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H∞ fault detection filter
Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World The International Federation of Congress Automatic Control Toulouse, France, July 2017 Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
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PapersOnLine 50-1 (2017) 8600–8605 A IFAC time-domain LPV H /H∞ A time-domain LPV H− /H∞ A time-domain LPV H− detection filter − /H∞ detection filter detection filter ∗∗ ∗
fault fault fault
L. Chen ∗ R. J. Patton ∗∗ L. Chen ∗ R. J. Patton ∗∗ L. Chen ∗ R. J. Patton ∗∗ ∗ College of Engineering, Mathematics and Physical Sciences, ∗ ∗ College of Engineering, Mathematics and Physical Sciences, ∗ University of Exeter, UK (e-mail: [email protected]). College of Engineering, Mathematics and Physical Sciences, ∗∗ University of Exeter, (e-mail: Science, [email protected]). School of Engineering andUK Computer University of Hull, ∗∗ ∗∗ School University of Exeter, UK (e-mail: [email protected]). of Engineering and Computer Science, University of Hull, UK (e-mail: [email protected]) ∗∗ School of Engineering and [email protected]) Computer Science, University of Hull, UK (e-mail: UK (e-mail: [email protected]) Abstract: In this paper, a time-domain linear parameter-varying H− /H∞ residual generator residualsensitivity generator Abstract: In thisfilter paper, a time-domain linear aparameter-varying H− or fault detection is proposed, which allows trade-off to be established, − /H∞ ∞ between /H residual Abstract: In this paper, a time-domain linear parameter-varying H − ∞ or fault detection filter is proposed, which allows a trade-off to be established, between sensitivity and robustness of the residual against the fault and disturbance, respectively. Also, the generator proposed or fault detection filter is proposed, allows a trade-off to beofrespectively. established, and robustness of the fits residual thetime-domain fault and disturbance, Also, thesensitivity proposed fault detection filter well against with which the property the linear between parameter-varying and robustness of the residual thetime-domain fault and disturbance, respectively. Also, the proposed fault detection filter fits well against withperformance the property the linear parameter-varying system. A special case when the of the residual ofgenerator approximates one of fault detection filter well withperformance theAtime-domain property ofgenerator theis linear parameter-varying system. Aestimators special case when the of the residual onethe of the fault isfits also discussed. turboshaft engine example usedapproximates to demonstrate system. A special case when the performance of the residual generator approximates one of the fault estimators is also discussed. A turboshaft engine example is used to demonstrate the effectiveness of the proposed method. the fault estimators is also discussed. effectiveness of the proposed method. A turboshaft engine example is used to demonstrate the effectiveness the proposed method. © 2017, IFAC of (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Linear parameter-varying; Fault detection and diagnosis; H− /H∞ robustness. Keywords: Linear parameter-varying; Fault detection and diagnosis; H− − /H∞ ∞ robustness. Keywords: Linear parameter-varying; Fault detection and diagnosis; H− /H∞ robustness. 1. INTRODUCTION time-domain H− index is originally defined. However, the time-domain indexdetection is originally defined. However, 1. INTRODUCTION discussion of H the− problem is not given. the − fault 1. INTRODUCTION time-domain indexdetection is originally defined. However, discussion of H the− fault problem is not given. the In recent years, linear parameter-varying (LPV) based The main contribution of this work is to propose a necesdiscussion of the fault detection problem is not given. In recent years,and linear parameter-varying (LPV) based main contribution of thisfor work isexistence to propose aa necesfault detection diagnosis (FDD) has been widely de- The sary and sufficient condition the of timeIn recent linear parameter-varying (LPV) based main contribution of thisfor work tofilter propose fault detection and diagnosis (FDD) has been widely de- The sary andLPV sufficient condition theisexistence of aguarana necestimeveloped in years, the literature (Bokor and Balas (2004); Henry domain optimal fault detection to fault detection andEdwards diagnosis (FDD) has been widely de- sary and sufficient condition for the existence of a veloped in theand literature (Bokor and Wei Balas (2004); Henry domain LPV optimal fault detection filter to guaran(2008); Alwi (2014); and Verhaegen tee H− /H∞ robustness. In addition, compared withtimethe veloped in the(2002); literature (Bokor and Balasand (2004); Henry (2008); Alwi and Edwards (2014); Wei Verhaegen LPV optimal fault detection filterHto/H guarantee H− In addition, compared with the (2011); Balas Casavola and Gagliardi (2014); Chen domain ∞ schemes proposed in literature where the ro− /H ∞ robustness. − ∞ (2008); Alwi (2002); and (2014); Wei and Verhaegen tee H− /H robustness. In addition, compared with the (2011); Balas Casavola and Gagliardi Chen ∞achieved schemes proposed in by literature wherethe theminimum H− roet al. (2016)). TheEdwards most obvious benefit of(2014); the use of bustness is maximizing − /H∞ ∞fault (2011); Balas (2002); Casavola and Gagliardi (2014); Chen schemes proposed inH literature where H− /H roet al.is(2016)). mostof obvious benefit ofand thestability, use of bustness ∞ is i.e. achieved by maximizing thethe minimum fault LPV that the The analysis the performance sensitivity, the index, which satisfies a given − et al.is(2016)). The mostof obvious benefit ofand thestability, useover of bustness is achieved by maximizing the minimum fault LPV that analysis the performance sensitivity, i.e. the H index, which satisfies a given together withthethe synthesis method are established fixed level of the H∞− − robustness against disturbance, LPV is that analysis of parameters. the performance and stability, together withthe the synthesis method areThe established over sensitivity, i.e.scheme, the HH index, which satisfies a given − robustness fixed level of the against disturbance, a wide range of changing LPV design ∞ the proposed using the coprime fac∞formulated with the synthesis method are established over fixed level of the H robustness against disturbance, atogether wide range of changing parameters. The LPV design proposed scheme, formulated thethe coprime facscheme can also be considered as an extension of the linear the torisation, ensures a∞ given fixed using level of minimum a wide range of changing parameters. The LPV design proposed scheme, formulated using thethe coprime facscheme can alsodesign be considered an extension the linear torisation, ensures a index given fixedthen level of minimum time-invariant scheme.asFor example, of based upon the fault sensitivity/H and maximizes the H − ∞ scheme can also be considered an extension of the torisation, ensures a given fixed level of the minimum time-invariant design scheme.asFor example, based upon fault sensitivity/H index and then maximizes the H the vertex property (Apkarian et al. (1995)), the linear LPV robustness − ∞ against disturbance. Furthermore, an optimal − ∞ time-invariant scheme. For example, based upon the vertex (Apkarian et al. (1995)), the linear LPV robustness fault space sensitivity/H index anddetection then maximizes the H ∞ against −disturbance. Furthermore, angiven optimal solution canproperty bedesign obtained by combining multiple state solution of the fault filter is and the (Apkarian et al. (1995)), LPV robustness solution canproperty be(LTI) obtained by combining multiple linear against disturbance. Furthermore, angiven optimal state space solution of the fault detection filter is and timevertex invariant solutions calculated on thethe vertices a specific H index is proposed, which allows the fault de− solution can be(LTI) obtained byacombining multiple linear state space of the detection filterperformance. is given time solutions calculated on the vertices a specific Hsolution index is proposed, which allows the faultand deof theinvariant polytope. In addition, trade-off between compu− tection filter achieve thefault fault estimation −to time invariant (LTI) solutions calculated on the vertices a specific H−to index is proposed, allowsperformance. the fault deof the polytope. In addition,and a trade-off between compufilter achieve the faultwhich estimation tational load, conservatism design performance can tection of polytope. In addition, a trade-off between compuoutline of achieve the paper as follows. First, notations tational load, by conservatism and design performance can The tection filter to the isfault estimation performance. be the established selecting various synthesis methods (e.g. The outline of the paper is as problem follows. First, notations tational load, conservatism and design performance can and definitions are given. The to be solved is be established by selecting various synthesis methods (e.g. polytopic LPV approach Apkarian et al. (1995), linear The outline of the paper is as follows. First, notations and definitions are given. The 4problem tothebestate solved is be established by selecting various synthesis methods (e.g. formulated in Section 3. Section presents space polytopic LPV approach Apkarian et al. (1995), linear fractional transform based design Scherer (2001), grid- and definitions are given. The problem to be solved is formulated in Section 3. Section 4 presents the state space polytopic LPV approach Apkarian et al. (1995), linear fractional transform based grid- solution of the time-domain H− /H∞ fault detection filter ding approach Apkarian anddesign AdamsScherer (1998))(2001), or selecting formulated in Section 3.toSection 4 presents the state space solution of the time-domain H /H fault detection filter fractional transform based design Scherer (2001), grid− ∞ which is then applied a turboshaft engine example as ding approach Apkarian and Adams (1998)) or selecting solution of the time-domain H− /H∞ fault detection filter whether the Lyapunov matrix is parameter-dependent. − some ∞ concluding is then applied5.toFinally, a turboshaft engine example as ding approach Apkarian and Adams (1998)) or selecting which illustrated in Section remarks whether the Lyapunov matrix is parameter-dependent. which is then applied to a turboshaft engine example as illustrated in Section 5. Finally, some concluding remarks The aim of the paper is to construct a robust residual genwhether the Lyapunov matrix is parameter-dependent. are given in Section 6. illustrated in Section 5. Finally, some concluding remarks The aimformulated of the paper construct a robust residual genare given in Section 6. erator, as isantoH /H optimization problem. − ∞ The aim of theofpaper construct a robust residual erator, formulated as isantoH problem. − ∞ The purpose combining H defined in genHou are given in Section 6. − /H ∞ optimization − (originally erator, formulated as an H /H optimization problem. − ∞ The purpose of combining H (originally defined in Hou 2. NOTATIONS AND DEFINITIONS − and Patton (1996)) with H∞ to − is to allow a trade-off The purpose of between combining H∞ (originally defined in Hou 2. NOTATIONS AND DEFINITIONS − and Patton (1996)) withsensitivity H is to allow a trade-off to be established, and robustness of the ∞ 2. NOTATIONS AND DEFINITIONS and Patton (1996)) with H is to allow a trade-off to ∞ be established, and robustness of the residual against between the faultsensitivity and disturbance, respectively. In The notations and definitions to be used in the note are be established, between sensitivity and robustness of the The notations and The definitions to be used the space note are residual against the fault and disturbance, respectively. In R n×m denotes thein real of the literature, H− /H∞ based approaches can be divided summarized here. n×m to be used in the note are n×m The notations and The definitions residual against the fault and disturbance, respectively. In summarized here. R denotes the real space of the literature, H /H based approaches can be divided the n × m matrices. For a matrix A with the appropriate − ∞ into two categories. − One ∞ category combines the H− index n×m summarized here. The RaAmatrix denotes the real space of T −1 † the H− /H based approaches can be divided the n × m matrices. For A with the appropriate ∞ into two categories. One category combines the H index dimension, A , A and denote its transpose, inverse − and literature, H performance as a multi-objective criterion (Ding − ∞ T −1 † T −1 † the n × m matrices. For a matrix A with the appropriate into two categories. One category combines the H index dimension, A , A and A denote its transpose, inverse − (Ding andal.H(2000); a multi-objective criterion and pseudo-inverse respectively. The A > 0 (A ≥ 0) ∞ et Wang etasal. (2007)). In another category, ∞ performance dimension, ATA, Ais−1positive and A† (semi-positive) denote its A transpose, andal. H(2000); a multi-objective (Ding and pseudo-inverse respectively. The >definite, 0 (Ainverse ≥that 0) ∞ performance et al. (2007)). In another category, that the mixed H−Wang /H∞ et isastransformed into acriterion uniform H∞ denotes and¯ pseudo-inverse respectively. 0 (A ≥that 0) et al. (2000); Wang et al. (2007)). In another category, denotes that A is positive (semi-positive) 1/2 The A 1/2>Tdefinite, ¯ the mixed H /H is transformed into a uniform H − ∞ ∞ problem (Henry is λ(A) > 0 (λ(A) ≥ 0). A1/2 = (A1/2 )T > 0 is the − and ∞ Zolghadri (2005); Li et al. (2012)), ∞ denotes that A is positive (semi-positive) definite, that ¯ ¯ 1/2 1/2 T the mixed H /H is transformed into a uniform H − and ∞ to ∞ problem (Henry Zolghadri Li etby al.Grenaille (2012)), = (A matrix ) >A.0 L is2ethe is λ(A)root > 0of(λ(A) ≥ 0).semi-definite A which was extended an LPV(2005); framework square is 1/2 T ¯ ¯aa positive problem (Henry and to Zolghadri etby al.Grenaille (2012)), ) wherein >A.0 L is2e the λ(A) > 0offinite-horizon (λ(A) ≥ 0).semi-definite A1/2 = (A which was extended an LPV framework square root positive matrix is et al. (2008); Henry (2012). In(2005); Li andLiLiu (2013), the is the extended Lebesgue 2-space, 2ethe which was extended an LPV Grenaille square root offinite-horizon a positive semi-definite matrixwherein A. L2ethe is et al. (2008); Henry to (2012). In framework Li and Liuby (2013), the the extended Lebesgue 2-space, et al. (2008); Henry (2012). In Li and Liu (2013), the the extended finite-horizon Lebesgue 2-space, wherein the
Copyright © 2017 IFAC 8934 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017, 2017 IFAC 8934Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 8934Control. 10.1016/j.ifacol.2017.08.1427
Proceedings of the 20th IFAC World Congress L. Chen et al. / IFAC PapersOnLine 50-1 (2017) 8600–8605 Toulouse, France, July 9-14, 2017
signal is continuous and norm bounded in a given finite time horizon, defined by L2e = {v|vτ ∈ L2 , ∀τ ∈ [0, ∞]} (1) and vτ is a truncation of v defined by v(t), 0 ≤ t ≤ τ vτ (t) = (2) 0, t>τ τ Let �v�22e = 0 v(t)T v(t)dt and Lebesgue 2-space becomes infinite-horizon when τ → ∞.
In this note, a system is denoted in boldface upper case, e.g. an LPV system G(ρ) : w �→ y = G(ρ)w is given by x˙ = A(ρ)x + B(ρ)w x(0) = 0 (3) y = C(ρ)x + D(ρ)w where x, w and y represent the system states, inputs and T outputs respectively. ρ = [ρ0 ρ1 . . . ρs ] ∈ Ωe ⊂ R s are the available varying parameters, where Ωe is a compact polytope in a given finite time horizon [0, τ ]. A state space realization of G(ρ) is given by A(ρ) B(ρ) G(ρ) (4) C(ρ) D(ρ) Definition 2.1. For a given LPV system G(ρ), a state space realization of the left inverse of G(ρ) is given by A(ρ) − B(ρ)D† (ρ)C(ρ) −B(ρ)D† (ρ) (5) G−1 (ρ) D† (ρ)C(ρ) D† (ρ)
where D(ρ) is full column rank for all ρ ∈ Ωe Definition 2.2. For a given LPV system G(ρ), a state space realization of the adjoint of G(ρ) is denoted by G∼ (ρ), given by −AT (ρ) −C T (ρ) ∼ (6) G (ρ) B T (ρ) DT (ρ) such that (G∼ (ρ))∼ = G(ρ) ∀ρ ∈ Ωe
(7)
and ∼ (G(ρ)G1 (ρ))∼ = (G∼ (8) 1 (ρ)G (ρ)) ∀ρ ∈ Ωe where G1 (ρ) and G(ρ) are systems with compatible dimension. Definition 2.3. For a given LPV system G(ρ), its timedomain H∞ performance and H− index in a given finite time horizon [0, τ ] are defined by �G(ρ)w�2e �G(ρ)�∞,e = sup , ∀ρ ∈ Ωe , ∀w ∈ L2e , w �= 0 �w�2e w,ρ (9) �G(ρ)w�2e , ∀ρ ∈ Ωe , ∀w ∈ L2e , w �= 0 �G(ρ)�−,e = inf w,ρ �w�2e (10) Remark 2.1. In the literature (e.g. in Hou and Patton (1996); Ding et al. (2000); Jaimoukha et al. (2006); Liu et al. (2005); Wang et al. (2007)), the H∞ performance and H− index have been defined by using a singular value property defined in frequency domain to measure the sensitivity of the residual against fault for a given linear time invariant system. The work in Wei and Verhaegen (2011) extended Wang et al. (2007) to one compatible with LPV systems. In Henry (2012), the H− index for LPV systems, defined based on the notion of the truncated L2
8601
norm of vectors, has the same formulation as (10). The H− index is defined in time-domain Li and Liu (2013) which represents the smallest system gain for all input signals in L2 . The same norm symbol for the time and frequency domain is used since both the Fourier transform and Laplace transform are Hilbert isomorphisms. Lemma 2.1. According to Definition 2.3, it is not hard to show that �G(ρ)G1 (ρ)�∞,e ≤ �G(ρ)�∞,e �G1 (ρ)�∞,e (11) �G(ρ)G1 (ρ)�−,e ≤ �G(ρ)�∞,e �G1 (ρ)�−,e (12) (13) �G(ρ)G1 (ρ)�∞,e ≥ �G(ρ)�−,e �G1 (ρ)�∞,e where G(ρ) and G1 (ρ) are systems with compatible dimensions, ∀ρ ∈ Ωe . Lemma 2.2. For a given LPV system G(ρ), there exists a left coprime factorization ˜ −1 (ρ)N(ρ) ˜ G(ρ) = M ∀ρ ∈ Ωe (14) such that
˜ ˜ M(ρ) N(ρ)
Xl (ρ) =I Yl (ρ)
(15)
˜ ˜ where M(ρ), N(ρ), Xl (ρ) and Yl (ρ) are stable systems with compatible dimensions. Provided (C(ρ), A(ρ)) is detectable for all ρ ∈ Ωe and L(ρ) with compatible dimension, a realization of the left coprime pair is given by ˜ ˜ M(ρ) N(ρ) (16) A(ρ) + L(ρ)C(ρ) L(ρ) B(ρ) + L(ρ)D(ρ) C(ρ) I D(ρ)
Lemma 2.3. For a stable LPV system G(ρ), with state space realization given by (4), let Po be the observability gramian, ∀ρ ∈ Ωe , if the realization satisfies Po A(ρ) + AT (ρ)Po + C T (ρ)C(ρ) = 0 (17) DT (ρ)C(ρ) + B T (ρ)Po = 0
(18)
T
(19) D (ρ)D(ρ) = I then G(ρ) is isometric on L2e , such that G∼ (ρ)G(ρ) = I and �G(ρ)w�2e = �w�2e . If the system is controllable, these conditions are also necessary. 3. PROBLEM FORMULATION Consider an LPV system, subject to disturbances, sensor and actuator faults, is described as x˙ = A(ρ)x + B(ρ)u + Bd (ρ)d + Bf (ρ)f (20) y = Cx + Du + Dd d + Df f where x ∈ R nx , u ∈ R nu , y ∈ R ny represent the system states, inputs and outputs, respectively. f ∈ R nf and d ∈ R nd are actuator and sensor fault signals and the unknown disturbances, respectively. The state space matrices A(ρ), B(ρ), the input fault and disturbance distribution matrices Bf (ρ), Bd (ρ) as well as the estimator gains L(ρ) are assumed to be affinely parameter-dependent with appropriate dimensions, in particular s A(ρ) = ζi (ρ)Ai (21) 1
where the functions ζi : Ωe → R are those to be selected by the designer and Ai are the decision matrices. Remark 3.1. To avoid evaluating an infinite number of the grid points in the LPV region, the matrices C and D and
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[ M(ρ) Nu (ρ) Nd (ρ) Nf (ρ) ] A(ρ) + L(ρ)C L(ρ) B(ρ) + L(ρ)D B(ρ) + L(ρ)Dd Bf (ρ) + L(ρ)Df C I D Dd Df
the output faults and disturbance distribution matrices Df and Dd are assumed to be fixed matrices in this paper. Nevertheless, all matrices in (20) can be assumed to be arbitrarily parameter-dependent and the parameter gridding can be used to obtain finite-dimensional matrix inequality conditions. The gridding-based synthesis approach is not addressed here. System (20) can be written as y = Gu (ρ)u + Gd (ρ)d + Gf (ρ)f (22) where Gu (ρ), Gd (ρ) and Gf (ρ) are parameter-dependent systems with compatible dimensions. The state space realization of [Gu (ρ) Gd (ρ) Gf (ρ)] is given by [ Gu (ρ) Gd (ρ) Gf (ρ) ] A(ρ) B(ρ) Bd (ρ) Bf (ρ) C D Dd Df
(23)
Assumption 3.1. (1) (C(ρ), A(ρ)) is detectable for all ρ ∈ Ωe . (2) Df is full column rank. Using Lemma 2.2 and ensuring Assumption 3.1, the left coprime factorization of the above system exists for all ρ ∈ Ωe , [ Gu (ρ) Gd (ρ) Gf (ρ) ] (24) = M−1 (ρ) [ Nu (ρ) Nd (ρ) Nf (ρ) ] where M(ρ), Nu (ρ), Nd (ρ) and Nf (ρ) are stable systems with compatible dimensions. Since (C, A(ρ)) is detectable for all ρ ∈ Ωe , the state space realization of systems M(ρ), Nu (ρ), Nd (ρ) and Nf (ρ) are given in (25) (as shown on the top of the page). In order to decouple the system inputs from the residual, the structure of the fault detection system can be written as r = F(ρ)(M(ρ)y − Nu (ρ)u) (26) = F(ρ)Nd (ρ)d + F(ρ)Nf (ρ)f where r is the residual signal and F(ρ) is the parameterdependent fault detection filter to be constructed. The structure for the general fault detection residual generator is showed in Fig. 2. u
N u( ρ )
plant
-
+
F( ρ )
y
(25)
ity of the residual signal to the faults can be established through finding a suitable fault detection filter F(ρ). Definition 3.1. For a given LPV system (20) with faults and let β > 0 be a given minimum fault sensitivity measure, the corresponding time-domain H− index satisfies �F(ρ)Nf (ρ)�−,e ≥ β, ∀ρ ∈ Ωe (27) Let z = F(ρ)Nf (ρ)f , the inequality in (27) is equivalent to an optimal control problem, given by τ (z T z − β 2 f T f ) ≥ ǫ�f �22e , ∀f ∈ L2e (28) 0
where ǫ is a positive scalar.
Based on Definition 3.1, the fault detection filter design problem is formulated in Problem 3.1: Problem 3.1. Find a stable LPV fault detection filter F(ρ) that satisfies (28), whilst minimizing �F(ρ)Nd (ρ)�∞,e for all ρ ∈ Ωe , i.e. min {�F(ρ)Nd (ρ)�∞,e : ∀ρ∈Ωe , ∀f ∈L2e τ (29) (z T z − β 2 f T f )dt ≥ ǫ�f �22e } 0
Remark 3.2. The objective function given in Problem 3.1 is different from the one proposed in Li and Zhou (2009) in which the H− /H∞ robustness is defined by finding the maximizer of �FNf �− that satisfies �FNd �∞ ≤ α, where α is a given scalar to represent the level of the H∞ robustness against disturbance. This represents one of the main contributions of this paper. 4. TIME-DOMAIN FAULT DETECTION FILTER In this section, the following Lemma 4.1 is proposed for constructing the time-domain fault detection filter. 1
Lemma 4.1. Provided (AR (ρ), (Bf (ρ) + L(ρ)Df )R− 2 ) is controllable, there always exists a stable system W(ρ), such that Nf (ρ)W−1 (ρ) is isometric, if and only if there exists a symmetric matrix P satisfying the following parameter-dependent Ricatti equation P AR (ρ) + AR (ρ)T P + CR (ρ)T CR (ρ) = 0 (30) where AR (ρ) = A(ρ) + L(ρ)C + (Bf (ρ) + L(ρ)Df )X(ρ) (31) CR (ρ) = Df X(ρ) + C
M( ρ )
X(ρ) = −R
−1
(DfT C
Proof. (Necessity) Let
f
Fig. 1. The structure of the fault detection system It is clear from (26) that a trade-off between the robustness of the residual signal to the disturbances and the sensitiv-
(32) T
+ (Bf (ρ) + L(ρ)Df ) P )
(33)
R = DfT Df (34) and suppose there exists a system W(ρ) (as shown in (35) on the top of the next page). Since R > 0, it follows 1 R 2 > 0. According to Definition 2.1, system W−1 (ρ) exists and can be written as (36) (as shown on the top of the next page). Then,
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W(ρ)
W
Nf (ρ)W
−1
(ρ)
−1
(ρ)
−R
− 12
8603
−(Bf (ρ) + L(ρ)Df ) A(ρ) + L(ρ)C 1 (DfT C + (Bf (ρ) + L(ρ)Df )T P ) R2
(35) 1
A(ρ) + L(ρ)C + (Bf (ρ) + L(ρ)Df )X(ρ) (Bf (ρ) + L(ρ)Df )R− 2 1 X(ρ) R− 2 1
AR (ρ) (Bf (ρ) + L(ρ)Df )R− 2 1 CR (ρ) Df R − 2
(37)
Suppose Nf (ρ)W−1 (ρ) is isometric and (AR (ρ), (Bf (ρ) + 1 L(ρ)Df )R− 2 ) is controllable, the necessary in Lemma 2.3 implies that P AR (ρ) + AR (ρ)T P + CR (ρ)T CR (ρ) = 0 (38) 1
1
(Df R− 2 )T CR (ρ) + ((Bf (ρ) + L(ρ)Df )R− 2 )T P = 0 (39) 1
1
(Df R− 2 )T Df R− 2 = I (40) Clearly, (38) is the Ricatti equation given in (30).
Theorem 4.1. Consider an LPV system in (20), there exists a state space realization of the optimal solution of the fault detection filter that satisfies Problem 3.1, given by 1 AR (ρ) (Bf (ρ) + L(ρ)Df )R− 2 F(ρ) (43) 1 βX(ρ) βR− 2 where AR (ρ) and X(ρ) are defined in (31) and (33), respectively. Additionally, the performance of the residual generator in (43) approximates the performance of the fault estimator when β = 1. Proof. (Sufficiency) Suppose there exists an optimal solution F(ρ), it follows from (36) and (43) that F(ρ) = βW−1 (ρ) (44) Define Q(ρ) = Nf (ρ)W−1 (ρ)(W−1 (ρ)Nf (ρ))−1 (45) and suppose Ξ(ρ) = (I − Q(ρ)QT (ρ)) (46) −1 −1 It is easy to verify that (W (ρ)Nf (ρ)) always exist since Df has full column rank. According to Lemma 4.1, Nf (ρ)W−1 (ρ) is isometric on L2e . It is easy to obtain that β1 Q(ρ)F(ρ)Nf (ρ) is also isometric on L2e . Hence,
(36)
1 (47) � Q(ρ)F(ρ)Nf (ρ)f �22e = �f �22e β According to Definition 3.1, z = F(ρ)Nf (ρ)f , (47) equates to �Q(ρ)z�22e = β 2 �f �22e (48) which can be written as τ (49) z T QT (ρ)Q(ρ)zdt = β 2 �f �22e 0
It is clear that (49) is equivalent to τ τ T z zdt − z T (I − Q(ρ)QT (ρ))zdt = β 2 �f �22e (50) 0
0
and therefore τ
T
z zdt −
τ
z T Ξ(ρ)zdt = β 2 �f �22e
(51)
0
0
(Sufficiency) Since the Ricatti equation (30) equates to (38), the controllability ensures that the states of the system Nf (ρ)W−1 (ρ) spans R n as system inputs ranges over L2e , (39) and (40) are then available. It follows from (39) and (40) that W∼ (ρ)W(ρ) = N∼ (41) f (ρ)Nf (ρ) Then −1 (ρ) = I (42) (W∼ )−1 (ρ)N∼ f (ρ)Nf (ρ)W −1 It is clear from (42) that Nf (ρ)W (ρ) is isometric.
1
Define η = Ξ 2 (ρ)z and the equality in (51) can be written as τ
(z T z − η T η)dt = β 2 �f �22e
(52)
0
Then
�z�22e − �η�22e = β 2 �f �22e
(53)
Suppose �z�22e = ζ�f �22e and let 1
ǫ = ζ�Ξ 2 (ρ)�−,e
(54)
Then, (53) can be written as �z�22e − �βf �22e = �η�22e 1
≥ ζ�Ξ 2 (ρ)�−,e �f �22e = ǫ�f �22e
(55)
which is equivalent to (28) for all f ∈ L2e . (Necessity) Using the fact that �F(ρ)W(ρ)�−,e ≥ �F(ρ)Nf (ρ)W−1 (ρ)W(ρ)�−,e
(56)
and �F(ρ)W(ρ)�−,e = �F(ρ)Nf (ρ)W−1 (ρ)W(ρ)�−,e , specially when Nf (ρ)W−1 (ρ) = I. It is clear from (56) that �F(ρ)W(ρ)�−,e ≥ �F(ρ)Nf (ρ)�−,e (57) As defined in (27), �F(ρ)Nf (ρ)�−,e ≥ β, then �F(ρ)W(ρ)�−,e ≥ β
(58)
Using (13) in Lemma 2.1, �F(ρ)Nd (ρ)�∞,e = �F(ρ)W(ρ)W−1 (ρ)Nd (ρ)�∞,e ≥ �F(ρ)W(ρ)�−,e �W−1 (ρ)Nd (ρ)�∞,e ≥ β�W−1 (ρ)Nd (ρ)�∞,e (59) It is clear from (59) that the optimal filter solution is F(ρ) = βW−1 (ρ). This proofs the necessity.
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Now a special case where the performance of the residual generator is able to approximate one of the fault estimator is discussed. Let γ and γ˜ be the minimizers of γ := min {�F(ρ)Nd (ρ)�∞,e : F(ρ)Nf (ρ) = I}
matrices have the affine structure (as shown in (21)), the LMIs in (66) and (67) are solved based on the vertex property (Apkarian et al. (1995); Wei and Verhaegen (2011)).
∀ρ∈Ωe ,f ∈L2e
(60) and γ˜ :=
min
∀ρ∈Ωe ,f ∈L2e
˜ ˜ {�F(ρ)N d (ρ)�∞,e : �F(ρ)Nf (ρ)�−,e ≥ 1}
(61) ˜ respectively, where F(ρ) and F(ρ) are two optimal stable detection filters and the term F(ρ)Nf (ρ) = I in (60) represents an ideal fault estimation.
5. DESIGN EXAMPLE Consider an LPV JetCat SPT5 turboshaft engine mode represented by
˜ Since F(ρ) and F(ρ) are stable systems, there always exists ˜ an invertible system X(ρ), such that F(ρ) = X(ρ)F(ρ). ˜ Since F(ρ)Nf (ρ) = I, it follows that X(ρ)F(ρ)Nf (ρ) = I −1 ˜ ˜ . Note that �F(ρ)N and X(ρ) = (F(ρ)N f (ρ)) f (ρ)�−,e ≥ −1 1 is equivalent to �X(ρ) �−,e ≥ 1. Then, �X(ρ)�∞,e ≤ 1 for square X(ρ), such that γ = �F(ρ)Nd (ρ)�∞,e ˜ = �X(ρ)F(ρ)N d (ρ)�∞,e ˜ ≤ �X(ρ)�∞,e �F(ρ)N d (ρ)�∞,e ˜ ≤ �F(ρ)Nd (ρ)�∞,e = γ˜
(62) (63) (64) (65)
It is clear that �I�− ≥ 1, which implies γ ≥ γ˜ . Using the fact that γ ≤ γ˜ in (65), it can be concluded that that γ = γ˜ . The optimal solution of the fault estimation filter can thus be deduced through assigning a specific H− index into (43), that is, β = 1.
1.6351 0.0261 −4.4209 −0.0090 A(ρ) = +ρ −1.3708 0.5112 2.9073 −2.0876
−1.2888 0.0151 1.2520 B(ρ) = 0.0652 −0.5850 0.1390 2.1541 −0.0102 0 +ρ 0.4538 0.4119 0
−1.2888 0.0151 2.1541 −0.0102 Bf (ρ) = +ρ 0.0652 −0.5850 0.4538 0.4119 −0.2973 0.0467 Bd = 0.0094 −0.1814 0.0865 0.1177 Dd = 0.3331 −0.4366 1 0 Df = 0 1
(72)
where the scheduling parameter ρ ∈ [0.3361 1.3810] is Based on Lemma 4.1, an LMI-based synthesis approach selected to be the Euclidean norm of the gas turbine engine for solving the fault detection filter gain matrix L(ρ) in spool speed. The LPV model is established based on a least square polynomial fitting of the multiple LTI models (43) is proposed in Proposition 4.1. provided in Pakmehr et al. (2013). Suppose the system is subject to two potential faults and two disturbances. Proposition 4.1. Suppose there exists parameter-dependent The disturbances distribution matrices Bd and Dd are matrices S(ρ) and M (ρ), a symmetric matrix P and a randomly generated. Let the first disturbance is a constant symmetric positive definite matrix Q such that the LMIs bias of amplitude simulated from 2sec and the second in (66) and (67) are feasible for all ρ ∈ Ωe with respect to disturbance is a white noise with mean zero and variance the decision variables P , S(ρ), M (ρ) and Q. 0.01. Two bias faults of 1 and −1 are simulated from 8sec and 15sec, respectively. A11 (ρ)T + A11 (ρ) + C T C A12 (ρ) >0 (66) R A12 (ρ)T Through solving LMIs (66) and (67), the matrix P and be calculated as QA(ρ) + M (ρ)C + (QA(ρ) + M (ρ)C)T < 0 (67) the gain matrix L(ρ) can 0.0967 0.0489 where P = (73) 0.0489 0.2125 A11 (ρ) = P A(ρ) + S(ρ)C (68) (69) A12 (ρ) = P Bf (ρ) + S(ρ)Df + C T Df −10.4103 2.6761 −2.1541 0.0102 L(ρ) = +ρ 2.6260 −4.7395 −0.4538 −0.4119 (70) R = DfT Df (74) Then the fault detection filter gain is given by L(ρ) = P −1 S(ρ) (71) Substituting L(ρ) into (31) and (33) to get AR (ρ) and X(ρ), the optimal filter F(ρ) in (43) can then be deduced. Proof. The LMI in (66) can be deduced by defining To simply demonstrate the effectiveness of the proposed S(ρ) = P L(ρ) and applying the Schur complement to the filter, the scheduling parameter is randomly chosen beparameter-dependent Ricatti equation in (30). The LMI tween 0.3361 and 1.3810 at the beginning of the simulation in (67) ensures (C, A(ρ)) is detectable for all ρ ∈ Ωe . and then remained fixed afterwards. Figure 2 shows the Remark 4.1. The inequality constraint shown in (67) en- residual signals (r1 and r2 ) before the filter, i.e. Gf (ρ)f + sures the fault detectability. Gd (ρ)d. The residual signals of the proposed optimal filter Remark 4.2. Using the fact that the scheduling parameter are depicted in Figure 3. It is clear from Figure 3 that the ρ varies inside a polytope and the parameter-dependent filtered residual is able to distinguish the fault from the 8938
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disturbance and the amplitudes of the faults are approximated when β = 1. 2
r1 r2
Residual signals
1.5 1 0.5 0 −0.5 −1 −1.5 0
2
4
6
8
10 12 Time (sec)
14
16
18
20
Fig. 2. Response of the unfiltered residual 1.5
r1 r2
Residual signals
1
0.5
0
−0.5
−1 0
2
4
6
8
10 12 Time (sec)
14
16
18
20
Fig. 3. Response of the filtered fault detection residual when β = 1 6. CONCLUSION In this paper, an LPV fault detection filter is developed based on a time-domain H− /H∞ robustness, which establishes a trade-off between the sensitivity of the residual signal to the fault and the robustness of the residual signal against the disturbances. In some special case, the residual generator can also be treated as a robust fault estimator. The proposed method is applied to an LPV turboshaft engine model to demonstrate the effectiveness. REFERENCES Alwi, H. and Edwards, C. (2014). Robust fault reconstruction for linear parameter varying systems using sliding mode observers. International Journal of Robust and Nonlinear Control, 24(14), 1947–1968. Apkarian, P. and Adams, R.J. (1998). Advanced gainscheduling techniques for uncertain systems. IEEE Transactions on Control Systems Technology, 6(1), 21– 32. Apkarian, P., Gahinet, P., and Becker, G. (1995). Selfscheduled H∞ control of linear parameter-varying systems: a design example. Automatica, 31(9), 1251–1261. Balas, G.J. (2002). Fault tolerant control and fault detection/isolation for linear parameter varying systems. International Journal of Robust and Nonlinear Control, 12(9), 763–796. Bokor, J. and Balas, G.J. (2004). Detection filter design for LPV systems − a geometric approach. Automatica, 40(3), 511–518.
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Casavola, A. and Gagliardi, G. (2014). Fault detection and isolation of electrical induction motors via LPV fault observer: A case study. International Journal of robust and nonlinear control, 25(5), 627–648. Chen, L., Patton, R., and Goupil, P. (2016). Robust fault estimation using an LPV reference model: ADDSAFE benchmark case study. Control Engineering Practice, 49, 194–203. Ding, S.X., Jeinsch, T., Frank, P.M., and Ding, E. (2000). A unified approach to the optimization of fault detection systems. Journal of Adaptive Control and Signal Processing, 17(4), 725–745. Grenaille, S., Henry, D., and Zolghadri, A. (2008). A method for designing fault diagnosis filters for LPV polytopic systems. J. Control Sci. Eng., 2008, 1–11. Henry, D. (2008). Fault diagnosis of the MICROSCOPE satellite actuators using H− /H∞ filters. AIAA Journal of Guidance, Control and Dynamics, 31(3), 699–711. Henry, D. (2012). Structured fault detection filters for LPV systems modeled in an LFT manner. International Journal of Adaptive Control and Signal Processing, 26(3), 190–207. Henry, D. and Zolghadri, A. (2005). Design and analysis of robust residual generations for systems under feedback control. Automatica, 41(2), 251–264. Hou, M. and Patton, R.J. (1996). An LMI approach to H− /H∞ fault detection observers. In Proceedings of the UKACC international conference on control, 305– 310. Exeter, UK, {available from IEEE Xplore}. Jaimoukha, I.M., Li, Z., and Papakos, V. (2006). A matrix factorization solution to the H− /H∞ fault detection problem. Automatica, 42(11), 1907–1912. Li, X. and Liu, H. (2013). Characterization of H− index for linear time-varying systems. Automatica, 49(5), 1449–1457. Li, X. and Zhou, K. (2009). A time domain approach to robust fault detection of linear time-varying systems. Automatica, 45(1), 94–102. Li, Z., Mazars, E., Zhang, Z., and Jaimoukha, I.M. (2012). State-space solution to the H− /H∞ fault-detection problem. International Journal of Robust and Nonlinear Control, 22(3), 282–299. Liu, J., Wang, J.L., and Yang, G.H. (2005). An LMI approach to minimum sensitivity analysis with application to fault detection. Automatica, 41(11), 1995–2004. Pakmehr, M., Fitzgerald, N., and M. Feron, E. (2013). Gain scheduling control of gas turbine engines: stability by computing a single quadratic lyapunov function. In Proceedings of ASME Turbo Expo, 1–14. San Antonio, Texas, USA. Scherer, C.W. (2001). LPV control and full block multipliers. Automatica, 37(3), 361–375. Shi, F. and Patton, R.J. (2014). Fault estimation and active fault tolerant control for linear parameter varying descriptor systems. International Journal of robust and nonlinear control, 25(5), 689–706. Wang, J.L., Yang, G.H., and Liu, J. (2007). An LMI approach to H− index and mixed H− /H∞ fault detection observer design. Automatica, 43(9), 1656–1665. Wei, X. and Verhaegen, M. (2011). LMI solutions to the mixed H− /H∞ fault detection observer design for linear parameter-varying systems. International Journal of Adaptive Control and Signal Processing, 25(2), 114–136.
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PapersOnLine 50-1 (2017) 8600–8605 A IFAC time-domain LPV H /H∞ A time-domain LPV H− /H∞ A time-domain LPV H− detection filter − /H∞ detection filter detection filter ∗∗ ∗
fault fault fault
L. Chen ∗ R. J. Patton ∗∗ L. Chen ∗ R. J. Patton ∗∗ L. Chen ∗ R. J. Patton ∗∗ ∗ College of Engineering, Mathematics and Physical Sciences, ∗ ∗ College of Engineering, Mathematics and Physical Sciences, ∗ University of Exeter, UK (e-mail: [email protected]). College of Engineering, Mathematics and Physical Sciences, ∗∗ University of Exeter, (e-mail: Science, [email protected]). School of Engineering andUK Computer University of Hull, ∗∗ ∗∗ School University of Exeter, UK (e-mail: [email protected]). of Engineering and Computer Science, University of Hull, UK (e-mail: [email protected]) ∗∗ School of Engineering and [email protected]) Computer Science, University of Hull, UK (e-mail: UK (e-mail: [email protected]) Abstract: In this paper, a time-domain linear parameter-varying H− /H∞ residual generator residualsensitivity generator Abstract: In thisfilter paper, a time-domain linear aparameter-varying H− or fault detection is proposed, which allows trade-off to be established, − /H∞ ∞ between /H residual Abstract: In this paper, a time-domain linear parameter-varying H − ∞ or fault detection filter is proposed, which allows a trade-off to be established, between sensitivity and robustness of the residual against the fault and disturbance, respectively. Also, the generator proposed or fault detection filter is proposed, allows a trade-off to beofrespectively. established, and robustness of the fits residual thetime-domain fault and disturbance, Also, thesensitivity proposed fault detection filter well against with which the property the linear between parameter-varying and robustness of the residual thetime-domain fault and disturbance, respectively. Also, the proposed fault detection filter fits well against withperformance the property the linear parameter-varying system. A special case when the of the residual ofgenerator approximates one of fault detection filter well withperformance theAtime-domain property ofgenerator theis linear parameter-varying system. Aestimators special case when the of the residual onethe of the fault isfits also discussed. turboshaft engine example usedapproximates to demonstrate system. A special case when the performance of the residual generator approximates one of the fault estimators is also discussed. A turboshaft engine example is used to demonstrate the effectiveness of the proposed method. the fault estimators is also discussed. effectiveness of the proposed method. A turboshaft engine example is used to demonstrate the effectiveness the proposed method. © 2017, IFAC of (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Linear parameter-varying; Fault detection and diagnosis; H− /H∞ robustness. Keywords: Linear parameter-varying; Fault detection and diagnosis; H− − /H∞ ∞ robustness. Keywords: Linear parameter-varying; Fault detection and diagnosis; H− /H∞ robustness. 1. INTRODUCTION time-domain H− index is originally defined. However, the time-domain indexdetection is originally defined. However, 1. INTRODUCTION discussion of H the− problem is not given. the − fault 1. INTRODUCTION time-domain indexdetection is originally defined. However, discussion of H the− fault problem is not given. the In recent years, linear parameter-varying (LPV) based The main contribution of this work is to propose a necesdiscussion of the fault detection problem is not given. In recent years,and linear parameter-varying (LPV) based main contribution of thisfor work isexistence to propose aa necesfault detection diagnosis (FDD) has been widely de- The sary and sufficient condition the of timeIn recent linear parameter-varying (LPV) based main contribution of thisfor work tofilter propose fault detection and diagnosis (FDD) has been widely de- The sary andLPV sufficient condition theisexistence of aguarana necestimeveloped in years, the literature (Bokor and Balas (2004); Henry domain optimal fault detection to fault detection andEdwards diagnosis (FDD) has been widely de- sary and sufficient condition for the existence of a veloped in theand literature (Bokor and Wei Balas (2004); Henry domain LPV optimal fault detection filter to guaran(2008); Alwi (2014); and Verhaegen tee H− /H∞ robustness. In addition, compared withtimethe veloped in the(2002); literature (Bokor and Balasand (2004); Henry (2008); Alwi and Edwards (2014); Wei Verhaegen LPV optimal fault detection filterHto/H guarantee H− In addition, compared with the (2011); Balas Casavola and Gagliardi (2014); Chen domain ∞ schemes proposed in literature where the ro− /H ∞ robustness. − ∞ (2008); Alwi (2002); and (2014); Wei and Verhaegen tee H− /H robustness. In addition, compared with the (2011); Balas Casavola and Gagliardi Chen ∞achieved schemes proposed in by literature wherethe theminimum H− roet al. (2016)). TheEdwards most obvious benefit of(2014); the use of bustness is maximizing − /H∞ ∞fault (2011); Balas (2002); Casavola and Gagliardi (2014); Chen schemes proposed inH literature where H− /H roet al.is(2016)). mostof obvious benefit ofand thestability, use of bustness ∞ is i.e. achieved by maximizing thethe minimum fault LPV that the The analysis the performance sensitivity, the index, which satisfies a given − et al.is(2016)). The mostof obvious benefit ofand thestability, useover of bustness is achieved by maximizing the minimum fault LPV that analysis the performance sensitivity, i.e. the H index, which satisfies a given together withthethe synthesis method are established fixed level of the H∞− − robustness against disturbance, LPV is that analysis of parameters. the performance and stability, together withthe the synthesis method areThe established over sensitivity, i.e.scheme, the HH index, which satisfies a given − robustness fixed level of the against disturbance, a wide range of changing LPV design ∞ the proposed using the coprime fac∞formulated with the synthesis method are established over fixed level of the H robustness against disturbance, atogether wide range of changing parameters. The LPV design proposed scheme, formulated thethe coprime facscheme can also be considered as an extension of the linear the torisation, ensures a∞ given fixed using level of minimum a wide range of changing parameters. The LPV design proposed scheme, formulated using thethe coprime facscheme can alsodesign be considered an extension the linear torisation, ensures a index given fixedthen level of minimum time-invariant scheme.asFor example, of based upon the fault sensitivity/H and maximizes the H − ∞ scheme can also be considered an extension of the torisation, ensures a given fixed level of the minimum time-invariant design scheme.asFor example, based upon fault sensitivity/H index and then maximizes the H the vertex property (Apkarian et al. (1995)), the linear LPV robustness − ∞ against disturbance. Furthermore, an optimal − ∞ time-invariant scheme. For example, based upon the vertex (Apkarian et al. (1995)), the linear LPV robustness fault space sensitivity/H index anddetection then maximizes the H ∞ against −disturbance. Furthermore, angiven optimal solution canproperty bedesign obtained by combining multiple state solution of the fault filter is and the (Apkarian et al. (1995)), LPV robustness solution canproperty be(LTI) obtained by combining multiple linear against disturbance. Furthermore, angiven optimal state space solution of the fault detection filter is and timevertex invariant solutions calculated on thethe vertices a specific H index is proposed, which allows the fault de− solution can be(LTI) obtained byacombining multiple linear state space of the detection filterperformance. is given time solutions calculated on the vertices a specific Hsolution index is proposed, which allows the faultand deof theinvariant polytope. In addition, trade-off between compu− tection filter achieve thefault fault estimation −to time invariant (LTI) solutions calculated on the vertices a specific H−to index is proposed, allowsperformance. the fault deof the polytope. In addition,and a trade-off between compufilter achieve the faultwhich estimation tational load, conservatism design performance can tection of polytope. In addition, a trade-off between compuoutline of achieve the paper as follows. First, notations tational load, by conservatism and design performance can The tection filter to the isfault estimation performance. be the established selecting various synthesis methods (e.g. The outline of the paper is as problem follows. First, notations tational load, conservatism and design performance can and definitions are given. The to be solved is be established by selecting various synthesis methods (e.g. polytopic LPV approach Apkarian et al. (1995), linear The outline of the paper is as follows. First, notations and definitions are given. The 4problem tothebestate solved is be established by selecting various synthesis methods (e.g. formulated in Section 3. Section presents space polytopic LPV approach Apkarian et al. (1995), linear fractional transform based design Scherer (2001), grid- and definitions are given. The problem to be solved is formulated in Section 3. Section 4 presents the state space polytopic LPV approach Apkarian et al. (1995), linear fractional transform based grid- solution of the time-domain H− /H∞ fault detection filter ding approach Apkarian anddesign AdamsScherer (1998))(2001), or selecting formulated in Section 3.toSection 4 presents the state space solution of the time-domain H /H fault detection filter fractional transform based design Scherer (2001), grid− ∞ which is then applied a turboshaft engine example as ding approach Apkarian and Adams (1998)) or selecting solution of the time-domain H− /H∞ fault detection filter whether the Lyapunov matrix is parameter-dependent. − some ∞ concluding is then applied5.toFinally, a turboshaft engine example as ding approach Apkarian and Adams (1998)) or selecting which illustrated in Section remarks whether the Lyapunov matrix is parameter-dependent. which is then applied to a turboshaft engine example as illustrated in Section 5. Finally, some concluding remarks The aim of the paper is to construct a robust residual genwhether the Lyapunov matrix is parameter-dependent. are given in Section 6. illustrated in Section 5. Finally, some concluding remarks The aimformulated of the paper construct a robust residual genare given in Section 6. erator, as isantoH /H optimization problem. − ∞ The aim of theofpaper construct a robust residual erator, formulated as isantoH problem. − ∞ The