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Sep 16, 1993 - vides a method for estimating the parameter error in the system. Keywords: robust fault detection, nonlinear sys- tems, sliding mode observers.
Second IEEE Conference on Control Applications, September 13 - 16, 1993 Vancouver, B.C.

Robust Fault Detection in Nonlinear Systems Using Sliding Mode Observers Ftajiv Sreedhar Benito Fernandez * Glenn Y. Masada Department of Mechanical Engineering University of Texas at Austin Austin Texas - 78712

Abstract A model-based scheme for robust detection and isolation of faults in a twin continuously-stirred tank reactor is presented. The scheme uses sliding mode observers for robust fault detection in the presence of parameter uncertainties in the system model. The fault detection and isolation scheme is validated by simulated faults in the sensors, actuators and plant operating parameters. The fault detection and isolation technique based on sliding mode observers is shown to be robust to parameter uncertainty in the model. The technique also provides a method for estimating the parameter error in the system. Keywords: robust fault detection, nonlinear systems, sliding mode observers.

1

Introduction

The early detection and isolation of system malfunctions is indispensable for the fail-safe operation of an automated chemical process. This requirement underscores the need for a robust on-line fault detection and isolation (FDI) technique. The developments in nonlinear control theory and the advances in computer systems have had a synergistic effect on the development of sophisticated on-line FDI techniques. A number of on-line FDI techniques have been developed of which the model-based FDI techniques have yielded the best results. The model-based techniques use the analytical redundancies in the system for fault detection and isolation. A review of model-based FDI techniques have been presented by Isserman [7], Gertler [6] and Frank (51. The FDI techniques, based on state estimation are widely used model-based FDI techniques. Most analytical redundancy techniques based on residual generation, have been shown to be special cases of observer-based techniques. The theory of observer-based FDI techniques for linear time invariant systems is well developed, successful implementations have been presented by Clark [2], Tylee 1171 and Patton 114). However most chemical processes are

nonlinear hence the use of linear observer based FDI techniques for such processes has severe limitations. Watanabe [18]presents an observer-based FDI scheme which is minimally sensitive to certain nonlinearities in the system. Frank [5] presents a nonlinear observer-based approach for nonlinear systems.

2

Robust FDI Using Sliding Mode Observers

The observer design techniques for nonlinear systems rely on the exact cancellation of the nonlinearities in the system to obtain linearized error dynamics. In order to use linear observer design techniques, the nonlinear system is often globally linearized with an appropriate coordinate transform and nonlinear feedback. In some cases the system is pseudo-linearized in the neighborhood of an operating point. Such linearizations of nonlinear systems are very sensitive t o parameter uncertainties in the system. Consider a nonlinear system described by the equations

where z = (zl, 22, ..., z,,)are the local coordinates for a smooth manifold M , and f and g are smooth vector fields on M , and h : M -+ R is a smooth mapping from the state space M to the output space. If the linearizability index of the output (r) is less than the order of the system (n), the nonlinear system can be transformed into an equivalent linear system of the form

i = Az y

=

cz

+ Bv

(3) (4)

where z represents a smooth, invertible nonlinear transform of the local coordinates spanning the state space M . The mapping from the input U to the input of the linear system v is a diffeomorphism represented by, =

'Corresponding Author

CH3243-3/93/0000-0715$1.00 0 1993 IEEE

((.)U

+

K(2)

(5)

The effect of parameter uncertainties, unmodeled disturbances, and unmodeled dynamics can be represented by unknown inputs to the system. The dynamics of the system is represented by

and,

j. =

y

respectively. Where

=

A x + U o ( z , u ) + ET]

(18) (19)

cz

respectively, where E is a constant known matrix and the unknown inputs vi are bounded by JqiJ< 1. Bastin and Gevers [l] and Marino [ l l ] present an adaptive observer scheme where the parameter is adapted online to give accurate state estimates. Frank [5] presents an application of the adaptive observer scheme for fault detection. The use of sliding mode observers for robust fault detection presented herein offers a distinct computational advantage over the adaptive observer scheme. The design procedure for sliding mode observers has been presented by Misawa [12]. The sliding mode observer dynamics are represented by

is the Lie derivative. The input-output linearized system then satisfies the relation (9)

The matrices A, B, and C are defined by

y o 1 o ... 0 1

j. =

e =

( A - LC)z + Ly + 'Po($,U ) + ZtZ, (i-i) = ( A - L C ) e + E q - K Z .

