Reconfigurable Fault Tolerant Control for Linear ...

Reconfigurable Fault Tolerant Control for Linear Stochastic Systems subject to Sequential Jumps Jean-Yves Keller CRAN - Université Henri Poincaré, Nancy I, BP 239 54506 Vandoeuvre les Nancy, France E-mail : [email protected] Abstract. For sequential jumps detection, isolation and estimation in discrete-time stochastic linear systems, Willsky and Jones (1976) have developed the Generalized Likelihood Ratio (GLR) test. For the treatment of sequential jumps, the jump-free Kalman filter is updated on-line after each detection of one jump by a direct state estimate and covariance incrementation using the informations produced by the GLR detector. This paper proposes another updating strategy based on a reference model updated on-line by the states of jumps declared to be occurred during the sequential processing. The augmented state Kalman filter designed on this updated reference model will be optimaly initialized at each detection time from information given by the GLR detector. We will show how the obtained Fault detection and Isolation (FDI) scheme can be integrated in a reconfigurable Fault-Tolerant Control System (FTCS) in order to ensure continued useful operation when sequential jumps occur.

1. Introduction Over the last two decades, growing demand for reliability, maintainability and survivability in dynamic systems has drawn significant research in the area of Fault-Tolerant Control System (FTCS). An active FTCS possesses the ability to accomodate system component failures automatically from information given by the Fault Detection and Isolation scheme (FDI) (see [1-2] and references therein). Compared to the passive FTCS obtained by standard robust control design on the worst case fault scenario which may never occur, the goal of this paper is to show how an observer based controller can be reconfigurated to compensate the sequential jumps. For sequential jumps detection in discrete-time stochastic linear systems, Willsky and Jones [3] have presented the GLR test used in a wide variety of applications including the detection of sensor and actuator failures [4-5], geophysical signal processing [6], freeways supervision [7] and more recently for air pollution prediction and control [8]. The standard GLR test consists in the following steps : 1: detect, isolate and estimate one jump from a GLR detector applied on the zero mean white innovation sequence of the jump-free Kalman filter, 2: update the jump-free Kalman filter using information given by the GLR detector, 3: go to the step 1 for the treatment of another jump. To ensure the stability of the updated filter allowing its integration in a FTCS, this paper proposes another updating strategy based on the Augmented State Kalman Filter (ASKF) designed on a reference model updated by the states of jumps declared to be occurred during the sequential processing. At the detection time of the first jump, the ( n + 1)-order ASKF will be optimaly initialized from the maximum likelihood jump estimate given by the GLR detector. After the detection time of the first jump, the treatment of another jump will be obtained by a GLR detector applied on the innovation sequence of the resulting ASKF. This sequential strategy will be repeated after each detection of one jump. The active FTCS derived to the obtained active GLR test will consist in the following steps 1: at the detection time of one jump, update the reference model with the new jump’s state 2: initialize the ASKF and replace the old control law by the new control law, 3: go to step one. Compared to a standard robust control design, the size of the disturbance vector will increase with

the number of jumps declared to be statistically significant by the active GLR test. We will show that the nominal system performances can be asymptotically recovered with no sacrifice of nominal performance for robustness before the occurrence of jumps. This paper is organized as follows: The section 2 presents the active GLR test in its recursive form. The section 3 presents the reconfigurable FTCS derived via the active GLR test before to conclude in section 4.

2. The active GLR test The model of the jump-free system h 0 is given by Xk0+1 = A 0 Xk0 + B 0 uk0 + Γ 0 wk

yk = C 0 Xk0 + vk

(1.a)

where Xk0 ∈ ℜ n is the state vector, yk ∈ ℜ m the measurement vector, uk0 ∈ ℜ d the input vector.   wk   W 0  The zero mean white Gaussian noises wk and vk satisfy E   w Tj v Tj  =  δ kj with W ≥ 0 ,   vk    0 V  Iz − A 0  jw 0 0 1/ 2 V > 0 and rank  (1.b)  = n, ∀z ∈ C, z ≥ 1 rank[ − e I + A Γ W ] = n, ∀w ∈[0 2 π] 0 C  

[

]