purpose combining H defined in genHou are given in Section 6. − /H ∞ optimization − (originally erator, formulated as an H /H optimization problem. − ∞ The purpose of combining H (originally defined in Hou 2. NOTATIONS AND DEFINITIONS − and Patton (1996)) with H∞ to − is to allow a trade-off The purpose of between combining H∞ (originally defined in Hou 2. NOTATIONS AND DEFINITIONS − and Patton (1996)) withsensitivity H is to allow a trade-off to be established, and robustness of the ∞ 2. NOTATIONS AND DEFINITIONS and Patton (1996)) with H is to allow a trade-off to ∞ be established, and robustness of the residual against between the faultsensitivity and disturbance, respectively. In The notations and definitions to be used in the note are be established, between sensitivity and robustness of the The notations and The definitions to be used the space note are residual against the fault and disturbance, respectively. In R n×m denotes thein real of the literature, H− /H∞ based approaches can be divided summarized here. n×m to be used in the note are n×m The notations and The definitions residual against the fault and disturbance, respectively. In summarized here. R denotes the real space of the literature, H /H based approaches can be divided the n × m matrices. For a matrix A with the appropriate − ∞ into two categories. − One ∞ category combines the H− index n×m summarized here. The RaAmatrix denotes the real space of T −1 † the H− /H based approaches can be divided the n × m matrices. For A with the appropriate ∞ into two categories. One category combines the H index dimension, A , A and denote its transpose, inverse − and literature, H performance as a multi-objective criterion (Ding − ∞ T −1 † T −1 † the n × m matrices. For a matrix A with the appropriate into two categories. One category combines the H index dimension, A , A and A denote its transpose, inverse − (Ding andal.H(2000); a multi-objective criterion and pseudo-inverse respectively. The A > 0 (A ≥ 0) ∞ et Wang etasal. (2007)). In another category, ∞ performance dimension, ATA, Ais−1positive and A† (semi-positive) denote its A transpose, andal. H(2000); a multi-objective (Ding and pseudo-inverse respectively. The >definite, 0 (Ainverse ≥that 0) ∞ performance et al. (2007)). In another category, that the mixed H−Wang /H∞ et isastransformed into acriterion uniform H∞ denotes and¯ pseudo-inverse respectively. 0 (A ≥that 0) et al. (2000); Wang et al. (2007)). In another category, denotes that A is positive (semi-positive) 1/2 The A 1/2>Tdefinite, ¯ the mixed H /H is transformed into a uniform H − ∞ ∞ problem (Henry is λ(A) > 0 (λ(A) ≥ 0). A1/2 = (A1/2 )T > 0 is the − and ∞ Zolghadri (2005); Li et al. (2012)), ∞ denotes that A is positive (semi-positive) definite, that ¯ ¯ 1/2 1/2 T the mixed H /H is transformed into a uniform H − and ∞ to ∞ problem (Henry Zolghadri Li etby al.Grenaille (2012)), = (A matrix ) >A.0 L is2ethe is λ(A)root > 0of(λ(A) ≥ 0).semi-definite A which was extended an LPV(2005); framework square is 1/2 T ¯ ¯aa positive problem (Henry and to Zolghadri etby al.Grenaille (2012)), ) wherein >A.0 L is2e the λ(A) > 0offinite-horizon (λ(A) ≥ 0).semi-definite A1/2 = (A which was extended an LPV framework square root positive matrix is et al. (2008); Henry (2012). In(2005); Li andLiLiu (2013), the is the extended Lebesgue 2-space, 2ethe which was extended an LPV Grenaille square root offinite-horizon a positive semi-definite matrixwherein A. L2ethe is et al. (2008); Henry to (2012). In framework Li and Liuby (2013), the the extended Lebesgue 2-space, et al. (2008); Henry (2012). In Li and Liu (2013), the the extended finite-horizon Lebesgue 2-space, wherein the
Copyright © 2017 IFAC 8934 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017, 2017 IFAC 8934Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 8934Control. 10.1016/j.ifacol.2017.08.1427
Proceedings of the 20th IFAC World Congress L. Chen et al. / IFAC PapersOnLine 50-1 (2017) 8600–8605 Toulouse, France, July 9-14, 2017
signal is continuous and norm bounded in a given finite time horizon, defined by L2e = {v|vτ ∈ L2 , ∀τ ∈ [0, ∞]} (1) and vτ is a truncation of v defined by v(t), 0 ≤ t ≤ τ vτ (t) = (2) 0, t>τ τ Let �v�22e = 0 v(t)T v(t)dt and Lebesgue 2-space becomes infinite-horizon when τ → ∞.
In this note, a system is denoted in boldface upper case, e.g. an LPV system G(ρ) : w �→ y = G(ρ)w is given by x˙ = A(ρ)x + B(ρ)w x(0) = 0 (3) y = C(ρ)x + D(ρ)w where x, w and y represent the system states, inputs and T outputs respectively. ρ = [ρ0 ρ1 . . . ρs ] ∈ Ωe ⊂ R s are the available varying parameters, where Ωe is a compact polytope in a given finite time horizon [0, τ ]. A state space realization of G(ρ) is given by A(ρ) B(ρ) G(ρ) (4) C(ρ) D(ρ) Definition 2.1. For a given LPV system G(ρ), a state space realization of the left inverse of G(ρ) is given by A(ρ) − B(ρ)D† (ρ)C(ρ) −B(ρ)D† (ρ) (5) G−1 (ρ) D† (ρ)C(ρ) D† (ρ)
where D(ρ) is full column rank for all ρ ∈ Ωe Definition 2.2. For a given LPV system G(ρ), a state space realization of the adjoint of G(ρ) is denoted by G∼ (ρ), given by −AT (ρ) −C T (ρ) ∼ (6) G (ρ) B T (ρ) DT (ρ) such that (G∼ (ρ))∼ = G(ρ) ∀ρ ∈ Ωe
(7)
and ∼ (G(ρ)G1 (ρ))∼ = (G∼ (8) 1 (ρ)G (ρ)) ∀ρ ∈ Ωe where G1 (ρ) and G(ρ) are systems with compatible dimension. Definition 2.3. For a given LPV system G(ρ), its timedomain H∞ performance and H− index in a given finite time horizon [0, τ ] are defined by �G(ρ)w�2e �G(ρ)�∞,e = sup , ∀ρ ∈ Ωe , ∀w ∈ L2e , w �= 0 �w�2e w,ρ (9) �G(ρ)w�2e , ∀ρ ∈ Ωe , ∀w ∈ L2e , w �= 0 �G(ρ)�−,e = inf w,ρ �w�2e (10) Remark 2.1. In the literature (e.g. in Hou and Patton (1996); Ding et al. (2000); Jaimoukha et al. (2006); Liu et al. (2005); Wang et al. (2007)), the H∞ performance and H− index have been defined by using a singular value property defined in frequency domain to measure the sensitivity of the residual against fault for a given linear time invariant system. The work in Wei and Verhaegen (2011) extended Wang et al. (2007) to one compatible with LPV systems. In Henry (2012), the H− index for LPV systems, defined based on the notion of the truncated L2
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norm of vectors, has the same formulation as (10). The H− index is defined in time-domain Li and Liu (2013) which represents the smallest system gain for all input signals in L2 . The same norm symbol for the time and frequency domain is used since both the Fourier transform and Laplace transform are Hilbert isomorphisms. Lemma 2.1. According to Definition 2.3, it is not hard to show that �G(ρ)G1 (ρ)�∞,e ≤ �G(ρ)�∞,e �G1 (ρ)�∞,e (11) �G(ρ)G1 (ρ)�−,e ≤ �G(ρ)�∞,e �G1 (ρ)�−,e (12) (13) �G(ρ)G1 (ρ)�∞,e ≥ �G(ρ)�−,e �G1 (ρ)�∞,e where G(ρ) and G1 (ρ) are systems with compatible dimensions, ∀ρ ∈ Ωe . Lemma 2.2. For a given LPV system G(ρ), there exists a left coprime factorization ˜ −1 (ρ)N(ρ) ˜ G(ρ) = M ∀ρ ∈ Ωe (14) such that
˜ ˜ M(ρ) N(ρ)
Xl (ρ) =I Yl (ρ)
(15)
˜ ˜ where M(ρ), N(ρ), Xl (ρ) and Yl (ρ) are stable systems with compatible dimensions. Provided (C(ρ), A(ρ)) is detectable for all ρ ∈ Ωe and L(ρ) with compatible dimension, a realization of the left coprime pair is given by ˜ ˜ M(ρ) N(ρ) (16) A(ρ) + L(ρ)C(ρ) L(ρ) B(ρ) + L(ρ)D(ρ) C(ρ) I D(ρ)
Lemma 2.3. For a stable LPV system G(ρ), with state space realization given by (4), let Po be the observability gramian, ∀ρ ∈ Ωe , if the realization satisfies Po A(ρ) + AT (ρ)Po + C T (ρ)C(ρ) = 0 (17) DT (ρ)C(ρ) + B T (ρ)Po = 0
(18)
T
(19) D (ρ)D(ρ) = I then G(ρ) is isometric on L2e , such that G∼ (ρ)G(ρ) = I and �G(ρ)w�2e = �w�2e . If the system is controllable, these conditions are also necessary. 3. PROBLEM FORMULATION Consider an LPV system, subject to disturbances, sensor and actuator faults, is described as x˙ = A(ρ)x + B(ρ)u + Bd (ρ)d + Bf (ρ)f (20) y = Cx + Du + Dd d + Df f where x ∈ R nx , u ∈ R nu , y ∈ R ny represent the system states, inputs and outputs, respectively. f ∈ R nf and d ∈ R nd are actuator and sensor fault signals and the unknown disturbances, respectively. The state space matrices A(ρ), B(ρ), the input fault and disturbance distribution matrices Bf (ρ), Bd (ρ) as well as the estimator gains L(ρ) are assumed to be affinely parameter-dependent with appropriate dimensions, in particular s A(ρ) = ζi (ρ)Ai (21) 1
where the functions ζi : Ωe → R are those to be selected by the designer and Ai are the decision matrices. Remark 3.1. To avoid evaluating an infinite number of the grid points in the LPV region, the matrices C and D and
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[ M(ρ) Nu (ρ) Nd (ρ) Nf (ρ) ] A(ρ) + L(ρ)C L(ρ) B(ρ) + L(ρ)D B(ρ) + L(ρ)Dd Bf (ρ) + L(ρ)Df C I D Dd Df
the output faults and disturbance distribution matrices Df and Dd are assumed to be fixed matrices in this paper. Nevertheless, all matrices in (20) can be assumed to be arbitrarily parameter-dependent and the parameter gridding can be used to obtain finite-dimensional matrix inequality conditions. The gridding-based synthesis approach is not addressed here. System (20) can be written as y = Gu (ρ)u + Gd (ρ)d + Gf (ρ)f (22) where Gu (ρ), Gd (ρ) and Gf (ρ) are parameter-dependent systems with compatible dimensions. The state space realization of [Gu (ρ) Gd (ρ) Gf (ρ)] is given by [ Gu (ρ) Gd (ρ) Gf (ρ) ] A(ρ) B(ρ) Bd (ρ) Bf (ρ) C D Dd Df
(23)
Assumption 3.1. (1) (C(ρ), A(ρ)) is detectable for all ρ ∈ Ωe . (2) Df is full column rank. Using Lemma 2.2 and ensuring Assumption 3.1, the left coprime factorization of the above system exists for all ρ ∈ Ωe , [ Gu (ρ) Gd (ρ) Gf (ρ) ] (24) = M−1 (ρ) [ Nu (ρ) Nd (ρ) Nf (ρ) ] where M(ρ), Nu (ρ), Nd (ρ) and Nf (ρ) are stable systems with compatible dimensions. Since (C, A(ρ)) is detectable for all ρ ∈ Ωe , the state space realization of systems M(ρ), Nu (ρ), Nd (ρ) and Nf (ρ) are given in (25) (as shown on the top of the page). In order to decouple the system inputs from the residual, the structure of the fault detection system can be written as r = F(ρ)(M(ρ)y − Nu (ρ)u) (26) = F(ρ)Nd (ρ)d + F(ρ)Nf (ρ)f where r is the residual signal and F(ρ) is the parameterdependent fault detection filter to be constructed. The structure for the general fault detection residual generator is showed in Fig. 2. u
N u( ρ )
plant
-
+
F( ρ )
y
(25)
ity of the residual signal to the faults can be established through finding a suitable fault detection filter F(ρ). Definition 3.1. For a given LPV system (20) with faults and let β > 0 be a given minimum fault sensitivity measure, the corresponding time-domain H− index satisfies �F(ρ)Nf (ρ)�−,e ≥ β, ∀ρ ∈ Ωe (27) Let z = F(ρ)Nf (ρ)f , the inequality in (27) is equivalent to an optimal control problem, given by τ (z T z − β 2 f T f ) ≥ ǫ�f �22e , ∀f ∈ L2e (28) 0
where ǫ is a positive scalar.
