A New Eigenstructure Fault Isolation Filter Zhenhai Li Supervised by Dr. Imad Jaimoukha Internal Meeting Imperial College, London 4 Aug 2005

2 Overviews 1 Introduction Introduction Model-based FDI Model-based FDI Simple Example and Conclusion Simple Example and Conclusion Fault Reverter - A Special Case Fault Reverter - A Special Case Observer-based FI Filter Observer-based FI Filter

3 Introduction 2 Sensor Failures in Dynamic Systems Sensor Failures in Dynamic Systems Modeling of sensor failures Sensor Faults ActuatorComponentSensors Input u Output y Pseudo-actuator faults Representation of Sensor Faults Direct representation (solid red line) Indirect representation (dotted blue line)

4 Introduction Fault Detection and Isolation (FDI) Fault Detection and Isolation (FDI)  Motivation Occurring faults are always possible to be indicated by exploring deeper knowledge of the system inputs and outputs. 3

5 Analytical Redundancy Analytical Redundancy 4 Model-based FDI GdGd GfGf Filter F faults residual disturbances ‘small’ gain‘large’ gain Decision threshold (via online/offline testing) post-fault configuration

6 Residual Generation Residual Generation  A LTI system  System input/output behaviour where  General residual generator 5 Model-based FDI

7 Residual Evaluation Residual Evaluation  Evaluation function  Threshold  Logic 6 Model-based FDI

8  The error between the original output and the observer output is regarded as the residual signal.  The observer can provide diagonalized faults in the residual via a suitable selection of L and H.  The effect of disturbances is also attenuated by using Linear Matrix Inequality (LMI) techniques. 7 Observer-based FI Filter Approach df d f y - u B B BfBf BdBd DdDd DfDf C A A C L - H r Real System Computer Aided Observer

9 Fault Isolation Residual Generation Fault Isolation Residual Generation  Problem 1 Assume that A is stable for simplicity. Let find the matrices L and H, if they exist, such that 1. T rf (s) is a given diagonal transfer matrix A+LC is stable  The first condition is called isolation condition.  This setup is also known as almost decoupling. 8 Observer-based FI Filter Approach

Observer-based FI Filter Approach Isolation Condition Isolation Condition  Lemma  Lemma Let all variables be as defined in above and assume that E:=CB f has full column rank, and denote E # =(E T E) -1 E T. Let and Then, if L and H satisfy and Furthermore, and

11 Isolation Condition Isolation Condition  Proof There exists a completion such that is nonsingular. Let Then, where T -1 T=I is used. 10 Observer-based FI Filter Approach

12 Stability Condition Stability Condition  Theorem 1  Theorem 1 Let all variables be as defined in Lemma 3. Then the following are equivalent. 1. There exist L and H such that. 2. The following transfer matrix is co-outer 3. The matrix CB f has full column rank and the pair (A+L 1 C,L 2 C) is detectable. 4. CB f has full column rank and G f (s) has no finite zeros in. 5. CB f has full column rank and there exists such that P>0 and 11 Observer-based FI Filter Approach

Observer-based FI Filter Approach Stability Condition Stability Condition  Proof (1=>2) Let. Note that (2=>1) Co-outer implies (A,C) is detectable, i.e., (A+L 1 C,L 2 C) is detectable. This can be shown via a contradiction.

Observer-based FI Filter Approach Fault Isolation Filter Fault Isolation Filter  Theorem 2  Theorem 2 Let all variables be as defined in Lemma 3 and Theorem 3, and assume that any of (2)-(5) is satisfied. Let. Then there exist R and S, with L and H, respectively, such that specifications in Problem 1 are satisfied if and only if that there exist such that P>0 and where R=P -1 Z.

Observer-based FI Filter Approach Summary of the Algorithm Summary of the Algorithm Stability Checking (Theorem 1)

16 Problem 2 Problem 2 If G f is co-outer, which means we can always find a filter F such that Then, the original problem can also be simplified to the following objectives: (stability)The closed-loop system is stable. (detection)The –norm of the sensitivity to disturbances is bounded by a small value. (isolation)Each potential fault signal is indicated by a unique component in the residual signal. 15 Fault Reverter – A Special Case Bounded by LMI u control input u d disturbance d y output y f fault f FDI Integrated Plant r residual r

17 Algorithm Algorithm 1. Choose with suitable choice of L 1, L 2 and H 1 to ensure Here, R and S are free matrices. 16 Fault Reverter – A Special Case 3. Construct the observer gain 2. Let all variables be defined as before. Then there exist R and S, with L and H, respectively, such that Problem 2 are satisfied if there exist Z and S such that P>0 and with

Simple Example and Conclusion Analysis of An Aircraft Analysis of An Aircraft  A modified F16XL aircraft sensor fault detection and isolation system (Douglas and Speyer, 1995) can detect pitch angle sensor failure and pitch rate sensor failure.  These faults may be difficult to distinguish from each other and the effect of wind gusts and deflector bias. Pitch angle Elevon deflector wind gusts

Simple Example and Conclusion Analysis of An Aircraft (contd.) Analysis of An Aircraft (contd.)  Suppose there are failures in pitch sensors. The FDI system will raise the alarm immediately despite the existence of disturbances from wind gusts. Longitudinal Dynamics Control System wind gust FDI System Indicator a constant deflector bias

20  Fast response to incipient faults.  An intuitive realization and numerically reliable.  Incorporate detection into a single observer without using banks of observers.  The robustness issue is partially covered by bounding the effect from disturbances to the residual.  Residual signal can be used for post fault handling or fault tolerant control. 19 Simple Example and Conclusion  Conservatism in stability conditions.  Only suboptimal solution achieved at this moment.  Further research needed to handle unstructured modelling errors. Benefits Benefits Limitations Limitations

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