Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain
AN IIR FILTER BASED PARITY SPACE APPROACH FOR FAULT DETECTION *
Hao Ye,
*
Guizeng Wang,
**
S.X. Ding and
***
Hongyu Su
*
Department of Automation, Tsinghua University, Beijing 100084, P.R. China. E-mail:
[email protected] ** Institute for Automatic Control and Complex Systems, Duisburg University, Duisburg 47048, Germany. E-mail:
[email protected] *** Tsinghua Tongfang Co. Ltd., Beijing 100084, P.R. China. E-mail:
[email protected]
Abstract: In traditional parity space approaches, a parity vector with a low order means a simple online realization but a bad performance, while that with a high order leads to a good performance but an unacceptable calculation amount. In this paper, by introducing a simple scalar IIR (Infinite Impulse Response) filter, a new kind of parity relation based residual generator and the corresponding optimization approach are proposed. With the new approach, a residual generator can deliver a very good performance index but with a parity vector of low order. Simulation results are given. Keywords: Fault detection, observers, optimal filtering, optimization problems, robustness
1. INTRODUCTION The parity relation based optimization approach is one of the important optimal design approaches for the robust fault detection systems (Patton et al., 1989; Wuennenberg, 1990; Gertler, 1991; Frank and Ding, 1997; Gertler, 1998; Chen and Patton, 1999). The problems related to the design of robust residual generator based on parity space approach is studied in this paper. Since the residual generator of parity relation based fault detection system is presented in an explicit form, and Ding et al. (1999) recently proved that the optimal performance index of the system gets better with the increase of the order of the parity vector, there is a tradeoff in the design of the parity vector,
i.e. a parity vector with a low order means an easy online implementation of the system but a bad performance index, while a parity vector with a high order brings a good performance index but an unacceptable online computation amount. Although Ye et al. (2000) proposes an approach which uses a wavelet based parity vector with a low order to deliver a high performance, the calculation of the wavelet transform of vectors and matrices also introduces some additional computation amount and makes the online realization of the residual generator more complex. In this paper, in order to get a good optimal performance but using a parity vector with a low order, i.e. to improve the system performance
without the increase in computation, a new kind of residual generator based on parity relation and IIR (Infinite Impulse Response) filter and the corresponding optimal design approach are proposed, because the traditional parity relation based residual generator is actually a FIR (Finite Impulse Response) filter, and it is well known that a FIR filter with high order can be realized with an IIR filter with low order. 2. A BRIEF REVIEW OF THE PARITY SPACE APPROACH Consider a linear time-invariant discrete systems described by x(k + 1) = Ax(k ) + Bu (k ) + E d d (k ) + E f f (k ) y (k ) = Cx (k ) + Du (k ) + Fd d (k ) + F f f (k )
Fd CE d H d ,s = # s−1 CA E d Ff CE f H f ,s = # s −1 CA E f
(1) ku
where x(k ) ∈ R is the state vector, u (k ) ∈ R the vector of control signals, y (k ) ∈ R vector, d (k ) ∈ R
kd
ky
the output
is called parity where vs = [vs , 0 , vs ,1 ! vs , s ] ∈ R y vector which should be selected from the parity space Ps defined by Ps = {v s | v s H 0, s = 0} , s is the order of the parity relation, and
u s (k ) = [u T (k − s ) u T (k − s + 1) ! u T (k )]T d s (k ) = [d T (k − s ) d T (k − s + 1) ! d T (k )]T f s (k ) = [ f T (k − s ) f T (k − s + 1) ! f T (k )]T
H 0,s = [C T AT C T ! ( AT ) s C T ]T H u ,s
0 ! D CB D " = # " " s−1 CA B ! CB
0 # 0 D
0
!
Ff
"
"
"
!
CE f
(4)
vs ∈Ps
vs ∈Ps
vs H d ,s H dT,s v sT vs H f ,s H Tf ,s vsT
(5)
(3)
whose dynamics is governed by
y s (k ) = [ y T (k − s ) y T (k − s + 1) ! y T (k )]T
! CE d
3. THE NEW PARITY SPACE APPROACH BASED ON IIR FILTER
A parity relation based residual generator is expressed by
k ( s +1)
"
0 # 0 Fd 0 # 0 F f
(2)
k
and f (k ) ∈ R f the vector of faults to be detected. A, B, C, D, Ed, Ef, Fd and Ff are known and of appropriate dimensions.
r (k ) = v s [ H d , s (k )d s (k ) + H f , s (k ) f s (k )]
"
Since the residual generator (2) is presented in an explicit form, its implementation is not ideal for online realization, and since Ding et al. (1999) proved that the optimal performance index Jtra will decrease with the increase of the order s, there is a tradeoff in the design of the parity vector, i.e. a parity vector vs with a low order s means an easy online implementation but leads to a bad performance index Jtra, while vs with a high order s brings a good performance index but an unacceptable online computation amount at the same time.
the unknown disturbance vector
r (k ) = v s [ y s (k ) − H u , s u s (k )]
!
