Hidden markov models for fault detection in dynamic systems

A method of monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said method performed in successive sampling intervals and comprising:

• defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time;
• observing a set of actual values of said parameters in a current one of said sampling intervals;
• obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b) wherein said obtaining an instantaneous probability comprises:
• observing the frequency of each failure state of said system and the corresponding parameter values over a period of time relatively long compared to said sampling intervals,
• constructing a training data set associating the frequency of each failure state with different sets of corresponding parameter values,
• using a classification algorithm operating on said training data to infer from the parameter values observed during the current sampling interval said instantaneous probabilities of the current sampling interval;
• computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• computing from the intermediate probability of the current sampling interval for each one of said states a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and, if said system has transitioned to one of said failure states, issuing an indication corresponding thereto.

A method of monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said method performed in successive sampling intervals and comprising:

• defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time;
• observing a set of actual values of said parameters in a current one of said sampling intervals;
• obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b) wherein said obtaining an instantaneous probability for a failure state is accomplished without training data related to that failure state and comprises:
• determining for each parameter of that failure state upper and lower bounds on the possible values thereof,
• computing the instantaneous probability of that failure state from said upper and lower bounds;
• computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• computing from the intermediate probability of the current sampling interval for each one of said states a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and, if said system has transitioned to one of said failure states, issuing an indication corresponding thereto.

A method of monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said method performed in successive sampling intervals and comprising:

• defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time;
• observing a set of actual values of said parameters in a current one of said sampling intervals wherein said observing comprises monitoring measurements of input commands and performance variables of said system and converting said measurements to parameters indicative of changes in said measurements, and wherein said observing further comprises observing parameters comprising one of (a) autoregressive coefficients of said measurements, (b) variances of said measurements and (c) mean values of said measurements;
• obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b);
• computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• computing from the intermediate probability of the current sampling interval for each one of said states a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and, if said system has transitioned to one of said failure states, issuing an indication corresponding thereto.

A method of monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said method performed in successive sampling intervals and comprising:

• defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time, wherein said defining plural transition probabilities comprises estimating a mean time between failures (MTBF) characteristic of each of said failure states and computing each corresponding transition probability therefrom;
• observing a set of actual values of said parameters in a current one of said sampling intervals;
• obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b), wherein said obtaining further comprises an instantaneous estimate of the probability that said system is in said one state given said set of actual measurements, divided by an unconditional probability of said system being in said one state;
• computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• computing from the intermediate probability of the current sampling interval for each one of said states a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and, if said system has transitioned to one of said failure states, issuing an indication corresponding thereto, wherein said computing from the intermediate probability comprises equating said posterior probability with the intermediate probability computed for the current sampling interval.

A method of monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said method performed in successive sampling intervals and comprising:

• defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time;
• observing a set of actual values of said parameters a current one of said sampling intervals;
• obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b), wherein said obtaining further comprises an instantaneous estimate of the probability that said system is in said one state given said set of actual measurements, divided by an unconditional probability of said system being in said one state wherein said obtaining an instantaneous probability further comprises:
• observing the frequency of each failure state of said system and the corresponding parameter values over a period of time relatively long compared to said sampling intervals;
• constructing a training data set associating the frequency of each failure state with different sets of corresponding parameter values;
• using a classification algorithm operating on said training data to infer from the parameter values observed during the current sampling interval said instantaneous probabilities of the current sampling interval;
• computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• computing from the intermediate probability of the current sampling interval for each one of said states a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and, if said system has transitioned to one of said failure states, issuing an indication corresponding thereto, wherein said computing from the intermediate probability comprises equating said posterior probability with the intermediate probability computed for the current sampling interval.

A method of monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said method performed in successive sampling intervals and comprising:

• defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time;
• observing a set of actual values of said parameters in a current one of said sampling intervals;
• obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b), wherein said obtaining further comprises an instantaneous estimate of the probability that said system is in said one state given said set of actual measurements, divided by an unconditional probability of said system being in said one state, wherein said obtaining an instantaneous probability for a failure state is accomplished without training data related to that failure state and comprises:
• determining for each parameter of that failure state upper and lower bounds on the possible values thereof;
• computing the instantaneous probability of that failure state from said upper and lower bounds;
• computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• computing from the intermediate probability of the current sampling interval for each one of said states a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and, if said system has transitioned to one of said failure states, issuing an indication corresponding thereto, wherein said computing from the intermediate probability comprises equating said posterior probability with the intermediate probability computed for the current sampling interval.

