1000 (/)

g

L

inT 275 K cpw 4200 (/ )JkgK

()RH−∆ 596619 (/ )Jmol jinT 250 K

ρ

1000 (/)gL

Table 1: Values of process parameters

The capital letters A, B, C, D and E are used to denote the chemical compounds 322 OSNa ,22 OH ,

632 OSNa , 422 OSNa , and OH 2. The reaction kinetic law is reported in the literature to be34

BAoo

BAA

CC

RT

EE

kk

CCTkr

∆+−

∆+=

=−

exp)(

)(

where o

k∆ and

E

∆ represent parametric uncertainties in the model. A mole balance for species

A and energy balances for the reactor and the cooling jacket result in the following nonlinear

process model

)(

)(

)(

)(

)(

)(

))()((

2)(

)(2)(

2

2

j

pwww

jjin

j

j

p

A

p

RR

in

AAAin

A

TT

cV

UAUA

TT

V

F

dt

dT

TT

cV

UAUA

CTk

c

HH

TT

V

F

dt

dT

CTkCC

V

F

dt

dC

−

∆+

+−=

−

∆+

−

∆−∆+∆−

+−=

−−=

ρ

ρρ

(28)

where

F

is the feed flow rate,V is the volume of the reactor, Ain

C is the inlet feed concentration,

in

T the inlet feed temperature, w

Vis the volume of the cooling jacket, jin

T is the inlet coolant

temperature, w

F

is the inlet coolant flow rate, p

cis the heat capacity of the reacting mixture, pw

c

is the heat capacity of the coolant,

ρ

is the density of the reacting mixture, U is the overall heat

transfer coefficient, and

A

is the area over which the heat is transferred. The process parameters

values are listed in Table 1.

Here, , , ( ), and

o

kE H UA∆∆∆∆ ∆ represent uncertainty in the pre-exponential factor, the

activation energy, the heat of reaction, and the overall heat transfer rate, respectively. When ,

o

k

∆

, ( ), and

EH UA∆∆∆ ∆ are all chosen equal to zero, the nominal nonlinear model exhibits

multiple steady states, of which the upper steady state, i.e.

15

( 0.0192076 /

Ass

CmolL=; 384.005

ss

TK=; 371.272

jss

TK

=

), is stable and chosen as the point

of operation. Since the activation energy appears exponentially in the state space description of

the process, the effect of uncertainty on the behavior of the system is significantly higher than for

the other parameters listed above. This observation has also been confirmed in simulations.

In order to validate the performance of the presented approach, it is in a first step

compared to the results derived from a fault detection scheme based upon a Luenberger observer

for the process under consideration. For now, the process parameters are assumed to be known

and given the values in Table 1. The system matrices obtained by linearizing the process model

(28) around the chosen steady state are

-123.7499724 -.07347363019 0

17408.48619 6.379943743 2.857142857

0 28.57142857 -31.57142857

A

=

TT

CC

=

=

1

0

0

,

0

1

0

21

(29)

With

{

}

( ) 112.94, 1.37, 34.63A

λ

=− − − . For performing fault isolation and identification it is

required to design observers for each of the two measurements as shown in Figure 1 and the

eigenvalues of the closed loop observers are placed at

{

}

6.85, 6.86, 6.87−−− . The observer gain

calculated for a measurement of the reaction temperature is

1

53.912

1.55 3

5.79 5

LE

E

−

=

+

+

and the gain corresponding to the coolant temperature is found to be

2

1.7 2

2.7 4

1.55 3

E

LE

E

+

=+

+

Both reaction temperature and coolant temperature sensor are induced with an additive fault

signal and random noise with normal distribution whose shape and size are shown in Figure 4.

