methodology for nonlinear fdi observer via nonlinear … et safepro… · hows: diagnosis principle...

irccyn

Methodology for nonlinear FDI Observer via Nonlinear Transformation :Application to a DC Series Motor

Ayan Mahamoud, Alain Glumineau, Ibrahim Souleiman

Institut de Recherche en Communications et Cybernetique de Nantes

Safeprocess’09, july 1st 2009

A. Mahamoud (IRCCyN) Methodology for nonlinear FDI Observer Safeprocess’09, july 1st 2009 1 / 39

irccyn

Introduction

Plan

1 Introduction


irccyn

Introduction

Plan

1 Introduction

2 Fault Detection and Isolation (FDI) - The howsHows: Diagnosis PrincipleHows:Techniques for FDI


irccyn

Introduction

Plan

1 Introduction


3 Observer-based FDIProblem StatementObserver DesignStability Proof


irccyn

Introduction

Plan

1 Introduction



4 Application to a DC Series Motor - Simulation ResultsThe DC series motor modelBenchmark definitionObserver Design - case 1Observer Design - case 2


irccyn

Introduction

Plan

1 Introduction




5 Conclusion


irccyn

Introduction

Introduction

The whys:

Automation of industrial systems ⇒ increasing complexity.

Need of safety, reliability and efficiency.

Difficulty: nonlinear behavior of most industrial systems complicates fault diagnosis.

Solution : to find an appropriate robust approach in order to monitor those industrialprocesses.

Objectives:

Use of observers for FDI purpose,

Study of the Observer Scheme stability,

Application to different cases of faults for a DC Series Motor,

Simulation validation of the Observer on a significant Diagnosis Benchmark.


irccyn

FDI FDI Principle

Plan

1 Introduction




5 Conclusion


irccyn

FDI FDI Principle

Diagnosis Principle

Twofold problem:

Detection thanks to Residual Generation ⇒ Residuala indicates fault occurrence,

Isolation ⇒ Residual identifies fault type.

Objectives:

Reduce the maintenance costs of industrial systems,

Ensure optimal efficiency and safety.

aComputed vector depending on both estimation and measure signals


irccyn

FDI Techniques for FDI

Plan

1 Introduction




5 Conclusion


irccyn


Different approaches for nonlinear diagnosis

The most recently developed methods include:


irccyn




parity relations (Cocquempot, Gertler, Isermann, Maquin, Ould Bouammama, Ragot,Staroswiecki),


irccyn





stochastic approach (Basseville, Frank, Isermann, Lesecq, Sauter, Zhang),


irccyn






parametric identification (Frank, Isermann, Schaeffer, Tnani, Trigeassou),


irccyn







observer-based approach (Alcorta Garcia, Besancon, Chen, DePersis, Ding, Edwards,Hammouri, Isidori, Kinnaert, Patton).


irccyn








Diverse Applications


irccyn









Power Systems (Cocquempot, Isermann, Kinnaert, Schaeffer, Staroswiecki, Tnani,Trigeassou),


irccyn










Aerospace Systems (Alcorta Garcia, Edwards, Isermann, Isidori, Zolghadri),


irccyn










Aerospace Systems (Alcorta Garcia, Edwards, Isermann, Isidori, Zolghadri),

Hydraulic Systems (Alcorta Garcia, Chen, Isermann, Kinnaert, Ould Bouammama, Patton,Zolghadri).


irccyn

Observer-based FDI Problem Statement

Plan

1 Introduction




5 Conclusion


irccyn


Problem Statement(1)

Consider the following nonlinear faulty system:

Σf :

η = f (η, u, y) + g(ǫ, d)y = h(η)

(1)

where η ∈ Rn is the state, u ∈ R

mis the input, y ∈ Rp is the output, f and h are analytic functions

of their arguments. g is the fault vector with d the faults to be detected and ǫ the disturbancesthat can also be considered as unknown inputs. Σf is supposed to be observable (Besancon 99).


irccyn




Σf :

η = f (η, u, y) + g(ǫ, d)y = h(η)

(1)




Remark

When no failure or perturbation have occurred (i.e g = 0), ǫ and d are both equal to zero.


