Model-Based Diagnosis of Hybrid Systems Papers by: Sriram Narasimhan and Gautam Biswas Presented by: John Ramirez

Introduction  Modern systems are complex, and include supervisory control that switches modes of behavior.  The controller is a software program and is not tightly meshed with the continuous plant dynamics. Plant Supervisory controller ActuatorsSensors Sensor valuesDiscrete Signals

Introduction  The continuous dynamics of the plant are defined by differential and algebraic equations. q(t) is the discrete model

Fault Detection and Isolation (FDI) The goal of this presentation is to briefly overview the study of FDI in hybrid systems with supervisory controllers. System faults may be component, actuator, sensor, and controller faults. (We do not deal with the later) The methodology we will cover combines qualitative and quantitative reasoning techniques to perform parameterized fault isolation of plant component faults.

Modeling for Diagnosis  Controller Model  Plant Model

Modeling for Diagnosis  Controller Model The primary model of the controller is implemented as a finite state machine (FSM). States of the FSM correspond to the states of the controller, which in turn define modes of the physical plant(q(t)). The Transitions determine the conditions for switching states.

Modeling for Diagnosis Controller Model t1 t3 t2 t4 t5 t6 t7 t8 t9 t10 t11 Controller Model for 3 tank system Tank 2 (C2) Tank 2 (C2) Tank 1 (C1) = Valve C = capacitance R = resistance Flow source 1Flow source 2 R1 R6 R2R4 R3R5 Three Tank system

Modeling for Diagnosis  Plant Model Hybrid Bond Graph Models (HBG). State equations and temporal causal graph (TCG) can be systematically derived from the bond graph representation of the system. State equations along with the TCG constitute our diagnosis models.

Methodology for Hybrid Diagnosis  Hybrid observer: follows the continuous dynamics of the plant and identifies discrete mode changes.  Fault detection mechanism: signals a fault when the observer cannot compensate for differences between observed and expected behavior.  Fault isolation mechanism: generates candidate faults and refines them with the hybrid model and measurement from the system.

Methodology for Hybrid Diagnosis Hybrid models Diagnosis models System Observer and mode detector Fault isolation Fault detection u y r Fault Hypotheses The following information is assumed to be available to all modules: -HBG -FSA -FSM -  A = all possible autonomous events in the system -U = inputs -Y = system outputs -Parameters nominal Diagnosis System Architecture

Methodology for Hybrid Diagnosis Algorithm 1:Diagnosis Module MODULE DIAGNOSE(M initial, X initial ) // Observe the system until a fault is detected =OBSERVER(M initial,X i nitial ); //Convert the quantitative residuals to qualitative values QualResidual current = SIGNAL_TO_SYMBOL(Y,Y estimated ); //Back propagate across modes to identify fault candidates BackHorizon=2; List candidates =HYBRID_BACK_PROP(Stack M,QualResidual current,BackHorizon); //Forward propagete across modes to isolate the fault List candidates =HYBRID_FAULT_OBSERVER(List candidates,Y estimated ); END DIAGNOSE

Hybrid Diagnosis Problem Time Line Mode 1 Mode 2Mode 3 Mode 4 Mode 5 Fault Occurs Fault Detected Tracked Trajectory Actual Trajectory T1 T2T3T4 T5 T6 Mode 6 Mode 7 Fault Hypothesis: Piecewise linear hybrid dynamical systems Presence of fault invalidates tracked mode trajectory Hypothesized fault mode Known Controlled Transition Hypothesized Autonomous Transition Possible current modes Hypothesized intermediate modes Roll Back to find fault hypotheses Roll Forward to confirm fault hypotheses Catch up to current system mode to verify hypotheses against measurements Note: Controller transitions known Autonomous transitions have to be hypothesized

Fault Isolation Background  The type of plant model employed determines the scheme to be employed.  Traditional schemes for the continuous domain use structured and directional residual approaches.  Extending these continuous methodologies to hybrid systems becomes intractable.

Fault Isolation  The approach we will follow involves hypotheses generation and hypotheses refinement.  Qualitative approach for hypotheses generation.  Qualitative-quantitative combined approach for hypotheses refinement.

Fault Isolation Hypotheses Generation  For initial hypotheses generation we have to back propagate across modes. The assumption that the controller model is correct implies that the observer predicted the correct mode sequence till the fault occurred. Therefore, the mode in which the fault occurred must be in the predicted trajectory of the observer.

Hypotheses Generation TCG generation Effort and flow variables are vertices Relation between variables as directed edges =implies that two variables associated with the edge take on equal values, 1 implies direct proportionality,-1 implies inverse proportionality. Edge associated with component represents the component’s constituent relation.

