µ Techniques for Uncertain Flight Control Systems

Copyright © IFAC Robust Control Design, Budapest, Hungary, 1997

ROBUST FAULT DETECTION OBSERVER DESIGN USING ILl ~ TECHNIQUES FOR UNCERTAIN FLIGHT CONTROL SYSTEMS M. A. Sadrnia, J. Chen and R. J. Patton Department of Electronic Engineering, The University of Hull, Hull, HU6 7RX, UK

Abstract: This paper presents a new approach to the design of a robust observer-based fault detection scheme for diagnosing incipient faults . The approach is based on robust H_ estimator which minimises the effect of disturbance on the estimation error and subsequently the diagnostic residual. The effect of faults on the diagnostic residual is maximised by the proper selection of the performance bound and the estimation weighting matrix of the H _ robust estimator. The approach has been applied to a flight control system and the results show that the fault detection scheme designed can detect incipient faults effectively even in the presence of disturbances and modelling errors. Keyword: Fault detection, observers, robustness, If" optimisation

1.

output is used as a diagnostic residual for FDI. The approach proposed in this paper is based on Hrobust FDI observer which is originated from robust H- filter/estimator [7-9] . The diagnostic scheme optimally satisfies robustness and sensitivity conditions required by FDI. This is achieving by using the freedom in the selection of the H_ estimator performance bound and the estimation weighting matrix. The most important contribution of this paper is the maximisation of the fault effect on the residual. This is different from earlier studies on the H- robust observer-based FDI in which only the minimisation of the disturbance effect is considered [13] . The minimisation of disturbance effect alone could desensitising fault effects as well. This problem is overcome by the design approach proposed in this paper. The approach involves the solution of Riccati equation which is relatively easy to solve comparing with the complicated factorisation method used in [10-12] . An extensive simulated assessment has been carried using a simulated flight control system to verify the effective of the approach developed.

INTRODUCTION

As flight control systems need to be very reliable, a fault detection and isolation (FDI) system is needed. Over the last two decades, the so-called model-based approach has been favoured in both research and applications. Inevitable modelling uncertainties in real complex systems have challenged model-based FDI techniques due to their undividable effects with faults [1-4]. In an attempt to solve this problem, the robustness problem has been the main challenge in FDI research for more than 10 years. The principle is to make the FDI algorithm sensitive to faults and insensitive to modelling uncertainty. A robust FDI design should satisfy the robustness-sensitivity property simultaneously. There is a trade-off between sensitivity and robustness that is an issue of fundamental problem. This paper presents a new approach for designing optimal residuals through H- robust fault detection observer. The basic idea is to estimate (reconstruct) the outputs of the process with the aid of observers and to use the estimation error or a function of it as a residual. In contrast to the state observer that is needed in feedback control system design in case of incomplete measurement of the state vector, .a diagnostic observer is an output observer. The weighted difference between actual and estimated

2.

PROBLEM FORMULATION

Consider the following linear uncertain MIMO faulty system:

491

time-invariant

x(t) = (A + M)x(t) + (B + LlS)u(t) + (E) + Llli) )d(t) + (R)

+~)

disturbance model are known . Figure 1 general input / output representation of a plant P with modelling uncertainties Ll estimator (filter) F. When there are uncertainty terms Llj, i=l, ... , m, then constructed with a block diagonal structure

(la)

)f(t)

yet) = (C + LlC)x(t) + (D + LlD)u(t) + (E 2 + Llli2 )d(t) +(R 2

+~2)f(t)

(Ib)

z(t) = Mx(t)

(le)

where x ER· is the state, u(t) = R r is the control input vector, y E R m is the measured output, fER r

Ll(s) =

represent the faults, d E R I denote for the disturbances. A, B, C, D, RI. R2, EI. E2 are constant matrices of appropriate dimensions that describe the nominal system and M, LlB, LlC, LlD, ~j, ~2' LlB), Llli2 represent the parameter uncertainties. M is a subset or combination of states to be estimated.

