- Email: [email protected]
Disturbance decoupling in fault detection of linear periodic systems
Automatica 43 (2007) 1410 – 1417 www.elsevier.com/locate/automatica
Brief paper
Disturbance decoupling in fault detection of linear periodic systems夡 Ping Zhang ∗ , Steven X. Ding Institute for Automatic Control and Complex Systems, University of Duisburg-Essen, Bismarckstr. 81 BB, 47057 Duisburg, Germany Received 17 March 2005; received in revised form 27 August 2006; accepted 19 January 2007 Available online 21 June 2007
Abstract This paper studies fault detection problems of linear discrete-time periodic systems. The aim is to design residual generators, which deliver a residual signal fully decoupled from unknown disturbances. First, a periodic parity relation based full decoupling residual generator with a periodically varying parity vector is established. Then, the relation between periodic parity vectors and periodic observer-based residual generators is investigated. It is shown that a periodic observer-based full decoupling residual generator can be obtained from a periodic full decoupling parity vector. Finally, the condition of disturbance decoupling is discussed and an example is given to illustrate the proposed approaches. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Fault detection; Linear periodic systems; Full decoupling; Parity space approach; Observer
1. Introduction and problem formulation During the last three decades, model based fault detection (FD) technology has attracted much attention (Chen & Patton, 1999; Frank, Ding, & Marcu, 2000; Gertler, 1998; Patton, Frank, & Clark, 2000). In the context of linear time-invariant (LTI) systems, a number of approaches have been proposed for the design of FD systems. Under certain conditions, the fault indicating signal, usually called residual, can be fully decoupled from unknown disturbances. To this aim, methods like eigenstructure assignment, parity space approach, unknown input observer have been developed (Chen & Patton, 1999; Gertler, 1998; Patton et al., 2000). Periodic systems are the simplest class of linear time-varying systems next to LTI systems and exist in different areas (Bittanti & Colaneri, 1999; Colaneri, 2000; Souza & Trofino, 2000). Periodic systems have also been used to describe multirate sampled-data systems (Bittanti & Colaneri, 1999), nonlinear 夡 A preliminary version of this paper was presented at the 16th IFAC World Congress held in Prague, 2005. This paper was recommended for publication in revised form by Associate Editor Michele Basseville under the direction of Editor Torsten Söderström. ∗ Corresponding author. Tel.: +49 203 3794295; fax +49 203 3792928. E-mail addresses: [email protected] (P. Zhang), [email protected] (S.X. Ding).
0005-1098/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.01.005
systems linearized along a periodic regime (Bittanti & Colaneri, 1999), and, more recently, networked control systems with periodic communication pattern (Rehbinder & Sanfridson, 2004). Our study is motivated not only by the continuous theoretical development in periodic control and filtering (Bittanti & Colaneri, 1999; Colaneri, 2000; Lampe & Rossenwasser, 2004; Souza & Trofino, 2000; Xie & Souza, 1993) but also by the increasing applications of periodic control in practice like helicopter vibration control (Arcara, Bittanti, & Lovera, 2000), satellite attitude control (Lovera, De Marchi, & Bittanti, 2002) as well as wind turbine (Stol, 2003). Extension of the FD technique to periodic systems will improve the safety and reliability of such applications and is increasingly receiving consideration (Fadali, Colaneri, & Nel, 2003; Fadali & Gummuluri, 2001; Varga, 2004; Zhang, Ding, Wang, & Zhou, 2003, 2005). This paper studies the full decoupling problem of linear discrete-time periodic systems described by f
x(k + 1) = Ak x(k) + Bk u(k) + Ekd d(k) + Ek f(k), f
y(k) = Ck x(k) + Dk u(k) + Fkd d(k) + Fk f(k),
(1)
where x ∈ Rn denotes the state vector, u ∈ Rnu the control input vector, y ∈ Rm the measured output vector, d ∈ Rnd the unknown disturbance vector, and f = [f1 f2 · · · fnf ]T ∈ Rnf the vector of faults to be detected, Ak , Bk , Ck , Dk , Ekd ,
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417 f
f
Ek , Fkd , Fk are known real bounded periodic matrices of period and with appropriate dimensions, i.e., for all integers k 0, they satisfy
f
Ak+
Bk+
d Ek+
Ek+
Ck+
Dk+
d Fk+
Fk+
f
=
f
Ak
Bk
Ekd
Ek
Ck
Dk
Fkd
Fk
f
where vectors U(k), D(k), Fi (k) and Y(k) contain the input and output sequences of system (1) over the moving horizon ⎡
(i) limk→∞ r(k) = 0, if f = 0 and no matter what the control inputs and the disturbances are; (ii) r(k) = 0 if fi (k) = 0, i = 1, . . . , nf . To the authors’ knowledge, there are only few publications on this topic, except the pioneering contributions of Fadali and Gummuluri (2001) and Varga (2004). In the scheme of Fadali and Gummuluri (2001), first a periodic observer is designed and then a bank of finite impulse response (FIR) filters are designed based on lifted LTI reformulations of the system over one period to achieve full disturbance rejection. Key of the decoupling approach proposed in Varga (2004) is the computation of a stable left annihilator of the periodic system. The first approach of this paper introduced in Section 2 is motivated by the observation that the parity space approach treats each time instant independently. In view of this, the full disturbance decoupling problem in periodic systems can be solved through a set of independent linear equations. Hence, a periodic parity relation based residual generator with a periodically varying parity vector is easily obtained. In Section 3, it will be shown that if given a periodic parity vector, then a periodic observer-based residual generator can be readily constructed. Moreover, if the periodic parity vector realizes a full decoupling, so does the resulting periodic observer-based residual generator. The freedom in the observer gain can be used to meet other FD performance specifications. The existence condition of full decoupling residual generators will be discussed in Section 4.
The essence of the parity space approach is to derive the so-called parity relations (Chow & Willsky, 1984). It is widely accepted thanks to its simple computation and straightforward implementation. In this section, we shall show that the parity space approach can be easily extended to periodic systems. At time instant k, consider the input–output relation of periodic system (1) during the moving horizon [k − s, k], where s is an integer and represents the length of the horizon. A parity relation is obtained as Y(k) = H0,k x(k − s) + Hu,k U(k)
i=1
Hfi ,k Fi (k),
(2)
y(k)
⎡
⎡
d(k − s)
u(k − s)
⎤
⎢ u(k − s + 1) ⎥ ⎢ ⎥ ⎥, U(k) = ⎢ .. ⎢ ⎥ ⎣ ⎦ .
⎤
u(k)
⎡
⎢ d(k − s + 1) ⎥ ⎢ ⎥ ⎥, D(k) = ⎢ .. ⎢ ⎥ ⎣ ⎦ .
⎤
fi (k − s)
⎢ f (k − s + 1) ⎥ ⎢ i ⎥ ⎥, Fi (k) = ⎢ .. ⎢ ⎥ ⎣ ⎦ .
d(k)
fi (k)
and matrices H0,k , Hu,k , Hd,k , Hfi ,k are as follows: ⎡
⎤
Ck−s
⎢ ⎢ H0,k = ⎢ ⎣
Ck−s+1 Ak−s .. .
⎥ ⎥ ⎥, ⎦
(3)
Ck Ak−1 · · · Ak−s+1 Ak−s ⎡ Hu,k
⎡
Hd,k
Dk−s
O
Ck−s+1 Bk−s .. .
Dk−s+1
Ck Ak−1 · · · Ak−s+1 Bk−s
···
d Fk−s
O
d Ck−s+1 Ek−s
d Fk−s+1
⎢ ⎢ ⎢ =⎢ ⎢ ⎣
⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣
Hfi ,k
⎢ ⎢ ⎢ =⎢ ⎢ ⎣
f
..
.
.. .
d Ck Ak−1 · · · Ak−s+1 Ek−s ⎡ fi Fk−s
2. Periodic parity space approach
nf
⎤
y(k − s)
⎢ y(k − s + 1) ⎥ ⎢ ⎥ ⎥, Y(k) = ⎢ .. ⎢ ⎥ ⎣ ⎦ .
.
The aim is to design a residual generator, so that the residual r satisfies:
+ Hd,k D(k) +
1411
f
··· ..
.
..
.
..
.
..
.
··· O f
i Ck−s+1 Ek−s .. .
i Fk−s+1
fi Ck Ak−1 · · · Ak−s+1 Ek−s
···
⎥ ⎥ ⎥ ⎥, ⎥ O⎦
Dk ···
⎤
O .. .
O
⎤
.. ⎥ ⎥ . ⎥ ⎥, ⎥ ⎥ O⎦
Fkd ··· ..
.
..