The jump hypotheses hi are modelized by Xk0+1 = A 0 Xk0 + B0 uk0 + fi (k, r) ν(k, r)+ Γ 0 wk , yk = C 0 Xk0 + vk

i ∈[1,.., N ]

(2.a)

where fi (k, r) is the fault distribution vector, ν(k, r) the scalar fault magnitude, r the unknown time of failure occurrence. Without lost of generality, the fault magnitude ν(k, r) is assumed to follow a constant bias model: ν(k, r)= ν {k ≥ r} with step profile: fi (k, r) = fi , {k ≥ r} and fi (k, r) = 0, {k < r} .  I − A [ f1 .. fi .. f N ] We assume rank  (2.b)  = n+ N 0  C  Concerning the sequential jumps scenario, we made the following assumptions: no a priori information is available concerning the jump magnitude ν and the jump occurrence time r . The first jump may occurre relatively infrequently and the time between the occurrence of two jumps is relatively large. After the detection of one or more sequential jumps, no further information will be available to focus our attention on a particular jump. The first jump may occurre relatively infrequently and the good reference model at the beginning of the processing is the jump-free system (1). A modified form of the standard GLR test [1] derived from the n-order jump-free Kalman filter 0 = A 0 Xˆ k0 + B0 uk0 + K k0 γ 0k Xˆ k+1 T T T T T Ω 0k +1 = A 0 Ω 0k A 0 + Γ 0 WΓ 0 − A 0 Ω 0k C 0 (C 0 Ω 0k C 0 + V ) −1 C 0 Ω 0k A 0 T T K 0 = A 0 Ω 0 C 0 (H 0 ) −1 γ 1 = y − C 0 Xˆ 0 H 0 = C 0Ω0C 0 + V k

k

k

k

k

k

k

(3.a) (3.b) (3.c)

k

is described as follows: k-α 0

If Tk0 = ∑ γ t0 T (Ht0 ) −1 γ t0 > λ 0 then ( j , rˆ ) = arg( t = k −β 0

max{Ti 0 (k, r )}

) i ∈[1,.., N ], r ∈[ k - β 0 ,.., k - α 0 ]

k

k

t =r

t =r

(3.d)

where Ti 0 (k, r) = bi0 ( k , r )2 ai0 ( k , r ) −1 , ai0 ( k , r )= ∑ ρi0T (t,r)(Ht0 ) −1 ρi0 (t,r) , bi ( k , r ) = ∑ ρi0T (t,r)(Ht0 ) −1 γ t0 are computed from the jump signatures ρi0 ( k , r )= C 0ζ i0 (k, r), ζ i0 (k + 1, r) = (A 0 - K k0 C 0 ) ζ i0 (k, r)+ fi , ζ i0 (r,r) = 0 ∀i ∈[1,.., N ] ( ζ i0 (k + 1, r) represents the additive effect of hypothesized jumps on the augmented state prediction error e 0 = X 0 − Xˆ 0 so that e 0 = e 0 + ζ 0 (k + 1, r) ν with e 0 = X 0 − Xˆ 0 ). If T 0 > k +1

k +1

k +1

k +1

k +1

i

k +1

k +1

k +1

k

λ 0 then a jump is detected at time k and isolated from ( j , rˆ ) = arg(max{Ti 0 (k, r )}) (to simplify the recursive presentation of the active GLR test, the first detected jump will be called h1 ). For the

treatment of second jump among, we propose the following strategy: update the reference model h 0  A 0 f1   B0  Γ 0  1 1 1 1 1 1 1 1 1 1 1 0 1 B =   , C = [C 0 ] , Γ =   as Xk +1 = A Xk + B uk + Γ wk , yk = C Xk + vk with A =  ,  0 1 0 0 0 X  where the augmented state Xk1 =  k  includes the jump’s state ν1 and rewritte the jump hypotheses  ν1  {h2 ,.., hN } w i t h r e s p e c t t o t h e n e w r e f e r e n c e m o d e l ( n o t e d h1 ) a s  fi (k, r) 1 1 1 1 Xk1+1 = A1 Xk1 + B1uk1 +   ν(k, r)+ Γ wk , yk = C Xk + vk ∀i ∈[ 2,.., N ] . On h , the ( n + 1) -order  0  ASKF 1 = A1 Xˆ k1 + B1uk1 + K k1 γ 1k Xˆ k+1 T T T T T Ω1k +1 = A1Ω1k A1 + Γ 1WΓ 1 − A1Ω 1k C1 (C 1Ω1k C1 + V ) −1 C1Ω1k A1 T T K 1 = A1Ω 1 C1 (H 1 ) −1 γ 1 = y − C1 Xˆ 1 H 1 = C1Ω1 C1 + V k