Residuals generated by the technique based upon a Luenberger observer with uncertainty

in the initial conditions are shown in Figure 5. Comparing Figure 4 and Figure 5, it is concluded

that the Luenberger observer-based fault diagnosis scheme is able to isolate and identify the

approximate nature of the fault in each sensor. Similar simulations have been carried out where

the process model includes

uncertainties ( 5% , 6% ,

oo

kkEE∆= ∆= ( ) 5% , and 5% )H H UA UA

∆

∆=∆ ∆ = . Figure 6 shows the

residual generated for the fault signal shown in Figure 4 for one specific case of parametric

uncertainty. From Figure 6 it is evident that while the shape of the fault is reproduced almost

perfectly, the bias in the residuals results from modeling uncertainties and can be misinterpreted

as a response to a step fault in the sensor. To illustrate this point, simulations of the fault

diagnosis scheme based upon the Luenberger observer are performed for a sufficiently large

number of scenarios (10,000) which include a random occurrence of faults in either or both the

sensors as well as randomly chosen parametric uncertainty within the given intervals in order to

16

determine the overall percentage of successfully identifying one or all the scenarios. The

scenarios denoted by "00 ", "01", "10", and "11" in the Tables 2-3 stand for no faults in both

sensors, no fault in reaction temperature sensor and fault in coolant temperature sensor, fault in

reaction temperature sensor and no fault in coolant temperature sensor, and fault in both sensor

respectively. Step faults starting at time 0t

=

, and of magnitude 5 K were added to the sensors.

Various thresholds are selected to determine whether or not a fault occurred in the sensors and

the fault isolation scheme (based upon a Luenberger observer) is tested in Monte Carlo

simulations where the parametric uncertainty is chosen at random within the given intervals. As

an example of this scheme, the scenario identifies the condition where no faults occur in both

sensors for a chosen threshold

α

if the following condition is satisfied:

a) If time average of ()

c

rt

α

<, where ( )

c

rtdenotes the coolant temperature residual, and

b) If time average of ()

T

rt

α

<, where ( )

T

rtdenotes the reactor temperature residual.

0 20 40 60 80 100 120 140 160 180 20

0

−8

−6

−4

−2

0

2

Reactor temperature fault

Temperature (K)

0 20 40 60 80 100 120 140 160 180 20

0

−2

0

2

4

6

8

Coolant temperature fault

Time (min)

Temperature (K)

Figure 4: Reactor and coolant temperature fault signal

17

0 20 40 60 80 100 120 140 160 180 20

0

−8

−6

−4

−2

0

2

Reactor temperature residual

Temperature (K)

0 20 40 60 80 100 120 140 160 180 20

0

−2

0

2

4

6

8

Coolant temperature residual

Time (min)

Temperature (K)

Figure 5: Reactor and coolant temperature residuals through Luenberger observer

scheme (no model uncertainty)

0 20 40 60 80 100 120 140 160 180 20

0

−10

−8

−6

−4

−2

0

Temperature (K)

Reactor temperature residual

0 20 40 60 80 100 120 140 160 180 20

0

−4

−2

0

2

4

6

Time (min)

Temperature (K)

Coolant temperature residual

Figure 6: Reactor and coolant temperature residuals through Luenberger observer

scheme (with model uncertainty)

18

1 2 3 4

00 3.92 9.51 35.80 67.52

01 24.01 8.82 44.98 16.80

10 55.38 55.81 42.20 39.04

11 76.25 58.18 44.36 16.32

Table 2: Monte Carlo simulation (Luenberger observer with model uncertainty)

The criteria used for the other (“01”, “10”, “11”) scenarios are chosen accordingly. Table

2 summarizes the results of how efficiently the fault isolation scheme was able to predict the

correct fault locations for random uncertainties in all the parameters within the range described

above. These results show that the parametric uncertainty can have a strong effect on robustness

properties of a fault diagnosis scheme and hence requires techniques that can cope with model

uncertainty. Because of these limitations, the nonlinear fault detection scheme presented in this

work is applied to the same scenario. Since the effect of uncertainty in the process parameters

other than the activation energy has been determined to be of lesser importance for fault isolation,

only uncertainty in the activation energy

{

}

: : 0.94 1.06

s

sss

EEEE∏= = ≤ ≤ is

considered ss

E=76534.704( / )Jmol. However, while the design is solely performed based upon

uncertainty in this one parameter, the evaluation of the fault diagnosis scheme will consider

uncertainty in all of the parameters to compare it to the Luenberger observer scheme. The

interval polynomial computed by a step-by-step procedure as discussed in section 4.2.2 is as

follows:

a) The Jacobian of the nonlinear dynamic model is symbolically evaluated around an

equilibrium point ),( ssss

x

θ

as a function of ),( ssss

x

θ

:

=

333231

232221

131211

),(

aaa

aaa

aaa

xA ssss

θ

where, the entries of the matrix ),( ssss

xA

θ

are nonlinear functions of ),( ssss

x

θ

.

b) The characteristic polynomial of the system is computed as shown in equation (15).

