1 Observer-Based Robust Fault Detection of Multirate Linear System ...
Observer-Based Robust Fault Detection of Multirate Linear System Using a Lift Reformulation M.S. Fadali Electrical Engineering Dept./260 University of Nevada, Reno Reno, NV 89557-0153 [email protected] Abstract This paper presents observer-based robust fault detection for multirate sampled-data linear system. It is assumed that all plant inputs are updated with a unique sampling period and that the plant outputs are sampled uniformly at integer multiples of the input sampling period. A state observer is constructed, as in the single rate case, and a residual generator is derived. The paper shows that the residual signals can be decoupled from noise and disturbances. An illustrative numerical example is provided. Keywords: Fault Detection, Observers, Multirate Sampling.
1. Introduction Advances in microcomputer technologies have made digital control increasingly popular. Digital control systems include multirate sampled-data control where each input and output is updated at a different sampling rate. Research in multirate sampled-data control systems has received considerable attention [1-7]. Due to technological constraints and economic benefits, multirate sampling is often required and beneficial. Furthermore, properly designed control systems using multirate sampled-data mechanism can significantly improve the performance of the controlled system [1]. As in the case of single rate systems [8,9], multirate systems are susceptible to a variety of faults that must be detected to avoid catastrophic fa ilure. To date, the only published study of
1
fault detection in multirate sampled-data control systems was presented by the authors in [10], [11]. In [10], we extended standard single rate residual generators to the multirate case. In [11], we presented a short preliminary version of this paper. This paper presents a new approach to robust fault detection in a multirate sampled-data linear system using state observers. Although the theory presented here is applicable to all multirate systems amenable to a lift reformulation, we restrict this study to the multirate sampled-data linear system introduced in [1]. All inputs are updated at a unique rate while each output is updated at an equal or slower rate. In addition, the output sampling periods are assumed integer multiples of the input sampling periods. It is assumed that we can summarize all uncertainties of a system as unknown inputs (disturbances) with known distribution matrix. The residual is an indicator signal of faults, and is usually based on a weighted comparison between estimated and measured variables. The residual expression based on the multirate sampled-data linear system is derived by the construction of a state observer. Our purpose is to de-couple the residual signals from the system uncertainties for robust fault detection. An example is presented to illustrate the approach. The paper is organized as follows: Section 2 is a review of multirate sampling. In Section 3, we discuss observers and fault detection. Section 4 presents a numerical example. Conclusions are given in Section 5.
2. Multirate Sampled-Data Linear System 2.1 System and Sampling Model Consider a linear time invariant system in the continuous or in the discrete-time form. Let us start from a continuous linear time- invariant model x& (t ) = Ax( t ) + Bu (t ) + D1 d (t ) + Q1 f a1 ( t )
(1a)
y (t ) = Cx (t ) + f s 1 (t )
(1b) 2
where A ∈ R n , n , B ∈ R n , p , D1 ∈ R n , q , Q1 ∈ R n , r and C ∈ R m ,n , x (t ) is the state vector, u (t ) is a vector of known inputs, d (t ) is the disturbance vector with distribution matrix D1 , f a1 (t ) is the actuator fault vector, and f s1 (t ) is the sensor fault vector. If the unique input sampling period of the system T0 is equal to the output sampling rates, we have the discrete-time linear time- invariant model x (k + 1) = Ad x( k ) + Bd u (k ) + Dd d (k ) + Qd f a1 (k )
(2a)
y (k ) = Cx( k ) + f s 1 ( k )
(2b)
Where x (k ), u ( k ), d (k ), f a1 ( k ), f s 1 ( k ) and the corresponding matrices have the same significance and order as their continuous counterparts. Note that Qd f a 1 (k ) and f s1 (k ) are unknown. Assume that the ith component y i (k ), i = 1,..., m of the output vector y (k ) is measure every TiT0 time instants, where Ti, i = 1, …, m, are positive integers. The sampling mechanism is introduced as follows. The measured output variables wi(k), i = 1, …, m, are defined as y (k ), k = k ′Ti + τi wi (k ) = i k ≠ k ′Ti + τi 0, where k ′ is a non-negative integer and the integers τi ,0 ≤ τi ≤ Ti , describe the skew sampling mechanism. Then w(k) is related to the system output y(k) by w(k) = N(k)y(k)
(3)
where y(k) =y(kT0 ) and 1, k = k ′Ti + τi m N (k ) := diag {vi (k )}i =1 , v i (k ) = 0, k ≠ k ′Ti + τi We can see that matrix N(k) is T-periodic, where
3
(4)
T = l .c.m.{Ti } i = 1,..., m .
