Robust Fault Detection H Filter for Markovian Jump

Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Robust Fault Detection H∞ Filter for Markovian Jump Linear Systems with Partial Information on the Jump Parameter L. Carvalho, A. M. de Oliveira, and O. L. V. Costa Departamento de Engenharia de Telecomunica¸c˜ oes e Controle Escola Polit´ ecnica da Universidade de S˜ ao Paulo, Brazil

Florian´ opolis, Brazil, September 02-05, 2018

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Contents

1 Preliminaries

Introduction and motivation Basic Concepts 2 Fault Detection Problem 3 Main results 4 Numerical Example 5 Conclusion

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Introduction and motivation Basic Concepts

Motivation

I Fault Detection in semi-reliable networks (Networked Control Systems) with Partial Information

System

y

yˆ Filter Channel

Intrinsic limitations of the channel: I Packet dropouts; I Delays etc;

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Introduction and motivation Basic Concepts

Basic Concepts

ρ21 I Markov Chain

ρ11

1

ρ12

2 ρ22

I Markov Jump Linear Systems (MJLS)

x(k + 1) = Aθk x(k) + Bθk w(k) z(k) = Cθk x(k) + Dθk w(k)

I M. Zhong, H. Ye, P. Shi, and G. Wang. Fault detection for Markovian jump systems. IEE Proceedings-Control Theory and Applications, 2005. I Alim PC Gon¸calves, Andr´e R Fioravanti, and Jos´e C Geromel. Filtering of discrete-time Markov jump linear systems with uncertain transition probabilities. Intern. Journal of Robust and Nonlinear Control, 2011.

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Fault Detection

Consider the following MJLS representation   x(k + 1) = Aθk x(k) + Bθk u(k) + Bdθk d(k) + Bf θk f (k) y(k) = Cθk x(k) + Ddθk d(k) + Df θk f (k) Ga :   x(0) = x0 , The weighting system  x (k + 1) = Awf xf (k) + Bwf f (k)   f Wf : fˆ(k) = Cwf xf (k) + Dwf f (k)   xf (0) = xf0

(1)

(2)

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Fault Detection

The Markovian filter that generates the residual signal  η(k + 1) = Aηθˆk η(k) + Mηθˆk u(k) + Bηθˆk y(k)    r(k) = Cηθˆk η(k) + Dηθˆk y(k) F:    η(0) = η

(3)

0

The following matrices must be calculated

Aηθˆk , Mηθˆk , Bηθˆk , Cηθˆk , Dηθˆk

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Block scheme

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Residual signal generation

We define the extended system composed by all the previous dynamics: ( ˜θ w(k) x ¯(k + 1) = A˜θk x ¯(k) + B k ¯ Gaug : ˜θ x ˜ θ w(k) re (k) = C ¯(k) + D ¯ k

(4)

k

where re (k) = r(k) − fˆ(k), x ¯(k) = [x0 (k) η 0 (k) x0f (k)]0 , and 0 0 0 0 ˆ w(k) ¯ = [u (k) d (k) f (k)] and "

A˜θk θˆk ˜ ˆ C θk θk

˜ ˆ B θk θk ˜ Dθk θˆk



# =

Aθk  Bηθˆ Cθk k   0 Dηθˆk Cθk

0 Aηθˆk 0 Cηθˆk

0 0 Awf −Cwf

Bθk Mηθˆk 0 0

Bdθk Bηθˆk Ddθk 0 Dηθˆk Ddθk

 Bf θk  Bηθˆk Df θk   Bwf Dηθˆk Df θk − Dwf

(5)

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Residual signal evaluation

We define L as the evaluation time. Then: ( for k ≥ L, r¯(k)0 = [r(k)0 r(k − 1)0 . . . r(k − L)0 ] for k < L, r¯(k)0 = [r(k)0 r(k − 1)0 . . . r(0)0 ]

(6)

and, given the discrepancy between the intervals, the evaluation functions for each case are set as  ( σ=k ) 21  X   0  for k − L ≥ 0, J(¯ r(k)) = r¯(σ) r¯(σ) ,    σ=k−L (7) (σ=k ) 12   X     r(k)) = r¯(σ)0 r¯(σ) . for k − L < 0, J(¯ σ=0

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Residual signal evaluation

The threshold is defined as Jth =

sup

E(J(¯ r(k))).

(8)

d∈L2 , f =0

The occurrence of faults can be detected by analyzing the value of J(¯ r(k)) as follow: ( J(¯ r(k)) < Jth , means that the system is in the nominal mode (9) J(¯ r(k)) ≥ Jth means that a fault occurred at the instant k.

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Mode dependent H∞ fault detection filter Theorem There exists a filter in form of (3) such that kGaug k∞ < γ if there exist ˜ il , N ˜il , S˜il , W , and the matrices ∆l , Ol , Fl , Gl symmetric matrices Zi , Xi , M with compatible dimensions that satisfy the LMI constraint (10), (10)  Zi Zi   0 0  0 0 Mil11  Mil21   Mil31   Nil11   Nil21   Nil31   ε (Z)A i i   Rl Ai + ∆l Ci + Ol  0 Gl Ci + Fl 

• Mil22 Mil32 Nil12 Nil22 Nil32 εi (Z)Ai Rl Ai + ∆l Ci 0 Gl Ci

• • Mil33 Nil13 Nil23 Nil33 0 0 εi (W )Awf −Cwf

• • • 11 Sil 21 Sil 31 Sil εi (Z)Bi Rl Bi + Hl 0 0

• Xi 0 0 0 0

• • Wi 0 0 0

• • • • 22 Sil 32 Sil εi (Z)Bdi Rl Bdi + ∆l Ddi 0 Gl Ddi

• • • γ2I 0 0

• • • • γ2I 0

• • • • • γ2I

       

• • • • • 33 Sil εi (Z)Bf i Rl Bf i + ∆l Df i εi (W )Bwf Gl Ddi − Dwf

>

 11 Mil Mil21  31 Mil P  l∈Ml ail  N 11  il21  Nil Nil31

• • • • • • εi (Z) 0 0 0

• Mil22 Mil32 Nil12 Nil22 Nil32

• • Mil33 Nil13 Nil23 Nil33

• • • • • • • 0 Rl + Rl + εi (Z) − εi (X) 0 0

• • • 11 Sil 21 Sil 31 Sil

• • • • • • • • εi (W ) 0

• • • • 22 Sil 32 Sil  • •  •  •  •  •  •  •  • I

 • •   •   •   •  33 Sil

>0

If a feasible solution is found, then a suitable RFD filter is given by Aηl = Rl−1 Ol , Bηl = Rl−1 ∆l , Mηl = Rl−1 Hl , Cηl = Fl , Dηl = Gl for all i ∈ K. 11 / 21

Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

System Values

Example extracted from Zhong et al. (2005). A1 Dd

= =

 0.1 0 0 0.1  0 0 0 0   0.2 0.4

1 0 0.2 0

 0 0.5  0 0.2   2 −1

, Df =



0.3

, A2 = −0.1 0 0

0 0.2 0 0

−1 0 −0.2 0

 0 −0.5  0  −0.5



0.8





1



   , Bf =  1 , C = , Bd = −2.4 2 1.6  0.8



0 1

1 0

0 1

1 0



,

−2

, Awf = 0.5, Bwf = 0.25, Cwf = 1, Dwf = 0.5.

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Norm behavior

 Ω1 =

0.9 0.1

  0.1 0.5 , Ω2 = 0.9 0.5

  0.5 0.1 , Ω3 = 0.5 0.9

  0.9 ρ1 Ψ= 0.1 1 − ρ2

1 − ρ1 ρ2



0.64 0.62 0.6 0.58 0.56 0.54 0.52 0.5 1 1

0.5

0.5 0

0

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Residual filter matrices For the temporal simulations the values of Ω and Ψ     0.8 0.2 0.9 0.1 Ω= ,Ψ = 0.7 0.3 0.7 0.3 By using the theorem, we get γ ≈ 0.5664: 

   0.23 0.66 0.46 0.71 −0.05 0.62  0.73 1.45  0.85 −0.53 0.26  , Aη2 =  0.20 ,  0.45 0.27 0.95  0.86 −0.20 0.60  −0.44 −0.40 −1.53 −0.63 −1.51 0.45 −0.92       −1.26 −0.73 −1.31 −0.97 0      −0.34 , Bη2 = −0.89 −0.01, Mη1 = Mη2 = 0, Bη1 = −1.12 −1.04 −0.48 0 −1.09 −0.37 1.82 0.48 1.79 0.63 0     −0.15 0 −0.01 0  0  0      , Cη2 = −0.06 , Dη1 = 0.32 , Dη2 = 0.11 . Cη1 = −0.37 −0.02 −0.13 0.11 0.03 −0.38 −0.06 0.08

Aη1 =  0.59 0.28 

0.80 1.15 1.08 −1.79

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Simulation data

I d(k) is assumed to be a white noise sequence E(d(k)) = 0 and E(d(k)2 ) = 0.7. I The evaluation time is taken as L = 300.

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Fault behavior

2

1.5

1

0.5

0

-0.5

0

50

100

150

200

250

300

instant k

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Evaluation function

4 w/o failure RFDF

J(r(k))

3

2

1

0 0

50

100

150

200

250

300

instant k

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Monte Carlo simulation

Average amount of time to detect the failure is k = 3.3930 ± 1.7840, for a Monte Carlo simulation of 5000 rounds. 2.5 w/o failure RFDF

J(r(k))

2 1.5 1 0.5 0

0

50

100

150

200

250

300

instant k

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Conclusion

The Robust Fault Detection problem associated with a Markov Jump Linear System in the discrete-time domain for the partial observation of the Markov chain is studied. An LMI formulation is proposed to design a filter satisfying an H∞ upper bound performance. A numerical example is presented to illustrate the method.

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

I We would like to thank to Coordena¸c˜ ao de Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES), S˜ ao Paulo Research Foundation (Funda¸c˜ ao de Amparo ` a Pesquisa do Estado de S˜ ao Paulo - FAPESP) and University of S˜ ao Paulo Foundation (FUSP) for the financial support of this research.

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Preliminaries Fault Detection Problem Main results Numerical Example Conclusion

Thank you for your attention!

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