()

()()

32

332211133112212211331132233322

221331231231321321331221322311332211

ssaaasaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaas

+−−−+−−++−+

++−−++−= …

δ

The coefficients of the characteristic polynomial are continuous nonlinear functions of

),( ssss

x

θ

.

c) ss

x is eliminated from the coefficients by using the equation 0),( =

ssss

xf

θ

for computation

of the upper and lower bounds of the coefficients of the above characteristic polynomial.

However, since analytic expression of the coefficients as a nonlinear function of only ss

θ

can

usually not be derived, ss

x is evaluated numerically by solving the equation 0),(

=

ssss

xf

θ

for ss

θ

. By varying the uncertain parameter vector in the set

∏

, the maximum and minimum

values of the coefficients of the characteristic polynomial are computed over the set

∏

.

Thresholds

Scenarios

19

This procedure is used to evaluate the interval family of polynomials given by equation

(30) for the nonlinear system described by equation (6). Figure 7 shows the plots of the

coefficients of the characteristic polynomial as the activation energy E varies in the

set

{

}

: : 0.94 1.06

s

sss

EEEE∏= = ≤ ≤ , 76534.704 /

ss

EJmol

=

. The interval polynomial family

thus computed takes the following form:

()

23

01 2

ssss

δδδδ

=+ + +

(30)

where

[]

11840,2143∈

o

δ

,

[

]

9090,1648

1

∈

δ

,

[

]

289,79

2

∈

δ

7 7.2 7.4 7.6 7.8 8 8.

2

x 10

4

0

5000

10000

15000

Plots of the coefficient of the characteristic polynomial

δ

0

7 7.2 7.4 7.6 7.8 8 8.

2

x 10

4

0

5000

10000

δ

1

7 7.2 7.4 7.6 7.8 8 8.

2

x 10

4

0

100

200

300

Activation energy (J/mol)

δ

2

Figure 7: Plots of the coefficients of the characteristic polynomial as a function of

the activation energy.

It can be verified by Theorem 1 that the interval polynomial family given by (30) is

Hurwitz stable, thereby verifying that the nonlinear system given by equation (28) is locally

stable around the operating points as E varies in the set

{}

ssss EEEE 06.194.0:: ≤≤==∏ ,molJEss / 704.76534

=

.

The detailed derivation of the observer gain computation is not presented here due to

space constraints, but the procedure has been provided in section 4.2.2. The observer gain

computed for the simultaneous state and parameter estimator from the reactor temperature is

1

5929

12970

(, ) (, ) 11347.5

6113

Lx T x

θθ

−

−

−

=

−

−

20

where,

()

1,Tx

θ

−

is the locally invertible transformation as shown in section 4.2.2. Similarly the

observer gains for the state estimator of the form equation (20) to be used for fault isolation are

computed to be

1

11

5929

ˆˆ

( , ) ( , ) 4143

878.5

Lx T x

θθ

−

−

=−

1

22

5929

ˆˆ

( , ) ( , ) 4143

878.5

Lx T x

θθ

−

−

=−

Using the presented technique and applying it to a system with uncertainty in all of the

model parameters, it is found that estimate of the activation energy converges to its true value

after 7 min in the absence of sensor faults. The condition that there is no initial sensor fault is a

reasonable assumption since one would like to have a certain level of confidence in the

measurements before a fault diagnosis procedure is invoked. Figure 8 shows the fault signal fs(t)

that is affecting the sensors. The corresponding coolant and reactor temperature residuals

generated by the Kharitonov theorem-based fault identification techniques are presented in

Figure 9. It is apparent that the residuals converge to the values of the faults even when

uncertainty exists in the model parameters. Additionally, the location, shape, and magnitude of

the faults are correctly reconstructed and sensor noise is filtered.