.
f
O
⎤
.. ⎥ ⎥ . ⎥ ⎥ ⎥ O ⎦ f
Fk i f
f
with Ek i , Fk i denoting the ith column of matrices Ek , Fk , respectively. Due to the periodicity of system matrices, in parity relation (2) matrices H0,k , Hu,k , Hd,k and Hfi ,k depend periodically on k. Based on parity relation (2), a residual generator can be constructed as r(k) = vkT (Y(k) − Hu,k U(k)),
(4)
where r ∈ R is the so-called residual signal, and the design parameter vkT is a -periodic row vector called parity vector that satisfies vkT H0,k = 0. If vkT can be selected in such a way that vkT [H0,k Hd,k ] = 0,
vkT Hfi ,k = 0, i = 1, . . . , nf
(5)
1412
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417
holds for any k, then r(k) =
nf
vkT Hfi ,k Fi (k).
i=1
The residual will be influenced neither by the initial state x(k − s) nor by the disturbance vector d or the control input vector u. As a result, the conditions (i)–(ii) are satisfied, a full decoupling is realized and each individual fault can be detected. Note that (5) is a set of independent linear equations and can be easily solved. This means a periodic parity relation based full decoupling residual generator can be simply designed by solving (5) for vkT . Moreover, as H0,k , Hd,k and Hfi ,k are periodic, (5) only needs to be solved over one period. A solution to (5) exists if and only if the following rank condition is satisfied: rank[H0,k Hd,k Hfi ,k ] > rank[H0,k Hd,k ], i = 1, . . . , nf
(6)
for any k. It is enough to check the rank condition for only one period. 3. Periodic observer-based full decoupling residual generator In this section, we propose an approach to design periodic observer-based full decoupling residual generators for periodic system (1). To the aim of FD, an observer-based residual generator is constructed as z(k + 1) = Gk z(k) + Hk u(k) + Lk y(k), r(k) = wkT z(k) + qkT u(k) + pTk y(k)
(7)
with z ∈ R and r ∈ R. The goal is to design the -periodic matrices Gk , Hk , Lk and row vectors wkT , qkT , pTk , so that the conditions (i)–(ii) are fulfilled. Let e(k) = z(k) − Tk x(k). If Gk is stable and the following equations: s
Tk+1 Ak − Gk Tk = Lk Ck ,
(8)
wkT Tk + pTk Ck = 0,
(9)
Hk = Tk+1 Bk − Lk Dk , qkT
(10)
= −pTk Dk
(11)
hold for any k, then the dynamics of residual generator (7) is governed by
f
+ (Lk Fk − Tk+1 Ek )f(k), f
r(k) = wkT e(k) + pTk Fkd d(k) + pTk Fk f(k)
T vT Theorem 1. Assume that the periodic vector vkT =[vk,0 k,1 · · · T ] satisfies vT H vk,s = 0. Then Eqs. (8)–(9) are solved by k 0,k
⎡
0
··· 0 . .. . ..
0
⎢ ⎢1 ⎢ ⎢ Gk = ⎢ ⎢ ... ⎢ ⎢ ⎣0
.
..
.
0
···
1
0
0
···
0
1
0 ..
(12)
and meets the basic requirement that ∀u, limk→∞ r(k) = 0, if d = 0, f = 0. Eqs. (8)–(11) are an extension of the well-known Luenberger condition in discrete-time periodic systems. If the unknowns
gk,1
⎤
⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ gk,s−1 ⎦ gk,2 .. . gk,s
= [ 0 0 · · · 0 −1 ] , ⎤ ⎡ ⎡ T vk+s,0 gk,1 ⎤ ⎥ ⎢ ⎢ T ⎢ vk+s−1,1 ⎥ ⎢ gk,2 ⎥ ⎥ T ⎥ ⎢ ⎢ vk,s , Lk = − ⎢ ⎥−⎢ . ⎥ .. ⎥ ⎣ .. ⎥ ⎢ ⎦ ⎦ ⎣ . wkT
T vk+1,s−1
⎡
T vk+s−1,1
⎢ T ⎢ vk+s−2,2 ⎢ Tk = ⎢ .. ⎢ ⎣ . ⎡ ⎢ ⎢ ×⎢ ⎢ ⎣
T vk,s
T pTk = vk,s ,
gk,s T · · · vk+s−1,s−1
···
T vk+s−1,s
T vk+s−2,s
0 .. .
···
0 Ck
⎤
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
0
⎥ ⎥ ⎥, ⎥ ⎦
Ck+1 Ak .. .
(13)
Ck+s−1 Ak+s−2 · · · Ak where periodic scalars gk,1 , . . . , gk,s appearing in matrices Gk , Lk are free parameters and should be selected in such a way that all characteristic multipliers, i.e., the eigenvalues of G−1 · · · G1 G0 , are inside the open unit disk of the complex plane. Proof. Note that vkT H0,k = 0 can be expanded as ⎡
e(k + 1) = Gk e(k) + (Lk Fkd − Tk+1 Ekd )d(k) f
Gk , Lk , Tk , wkT , pTk in (8)–(9) are found, then Hk and qkT follow immediately from (10) to (11). However, it is not easy to find a solution to Eqs. (8)–(9). Inspired by the fact that in the LTI case there is a one-to-one relationship between observer-based and parity relation based residual generators (Ding, Ding, & Jeinsch, 1998), we shall study below the construction of a periodic observer from a periodic parity vector.
⎢ ⎢ T T T ⎢ [vk,0 vk,1 · · · vk,s ]⎢ ⎣
Ck−s Ck−s+1 Ak−s .. .
⎤ ⎥ ⎥ ⎥ = 0. ⎥ ⎦
Ck Ak−1 · · · Ak−s+1 Ak−s Hence, the first row of Tk+1 Ak − Gk Tk equals the first row of Lk Ck . It is straightforward to show that the other rows of Tk+1 Ak − Gk Tk are identical with those of Lk Ck , respectively,
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417
if Gk , Tk and Lk are selected as (13). Therefore, Eq. (8) holds. T C and pT = vT , Eq. (9) holds. Since wkT Tk = −vk,s k k k,s Theorem 1 reveals that, given a periodically varying vector belonging to the parity space, a periodic observer-based residual generator satisfying (8)–(9) can be readily constructed according to (13). To ensure the stability of residual dynamics (12), a simple choice of gk,j is gk,j ≡ 0, j = 1, . . . , s. In this case, all characteristic multipliers will be placed at the origin and the residual signals obtained by residual generators (4) and (7) are identical. In general, no matter what gk,j is, residual generator (7) can always be rewritten as ¯ k z(k) + H ¯ k u(k) + L¯ k y(k) − gk r(k), z(k + 1) = G r(k) = wkT z(k) + qkT u(k) + pTk y(k)
(14)
and residual dynamics (12) can be expressed by ¯ k e(k) + (L¯ k Fd − Tk+1 Ed )d(k) e(k + 1) = G k k
f
r(k) = wkT e(k) + pTk Fkd d(k) + pTk Fk f(k),
0 .. . 0
0⎤ ⎥ 0⎥ ⎥ .. ⎥ , .⎦
0 ··· ⎡g ⎤
1
0
⎢ ⎢1 ¯k =⎢ G ⎢ .. ⎣.
⎡
T vk+s,0
⎤
Remark 3. A numerically stable algorithm based on the use of the periodic Schur form is proposed by Varga (2000) to solve ˆ k+1 A ˆ kT ˆ k, ˆ k −B ˆ k =C periodic Sylvester equations of the form T ˆ ˆ ˆ ˆ in which Tk is unknown and Ak , Bk , Ck are given.
pTk Fkd = 0,
(18)
Tk+1 Ekd − Lk Fkd = 0,
(19)
f
⎥ ⎢ T ⎢ vk+s−1,1 ⎥ ⎥ ⎢ L¯ k = − ⎢ ⎥, .. ⎥ ⎢ ⎦ ⎣ .
f
f
[Tk+1 Ek i − Lk Fk i pTk Fk i ] = 0,
i = 1, . . . , nf
(20)
are also fulfilled by Lk , Tk and pTk , then the dynamics of residual generator (7) satisfies the conditions (i)–(ii) and a full decoupling can be achieved. To solve (8)–(11) and (18)–(20) simultaneously, the following theorem is given.
T vk+1,s−1
k,1
⎢ gk,2 ⎥ ⎥ ⎢ ⎥ gk = ⎢ ⎢ .. ⎥ , ⎣ . ⎦
T vT Theorem 2. Assume that the periodic vector vkT =[vk,0 k,1 · · · T T T vk,s ] satisfies (5). Then Gk , Lk , Tk , wk and pk given by (13) satisfy (8)–(9) and (18)–(20) simultaneously.
gk,s ¯ k = Tk+1 Bk − L¯ k Dk . H ¯ k , L¯ k and H ¯ k are independent of column vector gk . Note that G Hence, gk can be interpreted as the gain of the implicit feedback in observer-based residual generator (7). Moreover, it can be shown that (14) and (15) are, respectively, equivalent to
+ gk−2,s−1 r(k − 2) + · · · + gk−s,1 r(k − s) nf
(16)
Hfi ,k Fi (k)) + gk−1,s r(k − 1)
i=1
+ gk−2,s−1 r(k − 2) + · · · + gk−s,1 r(k − s).
Proof. In view of vkT Hd,k =0, multiplying vkT with each column of Hd,k yields T Fkd = 0, vk,s T d vk,j Fk−s+j +
(21) s
T d vk,l Ck−s+l k−s+l,k−s+j +1 Ek−s+j = 0,
l=j +1
r(k) = vkT (Y(k) − Hu,k U(k)) + gk−1,s r(k − 1)
= vkT (Hd,k D(k) +
Remark 2. We would like to point out that an alternative way to derive solution (13) is to exploit the isomorphism between periodic systems and LTI systems with the help of the cyclic time-invariant representation (see Bittanti & Colaneri, 2000) of periodic system (1).
(15)
where ··· .. . .. .
Remark 1. Based on Theorem 1, Eqs. (8)–(11) can be solved by first solving algebraic equations vkT H0,k = 0 for vkT over one period, then making use of (13) to get a solution to Eqs. (8)–(9), and finally computing Hk and qkT by (10)–(11). In addition, this procedure is interesting in that it motivates the development of a so-called model-free approach to FD of periodic systems (Zhang & Ding, 2005).
