A frequency domain approach to residual generation for the industrial actuator benchmark

ControlEng. Practice,Vol. 3, No. 12, pp. 1747-1750, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0967-0661/95 $9.50 + 0.00

Pergamon 0967-0661 (95)00188-3

A FREQUENCY DOMAIN APPROACH TO RESIDUAL GENERATION FOR THE INDUSTRIAL ACTUATOR BENCHMARK E. Alcorta Garcla*, B. K~ppen-Seliger and P.M. Frank Department of Measurement and Control, University of Duisburg, Bismarckstr. 81 BB, D-47048 Duisburg, Germany

(Received June 1995; in final form July 1995) A b s t r a c t : This paper presents an application of a frequency domain approach to fault detection for the electro-mechanical test facility. An outline of the frequency domain design method is provided. The frequency domain residual generation is designed based on a linear model, and then tested on the various data sequences as given in the overview paper. Results of simulations, as well as a discussion of the method's capability, are also given. K e y w o r d s : Fault detection, analytical redundancy, Hoo-optimization, robustness, benchmark testing.

ble and proper is denoted by R H ~ . A double coprime factorization of G(s) is written as

1. I N T R O D U C T I O N This paper presents an application of the fault detection method in the frequency domain to the joint benchmark study on an electro-mechanical test facility. The main task is to design an algorithm that enables detection and isolation of the faults present in the system.

G(s) = N ( s ) M - l ( s ) - f4-1(s)lV(s)

(1)

where N(s), M - l ( s ) , M - l ( s ) and/V(s) are right and left coprime RHoo-matrices of G(s) respectively. These factorizations can be realized by using the algorithms provided in (Francis, 1987). A matrix Gi(s) G RHo~ is inner if GT(-s)Gi(s) = I. A matrix Go(s) E RH~o is outer if Gol(s) G R H ~ . For a RHoo matrix G(s) E Cm×p satisfying rank (G(s)) = p there exists a so-called innerouter factorization

In order to achieve the fault detection requirements, a frequency domain method is applied (Ding, 1992) (see also (Frank and Ding, 1994)); some difficulties related to the application of the frequency domain method to the benchmark are also pointed out.

G(s) = Go(,)Gi(,) The paper is organized as follows: in Section 2 basic concepts are reviewed. The design procedure for FDI in the frequency domain are presented in Section 3. The benchmark and the FDI design for it are described in Section 4. Results and simulations are shown in Section 5 and finally, a discussion is presented in Section 6.

(2)

where Go(s) E Cpxp is outer and Gi(s) is inner. When G(s) has zeros in C+, on the jw-axis or in infinity an extended inner-outer factorization (EIOF) is described by

G(s) = ao(s)Ge(s)ai(s)

(3)

where Go(s) is outer and invertible, Gi(s) is inner and Ge(s) has all zeros in infinity, in C+ and on the rio-axis.

2. PRELIMINARIES The set of all real-rational matrices which are sta*Sponsored by G e r m a n Academic Exchange Service (DAAD). 1747

E. Alcorta Garcfa et al.

1748

3. F R E Q U E N C Y DOMAIN APPROACH

G(s, L

U

The frequency domain approach to the design of fault-detection observers was introduced by (Viswanadham, el hi., 1987) and recently extended by (Frank and Ding, 1994). This approach offers powerful methods to tackle the robustness problem by using H~-optimization theory. The basic idea of the frequency domain approach is as follows: through a frequency characterization of all achievable residual dynamics, based on coprime stable factorizations and an analog result of the Youla parameterization (parameterization of all stabilizing controllers (Maciejowski, 1989)) to observers, the robust residual generation design is formulated as an optimization problem and solved by Ho~-optimization techniques. The frequency domain procedure of evaluation of the redundancy given by the mathematical model of a system can be divided into the following two steps (Frank, 1990): residual generation and

I R(s)

r(s) Fig. 1. Generalized residual generator

5. Set Q2(s) so that IIQ2(s)SM{Ga~(s)}II~ 6. Set R(s) as (5). 7. The residual is given by (4).

<_1

The second step of this procedure is an extra one, and the function of this is to facilitate fault isolation, even if uncertainties are present.