Hypotheses Generation Algorithm 2:Hybrid Back Propagation MODULE HYBRID_BACK_PROP(Stack M, QualR i, BackHorizon) //Generate candidates in each mode in the mode trajectory. =Pop(StackM); TCG current =GET_TCG(HBG, M current ) //Back propagate in selected mode for candidates in the mode F current =CONTINUOUS_BACK_PROP(TCG current,QualRi); Add(List candidates, ); Count=0; //Go back in the mode horizon upto BackHorizon number of nodes While(Count<BackHorizon) //Select next mode in mode trajectory and calculate TCG =Pop(StackM); TCG next, GET_TCG(HBG, M next ); // Propagate qualitative deviations across modes QualR next =BACK_PROP_ACROSS_MODES(M current, M next, QualRi) //Back propagate in selected mode for candidates in the mode F next =CONTINUOUS_BACK_PROP(TCG next, QualR next ); Add(List candidates, ); End While Return(List candidates ) END MODULE

Roll Back Process Qualitative Hypotheses Generation Back propagate through TCG in current mode to identify candidates Back propagate across mode transitions using transition conditions (need to account for reset conditions, and change in plant configuration – invert qualitatively) Repeat same process for previous modes to identify more candidates - Tank 1 Pressure - Tank 2 Pressure - Tank 3 Pressure Transition Fault Occurred Fault Detected System Autonomous Transition

Fault Isolation Hypotheses Refinement  First apply a qualitative forward propagation for each hypothesized fault candidate. To take into account mode changes, all possible modes changes from the current mode are hypothesized. A candidate is dropped when the predictions do not match the observations across all of the hypothesized modes  Apply a quantitative parameter estimation on remaining candidates. This approach works within a single continuous mode.

Hybrid Diagnosis Problem Time Line Mode 1 Mode 2Mode 3 Mode 4 Mode 5 Fault Occurs Fault Detected Tracked Trajectory Actual Trajectory T1 T2T3T4 T5 T6 Mode 6 Mode 7 Fault Hypothesis: Piecewise linear hybrid dynamical systems Presence of fault invalidates tracked mode trajectory Hypothesized fault mode Known Controlled Transition Hypothesized Autonomous Transition Possible current modes Hypothesized intermediate modes Roll Back to find fault hypotheses Roll Forward to confirm fault hypotheses Catch up to current system mode to verify hypotheses against measurements Note: Controller transitions known Autonomous transitions have to be hypothesized

Quick Roll Forward Goal: Get to current mode, so parameter estimation can be applied to refine faults and identify fault magnitude Lemma 2: Sequence of k mode transitions in any order drives the system to the same final model progressive monitoring Requires tracking of transients by progressive monitoring in continuous regions of space. Taylor series expansion defines qualitative fault signatures. Residual r(t) after fault can be described as: Progressive Monitoring: Match qualitative magnitude and slope of measurement signal transient against fault signature Fault signature: qualitative form of derivatives: Qualitative form of

Quick Roll Forward identify the current mode (roll forward)In continuous case, mismatch implies fault hypothesis is not consistent. However, in hybrid tracking, it may imply that we are not in the right mode. We need to identify the current mode (roll forward) All controlled transitions are known, but we have to hypothesize autonomous transitions since observer can no longer predict them correctly Use fault signatures to hypothesize mode transitions - Tank 1 Pressure - Tank 2 Pressure - Tank 3 Pressure Transition Fault Occurred Fault Detected System Autonomous Transition

Parameter Estimation (Real Time)  Derive transfer function model in current mode with only one unknown (fault parameter)  Initiate fault observer filter for each fault hypothesis least squares estimator for parameter estimation  Test for convergence identifies true fault candidate

Least Square Estimation from IOE

Parameter Estimation Example Plot of prediction error

Quantitative Parameter Estimation: Issues Deriving the simplified one unknown parameter equation for least square estimator Convergence to local minima – need good initial estimates Need for persistent excitation in input – mitigated to some extent by reducing it to a one parameter estimation problem Measurement noise leads to biased estimates – need to apply more sophisticated techniques: IVM methods Observation: What is good for qualitative FDI is not always good for quantitative identification using least squares methods

Summary  Model for Diagnosis Controller Model FSM Plant Model HBG  Fault Isolation Hypotheses Generation TCG Hypotheses Refinement Parameter Estimation

Conclusion  By having the supervisory controller model and assuming that our model is correct, we do not have to make the assumption that faults are detected in the mode in which they occur, and we still are able to avoid the intractability problem.  Combination of qualitative + quantitative approaches suitable for online diagnosis  Approach different from discrete-event approaches of Lunze and Sampath