(2a)

yet) = Cx(t) + Du(t)

(2b)

=Mx(t)

(2c)

r(t) = y(t)-y(t)

(2d)

z(t)

0

o

Ll 2(s)

.

[

o

...

(4) 0

Where each Ll i (s) represents an independent, linear norm-bounded modelling perturbation. To simplify the analysis, the d i (s) terms are generally normalised with frequency-dependent weights so that for each Ll i (s) ,

An H.. robust fault diagnosis observer has the following formulation:

~(t) = AX(t)+Bu(t) + L_(y(t)- yet»~

Lll (s)

shows a nominal and an multiple Ll(s) is

\;fro

(5)

The input to the plant, d, includes both process and measurement noise. £ and 11 represent the signals connecting the nominal plant and the perturbation.

In the standard form for robustness, the H_ robust £(s)

fault diagnosis system can be shown as Figure 1. The state space equations for the perturbed system can be rewritten as x = Ax+Qr1 +E\d+Bu +R)f

(3a)

£ = Sx + D 12 d + Dl3u + D)4f

(3b)

e=Mx-IZ

(3c)

r=y-y

(3d)

y = Cx+D3\11 +E 2d+Du+R2f

(3e)

Fig. 2 Closed-Loop Transfer Function

The closed-loop transfer function, G(s), is shown in Figure 2, and can be divided into four blocks, £(S)] = [G II (s) G I2 (S)][11(S)] [ e(s) G 2)(s) G 22 (s) des)

(5)

The nominal closed-loop system is given by G 22 (s) , and the perturbed closed-loop system is given by

£

G", (s) = G 22 (s) + G 21 (S)Ll(s)[I - G \I (S)Ll(S)r' G 12 (s)

(6)

y

The measure of robust performance is the maximum gain from the disturbance to the estimation error, IIG", (s)II- , in the presence of the norm-bounded

Fig. 1 Representation of the H_ Robust Fault Diagnosis, P(s) nominal system, 6(s) parameter uncertainty and F(s) output observer.

modelling uncertainty. To test the performance robustness, the inputs and / or outputs are scaled so that the robust performance condition is

Where 11 and £ are the fictitious input and output signals according to the plant perturbation, and Ll is a fictitious matrix. The goal in robust estimation is to minimise the estimation error for an entire fa.-nily of possible plants, input initial conditions and disturbances. This is in contrast to the Kalman filter, which assumes that the plant dynamics and

e 2 max IIlIdll1 = 11 G Cd 11 _ < 1 2

gi yen the modelling error LlE Bd..

492

(7)

A transfer function e.g. relating des) to res) - pes), can be placed into the standard fonn for robustness analysis and design, Fig. 1, by the following statespace representation:

e2 IIG cd 11 - = max IIIIdll11 < y 2

for minimum "11':>0

(l3)

,-

Where e is the estimation error: e=z-z=Mx-z

(14)

Using the results of robust H _ estimator from [5, 7,

8,9] an observer gain is given by L _ = P_C T which P_ comes from the following Riccati equation: and F(s) transfer function from y to

A

y has the state-

space representation· : , [A-L C C F(s) = M

I L0-] ' [~l = F(S{~]

AP_ +P_A'+p_[y-2(M M +S' S)-C'C)]P_

+ (QQ'+EJE J') = 0

(9) and

The state estimator error, x(t) = x(t) - x(t) , dynamic equation takes the fonn

r(t) = Cx(t) + D 3J 'T1(t) + E 2d(t) + R 2f(t)

Robust H.. observer design satisfies condition Eq. (l3) only. However, by iteration we can find a minimum y to satisfy condition (a) as well. By satisfying Eq. (l3), the perfonnance criterion (b) is also satisfied because:

(lOb)