irccyn




Σf :

η = f (η, u, y) + g(ǫ, d)y = h(η)

(1)




Remark


Aim

⇒ Propose an observer-based strategy based for fault detection and isolation such that theobserver is:


irccyn




Σf :

η = f (η, u, y) + g(ǫ, d)y = h(η)

(1)




Remark


Aim


1 sensitive w.r.t faults for the diagnosis,


irccyn




Σf :

η = f (η, u, y) + g(ǫ, d)y = h(η)

(1)




Remark


Aim



2 robust w.r.t faults, parametric uncertainties and disturbances for the control.


irccyn




Σf :

η = f (η, u, y) + g(ǫ, d)y = h(η)

(1)




Remark


Aim



2 robust w.r.t faults, parametric uncertainties and disturbances for the control.

Methodology

⇒ Observer design through a nonlinear transformation.


irccyn



Through the appropriate transformation, (1) is equivalent to the following state-affine system:

Σfa :

˙x = A(u, y)x + ϕ(u, y) + g(ǫ, d)y = Cx

(2)

where x ∈ Rn is the state, u ∈ R

m is the persistent input, y ∈ Rp is the output, g the fault vector

function of the possible disturbances ǫ and d the faults to be detected.


irccyn



Through the appropriate transformation, (1) is equivalent to the following state-affine system:

Σfa :

˙x = A(u, y)x + ϕ(u, y) + g(ǫ, d)y = Cx

(2)

where x ∈ Rn is the state, u ∈ R

m is the persistent input, y ∈ Rp is the output, g the fault vector

function of the possible disturbances ǫ and d the faults to be detected.

⇒ FDI observer could be designed. The conditions for this design are introduced in next section.


irccyn

Observer-based FDI Observer Design

Plan

1 Introduction




5 Conclusion


irccyn

Observer-based FDI Observer Design

Observer Design

For a faulty system which is affected by both disturbances and failures, to detect failures andisolate them from disturbances, it is wise to decouple them and split the system into twosubsystems, the first one being only affected by the fault vector d and the other one containingboth failure vectors d and ǫ.

⇒To set the right transformation T (x) such that (2) is equivalent to:

Σfa :

x1 = A1(u, y)x1 + ϕ1(u, y) + dy1 = C1x1

x2 = A2(u, y)x2 + ϕ2(u, y) + ǫ + dy2 = C2x2

(3)

Observer for (3) is called a residual generator for the particular case: the observer generates aresidual r(t) = f (y(t), y(t)).


irccyn

Observer-based FDI Stability Proof

Plan

1 Introduction




5 Conclusion


irccyn


Stability Proof (1)

Consider the following dynamical subsystem of Σfa:

Σd :

x1 = A1(u, y)x1 + ϕ1(u, y) + dy1 = C1x1

(4)

where x1 ∈ Rr (r < n) , u ∈ R

m, y1 ∈ Rs(s < p), where u is a persistent input, the term ϕ1(u, y)

is assumed uniformly bounded and where the matrix A1(u, y) ∈ IRr×r with A1(u, y) =

0 α1(u, y) 0 · · · 00 0 α2(u, y) · · · 0...

.

.

....

. . ....

0 0 0. . . αr−1(u, y)

0 0 0 · · · 0

.

A1. There exists a constant ν such that ‖d‖ ≤ ν i.e ‖d‖ is norm bounded. Meanwhile, thesystem perturbation is also bounded; there exists a constant γ such that ‖ǫ‖ ≤ γ with ν ≥ γ.


irccyn


Stability Proof (2)

When d = 0, a Kalman-like observer for the equivalent system Σd would be (Besancon 99):

O :

z1 = A1(u, y)z1 + ϕ1(u, y) − KC(z1 − x1)y1 = C1z1

r1 = y1 − y1

(5)

where z1 ∈ Rr is the state estimate, K = S−1CT the observer gain matrix selected such that S is

an SPD matrix, solution of S = −θS − AT1 (u, y)S − SA1(u, y) + CT C , with S0 = ST

0 > 0 and θ

a scalar parameter for the observer adjustement.