(5)
Example 1 Consider an output y2 with sampling period 2T0 (T2 = 2) and sampling skew τi =1. The actual measurement is taken every two of the measurements for the system with sampling period T0 . The skew in the sampling means that the sampling is shifted by one period so that the first sample is at k = 1 rather than k = 0. We summarize the variables k', v 2 (k), and w2 (k), for a few sample points in Table 1. Note that k' is not listed in the equality in (3) is not satisfied and that v 2 (k) and w2 (k) are then zero. Table 1. Sampling mechanism of Example 1 k
0
1
2
3
k'
-
0
-
1
v2 (k)
0
1
0
1
w2 (k)
0
y2 (1)
0
y2 (3)
For a system with T1 = 1, N(k) is given in Table 2. Clearly, N(k) is periodic with period T = l.c.m.{T1 , T2 } = 2. Table 2. Matrix N(k) of Example 1 k
0
1
2
3
N(k)
1 0 0 0
1 0 0 1
1 0 0 0
1 0 0 1
♦ By combining model (2) and the multirate sampling mechanism (3)-(5), we have a T-periodic system in the output transformation.
4
Note that the possible presence of time delays in the output measurements can be dealt with by augmenting the dimension of the model state space. In order to simplify our analys is, we let τi, i=1, …, m, be zero.
2.2
Time-invariant reformulation
Consider the solution of the discrete-time state equation (2a). For any positive integer i, the solution is x (kT + i ) = Adi x( kT ) +
kT + i −1
kT + i −1
kT + i −1
j = kt
j =kt
j = kt
∑ AdkT +i −1− j Bd u ( j ) +
∑ AdkT + i−1− j Dd d ( j) +
∑A
kT +i −1 − j d
Qd f a1 ( j )
To obtain a time- invariant formulation for the system, we need to rewrite the solution for i = T in the form
[
x (kT + T ) = AdT x( kT ) + AdkT +T −1 Bd
[
+ AdkT + T −1 Dd
[
+ AdkT + T −1 Qd
u( kT ) M Bd u( kT + T − 2) u ( kT + T − 1)
]
L Ad Bd
L Ad D d
d ( kT ) M Dd d ( kT + T − 2) d ( kT + T − 1)
L Ad Q d
f a1 ( kT ) M Qd f a 1 ( kT + T − 2) f a 1 ( kT + T − 1)
]
]
x (k + 1) = Ad x( k ) + Bd u (k ) + Dd d (k ) + Qd f a1 (k )
(2a)
y (k ) = Cx( k ) + f s 1 ( k )
(2b)
5
Let k be a new discrete-time variable and define x (k ) := x(kT ) u (k ) := col {u (kT ), u (kT + 1), L , u (kT + T − 1)} f a (k ) := col { f a 1 (kT ), f a1 (kT + 1), L , f a 1 (kT + T − 1)} d (k ) := col {d (kT ), d (kT + 1), L , d (kT + T − 1)} f s (k ) := col { f s1 (kT ), f s 1 (kT + 1), L , f s 1 (kT + T −1)} We also define a new output vector w (k ) := col {w(kT ), w(kT + 1), L, w(kT + T − 1)} N ( 0) y (kT ) N (1) y ( kT + 1) = M N (T − 1) y ( kT + T − 1) Substituting from the output equation (2b) gives w (k ) = N (T = N (T
N ( 0){Cx( kT ) + f s1 ( kT )} N (1){Cx( kT + 1) + f s 1 ( kT + 1)} M − 1){Cx( kT + T − 1) + f s1 (kT + T − 1)}
N (0){Cx (kT ) + f s 1 ( kT )} N (1){C[ Ad x( kT ) + Bd u (kT ) + Dd d ( kt) + Qd f a 1 ( kT ) ] + f s1 ( kT + 1)} M kT + T − 2 T −1 kT +T −2 − j Bd u ( j ) + Ad x ( kT ) + ∑ Ad j = kt − 1)C kT +T −2 + f s1 ( kT + T − 1) kT +T −2 kT + T − 2 − j kT + T − 2 − j Ad Dd d ( j ) + ∑ Ad Qd f a 1 ( j ) ∑ j = kt j = kT
Then a T-sampled representation of the system (3)-(5) is x (k +1) = Fx (k ) + Gu (k ) + Dd (k ) + Qf a (k ) w (k ) = Hx (k ) + Eu (k ) + Db d (k ) + f s (k )
(6a) (6b)
where F = AdT
(6c)
6
G = [G0
G1 L GT −1 ]
Gi = A
T −i −1 d
(6d)
Bd , i = 0, L, T − 1
H = H 0 ′ N ′(0 ) H 1′ N ′(1) L H T −1 ′ N ′(T − 1) i H i = CAd , i = 0,L, T − 1
