Lecture 1 Model-based Diagnosis of Continuous Systems Gautam Biswas Dept. of EECS and ISIS Vanderbilt University gautam.biswas@vanderbilt.edu http://www.vuse.vanderbilt.edu/~biswas Acknowledge Pieter Mosterman, Sriram Narasimhan, Eric Manders Supported by DARPA SEC, NASA-IS, NASA-ALS, & NSF-ITR Copyright © Vanderbilt University, 2006.

Overview Context (History) of the work Initially extend qualitative consistency-based diagnosis to diagnosis of continuous (dynamic) systems (Lecture 1) Extend to diagnosis of hybrid systems to accommodate more real-world applications (physical processes with supervisory (discrete) control) (Lecture 2) Online diagnosis? What’s the use – extend to fault-adaptive control (Lecture 3) Look at the bigger picture – fault-adaptive control, different fault profiles, prognosis, maintenance, safety, etc. … (Lecture 3) Some related problems: distributed diagnosis, multi-level diagnosis, …. 9/17/2018

The need for fault-adaptivity In complex systems even simple failures lead to complex cascades of events that are difficult to understand and manage. How to detect and isolate faults? react to faults to mitigate their effect? 9/17/2018

Overview FACT – Fault Adaptive Control Technology Goal: systematic model-based approaches to maintain system operations under degraded and failure conditions Expanded goals: reliable, safe, autonomous operation for long-duration missions Approach: Develop the technology and required tool-suite using Model-Integrated Computing approach to achieve this Components: Modeling Approaches – hybrid dynamic processes of the plant + reconfigurable controllers Online monitoring of system behavior Online fault detection, isolation, and identification Adaptive models – update plant model after failure Model-predictive fault-adaptive control 9/17/2018

Model-based Tools and System Development Visual modeling tool for creating: Hybrid bond-graph models Timed failure propagation graph models Controller models (including reconfiguration) Controller Models Strategy Models Hybrid Diagnostics Active Model Modular run-time environment contains: Hybrid observer and fault detectors Hybrid and Discrete diagnostics modules Controller and reconfiguration strategy model library Controller selector and reconfiguration manager Active controller modules Modules can be used standalone Can use an RTOS as the platform Failure Propagation Controller Diagnostics Selector Fault Detector Plant Models Hybrid Observer Interface & Controllers Reconfiguration Manager Run-time Platform (RTOS) 9/17/2018

FACT Components Modeling environment: Plant + controllers hierarchical, multi-aspect Three views HBG view Plant I/O view: Sensors + Actuators Controller view (finite state machine, extend to MPC) parameterize faults TFPG view: specialized discrete-event diagnosis approach with time intervals Simulation environment: simulates physical system behavior including fault scenarios Run-time environment: implements fault detection, isolation, identification, and fault adaptive control algorithms 9/17/2018

Lecture 1 Model-based Diagnosis of Continuous Systems Modeling of Continuous, Dynamic Systems Fault Detection and Isolation (FDI) system architecture Components of FDI system Observer Fault Detector Fault Isolation Summary and Conclusions

Model-Based Diagnosis of Dynamic Systems FDI Models: examples Continuous Discrete Quantitative State Estimation Parameter Estimation GDE (static) Qualitative Fault Signatures & TCGs (Mosterman and Biswas) Sampath, et al. Lunze, et al. Values: Qualitative vs. Quantitative computational qualitative models do not require numerical parameter values but diagnosis is less precise run time complexity of qualitative methods may be less Temporal Behavior: Discrete vs. Continuous discrete methods may be computationally simpler at run time but are less precise, coarser complete analysis of faulty behavior up front or spurious results possible 9/17/2018

Models Why do we need models? Suppose, we were doing diagnosis of system in steady-state Nominal behavior of system known, don’t need model to track system behavior Once deviations recorded, use model of system to analyze cause for deviations If system operating in dynamic regions Need model to track behavior of system – to determine when system behavior deviates Use model to analyze cause for deviations 9/17/2018

Model-based FDI of Continuous Systems (TRANSCEND) Plant (Process) Extended Kalman filter Statistical Hypothesis Testing + r r Fault Detector Observer  ŷ Symbol Generator Observer Model rs m Hypothesis Generation fh, p Hypothesis Refinement fr Diagnosis Model State Space Eqns Temporal Causal Graph (TCG) Qualitative 9/17/2018