Theorem Assume that system (4) satisfies Assumption 1. Then, for d = 0, system (5) is anexponential observer for system (4). Moreover, the convergence of the error dynamics of thisobserver can be made arbitrarily fast. Else, when d 6= 0, system (5) is a residual generator forsystem (4).


irccyn


Stability Proof (2)

When d = 0, a Kalman-like observer for the equivalent system Σd would be (Besancon 99):

O :

z1 = A1(u, y)z1 + ϕ1(u, y) − KC(z1 − x1)y1 = C1z1

r1 = y1 − y1

(5)

where z1 ∈ Rr is the state estimate, K = S−1CT the observer gain matrix selected such that S is

an SPD matrix, solution of S = −θS − AT1 (u, y)S − SA1(u, y) + CT C , with S0 = ST

0 > 0 and θ

a scalar parameter for the observer adjustement.

Theorem Assume that system (4) satisfies Assumption 1. Then, for d = 0, system (5) is anexponential observer for system (4). Moreover, the convergence of the error dynamics of thisobserver can be made arbitrarily fast. Else, when d 6= 0, system (5) is a residual generator forsystem (4).

Stability Guarantee

⇒ provided that, for a candidate positive definite Lyapunov function V (e), V (e) ≤ 0


irccyn


Stability Proof (3)

Sketch of the proof.

Defining the estimation error as e = z − x , from (4) and (5), the error dynamics is given by

e =(

A(u, y) − S−1CT C)

e + d . (6)

Now, let be V (e) = eT Se = ‖e‖2S a candidate Lyapunov function for system (6).

• If d = 0 :Then V (e) = −eT [θS + CT C ]e = −θeT Se − ‖Ce‖2 and V (e) ≤ −θeT Se.

V (e) ≤ −θV ⇒ V (e) ≤ e−θt .

This ends the proof that system (5) is an exponential observer for system (4) when d = 0.


irccyn


Stability Proof (4)

• If d 6= 0 :The time derivative of V along (9) is

V (e) = −eT [θS + CT C ]e + dT Se + eT Sd .

With Assumption 1 ( ‖d‖ ≤ ν), this yields:

V (e) ≤ −θ ‖e‖2S + 2 ‖e‖S ν. (7)

Then, we have

V (e) ≤ −‖e‖ (θ ‖e‖S − 2ν). (8)

With an appropriate choice of θ, then ‖e‖ > 2ν

θ. This yields V (e) ≤ 0 and ends the proof of

theorem 1.


irccyn

Application to a DC Series Motor Dynamics of DC series motor

Plan

1 Introduction




5 Conclusion


irccyn


The DC series motor model

The dynamics of the DC series Motor can de described by ([15]):

Σ :

Li = −Ri − KmLf iΩ + u

JΩ = −DΩ − KmLf i2 − τl

y = [i , Ω]T .

(9)

with Ω : angular speed i.e measured outputi : current i.e measured outputu: voltage i.e. controlled inputR: resistanceL: self-inductanceJ: inertiaD: Viscous friction coefficientτl : Load Torque, seen as a disturbance.


irccyn




Σ :



y = [i , Ω]T .

(9)


(9) is a multi-output model, similar to (2) with x = [i , Ω]T but faults, abrupt variations on R, aremultiplicative.Aim:


irccyn




Σ :



y = [i , Ω]T .

(9)



Validation of the nonlinear Observer-based FDI made for two cases of multiplicativeparameter failures.


irccyn




Σ :



y = [i , Ω]T .

(9)



Validation of the nonlinear Observer-based FDI made for two cases of multiplicativeparameter failures.

Transformation applied to the system equation to increase the observer sensitivity to failuresand disturbances. ⇒ faults which were multiplicative to the original system equation becomeadditive ones.


irccyn


Plan

1 Introduction




5 Conclusion


irccyn


Motor Parameters

The FDI algorithm is applied to a DC series motor with the parameters given in table 1 (MehtaChiasson 98).

Resistance R 7.2 ΩSelf-inductance L 0,0917 H

Inertia J 0.0007046 Nm/rad/s2

Viscous friction coefficient D 0.0004 Nm/rad/sKmLf 0.1236 Nm/Wb.A

Table: 1. DC series motor parameters


irccyn


Benchmark definition

Simulation sampling period: 0.5ms.Torque disturbance from t = 1.5s to t = 3.5s (low speed) andat high speed from t = 6s.