Since the performed simulation has only used uncertainty in the activation energy, Monte

Carlo simulations have a 100% success rate for the scenarios considered in Table 2. However,

since this is not a very realistic assumption and in order to compare the presented fault detection

scheme to the Luenberger observer-based one, Monte Carlo simulations are performed taking

uncertainty in all the parameters

(5%, 6%,

oo

kkEE∆= ∆= ( ) 5% , and 5% )HHUAUA∆∆ = ∆ ∆ = into account. The results are

summarized in Table 3 and clearly show that the fault detection, isolation, and identification

scheme performs very well even under the influence of uncertainty in all the model parameters.

It should also be noted that the assumption that only the activation energy has a major impact on

the fault diagnosis was a good one, since the fault identification was only designed for

uncertainty in this parameter; nevertheless, reliable fault diagnosis is possible even under the

influence of uncertainty in several other parameters. Additionally, it can be concluded that it is

an important task to choose an appropriate threshold for determining a fault.

1 2 3 4

00 100 100 100 100

01 100 100 100 96.45

10 89.9189 100 100 100

11 100 100 100 100

Table 3: Monte Carlo simulation (presented approach with model uncertainty)

Thresholds

Scenarios

21

0 20 40 60 80 100 120 140 160 180 200 22

0

−8

−6

−4

−2

0

2

Reactor temperature fault

Temperature (K)

0 20 40 60 80 100 120 140 160 180 200 22

0

−2

0

2

4

6

8

Coolant temperature fault

Time (min)

Temperature (K)

Figure 8: Reactor and coolant temperature fault signal

0 20 40 60 80 100 120 140 160 180 200 22

0

−8

−6

−4

−2

0

2

Reactor temperature residual

Temperature (K)

0 20 40 60 80 100 120 140 160 180 200 22

0

−2

0

2

4

6

8Coolant temperature residual

Time (min)

Temperature (K)

Figure 9: Reactor and coolant temperature residual signal through presented scheme

(with model uncertainty)

5.2 Fault diagnosis of a reactor with uncertain and time-varying parameters

In this section, the performance of the proposed fault diagnosis scheme is evaluated for the non-

isothermal CSTR problem as introduced in section 5.1 but with model parameters varying with

22

time. Since activation energy affects the behavior of the system significantly stronger than any

other parameter, it is assumed that only the activation energy varies with time possibly due to

catalyst deactivation or coking. Figure 10 shows the plot of the activation energy and its estimate

over the simulated time span and Figure 11 presents the fault signal fs(t) that is affecting the

sensors. The corresponding coolant and reactor temperature residuals generated by the

Kharitonov theorem-based fault identification technique are shown in Figure 12. The time period

during which the parameter is identified within acceptable limits ranges from t=0 to 10 min.

These times were determined by comparing the measured output and the predicted output. The

first long time period during which fault detection and identification is invoked ranges from 10

min to 200 min. The parameter is adapted from 200 min to 210 min. This is followed by another

fault detection period ranging from t=210 to 400 min. It can be concluded from Figure 12 that

the fault identification scheme is effective even in the presence of time-varying uncertain

parameters. It should be noted that the system would not work as well if the parameters are not

periodically re-identified, as can be seen from Figure 12 during the time period just before 200

min.

0 50 100 150 200 250 300 350 400 45

0

1.02

1.03

1.04

1.05

1.06

1.07

1.08

Activation energy

Time (min)

Activation energy/76534.704 (J/mol)

Actual

Estimate

Figure 10: Activation energy change with time

23

0 50 100 150 200 250 300 350 400 45

0

−8

−6

−4

−2

0

2

Reactor temperature fault

Temperature (K)

0 50 100 150 200 250 300 350 400 45

0

−2

0

2

4

6

8

Coolant temperature fault

Time (min)

Temperature (K)

Figure 11: Reactor and coolant temperature fault signal

0 50 100 150 200 250 300 350 400 45

0

−8

−6

−4

−2

0

2

Reactor temperature residual

Temperature (K)

0 50 100 150 200 250 300 350 400 45

0

−2

0

2

4

6

8

Coolant temperature residual

Temperature (K)

Time (min)

Short time period(1) Short time period(3)

Long time period(2)

Long time period(2)

Long time period(2)

Long time period(2)

Figure 12: Reactor and coolant temperature residual signal through presented scheme