′ (6e)
E = {E ij } N (i − 1)CAdi − j −1 Bd Eij = 0
i> j i≤ j
i = 1,L , T j = 1, L, T
(6f)
i> j i≤ j
i = 1,L , T j = 1, L, T
(6g)
Db = {Dij } N (i − 1)CAdi − j −1 Bd Dij = 0
The matrices D and Q are defined similarly to G using Bd and Bf respectively. Note that Q and f s(k) are unknown. 3. State Observer and Fault Detection We construct a state observer for the time invariant system (6):
[
]
xˆ (k + 1) = Fxˆ (k ) + Gu (k ) + Lb w (k ) − Hxˆ (k ) − Eu (k ) = (F − Lb H )xˆ (k ) + (G − Lb E )u (k ) + Lb w (k )
(7a)
wˆ (k ) = Hxˆ (k ) + Eu (k )
(7b)
The state estimation errors are e(k ) = x (k ) − xˆ (k )
(8)
Their dynamics are e(k + 1) = x ( k + 1) − xˆ ( k + 1)
= Fx (k ) + Gu (k ) + Dd (k ) + Qf a (k ) − (F − Lb H )xˆ (k ) − (G − Lb E )u (k )
[
]
− Lb Hx + Eu (k ) + Db d (k ) + f s (k )
7
= ( F − Lb H )x (k ) + (G − Lb E )u (k ) − ( F − Lb H )xˆ (k ) − (G − Lb E )u (k ) + (D − Lb Db )d (k ) + Qf a (k ) − Lb f s (k )
= Fc e(k ) + ( D − Lb Db )d (k ) + Qf s (k ) − Lb f s (k )
(9)
where Fc = F − Lb H. A residual vector is generated from the difference between the actual and estimated measurements ε(k ) = Weγ (k ) = W (γ (k ) − γˆ (k ))
(10)
ε(k ) = WHe (k ) + WD b ξ(k ) + Wf s (k )
(11)
where W is a weighting matrix. From (9) and (11), we have the complete response of the residual vector ε(z ) := Grs ( z ) f s (z ) + Gra ( z ) f a ( z ) + Grd ( z ) d ( z )
(12)
with Grs (z), Gra (z), and Grd (z) the transfer functio ns given by
[
Grs ( z ) = W ( z ) I − H ( zI − Fc ) Lb −1
]
Gra ( z ) = WH ( zI − Fc ) Q −1
[
Grd (z ) = W ( z ) Db + H ( zI − Fc )
−1
( D − Lb Db )]
To nullify the effect of the disturbances, we require [8] Grd (z ) = W ( z ) M ( z ) = 0 where
[
(13)
]
M = Db + H ( zI − Fc ) −1 (D − Lb Db )
We know that Fc ∈ R n , s 1 , D ∈ R n ,s 2 , Db ∈ R s1× s 2 , H ∈ R s1× n , s1 = mT, s2 = qT. In the period T T0 , s1 is the maximum number of the measured outputs and s2 is the number of unknown input variables.
8
Observe that the matrix M ∈ R s 1× s 2 . Usually, we can remove the common zero rows of H , E , Db and f s (k ) since the number of the measured outputs s3 is less than s1, M ∈ R s 3× s 2 , Therefore, the conditions for the weighting matrix W to exist is s3 > s 2 , and rank (M ) ≤ s 2 . Here the elements of weighting matrix W are z-domain polynomials. It is hard to find a constant matrix W using the eigenstructure assignment approach of [12]. This is due to the involvement of the disturbances in the measured outputs expression. There are more constraints on the eigenstructure if the eigenstructure assignment approach is applied to (13). For example, once Lb is specified, the eigenstructure cannot be freely assigned. These ideas are explored further in the following example.
4. Example Consider a continuous system given by 0 0 − 1 1 − 1 0 0 0 A= 0 −1 −1 0 0 0 − 1 1
1 1 0 1 C = 0 1 1 0 1 0 0 1
− 1 0 D1 = 0 0
The inputs are updated every 0.1s and the three outputs are updated every 0.1, 0.2, and 0.2s, respectively. Hence we have T = 2. The corresponding discrete-time matrices are 0.0950 0 0 0.9002 − 0.0950 0.9952 0 0 AD = 0.0047 − 0.0950 0.9048 0 0.0047 0 0.9048 0.0903 N (0) = I 3
N (1) = diag {1,0,0}
− 0.0950 0 .0048 DD = − 0.0002 − 0.0047
N (i + 2) = N (i ), i = 0,1,2, ...