(20) (21)

respectively. The vector Z,is obtained by scaling the estimation error, projecting it onto the range of matrix E and then taking the tanh of each element.

roi

respectively. The system represented in Equation 3 is linear with respect to input v, and is in observer canonical form. The matrices ( A , C) form an observable pair. The transformed states z can thus be observed by an observer with linear error dynamics. The observer dynamics and the error dynamics are represented by the following equations. i = (A-LC)i+Ly+Bv i = ( i - i )= ( A - L C ) ~

where

R(i) = E ( E T E ) - ' E T ( z- i) (23) respectively. The function tanh is chosen instead of the signum function to obtain a smooth differentiable mapping. The sliding surface is chosen as

(12) (13)

S ( t ) = { r ( t ) 1 S(X) = E ( E T E ) - ' E T ( x ( t )- i ( t ) ) = 0) (24) The sliding surface s ( t ) is attractive if the condition

The observer error asymptotically approaches zero if ( A LC) is stable. The extension of this approach to multiinput, multi-output cases is presented by Isidori [SI and Nijmeijer [13]. In the pseudo-linearization approach, the dynamics of the system are linearized in the neighborhood of an operating point (zo,u o ) . The linearized system dynamics are then represented by equations of the form X

=

A X + U,(X,U)

y

=

cx

lim sT(t).i(t)=

a(t)-O

E(ETE)-'ET lim .(+O

(x;- i,)(i, - &)

< 0; Vi (25)

is satisfied (Fernandez [4, 31). The matrix I< is a diagonal matrix with elements IC;, large enough to make the surface attractive (Equation 25). The scaling factor p, determines the thickness of the boundary layer associated with the sliding surface s ( t ) = 0. It should be noted that this design procedure is valid only when full state measurement is available. In the absence of full state measurement, the design of the observer is considerably more difficult (Misawa [12]). If the attractiveness condition (Equation 25) is satisfied the system is robust to the disturbance Eq and remains within a boundary layer enveloping the sliding surface s ( t ) = 0. The dynamics of the system with faults are represented by

(14) (15)

where A , C are constant matrices and ~ k o ( x ,is ~a)known function. Linear design techniques are used to design observers for the pseudo-linearized system. The observer dynamics are represented by

respectively.

j. =

716

Ax

+ U,(S,U) + Ea -+ M J

(26)

where M is the fault distribution matrix and f is the fault vector. The faults M f in the system are detected by measuring the deviation of the system trajectory from the sliding surface. The relations between the faults M f in the system and the observer innovations are represented by

18

The states 1ltoz4 correspond to tank 1, while states zStoz8 correspond to tank 2. The control variables are u1 = F/10 : Feedstream flow (lbm/min),

( I - E ( E ~ E ) - ' E ~ ) L ( =- i) = ( I - E ( E ~ E ) - ' E ~ ) M ~ (27) Since the observer is robust to disturbances of the form Eq, the innovation vector L ( z - i) is contained in the nullspace of E, therefore the innovations satisfy the relation E ( E ~ E ) - ' E ~ L (-~i) = o

= To2/10 : Sensed output temperature ( O F ) of tank 2.

u2 : Feedstream heat flow (BTU/min) of tank 1, u3 : Feedstream heat flow (BTU/min) of tank 2. respectively. The following nonlinear state equations represent the dynamics of the system

(28)

Thus Equation 27 is equivalent to

q5- i) = ( I - E ( E ~ E ) - ' E ~ ) M ~

(29)

Thc lcast squarc solution or the fault vcctor f is determined using Equation 29. If M is full rank and if f is of dimension less than R - rank(E), a unique solution for f can be determined. The term KZ,,(Equation 21) exactly cancels the disturbance E9. A least square solution for 9 can be obtained from the relation ItZ, = E T . If JI corresponds to modeling errors induced by faults, they can be detected by placing bounds on the magnitude and/or the rate of change of the parameter 0. The least square solution for 9 is given by =(E~E)-'E~(K~~)

(30)

xs =

3

System Description

The following example of a continuously stirred-tank reactor (CSTR) illustrates the robust FDI technique. The CSTR system is reviewed by Perry and Chilton (151 and the details are presented by Levenspiel [lo] and Smith (161. Consider the CSTR shown in Figure 1. In the reactor, chemical A reacts to yield chemical B (product) which itself reacts to form chemical C (by-product).

where

-E2

= exp(P2 - R(Tol

-E3 R3 = K3 exp( )zi

= exp(P3 -

RT02

Ac-.B++C

(31)

-E4

R4 = K4 exp( -)x6 RT02

The contents of the CSTR are assumed to be perfectly mixed, so the output concentrations COand temperature Toare the same as those in the tank. The input stream consists of a mixture of reactants with concentrations C,A,B,C (moles/Kmole) and temperature T, ( " F ) . The reaction is exothermic and heat is removed by a coolant flowing through a jacket around the reactor. The state variables are 11, 5 5 : Concentration of A (moles of A/Kmoles of mixture),

E3 R(To2 46)

(41) (42)

E4

(43)

Cik, k = 1,2,3

(moles of A,B or C per Kmole of mixture) (44) Ti = (Feedstream temp.(OF)/lo The nominal values for the constants used in this example are

% = 2800.0; % = 3750.0; % = 2800.0; % = 3750.0 V, = 1000.0; & = 200.0; 01 = 1.65; a2 = 3.25 a3 = 1.65; a 4 = 3.25; 81 = 26.427; Pz = 33.283 P3 = 26.427; P 4 = 33.283; q1 = 15.0; 92 = 1.0 43 = 0.0; Ti = 10.0; 210 = 15.0; 2 2 0 = 2.0 530 = 0.5; 240 = 15.0; rc = 0.2 (45) respectively (adapted from Fernandez [3] and Kravaris (91).

: Concentration of B (moles of B/Kmoles of mixture),

14

+ 46))"

+ = exp(D4 R(To2+ 46)lz6 q k = 1000 x

5 2 , 16

23,

Ez

RTo1 Rz = KZexp(-)zz

: Concentration of C (moles of C/Kmoles of mixture),

27

= Tol/10 : Sensed output temperature ( O F ) of tank 1,

717

{

}'

qT = AR1 ARz AR3 AR4 (50) respectively. A sliding mode observer is designed for the linearized system (Equation 18). The dynamics of the sliding mode observer are represented by Equations 20 and 21. The matrix L (Equation 51) is arbitrarily chosen so that the poles of ( A - LC) are at -0.3. The elements of the diagonal matrix ' h are chosen large enough to make the sliding surface a ( t ) attractive (Equation 25). The diagonal elements kii are chosen equal to 15.0, which is approximately five times the largest singular value of matrix E . The parameter ,u; determines the thickness of the boundary layer around the sliding surface and is chosen as small as possible. However very small values of p, cause the state estimates to chatter, so the boundary layer thickness is increased until the chatter vanishes. For this system p , = 0.15 yields satisfactory results.

rC*F

p

0 o

L= The states z1- IS of the twin CSTR system are measured. The state equations (Equations 32 - 39) are linearized in the neighborhood of 211 = 1.0. It is assumed that only the nominal values of RI-4 (Equations 40 - 43) can be calculated. Thus the actual values of RI-4 are represented by

R, = hi + AR,; i = 1, 2, , 3 , 4

L

o

o

0 0 PL

rL

(46)

q

o

o

T

p

0 L

O

o

0

o

L

o

QL SL

o

0

O q L O o

o

q

0 s L O 0 0 O S L O

O r L O 0 0 r

2.988 x lo-'; = 6.0 x

L

O q L O

O P L O

r L 0 O

o

0

O P L O

Figure 1: Twin Continuously Stirred Tank Reactors with Recirculation

O

O

~

O

s

L

L

0

0 O

O

O

s

~

= 2.0 x = 2.94 x lo-'