In the sequel, we shall discuss how to realize full decoupling with periodic observer-based residual generator (7). If, besides (8)–(11), the following conditions:
f f + (L¯ k Fk − Tk+1 Ek )f(k) − gk r(k),
⎡0
1413
(17)
It means that gk = 0 will lead to a closed-loop structured implementation. The freedom provided by gk could be used to meet additional specifications on the residual dynamics, for instance, to modulate the frequency domain behaviour of the residual generator (Ye, Wang, Ding, & Su, 2002).
j = 0, 1, . . . , s − 1,
(22)
where the state transition matrix k2 ,k1 is defined by I if k2 = k1 , k2 ,k1 = Ak2 −1 Ak2 −2 · · · Ak1 if k2 > k1 . T , Eq. (18) follows immediately from (21). Eq. (22) As pTk = vk,s can be rewritten as T vk+s−j,j Fkd +
s l=j +1
T vk+s−j,l Ck+l−j k+l−j,k+1 Ekd = 0
1414
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417
by substituting k with k + s − j . Thus, (19) holds. In a similar manner it can be proven that vkT Hfi ,k = 0 ensures (20). Theorem 2 states that if the periodic parity vector vkT realizes a full decoupling, so does periodic observer-based residual generator (7) with coefficients (13). Thus a periodic observerbased full decoupling residual generator can be obtained from a periodic full decoupling parity vector. The order of the periodic observer is equal to the order of the parity relation s. It is also interesting to note that matrices Lk and Tk of periodic observer-based residual generator (7) at each time are related to the periodic parity vector over one period, which indicates the importance of correct information of period time . In summary, the proposed procedure of designing a periodic observer-based full decoupling residual generator in the form of (7) is as follows: • Set the value of s and construct matrices H0,k , Hd,k , Hfi ,k , i = 1, . . . , nf , by (3). • Solve (5) for the periodic row vector vkT over a period. 1×m T vT T T , • Partition vkT as vkT =[vk,0 k,1 · · · vk,s ] with vk,j ∈ R j = 0, 1, . . . , s. • Get Gk , Lk , Tk , wkT , pTk by (13) with gk,1 , . . . , gk,s ensuring the stability of the residual dynamics. • Compute Hk and qkT from (10) to (11). On the other side, provided that a periodic observer-based residual generator (7) satisfying Eqs. (8)–(11) with Gk , Lk , wkT of the form ⎡0 · · · 0 g ⎤ ⎡ T ⎤ lk,1 k,1 ⎥ ⎢ ⎢ T ⎥ . . ⎢ 1 . . .. g ⎥ ⎢ lk,2 ⎥ k,2 ⎥ ⎥ ⎢ ⎢ Gk = ⎢ ⎥ , Lk = ⎢ . ⎥ , ⎢ .. . . ⎢ . ⎥ .. ⎥ ⎣. ⎣ . ⎦ . 0 . ⎦ T lk,s 0 · · · 1 gk,s wkT = [0 · · · 0 − 1] is given, the row vector T T vkT = [lk−s,1 + gk−s,1 pTk−s lk−s+1,2 + gk−s+1,2 pTk−s+1 T · · · lk−1,s + gk−1,s pTk−1 − pTk ]
(23)
can be proven to be a periodic parity vector, i.e., vkT in (23) satisfies vkT H0,k = 0, as shown below. For convenience, let T Tk,0 = (lk−s,1 + gk−s,1 pTk−s )Ck−s , . . . , T Tk,s−1 = (lk−1,s + gk−1,s pTk−1 )Ck−1 Ak−2 · · · Ak−s ,
Tk,s = −pTk Ck Ak−1 Ak−2 · · · Ak−s , T ¯ k−i , ¯ k−1 · · · G wk,i = wkT G
i = 1, . . . , s − 1.
T ¯ k−1 +gk−1,s wT , wkT Gk−1 =wkT G Note that wkT Lk−1 =−lk−1,s k−1 T T T T and pTk−1 Ck−1 = −wk−1 k−1 , there is k,s−1 + k,s = T T wk,1 k−1 Ak−2 · · · Ak−s . Repeating the above derivation, we get s
T Tk,j = wk,i Tk−i Ak−i−1 · · · Ak−s .
j =s−i T Since wk,s−1 = [−1 0 · · · 0], there is
vkT H0,k =
s
T Tk,j = Tk,0 + wk,s−1 Tk−s+1 Ak−s
j =0 T = Tk,0 + wk,s−1 (Gk−s Tk−s + Lk−s Ck−s ) = 0.
Similarly, it can be proven that, if the given residual generator (7) is fully decoupled from the unknown disturbances, then vector (23) satisfies (5) and also realizes a full decoupling. 4. Existence condition In this section, we discuss the existence of solutions to the disturbance decoupling problem. As mentioned in Section 2, a full decoupling residual generator in the form of (4) or (7) can be designed as long as rank condition (6) is satisfied. Along with the computational decoupling approach presented in Varga (2004), a necessary and sufficient condition is given for the existence of full decoupling residual generators as (Varga, 2004) rank [Gd (z) Gfi (z)] > rank Gd (z),
i = 1, . . . , nf ,
(24)
where Gd (z) is the transfer function matrix of the stacked timeinvariant representation (see Grasselli & Longhi, 1991; Misra, 1996; Varga, 2004) of subsystem (Ak , Ekd , Ck , Fkd ) with initial f f time 1, and Gfi (z) is that of subsystem (Ak , Ek i , Ck , Fk i ). In the next, we shall briefly discuss the relation between (6) and (24). ˜ , E˜ d , C ˜ , F˜ d ) and (A ˜ , E˜ fi , C ˜ , F˜ fi ) are, reAssume that (A spectively, the lifted time-invariant representation (see Bittanti & Colaneri, 2000; Grasselli & Longhi, 1991; Misra, 1996) of f f (Ak , Ekd , Ck , Fkd ) and (Ak , Ek i , Ck , Fk i ) with initial time . De˜ d, (z) and G ˜ fi , (z) the associated transfer function note by G matrices. According to Grasselli and Longhi (1991), the rank ˜ d, (z) and G ˜ fi , (z) is independent of , and for any z in of G the complex plane there is ˜ d,1 (z), Gd (z) = G
˜ fi ,1 (z). Gfi (z) = G
(25)
Therefore, condition (24) is equivalent to ˜ fi , (z)] > rank G ˜ d, (z), ˜ d, (z) G rank[G
i = 1, . . . , nf , (26)
Considering (8)–(9), we have
for any . On the other side, it is well-known (Frank et al., 2000) that, for LTI systems, condition (26) is again equivalent to
Tk,s = wkT (Gk−1 Tk−1 + Lk−1 Ck−1 )Ak−2 · · · Ak−s .
˜ d, H ˜ fi , ] > rank[H ˜ d, ], ˜ 0, H ˜ 0, H rank[H
(27)
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417
where
⎡ ⎢ ⎢ ˜ fi , = ⎢ H ⎢ ⎢ ⎣
F˜ d
O
···
˜ E˜ d C .. . ˜ A ˜ s˜−1 E˜ d C ⎤ ··· O .. ⎥ .. . . ⎥ ⎥ ⎥ ⎥ .. . O ⎦
F˜ d
..
.
..
.
⎢ ⎢ ˜ d, = ⎢ H ⎢ ⎢ ⎣
f F˜ i
˜ E˜ fi C .. . s ˜ ˜ A ˜ −1 E˜ fi C
O f F˜ i
.. ⎥ . ⎥ ⎥ ⎥, ⎥ O⎦ F˜ d
···
5. Design example To illustrate the proposed design procedures, we consider a periodic system of period = 2 described by (1) with (Fadali & Gummuluri, 2001) ⎡ ⎤ ⎡ 0.5 ⎤ 0.25 0.25 0.1 −0.1 ⎢ 0.5 ⎢ 0.1 ⎥ 0.1 0.1 0.5 ⎥ ⎢ ⎥ ⎥, A0 = ⎢ ⎥ , B0 = ⎢ ⎣ ⎦ ⎣ 0.5 −0.2 0.2 0.25 ⎦ 0.1 0.25 0.1 0 0.25 0.1 ⎡ ⎤ ⎡ ⎤ 0.1 0.2 0.1 −0.1 0.1 ⎢ −0.1 0.5 0 ⎢ 0.5 ⎥ 0.5 ⎥ ⎢ ⎥ ⎢ ⎥ A1 = ⎢ ⎥ , B1 = ⎢ ⎥, ⎣ 0.5 0.5 0.1 0.25 ⎦ ⎣ 0.1 ⎦ 0
0.1
0.25 0.1 ⎢ C0 = ⎣ −0.1 0.5
0.1
0.25
0.2 0.2
0.5
−0.1
0.1 0.25 ⎢ C1 = ⎣ 0.25 0.1
0.1
⎡
0.25
0.1
0.5
⎤
⎥ 0.5 ⎦ , 0.1 −0.1
0.1
⎤
F1d = 0,
0.1
⎤
⎢ −1 ⎥ ⎢ ⎥ f E0 = ⎢ ⎥, ⎣ 0.2 ⎦ 0.1
0
10
20
30
40 50 60 Discrete time
70
80
90
100
Fig. 1. Fault signal.
Letting s = 1, we obtain matrices H0,k , Hu,k , Hd,k , Hf,k by (3). As for any k, rank[H0,k Hd,k Hf,k ] = 6,
rank[H0,k Hd,k ] = 5,
condition (6) is satisfied. To decouple the residual from the unknown disturbances, we then solve (5) for vkT , k = 0, 1, respectively. As a result, the periodic parity relation based full decoupling residual generator is r(k) = vkT
y(k − 1) y(k)
− Hu,k
u(k − 1) u(k)
,
(28)
where
In the simulation, it is assumed that the control input is a step signal (step time at 0) of amplitude 1, the disturbance d(k) = sin(0.01k), and the fault appears at the 40th discrete time as illustrated in Fig. 1. The residual signal obtained by residual generator (28) is shown in Fig. 2. It can be seen that the residual signal r is not influenced by u, d and changes only if f = 0. This means residual generator (28) has achieved a full decoupling. Now let g0 = −0.2, g1 = −0.3. From the periodic full decoupling parity vector got above, a periodic observer-based full decoupling residual generator can be readily obtained as z(k + 1) = Gk z(k) + Hk u(k) + Lk y(k),
0.1 F0d = 0,
D1 = 0, ⎡
−2
0.32 ⎤ 0.1 ⎢ −1 ⎥ ⎢ ⎥ f E1 = ⎢ ⎥, ⎣ 0.2 ⎦ ⎡
-0.2
v1T =[−0.0631 − 0.1348 0.0314 0.2316 −0.5703 0.7733].
⎤
0.25 −0.2 0.5 ⎡ ⎤ 1.3 3.2 ⎢ 1.8 ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ E0d = ⎢ ⎥ , E1d = ⎢ ⎥, ⎣ 1.6 ⎦ ⎣ −1 ⎦ ⎡
0.1
v0T = [0.3535 0.2589 0.1962 − 0.8290 − 0.1421 0.2491],
D0 = 0,
⎥ 0.1 ⎦ ,
0.2
0.2
-0.1
˜ 0, , Hd,k = with s˜ being an integer. By noticing that H0,k = H ˜ fi , , if s = (˜s + 1) − 1 and k = j + + s, ˜ d, and Hfi ,k = H H condition (6) is equivalent to condition (27). Therefore, it is evident that condition (6) is equivalent to condition (24) given by Varga (2004).
⎡
0.3
0
f F˜ i
···
0.4
O⎤
Fault
⎡ ˜ ⎤ C ⎢C ˜⎥ ⎥ ⎢ ˜ A ⎢ ˜ H0, = ⎢ . ⎥ ⎥, ⎣ .. ⎦ ˜ s˜ ˜ A C
⎡
1415
f F0
= 0,
f F1
= 0.
r(k) = wkT z(k) + qkT u(k) + pTk y(k)
(29)
1416
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417
seen, the influence of initial estimation error disappears after several time points and the residual signal is not influenced by u, d. Hence, residual generator (29) also allows a full decoupling and a reliable detection of the fault.
0.2 0.15 0.1
Residual
0.05
6. Conclusion
0 -0.05 -0.1 -0.15 -0.2 0
10
20
30
40 50 60 Discrete time
70
80
90
100
Fig. 2. Residual signal generated by periodic parity relation based residual generator (28).
0.2 0.1
Residual
0 -0.1 -0.2
In this paper, approaches to design residual generators fully decoupled from unknown disturbances in linear discrete-time periodic systems have been presented. The periodic parity space approach needs to solve only a set of independent linear algebraic equations. Then, by exploring the relationship between periodic parity vectors and periodic observer-based residual generators, we obtain a periodic observer-based full decoupling residual generator. At last, the condition of full disturbance decoupling is investigated. We should point out that numerical problems may be met for some periodic systems, as the computation of matrix products is needed to determine matrices H0,k , Hd,k and Hfi ,k . The proposed approach can be used for the aim of fault isolation by designing a bank of residual generators, each of which is decoupled from a part of faults and sensitive to the other part of faults. The full decoupling problem of linear continuous-time periodic systems is much more difficult and currently we are investigating the FD system design for periodic systems with multiplicative disturbances and faults. References
-0.3 -0.4 -0.5 -0.6 0
10
20
30
40 50 60 Discrete time
70
80
90
100
Fig. 3. Residual signal generated by periodic observer-based residual generator (29).
with G0 = −0.2,
G1 = −0.3,
H0 = 0.0504,
H1 = −0.1142,
L0 = [−0.1027 0.1064 0.0184], L1 = [−0.2840 − 0.4300 0.0358], pT0 = [−0.8290 − 0.1421 0.2491], pT1 = [0.2316 − 0.5703 0.7733], w0T = −1,
w1T = −1,
q0T = 0,
q1T = 0.
(30)
It is worth noticing that periodic observer (29) is only of first order. By changing the value of s, the order of the periodic observer could be adjusted. Under the same simulation conditions as before, the residual signal obtained by periodic observerbased residual generator (29) is presented in Fig. 3. As can be
Arcara, P., Bittanti, S., & Lovera, M. (2000). Periodic control of helicopter rotors for attenuation of vibrations in forward flight. IEEE Transactions on Control Systems Technology, 8(6), 883–894. Bittanti, S., & Colaneri, P. (1999). Periodic control. In J. Webster (Ed.), John Wiley Encyclopaedia on Electrical and Electronic Engineering (Vol. 16, pp. 2–16), New York: Wiley. Bittanti, S., & Colaneri, P. (2000). Invariant representations of discrete-time periodic systems. Automatica, 36, 1777–1793. Chen, J., & Patton, R. J. (1999). Robust model-based fault diagnosis for dynamic systems. Boston: Kluwer Academic Publishers. Chow, E. Y., & Willsky, A. S. (1984). Analytical redundancy and the design of robust failure detection systems. IEEE Transactions on Automatic Control, 29, 603–614. Colaneri, P. (2000). Continuous-time periodic systems in H2 and H∞ , Part I: Theoretical aspects. Kybernetika, 36(2), 211–242. Ding, S. X., Ding, E. L., & Jeinsch, T. (1998). A numerical approach to optimization of FDI systems. In Proceedings of the 37th IEEE CDC (pp. 1137–1142), Tampa, USA. Fadali, M. S., Colaneri, P., & Nel, M. (2003). H2 robust fault estimation for periodic systems. In Proceedings of the 2003 American control conference (pp. 2973–2978). Fadali, M. S., & Gummuluri, S. (2001). Robust observer-based fault detection for periodic systems. In Proceedings of the 2001 American control conference (pp. 464–469). Frank, P. M., Ding, S. X., & Marcu, T. (2000). Model-based fault diagnosis in technical processes. Transactions of the Institute of Measurement and Control, 22, 57–101. Gertler, J. J. (1998). Fault detection and diagnosis in engineering systems. New York: Marcel Dekker. Grasselli, O. M., & Longhi, S. (1991). Finite zero structure of linear periodic discrete-time systems. International Journal of Systems Science, 22(10), 1785–1806.
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417 Lampe, B. P., & Rossenwasser, E. N. (2004). Closed formulae for the L2norm of linear continuous-time periodic systems. In Proceedings of the PSYCO (pp. 231–236), Yokohama, Japan. Lovera, M., De Marchi, E., & Bittanti, S. (2002). Periodic attitude control techniques for small satellites with magnetic actuators. IEEE Transactions on Control Systems Technology, 10(1), 90–95. Misra, P. (1996). Time-invariant representation of discrete periodic systems. Automatica, 32, 267–272. Patton, R. J., Frank, P. M., & Clark, R. N. (Eds.). (2000). Issues of fault diagnosis for dynamic systems. Berlin: Springer. Rehbinder, H., & Sanfridson, M. (2004). Scheduling of a limited communication channel for optimal control. Automatica, 40, 491–500. Souza, C. E. De, & Trofino, A. (2000). An LMI approach to stabilization of linear discrete-time periodic systems. International Journal of Control, 73, 696–703. Stol, K. A. (2003). Time-varying control of wind turbines. In Proceedings of the 2003 ACC (p. 3796). Varga, A. (2000). Robust and minimum norm pole assignment with periodic state feedback. IEEE Transactions on Automatic Control, 45(5), 1017–1022. Varga, A. (2004). Design of fault detection filters for periodic systems. In Proceedings of the 43rd IEEE CDC, Atlantis, Bahamas. Xie, L. H., & Souza, C. E. De (1993). H∞ state estimation for linear periodic systems. IEEE Transactions on Automatic Control, 38(11), 1704–1707. Ye, H., Wang, G. Z., Ding, S. X., & Su, H. Y. (2002). An IIR filter based parity space approach for fault detection. In Proceedings of the 15th IFAC world congress, Barcelona, Spain. Zhang, P., & Ding, S. X. (2005). A model-free approach to fault detection of periodic systems. In Proceedings of the 2005 IEEE international symposium on intelligent control (pp. 843–848), Limassol, Cyprus. Zhang, P., Ding, S. X., Wang, G. Z., & Zhou, D. H. (2003). Fault detection in uncertain linear continuous-time periodic systems. In Proceedings of the European control conference 2003, Cambridge, UK.
1417
Zhang, P., Ding, S. X., Wang, G. Z., & Zhou, D. H. (2005). Fault detection of linear discrete-time periodic systems. IEEE Transactions on Automatic Control, 50(2), 239–244.
Ping Zhang received B.Sc. degree in Control Engineering from Huazhong University of Science and Technology, Wuhan, China, in 1997. From 1999 to 2001, she was with the University of Applied Science Lausitz in Senftenberg, Germany, in the framework of the DAAD-Sandwich-Program. She received M.Sc. and Ph.D. degrees in Control Engineering from Tsinghua University, Beijing, China, in 2002. Since then, she has been working in the Institute for Automatic Control and Complex Systems (AKS) at the University of Duisburg-Essen, Germany. Her research interests are model based fault diagnosis, fault tolerant control, identification for fault diagnosis, periodic and time-varying systems and networked control systems.
Steven X Ding received Ph.D. degree in electrical engineering from the Gerhard-Mercator University of Duisburg, Germany, in 1992. From 1992 to 1994, he was a R&D engineer at Rheinmetall GmbH. From 1995 to 2001, he was a professor of control engineering at the University of Applied Science Lausitz in Senftenberg, Germany, and served as vice president of this university during 1998–2000. He is currently a professor of automatic control and the chair of the Institute for Automatic Control and Complex Systems (AKS) at the University of Duisburg-Essen, Germany. His research interests are model based fault diagnosis, fault tolerant systems and their application in industry with a focus on automotive systems.