residual evaluation 3.3 Residual Evaluation 3.1 Residual Generation It is well known that the construction of the residual r(s) is based on the application of an output observer. The residual is generated using a comparison between the estimated and the measured output y(l). A generalized form of the residual is given in (Ding, 1992):

r(s) = R ( s ) ( M . ( s ) u ( s )

- g~u(s))

(4)

where R(s) E R H ~ and is defined as:

(5)

a(s) = Q2(s)UL (s)c 2(s)Ql(s) with

the

matrix

Q2(s)

such

that

IIQ2(s)SM{Gd~(S)}II~ <_ 1; SM{Gd~(s)) is the Smith-McMillan form of Gd¢(s); Uae(s) is defined through Gae(s) = Ua,(s)SM{Gd,(s)}Vae(s); Gao(s) is defined like Go(s) in (3) and Ql(s) such that Ql(s)IQ~(s)Gl(s) = diag(7~) (see Fig. 1). A possible selection of Ql(s) is Q,(s) = adj(G! (s)) (Vidyasagar, 1985) when ]VIu(s)G! (s)is a square matrix.

3.2 Construction of Residuals The procedure to build residuals is summarized as follows (Frank and Ding, 1994): 1. Do a coprime factorization of Gu(s). 2. Find Ql(S) such that Ql(s)M~(s)Gl(s ) =

diag(7~ ). 3. Do an EIOF of Ql(s)JVI~(s)Ga(s) 4. Find the Smith-McMillan form of Gde(s).

The fault-detection decisions are based on determining the characteristics of the residuals. The consequence of model uncertainties and perturbations on the residual is that it will not be zero even if there are no faults present in the system. One is forced to use thresholds because of the existence of model uncertainties and perturbations acting in the same way as the faults to be detected. The use of thresholds allow one to decide when a fault is in the system. An adaptive threshold in the frequency domain is determined by (Frank and Ding, 1994): t~ 2

J?h(0

'~d ] . . ~.. . . s u p - - j Gd()W)M,(3w)Q (3w) d

271"• o21

.Q(jw))fI~(jw)Gd(jw)dw

(6)

where )~d > [d(jw)[. The problem here is the upper bound of the unknown input d because this value is usually unknown. 4. T H E INDUSTRIAL A C T U A T O R BENCHMARK T E S T The industrial actuator benchmark test is based on an electro-mechanical test facility, which has been built at Aalborg University in Denmark (Blanke, el al., 1995). The system model is described in the overview paper (Blanke, el al., 1995) eq. (3) and the corre-

A Frequency Domain Approach to Residual Generation sponding matrix transfer functions are given by:

Gu(s) = C(sI- A¢)-IB, +D Gl(s ) = [ C(sI-Ac)-IFa, F, ]

(7) (8)

Gd(s) =

(9)

C(sI - A~)-X E~

where,=["] L

"

4.I Residual Generation and Evaluation Design for the Benchmark Problem For the residual generation design, it is only necessary to follow the given steps (1-7). 1) Coprime factorization of Gu(s) is given by: s 3 + 171s 2 + 18554si2 (s + 100) 3

+

g.(~)

2) Q l ( s ) =

163s 2 18554s ) (s + 100) 3 1.79s + 203.8 (s + 100) a

=

(1 0) 1.99 181s

(s + 100) ~ 0

Gdi(s)

0 0

)

s + 100 100

"-

;¥1--66

4) T h e SM{Gd~(S)} = adds)

5) Q2(s)= 6) R ( s ) =

200

0

0

1

)