k

k

k

k

k

k

(4.a) (4.b) (4.c)

k

is optimaly initialized as  Xˆ 0 + ζ 0 (k, r) ˆ Σ10 ( k , rˆ )ζ10 (k, r) ˆT Ω 0 + ζ10 (k, r) ˆ 10 ( k , rˆ ) 1 ˆ ν 1 ˆ) =  k Ω ( k , r Xˆ 1 ( k , rˆ ) =  k  ˆT Σ10 ( k , rˆ )ζ10 (k, r) ˆ 10 ( k , rˆ ) ν   

ˆ Σ10 ( k , rˆ ) ζ10 (k, r)  Σ10 ( k , rˆ ) 

(4.d)

where ˆν10 ( k , rˆ ) = a10 ( k , rˆ ) −1 b10 ( k , rˆ ) and Σ10 ( k , rˆ ) = a10 ( k , rˆ ) −1 represents the maximum likelihood jump estimate given by the detector. After that the above updating strategy is applied at the detection time of the first jump, the treatment of a second jump is obtained from the zero mean white innovation sequence γ 1k of the ( n + 1) -order ASKF as follows k-α 1

If Tk1 = ∑ γ 1t T (Ht1 ) −1 γ 1t > λ1 then ( j , rˆ ) = arg( t = k −β1

where

max{Ti 1 (k, r )} ) i ∈[ 2,.., N ], r ∈[ k - β1 ,.., k - α1 ]

k

−1

ai1 ( k , r )= ∑ ρi1 T (t,r)Ht1 ρi1 (t,r) ,

Ti 1 (k, r) = bi1 ( k , r )2 ai1 ( k , r ) −1 ,

t =r

(4.e)

k

−1

bi1 ( k , r )= ∑ ρi1 T (t,r)Ht1 γ 1t t =r

are

 fi  computed from the jump signatures ρ1i ( k , r ) = C1 ζ1i (k, r), ζ1i (k + 1, r) = (A1 - K k1C 1 ) ζ1i (k, r)+   , ζ1i (r,r) 0 1 = 0, ∀i ∈[ 2,.., N ] ( ζ i (k + 1, r) represents the additive effect of hypothesized jumps on the augmented state prediction error e1 = X 1 − Xˆ 1 ). If T 1 > λ then a second jump is detected at time k , isolated k +1

k +1

k +1

k

1

from ( j , rˆ ) = arg(max{Ti 1 (k, r )}) and estimated by ˆν12 ( k , rˆ ) = a21 ( k , rˆ ) −1 b21 ( k , rˆ ) and Σ12 ( k , rˆ ) = a21 ( k , rˆ ) −1 (the second detected jump is called h2 ). To derive the recursive algorithm of the active GLR test by

[

induction, assume the presence of q jumps ν1 .. ν q hypotheses.

From

Xkq = [( Xkq−1 )T

ν

],

q T k

the

]

T

{

where hq +1 , h3 ,.., hN

reference

} is the subset of jump hq

model

q −1

is

expressed

fq  A B  q q q −1 Xkq+1 = A q Xkq + B q ukq + Γ q wk and yk = C q Xkq + vk with A q =  0] and , B =   , C = [C 0 1 0     Γ q−1  q Γq =  as  and the jump hypotheses rewritten with respect to h 0    fi (k, r) q q q Xkq+1 = A q Xkq + B q ukq +   ν(k, r)+ Γ wk and yk = C Xk + vk . The treatment of the ( q + 1) th jump 0   is then obtained from the innovation sequence γ qk of the ( n + q ) -order ASKF as k-α q

If Tkq = ∑ γ tqT (Htq ) −1 γ tq > λ q then ( j , rˆ ) = arg( t = k −β q

q−1

max{Ti q (k, r )}

[

i ∈[q + 1,.., N ], r ∈ k - β q ,.., k - α q

]

)

(5.a)

k

k

t =r

t =r

where Ti q (k, r) = biq ( k , r )2 aiq ( k , r ) −1 , aiq ( k , r )= ∑ ρiq T (t,r)(Htq ) −1 ρiq (t,r) , biq ( k , r )= ∑ ρiq T (t,r)(Htq ) −1 γ tq are  fi  computed from the jump signatures ρiq (t,r)= Cζ iq (k, r) , ζ iq (k + 1, r)= (A q - K kq C q ) ζ iq (k, r)+   , ζ iq (r,r) 0 = 0, ∀i ∈[q + 1,.., N ] . During the sequential processing, the successive initializations of the ( n + 1) , .. , ( n + q ) -order ASKF at each detection time are realized by the recursion  Xˆ q−1 + ζ q−1 (k, r) ˆ Σ qq−1 ( k , rˆ )ζ qq−1 (k, r) ˆ T ζ qq−1 (k, r) ˆ Σ qq−1 ( k , rˆ ) Ω q−1 + ζ qq−1 (k, r) ˆ ˆν qq−1 ( k , rˆ ) q q ˆ) =  k Ω ( k , r Xˆ q ( k , rˆ ) =  k   ˆT Σ qq−1 ( k , rˆ )ζ qq−1 (k, r) Σ qq−1 ( k , rˆ ) ˆ qq−1 ( k , rˆ ) ν     −1 −1 q −1 q−1 q−1 q −1 q−1 ˆ with ν q ( k , rˆ ) = aq ( k , rˆ ) bq ( k , rˆ ) and Σ q ( k , rˆ )= aq ( k , rˆ ) . Now completely presented in its recursive form, the active GLR test solves on-line a dynamic statistical decoupling problem by rejecting the nuisance parameters which are statistically significant, i.e. the size of the nuisance T parameters ν1 .. ν q increases with the number of jumps declared to be statistically significant during the processing. The convergence and stability conditions of the adaptive Kalman filter used in the standard GLR test are not established. For our active GLR test, we can derive the following convergence and stability conditions:

[

]

Theorem 1: Under (1.b) and (2.b), the ( n + q ) -order ASKF has a strong solution (i.e. eig( A q − K q C q ) ≤ 1) for any jump scenario. Proof. Under (1.b), the jump-free Kalman filter has a stabilizing solution (i.e. eig( A 0 − K 0 C 0 ) < 1 ). The pair ( A q , Γ q W 1 / 2 ) has q unreachable modes on the unit circle and the detectability of ( A q , C q ) ensures the existence of a strong solution for the augmented state Ricatti difference equation T T T T T Ω qk +1 = A q Ω qk A q + Γ q WΓ q − A q Ω qk C q (C q Ω qk C q + V ) −1 C q Ω qk A q . Under (1.b), the detectability of the pair ( A q , C q ) (corresponding to the geomatrical jump detectability condition derived in [10]) is guaranted by (2.b) (or by the non existence of invariant zero at zero frequency on the transfer H N ( z ) = C 0 ( Iz − A 0 ) −1 F N ).

[

Theorem 2: The inferior bound α q of the sliding window k - β q ,.., k - α q

]

must correspond to the

degree of the unitary interactor matrix ξ (z) (a polynomial matrix so that ξ q (z)[ ξ q (z)]* = I ) satisfying rank lim H N −q ( z )ξ q (z) = N − q with H N −q ( z ) = C 0 ( Iz − A 0 ) −1 fq +1 .. f N (5.b) q

{

}

z→∞

[

]

q

Proof. From the reference model h , define the least favorable jumps scenario as the case where all the jumps hq +1 ,.., hN appear simultaneously at time r . On γ qk , the signature ρ N −q ( k , r )= C qζ N −q ( k , r )

{

}

[

]

 f .. f  of this least favorable scenario is expressed from ζ N −q ( k + 1, r ) = (A q - K q C q ) ζ N −q ( k , r )+  q +1 N  0   T with ζ N −q ( r, r ) = 0 and the Kullback divergence is expressed as δ N −q ( k , r ) = ν an−q ( k ) ν with aN −q ( k , r ) k