Brief paper
Disturbance decoupling in fault detection of linear periodic systems夡 Ping Zhang ∗ , Steven X. Ding Institute for Automatic Control and Complex Systems, University of Duisburg-Essen, Bismarckstr. 81 BB, 47057 Duisburg, Germany Received 17 March 2005; received in revised form 27 August 2006; accepted 19 January 2007 Available online 21 June 2007
Abstract This paper studies fault detection problems of linear discrete-time periodic systems. The aim is to design residual generators, which deliver a residual signal fully decoupled from unknown disturbances. First, a periodic parity relation based full decoupling residual generator with a periodically varying parity vector is established. Then, the relation between periodic parity vectors and periodic observer-based residual generators is investigated. It is shown that a periodic observer-based full decoupling residual generator can be obtained from a periodic full decoupling parity vector. Finally, the condition of disturbance decoupling is discussed and an example is given to illustrate the proposed approaches. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Fault detection; Linear periodic systems; Full decoupling; Parity space approach; Observer
1. Introduction and problem formulation During the last three decades, model based fault detection (FD) technology has attracted much attention (Chen & Patton, 1999; Frank, Ding, & Marcu, 2000; Gertler, 1998; Patton, Frank, & Clark, 2000). In the context of linear time-invariant (LTI) systems, a number of approaches have been proposed for the design of FD systems. Under certain conditions, the fault indicating signal, usually called residual, can be fully decoupled from unknown disturbances. To this aim, methods like eigenstructure assignment, parity space approach, unknown input observer have been developed (Chen & Patton, 1999; Gertler, 1998; Patton et al., 2000). Periodic systems are the simplest class of linear time-varying systems next to LTI systems and exist in different areas (Bittanti & Colaneri, 1999; Colaneri, 2000; Souza & Trofino, 2000). Periodic systems have also been used to describe multirate sampled-data systems (Bittanti & Colaneri, 1999), nonlinear 夡 A preliminary version of this paper was presented at the 16th IFAC World Congress held in Prague, 2005. This paper was recommended for publication in revised form by Associate Editor Michele Basseville under the direction of Editor Torsten Söderström. ∗ Corresponding author. Tel.: +49 203 3794295; fax +49 203 3792928. E-mail addresses: [email protected] (P. Zhang), [email protected] (S.X. Ding).
0005-1098/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.01.005
systems linearized along a periodic regime (Bittanti & Colaneri, 1999), and, more recently, networked control systems with periodic communication pattern (Rehbinder & Sanfridson, 2004). Our study is motivated not only by the continuous theoretical development in periodic control and filtering (Bittanti & Colaneri, 1999; Colaneri, 2000; Lampe & Rossenwasser, 2004; Souza & Trofino, 2000; Xie & Souza, 1993) but also by the increasing applications of periodic control in practice like helicopter vibration control (Arcara, Bittanti, & Lovera, 2000), satellite attitude control (Lovera, De Marchi, & Bittanti, 2002) as well as wind turbine (Stol, 2003). Extension of the FD technique to periodic systems will improve the safety and reliability of such applications and is increasingly receiving consideration (Fadali, Colaneri, & Nel, 2003; Fadali & Gummuluri, 2001; Varga, 2004; Zhang, Ding, Wang, & Zhou, 2003, 2005). This paper studies the full decoupling problem of linear discrete-time periodic systems described by f
x(k + 1) = Ak x(k) + Bk u(k) + Ekd d(k) + Ek f(k), f
y(k) = Ck x(k) + Dk u(k) + Fkd d(k) + Fk f(k),
(1)
where x ∈ Rn denotes the state vector, u ∈ Rnu the control input vector, y ∈ Rm the measured output vector, d ∈ Rnd the unknown disturbance vector, and f = [f1 f2 · · · fnf ]T ∈ Rnf the vector of faults to be detected, Ak , Bk , Ck , Dk , Ekd ,
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417 f
f
Ek , Fkd , Fk are known real bounded periodic matrices of period and with appropriate dimensions, i.e., for all integers k 0, they satisfy
f
Ak+
Bk+
d Ek+
Ek+
Ck+
Dk+
d Fk+
Fk+
f
=
f
Ak
Bk
Ekd
Ek
Ck
Dk
Fkd
Fk
f
where vectors U(k), D(k), Fi (k) and Y(k) contain the input and output sequences of system (1) over the moving horizon ⎡
(i) limk→∞ r(k) = 0, if f = 0 and no matter what the control inputs and the disturbances are; (ii) r(k) = 0 if fi (k) = 0, i = 1, . . . , nf . To the authors’ knowledge, there are only few publications on this topic, except the pioneering contributions of Fadali and Gummuluri (2001) and Varga (2004). In the scheme of Fadali and Gummuluri (2001), first a periodic observer is designed and then a bank of finite impulse response (FIR) filters are designed based on lifted LTI reformulations of the system over one period to achieve full disturbance rejection. Key of the decoupling approach proposed in Varga (2004) is the computation of a stable left annihilator of the periodic system. The first approach of this paper introduced in Section 2 is motivated by the observation that the parity space approach treats each time instant independently. In view of this, the full disturbance decoupling problem in periodic systems can be solved through a set of independent linear equations. Hence, a periodic parity relation based residual generator with a periodically varying parity vector is easily obtained. In Section 3, it will be shown that if given a periodic parity vector, then a periodic observer-based residual generator can be readily constructed. Moreover, if the periodic parity vector realizes a full decoupling, so does the resulting periodic observer-based residual generator. The freedom in the observer gain can be used to meet other FD performance specifications. The existence condition of full decoupling residual generators will be discussed in Section 4.
The essence of the parity space approach is to derive the so-called parity relations (Chow & Willsky, 1984). It is widely accepted thanks to its simple computation and straightforward implementation. In this section, we shall show that the parity space approach can be easily extended to periodic systems. At time instant k, consider the input–output relation of periodic system (1) during the moving horizon [k − s, k], where s is an integer and represents the length of the horizon. A parity relation is obtained as Y(k) = H0,k x(k − s) + Hu,k U(k)
i=1
Hfi ,k Fi (k),
(2)
y(k)
⎡
⎡
d(k − s)
u(k − s)
⎤
⎢ u(k − s + 1) ⎥ ⎢ ⎥ ⎥, U(k) = ⎢ .. ⎢ ⎥ ⎣ ⎦ .
⎤
u(k)
⎡
⎢ d(k − s + 1) ⎥ ⎢ ⎥ ⎥, D(k) = ⎢ .. ⎢ ⎥ ⎣ ⎦ .
⎤
fi (k − s)
⎢ f (k − s + 1) ⎥ ⎢ i ⎥ ⎥, Fi (k) = ⎢ .. ⎢ ⎥ ⎣ ⎦ .
d(k)
fi (k)
and matrices H0,k , Hu,k , Hd,k , Hfi ,k are as follows: ⎡
⎤
Ck−s
⎢ ⎢ H0,k = ⎢ ⎣
Ck−s+1 Ak−s .. .
⎥ ⎥ ⎥, ⎦
(3)
Ck Ak−1 · · · Ak−s+1 Ak−s ⎡ Hu,k
⎡
Hd,k
Dk−s
O
Ck−s+1 Bk−s .. .
Dk−s+1
Ck Ak−1 · · · Ak−s+1 Bk−s
···
d Fk−s
O
d Ck−s+1 Ek−s
d Fk−s+1
⎢ ⎢ ⎢ =⎢ ⎢ ⎣
⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣
Hfi ,k
⎢ ⎢ ⎢ =⎢ ⎢ ⎣
f
..
.
.. .
d Ck Ak−1 · · · Ak−s+1 Ek−s ⎡ fi Fk−s
2. Periodic parity space approach
nf
⎤
y(k − s)
⎢ y(k − s + 1) ⎥ ⎢ ⎥ ⎥, Y(k) = ⎢ .. ⎢ ⎥ ⎣ ⎦ .
.
The aim is to design a residual generator, so that the residual r satisfies:
+ Hd,k D(k) +
1411
f
··· ..
.
..
.
..
.
..
.
··· O f
i Ck−s+1 Ek−s .. .
i Fk−s+1
fi Ck Ak−1 · · · Ak−s+1 Ek−s
···
⎥ ⎥ ⎥ ⎥, ⎥ O⎦
Dk ···
⎤
O .. .
O
⎤
.. ⎥ ⎥ . ⎥ ⎥, ⎥ ⎥ O⎦
Fkd ··· ..
.
..
.
f
O
⎤
.. ⎥ ⎥ . ⎥ ⎥ ⎥ O ⎦ f
Fk i f
f
with Ek i , Fk i denoting the ith column of matrices Ek , Fk , respectively. Due to the periodicity of system matrices, in parity relation (2) matrices H0,k , Hu,k , Hd,k and Hfi ,k depend periodically on k. Based on parity relation (2), a residual generator can be constructed as r(k) = vkT (Y(k) − Hu,k U(k)),
(4)
where r ∈ R is the so-called residual signal, and the design parameter vkT is a -periodic row vector called parity vector that satisfies vkT H0,k = 0. If vkT can be selected in such a way that vkT [H0,k Hd,k ] = 0,
vkT Hfi ,k = 0, i = 1, . . . , nf
(5)
1412
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417
holds for any k, then r(k) =
nf
vkT Hfi ,k Fi (k).
i=1
The residual will be influenced neither by the initial state x(k − s) nor by the disturbance vector d or the control input vector u. As a result, the conditions (i)–(ii) are satisfied, a full decoupling is realized and each individual fault can be detected. Note that (5) is a set of independent linear equations and can be easily solved. This means a periodic parity relation based full decoupling residual generator can be simply designed by solving (5) for vkT . Moreover, as H0,k , Hd,k and Hfi ,k are periodic, (5) only needs to be solved over one period. A solution to (5) exists if and only if the following rank condition is satisfied: rank[H0,k Hd,k Hfi ,k ] > rank[H0,k Hd,k ], i = 1, . . . , nf
(6)
for any k. It is enough to check the rank condition for only one period. 3. Periodic observer-based full decoupling residual generator In this section, we propose an approach to design periodic observer-based full decoupling residual generators for periodic system (1). To the aim of FD, an observer-based residual generator is constructed as z(k + 1) = Gk z(k) + Hk u(k) + Lk y(k), r(k) = wkT z(k) + qkT u(k) + pTk y(k)
(7)
with z ∈ R and r ∈ R. The goal is to design the -periodic matrices Gk , Hk , Lk and row vectors wkT , qkT , pTk , so that the conditions (i)–(ii) are fulfilled. Let e(k) = z(k) − Tk x(k). If Gk is stable and the following equations: s
Tk+1 Ak − Gk Tk = Lk Ck ,
(8)
wkT Tk + pTk Ck = 0,
(9)
Hk = Tk+1 Bk − Lk Dk , qkT
(10)
= −pTk Dk
(11)
hold for any k, then the dynamics of residual generator (7) is governed by
f
+ (Lk Fk − Tk+1 Ek )f(k), f
r(k) = wkT e(k) + pTk Fkd d(k) + pTk Fk f(k)
T vT Theorem 1. Assume that the periodic vector vkT =[vk,0 k,1 · · · T ] satisfies vT H vk,s = 0. Then Eqs. (8)–(9) are solved by k 0,k
⎡
0
··· 0 . .. . ..
0
⎢ ⎢1 ⎢ ⎢ Gk = ⎢ ⎢ ... ⎢ ⎢ ⎣0
.
..
.
0
···
1
0
0
···
0
1
0 ..
(12)
and meets the basic requirement that ∀u, limk→∞ r(k) = 0, if d = 0, f = 0. Eqs. (8)–(11) are an extension of the well-known Luenberger condition in discrete-time periodic systems. If the unknowns
gk,1
⎤
⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ gk,s−1 ⎦ gk,2 .. . gk,s
= [ 0 0 · · · 0 −1 ] , ⎤ ⎡ ⎡ T vk+s,0 gk,1 ⎤ ⎥ ⎢ ⎢ T ⎢ vk+s−1,1 ⎥ ⎢ gk,2 ⎥ ⎥ T ⎥ ⎢ ⎢ vk,s , Lk = − ⎢ ⎥−⎢ . ⎥ .. ⎥ ⎣ .. ⎥ ⎢ ⎦ ⎦ ⎣ . wkT
T vk+1,s−1
⎡
T vk+s−1,1
⎢ T ⎢ vk+s−2,2 ⎢ Tk = ⎢ .. ⎢ ⎣ . ⎡ ⎢ ⎢ ×⎢ ⎢ ⎣
T vk,s
T pTk = vk,s ,
gk,s T · · · vk+s−1,s−1
···
T vk+s−1,s
T vk+s−2,s
0 .. .
···
0 Ck
⎤
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
0
⎥ ⎥ ⎥, ⎥ ⎦
Ck+1 Ak .. .
(13)
Ck+s−1 Ak+s−2 · · · Ak where periodic scalars gk,1 , . . . , gk,s appearing in matrices Gk , Lk are free parameters and should be selected in such a way that all characteristic multipliers, i.e., the eigenvalues of G−1 · · · G1 G0 , are inside the open unit disk of the complex plane. Proof. Note that vkT H0,k = 0 can be expanded as ⎡
e(k + 1) = Gk e(k) + (Lk Fkd − Tk+1 Ekd )d(k) f
Gk , Lk , Tk , wkT , pTk in (8)–(9) are found, then Hk and qkT follow immediately from (10) to (11). However, it is not easy to find a solution to Eqs. (8)–(9). Inspired by the fact that in the LTI case there is a one-to-one relationship between observer-based and parity relation based residual generators (Ding, Ding, & Jeinsch, 1998), we shall study below the construction of a periodic observer from a periodic parity vector.
⎢ ⎢ T T T ⎢ [vk,0 vk,1 · · · vk,s ]⎢ ⎣
Ck−s Ck−s+1 Ak−s .. .
⎤ ⎥ ⎥ ⎥ = 0. ⎥ ⎦
Ck Ak−1 · · · Ak−s+1 Ak−s Hence, the first row of Tk+1 Ak − Gk Tk equals the first row of Lk Ck . It is straightforward to show that the other rows of Tk+1 Ak − Gk Tk are identical with those of Lk Ck , respectively,
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417
if Gk , Tk and Lk are selected as (13). Therefore, Eq. (8) holds. T C and pT = vT , Eq. (9) holds. Since wkT Tk = −vk,s k k k,s Theorem 1 reveals that, given a periodically varying vector belonging to the parity space, a periodic observer-based residual generator satisfying (8)–(9) can be readily constructed according to (13). To ensure the stability of residual dynamics (12), a simple choice of gk,j is gk,j ≡ 0, j = 1, . . . , s. In this case, all characteristic multipliers will be placed at the origin and the residual signals obtained by residual generators (4) and (7) are identical. In general, no matter what gk,j is, residual generator (7) can always be rewritten as ¯ k z(k) + H ¯ k u(k) + L¯ k y(k) − gk r(k), z(k + 1) = G r(k) = wkT z(k) + qkT u(k) + pTk y(k)
(14)
and residual dynamics (12) can be expressed by ¯ k e(k) + (L¯ k Fd − Tk+1 Ed )d(k) e(k + 1) = G k k
f
r(k) = wkT e(k) + pTk Fkd d(k) + pTk Fk f(k),
0 .. . 0
0⎤ ⎥ 0⎥ ⎥ .. ⎥ , .⎦
0 ··· ⎡g ⎤
1
0
⎢ ⎢1 ¯k =⎢ G ⎢ .. ⎣.
⎡
T vk+s,0
⎤
Remark 3. A numerically stable algorithm based on the use of the periodic Schur form is proposed by Varga (2000) to solve ˆ k+1 A ˆ kT ˆ k, ˆ k −B ˆ k =C periodic Sylvester equations of the form T ˆ ˆ ˆ ˆ in which Tk is unknown and Ak , Bk , Ck are given.
pTk Fkd = 0,
(18)
Tk+1 Ekd − Lk Fkd = 0,
(19)
f
⎥ ⎢ T ⎢ vk+s−1,1 ⎥ ⎥ ⎢ L¯ k = − ⎢ ⎥, .. ⎥ ⎢ ⎦ ⎣ .
f
f
[Tk+1 Ek i − Lk Fk i pTk Fk i ] = 0,
i = 1, . . . , nf
(20)
are also fulfilled by Lk , Tk and pTk , then the dynamics of residual generator (7) satisfies the conditions (i)–(ii) and a full decoupling can be achieved. To solve (8)–(11) and (18)–(20) simultaneously, the following theorem is given.
T vk+1,s−1
k,1
⎢ gk,2 ⎥ ⎥ ⎢ ⎥ gk = ⎢ ⎢ .. ⎥ , ⎣ . ⎦
T vT Theorem 2. Assume that the periodic vector vkT =[vk,0 k,1 · · · T T T vk,s ] satisfies (5). Then Gk , Lk , Tk , wk and pk given by (13) satisfy (8)–(9) and (18)–(20) simultaneously.
gk,s ¯ k = Tk+1 Bk − L¯ k Dk . H ¯ k , L¯ k and H ¯ k are independent of column vector gk . Note that G Hence, gk can be interpreted as the gain of the implicit feedback in observer-based residual generator (7). Moreover, it can be shown that (14) and (15) are, respectively, equivalent to
+ gk−2,s−1 r(k − 2) + · · · + gk−s,1 r(k − s) nf
(16)
Hfi ,k Fi (k)) + gk−1,s r(k − 1)
i=1
+ gk−2,s−1 r(k − 2) + · · · + gk−s,1 r(k − s).
Proof. In view of vkT Hd,k =0, multiplying vkT with each column of Hd,k yields T Fkd = 0, vk,s T d vk,j Fk−s+j +
(21) s
T d vk,l Ck−s+l k−s+l,k−s+j +1 Ek−s+j = 0,
l=j +1
r(k) = vkT (Y(k) − Hu,k U(k)) + gk−1,s r(k − 1)
= vkT (Hd,k D(k) +
Remark 2. We would like to point out that an alternative way to derive solution (13) is to exploit the isomorphism between periodic systems and LTI systems with the help of the cyclic time-invariant representation (see Bittanti & Colaneri, 2000) of periodic system (1).
(15)
where ··· .. . .. .
Remark 1. Based on Theorem 1, Eqs. (8)–(11) can be solved by first solving algebraic equations vkT H0,k = 0 for vkT over one period, then making use of (13) to get a solution to Eqs. (8)–(9), and finally computing Hk and qkT by (10)–(11). In addition, this procedure is interesting in that it motivates the development of a so-called model-free approach to FD of periodic systems (Zhang & Ding, 2005).