0 2 4 6 8 10 12 14 16 18 200

50

100

150Benchmark trajectories (Speed, Fault, Disturbance)

Spe

ed (

rad/

s)

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

Faul

t d (O

hm)

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

Time(s)

Load

Tor

que

Tl (N

.m)

Figure: 1. Benchmark trajectories for speed Ω, fault d and disturbance τl

.The control strategy is a classical PI one with the speed reference trajectory.


irccyn

Application to a DC Series Motor Case 1: a single fault and a disturbance

Plan

1 Introduction




5 Conclusion


irccyn


Observer Design (1)

In this case, a resistance variation is considered as a fault to detect, and the load torque is seen asa disturbance.

Through the following transformation x = T (x) with x =

(

ln x1

x2

)

, system (9) is equivalent

to:

Σ :

Lx1 = −R − KmLf y2 + uy1

Jx2 = −Dx2 − KmLf y21 − τl

(10)

Methodology:

Measured signals contain the fault ⇒ need of a transformation to detect the parameter faulton R independently of the disturbance τl .

Observer will provide a fault-free state vector, which will be compared to the measuredsignals for residual generation.


irccyn


Observer Design (2)

From the DC series motor model, the goal is to write this model as different subsystems incascade, each of them satisfying the previously defined properties for observer design. For the DCmotor, the electric and mechanical parts can be easily separated in two subsystems:

x1e = A1e(u, y)x1e + ϕ1e(u, y)

y1e = C1ex1e


y2e = C2ex2e

where the electrical subsystem has the following form


where x1e = (x1, R)T , is the extended state vector, R being the fault d (4),

A1e(u, y) =

(

0 − 1L

0 0

)

;

ϕ1e(u, y) =

(

−KmLfL

y2 + uLy1

0

)

.

The measurable output is y1 = i , (C1e = [1 0]).


irccyn


Observer Design (3)

From this first sub-system, an electrical observer can be designed by

z1 =

(

0 − 1L

0 0

)

z1 +

(

−KmLfL

y2 + uLy1

0

)

−

(

k1 0k2 0

)

(z1 − x1e).


irccyn


Observer Design (4)

On the other hand, the mechanical subsystem is given by

x2e =

(

−DJ

− 1J

0 0

)

x2e +

(

KmLfJ

y21

0

)

where x2e = (Ω, τl )T , is the extended state vector, τl being the disturbance ǫ (3),

A2e(u, y) =

(

−DJ

− 1J

0 0

)

, ϕ2e(u, y) =

(

KmLfJ

y21

0

)

, and the measurable output is given by

y2 = Ω, (C2e = [1 0]).

Then for the mechanical subsystem, an observer can be designed by

z2 =

(

−DJ

− 1J

0 0

)

z2 +

(

KmLfJ

y21

0

)

−

(

l1 0l2 0

)

(z2 − x2e).


irccyn


Simulation results (1)

For the first test, (Fig. 2, 3) the failure to detect is a sudden 50% fast variation of the resistanceR. The new resistance value Rd is therefore Rd = R + ∆R, ∆R being the variation. The fault isapplied at t = 4s. Furthermore, the load torque is seen as a disturbance.

0 5 10 15−2

−1

0

1

2

3Current i

Cur

rent

(A)

i mesi obs

0 5 10 150

50

100

150

Speed w

Time(s)

Spee

d (ra

d/s)

w mesw obs

Figure: 2. Measured, observed current vector i and speed Ω

Figure 2 represents the observed speed, the measured speed and the current trajectories.The observer used in case 1 does not estimate the DC motor parameters.


irccyn



This bank of observers is designed with the fault-free system equation while the measurementsgive the fault information. Residuals are then computed such that r(t) = y(t) − y(t).

