Linear Matrix Inequality Solution To The Fault Detection Problem Emmanuel Mazars co-authors Zhenhai li and Imad Jaimoukha Imperial College IASTED International Conference Cancun 19 May 2005

2 Overview 1 Introduction Introduction Problem definition Problem definition Solution using LMIs Solution using LMIs Numerical example Numerical example Conclusion Conclusion

3 Introduction Target identification and tracking systems involve a large number of actuators and sensors. Target identification and tracking systems involve a large number of actuators and sensors. An actuator failure implies actuator output is degraded by bias, drift or physical damage. An actuator failure implies actuator output is degraded by bias, drift or physical damage. Actuator or sensor failures can cause rapid breakdown in control systems. Actuator or sensor failures can cause rapid breakdown in control systems. Design objective : Design objective : Design and implement a fault detection and isolation (FDI) filter for large scale systems that is insensitive to disturbances 2

4 Introduction To enhance the reliability of sensor systems in tough conditions. To enhance the reliability of sensor systems in tough conditions. To act as an aid to human operator in fast changing situations. To act as an aid to human operator in fast changing situations. Domain of applications : Domain of applications : Noisy control and monitoring systems that involve a large number of sensors when : 3  The dynamic model is known  The sensors are prone to failure  Disturbance are acceptable, but faults may cause performance degradation Pitch angle Elevon deflector wind gusts

5 Problem definition A LTI system A LTI system System input/output behaviour Where 4

6 Problem definition Fault detection and isolation observer/filter Fault detection and isolation observer/filter 5 df d f y - u B B BfBf BdBd DdDd DfDf C A A C L - H r Real System Computer Aided Observer

7 Problem definition State estimation error : State estimation error : The residual dynamics are given by : The residual dynamics are given by : By taking Laplace transforms, we have : By taking Laplace transforms, we have :where 6

8 Problem definition Problem : Problem : Assume that and that has full column rank on the extended imaginary axis. Find and an optimal filter (which has the previous form) that achieves the infimum. Remark : 7

9 Solution using LMIs Problem 1: Problem 1: Assume that the pair (C,A) is detectable and is a co-outer function. The optimal FDI filter design is to find L and H to minimize a such that (stability) is stable (stability) is stable (detection) (detection) (isolation) (isolation)Where 8

10 Solution using LMIs Lemma 1: Lemma 1: Let. There exist and such that is stable and if and only if there exist, and such that and 9

11 Solution using LMIs We want to achieve isolation We want to achieve isolation Assume that has full column rank let (Moore Penrose Generalized inverse) Assume that has full column rank let (Moore Penrose Generalized inverse) Let With Let With 10 We get and We get and and are free matrices and are free matrices

12 Solution using LMIs Theorem 1: Theorem 1: Assume that is detectable, and has full column rank, let as defined previously. There exist and such that the problem 1 is solved if there exist, and such that and If these LMIs are solved, we can construct and as 11

13 Solution using LMIs Remark 1 : Remark 1 : In the case that 12  and are unique  Isolation if is stable Remark 2 : Remark 2 : The assumption that is co-outer can be relaxed by effecting a co-outer-inner factorization

14 Numerical example Randomly generated state-space plant with : Randomly generated state-space plant with : 13 The solutions given by LMIs are : The solutions given by LMIs are :

15 Numerical example Simulation with : Simulation with : 14  fault in actuator1 simulated by a soft bias at the 2th second  fault in actuator2 simulated by a negative jump at the 6th second

16 Conclusion Optimal FD filter scheme is maximally insensitive to disturbances with acceptable sensitivity to faults We have incorporated fault isolation into our scheme without the need for using a bank of observers. The numerical algorithm is much simpler than solving a model-matching problem 15

17 Thank You