The satisfaction of Eq. (l3) can not provide good FDI perfonnance. This is because the fault effect on the residual may also be reduced by minimising modelling uncertainty effect. To overcome this problem, the robust perfonnance bound, y, should be relaxed (i.e. increased). The satisfaction of design criterion (c) and the trade-off with the criterion (b) will be achieved by taking the advantage of freedom in choosing M and y. By finding Y min' the robust H_ observer design is only half completed. The

Where r(t) is the so-called residual and the design must satisfy the following perfonnance criteria, [6]: Ac=A-L _ Cisstable,

(b)

supcr ••.JGrd(jro)] := IIGrd(jro)jL < ex

(c)

ir! cr mm [G" (jro)] : = IIG" (jro)IL > ~

where

.,

~

> ex >0 and

Grd(s) = C(sl-Ac>-I[E\ +Llli J -L_(E2 +Llli 2 )] +E2 +Llli2 (11)

appropriate choice of M and finding the optimal

YOf'

can give significant increase in fault sensitivity.

Grf(jro)=C(sl-Ac)-J[R J +dR. J -L_(R2 +dR. 2)] +R2+dR. 2 (12)

3.

(16)

(lOa)

Where the design of a H.. robust fault diagnosis observer is to design of an observer gain matrix L.. that stabilise observer, rrunlIIDse effects of disturbances to the residual in the H.. nonn sense and maximise (maintain) effects of faults to the residual in the whole range of modelling error.

(a)

P_ >0

Q and S are the effects of parameter uncertainty onto the nominal system. The Appendix presents how Q and S can be obtained from parameter uncertainties.

x(t)=(A-L_C)x(t)+(B\-L_D 3J )J1(t) +(E 1 -L_E2)d(t)+(R\ -L_R2)f(t)

y> 0;

(15)

Finally, the robust FDI design will achieve the following conditions:

SOLUTION OF THE H_ ROBUST

(18)

FAULT DIAGNOSIS OBSERVER where ex l >cx 2 >0 are design parameters. The first one

H _ observer is a dual to the full infonnation H control problem. It produces the best estimation z based on y, such that minimise

illustrates the expected case of residual robust perfonnance and the second one illustrates the worst case of the residual robust perfonnance.

Z of

4.

• In H _ control tenninoJogy a transfer matrix in terms of stale-space data is denoted

[~I ~] :=C(sl-ArIB+D'

A DESIGN EXAMPLE

The proposed approach is applied to an incipient fault diagnosis of a flight control system for an 493

unmanned aircraft; the Machan, Spurgeon and Patton [14]. The robust performance is illustrated in some graphs. To assess the effectiveness of the developed approach, the robust design is compared with the non-robust design based on pole placement. In a trim setting for rudder and aileron of 1 't = 0, ~ = 0 and VT = 33ms- the linear system matrices are: x(t) = A Ia"""x(t) + Bu(t) + E 1d(t) y(t) =CIaIcr.,X(t) + Du(t) + Rzf(t)

[-6.3962 ±4.6081 -0.7930 -1.3673 -4.9717 - 8.0806 -8.4843t·

. Cl I th eary e ratIos

11

0

II0nlll_

I O ,rIL II0nll!.

d

n "-

an

are much

bigger for the H ~ design than pole placement design. Correspondingly, in the time domain the response of the H. residual to an incipient fault is clearly more sensitive than pole placement design. Notice how well the H robust fault detection observer has maximised the effect of the incipient fault on the residual, Figure 4. Figure 5 shows how difficult is to see the effect of the fault on the output.

where

v = Side Slip Velocity IIGrllllllGrdll for Robust H_inf. and Ploe Place. Design

P =Roll Rate

1.5

R = Yaw Rate ~

x=

;:;: Roll Angle

&u= {

= Yaw Angle 't = Rudder Angle

'I' ~

Alatcr.ll=

B=[l o

't = ~

r------~---...._-------__.

