Proceedings on the 37th IEEE Conference on Decision & Control • Tampa, Florida USA • December 1998
WM03-3 14:00
Finite Memory Generalised State Observer For Failure Detection In Dynamic Systems Walter
NUNINGER,
Fr6d6ric
KRATZ,
JOS6 RAGOT
Centre de Recherche en Automatique de Nancy - CNRS UPRES-A 7039 Institut National Polytechnique de Lorraine -2, Avenue de la Fori% de Haye. 1?54516 Vandoeuvre-16s-Nancy Cedex, France Tel. : (33) 383595959Fax : (33) 383595644 E-mail : { wnuninge, fkratz, jragot}@ensem.u-nancy .ti
ABSTRACT To solve the Fault Detection and Isolation problem, many techniques are based on state estimation. The divergence phenomenon of the state estimation error due to the accumulation of model uncertainties (as all the process history is taken into account [1]) leads to the design of finite memory observer for both discrete and continuous time representations. Such observers are therefore more efficient for fault detection. This paper presents an extension of the finite memory observer so that both the input and the state of the system (so-called generalised state) are estimated on a fwed number of measurements. The relevance is that the estimation can be performed even if the data are not given with the same sampling period. In addition, actuator and sensor fault detection and isolation can be performed within bank of observers driven by different subsets of input and output measurements. Further, a new residual can be defmed so that the residual robustness to model uncertainties might be improved as it has already been shown and applied by Nuninger et al. within finite memory state estimation [2][3]. Key-word: finite memory, observer, fault detection, sliding window, recursive estimation, no constant sampling.
1. INTRODUCTION The Fault Detection and Isolation problem is usually solved within a diagnostic procedure generally composed by two steps: residuals generation followed by residuals evaluation [4][5~6] [7]. Within analytical redundancy, many techniques are based on output estimation to generate indirect residuals; on the other hand parity space approaches generate the so-called direct residuals. In the literature, both types of residuals are known to be structurally equivalent under some hypothesis [8] [9]. This result was also proved by Nuninger et al. for unknown input observers [10] and the works by Delmaire et al. [11] shows a link between parity space and parameter the Nevertheless, identification based residuals. accumulation of model uncertainties leads to the nonconvergence to zero of the state estimation error (the residual) that cause the robustness degree of the detection
0-7803-4394-8/98 $10.00 (c) 1998 IEEE
system to decrease. Such a phenomenon was studied by Toda et al, [1] for Luenberger observers and by Heffes [12] for Kahnan filters. In addition, state estimations based on all the process history (i.e. iufmite memory) are less sensitive to recent measurements that might contain information about incipient faults. Some solutions were proposed as the fading filter by Sorenson et al. [13] which is a weighted Kalman filter and the finite memory observer by Medvedev et al. [14][15]. This last observer is more efticient because the estimation is based on a limited number of data. In addition, a residual defined as the difference of two estimations based on overlapping finite observation horizons can improve the robustness degree with respect to model uncertainties [16] [2][3]. This degree is the result of the compromise between small no-detection and false alarm rates and can therefore be defined as the ratio of both sensitivities of the residual with respect to fault and to model uncertainties. The relevance of this paper is to develop an extension of the finite memory observer in order to estimate the generalised state of the system: i.e. the simultaneous estimation of the state and the input of the system. As a consequence, the proposed generalised state observer with finite memory filled the gap between continuous and discrete time representations as a similar solution has already been proposed for discrete time representations by Bousghiri et al. [17]. In addition, this paper stresses on the fact that within the particular previously defined residwdl less sensitivity to model uncertainties might be obtained. This paper is organised as follows. First the problem formulation is given. Second, the generalised state finite memory observer is presented under the assumption that not all the input are measured. The formulation of the observer is fiuther extended under the assumption that the inputs and outputs are not simultaneously measured. Note that this case stands for the most interesting one in comparison with the solution for discrete time representation [7] that requires simultaneous measurements. The fault detection problem is fhrther tackled within the definition of a new residual and a fault detection scheme based on different subsets of input and output is presented so that isolation is performed. Finally a conclusion is given on fiture studies.
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Proceedings on the 37th IEEE Conference on Decision & Control • Tampa, Florida USA • December 1998
2. PROBLEM Linear time-invariant time representation: x(t) = Ax(t)+
FORMULATION
systems are considered of continuous-
y(t) = Cx(t) +V(t)
(la) (lb)
z(t) =Du(t)+w(t)
(1.C)
where x is the state of dimension, u the input of dimension m, y the output measurement vector of dimension p and z the input measurement vector of dimension q. A, B and C are matrices of real coefilcients and right dimensions. Vectors w and v stand for the measurement noises on the input and output respectively that satis~ gaussian distributions of zero means and known constant covarimce matrices: VYand V. respectively. At the fi-ozen instant t, a finite observation horizon is considered that contains (k-t 1) output measurements and The considered outputs at (r+l) input measurements. instant t-~i are noted y(t – ~i ) for i from O to k whereas z(t – y, ) stands for the inputs
at instant t-~j for j horn
1 to r. Note that the (k+l) real c and the (r+l) real not bound to be consistent but satis~:
