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Lecture 3 Hierarchical, Limited Look ahead Control + Extensions to diagnosis work:  Lecture 3 Hierarchical, Limited Look ahead Control + Extensions to diagnosis work G. Biswas Dept. of EECS and ISIS Vanderbilt University gautam.biswas@vanderbilt.edu http://www.vuse.vanderbilt.edu/~biswas Acknowledge Sherif Abdelwahed, Jian Wu, Eric Manders, Xenofon Koutsoukos, Indranil Roychoudhury, Matt Daigle Supported by DARPA SEC, NASA-IS, NASA-ALS, & NSF-ITR Copyright © Vanderbilt University, 2006. Overview:  Overview Context (History) of the work Lecture 1: Diagnosis of continuous (dynamic) systems Lecture 2: Diagnosis of hybrid systems + simulation test beds Lecture 3: Fault-adaptive control + Incipient fault diagnosis + Distributed Diagnosis Lecture 3:  Lecture 3 Limited Lookahead + Hierarchical Control Distributed Diagnosis Diagnosis of Incipient Faults FACT – Fault Adaptive Control Technology:  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 Model-based Tools and System Development:  Model-based Tools and System Development Run-time Platform (RTOS) Interface & Controllers Hybrid Observer Hybrid Diagnostics Failure Propagation Diagnostics Active Model Controller Selector Reconfiguration Manager Fault Detector Plant Models Visual modeling tool for creating: Hybrid bond-graph models Timed failure propagation graph models Controller models (including reconfiguration) Controller Models Strategy Models Slide6:  Fault-Adaptive Control Architecture Key Contributions:  Key Contributions Common Modeling Framework Graphical, Hierarchical Component-based  Hybrid Bond Graphs: State Space, Temporal Causal Graphs (TCGs), discrete-time models Model-Integrated Computing Meta language  Domain Specific Modeling Languages  Interpreters for model generation & analysis tools + run time components Integrated Monitoring, FDI, and Control Architecture Run-time Environment Data Flow Architecture Qualitative TCG-based analysis + Quantitative Parameter estimation + Utility-based online supervisory control Water Recovery System:  Water Recovery System Reverse Osmosis Model Building Environment :  Model Building Environment  Graphical Component-based Modeling environment (GME)  Modeling environment: compositional tools for generating simulation models, hybrid observers, & diagnosis models  Under each component model – hybrid bond graph model ALS – HBG model:  ALS – HBG model Mechanical Hydraulic Conductivity Feed Pump Recirc Pump Membrane Feedback Loop Purge to AES Conductivity calculations Real & Simulated Data:  Real & Simulated Data Data from JSC Test Bed Data Generated by simulating HBG model FACT: Runtime Environment:  FACT: Runtime Environment Plant HO + SG FD QFI PE ŷ r y u s Active State Model State Space Temporal Causal Graph Residual Evaluation (Qualitative) Quantitative Fault Adaptive Control Residual Generation (H1,..,Hn) Hi=(Pi,mi) (H1,..,Hk) (V1,..,Vk) mode (Hi,Vi) FDI Results Selected Faults:  FDI Results Selected Faults Self-Managing Systems:  Definition Artificial systems that can manage their resources efficiently to achieve their objectives in a dynamic environment and under varying operation requirements Advantages Rapid adaptation to dynamic operating conditions Autonomy Automatic recovery from certain class of failures Application Domain Space exploration systems Manufacturing, Avionics and Automation systems Self-Managing Systems Modeling for Online Control:  Modeling for Online Control System dynamics modeled as switching hybrid system — a class of hybrid systems with finite control set Performance Specification can be in either set point or utility optimization form. Constraints about the system variables and inputs can be part of the performance specification Control objective is to minimize the distance between the current state and the optimal state (the set-point), or maximize the system utility Hierarchical control Resource Management + Performance:  Hierarchical control Resource Management + Performance Controller AES System Controller Global Controller Utility-based Optimize performance Constraint-based Distribution of resources Weekly crew schedule WRS System WRS Controller ARS System AES Controller Global Controller Power Generation Crew Chamber SABATIER CDRA RO AES OGA LC-BWP LC-RO LC-AES LC-CDRA LC-SAB LC-OGA WRS System BWP Crew Scheduler Hierarchical control:  Hierarchical control Utility-based Optimize performance Constraint-based Distribution of resources Set point Control Online Supervisory Control:  Online Supervisory Control Filter estimates future environment parameters using past values Analytical model captures current and future system behavior Optimizer provides the appropriate control actions based on the given utility or set point Predictive filter Physical system Environment Parameters Active Inputs Forecast values State Control inputs Future inputs Predicted state The Limited Lookahead Control Approach:  Use behavioral model to estimate future system states over the prediction horizon t 0 Time Prediction horizon Start state Current state Obtain the sequence of control inputs that optimize desired QoS Apply the first control input in the sequence at time t; discard the rest The Limited Lookahead Control Approach Repeat the process at each time step Limited Lookahead Control Design:  Limited Lookahead Control Design Discrete time model of plant + transitions To choose best action, perform look ahead search up to L steps Define utility function Repeat for next time step – accommodates for faults and disturbances in system Global Controller Design:  Global Controller Design Arrrival Rates to system (given profile based on crew activity and structure) High level model of dynamics Resource constraints Abstract input/output models of subsystem behavior (e.g., power consumed to production of each subsystem) Cost function that has to be optimized – based on set point Experimental Results RO system:  Experimental Results RO system Four high level modes (power consumption, flow rate): 1 – off; 2 – (low, low); 3 – (medium, medium); 4 – (high, high) Slide23:  Performance Evaluation Global Prediction horizon = 1 step Global time step = 1 hour Global controller switches mode for local modules based on expected signal rate Local Prediction horizon = 3 steps Local time step = 1 hour Local controller optimize performance for a given mode Experimental Results Fault situation: block in pipe:  Experimental Results Fault situation: block in pipe 35% increases its resistance occurred at t = 400 sec Isolated at t = 430 sec. Online controller managed to compensate for the fault by increasing the time spent in the primary loop Overall average utility in this case was only 0.93% less than the utility in the non-faulty situation. Experiment results:  Experiment results Slide26:  Technical Results To find B(xs) find (NLP) where Theorem: B(r,xs) is the minimal containable region of xs To determine finite reachability Theorem: B(r,xs)  Q  B(r,xs) is finitely reachable from xRn System Dynamics Single-Mode Discrete-Time One-step online control policy Stability Analysis for Online Control Objective For a domain D and an initial state xsD, decide if there is a neighbor- hood B(r,xs)  D of xs such that: B(r,xs) is finitely reachable from any point in D The system remains in B(xs) under the online control law Q set of all states from which a control action is available to move the system closer to xs xs B(r,xs) Application Projects:  Application Projects Monitoring, Control, and Fault-Adaptation in Three-Tank system Testbed Worked on NASA Challenge problem: 90 day surface Habitat Lander of Lunar South Pole (8 day and 2 night cycles) with crew of four Our focus: Air, Water, Thermal, Crew Chamber, Power Generation and Consumption Deal with flexible crew schedules & EVA activities Three Tank Experimental Testbed:  Three Tank Experimental Testbed NCAP/STIM 1 & 2: Fill, Transfer and Drain valves for Tanks 1 and 2 NCAP/STIM 3: Transfer and Drain valve for Tank 3 NCAP/STIM 4: Pump controller Controller: 3 Parallel FSMs:  Controller: 3 Parallel FSMs Surface Habitat -- Architecture:  Surface Habitat -- Architecture ALS: Data flow + Control :  ALS: Data flow + Control O2 Reg. WRS Controller Global Controller WWT CWT AES RO BWP PPS RO_mode, RO_time AES_mode, AES_time CW_FO_WRS WW_FI_WRS Crew Controller Crew Chamber Crew WW_L_WRS CW_L_WRS day_schedule Estimation module eCW_FI_CRW eWW_FO_CRW eCW_FO_ARS eWW_FI_ARS CW_FI_CRW CRW_state CCH_state week_schedule WRS_mode ARS Controller CDRA SABATIER OGA CW_FI_ARS H2T O2T CO2 WW_FO_CRW WW_FO_AES ARS_mode O2T_L_ARS CO2T_L_ARS H2T_L_ARS HCA_FO_CRW LCA_FO_ARS System Resources Monitor CDRM_mode, CDRM_time OGS_mode, OGS_time PA_FI_CRW O2_FO_ARS H2_FO_ARS water_level O2_level power_level CO2_FI_ARS CO2_FO_ARS H2_FI_ARS Control Goal(s):  Control Goal(s) For appropriate size of buffers maintain cabin O2 and CO2 levels + temperature & provide adequate clean water supply at specified levels to support crew habitat + EVA activities Ensure closed loop operation (minimum waste) of resources while not exceeding power (energy) requirements Ensure enough resources (air + water) to last crew for 90 days Evaluating System Performance 90 Day Mission:  Evaluating System Performance 90 Day Mission Potable water: Initial: 650 liters; End: 200 liters Energy stored: Min: 200 kW-hour; Max: 1300 kW-hour Oxygen tank: Initial = 9.9 kg; Max = 10 kg; Min = 9.9 kg CO2 tank: Initial = 0 kg; Max = 2.6 kg; Min = 1.4 kg Crew Chamber:  Crew Chamber Oxygen level in crew chamber: 28.7% CO2 level in crew chamber: 0.25% Conclusions:  Conclusions Successful in building multi-level controller that operates online for interacting subsystems global system operates system within resource constraints while optimizing performance of subsystems Safety, robustness, and reliability based on dynamic models Tighter bounds on sizing problems at design time Future Work/ Conjectures Formalize controller modeling paradigm Systematic methods for building abstract dynamic models for online control Simulation test-bed for what if analysis Designing Distributed Diagnosers for Complex Physical Systems :  Designing Distributed Diagnosers for Complex Physical Systems Motivation:  Motivation Most engineered systems today are made up of a large number of complex subsystems. Centralized fault diagnosis is computationally very expensive for such large systems. Fault effects eventually propagate to all parts of a continuous system and hence distributed diagnosis is difficult. Related Work:  Related Work Debouk et al (2000): Assumes knowledge of global model. Local diagnosers communicate with a central coordinator. Baroni et al (1999) and Pencolé et al (2005): Do not assume knowledge of global model. Global diagnosis obtained by merging local diagnosis. Diagnoser 1 Diagnoser 3 Diagnoser 2 Coordinator Diagnoser n … R. Debouk, S. Lafortune, D. Teneketzis (2000) Our Approach:  Our Approach Partition the fault set into independent subsets. For each subset build a diagnoser Independence of diagnosers  Local diagnosis  Global diagnosis Independence  No Coordinator required. Two notions of Independence: Strong: No information is shared between diagnosers Weak: Information may be shared between diagnosers Transcend Diagnosis Methodology :  Plant Observer Diagnosis Model Symbol Generation Fault Detection Hypothesis Generation Hypothesis Refinement Transcend Diagnosis Methodology + - u y residual r y ^ Temporal Causal Graph (TCG) One tank System Kalman Filter based on State Space Equations dx/dt = Ax + Bu y = Cx + Du Progressive Monitoring Scheme  Track measurement residuals in time Statistical methods symbol  magnitude and slope of residual signal Bond Graph dx/dt = Ax + Bu y = Cx + Du State Space Equations Temporal Causal Graph (TCG) Transcend: Single, abrupt fault Temporal Causal Graph Backward Propagation along TCG: Generate Fault Hypothesis Forward Propagation along TCG: Generate Fault Signatures Slide41:  Fault Signatures Fault Signature: Qualitatively captures the characteristic of the signal. k+1 feature values computed from the signal residual the magnitude and 1st through kth order derivative +  increasing, -  decreasing, 0  steady Discriminatory Power of Fault Signatures: If there is an abrupt change {+ -}, {- +}, {+ +}, {- -} If no abrupt change {0 +}, {0 -} Slide42:  Complete Diagnosability Given A set of faults F = {f1, … ,fl} A set of measurements M = {m1, … ,mn}, A fault fi  F is diagnosable for M: At least one distinguishing fault signature between fi and all other faults in the system. Set of faults F = {f1, f2, … ,fl} is diagnosable for M: All faults if F are diagnosable for measurements in M. Formally, (i,j  [1,l], i ≠ j) (mk  M) FS(fi,mk) ≠ FS(fj,mk) FS(fi,mk)  fault signature for measurement mk if fault fi occurs. Slide43:  Distributed Diagnosis Partition the fault set F into independent subsets P1, P2, …, Pk Each Pi has a corresponding measurement subset Qi  M All faults f  Pi are globally diagnosable with Qi. Therefore, Local diagnosis result  Global diagnosis. No coordinator needed Independence Strong: do not share measurements, i.e.,  i,j s.t. i ≠ j, Qi ∩ Qj = Weak: allow sharing of measurements, i.e.,  i,j s.t. i ≠ j, Qi ∩ Qj ≠ Slide44:  Fault Independence: Example Strongly Independent Fault Partition: If P1 = {f1}, P2 = {f3} Q1 = {m1} , Q2 = {m2} Weakly Independent Fault Partition If P1 = {f1} , P2 = {f2, f3} Q1 = {m1}, Q2 = {m1, m2} Algorithm for Partitioning Faults Strong Independence:  Algorithm for Partitioning Faults Strong Independence Input: Set of l faults F = {fi| i = 1, 2, …, l} Set of n measurements M = {mi| j = 1, 2, …, n} Fault Signature Matrix for the given faults and measurements Init nodeList = rootNode; goalFoundFlag = 0; h = 0; REPEAT UNTIL goalFoundFlag == 1: REPEAT UNTIL nodeList == EMPTY: node = nodeList.pop() IF node == goalNode: Output node; goalFoundFlag == 1; BREAK; ELSE: Calculate h for node; IF h is maximum h calculated till now: nextNode = node; IF goalFoundFlag == 0: nodeList.push (EXPAND nextNode); h = 0 ELSE: BREAK; max h max h Solution (Goal Node): (Pi, Qi), i = 1,2,…,N pairs (P1 ∪ P2 ∪ … ∪ PN) = F Heuristic: maximize h = |(P1 ∪ P2 ∪ … ∪ PN)| Performance Evaluation:  Performance Evaluation Exhaustive search: Exponential and Optimal Our Algorithm : Polynomial and Sub-optimal Runtime Complexity: O(l2n4) where l = number of faults and n = number of measurements Slide47:  Example: Six Tank System Faults Measurements Pressure Flow rate Slide48:  Six Tank System: Result Strong Independence ({C1},{e1}), ({C2, R12},{e6}), ({C3, R23},{e11, f7}), ({C4, R34},{e16, f12}) ({C5, R45},{e21, f17}) ({C6},{f27}) ({R56},{e26, f22}) Algorithm for Partitioning Faults Weak Independence:  Algorithm for Partitioning Faults Weak Independence M1 = {m1, m2, m3} F1 = {f1, f2} M3 = {m6, m7, m8} F3 = {f6, f7, f8, f9} M2 = {m4, m5} F2 = {f3, f4, f5} M4 = {m9, m10, m11} F4 = {f10, f11, f12, f13} remFaults1 =  remFaults2 = {f4} remFaults3 = {f6, f7} remFaults4 = {f11, f12,f13} d = 1 d ≤ 2 d ≤ 3 M = {m1, m2, m3, m4, m5, m6, m7, m8, m9, m10, m11} F = {f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, f12, f13} Slide50:  Six Tank System: Result Weak Independence ({C1, R12},{e1, f2, e6}), ({C2, R23},{e6, f7, e11}), ({C3, R34},{e11, f12, e16}), ({C4, R45},{e16, f17, e21}) ({C5, R56},{e21, f22, e26}) ({C6},{e26, f27}) Another Example: Reverse Osmosis System:  Another Example: Reverse Osmosis System Strong Independence: ({Ifp, Rfp},{f30}), ({Rmemb, Irp, Rrp},{e37, e16}), ({Cmemb},{e23}) Weak Independence: ({Ifp, Rfp},{f30}), ({Irp, Rrp},{e37, e16}), ({Cmemb, Rmemb},{e23, e16}) f30 e37 e16 e23 Feed Pump Recirculation Pump Membrane Summary/Future Work:  Summary/Future Work Developed a methodology for designing distributed diagnosers for complex continuous systems by partitioning the fault set. Algorithms guarantee Local Diagnosis  Global Diagnosis. No coordinator needed. But need to have global model of system. Partitioning of fault set Strongly independent sets  polynomial in complexity. Weakly independent sets  exponential in complexity. Apply the approach to a large real world system Advance life support system with multiple subsystems Relax the assumption local diagnosis  global diagnosis What if the global model is not available? Incipient Fault Diagnosis A Dynamic Bayes Net Approach:  Incipient Fault Diagnosis A Dynamic Bayes Net Approach Motivation:  Motivation Degradations in system components Modeled as incipient faults i.e. slow drifts in system parameters Dynamic Bayesian Network (DBN) based diagnosis approaches Robust fault diagnosis in the presence of measurement noise and modeling error Suffer from computational intractability Our approach efficiently uses DBNs for fault diagnosis by combining A qualitative fault isolation (QFI) scheme refine the number of fault hypotheses A quantitative DBN based diagnosis scheme Uses a DBN model built using only the small number of fault hypotheses that remain after the QFI scheme instead of the entire candidate space Quantitatively isolates the true fault if multiple hypotheses remain after QFI step Estimates the rate of change of the true fault parameter Methodology:  Methodology Fault profile: Detection of Incipient Faults Nominal DBN is used as an observer Expected behavior is compared with actual measurements Z-test for difference of means for robust fault detection Qualitative Fault Isolation (QFI) Uses symbolic deviations of measurement and qualitative fault signatures Quantitative Fault Isolation and Identification (FII) using DBNs The QFI refines the fault candidates to a small number This makes the DBN based quantitative FII efficient Time of fault detection System Architecture:  System Architecture Modeling – DBN Observer:  Modeling – DBN Observer Estimates nominal system behavior DBN Observer Components Regular Bayesian network Captures variable dependencies at any given time slice Two slice temporal Bayesian network First order Markov assumption Captures across time relations defined by the state equation model of the dynamic system Modeling – DBN Diagnoser:  Modeling – DBN Diagnoser Tracks faulty system behavior DBN Diagnoser components Nodes present in DBN observer Continuous nodes Parameters corresponding to incipient fault hypotheses Discrete nodes One-to-one correspondence with fault parameters Indicate the absence/presence of a fault for that parameter Logical variables: 1  linked parameter has a fault 0  linked parameter has no faults Modeling – DBN Diagnoser:  Modeling – DBN Diagnoser DBN based diagnosis scheme by [Lerner et al, 2000] Includes all possible faults in the system Number of possible faults can be really large in complex systems Has computational complexity issues Retains few of the most probably hypotheses to tackle intractability May result in dropping of the actual fault when its probability is low early on In our work set of possible candidates are reduced using QFI scheme This reduces the size of the DBN diagnoser for the quantitative FII All active fault candidates are retained Considerable improvement in the efficiency of diagnosis Diagnosis of Incipient Faults Tracking Nominal Behavior Using a DBN:  Diagnosis of Incipient Faults Tracking Nominal Behavior Using a DBN Uses the DBN observer Set of nodes and their distribution provide snapshot of system state Yt: Measured variables at time step t Xt: State variables at time step t Tracking problem: First order Markov assumption for state space model P(Xt|X0:t-1) = P(Xt|Xt-1) Evidence model P(Yt|X0:t,Y0:t-1) = P(Yt|Xt) Combining the above equations Diagnosis of Incipient Faults - Qualitative Fault Isolation:  Diagnosis of Incipient Faults - Qualitative Fault Isolation QFI is run for at most s steps Single fault – DBN diagnoser used for fault identification Multiple faults – DBN diagnoser used for fault isolation and identification Once QFI is terminated, quantitative FII is initiated Refined hypothesis, therefore DBN diagnoser more efficient s must be carefully chosen If s is large, delay in isolation and identification If s is too small, few candidates will be dropped and ensuing DBN diagnoser will be not be efficient Diagnosis of Incipient Faults Quantitative Fault Isolation and Identification:  Diagnosis of Incipient Faults Quantitative Fault Isolation and Identification Uses the DBN diagnoser DBN observer Continuous nodes denoting remaining fault hypothesis after QFI Discrete valued nodes representing absence/presence of fault Initialized to system state at td: time of fault detection All observations have been cached from td onwards DBN diagnoser unfolded using the same procedure used for unfolding DBN observer Adjust weights and parameters of the multivariate Gaussian distributions as each hypothesis is conditioned on the new measurements Yt+1 Diagnosis of Incipient Faults Quantitative Fault Isolation and Identification:  Diagnosis of Incipient Faults Quantitative Fault Isolation and Identification As the estimates are conditioned on more evidences The mean of true fault parameter changes gradually The mean of the non-faulty parameters do not change Variances of each distribution decreases Observations estimated with true fault hypothesis converge to observed measurements Observations estimated with incorrect fault hypothesis do not converge to observed measurements Diagnosis of Incipient Faults Quantitative Fault Isolation and Identification:  Diagnosis of Incipient Faults Quantitative Fault Isolation and Identification Fault Isolation Z-test applied to the measured flow estimates for each hypothesis If Z-test determines a deviation in the measurement residuals for that hypothesis, that particular hypothesis is dropped If no deviation is detected, that is the true fault hypothesis Fault Identification The rate of change of the true fault parameter is estimated by observing the sequence of means at previous time steps Results assumption everything is gaussian:  Results assumption everything is gaussian Experimental Set: Two tank System Fault Signatures DBN update equations Results:  Results Results:  Results Future Work:  Future Work Further investigate the observability of the DBN diagnoser and its impact on diagnosis Pressure in tank 2 and outflow from tank 1 can uniquely isolate the fault hypotheses However, for quantitative FII, we need all three flow measurements An interesting research issue is the problem of identifying the correct set of measurements so that system is diagnosable as well as the DBN is observable Relax the assumption that the parameters of the DBN are Gaussian distributions Adapt this approach for the diagnosis of multiple incipient faults Extend this Bayesian approach to the diagnosis of both incipient and abrupt faults. Future Work:  Future Work Consider hybrid system made up of interacting distributed subsystems Physical subsystems coupled through a backbone Each unit includes ECDs that implement the control, monitoring, and fault diagnosis tasks Subsystem interactions at two levels: physical – energy-based logical – information based, facilitated by LANs Levels are not independent Question: How does one systematically model the interactions between the subsystems efficiently while avoiding the computational complexity of generating global hybrid models? Implications: Reachability Analysis, design, control, and fault diagnosis References:  References G. Biswas, E.J. Manders, J.W. Ramirez, N. Mahadevan, and S. Abdelwahed, “Online Model-Based Diagnosis to Support Autonomous Operation of an Advanced Life Support System,” Habitation: International Journal of Human Support Research, vol. 10, no. 1, pp. 21-38, 2004. S. Abdelwahed, J. Wu, G. Biswas, J.W. Ramirez, and E.J. Manders, “Online Hierarchical Fault-Adaptive Control for Advanced Life-Support Systems,” Proc. 34th Annual Meeting of Intl. Conf. on Environmental Systems (ICES), paper number 2004-01-2441, Colorado Springs, CO, July 2004. S. Abdelwahed, J. Wu, G. Biswas, J. Ramirez, and E.J. Manders, “Online Fault-Adaptive Control for Efficient Resource Management in Advanced Life Support Systems,” Habitation: International Journal of Human Support Research, vol. 10, no. 2, pp. 105-115, 2005. I. Roychoudhury, G. Biswas, X. Koutsoukos, and S. Abdelwahed, "Designing Distributed Diagnosers for Complex Physical Systems". 16th International Workshop on Principles of Diagnosis, Monterey, CA, pp. 31-36, June 2005. S. Abdelwahed, J. Wu, G. Biswas, and E. J.-Manders,” Hierarchical Online Control Design for Autonomous Resource Management in Advanced Life Support Systems,’’ International Conference on Environmental Systems, Paper no. 2005-01-2965, Rome, Italy, July 2005. G. Biswas, P. Bonasso, S. Abdelwahed, E.J. Manders, J. Wu, D. Kortenkamp, and S. Bell, “Requirements for an Autonomous Control Architecture for Advanced Life Support Systems,’’ International Conference on Environmental Systems, Paper no.2005-01-3010, Rome, Italy, July 2005. I. Roychoudhury, G. Biswas, and X. Koutsoukos, “A Bayesian Approach to Efficient Diagnosis of Incipient Faults,” 17th International Workshop on Principles of Diagnosis, Penaranda de Duero, Spain, June 2006. Thank you very much!!!:  Thank you very much!!! Look forward to further interactions

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