The innovation vector L ( x - i) is an element of the null-space of E. The structure of E (Equation 49) indicates that the null-space of E can be partitioned into two orthogonal subspaces, each of dimension two, associated with the dynamics of CSTR 1 and CSTR 2, respectively. It is therefore possible to uniquely detect and identify only two faults each, in CSTRs 1 and 2, respectively. The innovations used for fault detection are based on the innovations associated with states 2 2 , 5 4 , 26, and x s , respec-

respectively. The dynamics of the actual plant are then represented by Equation 18, where

tively. Let G = [g2ig4Ige!gs] be a matrix consisting of the 2nd, 4'", 6**,and €Jth columns of ( I - E ( E T E ) - ' E T ) .The robust residues associated with states 5 2 , 5 4 , 2 6 , and zS, respectively, are defined as r = (G=G)-'G=L($- 2 )

(52)

The residues are equal to zero in the absence of a fault. Typically a fault is signaled if the residue crosses a certain threshold value. A threshold value of 0.2 is chosen for the simulations shown below.

4

Simulations

Sensor faults, actuator failure and some process faults are simulated. The twin CSTR model is assumed to have modeling errors.The nominal values for the constants used in the model are given in Equation 45. The actual values of the constants in the CSTR are

L o

0

0 0

0

-1

9 = 2500.0; 9 = 2700.0;

l a ; ]

@z = 33.0;

@4

= 33.1

(53)

718

0.5

-

1

0.5

-0.5

-

0

-0.5

1

0

100

200

tlr.

300 I..=.)

400

100

0

600

300

200

..it

respectively.

500

600

0

Abrupt Sensor Faults

... -

-0.1

The detection of abrupt faults in the sensors measuring states 52, 1 4 , 2 6 , and zg is simulated. Bias faults of -1 units in the measurements of states 5 2 and 56 are introduced at time t = 200, and t = 375, respectively. Similarly bias faults of $1 units are introduced in the measurements of states z4 and x g at time t = 300 and t = 500, respectively. The duration of each fault is 50 seconds. Figure 2 shows a plot of the residues (Equation 52) using a sliding mode observer (Equation 20). Sharp spikes mark the time at which the faults occur and disappear. The direction and magnitude of the spike indicate the sign and magnitude of the bias error. Figure 3 shows the same residues obtained using a linear observer designed for the linearized plant. The dynamics of the linear observer are represented by Equation 16. The observer gain matrix L for the linear observer is the same as that for the sliding mode observer. The figures show that the residues obtained from the linear observer are significant even when there is no fault in the system. However the residues obtained from the sliding mode observer are significant only when there is a fault in the system. Thus the FDI technique based on sliding mode observers is robust to parameter uncertainty and is less likely to signal false alarms. It must be noted that there are some choices of the linear observer gain matrix L which may make it minimally sensitive to the parameter errors. Robust fault detection results based on such observers has been presented by Frank [5], Patton et-al. [14] and Watanabe et-al. [18].

4.2

400

(..C.l

Figure 3: FDI of Sensor Faults using Linear Observers

Figure 2: FDI of Sensor Faults using Sliding Mode Observers

4.1

100

I

-0.2

4

-0.3

-0.4

-0.5

-0.6

-0.7

-0.0

I tlm.

I.eC.1

Figure 4: FDI of Actuator Faults using Sliding Mode Obsewers efficiency of heat exchanger in tank 2 is degraded to 90% between time t = 300 and t = 500 seconds. Figures 4 and 5 show the residues obtained using a sliding mode observer and a linear observer for the linearized plant, respectively. A fault in heat exchangers 1 and 2 cause bias changes in the residues associated with state 2 4 and x g , respectively. The residues obtained from the sliding mode observer clearly show the presence of a fault in heat exchangers 1 and 2. When a linear observer is used the fault signature is superimposed on the residues due to modeling errors.

4.3

Process Faults

The fouling of the recirculation tube connecting tank 1 and tank 2 is simulated. The recirculation changes from 20% of the feed stream flow to zero a t t = 100 seconds. It can be seen from the state equations (Equations 32 - 39), that a change in the recirculation flow causes residues associ= ated with 1 2 16 and 5 4 x8 to be in the ratio Figures 6 and 7 show the residues obtained using a slid-

Actuator Faults

A decrease in efficiency in the heat exchangers in tank 1 and 2 is simulated. The system is operating at steady state with states z4,z6,and z g equal to 28,4 and 35, respectively. The efficiency of heat exchanger in tank 1 is degraded to 80% between time t = 100 and t = 250 seconds, and the

2 2.