The time invariant model for the system is described by 9
0.1801 0 0 0.8013 − 0.1801 0.9813 0 0 F= 0.0175 − 0.1801 0.8187 0 0.0175 0 0.8187 0.1626 1 1 0 1 H = 1 0 0.8955 1.0948
− 0.0851 − 0.0950 0.0138 0.0048 DD = − 0.0010 − 0.0002 − 0.0128 − 0.0047
0 1 1 0 0 3×1 0 3×1 Db = 0 1 − 0.0948 0 0 0.9048
The system resulting from time invariant reformulation is observable. The eigenvalues of the closed- loop observer can be freely assigned by choosing the gain matrix Lb . We select the observer eigenvalues {0.5±j0.5, 0.6±j0.6} and obtain the observer gain matrix 0 − 3.6702 − 21.5262 23.1480 49.5549 0 − 8.9153 − 44.9137 Lb = 60.0951 0.3187 − 10.5172 − 55.3447 0.5 3.9742 16.6988 − 18.7651
M=
2. 601 4. 367 .689 2.688
0.059 − 6.815 − . 165 8. 897 .212 − 4.802 − 0. 095 0 0.025 − 4 −13. 82 − 0.049 −3 17.78 0. 047 − 2 − 9.496 0. 005 −1 0 z + z + z + z + − . 868 − . 114 .810 .163 − . 556 0 0.032 − 0.1 0.058 0. 2 − 7. 173 − . 158 9.33 − 4. 923 − 0. 084 − . 095 −1 −2 −3 −4 1 − 2.2 z + 2.42 z − 1.32 z + 0. 36 z
The weighting matrix W(z) is 0. 536 0.009 − 0.045 − 0.522 − 2 0.494 − 0.058 − 0. 235 − 0. 325 −1 W = z z + − 0.057 0.230 0. 558 − 0. 425 0. 195 − 0. 051 − 0.239 − 0.044 0. 115 0 0.117 0.056 + 0.406 − 0. 178 − 0. 395 0
The residuals are of the form
[
]
ε( z ) = W I − H ( zI − Fc )−1 Lb f s ( z ) + WH ( zI − Fc ) −1 Qf a ( z ) Clearly, the unknown inputs are decoupled from the residuals by choice of the matrix W.
10
0 0 0 0
Conclusion Fault detection for multirate sampled-data systems is an emerging research topic. While there are numerous results available for both multirate systems and for single-rate fault detection, there have been surprisingly few attempts to combine the two. This paper develops a fault detection scheme for a class of multirate systems based on observer design. Future efforts will involve extension to other classes of multirate sampled data systems and optimization of the observer design. References 1. P. Colaneri, R. Scattolini and N. Schiavoni, "Stabilization of Multirate Sampled-Data Linear Systems", Automatica, 26, 377-380, 1990. 2. P. Colaneri, R. Scattolini and N. Schiavoni, "Stabilization, Regulation and Optimization of Multirate Sampled-Data Systems", Control and Dynamic Systems (C.T. Leondes, Ed.), Academic Press, Orlando. 3. R. Scattolini and N. Schiavoni, "Decentralized Control of Multirate Systems Subject to Arbitrary Exogenous Signals", Proceedings of the 34th Conference on Decision & Control, New Orleans, LA-December, 1995. 4. M. Araki and K. Yamamoto, "Multivariable Multirate Sampled-Data Systems: State Space Description, Transfer Characteristics, and Nyquist Criterion", IEEE Trans. Automat. Contr. AC-31, 145-154, 1986. 5. T. Hagiwara and M. Araki, "Design of a Stable State Feedback Controller Based on the Multirate Sampling of the Plant Output", IEEE Transactions on Automat. Contr., Vol. 33, No. 9, September, 1988. 6. R. A. Meyer and C.S. Burrus, "A Unified Analysis of Multirate and Periodically TimeVarying Digital Filters", IEEE Trans. Ccts Syst., CAS-22, 162-168, (1975). 7. G. M. Kranc, "Input-Output Analysis of Multirate Feedback System", IRE Trans. Automat. Contr., 3, 21-28, 1957. 8. R. J. Patton and J. Chen, "Robust Fault Detection Using Eigenstructure Assignment: A Tutorial Consideration and Some New Results", Proceedings of the 30th Conference on Decision and Control, Brighton, England, December 1991. 9. J. J. Gertler, "Survey of Model- Based Diagnosis and Isolation in Complex Plants", IEEE Control System Magazine, 8(6), 3-11, 1988. 10. M. S. Fadali and W. Liu, "Observer-Based Robust Fault Detection for a Class of Multirate Sampled-Data Linear System", Proc. 1998 American Control Conf., Philadelphia, PA, June, 1998. 11. M. S. Fadali and W. Liu, "Fault Detection for Systems With Multirate Sampling", Proc. 1999 American Control Conf., San Diego, CA, June, 1999. 12. M. M. Fahmy and J. O'Reilly, "On Eigenstructure Assignment in Linear Multivariable Systems", IEEE Trans. Automat. Contr., Vol. AC-27, No. 3, pp. 690-693, June 1982.