In the sequel, we shall discuss how to realize full decoupling with periodic observer-based residual generator (7). If, besides (8)–(11), the following conditions:
f f + (L¯ k Fk − Tk+1 Ek )f(k) − gk r(k),
⎡0
1413
(17)
It means that gk = 0 will lead to a closed-loop structured implementation. The freedom provided by gk could be used to meet additional specifications on the residual dynamics, for instance, to modulate the frequency domain behaviour of the residual generator (Ye, Wang, Ding, & Su, 2002).
j = 0, 1, . . . , s − 1,
(22)
where the state transition matrix k2 ,k1 is defined by I if k2 = k1 , k2 ,k1 = Ak2 −1 Ak2 −2 · · · Ak1 if k2 > k1 . T , Eq. (18) follows immediately from (21). Eq. (22) As pTk = vk,s can be rewritten as T vk+s−j,j Fkd +
s l=j +1
T vk+s−j,l Ck+l−j k+l−j,k+1 Ekd = 0
1414
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417
by substituting k with k + s − j . Thus, (19) holds. In a similar manner it can be proven that vkT Hfi ,k = 0 ensures (20). Theorem 2 states that if the periodic parity vector vkT realizes a full decoupling, so does periodic observer-based residual generator (7) with coefficients (13). Thus a periodic observerbased full decoupling residual generator can be obtained from a periodic full decoupling parity vector. The order of the periodic observer is equal to the order of the parity relation s. It is also interesting to note that matrices Lk and Tk of periodic observer-based residual generator (7) at each time are related to the periodic parity vector over one period, which indicates the importance of correct information of period time . In summary, the proposed procedure of designing a periodic observer-based full decoupling residual generator in the form of (7) is as follows: • Set the value of s and construct matrices H0,k , Hd,k , Hfi ,k , i = 1, . . . , nf , by (3). • Solve (5) for the periodic row vector vkT over a period. 1×m T vT T T , • Partition vkT as vkT =[vk,0 k,1 · · · vk,s ] with vk,j ∈ R j = 0, 1, . . . , s. • Get Gk , Lk , Tk , wkT , pTk by (13) with gk,1 , . . . , gk,s ensuring the stability of the residual dynamics. • Compute Hk and qkT from (10) to (11). On the other side, provided that a periodic observer-based residual generator (7) satisfying Eqs. (8)–(11) with Gk , Lk , wkT of the form ⎡0 · · · 0 g ⎤ ⎡ T ⎤ lk,1 k,1 ⎥ ⎢ ⎢ T ⎥ . . ⎢ 1 . . .. g ⎥ ⎢ lk,2 ⎥ k,2 ⎥ ⎥ ⎢ ⎢ Gk = ⎢ ⎥ , Lk = ⎢ . ⎥ , ⎢ .. . . ⎢ . ⎥ .. ⎥ ⎣. ⎣ . ⎦ . 0 . ⎦ T lk,s 0 · · · 1 gk,s wkT = [0 · · · 0 − 1] is given, the row vector T T vkT = [lk−s,1 + gk−s,1 pTk−s lk−s+1,2 + gk−s+1,2 pTk−s+1 T · · · lk−1,s + gk−1,s pTk−1 − pTk ]
(23)
can be proven to be a periodic parity vector, i.e., vkT in (23) satisfies vkT H0,k = 0, as shown below. For convenience, let T Tk,0 = (lk−s,1 + gk−s,1 pTk−s )Ck−s , . . . , T Tk,s−1 = (lk−1,s + gk−1,s pTk−1 )Ck−1 Ak−2 · · · Ak−s ,
Tk,s = −pTk Ck Ak−1 Ak−2 · · · Ak−s , T ¯ k−i , ¯ k−1 · · · G wk,i = wkT G
i = 1, . . . , s − 1.
T ¯ k−1 +gk−1,s wT , wkT Gk−1 =wkT G Note that wkT Lk−1 =−lk−1,s k−1 T T T T and pTk−1 Ck−1 = −wk−1 k−1 , there is k,s−1 + k,s = T T wk,1 k−1 Ak−2 · · · Ak−s . Repeating the above derivation, we get s
T Tk,j = wk,i Tk−i Ak−i−1 · · · Ak−s .
j =s−i T Since wk,s−1 = [−1 0 · · · 0], there is
vkT H0,k =
s
T Tk,j = Tk,0 + wk,s−1 Tk−s+1 Ak−s
j =0 T = Tk,0 + wk,s−1 (Gk−s Tk−s + Lk−s Ck−s ) = 0.
Similarly, it can be proven that, if the given residual generator (7) is fully decoupled from the unknown disturbances, then vector (23) satisfies (5) and also realizes a full decoupling. 4. Existence condition In this section, we discuss the existence of solutions to the disturbance decoupling problem. As mentioned in Section 2, a full decoupling residual generator in the form of (4) or (7) can be designed as long as rank condition (6) is satisfied. Along with the computational decoupling approach presented in Varga (2004), a necessary and sufficient condition is given for the existence of full decoupling residual generators as (Varga, 2004) rank [Gd (z) Gfi (z)] > rank Gd (z),
i = 1, . . . , nf ,
(24)
where Gd (z) is the transfer function matrix of the stacked timeinvariant representation (see Grasselli & Longhi, 1991; Misra, 1996; Varga, 2004) of subsystem (Ak , Ekd , Ck , Fkd ) with initial f f time 1, and Gfi (z) is that of subsystem (Ak , Ek i , Ck , Fk i ). In the next, we shall briefly discuss the relation between (6) and (24). ˜ , E˜ d , C ˜ , F˜ d ) and (A ˜ , E˜ fi , C ˜ , F˜ fi ) are, reAssume that (A spectively, the lifted time-invariant representation (see Bittanti & Colaneri, 2000; Grasselli & Longhi, 1991; Misra, 1996) of f f (Ak , Ekd , Ck , Fkd ) and (Ak , Ek i , Ck , Fk i ) with initial time . De˜ d, (z) and G ˜ fi , (z) the associated transfer function note by G matrices. According to Grasselli and Longhi (1991), the rank ˜ d, (z) and G ˜ fi , (z) is independent of , and for any z in of G the complex plane there is ˜ d,1 (z), Gd (z) = G
˜ fi ,1 (z). Gfi (z) = G
(25)
Therefore, condition (24) is equivalent to ˜ fi , (z)] > rank G ˜ d, (z), ˜ d, (z) G rank[G
i = 1, . . . , nf , (26)
Considering (8)–(9), we have
for any . On the other side, it is well-known (Frank et al., 2000) that, for LTI systems, condition (26) is again equivalent to
Tk,s = wkT (Gk−1 Tk−1 + Lk−1 Ck−1 )Ak−2 · · · Ak−s .
˜ d, H ˜ fi , ] > rank[H ˜ d, ], ˜ 0, H ˜ 0, H rank[H
(27)
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417
where
⎡ ⎢ ⎢ ˜ fi , = ⎢ H ⎢ ⎢ ⎣
F˜ d
O
···
˜ E˜ d C .. . ˜ A ˜ s˜−1 E˜ d C ⎤ ··· O .. ⎥ .. . . ⎥ ⎥ ⎥ ⎥ .. . O ⎦
F˜ d
..
.
..
.
⎢ ⎢ ˜ d, = ⎢ H ⎢ ⎢ ⎣
f F˜ i
˜ E˜ fi C .. . s ˜ ˜ A ˜ −1 E˜ fi C
O f F˜ i
.. ⎥ . ⎥ ⎥ ⎥, ⎥ O⎦ F˜ d
···
5. Design example To illustrate the proposed design procedures, we consider a periodic system of period = 2 described by (1) with (Fadali & Gummuluri, 2001) ⎡ ⎤ ⎡ 0.5 ⎤ 0.25 0.25 0.1 −0.1 ⎢ 0.5 ⎢ 0.1 ⎥ 0.1 0.1 0.5 ⎥ ⎢ ⎥ ⎥, A0 = ⎢ ⎥ , B0 = ⎢ ⎣ ⎦ ⎣ 0.5 −0.2 0.2 0.25 ⎦ 0.1 0.25 0.1 0 0.25 0.1 ⎡ ⎤ ⎡ ⎤ 0.1 0.2 0.1 −0.1 0.1 ⎢ −0.1 0.5 0 ⎢ 0.5 ⎥ 0.5 ⎥ ⎢ ⎥ ⎢ ⎥ A1 = ⎢ ⎥ , B1 = ⎢ ⎥, ⎣ 0.5 0.5 0.1 0.25 ⎦ ⎣ 0.1 ⎦ 0
0.1
0.25 0.1 ⎢ C0 = ⎣ −0.1 0.5
0.1
0.25
0.2 0.2
0.5
−0.1
0.1 0.25 ⎢ C1 = ⎣ 0.25 0.1
0.1
⎡
0.25
0.1
0.5
⎤
⎥ 0.5 ⎦ , 0.1 −0.1
0.1
⎤
F1d = 0,
0.1
⎤
⎢ −1 ⎥ ⎢ ⎥ f E0 = ⎢ ⎥, ⎣ 0.2 ⎦ 0.1
0
10
20
30
40 50 60 Discrete time
70
80
90
100
Fig. 1. Fault signal.
Letting s = 1, we obtain matrices H0,k , Hu,k , Hd,k , Hf,k by (3). As for any k, rank[H0,k Hd,k Hf,k ] = 6,
rank[H0,k Hd,k ] = 5,
condition (6) is satisfied. To decouple the residual from the unknown disturbances, we then solve (5) for vkT , k = 0, 1, respectively. As a result, the periodic parity relation based full decoupling residual generator is r(k) = vkT
y(k − 1) y(k)
− Hu,k
u(k − 1) u(k)
,
(28)
where
In the simulation, it is assumed that the control input is a step signal (step time at 0) of amplitude 1, the disturbance d(k) = sin(0.01k), and the fault appears at the 40th discrete time as illustrated in Fig. 1. The residual signal obtained by residual generator (28) is shown in Fig. 2. It can be seen that the residual signal r is not influenced by u, d and changes only if f = 0. This means residual generator (28) has achieved a full decoupling. Now let g0 = −0.2, g1 = −0.3. From the periodic full decoupling parity vector got above, a periodic observer-based full decoupling residual generator can be readily obtained as z(k + 1) = Gk z(k) + Hk u(k) + Lk y(k),
0.1 F0d = 0,
D1 = 0, ⎡
−2
0.32 ⎤ 0.1 ⎢ −1 ⎥ ⎢ ⎥ f E1 = ⎢ ⎥, ⎣ 0.2 ⎦ ⎡
-0.2
v1T =[−0.0631 − 0.1348 0.0314 0.2316 −0.5703 0.7733].
