This paper addresses the problem of fault detection for linear parameter-varying systems in the presence of measurement noise and exogenous disturbances. The applicability of current methods is limited in the sense that, to increase accuracy, the detection requires a large number of past measurements and the boundedness of the set-valued estimates is only guaranteed for stable systems. In order to widen the class of systems to be modeled and also to reduce the associated computational cost, the aforementioned issues must be addressed. A solution involving left-coprime factorization and deadbeat observers is proposed in order to reduce the required number of past measurements without compromising accuracy and allowing the design of Set-Valued Observers (SVOs) for fault detection of unstable systems by using the resulting stable subsystems of the coprime factorization. The algorithm is shown to produce bounded set-valued estimates and an example is provided. Performance is assessed through simulations, illustrating, in particular that small-magnitude faults (compared to exogenous disturbances) can be detected under mild assumptions

Hespanha, J.P.

Rosa, P.

Silvestre, C.

Silvestre, Daniel

Systems & Control Letters

English

Camões - Repositório Institucional da Universidade Autónoma de Lisboa

Fault Detection for LPV Systems using Set-ValuedObservers: A Coprime Factorization ApproachDaniel Silvestrea,∗, Paulo Rosa1, João P. Hespanha1, Carlos Silvestre1aContact the author D. Silvestre through dsilvestre@isr.ist.utl.pt for a freeunformatted copy of this article or go tohttps://www.sciencedirect.com/science/article/pii/S0167691117301019 for the finalformatted version.AbstractThis paper addresses the problem of fault detection for linear parameter-varyingsystems in the presence of measurement noise and exogenous disturbances usingSet-Valued Observers (SVOs). The applicability of current methods is limitedin the sense that, to increase accuracy, the detection requires a large numberof past measurements and the boundedness of the set-valued estimates is onlyguaranteed for stable systems. In order to widen the class of systems to be mod-eled and also to reduce the associated computational cost, the aforementionedissues must be addressed. A solution involving left-coprime factorization anddeadbeat observers is proposed that reduces the required number of past mea-surements without compromising accuracy and allowing the design of SVOs forfault detection of unstable systems by using the resulting coprime factorizationstable subsystems. The algorithm is shown to produce bounded set-valued esti-mates and an example is provided. Performance is assessed through simulations,illustrating, in particular that small-magnitude faults (compared to exogenousdisturbances) can be detected under mild assumptions.1. IntroductionThe problem of detecting faults in the context of Linear Parameter-Varying(LPV) systems relates to that of determining if the current observations of thetrue system are compatible with the theoretical fault-free model. In particular,the framework of LPV systems is considered in this paper since applications offault detection mechanisms for LPV systems are commonly found in industrialprocesses (see examples in the survey in [1]). In addition, distributed algorithmscan also be viewed as LPV systems driven by dynamics dependent on stochasticor deterministic actions that can be measured only at the current time instant[2].∗Corresponding authorEmail address: dsilvestre@isr.ist.utl.pt (Daniel Silvestre)Preprint submitted to Systems and Control Letters February 6, 2018The study of fault detection problems has been a long standing researchtopic, since the early 70’s (see [3]), but still poses remarkable challenges toboth the scientific community and the industry (see, for example, the surveyin [4] and the references therein). Classical fault detection methods such asthe ones proposed in [3], [5], [6], [7], [8], [9] and [10], rely on designing filtersthat generate residuals that should be large under faulty conditions. Thesestrategies aim to derive bounds (or thresholds) on these residuals that can beused to decide whether a fault has occurred or not. However, the computation ofthese thresholds is typically cumbersome or poses stringent assumptions on theexogenous disturbances and measurement noise acting upon the system. Manyimplementations of residual-based fault detection techniques are available in theliterature such as [11], [12], [13] and [14].In [15], [16], the authors develop the idea of using Set-Valued Observers(SVOs), whose concept was introduced in [17] and [18] (further informationcan be found in [19] and [20] and the references therein), for fault detectionby resorting to a model invalidation (falsification) approach. The method isparticularly interesting in the sense that it is able to handle a relatively largeclass of dynamic models, while also reducing the conservatism of the resultsby incorporating the information of past observations in the construction ofthe current set-valued state estimates. However, two main drawbacks of theapproach can be identified: the convergence properties have only been proven forstable systems [16], and the calculation of the set-valued state estimates requiresa significant computational effort. The latter limitation is a consequence of theneed to increase the horizon of the observations to produce accurate results.The aim of this paper is to extend the SVO-based fault detection method inorder to cope with unstable systems and to reduce the necessary horizon valuefor the class of LPV systems. This paper builds on existing results that exploita left-coprime factorization-based approach into the design of SVOs for LinearTime-Invariant systems (LTI) [21].The choice for representing the set of possible states depends on a math-ematical formulation that enables fast and non-conservative intersections andunions of sets, as those are time-consuming operations when implemented in acomputer. One alternative is to use the concept of zonotopes, described in [22]and further developed in [23],[24] and [25]. However, it is normally the case thateach proposal represents a compromise between the computational complexityof the unions and intersections. Alternatively, the idea of interval analysis [26]may also be adopted, although it introduces conservatism by not consideringhorizon values larger than one in their typical formulation, unlike SVOs [16].In [27], interval observers for linear and nonlinear systems are proposed withmild assumptions such as the boundedness of the disturbances and measurementnoise (similar to the assumptions in this paper).The main contributions of this paper can be summarized as follows:• The use of a left coprime factorization for LPV systems enables SVO-basedfault detection, even when the plant is unstable;• The convergence proof of the method is provided for a broad class of LPV2systems and for any horizon greater than nx, the size of the state space,by exploiting the properties of deadbeat observers.[1] W. J. Rugh, J. S. Shamma, Research on gain scheduling, Automatica36 (10) (2000) 1401 – 1425.[2] D. Silvestre, P. Rosa, J. ao P. Hespanha, C. Silvestre, Stochastic anddeterministic fault detection for randomized gossip algorithms, Automatica78 (2017) 46 – 60. doi:http://doi.org/10.1016/j.automatica.2016.12.011.URL http://www.sciencedirect.com/science/article/pii/S0005109816305192[3] A. S. Willsky, A survey of design methods for failure detection in dynamicsystems, Automatica 12 (6) (1976) 601 – 611.[4] I. Hwang, S. Kim, Y. Kim, C. Seah, A survey of fault detection, isolation,and reconfiguration methods, Control Systems Technology, IEEE Transac-tions on 18 (3) (2010) 636 –653.[5] J. D. Barrett, Diagnosis and fault-tolerant control, Technometrics 49 (4)(2007) 493–494.[6] J. Bokor, G. Balas, Detection filter design for LPV systems – a geometricapproach, Automatica 40 (2004) 511–518.[7] G. Ducard, Fault-tolerant Flight Control and Guidance Systems: Practi-cal Methods for Small Unmanned Aerial Vehicles, Advances in industrialcontrol, Springer, 2009.[8] A. Marcos, S. Ganguli, G. J. Balas, An application of H∞ fault detectionand isolation to a transport aircraft, Control Engineering Practice 13 (1)(2005) 105 – 119.[9] X. Ding, P. M. Frank, Fault detection via factorization approach, Systemsand Control Letters 14 (5) (1990) 431 – 436.[10] S. Narasimhan, P. Vachhani, R. Rengaswamy, New nonlinear residual feed-back observer for fault diagnosis in nonlinear systems, Automatica 44(2008) 2222–2229.[11] H. Hammouri, M. Kinnaert, E. E. Yaagoubi, Fault detection and isolationfor state affine systems, European Journal of Control 4 (1) (1998) 2 – 16.[12] D. Sauter, Diagnosis and fault-tolerant control m. blanke, m. kinnaert, j.lunze and m. staroswiecki, springer-verlag: Berlin, 2003, 571 pp, isbn 3-540-01056-4, International Journal of Robust and Nonlinear Control 15 (3)(2005) 151–154.[13] J. Chen, R. J. Patton, Robust model-based fault diagnosis for dynamicsystems, Vol. 3, Springer Science & Business Media, 2012.3[14] G. Ducard, Actuator fault detection in uavs, in: K. P. Valavanis, G. J.Vachtsevanos (Eds.), Handbook of Unmanned Aerial Vehicles, SpringerNetherlands, 2015, pp. 1071–1122.[15] P. Rosa, C. Silvestre, J. Shamma, M. Athans, Fault detection and isola-tion of LTV systems using set-valued observers, 49th IEEE Conference onDecision and Control (2010, Atlanta, Georgia, USA.) 768–773.[16] P. Rosa, C. Silvestre, Fault detection and isolation of LPV systems us-ing set-valued observers: An application to a fixed-wing aircraft, ControlEngineering Practice 21 (3) (2013) 242 – 252.[17] H. Witsenhausen, Sets of possible states of linear systems given perturbedobservations, IEEE Transactions on Automatic Control 13 (5) (1968) 556– 558.[18] F. Schweppe, Recursive state estimation: Unknown but bounded errorsand system inputs, IEEE Transactions on Automatic Control 13 (1) (1968)22 – 28.[19] F. Schweppe., Uncertain Dynamic Systems, Prentice-Hall, 1973.[20] M. Milanese, A. Vicino, Optimal estimation theory for dynamic systemswith set membership uncertainty: An overview, Automatica 27 (6) (1991)997 – 1009.[21] P. Rosa, C. Silvestre, M. Athans, Model falsification using set-valued ob-servers for a class of discrete-time dynamic systems: a coprime factorizationapproach, International Journal of Robust and Nonlinear Control 24 (17)(2014) 2928–2942.[22] D. Bertsekas, I. Rhodes, Recursive state estimation for a set-membershipdescription of uncertainty, IEEE Transactions on Automatic Control 16 (2)(1971) 117 – 128.[23] C. Combastel, A state bounding observer for uncertain non-linearcontinuous-time systems based on zonotopes, in: 44th IEEE Conferenceon Decision and Control, 2005 and 2005 European Control Conference.CDC-ECC ’05., 2005, pp. 7228 – 7234.[24] T. Alamo, J. Bravo, E. Camacho, Guaranteed state estimation by zono-topes, Automatica 41 (6) (2005) 1035 – 1043.[25] J. K. Scott, D. M. Raimondo, G. R. Marseglia, R. D. Braatz, Constrainedzonotopes: A new tool for set-based estimation and fault detection, Auto-matica 69 (2016) 126 – 136.[26] R. E. Moore, Interval analysis, Prentice-Hall series in automatic computa-tion, Prentice-Hall, Englewood Cliffs, NJ, 1966.4[27] T. Raissi, D. Efimov, A. Zolghadri, Interval state estimation for a class ofnonlinear systems, Automatic Control, IEEE Transactions on 57 (1) (2012)260–265.5

Fault detection for LPV systems using Set-Valued Observers: A coprime factorization approach

Repositório Institucional da Universidade Autónoma de Lisboa

Fault Detection for LPV Systems using Set-Valued
Observers: A Coprime Factorization Approach
Daniel Silvestrea,∗, Paulo Rosa1, Joa˜o P. Hespanha1, Carlos Silvestre1
aContact the author D. Silvestre through dsilvestre@isr.ist.utl.pt for a free
unformatted copy of this article or go to
https://www.sciencedirect.com/science/article/pii/S0167691117301019 for the final
formatted version.
Abstract
This paper addresses the problem of fault detection for linear parameter-varying
systems in the presence of measurement noise and exogenous disturbances using
Set-Valued Observers (SVOs). The applicability of current methods is limited
in the sense that, to increase accuracy, the detection requires a large number
of past measurements and the boundedness of the set-valued estimates is only
guaranteed for stable systems. In order to widen the class of systems to be mod-
eled and also to reduce the associated computational cost, the aforementioned
issues must be addressed. A solution involving left-coprime factorization and
deadbeat observers is proposed that reduces the required number of past mea-
surements without compromising accuracy and allowing the design of SVOs for
fault detection of unstable systems by using the resulting coprime factorization
stable subsystems. The algorithm is shown to produce bounded set-valued esti-
mates and an example is provided. Performance is assessed through simulations,
illustrating, in particular that small-magnitude faults (compared to exogenous
disturbances) can be detected under mild assumptions.
1. Introduction
The problem of detecting faults in the context of Linear Parameter-Varying
(LPV) systems relates to that of determining if the current observations of the
true system are compatible with the theoretical fault-free model. In particular,
the framework of LPV systems is considered in this paper since applications of
fault detection mechanisms for LPV systems are commonly found in industrial
processes (see examples in the survey in [1]). In addition, distributed algorithms
can also be viewed as LPV systems driven by dynamics dependent on stochastic
or deterministic actions that can be measured only at the current time instant
[2].
∗Corresponding author
Email address: dsilvestre@isr.ist.utl.pt (Daniel Silvestre)
Preprint submitted to Systems and Control Letters February 6, 2018
The study of fault detection problems has been a long standing research
topic, since the early 70’s (see [3]), but still poses remarkable challenges to
both the scientific community and the industry (see, for example, the survey
in [4] and the references therein). Classical fault detection methods such as
the ones proposed in [3], [5], [6], [7], [8], [9] and [10], rely on designing filters
that generate residuals that should be large under faulty conditions. These
strategies aim to derive bounds (or thresholds) on these residuals that can be
used to decide whether a fault has occurred or not. However, the computation of
these thresholds is typically cumbersome or poses stringent assumptions on the
exogenous disturbances and measurement noise acting upon the system. Many
implementations of residual-based fault detection techniques are available in the
literature such as [11], [12], [13] and [14].
In [15], [16], the authors develop the idea of using Set-Valued Observers
(SVOs), whose concept was introduced in [17] and [18] (further information
can be found in [19] and [20] and the references therein), for fault detection
by resorting to a model invalidation (falsification) approach. The method is
particularly interesting in the sense that it is able to handle a relatively large
class of dynamic models, while also reducing the conservatism of the results
by incorporating the information of past observations in the construction of
the current set-valued state estimates. However, two main drawbacks of the
approach can be identified: the convergence properties have only been proven for
stable systems [16], and the calculation of the set-valued state estimates requires
a significant computational effort. The latter limitation is a consequence of the
need to increase the horizon of the observations to produce accurate results.
The aim of this paper is to extend the SVO-based fault detection method in
order to cope with unstable systems and to reduce the necessary horizon value
for the class of LPV systems. This paper builds on existing results that exploit
a left-coprime factorization-based approach into the design of SVOs for Linear
Time-Invariant systems (LTI) [21].
The choice for representing the set of possible states depends on a math-
ematical formulation that enables fast and non-conservative intersections and
unions of sets, as those are time-consuming operations when implemented in a
computer. One alternative is to use the concept of zonotopes, described in [22]
and further developed in [23],[24] and [25]. However, it is normally the case that
each proposal represents a compromise between the computational complexity
of the unions and intersections. Alternatively, the idea of interval analysis [26]
may also be adopted, although it introduces conservatism by not considering
horizon values larger than one in their typical formulation, unlike SVOs [16].
In [27], interval observers for linear and nonlinear systems are proposed with
mild assumptions such as the boundedness of the disturbances and measurement
noise (similar to the assumptions in this paper).
The main contributions of this paper can be summarized as follows:
• The use of a left coprime factorization for LPV systems enables SVO-based
fault detection, even when the plant is unstable;
• The convergence proof of the method is provided for a broad class of LPV
2
systems and for any horizon greater than nx, the size of the state space,
by exploiting the properties of deadbeat observers.
[1] W. J. Rugh, J. S. Shamma, Research on gain scheduling, Automatica
36 (10) (2000) 1401 – 1425.
[2] D. Silvestre, P. Rosa, J. ao P. Hespanha, C. Silvestre, Stochastic and
deterministic fault detection for randomized gossip algorithms, Automatica
78 (2017) 46 – 60. doi:http://doi.org/10.1016/j.automatica.2016.12.011.
URL http://www.sciencedirect.com/science/article/pii/
S0005109816305192
[3] A. S. Willsky, A survey of design methods for failure detection in dynamic
systems, Automatica 12 (6) (1976) 601 – 611.
[4] I. Hwang, S. Kim, Y. Kim, C. Seah, A survey of fault detection, isolation,
and reconfiguration methods, Control Systems Technology, IEEE Transac-
tions on 18 (3) (2010) 636 –653.
[5] J. D. Barrett, Diagnosis and fault-tolerant control, Technometrics 49 (4)
(2007) 493–494.
[6] J. Bokor, G. Balas, Detection filter design for LPV systems – a geometric
approach, Automatica 40 (2004) 511–518.
[7] G. Ducard, Fault-tolerant Flight Control and Guidance Systems: Practi-
cal Methods for Small Unmanned Aerial Vehicles, Advances in industrial
control, Springer, 2009.
[8] A. Marcos, S. Ganguli, G. J. Balas, An application of H∞ fault detection
and isolation to a transport aircraft, Control Engineering Practice 13 (1)
(2005) 105 – 119.
[9] X. Ding, P. M. Frank, Fault detection via factorization approach, Systems
and Control Letters 14 (5) (1990) 431 – 436.
[10] S. Narasimhan, P. Vachhani, R. Rengaswamy, New nonlinear residual feed-
back observer for fault diagnosis in nonlinear systems, Automatica 44
(2008) 2222–2229.
[11] H. Hammouri, M. Kinnaert, E. E. Yaagoubi, Fault detection and isolation
for state affine systems, European Journal of Control 4 (1) (1998) 2 – 16.
[12] D. Sauter, Diagnosis and fault-tolerant control m. blanke, m. kinnaert, j.
lunze and m. staroswiecki, springer-verlag: Berlin, 2003, 571 pp, isbn 3-
540-01056-4, International Journal of Robust and Nonlinear Control 15 (3)
(2005) 151–154.
[13] J. Chen, R. J. Patton, Robust model-based fault diagnosis for dynamic
systems, Vol. 3, Springer Science & Business Media, 2012.
3
[14] G. Ducard, Actuator fault detection in uavs, in: K. P. Valavanis, G. J.
Vachtsevanos (Eds.), Handbook of Unmanned Aerial Vehicles, Springer
Netherlands, 2015, pp. 1071–1122.
[15] P. Rosa, C. Silvestre, J. Shamma, M. Athans, Fault detection and isola-
tion of LTV systems using set-valued observers, 49th IEEE Conference on
Decision and Control (2010, Atlanta, Georgia, USA.) 768–773.
[16] P. Rosa, C. Silvestre, Fault detection and isolation of LPV systems us-
ing set-valued observers: An application to a fixed-wing aircraft, Control
Engineering Practice 21 (3) (2013) 242 – 252.
[17] H. Witsenhausen, Sets of possible states of linear systems given perturbed
observations, IEEE Transactions on Automatic Control 13 (5) (1968) 556
– 558.
[18] F. Schweppe, Recursive state estimation: Unknown but bounded errors
and system inputs, IEEE Transactions on Automatic Control 13 (1) (1968)
22 – 28.
[19] F. Schweppe., Uncertain Dynamic Systems, Prentice-Hall, 1973.
[20] M. Milanese, A. Vicino, Optimal estimation theory for dynamic systems
with set membership uncertainty: An overview, Automatica 27 (6) (1991)
997 – 1009.
[21] P. Rosa, C. Silvestre, M. Athans, Model falsification using set-valued ob-
servers for a class of discrete-time dynamic systems: a coprime factorization
approach, International Journal of Robust and Nonlinear Control 24 (17)
(2014) 2928–2942.
[22] D. Bertsekas, I. Rhodes, Recursive state estimation for a set-membership
description of uncertainty, IEEE Transactions on Automatic Control 16 (2)
(1971) 117 – 128.
[23] C. Combastel, A state bounding observer for uncertain non-linear
continuous-time systems based on zonotopes, in: 44th IEEE Conference
on Decision and Control, 2005 and 2005 European Control Conference.
CDC-ECC ’05., 2005, pp. 7228 – 7234.
[24] T. Alamo, J. Bravo, E. Camacho, Guaranteed state estimation by zono-
topes, Automatica 41 (6) (2005) 1035 – 1043.
[25] J. K. Scott, D. M. Raimondo, G. R. Marseglia, R. D. Braatz, Constrained
zonotopes: A new tool for set-based estimation and fault detection, Auto-
matica 69 (2016) 126 – 136.
[26] R. E. Moore, Interval analysis, Prentice-Hall series in automatic computa-
tion, Prentice-Hall, Englewood Cliffs, NJ, 1966.
4
[27] T. Raissi, D. Efimov, A. Zolghadri, Interval state estimation for a class of
nonlinear systems, Automatic Control, IEEE Transactions on 57 (1) (2012)
260–265.
5


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Fault detection for LPV systems using Set-Valued Observers: A coprime factorization approach

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