(with time-varying parametric uncertainty)

6 Conclusions

A new observer-based fault diagnosis scheme for nonlinear dynamic systems with

parametric uncertainty was presented. This approach is centered around two main components:

the design of a nonlinear observer, which includes uncertain parameters as augmented states, and

the choice of an appropriate fault isolation and identification filter for reconstructing the location

24

and nature of the fault. The observer design was performed based upon Kharitonov’s theorem

but takes into account the effect that changes in the parameters have on the steady state of the

system. This resulted in a nonlinear, augmented observer, which has the property that it is locally

stable for parametric uncertainty within a specified range. The fault isolation and identification

filter was designed based upon a linearization of the nonlinear model at each time step.

Repeatedly computing linearization of the model does not pose a problem in practice since it is

computationally inexpensive.

Since it is not possible to simultaneously perform parameter estimation and fault

detection, these two tasks were implemented at different time scale. The parameters were

estimated at periodic intervals where the fault was either assumed to be zero or known and

constant, whereas the fault detection scheme was invoked at all times with the exception of the

short periods used for parameter estimation.

The performance of the proposed fault diagnosis method was evaluated using a numerical

example of an exothermic CSTR and by performing Monte Carlo simulations on a bounded set

of parametric uncertainties for a series of faults in both of the available measurements. The faults

were reconstructed correctly even in the presence of severe uncertainties in the model parameters

and measurement noise.

Notation

CA, System matrices

CA, System matrices for the augmented system

CA

, Canonically transformed matrices

s

f Vector of faults

)(),( xhxf Vector fields in state space description of a continuous time nonlinear system

l Constant observer gain vector.

L Constant observer gain matrix

)

~

,

~

(

θ

xL Nonlinear observer gain for augmented system

)

~

,

ˆ

(

θ

xL Nonlinear observer gain for original nonlinear system

hL f Lie derivative of )(xh w.r.t )(xf

)(tQ Fault reconstruction filter

)(tr Difference between the actual and estimated output.

t Time

,TT Invertible transformation matrix for augmented and original system, respectively

)(xWo Observability matrix

x

Vector of state variables

x

ˆ Estimate of state variables of the original nonlinear system

x

Estimate of state variables of the augmented system

x

Augmented state variables

y Vector of output variables

y

ˆ Estimate of output variables

z Transformed state vector

25

Greek letters

)(s

δ

Open loop interval polynomial family

()s

γ

Closed loop interval polynomial family

Ω Hyperrectangle of coefficients of an interval polynomial family

θ

Uncertain parameter vector

θ

~ Estimate of parameter vector

ss

θ

Nominal parameter value

Π Hyperrectangle within which the uncertain parameter varies.

)(x

φ

Nonlinear map

)( A

λ

Eigen values of the matrix A

ξ

Vector of state variables

,,

η

εα

Positive scalars

(.)S Unit step function

Other symbols

2

. Euclidean norm

n

R

n-dimensional Euclidean space

Complex plane

−

Left half complex plane.

Appendix A. Observer form state transformation32

The following state space representation of a LTI system

Cxy

Axx

=

=

is given where n

x

R∈ and 1

yR∈. The characteristic polynomial of the matrix A results in:

21

01 2 1

() nn

n

sss ss

δδδδ δ

−

−

=+ + + + +………

The aim is to find a coordinate transformation matrix

T

, which transforms the aforementioned

LTI system into the following one:

zAz

yCz

=

=

, Txz

=

where,

26

1

2

2

1

0

10 00

01 00

0010

0001

0000

n

n

A

δ

δ

δ

δ

δ

−

−

=

[

]

10 000C=

The transformation matrix that transforms the original system into an observable canonical form

is designed as follows:

1) Let the transformation matrix

T

be presented by the row vector as follows:

=

−

n

n

t

t

t

t

t

T

1

3

2

1

where

1

1

ATAT

CCT

−

−

=

=

2) The first row of the matrix

T

can be obtained from the following relation

[]

==

−

1

1

3

2

1

0001 t

t

t

t

t

t

n

n

C

3) The remaining rows of the matrix

T

can then be computed by the following recursive relation

2111

3212

11nn n

tttA

tttA

tttA

δ

δ

δ

−

=− +

=− +

=− +

27

It can be shown that the invertibility of the transformation matrix

T

is guaranteed if the matrix

pair

}{

CA, is observable.