11
1. Introduction Advances in microcomputer technologies have made digital control increasingly popular. Digital control systems include multirate sampled-data control where each input and output is updated at a different sampling rate. Research in multirate sampled-data control systems has received considerable attention [1-7]. Due to technological constraints and economic benefits, multirate sampling is often required and beneficial. Furthermore, properly designed control systems using multirate sampled-data mechanism can significantly improve the performance of the controlled system [1]. As in the case of single rate systems [8,9], multirate systems are susceptible to a variety of faults that must be detected to avoid catastrophic fa ilure. To date, the only published study of
1
fault detection in multirate sampled-data control systems was presented by the authors in [10], [11]. In [10], we extended standard single rate residual generators to the multirate case. In [11], we presented a short preliminary version of this paper. This paper presents a new approach to robust fault detection in a multirate sampled-data linear system using state observers. Although the theory presented here is applicable to all multirate systems amenable to a lift reformulation, we restrict this study to the multirate sampled-data linear system introduced in [1]. All inputs are updated at a unique rate while each output is updated at an equal or slower rate. In addition, the output sampling periods are assumed integer multiples of the input sampling periods. It is assumed that we can summarize all uncertainties of a system as unknown inputs (disturbances) with known distribution matrix. The residual is an indicator signal of faults, and is usually based on a weighted comparison between estimated and measured variables. The residual expression based on the multirate sampled-data linear system is derived by the construction of a state observer. Our purpose is to de-couple the residual signals from the system uncertainties for robust fault detection. An example is presented to illustrate the approach. The paper is organized as follows: Section 2 is a review of multirate sampling. In Section 3, we discuss observers and fault detection. Section 4 presents a numerical example. Conclusions are given in Section 5.
2. Multirate Sampled-Data Linear System 2.1 System and Sampling Model Consider a linear time invariant system in the continuous or in the discrete-time form. Let us start from a continuous linear time- invariant model x& (t ) = Ax( t ) + Bu (t ) + D1 d (t ) + Q1 f a1 ( t )
(1a)
y (t ) = Cx (t ) + f s 1 (t )
(1b) 2
where A ∈ R n , n , B ∈ R n , p , D1 ∈ R n , q , Q1 ∈ R n , r and C ∈ R m ,n , x (t ) is the state vector, u (t ) is a vector of known inputs, d (t ) is the disturbance vector with distribution matrix D1 , f a1 (t ) is the actuator fault vector, and f s1 (t ) is the sensor fault vector. If the unique input sampling period of the system T0 is equal to the output sampling rates, we have the discrete-time linear time- invariant model x (k + 1) = Ad x( k ) + Bd u (k ) + Dd d (k ) + Qd f a1 (k )
(2a)
y (k ) = Cx( k ) + f s 1 ( k )
(2b)
Where x (k ), u ( k ), d (k ), f a1 ( k ), f s 1 ( k ) and the corresponding matrices have the same significance and order as their continuous counterparts. Note that Qd f a 1 (k ) and f s1 (k ) are unknown. Assume that the ith component y i (k ), i = 1,..., m of the output vector y (k ) is measure every TiT0 time instants, where Ti, i = 1, …, m, are positive integers. The sampling mechanism is introduced as follows. The measured output variables wi(k), i = 1, …, m, are defined as y (k ), k = k ′Ti + τi wi (k ) = i k ≠ k ′Ti + τi 0, where k ′ is a non-negative integer and the integers τi ,0 ≤ τi ≤ Ti , describe the skew sampling mechanism. Then w(k) is related to the system output y(k) by w(k) = N(k)y(k)
(3)
where y(k) =y(kT0 ) and 1, k = k ′Ti + τi m N (k ) := diag {vi (k )}i =1 , v i (k ) = 0, k ≠ k ′Ti + τi We can see that matrix N(k) is T-periodic, where
3
(4)
T = l .c.m.{Ti } i = 1,..., m .