"(OP' = 22.4

Rudder angle

= Aileron Angle

;:;: Aileron Angle

[

~ ~~~ -8 ~Z48 -:27~07 98~OO ~ -5~324 _28~] o J6SO

0

010000

-0 '139

0

0

o

0

1.0000

o

0

0

o

0

-9.4170

00

0

0

0

0 -10.0000 0 0

0 0

0

Frequency

0

Fig. 3 Comparison of H. design with Pole Placement

-5.0000

-[G rf (j ro)]

design (Top: cr

!]. D=[~ ~l

a [Gro(jW)]

denotes for H. design and 0 denotes for pole placement design).

10

100 0 00 0]

Residual comparison

0100000

E,

=

cr [a rfUro)] , x cr[Onj(jro)]

and bottom: -

o

0 1 0 0 0 0 0 0 0 1 0 0 0 CIaI=I 0000100 o 0 0 0 0 g 0

[

[0 0 1 0 0 0 0 0

=

0 0 0] 1 0 0 RZ

[, 0] 0 1

=

0.OO2r------------~-.......,

H

0000004

We compare the observer gain obtained using the H~ robust fault diagnosis observer design with the Pole Placement design by examining the two performance criteria in (18). Figure 3 shows that the performance indices for H ~ design is much greater than pole placement design (Yopt = 22.4 ). The observer gain designed by the H. robust fault diagnosis and pole placement are

-;;'"

0.001

L '1

::I

~

01 =:'"

pp

11 I

0

~.OOI~--~----------------~--~

o

10

20

30

40

50

Fig. 4 Time response of residuals

As discussed previously, robust observer design may desensitise fault effect. To illustrate this aspect another design is used. In this design, optimal modelling uncertainty robustness is attempted, i.e. a minimum robust performance bound 'Y min = 21.097 is chosen. The fault sensitivity has not been considered. The fault detection performance indices are shown in

L = [117.0783 1.4955 10.9628 10.3605 0.8534 -{I.7064 -{I.0587]T -{I.5I77 0.3600 0.8534 1.2462 1.4105 -{I.01l3 -{I.OO22

and T

-2.8855 -11.8727 4.4402 3.5963 1.3442 -30549 0.0543]. L", = [ 18.6966 6.6981 0.0376 1.0617 7.9331 0.7208 -0.1477

These gains place the poles of the observer at 494

[2]

Fig. 6. It can be seen that, in the case of Ymill = 21.097 produced by the robust H_ filter, the worst case fault detectability is zero. This is not suitable for robust fault diagnosis application.

[3] Fault and measurements

faulty

measuremeDt

~//

-0.2

-0.6

/,

/,

"

""

[4]

, ,

" DOnnal

measuremeDt

/,

[5]

[6]

Fig. 5 Normal and faulty outputs

IIGrflllllGrdll for Robust H inf. and Pole Place. Desil!:ll 1.5

)/

Ymin = 21.097

rill

[7]

[8]

[9]

0'-J(T'

_"xx 10'

10'

Frequency

[IOJ

Fig. 6 As for figure 3 but with YnUn = 21.097 .

5.

CONCLUSION [11]

This paper presents a new H_ robust FDI observer design approach. The significance of the approach is based on the simultaneous achievement of the robustness-sensitivity property. Solved with only one Riccati equation, in comparison with the existing robust observer-based fault diagnosis methods, the method presented is more direct and simple. The simulation results show the effectiveness of the developed method.

6. [1]

[12]

REFERENCES

[13]

Patton R. J., Frank P. and Clark R.(1989), Fault Diagnosis in Dynamic Systems, Theory and Application. Prentice Hall (Control Engineering Series), London.