719

... 1.1

1.4

k

~

4

- ...

................... ............................ L

0.2

3.5

-

r... r... ...I

2

3 4

..... -

......

3

........

J

2.5

..................................................................................... -0.2

2

-

1.5

-0.4

0

100

200

300 th.

400

500

600

I..C.)

1

Figure 5: FDI of Actuator Faults using Linear Observers

1.6 1.4 1.2

-

Q I

-0.2

-

1

r...

2 3 4

C...

r...

i!

-. ..... -

0

100

200

300

tim.

400

500

600

m.s.1

Figure 7: FDI of Recirculation Fouling using Linear Observers

_____

L--

-0.4

2x8.

0.5

~

Figure 6: FDI of Recirculation Fouling using Sliding Mode Observers 0.16

ing mode observer and a linear observer for the linearized plant. Figure 6 shows that the residue 7-4 associated with state zg is approximately -5 times the residue r2, associated with state q.This relationship is not evident between residues T I and 7-3, because the domain in which the plant is operating does not cause significant changes in r1 and 73. When linear observers are used this relationship is not noticeable, as the residuals are corrupted by artifacts due to modeling errors. When sliding mode observers are used it is also possible to evaluate the parameters {AR}= using the relation

{ A R } = (ETE)-'ET(ZtzS)

0.14

0.12

0.1 0.Ol

0.06 0.04

0.02

0

-0.02

(54)

I

0

In the simulation shown in Figure 6, the CSTR is simulated with the nominal values and a t time t = 200 seconds t.he parameter is changed from 2800.0 to 2450.0, and at time t = 400 seconds the parameter Q is changed from 2600.0 to 2700.0. This simulates a change in the rate of reB in tanks 1 and 2, action associated with reaction A respectively. The change in reaction rates causes errors in

100

200

300 t h s l..E.l

400

500

I

600

Figure 6: Detection of Parameter Changes due to Faults

-

720

parameters RI and RJ, respectively (Equation 40,42). The resulting errors AR1 and AR3 are detected using Equation 54. The results are shown in Figure 8.

[9) Kravaris, C. and Chung, C., “Nonlinear State Feed-

back Synthesis by Global Input/Output Linearization”, American Control Conference, Seattle, Washington, pp 997-1005, June 1986.

Conclusions

5

[lo] Levenspiel, O., Chemical Reactor Engineering, John Willey and sons, New York, 1962.

The robust detection of a subset of sensor, actuator and process faults using sliding mode observers has been presented. The performance of sliding mode observer-based FDI technique was shown to be robust to parameter uncertainty in the system model. The sliding mode observerbased FDI technique also provides a method for estimating the parameter error in the system.

6

[ll] Marino, R., (1990), ”Adaptive Observers for Single Output Nonlinear Systems”, IEEE Transactions on Automatic Control, AC 35, pp 1054-1058, September 1990. [12] Misawa, E.A., “Nonlinear State Estimation Using

Sliding Observers”, PhD. Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, February 1988.

Acknowledgments

[13] Nijmeijer, H. and van der Schaft, A.J., Nonlinear Dynamical Control Systems, Springer-Verlag 1990.

The authors appreciate the financial support of the Texas Higher Education Coordinating Board Energy Research Applications Program #003658-078.

[14] Patton, R.J. and Kangethe, S.M., “Robust Fault Diagnosis Using Eigenstructure Assignment of Observers”, in: Patton, R.J., Frank, P.M., Clark,R.N. (Eds), Fault

References

Diagnosis in Dynamic Systems, Theory and Applications, Prentice Hall, 1989.

[ I ] Bastin, B. and Gevers, M.R., “Stable adaptive Observers for Nonlinear Time varying Systems,” IEEE Transactions on Automatic Control, AC-33, pp 650658, July 1988.

[15] Perry, R.H. and Chilton, C.H., Chemical Engineers

Handbook, MacGraw-Hill Kogakusha, Tokyo, Japan. (161 Smith,

J.M., Chemical Engineering I