Based on Definition 3.1, the fault detection filter design problem is formulated in Problem 3.1: Problem 3.1. Find a stable LPV fault detection filter F(ρ) that satisfies (28), whilst minimizing �F(ρ)Nd (ρ)�∞,e for all ρ ∈ Ωe , i.e. min {�F(ρ)Nd (ρ)�∞,e : ∀ρ∈Ωe , ∀f ∈L2e τ (29) (z T z − β 2 f T f )dt ≥ ǫ�f �22e } 0
Remark 3.2. The objective function given in Problem 3.1 is different from the one proposed in Li and Zhou (2009) in which the H− /H∞ robustness is defined by finding the maximizer of �FNf �− that satisfies �FNd �∞ ≤ α, where α is a given scalar to represent the level of the H∞ robustness against disturbance. This represents one of the main contributions of this paper. 4. TIME-DOMAIN FAULT DETECTION FILTER In this section, the following Lemma 4.1 is proposed for constructing the time-domain fault detection filter. 1
Lemma 4.1. Provided (AR (ρ), (Bf (ρ) + L(ρ)Df )R− 2 ) is controllable, there always exists a stable system W(ρ), such that Nf (ρ)W−1 (ρ) is isometric, if and only if there exists a symmetric matrix P satisfying the following parameter-dependent Ricatti equation P AR (ρ) + AR (ρ)T P + CR (ρ)T CR (ρ) = 0 (30) where AR (ρ) = A(ρ) + L(ρ)C + (Bf (ρ) + L(ρ)Df )X(ρ) (31) CR (ρ) = Df X(ρ) + C
M( ρ )
X(ρ) = −R
−1
(DfT C
Proof. (Necessity) Let
f
Fig. 1. The structure of the fault detection system It is clear from (26) that a trade-off between the robustness of the residual signal to the disturbances and the sensitiv-
(32) T
+ (Bf (ρ) + L(ρ)Df ) P )
(33)
R = DfT Df (34) and suppose there exists a system W(ρ) (as shown in (35) on the top of the next page). Since R > 0, it follows 1 R 2 > 0. According to Definition 2.1, system W−1 (ρ) exists and can be written as (36) (as shown on the top of the next page). Then,
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W(ρ)
W
Nf (ρ)W
−1
(ρ)
−1
(ρ)
−R
− 12
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−(Bf (ρ) + L(ρ)Df ) A(ρ) + L(ρ)C 1 (DfT C + (Bf (ρ) + L(ρ)Df )T P ) R2
(35) 1
A(ρ) + L(ρ)C + (Bf (ρ) + L(ρ)Df )X(ρ) (Bf (ρ) + L(ρ)Df )R− 2 1 X(ρ) R− 2 1
AR (ρ) (Bf (ρ) + L(ρ)Df )R− 2 1 CR (ρ) Df R − 2
(37)
Suppose Nf (ρ)W−1 (ρ) is isometric and (AR (ρ), (Bf (ρ) + 1 L(ρ)Df )R− 2 ) is controllable, the necessary in Lemma 2.3 implies that P AR (ρ) + AR (ρ)T P + CR (ρ)T CR (ρ) = 0 (38) 1
1
(Df R− 2 )T CR (ρ) + ((Bf (ρ) + L(ρ)Df )R− 2 )T P = 0 (39) 1
1
(Df R− 2 )T Df R− 2 = I (40) Clearly, (38) is the Ricatti equation given in (30).
Theorem 4.1. Consider an LPV system in (20), there exists a state space realization of the optimal solution of the fault detection filter that satisfies Problem 3.1, given by 1 AR (ρ) (Bf (ρ) + L(ρ)Df )R− 2 F(ρ) (43) 1 βX(ρ) βR− 2 where AR (ρ) and X(ρ) are defined in (31) and (33), respectively. Additionally, the performance of the residual generator in (43) approximates the performance of the fault estimator when β = 1. Proof. (Sufficiency) Suppose there exists an optimal solution F(ρ), it follows from (36) and (43) that F(ρ) = βW−1 (ρ) (44) Define Q(ρ) = Nf (ρ)W−1 (ρ)(W−1 (ρ)Nf (ρ))−1 (45) and suppose Ξ(ρ) = (I − Q(ρ)QT (ρ)) (46) −1 −1 It is easy to verify that (W (ρ)Nf (ρ)) always exist since Df has full column rank. According to Lemma 4.1, Nf (ρ)W−1 (ρ) is isometric on L2e . It is easy to obtain that β1 Q(ρ)F(ρ)Nf (ρ) is also isometric on L2e . Hence,
(36)
1 (47) � Q(ρ)F(ρ)Nf (ρ)f �22e = �f �22e β According to Definition 3.1, z = F(ρ)Nf (ρ)f , (47) equates to �Q(ρ)z�22e = β 2 �f �22e (48) which can be written as τ (49) z T QT (ρ)Q(ρ)zdt = β 2 �f �22e 0
It is clear that (49) is equivalent to τ τ T z zdt − z T (I − Q(ρ)QT (ρ))zdt = β 2 �f �22e (50) 0
0
and therefore τ
T
z zdt −
τ
z T Ξ(ρ)zdt = β 2 �f �22e
(51)
0
0
(Sufficiency) Since the Ricatti equation (30) equates to (38), the controllability ensures that the states of the system Nf (ρ)W−1 (ρ) spans R n as system inputs ranges over L2e , (39) and (40) are then available. It follows from (39) and (40) that W∼ (ρ)W(ρ) = N∼ (41) f (ρ)Nf (ρ) Then −1 (ρ) = I (42) (W∼ )−1 (ρ)N∼ f (ρ)Nf (ρ)W −1 It is clear from (42) that Nf (ρ)W (ρ) is isometric.
1
Define η = Ξ 2 (ρ)z and the equality in (51) can be written as τ
(z T z − η T η)dt = β 2 �f �22e
(52)
0
Then
�z�22e − �η�22e = β 2 �f �22e
(53)
Suppose �z�22e = ζ�f �22e and let 1
ǫ = ζ�Ξ 2 (ρ)�−,e
(54)
Then, (53) can be written as �z�22e − �βf �22e = �η�22e 1
≥ ζ�Ξ 2 (ρ)�−,e �f �22e = ǫ�f �22e
(55)
which is equivalent to (28) for all f ∈ L2e . (Necessity) Using the fact that �F(ρ)W(ρ)�−,e ≥ �F(ρ)Nf (ρ)W−1 (ρ)W(ρ)�−,e
(56)
and �F(ρ)W(ρ)�−,e = �F(ρ)Nf (ρ)W−1 (ρ)W(ρ)�−,e , specially when Nf (ρ)W−1 (ρ) = I. It is clear from (56) that �F(ρ)W(ρ)�−,e ≥ �F(ρ)Nf (ρ)�−,e (57) As defined in (27), �F(ρ)Nf (ρ)�−,e ≥ β, then �F(ρ)W(ρ)�−,e ≥ β
(58)
Using (13) in Lemma 2.1, �F(ρ)Nd (ρ)�∞,e = �F(ρ)W(ρ)W−1 (ρ)Nd (ρ)�∞,e ≥ �F(ρ)W(ρ)�−,e �W−1 (ρ)Nd (ρ)�∞,e ≥ β�W−1 (ρ)Nd (ρ)�∞,e (59) It is clear from (59) that the optimal filter solution is F(ρ) = βW−1 (ρ). This proofs the necessity.
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Now a special case where the performance of the residual generator is able to approximate one of the fault estimator is discussed. Let γ and γ˜ be the minimizers of γ := min {�F(ρ)Nd (ρ)�∞,e : F(ρ)Nf (ρ) = I}
matrices have the affine structure (as shown in (21)), the LMIs in (66) and (67) are solved based on the vertex property (Apkarian et al. (1995); Wei and Verhaegen (2011)).