⎤
0.25 −0.2 0.5 ⎡ ⎤ 1.3 3.2 ⎢ 1.8 ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ E0d = ⎢ ⎥ , E1d = ⎢ ⎥, ⎣ 1.6 ⎦ ⎣ −1 ⎦ ⎡
0.1
v0T = [0.3535 0.2589 0.1962 − 0.8290 − 0.1421 0.2491],
D0 = 0,
⎥ 0.1 ⎦ ,
0.2
0.2
-0.1
˜ 0, , Hd,k = with s˜ being an integer. By noticing that H0,k = H ˜ fi , , if s = (˜s + 1) − 1 and k = j + + s, ˜ d, and Hfi ,k = H H condition (6) is equivalent to condition (27). Therefore, it is evident that condition (6) is equivalent to condition (24) given by Varga (2004).
⎡
0.3
0
f F˜ i
···
0.4
O⎤
Fault
⎡ ˜ ⎤ C ⎢C ˜⎥ ⎥ ⎢ ˜ A ⎢ ˜ H0, = ⎢ . ⎥ ⎥, ⎣ .. ⎦ ˜ s˜ ˜ A C
⎡
1415
f F0
= 0,
f F1
= 0.
r(k) = wkT z(k) + qkT u(k) + pTk y(k)
(29)
1416
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417
seen, the influence of initial estimation error disappears after several time points and the residual signal is not influenced by u, d. Hence, residual generator (29) also allows a full decoupling and a reliable detection of the fault.
0.2 0.15 0.1
Residual
0.05
6. Conclusion
0 -0.05 -0.1 -0.15 -0.2 0
10
20
30
40 50 60 Discrete time
70
80
90
100
Fig. 2. Residual signal generated by periodic parity relation based residual generator (28).
0.2 0.1
Residual
0 -0.1 -0.2
In this paper, approaches to design residual generators fully decoupled from unknown disturbances in linear discrete-time periodic systems have been presented. The periodic parity space approach needs to solve only a set of independent linear algebraic equations. Then, by exploring the relationship between periodic parity vectors and periodic observer-based residual generators, we obtain a periodic observer-based full decoupling residual generator. At last, the condition of full disturbance decoupling is investigated. We should point out that numerical problems may be met for some periodic systems, as the computation of matrix products is needed to determine matrices H0,k , Hd,k and Hfi ,k . The proposed approach can be used for the aim of fault isolation by designing a bank of residual generators, each of which is decoupled from a part of faults and sensitive to the other part of faults. The full decoupling problem of linear continuous-time periodic systems is much more difficult and currently we are investigating the FD system design for periodic systems with multiplicative disturbances and faults. References
-0.3 -0.4 -0.5 -0.6 0
10
20
30
40 50 60 Discrete time
70
80
90
100
Fig. 3. Residual signal generated by periodic observer-based residual generator (29).
with G0 = −0.2,
G1 = −0.3,
H0 = 0.0504,
H1 = −0.1142,
L0 = [−0.1027 0.1064 0.0184], L1 = [−0.2840 − 0.4300 0.0358], pT0 = [−0.8290 − 0.1421 0.2491], pT1 = [0.2316 − 0.5703 0.7733], w0T = −1,
w1T = −1,
q0T = 0,
q1T = 0.
(30)
It is worth noticing that periodic observer (29) is only of first order. By changing the value of s, the order of the periodic observer could be adjusted. Under the same simulation conditions as before, the residual signal obtained by periodic observerbased residual generator (29) is presented in Fig. 3. As can be
Arcara, P., Bittanti, S., & Lovera, M. (2000). Periodic control of helicopter rotors for attenuation of vibrations in forward flight. IEEE Transactions on Control Systems Technology, 8(6), 883–894. Bittanti, S., & Colaneri, P. (1999). Periodic control. In J. Webster (Ed.), John Wiley Encyclopaedia on Electrical and Electronic Engineering (Vol. 16, pp. 2–16), New York: Wiley. Bittanti, S., & Colaneri, P. (2000). Invariant representations of discrete-time periodic systems. Automatica, 36, 1777–1793. Chen, J., & Patton, R. J. (1999). Robust model-based fault diagnosis for dynamic systems. Boston: Kluwer Academic Publishers. Chow, E. Y., & Willsky, A. S. (1984). Analytical redundancy and the design of robust failure detection systems. IEEE Transactions on Automatic Control, 29, 603–614. Colaneri, P. (2000). Continuous-time periodic systems in H2 and H∞ , Part I: Theoretical aspects. Kybernetika, 36(2), 211–242. Ding, S. X., Ding, E. L., & Jeinsch, T. (1998). A numerical approach to optimization of FDI systems. In Proceedings of the 37th IEEE CDC (pp. 1137–1142), Tampa, USA. Fadali, M. S., Colaneri, P., & Nel, M. (2003). H2 robust fault estimation for periodic systems. In Proceedings of the 2003 American control conference (pp. 2973–2978). Fadali, M. S., & Gummuluri, S. (2001). Robust observer-based fault detection for periodic systems. In Proceedings of the 2001 American control conference (pp. 464–469). Frank, P. M., Ding, S. X., & Marcu, T. (2000). Model-based fault diagnosis in technical processes. Transactions of the Institute of Measurement and Control, 22, 57–101. Gertler, J. J. (1998). Fault detection and diagnosis in engineering systems. New York: Marcel Dekker. Grasselli, O. M., & Longhi, S. (1991). Finite zero structure of linear periodic discrete-time systems. International Journal of Systems Science, 22(10), 1785–1806.
P. Zhang, S.X. Ding / Automatica 43 (2007) 1410 – 1417 Lampe, B. P., & Rossenwasser, E. N. (2004). Closed formulae for the L2norm of linear continuous-time periodic systems. In Proceedings of the PSYCO (pp. 231–236), Yokohama, Japan. Lovera, M., De Marchi, E., & Bittanti, S. (2002). Periodic attitude control techniques for small satellites with magnetic actuators. IEEE Transactions on Control Systems Technology, 10(1), 90–95. Misra, P. (1996). Time-invariant representation of discrete periodic systems. Automatica, 32, 267–272. Patton, R. J., Frank, P. M., & Clark, R. N. (Eds.). (2000). Issues of fault diagnosis for dynamic systems. Berlin: Springer. Rehbinder, H., & Sanfridson, M. (2004). Scheduling of a limited communication channel for optimal control. Automatica, 40, 491–500. Souza, C. E. De, & Trofino, A. (2000). An LMI approach to stabilization of linear discrete-time periodic systems. International Journal of Control, 73, 696–703. Stol, K. A. (2003). Time-varying control of wind turbines. In Proceedings of the 2003 ACC (p. 3796). Varga, A. (2000). Robust and minimum norm pole assignment with periodic state feedback. IEEE Transactions on Automatic Control, 45(5), 1017–1022. Varga, A. (2004). Design of fault detection filters for periodic systems. In Proceedings of the 43rd IEEE CDC, Atlantis, Bahamas. Xie, L. H., & Souza, C. E. De (1993). H∞ state estimation for linear periodic systems. IEEE Transactions on Automatic Control, 38(11), 1704–1707. Ye, H., Wang, G. Z., Ding, S. X., & Su, H. Y. (2002). An IIR filter based parity space approach for fault detection. In Proceedings of the 15th IFAC world congress, Barcelona, Spain. Zhang, P., & Ding, S. X. (2005). A model-free approach to fault detection of periodic systems. In Proceedings of the 2005 IEEE international symposium on intelligent control (pp. 843–848), Limassol, Cyprus. Zhang, P., Ding, S. X., Wang, G. Z., & Zhou, D. H. (2003). Fault detection in uncertain linear continuous-time periodic systems. In Proceedings of the European control conference 2003, Cambridge, UK.
1417
Zhang, P., Ding, S. X., Wang, G. Z., & Zhou, D. H. (2005). Fault detection of linear discrete-time periodic systems. IEEE Transactions on Automatic Control, 50(2), 239–244.
Ping Zhang received B.Sc. degree in Control Engineering from Huazhong University of Science and Technology, Wuhan, China, in 1997. From 1999 to 2001, she was with the University of Applied Science Lausitz in Senftenberg, Germany, in the framework of the DAAD-Sandwich-Program. She received M.Sc. and Ph.D. degrees in Control Engineering from Tsinghua University, Beijing, China, in 2002. Since then, she has been working in the Institute for Automatic Control and Complex Systems (AKS) at the University of Duisburg-Essen, Germany. Her research interests are model based fault diagnosis, fault tolerant control, identification for fault diagnosis, periodic and time-varying systems and networked control systems.
Steven X Ding received Ph.D. degree in electrical engineering from the Gerhard-Mercator University of Duisburg, Germany, in 1992. From 1992 to 1994, he was a R&D engineer at Rheinmetall GmbH. From 1995 to 2001, he was a professor of control engineering at the University of Applied Science Lausitz in Senftenberg, Germany, and served as vice president of this university during 1998–2000. He is currently a professor of automatic control and the chair of the Institute for Automatic Control and Complex Systems (AKS) at the University of Duisburg-Essen, Germany. His research interests are model based fault diagnosis, fault tolerant systems and their application in industry with a focus on automotive systems.