Appendix B. Observer gain computation6

Consider a polynomial

nn

no ssssss ++++++= −

−

1

1

3

3

2

21 ....)(

δδδδδδ

whose coefficients can vary independently within a given uncertainty range as follows

−=

∆

+≤≤

∆

−1,....,2,1,0,

22

:00 ni

i

ii

i

i

δ

δδ

δ

δδ

, , 0,1, , 1

ii i

in

δδ δ

+−

∆

=− = −……

The aim is to find a constant vector

(

)

121 ,,,, −

=

no lllll to transform the interval polynomial

family ()s

δ

into another interval polynomial family described by

21

00 11 22 1 1

() ( ) ( ) ( ) ( )nn

nn

sl lsls lss

γδ δ δ δ

−

−−

=+++++ + + + +………

such that, entire family ( )s

γ

remains Hurwitz.

1) Consider any stable polynomial ( )

R

s of degree 1

−

n. Let

(

)

()

R

s

ρ

be the radius of the

largest stability hypersphere6 around ()

R

s. It can be shown that for any positive real number

α

,

()

()

()

()

R

sRs

ρα αρ

=6.

2) Thus it is possible to find a polynomial

(

)

R

s

α

such that

()

()

2

1

0

()

4

n

i

i

Rs

δ

αρ

−

=

∆

>∑

3) Denoting 21

01 2 1

( ) ... nn

n

R

srrsrs rs s

−

−

=+ + ++ +, the constant vector l is calculated as follows:

{

}

0

: , 0,1, 2,..., 1

iii

ll r i n

αδ

=

−= −

It can be seen from above calculations that for a given interval family ()s

δ

with associated

uncertainty ranges there is an infinite number of possibilities for the constant gain vector l that

transform the given interval family ( )s

δ

into ( )s

γ

such that ( )s

γ

is Hurwitz.

Acknowledgements

The authors would like to thank Professor Shankar Bhattacharyya for his comments in

preparation of this manuscript.

References

[1] Doyle, F.J. Nonlinear inferential control for process applications. Journal of Process

Control, 1998, 8, 339.

28

[2] Soroush, M. State and parameter estimations and their applications in process control.

Computers and Chemical Engineering, 1998, 23, 229.

[3] Chen, J.; and Patton, R. Robust Model based fault diagnosis for dynamic systems. Kluwer

Academic Publishers, 1999.

[4] Garcia, E.A.; and Frank, P.M. Deterministic nonlinear-observer based approaches to fault

diagnosis: A survey, Control Engineering Practice, 1997, 5, 663.

[5] Frank, P.M.; and Ding, X. Survey of robust residual generation and evaluation methods

in observer-based fault detection systems. Journal of Process Control, 1997, 7, 403.

[6] Bhattacharyya, S.P.; Chappellat, H.; and Keel, L.H. Robust Control: The Parametric

Approach. Prentice Hall PTR, Upper Saddle River, NJ, 1995.

[7] Venkatasubramanian, V.; Rengaswamy, R.; Kavuri, S.N.; and Kewen, Yin. A review of

process fault detection and diagnosis: Part III: Process history based methods. Computers

and Chemical Engineering, 2003, 27, 327.

[8] Massoumia, M.; Verghese, G.C.; and Willsky AS. Failure detection and identification.

IEEE Transactions on Automatic Control, 1989, 34, 316.

[9] Kruger, U.; Chen, Q.; McFarlane, R.C.; and Sandoz D.J. Extended PLS approach for

Enhanced Condition Monitoring for Industrial Processes. AIChE Journal, 2001, 47(9)

2076.

[10] Qin, S.J. Statistical process monitoring: basics and beyond. Journal of Chemometrics,

2003, 17(8-9), 480.

[11] Soderstrom, T.A; Himmelblau, D.M.; and Edgar, T.F. The Extension of a Mixed-

Integer Optimization-based Approach to Simultaneous Data Reconciliation and Bias

Identification. FOCAPO 2003, Boca Raton, FL, January, 2003.