0 5 10 15−1

0

1

2Current Residual

Res

idua

l (A

)

0 5 10 150

0.5

1Speed Residual

Res

idua

l (ra

d/s)

0 5 10 150

0.5

1Load Torque Tl

Load

Tor

que

(N.m

)

Time(s)

TlTl obs

Figure: 3. Current and speed residual, measured and observed load torque τl

Thanks to the transformation applied to system equations (10), the failure is detected and in thiscase, the current residual (Fig. 3) indicates the fault’s occurrence and its time of appearance.This indication is given thanks to the difference between the estimated and the measured currentsignals. Both speed and load torque are well-estimated (Fig(3)).


irccyn

Application to a DC Series Motor Case 2 : multiple multiplicative faults and a disturbance

Plan

1 Introduction




5 Conclusion


irccyn


Observer Design (1)

In addition to the previous resistance variation, we consider now a self-inductance variation andthe load torque is still seen as a disturbance. Through the right transformation x = T (x) with

x =

(

L ln x1

x2

)

, system (9) is equivalent to:

Σ :

x1 = −R − KmLf y2 + uy1

Jx2 = −Dx2 − KmLf y21 − τl .

(11)


irccyn


Observer Design (2)

Now the system may be seen as two subsystems in cascade, each of them satisfying the requiredproperties for observer design (Theorem 1). It can be written in the following form:


y1e = C1ex1e


y2e = C2ex2e

where the electrical subsystem has the following form


where x1e = (i , R)T ,

A1e(u, y) =

(

0 −10 0

)

;

ϕ1e(u, y) =

(

−KmLfL

y2 + uLy1

0

)

. The measurable output is y1 = i


irccyn


Observer Design (3)

Then the electrical subsystem’s observer is given by

z1 =

(

0 −10 0

)

z1 +

(

−KmLfL

y2 + uLy1

0

)

−

(

k1 0k2 0

)

(z1 − x1e).

An unknown input observer is designed for the second fault. Hence, the observer for L is given by :

ˆL = −θ(z1 − x1e), θ the observer gain.

The mechanical subsystem is given by

x2e =

(

−DJ

− 1J

0 0

)

x2e +

(

KmLfJ

y21

0

)

where x2e = (Ω, τl )T , A2e(u, y) =

(

−DJ

− 1J

0 0

)

, ϕ2e(u, y) =

(

KmLfJ

y21

0

)

, and the

measurable output is given by y2 = Ω.


irccyn


Observer Design (4)

The mechanical subsystem’s observer is given by

z2 =

(

−DJ

− 1J

0 0

)

z2 +

(

KmLfJ

y21

0

)

+

(

l1 0l2 0

)

(z2 − x2e).


irccyn



In case 2 (several faults and a disturbance) (Fig. 4, 5, 6), we apply a 10% variation of theinductance L in addition to the previous 50% variation of the resistance R. Along with thosefaults, the load torque is considered as a disturbance.

0 5 10 15

0

1

2

3

4

5Current i

Cur

rent

(A)

0 5 10 150

50

100

150

Speed w

Spe

ed (r

ad/s

)

Time(s)

i mesi obs

w mesw obs

Figure: 4. Measured, observed current i and speed Ω

The speed (Fig. 4), as well as the load torque (Fig. 5), are once again well-estimated. Hence,despite all faults and disturbance applied, no variations are noticed between the estimated andthe measured current signal. The observed and the measured current are similar (Fig. 4).


irccyn



This time, the designed observer gives also an accurate estimation of the motor parameters (seeFig. 5 and 6).

0 5 10 150

2

4

6

8Resistance R

Res

itanc

e (O

hm)

0 5 10 150

0.05

0.1

0.15

0.2Self−inductance L

Sel

f−in

duct

ance

(H)

0 5 10 150

0.1

0.2

0.3

Load Torque Tl

Load

Tor

que

(N.m

)

Time(s)

R obs

L obs

TlTl obs

Figure: 5. Observed parameters vector R, L and τl


irccyn



0 5 10 15−0.1

−0.05

0

0.05

0.1Error on R

Err

or R

(Ohm

)

0 5 10 15−0.1

−0.05

0

0.05

0.1Error on L

Err

or L

(H)

0 5 10 15−0.1

−0.05

0

0.05

0.1Error on Tl

Err

or T

l (N

.m)

Time(s)

Figure: 6. Error vector for motor parameters R, L and τl


irccyn

Conclusion

Conclusions

Design of a FDI observer thanks to a state transformation coupled with an outputtransformation of the system equations,

Two opposite objectives : sensitivity w.r.t faults for the diagnosis and robustness w.r.t faults,parametric uncertainties and disturbances for the control.

