April 6, 2005Massachusetts Institute of Technology Hybrid Estimationfor Fault Detection and MoreLars Blackmore2Overview• Background– Kalman Filter review– Fault detection using Kalman Filters(Multiple Model method)• Fault detection (and more) using Hybrid Estimation– Why Hybrid?– Modeling– Technical challenges– Current state of the art60s-80s90s - present3Fault Detection4Kalman Filter Review•Problem: – Given a continuous dynamic system model:– And a set of noisy observations:– Estimate the ‘hidden’ state of the system)(),()(),,(111tgttfttttttωυ+=+=+++uxyuxxθVxVy5Kalman Filter Review• Solution: Kalman Filters–Calculate belief state about hidden variables– Approximate as Gaussian– Predict/update cycle:1. Start with belief state at t-12. Predict belief state at t using system model3. Use measurement at t to adjust the belief state),|(:1:1 tttp uyx),ˆ(~),|(:1:1 tttttNp Pxuyx6Kalman Filter Review• Details– KF equations:– Linear systems:• State distribution is Gaussian, KF is exact– Nonlinear systems:• Gaussian assumption is an approximation• Extended Kalman Filter accurate to 1storder• Unscented Kalman Filter accurate to 2ndorderTtttTtttttttfWQWAPAPuxx1111),ˆ(ˆ−−−−−−+==Prediction step()TttttttttttgAPHKIPxyKxx−−−−=−+=)()ˆ(ˆˆMeasurement update step7Overview• Background– Kalman Filter review– Fault detection using Kalman Filters(Multiple Model method)• Fault detection (and more) using Hybrid Estimation– Why Hybrid?– Modeling– Technical challenges– Current state of the art60s-80s90s - present8Multiple Model Fault Detection• Fault detection:– Is the system operating nominally or is it faulty?• Assume models known for both cases– Which model most likely given observations and inputs?• How can we use Kalman Filters here?9Multiple Model Fault Detection• KF predicts distribution of observation given inputs at each time step• KF ‘innovation’ is discrepancy between expectation and actual observation• Can use this to determine agreement between model and observationsinnovation)ˆ(−tg xty10Multiple Model Fault Detection• Idea:– Use a Kalman Filter for each model– Small innovations Æ model and observations agree– Large innovations Æ model and observations disagree– So compare innovations from faulty and nominal KF– If innovations smaller for faulty KF, diagnose a fault11Multiple Model Fault Detection• More formally– Let each model be denoted by Hi(e.g. H0=nominal, H1=faulty)– Assume some belief about each model at time t-1:– We want posterior probability at time t:– Use Bayes’ Rule:– We can calculate )|()(:1 tiiHptp y=∑=−−−−=njjtjtitititpHptpHptp11:11:1)1(),|()1(),|()(yyyy)|()1(1:1 −=−tiiHptp y),|(1:1 −titHp yy12Multiple Model Fault Detection• We can calculate– using the Kalman Filter innovation• So by tracking n Kalman Filters we can calculate the probability of each model given the observations– Problem solved?),|(1:1 −titHp yyttTteCHpttitiViyy21:1),|(−=Innovation at time tKFn…KF2KF1observationsinnovationsP(model i)Probabilitycalculation13Multiple Model Fault DetectionSimulation of‘faulty’ modelSimulation of‘nominal’ model14Multiple Model Fault DetectionTime(s)Simulation of‘faulty’ modelGround truthSimulation of‘nominal’ modelFault occurs here15Multiple Model Fault Detection• Main problem:– Our model of the failure-prone system is inadequate• Challenges (rest of the lecture):– Model failure-prone systems– Reason about failure-prone systems(detect faults and more)16Overview• Background– Kalman Filter review– Fault detection using Kalman Filters• Fault detection (and more) using Hybrid Estimation– Why Hybrid?– Modeling– Technical challenges– Current state of the art60s-70s80s - present17Hybrid System Models• Better model for our system:– Discrete modes and transitions between them– Continuous dynamics corresponding to each modenominalfailed0.00100.9991)(),()(),,(nominal1nominal1nominalnominal1tgttfttttttωυ+=+=+++uxyuxx)(),()(),,(failed1failed1failedfailed1tgttfttttttωυ+=+=+++uxyuxx18Hybrid System Models• System now has hybrid state x={xc,txd,t}– Continuous state xc,t– Discrete modes xd,t• Our model is a Hidden Markov Model:ut-1utxd,t-1xd,txc,t-1xc,tyt-1yt19Hybrid System Models• Hybrid models not just for failure detection• Many systems have both discrete and continuous state even in normal operation:– Hardware/software controlling physical system• e.g. Mars rover, robot manipulator – Systems with valves, switches, doors• Lunar habitat20Hybrid System Models• Even better model:– Model each component as Probabilistic Hybrid Automaton• Main difference: guards on discrete transitions– Transition probability conditioned on component inputs and component continuous state• Example:– More likely to fail if temperature over safe thresholdactuatorinputs outputsokfailedg1g20.0010.9990.90.1121Hybrid System Models• ‘Compose’ PHA components to form overall system•With m modes per PHA, composed PHA has O(mn) modes overalluc1ud1ud2yc2yc1PHA1PHA2wc1PHA3n components22Reasoning about Hybrid System Models• Now we have trajectories of discrete modes– Example of a possible mode trajectory of the system:• Known system dynamics for each trajectoryok okokok okfailedfailed failedt=0t=1t=2 t=3 t=4 t=5 t=6 t=7Time(s)23Hybrid Estimation•Problem:– Estimate the hybrid state of a system– Given model, observations and system inputs),|,(:1:1,, tttctdp uyxx24Hybrid EstimationApproach:1. Each time t, consider all possible mode trajectories2. Calculate the distribution over trajectories and 3. Sum over the trajectoriesokokfailedokfailedfailedok),|,(:1:1,:1, tttctdp uyxx∑−1:1,),|,(:1:1,:1,tdtttctdpxuyxx=),|,(:1:1,, tttctdp uyxxt=0 t=1 t=2tc,x25Hybrid Estimation• How to calculate distribution for a trajectory?• Belief state update:),|()|()|,(:1:1,,:1:1,:1,:1, ttdtcttdttctdppp yxxyxyxx=Probability of trajectorygiven observationsDistribution of continuous state givenmode trajectory and observationsÆgiven by kalman filterCalculate using belief state update)()|(:1,:1:1, tdttdbp xyx=)()()(1:1,, −⋅⋅=tdtdTtObPP xxy26Hybrid Estimation• Observation function PO– Track a Kalman Filter for the trajectory– Use KF innovation (as MM method)),|(1:1:1, −=ttdtOpP yxytd :1,xttTteCptttdtiViyxy211:1:1,),|(−−=27Hybrid Estimation• Transition
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