= ∑ ρTN −q (t , r )( Htq ) −1 ρ N −q (t , r ) . The Kullback divergence δ N −q ( k , r ) cannot satisfy δ N −q ( k , r ) = 0 with t =r

[

ν = ν q +1 .. ν j

{

νN

]

T

≠ 0 if and only if an−q ( k , r ) > 0 and the structural detection delay α q for

}

{

}

jumps hypotheses hq +1 ,.., hN can be defined as α q = min t : rank [ ρ N −q (t , r )] = N − q, t = r + 1, r + 2.. or equivalently by the order of the minimal polynomial matrix ξ q (z) satisfying (5.b) (Wolowich and Falb [11]) describing the structure at infinity of the transfer H N −q ( z ). For a given sequence h 0 ,.., h j ,.., h q (depending to the jump scenario), the inferior bounds of sliding data windows k - β j ,.., k - α j may decrease (i.e. α 0 ≥ .. ≥ α j ≥ .. ≥ α q ), the main structural result of theorem 2.

[

]

Under h j , the detection variables Tk j follow a Chi-squared distribution χ 2 ( dl j , 0 ) with dl j = (β j − α j + 1)m degrees of freedom and the threshold levels λ j can be chosen by fixing a low ∞ false alarms probability PjFA = ∫λ j χ 2 ( dlq , 0 )dx . The missed alarms probability PjND can be bounded as λ

PjND ≤ ∫0 j χ 2 ( dl j , δ j ( νcj )2 )dx where χ 2 ( dl j , δ j ( νcj )2 ) is a Chi-squared distribution with dl j degrees of

{a (r + α , r )}( ν ) minimum jump that is required to be detected for jump hypotheses {h

freedom and noncentrality parameter δ j = min

i∈[ j+1,..,N ]

j i

c 2 j

j

(where νcj represents the

}

,.., hN under r = k − α j ). To ensure a good reference model h during the processing, the statistical goal PjND ≤ PjFA ∀j ∈[0,.., q ] is j +1

q

2λ j

reached if and only if ν ≥ νcj ∀j ∈[0,.., q ] with νcj = thus ν is great with Pj c j

min

i∈[ j+1,..,N FA

(the threshold λ j and

{a (r + α , r )} ] j i

j

small).

3. The active reconfigurable Fault-tolerant control system The purpose of this section is to briefly show how the active GLR test restricted to the treatment of severe jumps can be integrated in a reconfigurable FTCS in order to ensure the asymptotical rejection of detected jumps on the regulated output yk . We assume that the number of regulated output m is equals to the number of controlled input d (no actuator rendundancy). The nominal control law uk0 = − L0 Xˆ k0 for h 0 is available (the pair ( A 0 , B0 ) is controlable).  I − A0 B0  I − A0 B0  f1 .. fq  = rank Theorem 3.1: With I − A 0 invertible and rank  0   0  , the 0 0 0   C  C reconfigurable control law u q = −Lq Xˆ q for h q which asymptotically recover the nominal

[

k

]

k

performances of u = − L Xˆ k0 is described by 0 k

Lq = [ L0

0

L0 T q + G q ]

with G = [ C ( I − A ) B ] C ( I − A ) q

0 −1

0

0

−1

0 −1

0

[ f .. f ] and T 1

q

q

[

(6.a)

]

= ( I − A ) ( B G − f1 .. fq ) 0 −1

0

q

(6.b)

 X 0   I T q   Xk0  Proof. Let ukq = uk0 − G q ν k with uk0 = − L0 Xk0 and the state transformation  k  =     . The  ν k  0 I   ν k  reference model h q controlled by ukq = uk0 − G q ν k can then be equivalently rewritten

[

]

 Xk0+1   A 0 ( I − A 0 )T q + f1 .. fq   Xk0   B0  0 X 0  q yk = [C 0 − C 0 T q ] k  (7)    +  (uk − G ν k ) =  I  ν k +1   0  νk    νk   0  T On (7), the influence of the jump’s state ν k = ν1 ..ν q is zero if and only if T q and G q satisfy

[

[

]