In the case that a perfect decoupling from d(k) is impossible, designing a robust residual generator under a certain optimization sense becomes necessary. Patton et al. (1989) and Wuennenberg (1990) proposed the following performance index min J tra = min
n
0 Fd "
3.1 The Residual Generator The basic idea of this paper is based on a new conclusion proposed by Ye et al. (2000), i.e. let Vs(z) denote the Z-Transform of the parity vector vs, then for a high order s, the optimal Vs(z) of (5) is a narrowband filter, and its frequency band becomes narrower with the increase of s. Since a parity vector with a higher order means a better performance, the conclusion means that the necessary condition of getting a high performance is that the parity vector is a narrowband filter. In traditional parity space approach, since the parity vector vs is a FIR filter, it has to have a high order in order to be narrow in frequency domain. While it is well known that a narrowband FIR filter of a high order can be realized with an IIR filter, which is also narrow in frequency domain and very long (actually has infinite length) in time domain, but has a very low order.
In this paper, a parity vector vs with a low order is firstly used in the residual generator which is not narrow in frequency domain but can remain the basic function of the parity vector, then a narrowband IIR filter 1/A(z) with a low order is introduced to make the IIR filter Vs(z)/A(z) become narrow and so that to improve the optimal performance index of the residual generator, where A( z ) = 1 − a1 z −1 ! − a p z − p . Since in real applications, usually a 2-order filter 1/A(z) is good enough to realize a narrow bandpass filter at any frequency, and a 1-order system 1/A(z) can be used to realize a low pass or a high pass filter, in this paper, let p=2, i.e. A( z ) = 1 − a1 z −1 − a 2 z −2 , and use 1/A(z) with a2=0 to denote a 1-order IIR system. So a new kind of parity relation based residual generator is proposed as following r (k ) = a1r (k − 1) + a2 r (k − 2) + vs [ y s (k ) − H u ,s u s (k )] (6) It is worth to notice that, since r(k) is a scalar signal, the introduction of 1/A(z) in (6) brings almost no additional calculation work compared with (2) when they have the same order s. But since the 1-order or 2 order filter 1/A(z) can be very narrow, the optimal performance index of the system based on (6) with a very low order s might be much better than that based on (2) with the same order.
where 0 d ∈ R kd ( s +1)×kd and I d ∈ R kd ( s +1)×kd ( s +1) denote a zero matrix and an identity matrix of the appropriate dimensions respectively, M d (i ) = [0 d !0 d
d s + m (k ) = [d (k − s − m) ! d T (k )]T
(7)
h(i )v s H f ,s f s (k − i ) = vs H f ,s M f (i ) f s + m (k ) M f (i ) = [ 0 f ! 0 f
h(i ) I f
f s+ m (k ) = [ f (k − s − m) ! f T (k )]T where 0 f ∈ R
k ( s +1)×k ( s +1)
k f ( s +1)×k f
Then according to (9) and (11), (8) further leads to m r (k ) = v s {H d ,s ∑ M d (i ) d s + m (k ) i =0 m + H f ,s ∑ M f (i ) f s + m (k )} i =0 = vs {H d ,s N d d s+ m (k ) + H f ,s N f f s+m (k )}
i =0
m
m
i =0
i=0
Comparing (13) with (3), the following optimization objective similar to (5) can be defined to determine the optimal vs and A(z) min J IIR = min
= h(i )vs H d ,s [d T (k − s − i ) ! d T (k − i )]T h(i ) I d
0 d !0 d ]
T
T
d (k − i ) ! d (k )] = v s H d ,s M d (i )d s+ m (k )
vs H f ,s N f N Tf H Tf ,s vsT
(15)
3.3 Analysis of the Optimization Objective
× [d T ( k − s − m) ! d T (k − s − i ) ! T
a1 , a2 ,vs∈Ps
vs H d ,s N d N dT H dT,s vsT
It can be seen that the performance index Jtra defined by (5) is a special case of JIIR if a1=a2=0.
h(i )vs H d ,s d s (k − i ) = v s H d ,s [0 d !0 d
(14)
are matrices related to the coefficients a1 and a2, because the impulse response h(i) is determined by a1 and a2.
a1 ,a2 ,vs∈Ps
It can be easily proved that for any 0 ≤ i ≤ m ,
(13)
where
m
r (k ) = ∑ h(i )vs [ H d ,s d s (k − i ) + H f ,s f s (k − i )] (8)
(12)
f denote and I f ∈ R f a zero matrix and an identity matrix of the appropriate dimensions respectively, and the numbers of 0f before and after h(i)If are (m − i ) and i respectively.
3.2 The Optimization Objective
Suppose vs ∈ Ps , then the dynamic (7) can be further expressed in an explicit form
(11)
0 f !0 f ]
T
N d = ∑ M d (i ) , N f = ∑ M f (i ) Let the scalar signal h(i ) (i = 0 ! m) denote the impulse response corresponding to 1/A(z), and m the settling time of h(i), i.e. h(i ) → 0 when i ≥ m .
(10)
and the numbers of 0d before and after h(i)Id are (m − i ) and i respectively. Similarly,
According to (2) and (3), it can be easily proved that when vs ∈ Ps , the dynamic of (6) is given by r (k ) = a1r (k − 1) + a2 r (k − 2) + vs [ H d ,s d s (k ) + H f ,s f s (k )]
0 d !0 d ]
h (i ) I d
T
(9)
It is necessary to discuss for the same order s, whether the introduction of 1/A(z) can surely make the optimal JIIR of the new approach become better
than the optimal Jtra of the traditional parity space approach. Suppose
obtained by the traditional approach, a narrow band filter 1/A(z) can be found to minimize JIIR and so that to make JIIR