[14]

495

Frank P. M. (1990), Fault diagnosis in dynamic system using analytical and knowledge based redundancy - A survey and some new results. Automatica, Vol.26, No.3, 459-474. Patton R. J. and Chen J. (1994), A review of parity space approaches to fault diagnosis applicable to aerospace systems. 1. of Guidance, Control and Dynamics, VoLl7, No.2, 278-285. Gertler J. (1991), Analytical redundancy methods in failure detection and isolation. Preprints of IFACIIMACS Symposium SAFEPROCESS'91, Baden-Baden, YoU, 921. Appleby B. D. (1990), Robust state estimator design using the H _ nonn and !l synthesis, Ph.D. thesis, Dept. of Areo. and Astro., MIT. Patton R J, & Hou M (1996), A matrix pencil approach to fault detection and isolation observers, to be presented at the 13th IFAC World Congress, San Francisco, 30 June -5 July 1996. Shaked U. (1990), H_ -minimum error state estimation of linear stationary process. IEEE Trans. on Auto. Cont., Vol. 35, No. 5, pp. 554-558. Shaked U. and Theodor Y. (1992), H_optimal estimation: A tutorial. Proc. of the 31st CDC, Arizona, pp. 2278-2286. Yao Y. X., Darouach M. and Schaefers J. (1994), A robust H_ estimator design for linear uncertain systems. Amer. Cont. Con., Maryland, pp. 3568-3569. Ding X. and Frank P. M. (1990). Fault detection and identification via frequency domain observation approaches. IMACS ANNALS on Computing and Applied Mathematics Proceedings MIM-S2, Sept. 37. Ding X. and Frank P. M. (1991). Frequency domain approach and threshold selector for robust model-based fault detection and isolation. Preprint of IFAClIMACS Symp. SAFEPROCESS'91, Baden-Baden, pp. 307312. Qiu Z and Gertler J. (1994). Robust FDI systems and H_ -optimisation-Tall fault systems and example. Preprints of IFAC Symposium SAFEPROCESS '94, Finland, YoU, pp. 260-265. Edelmayer A., Bokor J. and Keviczky L. (1994). An H~ filtering approach to robust detection of failures in dynamical systems. Proc. of 33rd CDC. pp. 3037-3039. Spurgeon Sand Patton R. J. (1990). Robust variable structure control of model reference

k

systems, lEE Proceedings-D Control Theory and Application, 137, (6), 341-348.

7.

k

x(t)=(A+ LQ;O;In;S ;)x(t)+(E J + LQ;O)njT.)d(t) j=l

i=l

ApPENDIX

Parameter uncertainty in the standard form Fig. 1 k

yet) = (C+ LR;O)n;SJx(t)+(E2 +

A state-space model of a typical plant is fonned from a number of physical parameters. These parameters often affect many elements of the state-space model. For a plant with a nominal state-space representation

i=l

k

L R;O;InjTJd(t) i =l

x(t) = Ax(t) + EJd(t) yet) = Cx(t) + E 2d(t) z(t) = Mx(t) where x is the plant state vector, y is the measurement vector, and z is a subset or combination of states to be estimated. Parameter errors can be modelled as k

The transfer function can be placed into the standard fonn shown in Figure 1, where pes) has the following state-space representation, [5]:

k

x(t) = (A+ LLlA»Jx(t)+(E 1 + LLlEJjOJd(t) i=l

i=l

k

k

yet) = (C + LLlC;OJX(t)+ (E 2 + LLlE2;O;)d(t) i_ I

i=l

Each

0 i represents

nonnalised as -1 < 0i < 1

a parameter error that is

V1

The matrices associated with each uncertain parameter can then be combined into a matrix

N;

M

~l; ] E R(Dx+ny )X(nx+n~ )

=[ LlC,' ~

2;

where n x' n y' nd are the dimensions of the vectors x, y, and d. Generally this matrix will not be of full rank since one parameter rarely will affect all of the states and outputs. N; can therefore be written in the fonn

N;=

[~Js; TJ where

and n i is the rank of the matrix Ni. The statespace model of the perturbed system can be rewritten as

496