Bond Graphs Overview 9/17/2018

Modeling Approach Bond Graphs What are Bond Graphs? Topological, lumped parameter domain-independent modeling scheme for physical systems Based on concept of reticulation Properties of system lumped into processes with distinct parameter values Dynamic System Behavior: function of energy exchange between components State of physical system – defined by distribution of energy at any particular time Dynamic Behavior: Current State + Energy exchange mechanisms (Ref: physical systems dynamics – Rosenberg and Karnopp, 1983, 2003) 9/17/2018

Example of Reticulation Suspension system of automobile Input : Velocity at bottom of tire (function of road conditions) V g K B v k V0(t) M m Mass of car Mass of tire Tire stiffness Suspension system parameters 9/17/2018

Bond Graphs Modeling language Domain independent Compact representation: Generic components: Energy storage (C, I), dissipative (R), source (Se, Sf), Transformers (TF, GY) Idealized junctions to allow connections among components: (1- series, 0 – parallel) Energy exchange between components through mechanisms called bonds Dynamic system behavior governed by conservation of energy and continuity of power A B e f Two associated variables with bond: e: effort; f : flow; such that e  f = power 9/17/2018

Modeling with Bond Graphs Modeling language (based on small number of primitives) Dissipative elements: R Energy storage elements: C, I Source elements: Se, Sf Junctions: 0, 1 Transformer Elements: TF, GY physical system mechanisms R C, I Se, Sf 0,1 uniform network–like representation domain independent forces the modeler to make assumptions explicit 9/17/2018

Building Bond Graphs: Examples Impacting Trains Tank2 C2 R12 Tank1 C1 R2 R1 Sf1 Two tank system Lumped parameter Topological Modeling C: C1 R: R12 C: C2 Sf: Sf1 1 Compact Representation across domains R: R1 R: R2 Two Tank System 9/17/2018

Bond Graphs Modeling non linear behaviors For linear systems, BG elements are constant values Non linear systems parameter value = f (effort, flow, external variables) Sf: Sf1 R: R1 1 R: R12 C: C2 R: R2 Tank2 C2 R12 Tank1 C1 R2 R1 Sf1 Two tank system p1 p2 9/17/2018

Bond Graphs Building Models for Diagnosis 9/17/2018

TRANSCEND Architecture y Plant (Process) Extended Kalman filter Statistical Hypothesis Testing + r r Fault Detector Observer  ŷ Symbol Generator Observer Model rs m Hypothesis Generation fh, p Hypothesis Refinement fr Diagnosis Model State Space Eqns Temporal Causal Graph (TCG) Qualitative 9/17/2018

Bond Graphs Different Model Forms Deriving different model forms State Space equations for tracking dynamic behavior Causal models for diagnostic analysis Central to this Causality Information that can be derived from the Topological Bond Graph Model 9/17/2018

Causality in Bond Graphs To aid equation generation, use causality relations among variables Bond graph looks upon system variables as interacting variable pairs Cause effect relation: effort pushes, response is a flow Indicated by causal stroke on a bond A B e f 9/17/2018

Causality for basic multiports Note that a lot of the causal considerations are based on algebraic relations 9/17/2018

Causality Assignment Procedure 9/17/2018

Causality Assignment: Example 9/17/2018

Causality Assignment: Example 2 Tank2 C2 R12 Tank1 C1 R2 R1 Sf1 Sf: Sf1 C: C1 R: R1 1 R: R12 C: C2 R: R2 Two Tank System Causality assignment important – for building TemporalCausal Graphs (TCGs) that are used for diagnosis from transients 9/17/2018

Generating State Space Models Step 1: Augment bond graph by adding Numbers to bonds Reference power direction to each bond A causal sense to each e,f variable of bond 9/17/2018

Generating State Space Models Step 2: Identify state and input variables 9/17/2018

Model-based Diagnosis TRANSCEND approach Fault Detection + Fault Isolation (Qualitative)

TRANSCEND architecture y Plant (Process) Extended Kalman filter Statistical Hypothesis Testing + r r Fault Detector Observer  ŷ Symbol Generator Observer Model rs m Hypothesis Generation fh, p Hypothesis Refinement fr Diagnosis Model State Space Eqns Temporal Causal Graph (TCG) Qualitative 9/17/2018

Qualitative Approach to Fault Isolation Why Qualitative ? Accuracy of models: structural + difficulty in estimating parameters Imprecision of real world numeric models computational issues, e.g., convergence problems Qualitative Constraints magnitude deviations + higher order derivatives. (currently +, 0, -) Topological Models Graph-based, compositional 9/17/2018