[12] Wattanabe, K.; and Himmelblau, D.M. Instrument fault detection in system with

uncertainties. International Journal of System Science, 1982, 13(2), 137.

[13] Wunnenberg, J.; and Frank, P.M. Sensor fault detection via robust observers, in

Tzafestas, S.G.; Singh, M.G.; and Schmidt, G. (eds), 147-160. System Fault Diagnostics,

Reliability & Related Knowledge-Based Approaches. D. Riedel Press, Dordrecht, 1987.

[14] Xiong, Y.; and Saif, M. Robust fault detection and isolation via a diagnostic observer.

International Journal of Robust Nonlinear Control, 2000, 10, 1175.

[15] Patton, R.J.; and Kangethe, S.M. Robust fault diagnosis using eigen-structure

assignment of observers, chapter 4, 99-154. Fault Diagnosis in Dynamic Systems, Theory

and Application. Prentice Hall, 1989.

[16] Seliger, R.; and Frank, P.M. Robust fault detection and isolation in nonlinear dynamical

systems using nonlinear unknown input observers. In Preprints of the IFAC/IMACS

Symposium SAFEPROCESS’ 91, 1991, 1, 313, Baden-Baden.

[17] Ding, X.; and Frank, P.M. Frequency domain approach and threshold selector for

robust model-based fault detection and isolation. In Preprints of IFAC/IMACS Symp.

SAFEPROCESS’91, 1991, Baden-Baden.

[18] Frank, P.M.; and Ding, X. Frequency domain approach to optimally robust residual

generation and evaluation for model-based fault diagnosis. Automatica, 1994, 30(4), 789.

[19] Marquez, H.J.; and Diduch, C.P. Sensitivity robustness in failure detection: A

frequency domain approach. In Proceedings 29th IEEE CDC, Honolulu, USA, 1990.

[20] Basseville, M. Detecting changes in signals and systems – a survey. Automatica, 1988,

3, 309.

29

[21] Ding, X.; and Frank, P.M. On-line fault detection in uncertain systems using adaptive

observers. European Journal of Diagnosis and Safety in Automation, 1993, 3, 9.

[22] Frank, P.M.; Ding, X.; and Guo, L. An adaptive observer based fault detection system

for uncertain nonlinear systems. In Proceedings of 12th IFAC World Congress, 1993.

[23] Tortora, G. Fault–tolerant control and intelligent instrumentation. IEEE Computing &

Control Journal, 2002, 13, 259.

[24] Francis, B.A. A course in H

∞

control theory. Springer Verlag, Berlin-New York, 1987.

[25] Ding, X.; and Frank, P.M. Fault detection via factorization approach. Systems &

Control Letters, 1990, 14, 431.

[26] Bestle, D.; and Zeitz, M. Canonical form observer design for nonlinear time-varying

system. International Journal of Control, 1988, 47(6), 1823.

[27] Othman, S.; Gauthier, J.P.; and Hammouri, H. A simple observer for nonlinear systems:

Applications to bioreactors. IEEE Trans. Automatic Control, 1992, AC-7, 875.

[28] Bastin, G.; and Gevers, M.R. Stable adaptive observers for non-linear time-varying

systems. IEEE Trans. Automatic Control, 1988, 7, 650.

[29] Krener, A.J.; and Isidori, A. Linearization by output injection and nonlinear observers.

Systems & Control Letters, 1998, 34, 241.

[30] Kazantzis, N.; and Kravaris, C. Nonlinear observer design using Lyapunov’s auxiliary

theorem. Systems & Control Letters, 1988, 34, 241.

[31] Hermann, R.; and Krener, A.J. Nonlinear controllability and observability. IEEE Trans.

Autom. Control, 1977, AC-22, 728.

[32] Fairman, F.W. Linear Control Theory. John Wiley and Sons, New York, 1998.

[33] Zeitz, M. The extended Luenberger observer for nonlinear systems. Systems & Control

Letters, 1987, 9, 149.

[34] Vejtasa, S.A.; and Schmitz, R.A. An experimental study of steady-state multiplicity and

stability in an adiabatic stirred reactor. AIChE Journal, 1970, 3,410.

[35] Fogler, H.S. Elements of Chemical Reaction Engineering; Prentice Hall, Englewoods

Cliffs, NJ, 1992.