(5)
Example 1 Consider an output y2 with sampling period 2T0 (T2 = 2) and sampling skew τi =1. The actual measurement is taken every two of the measurements for the system with sampling period T0 . The skew in the sampling means that the sampling is shifted by one period so that the first sample is at k = 1 rather than k = 0. We summarize the variables k', v 2 (k), and w2 (k), for a few sample points in Table 1. Note that k' is not listed in the equality in (3) is not satisfied and that v 2 (k) and w2 (k) are then zero. Table 1. Sampling mechanism of Example 1 k
0
1
2
3
k'
-
0
-
1
v2 (k)
0
1
0
1
w2 (k)
0
y2 (1)
0
y2 (3)
For a system with T1 = 1, N(k) is given in Table 2. Clearly, N(k) is periodic with period T = l.c.m.{T1 , T2 } = 2. Table 2. Matrix N(k) of Example 1 k
0
1
2
3
N(k)
1 0 0 0
1 0 0 1
1 0 0 0
1 0 0 1
♦ By combining model (2) and the multirate sampling mechanism (3)-(5), we have a T-periodic system in the output transformation.
4
Note that the possible presence of time delays in the output measurements can be dealt with by augmenting the dimension of the model state space. In order to simplify our analys is, we let τi, i=1, …, m, be zero.
2.2
Time-invariant reformulation
Consider the solution of the discrete-time state equation (2a). For any positive integer i, the solution is x (kT + i ) = Adi x( kT ) +
kT + i −1
kT + i −1
kT + i −1
j = kt
j =kt
j = kt
∑ AdkT +i −1− j Bd u ( j ) +
∑ AdkT + i−1− j Dd d ( j) +
∑A
kT +i −1 − j d
Qd f a1 ( j )
To obtain a time- invariant formulation for the system, we need to rewrite the solution for i = T in the form
[
x (kT + T ) = AdT x( kT ) + AdkT +T −1 Bd
[
+ AdkT + T −1 Dd
[
+ AdkT + T −1 Qd
u( kT ) M Bd u( kT + T − 2) u ( kT + T − 1)
]
L Ad Bd
L Ad D d
d ( kT ) M Dd d ( kT + T − 2) d ( kT + T − 1)
L Ad Q d
f a1 ( kT ) M Qd f a 1 ( kT + T − 2) f a 1 ( kT + T − 1)
]
]
x (k + 1) = Ad x( k ) + Bd u (k ) + Dd d (k ) + Qd f a1 (k )
(2a)
y (k ) = Cx( k ) + f s 1 ( k )
(2b)
5
Let k be a new discrete-time variable and define x (k ) := x(kT ) u (k ) := col {u (kT ), u (kT + 1), L , u (kT + T − 1)} f a (k ) := col { f a 1 (kT ), f a1 (kT + 1), L , f a 1 (kT + T − 1)} d (k ) := col {d (kT ), d (kT + 1), L , d (kT + T − 1)} f s (k ) := col { f s1 (kT ), f s 1 (kT + 1), L , f s 1 (kT + T −1)} We also define a new output vector w (k ) := col {w(kT ), w(kT + 1), L, w(kT + T − 1)} N ( 0) y (kT ) N (1) y ( kT + 1) = M N (T − 1) y ( kT + T − 1) Substituting from the output equation (2b) gives w (k ) = N (T = N (T
N ( 0){Cx( kT ) + f s1 ( kT )} N (1){Cx( kT + 1) + f s 1 ( kT + 1)} M − 1){Cx( kT + T − 1) + f s1 (kT + T − 1)}
N (0){Cx (kT ) + f s 1 ( kT )} N (1){C[ Ad x( kT ) + Bd u (kT ) + Dd d ( kt) + Qd f a 1 ( kT ) ] + f s1 ( kT + 1)} M kT + T − 2 T −1 kT +T −2 − j Bd u ( j ) + Ad x ( kT ) + ∑ Ad j = kt − 1)C kT +T −2 + f s1 ( kT + T − 1) kT +T −2 kT + T − 2 − j kT + T − 2 − j Ad Dd d ( j ) + ∑ Ad Qd f a 1 ( j ) ∑ j = kt j = kT
Then a T-sampled representation of the system (3)-(5) is x (k +1) = Fx (k ) + Gu (k ) + Dd (k ) + Qf a (k ) w (k ) = Hx (k ) + Eu (k ) + Db d (k ) + f s (k )
(6a) (6b)
where F = AdT
(6c)
6
G = [G0
G1 L GT −1 ]
Gi = A
T −i −1 d
(6d)
Bd , i = 0, L, T − 1
H = H 0 ′ N ′(0 ) H 1′ N ′(1) L H T −1 ′ N ′(T − 1) i H i = CAd , i = 0,L, T − 1
′ (6e)