Application to a DC series motor in different cases of multiplicative parameters failure,

Multiplicative faults transformed into additive ones through a logarithmic transformation,

For the case of single fault and a disturbance designed observers used as residualsgenerators. Residuals are function of the estimated signals provided by the observers and themeasured signals,

Estimation of the system parameters along with the speed and the current estimation for thecase of multiple faults and a disturbance,

The method exposed has proved to be efficient when it comes to residual generation and failurediagnosis. Indeed, both control and observation are made possible thanks to the bank ofUnknown Input Observers.

Perspectives

Extend this result to the induction motor observer-based diagnosis:⇒ with mechanical sensor⇒ sensorless


irccyn

Conclusion

Questions

?


irccyn

Conclusion

A. Akhenak, M. Chadli, J. Ragot and D. Maquin,“State estimation of uncertain multiplemodel with unknown inputs”,In Proc. of IEEE Conference on Decision and Control,Bahamas, 2004.

E. Alcorta Garcia, P. M. Frank, “A novel design of structured observer-based residuals forFDI”, In Proc. of American Control Conference: San Diego, USA 1999.

J. Aslund, E. Frisk, “An observer for nonlinear differential algebraic systems”, Automatica,42, 959-965, 2006.

M. Basseville, “Information criteria for residual generation and fault detection and isolation”,Automatica, 33(5): 783-803, 1997.

G. Besancon, “A viewpoint on observability and observer design for nonlinearsystems”,Lecture Notes in Control and Information Science (244), Springer-Verlag, 1999.

G. Besancon, “High-gain observation with disturbance attenuation and application to robustfault detection”, Automatica, 39, 1095-1102, 2003.

C. DePersis, A. Isidori,“A geometric approach to nonlinear fault detection and isolation”IEEE TAC, 46(6): 853-865, 2001.

P.M. Frank, “Fault diagnosis in dynamic systems using analytical and knowlegde-basedredundacy. A survey and some new results”, Automatica, 26(3), 459-474, 1990.

P.M Frank, X. Ding, “Survey of robust residual generation and evaluation methods inobserver-based fault detection systems”, Journal of Proc. Cont., 7(6), 403-424, 1997


irccyn

Conclusion

H. Hammouri, M. Kinnaert, E. H. El Yacoubi, “Observer-based approach to fault detectionand isolation for nonlinear systems”, IEEE TAC, 44(10):1879-1884, 1999.

R. Isermann, “Supervision, fault detection and fault diagnosis methods. An introduction”,Control Eng. Practice, 5(5), 639-652, 1997.

B. Jiang, F. N. Chowdhury, “Parameter fault detection end estimation of a class of nonlinearsystems using observers”, Journal of the Franklin Institute, 342, 725-736, 2005.

T. F Lootsma, “Observer-based fault detection and isolation for nonlinear systems”, PhDthesis, Aalborg University, Aalborg, Danemark, 2001.

M-A Massoumnia, “A Geometric approach to the synthesis of failure detection filters”, IEEETAC, AC-31(9): 836-846, 1986.

S. Mehta and J. Chiasson, “Nonlinear Control of a Series DC Motor: Theory andExperiment”, IEEE Transactions on Industrial Electronics,45(1),1998.

M. Staroswiecki, G. Comtet-Varga, “Analytical redundacy relations for fault detection andisolation in algebraic dynamic systems”, Automatica, 37, 687-699, 2001.

A. Xu, “Nonlinear system fault diagnosis based on adaptive estimation”, Automatica, 40(7):1181–1993, 2004.

Q. Zhang, M. Basseville, A. Benveniste, “Fault detection and isolation in nonlinear dynamicssystems: a combined input-output and local approach”, Automatica, 34(11): 1359-1373,1998.

www.irccyn.ec-nantes.fr/hebergement/BancEssai/


methodology for nonlinear fdi observer via nonlinear … et safepro… · hows: diagnosis principle...

Documents