]

 f1 .. fq  I − A B T   . Its unique solution is given by (6.b). Under (6.b), (7) gives  0  q  = − 0  G   C  0  Xk0+1 = ( A 0 − B0 L0 ) Xk0 , yk = C 0 Xk0 corresponding to h 0 controlled by uk0 = − L0 Xk0 . So, Xq X 0   I T q   I − T q   Xk0  q q  k  (8) ukq = −[ L0 G q ] k  = −[ L0 G q ] L LT G + ]      = −[ I   νk  0 I  0  νk   νk  0

0

q

is a stabilizing control law rejecting the uncontrolable jump’s state ν k (the pair ( Aq , Bq ) has q uncontrolable modes represented by the q jump’s states under ( A 0 , B0 ) controlable). From theorem 1, the ( n + q ) -order ASKF is stable for any jump scenario and from the separation principle, we conclude that the reconfigurable control law u 0 = − L0 Xˆ 0 is a stabilizing control law which k

k

ˆ k = ν k is obtained by the ASKF, the key point of the solution). asymptotically reject ν k (since lim ν

[

We have also lim K = K k →∞

q k

0T

0

k →∞ T

]

= K q ( K 0 is the steady state gain of the jump-free filter) leading to

K q ( z ) = Lq [ zI − ( Aq − Bq Lq − K q C q )]−1 K q = L0 [ zI − ( A 0 − B0 L0 − K 0 C 0 )] −1 K 0 = K 0 ( z ) (9) where K q ( z ) and K 0 ( z ) represent the transfer functions of ukq = − Lq Xˆ kq and uk0 = − L0 Xˆ k0 , respectively. We conclude that the nominal closed-loop performances (fixed by K 0 ( z ) on h 0 ) are asymptotically recovered. These completes the proof. At the beginning of the processing, the reference model is h 0 and the nominal control law uk0 = − L0 Xˆ k0 is active. At the detection time of a first abrupt change, the nominal control law is replaced by the new control law u1 = − L1 Xˆ 1 where the k

k

internal state Xˆ k1 of the controller K 1 ( z ) = L1 [ zI − ( A1 − B1 L1 − K 1C1 )] −1 K 1 is initialized from information given by the GLR detector ( β 0 represents the reconfiguration delay of uk0 ). We repeat the algorithm described above until no further jumps are detected where β j corresponds to the reconfiguration delay of the control law ukj (designed on h j ) updated on the new reference model h j+1 at the detection time of the ( j + 1) th jump. In the area of fault tolerant control, β j cannot be too large in order to quickly reconfigurate the control law. On a constant data window so that dl = (β j − α j + 1)m implies λ = λ j under P FA = PjFA ∀j ∈[0,.., q ] , the tuning parameters β j will satisfy β 0 ≥ .. ≥ β j ≥ .. ≥ β q signifying that the reconfiguration delays can decrease after each detection. We have also νc0 ≥ .. ≥ νcj ≥ .. ≥ νcq signifying that the assumption PjND ≤ P FA with ν ≥ νcj ∀j ∈[0,.., q ] (ensuring a good reference model h q under a low false alarms probability P FA ) can become least restrictive after each detection. We concluse that our FTCS can be applied if νc0 ≤ νc and β 0 ≤ β where νc is the minimum jump that is required to be detected and β the maximum admissible reconfiguration delay depending to the practical application of this work. A great number of papers focussed in the field of FTCS consider the treatment of faults which are perhaps not statistically significant by confusing disturbances always present on the system and faults occurring infrequently leading to a robust control design based on the worst case scenario which may never occur. Compared to such robust control, the performance of our FTCS is drastically improved since there is no sacrifice of nominal performance for robustness before the occurrence of a first jump. After the detection of one or more jumps, the controller stucture is reconfigurated by adding a robustness loop to compensate the occurred jumps in order to asymptotically recover the nominal system performances.

4. Conclusion Based on a reference model updated by the state of jumps declared to be statistically significant during the processing, this paper has presented the active GLR test for the sequential treatment of severe jumps in stochastic discrete-time linear systems. We have studied its integration in a reconfigurable fault tolerant control system where the FDI results drive the design model of the reconfigurable control law.

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