Fault Characterization Model parameters in TCG correspond to system components. Fault – model parameter that is deviated from its normal operating value Abrupt Fault – instantaneous and persistent parameter value change (modeled as a step change). Fault Characterization Additive: sensor and actuator faults Multiplicative: component parameter faults multiplicative faults directly affect dynamic response of system, therefore, much harder to analyze residuals 9/17/2018

Example Two-Tank System Rb1 Tank2 C2 R12 Tank1 C1 Rb2 f1 f3 f8 f5 e2 e7 Focus on process faults multiplicative Parameterized Model Possible faults: Bond Graph component parameters: Sf, C1, C2, Rb1, Rb2, R12 Fault profile abrupt change in parameter value: increase or decrease Note – partial change not complete failure Bond Graph Model 9/17/2018

Diagnosis from Transients Abrupt Faults cause transients in observed measurements. Goal: Isolate fault as quickly as possible after occurrence of transient. Two primary tasks: Reliable detection of transient Isolation of fault based on transient characteristics C1 C2 Rb1 R12 Rb2 f5 (flowrate through connecting pipe): Faults: Rb1, Rb2, R12 Faults: C1, C2 Discontinuity 9/17/2018

Two-Tank System Diagnosis Model Rb1 Tank2 C2 R12 Tank1 C1 Rb2 f1 f3 f8 f5 e2 e7 Bond Graph Model Important Characteristics: Algebraic and temporal – cause effect relations Note the state loops – there are three in this TCG Temporal Causal Graph 9/17/2018

Transient Analysis Our approach is to analyze measurements individually. Transient Response of residual of a signal (can be approximated by Taylor series of order k) r(t) = r(t0) + r'(t0)(t- t0)/ 1! + r''(t0)(t- t0)2/ 2! + …… + r(k)(t0)(t- t0)k/ k! + Rk(t), where Rk(t) is the remainder term based on y(k+1)(t). Signal transient due to a fault at t0 can be expressed as discontinuous magnitude change, r(t0), plus first and higher order derivative changes, r'(t0), r''(t0), ….., r(k)(t0). 9/17/2018

Qualitative Transient Analysis As order of derivative increases accuracy of match improves Matching the measured magnitude and slope of a signal to the signature Signature: <+ - + -> step Measured signal + . . . + - + - 1 + - . . + - + - 2 - - . . + - + - - - + - 3 - - . . - - + - 4 - ? . . - - + - 5 - + . . - - + - - + - . Signature Transient Signal from 2nd order system (1st to 4th order Taylor’s series expansion shown as dashed lines) Tracking a signal when only the magnitude change and slope of signal are measured 9/17/2018

Fault Signature Signal feature vector in response to a fault expressed as a sequence of qualitative derivative values at the point of failure. Qualitative Fault Signature of order k: fault signature that includes derivatives up to order k. Assumptions – (I) Abrupt faults, (ii) The sampling rate of the measured signals is set to be fast enough so that no qualitative change in the transient dynamics is missed. . The Diagnosis Process 1. Generate Fault Hypotheses – Backward Propagation 2. Predict Behaviors – Forward Propagation (Individual Fault Signatures) 3. Monitoring - compare signatures to observations: Progressive Monitoring & Discontinuity Detection 9/17/2018

Analysis of Fault Transients Analyze behavior immediately after fault onset Time point of failure occurrence important Detection + estimation problem Transient Dynamics Captured as fault signatures: Magnitude First and higher order derivatives Progressive Monitoring Higher order derivatives progressively affect magnitude and slope of signal For details of algorithms, see Mosterman & Biswas (1999) 9/17/2018

Run-time System Fault Detection If there is no fault, then the Kalman filter will track – Residual: Gaussian white with zero mean To detect a fault, the generalized likelihood ratio test (LR) is used for hypothesis testing  n Faults System + + + residual Model  Kalman Filter  Residuals deviate from zero because of Noise (n) Separation of effects necessary! Modeling errors () X Sensor inaccuracies () Faults ! 9/17/2018

Fault Detection and Symbol Generation Approach: model based Residuals: difference between ‘ideal’ and measured behavior Fault detection: residual evaluation Fault detection says “yes” if ‘significant’ deviation is detected Generated symbols: Sign of deviation Sign of first derivative r + – t 9/17/2018

Fault Detection Assumptions (model based again…) Noise is white, Gaussian N(0, ) Variance is constant (slowly changing) Modeling errors and sensor inaccuracies are treated as noise ‘Significant’ residual deviation  Fault Significance is determined by a statistical hypothesis test (approximate Z-test): Mean value and variance used 9/17/2018

Z-test Fault detection Z-test Mapping the confidence level (1-) to z: time Variance estimation Mean estimation VarDelay k N2 N1 Mean: info on the transient residual: Variance: info on the nominal dynamics Typical values: N1=50; N2=5 VarDelay=150 Fault detection Z-test Z-value distn.: Decision making: where N2 is the number of samples and  is the standard deviation. The  is also estimated, but from a larger set of samples (N1 >> N2) Z /2 z- z+ Mapping the confidence level (1-) to z: 9/17/2018

Symbol Generation Decision: 1st symbol: sign of deviation after fault detection  2nd symbol: sign of slope after fault detection Is difference significant? N3 r Estimation of the ‘initial value’ Estimation of the ‘later value’ Fault detection t k Additional Decision If 1 & 2 have opposite signs, then change is discontinuous Decision: 9/17/2018

Slope Symbol Generation Example 20 40 60 80 100 120 140 160 -0.5 0.5 N3=1 N3=10 N3=20 m d +z + s -z r /sqrt(N 3 ) detection time 20 40 60 80 100 120 140 160 0.5 1 1.5 Residual Choice of N3 Small value  large threshold, thus long detection time Large value  significant delay, may suppress short transients Typical values of N3 are 5…20 Better solution: adaptive settings … 9/17/2018

Detection, Symbol Generation Complete Onto Qualitative Fault Isolation

Generate Fault Hypotheses – Back Propagation Rb1 Tank2 C2 R12 Tank1 C1 Rb2 f1 f3 f8 f5 e2 e7 9/17/2018

Prediction by Forward Propagation Fault Signatures Qualitative Signature: magnitude + first and higher order derivative changes expressed as +,0,- values. How to generate signatures from TCG ? Temporal links imply integrating edges, affects derivative of variable on the effect side Start with 0-order changes Every integrating edge increases order by one Rb2+ 2nd Order signature of e7: < 0,+,- > 9/17/2018

Progressive Monitoring track system behavior after failure 1 2 3 5 4 6 1 2 3 4 5 6 System Behavior Convolutes the Predicted Transient at Time of Failure dynamically change the signature: Justified by Taylor’s series measured k 1 + 2 3 + + 4 + 0 5 + - 6 + - - k signature1 k signature2 0 + - + - + 1 + + - 2 + + - no match! 3 + + - match! 9/17/2018

Three Tank Results 9/17/2018

Analysis of Qualitative Fault Signatures Discriminatory Power of Qualitative Fault Signatures Abrupt change – direction of abrupt change + direction of change immediately following abrupt change, (+,+), (+,-), (-,+), and (-,-) No abrupt change – first direction of change of the signal only, (0,+), and (0,-) Problem: further +,- changes provide no discriminatory evidence because qualitative information contains no time constant information. Ways to handle this problem: measurement selection – end up needing many more measurements than log4[k], where k – number of fault hypotheses. estimate fault parameters values; true fault is the one whose parameter is consistent across multiple measurements. Use gradient descent methods for parameter estimation. 9/17/2018

Summary Modeling for Diagnosis Lumped Parameter Modeling of Dynamic Systems Topological Models have their advantages: can derive causal models that facilitate diagnosis FDI of continuous systems by qualitative analysis of transients Fault detection and Fault Isolation algorithms Advantages and limitations of qualitative analysis 9/17/2018

References Dean C. Karnopp, Donald L. Margolis, and Ronald C. Rosenberg, System Dynamics: Modeling and Simulation of Mechatronic Systems, 4th Edition , John Wiley, New York, NY. P.J. Mosterman and G. Biswas, “Diagnosis of Continuous Valued Systems in Transient Operating Regions,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 29, no. 6, pp. 554-565, Nov. 1999. E.J. Manders, P.J. Mosterman, and G. Biswas, “Signal to Symbol Transformation Techniques for Robust Diagnosis in TRANSCEND,” Tenth Intl. Workshop on Principles of Diagnosis, Loch Awe, Scotland, pp. 155-165, June 1999. E. J. Manders, S. Narasimhan, G. Biswas, and P. J. Mosterman, ``A Combined Qualitative/Quantitative Approach for Fault Isolation in Continuous Dynamic Systems,’’ 4th Symposium on Fault Detection, Supervision and Safety for Technical Processes (Safeprocess 2000), Budapest, Hungary, pp. 512-517, June 2000. 9/17/2018

Next Lecture 2 Hybrid Modeling and Diagnosis of Hybrid Systems Combining Qualitative FDI with Quantitative parameter estimation methods for more informed and more refined diagnosis Example Applications Generation of Toolsuite Simulation experiments Run time system 9/17/2018