(45.0470) 1.99 1 181s

7) Finally, the residuals for the implementation are given by: rl(s) =

The residuals for the analysis are given by: 181.4s 2 4.4s 2 (s + 100) aF"(s) + (s + 100) 3Dis)'

r~(s) = 0.978 s3 + 171s2 + 18554s ,~, , It can be seen in the last equations, that the evaluation of the residual rl(s) requires the use of a threshold due to the presence of the uncertainty D(s). The design, however, is not possible, because an upper bound of the uncertainty (load) D(s) is unknown. With respect to the residual r~(s), the fault signal is obtained directly. A threshold is needed because of the numerical rounding and noise. For the design of a threshold, additional information is required (Hoefling, et ai., 1994), for example a maximal value of the fault-free residual or a heuristic knowledge of the fault-free residual's behavior. The thresholds have not been carried out in the work described in this paper. 5. SIMULATION RESULTS

= (,.,4o o)1 =

0.0106s 2 + 1.8s + 196.7 y2(s) (s + 100) 3

1

3) An EIOF of Q~(s)M,,Gd(s) is as follows:

ad~(S)

s 3 + 171s 2 + 18554s , , . . . . T5 yl(s) (s + 100)

--

+

rl(s)

With the system model in this form, the application of the frequency domain method described above is straightforward.

=

r2(s)

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45s a + 7707s 2 + 836228s u,(s) (s + 100) 3 7346s 2 + 836228s , ,

A simulation of the FDI design for the benchmark test has been carried out. The simulation sampling period was 0.01 s. The faults' occurrence times are: 0 . 7 - 0.9s for the sensor fault (f~) and 2 . 7 - 3s for the current fault (fa). All of the simulations were performed using SIMULINK, and the results are shown in Figs. 2-7, with the different input-output data sequences. In Figs. 3, 5 and 7 the residual r2 (sensor fault) is shown corresponding to different data sequences; these indicate that with this approach and a fixed threshold it is possible to detect and isolate this fault. The residual rl with the different data sequences is shown in Figs. 2, 4 and 6. Here it is clear that the linear model design with a fixed threshold is not sufficient to detect this fault when nonlinearities (or uncertainties) are present (see Fig. 6), without false alarms.

6. CONCLUSIONS This paper has presented an application of a frequency domain approach to fault detection and isolation on the benchmark study. The procedure to design a residual generator and a residual evaluator in the frequency domain have been shown.

E. Alcorta Garcfa et al.

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Fig. 2. rl(fa) with DS2 The residual generator has been carried out for the industrial actuator benchmark test. For this study it has not been possible to design an adaptive threshold, because the upper bound of the uncertainties is unknown. It has been shown that the frequency domain approach can be used effectively in treating the linear and nonlinear problems with small signals when an upper bound of the uncertainties is given.

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Fig. 4. rl(fa REFERENCES Blanke, M., S. Bogh., R. B. JOrgensen and R. J. Patton (1995). 'Fault detection for diesel engine actuator - a benchmark for fdi'. Control Engineering Practice, in this issue. Ding, X. (1992). Frequenzbereichsverfahren zur beobachtergestuetzten Fehlerentdeckung. VDI-Verlag, Reihe 8, Nr. 295. Duesseldorf, Germany. Francis, B. A. (1987). A Coarse in H¢¢ Control Theory. Springer-Verlag. Berlin-New York. Frank, P. M. (1990). 'Fault diagnosis in dynamic systems using analytical and knowledgebased redundancy - a survey'. Automatica 26,459-474. Frank, P. M. and X. Ding (1994). 'Frequency domain approach to optimally robust residual generation and evaluation for model-based fault diagnosis'. Automatica 30(5), 789-804. Hoefling, T., T. Pfeufer, R. Deibert and R. Isermann (1994). 'Industrial actuator benchmark test - a systematic approach to an optimal fdi'. 2nd IFAC SAFEPROCESS 94 pp. 519524. Maciejowski, J. M. (1989). Multivariable Feedback Design. Adison-Wesley. Vidyasagar, M. (1985). Control Systems Synthesis: A Factorization Approach. MIT press. Viswanadham, N., J. H. Taylor and E. C. Lute (1987). 'Frequency domain approach to failure detection and isolation with application to ge-21 turbine engine control systems'. Control-Theory and Advanced technology 3, 45-72.

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Fig. 6. rl(fa) with DS4 0.O4 J

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