E = {E ij } N (i − 1)CAdi − j −1 Bd Eij = 0
i> j i≤ j
i = 1,L , T j = 1, L, T
(6f)
i> j i≤ j
i = 1,L , T j = 1, L, T
(6g)
Db = {Dij } N (i − 1)CAdi − j −1 Bd Dij = 0
The matrices D and Q are defined similarly to G using Bd and Bf respectively. Note that Q and f s(k) are unknown. 3. State Observer and Fault Detection We construct a state observer for the time invariant system (6):
[
]
xˆ (k + 1) = Fxˆ (k ) + Gu (k ) + Lb w (k ) − Hxˆ (k ) − Eu (k ) = (F − Lb H )xˆ (k ) + (G − Lb E )u (k ) + Lb w (k )
(7a)
wˆ (k ) = Hxˆ (k ) + Eu (k )
(7b)
The state estimation errors are e(k ) = x (k ) − xˆ (k )
(8)
Their dynamics are e(k + 1) = x ( k + 1) − xˆ ( k + 1)
= Fx (k ) + Gu (k ) + Dd (k ) + Qf a (k ) − (F − Lb H )xˆ (k ) − (G − Lb E )u (k )
[
]
− Lb Hx + Eu (k ) + Db d (k ) + f s (k )
7
= ( F − Lb H )x (k ) + (G − Lb E )u (k ) − ( F − Lb H )xˆ (k ) − (G − Lb E )u (k ) + (D − Lb Db )d (k ) + Qf a (k ) − Lb f s (k )
= Fc e(k ) + ( D − Lb Db )d (k ) + Qf s (k ) − Lb f s (k )
(9)
where Fc = F − Lb H. A residual vector is generated from the difference between the actual and estimated measurements ε(k ) = Weγ (k ) = W (γ (k ) − γˆ (k ))
(10)
ε(k ) = WHe (k ) + WD b ξ(k ) + Wf s (k )
(11)
where W is a weighting matrix. From (9) and (11), we have the complete response of the residual vector ε(z ) := Grs ( z ) f s (z ) + Gra ( z ) f a ( z ) + Grd ( z ) d ( z )
(12)
with Grs (z), Gra (z), and Grd (z) the transfer functio ns given by
[
Grs ( z ) = W ( z ) I − H ( zI − Fc ) Lb −1
]
Gra ( z ) = WH ( zI − Fc ) Q −1
[
Grd (z ) = W ( z ) Db + H ( zI − Fc )
−1
( D − Lb Db )]
To nullify the effect of the disturbances, we require [8] Grd (z ) = W ( z ) M ( z ) = 0 where
[
(13)
]
M = Db + H ( zI − Fc ) −1 (D − Lb Db )
We know that Fc ∈ R n , s 1 , D ∈ R n ,s 2 , Db ∈ R s1× s 2 , H ∈ R s1× n , s1 = mT, s2 = qT. In the period T T0 , s1 is the maximum number of the measured outputs and s2 is the number of unknown input variables.
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Observe that the matrix M ∈ R s 1× s 2 . Usually, we can remove the common zero rows of H , E , Db and f s (k ) since the number of the measured outputs s3 is less than s1, M ∈ R s 3× s 2 , Therefore, the conditions for the weighting matrix W to exist is s3 > s 2 , and rank (M ) ≤ s 2 . Here the elements of weighting matrix W are z-domain polynomials. It is hard to find a constant matrix W using the eigenstructure assignment approach of [12]. This is due to the involvement of the disturbances in the measured outputs expression. There are more constraints on the eigenstructure if the eigenstructure assignment approach is applied to (13). For example, once Lb is specified, the eigenstructure cannot be freely assigned. These ideas are explored further in the following example.
4. Example Consider a continuous system given by 0 0 − 1 1 − 1 0 0 0 A= 0 −1 −1 0 0 0 − 1 1
1 1 0 1 C = 0 1 1 0 1 0 0 1
− 1 0 D1 = 0 0
The inputs are updated every 0.1s and the three outputs are updated every 0.1, 0.2, and 0.2s, respectively. Hence we have T = 2. The corresponding discrete-time matrices are 0.0950 0 0 0.9002 − 0.0950 0.9952 0 0 AD = 0.0047 − 0.0950 0.9048 0 0.0047 0 0.9048 0.0903 N (0) = I 3
N (1) = diag {1,0,0}
− 0.0950 0 .0048 DD = − 0.0002 − 0.0047
N (i + 2) = N (i ), i = 0,1,2, ...