∀ρ∈Ωe ,f ∈L2e
(60) and γ˜ :=
min
∀ρ∈Ωe ,f ∈L2e
˜ ˜ {�F(ρ)N d (ρ)�∞,e : �F(ρ)Nf (ρ)�−,e ≥ 1}
(61) ˜ respectively, where F(ρ) and F(ρ) are two optimal stable detection filters and the term F(ρ)Nf (ρ) = I in (60) represents an ideal fault estimation.
5. DESIGN EXAMPLE Consider an LPV JetCat SPT5 turboshaft engine mode represented by
˜ Since F(ρ) and F(ρ) are stable systems, there always exists ˜ an invertible system X(ρ), such that F(ρ) = X(ρ)F(ρ). ˜ Since F(ρ)Nf (ρ) = I, it follows that X(ρ)F(ρ)Nf (ρ) = I −1 ˜ ˜ . Note that �F(ρ)N and X(ρ) = (F(ρ)N f (ρ)) f (ρ)�−,e ≥ −1 1 is equivalent to �X(ρ) �−,e ≥ 1. Then, �X(ρ)�∞,e ≤ 1 for square X(ρ), such that γ = �F(ρ)Nd (ρ)�∞,e ˜ = �X(ρ)F(ρ)N d (ρ)�∞,e ˜ ≤ �X(ρ)�∞,e �F(ρ)N d (ρ)�∞,e ˜ ≤ �F(ρ)Nd (ρ)�∞,e = γ˜
(62) (63) (64) (65)
It is clear that �I�− ≥ 1, which implies γ ≥ γ˜ . Using the fact that γ ≤ γ˜ in (65), it can be concluded that that γ = γ˜ . The optimal solution of the fault estimation filter can thus be deduced through assigning a specific H− index into (43), that is, β = 1.
1.6351 0.0261 −4.4209 −0.0090 A(ρ) = +ρ −1.3708 0.5112 2.9073 −2.0876
−1.2888 0.0151 1.2520 B(ρ) = 0.0652 −0.5850 0.1390 2.1541 −0.0102 0 +ρ 0.4538 0.4119 0
−1.2888 0.0151 2.1541 −0.0102 Bf (ρ) = +ρ 0.0652 −0.5850 0.4538 0.4119 −0.2973 0.0467 Bd = 0.0094 −0.1814 0.0865 0.1177 Dd = 0.3331 −0.4366 1 0 Df = 0 1
(72)
where the scheduling parameter ρ ∈ [0.3361 1.3810] is Based on Lemma 4.1, an LMI-based synthesis approach selected to be the Euclidean norm of the gas turbine engine for solving the fault detection filter gain matrix L(ρ) in spool speed. The LPV model is established based on a least square polynomial fitting of the multiple LTI models (43) is proposed in Proposition 4.1. provided in Pakmehr et al. (2013). Suppose the system is subject to two potential faults and two disturbances. Proposition 4.1. Suppose there exists parameter-dependent The disturbances distribution matrices Bd and Dd are matrices S(ρ) and M (ρ), a symmetric matrix P and a randomly generated. Let the first disturbance is a constant symmetric positive definite matrix Q such that the LMIs bias of amplitude simulated from 2sec and the second in (66) and (67) are feasible for all ρ ∈ Ωe with respect to disturbance is a white noise with mean zero and variance the decision variables P , S(ρ), M (ρ) and Q. 0.01. Two bias faults of 1 and −1 are simulated from 8sec and 15sec, respectively. A11 (ρ)T + A11 (ρ) + C T C A12 (ρ) >0 (66) R A12 (ρ)T Through solving LMIs (66) and (67), the matrix P and be calculated as QA(ρ) + M (ρ)C + (QA(ρ) + M (ρ)C)T < 0 (67) the gain matrix L(ρ) can 0.0967 0.0489 where P = (73) 0.0489 0.2125 A11 (ρ) = P A(ρ) + S(ρ)C (68) (69) A12 (ρ) = P Bf (ρ) + S(ρ)Df + C T Df −10.4103 2.6761 −2.1541 0.0102 L(ρ) = +ρ 2.6260 −4.7395 −0.4538 −0.4119 (70) R = DfT Df (74) Then the fault detection filter gain is given by L(ρ) = P −1 S(ρ) (71) Substituting L(ρ) into (31) and (33) to get AR (ρ) and X(ρ), the optimal filter F(ρ) in (43) can then be deduced. Proof. The LMI in (66) can be deduced by defining To simply demonstrate the effectiveness of the proposed S(ρ) = P L(ρ) and applying the Schur complement to the filter, the scheduling parameter is randomly chosen beparameter-dependent Ricatti equation in (30). The LMI tween 0.3361 and 1.3810 at the beginning of the simulation in (67) ensures (C, A(ρ)) is detectable for all ρ ∈ Ωe . and then remained fixed afterwards. Figure 2 shows the Remark 4.1. The inequality constraint shown in (67) en- residual signals (r1 and r2 ) before the filter, i.e. Gf (ρ)f + sures the fault detectability. Gd (ρ)d. The residual signals of the proposed optimal filter Remark 4.2. Using the fact that the scheduling parameter are depicted in Figure 3. It is clear from Figure 3 that the ρ varies inside a polytope and the parameter-dependent filtered residual is able to distinguish the fault from the 8938
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disturbance and the amplitudes of the faults are approximated when β = 1. 2
r1 r2
Residual signals
1.5 1 0.5 0 −0.5 −1 −1.5 0
2
4
6
8
10 12 Time (sec)
14
16
18
20
Fig. 2. Response of the unfiltered residual 1.5
r1 r2
Residual signals
1
0.5
0
−0.5
−1 0
2
4
6
8
10 12 Time (sec)
14
16
18
20
Fig. 3. Response of the filtered fault detection residual when β = 1 6. CONCLUSION In this paper, an LPV fault detection filter is developed based on a time-domain H− /H∞ robustness, which establishes a trade-off between the sensitivity of the residual signal to the fault and the robustness of the residual signal against the disturbances. In some special case, the residual generator can also be treated as a robust fault estimator. The proposed method is applied to an LPV turboshaft engine model to demonstrate the effectiveness. REFERENCES Alwi, H. and Edwards, C. (2014). Robust fault reconstruction for linear parameter varying systems using sliding mode observers. International Journal of Robust and Nonlinear Control, 24(14), 1947–1968. Apkarian, P. and Adams, R.J. (1998). Advanced gainscheduling techniques for uncertain systems. IEEE Transactions on Control Systems Technology, 6(1), 21– 32. Apkarian, P., Gahinet, P., and Becker, G. (1995). Selfscheduled H∞ control of linear parameter-varying systems: a design example. Automatica, 31(9), 1251–1261. Balas, G.J. (2002). Fault tolerant control and fault detection/isolation for linear parameter varying systems. International Journal of Robust and Nonlinear Control, 12(9), 763–796. Bokor, J. and Balas, G.J. (2004). Detection filter design for LPV systems − a geometric approach. Automatica, 40(3), 511–518.
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Casavola, A. and Gagliardi, G. (2014). Fault detection and isolation of electrical induction motors via LPV fault observer: A case study. International Journal of robust and nonlinear control, 25(5), 627–648. Chen, L., Patton, R., and Goupil, P. (2016). Robust fault estimation using an LPV reference model: ADDSAFE benchmark case study. Control Engineering Practice, 49, 194–203. Ding, S.X., Jeinsch, T., Frank, P.M., and Ding, E. (2000). A unified approach to the optimization of fault detection systems. Journal of Adaptive Control and Signal Processing, 17(4), 725–745. Grenaille, S., Henry, D., and Zolghadri, A. (2008). A method for designing fault diagnosis filters for LPV polytopic systems. J. Control Sci. Eng., 2008, 1–11. Henry, D. (2008). Fault diagnosis of the MICROSCOPE satellite actuators using H− /H∞ filters. AIAA Journal of Guidance, Control and Dynamics, 31(3), 699–711. Henry, D. (2012). Structured fault detection filters for LPV systems modeled in an LFT manner. International Journal of Adaptive Control and Signal Processing, 26(3), 190–207. Henry, D. and Zolghadri, A. (2005). Design and analysis of robust residual generations for systems under feedback control. Automatica, 41(2), 251–264. Hou, M. and Patton, R.J. (1996). An LMI approach to H− /H∞ fault detection observers. In Proceedings of the UKACC international conference on control, 305– 310. Exeter, UK, {available from IEEE Xplore}. Jaimoukha, I.M., Li, Z., and Papakos, V. (2006). A matrix factorization solution to the H− /H∞ fault detection problem. Automatica, 42(11), 1907–1912. Li, X. and Liu, H. (2013). Characterization of H− index for linear time-varying systems. Automatica, 49(5), 1449–1457. Li, X. and Zhou, K. (2009). A time domain approach to robust fault detection of linear time-varying systems. Automatica, 45(1), 94–102. Li, Z., Mazars, E., Zhang, Z., and Jaimoukha, I.M. (2012). State-space solution to the H− /H∞ fault-detection problem. International Journal of Robust and Nonlinear Control, 22(3), 282–299. Liu, J., Wang, J.L., and Yang, G.H. (2005). An LMI approach to minimum sensitivity analysis with application to fault detection. Automatica, 41(11), 1995–2004. Pakmehr, M., Fitzgerald, N., and M. Feron, E. (2013). Gain scheduling control of gas turbine engines: stability by computing a single quadratic lyapunov function. In Proceedings of ASME Turbo Expo, 1–14. San Antonio, Texas, USA. Scherer, C.W. (2001). LPV control and full block multipliers. Automatica, 37(3), 361–375. Shi, F. and Patton, R.J. (2014). Fault estimation and active fault tolerant control for linear parameter varying descriptor systems. International Journal of robust and nonlinear control, 25(5), 689–706. Wang, J.L., Yang, G.H., and Liu, J. (2007). An LMI approach to H− index and mixed H− /H∞ fault detection observer design. Automatica, 43(9), 1656–1665. Wei, X. and Verhaegen, M. (2011). LMI solutions to the mixed H− /H∞ fault detection observer design for linear parameter-varying systems. International Journal of Adaptive Control and Signal Processing, 25(2), 114–136.
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