The time invariant model for the system is described by 9
0.1801 0 0 0.8013 − 0.1801 0.9813 0 0 F= 0.0175 − 0.1801 0.8187 0 0.0175 0 0.8187 0.1626 1 1 0 1 H = 1 0 0.8955 1.0948
− 0.0851 − 0.0950 0.0138 0.0048 DD = − 0.0010 − 0.0002 − 0.0128 − 0.0047
0 1 1 0 0 3×1 0 3×1 Db = 0 1 − 0.0948 0 0 0.9048
The system resulting from time invariant reformulation is observable. The eigenvalues of the closed- loop observer can be freely assigned by choosing the gain matrix Lb . We select the observer eigenvalues {0.5±j0.5, 0.6±j0.6} and obtain the observer gain matrix 0 − 3.6702 − 21.5262 23.1480 49.5549 0 − 8.9153 − 44.9137 Lb = 60.0951 0.3187 − 10.5172 − 55.3447 0.5 3.9742 16.6988 − 18.7651
M=
2. 601 4. 367 .689 2.688
0.059 − 6.815 − . 165 8. 897 .212 − 4.802 − 0. 095 0 0.025 − 4 −13. 82 − 0.049 −3 17.78 0. 047 − 2 − 9.496 0. 005 −1 0 z + z + z + z + − . 868 − . 114 .810 .163 − . 556 0 0.032 − 0.1 0.058 0. 2 − 7. 173 − . 158 9.33 − 4. 923 − 0. 084 − . 095 −1 −2 −3 −4 1 − 2.2 z + 2.42 z − 1.32 z + 0. 36 z
The weighting matrix W(z) is 0. 536 0.009 − 0.045 − 0.522 − 2 0.494 − 0.058 − 0. 235 − 0. 325 −1 W = z z + − 0.057 0.230 0. 558 − 0. 425 0. 195 − 0. 051 − 0.239 − 0.044 0. 115 0 0.117 0.056 + 0.406 − 0. 178 − 0. 395 0
The residuals are of the form
[
]
ε( z ) = W I − H ( zI − Fc )−1 Lb f s ( z ) + WH ( zI − Fc ) −1 Qf a ( z ) Clearly, the unknown inputs are decoupled from the residuals by choice of the matrix W.
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0 0 0 0
Conclusion Fault detection for multirate sampled-data systems is an emerging research topic. While there are numerous results available for both multirate systems and for single-rate fault detection, there have been surprisingly few attempts to combine the two. This paper develops a fault detection scheme for a class of multirate systems based on observer design. Future efforts will involve extension to other classes of multirate sampled data systems and optimization of the observer design. References 1. P. Colaneri, R. Scattolini and N. Schiavoni, "Stabilization of Multirate Sampled-Data Linear Systems", Automatica, 26, 377-380, 1990. 2. P. Colaneri, R. Scattolini and N. Schiavoni, "Stabilization, Regulation and Optimization of Multirate Sampled-Data Systems", Control and Dynamic Systems (C.T. Leondes, Ed.), Academic Press, Orlando. 3. R. Scattolini and N. Schiavoni, "Decentralized Control of Multirate Systems Subject to Arbitrary Exogenous Signals", Proceedings of the 34th Conference on Decision & Control, New Orleans, LA-December, 1995. 4. M. Araki and K. Yamamoto, "Multivariable Multirate Sampled-Data Systems: State Space Description, Transfer Characteristics, and Nyquist Criterion", IEEE Trans. Automat. Contr. AC-31, 145-154, 1986. 5. T. Hagiwara and M. Araki, "Design of a Stable State Feedback Controller Based on the Multirate Sampling of the Plant Output", IEEE Transactions on Automat. Contr., Vol. 33, No. 9, September, 1988. 6. R. A. Meyer and C.S. Burrus, "A Unified Analysis of Multirate and Periodically TimeVarying Digital Filters", IEEE Trans. Ccts Syst., CAS-22, 162-168, (1975). 7. G. M. Kranc, "Input-Output Analysis of Multirate Feedback System", IRE Trans. Automat. Contr., 3, 21-28, 1957. 8. R. J. Patton and J. Chen, "Robust Fault Detection Using Eigenstructure Assignment: A Tutorial Consideration and Some New Results", Proceedings of the 30th Conference on Decision and Control, Brighton, England, December 1991. 9. J. J. Gertler, "Survey of Model- Based Diagnosis and Isolation in Complex Plants", IEEE Control System Magazine, 8(6), 3-11, 1988. 10. M. S. Fadali and W. Liu, "Observer-Based Robust Fault Detection for a Class of Multirate Sampled-Data Linear System", Proc. 1998 American Control Conf., Philadelphia, PA, June, 1998. 11. M. S. Fadali and W. Liu, "Fault Detection for Systems With Multirate Sampling", Proc. 1999 American Control Conf., San Diego, CA, June, 1999. 12. M. M. Fahmy and J. O'Reilly, "On Eigenstructure Assignment in Linear Multivariable Systems", IEEE Trans. Automat. Contr., Vol. AC-27, No. 3, pp. 690-693, June 1982.
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