A method of monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said method performed in successive sampling intervals and comprising:

• defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time;
• observing a set of actual values of said parameters in a current one of said sampling intervals, wherein said observing comprises monitoring measurements of input commands and performance variables of said system and converting said measurements to parameters indicative of changes in said measurements and wherein said observing further comprises observing parameters comprising one of (a) autoregressive coefficients of said measurements, (b) variances of said measurements and (c) mean values of said measurements;
• obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b), wherein said obtaining further comprises an instantaneous estimate of the probability that said system is in said one state given said set of actual measurements, divided by an unconditional probability of said system being in said one state;
• computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• computing from the intermediate probability of the current sampling interval for each one of said states a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and, if said system has transitioned to one of said failure states, issuing an indication corresponding thereto, wherein said computing from the intermediate probability comprises equating said posterior probability with the intermediate probability computed for the current sampling interval.

A method of monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said method performed in successive sampling intervals and comprising:

• defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time, wherein said defining further comprises estimating a mean time between failures (MTBF) characteristic of each of said failure states and computing each corresponding transition probability therefrom;
• observing a set of actual values of said parameters in a current one of said sampling intervals;
• obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b), wherein said obtaining further comprises obtaining a probability of the actual values of the current sampling interval being observed given said system being in said one state, and wherein said obtaining further comprises employing a classifier trained to output the instantaneous probability for each state in response to said set of actual values;
• computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• computing from the intermediate probability of the current sampling interval for each one of said states a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and, if said system has transitioned to one of said failure states, issuing an indication corresponding thereto, wherein said computing from the intermediate probability comprises dividing said intermediate probability by an unconditional probability of observing the sets of actual values of the current and previous sampling intervals.

A method of monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said method performed in successive sampling intervals and comprising:

• defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any
• observing a set of actual values of said parameters in a current one of said sampling intervals;
• obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b), wherein said obtaining further comprises obtaining a probability of the actual values of the current sampling interval being observed given said system being in said one state, wherein said obtaining further comprises employing a classifier trained to output the instantaneous probability for each state in response to said set of actual values, and wherein said obtaining further comprises:
• observing the frequency of each failure state of said system and the corresponding parameter values over a period of time relatively long compared to said sampling intervals;
• constructing a training data set associating the frequency of each failure state with different sets of corresponding parameter values;
• using a classification algorithm operating on said training data to infer from the parameter values observed during the current sampling interval said instantaneous probabilities of the current sampling interval;
• computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• computing from the intermediate probability of the current sampling interval for each one of said states a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and, if said system has transitioned to one of said failure states, issuing an indication corresponding thereto, wherein said computing from the intermediate probability comprises dividing said intermediate probability by an unconditional probability of observing the sets of actual values of the current and previous sampling intervals.

A method of monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said method performed in successive sampling intervals and comprising:

• defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time;
• observing a set of actual values of said parameters in a current one of said sampling intervals;
• obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b), wherein said obtaining further comprises obtaining a probability of the actual values of the current sampling interval being observed given said system being in said one state, wherein said obtaining further comprises employing a classifier trained to output the instantaneous probability for each state in response to said set of actual values, wherein said obtaining an instantaneous probability for a failure state is accomplished without training data related to that failure state and comprises:
• determining for each parameter of that failure state upper and lower bounds on the possible values thereof;
• computing the instantaneous probability of that failure state from said upper and lower bounds;
• computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• computing from the intermediate probability of the current sampling interval for each one of said states a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and, if said system has transitioned to one of said failure states, issuing an indication corresponding thereto, wherein said computing from the intermediate probability comprises dividing said intermediate probability by an unconditional probability of observing the sets of actual values of the current and previous sampling intervals.

A method of monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said method performed in successive sampling intervals and comprising:

• defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time;
• observing a set of actual values of said parameters in a current one of said sampling interval, wherein said observing comprises monitoring measurements of input commands and performance variables of said system and converting said measurements to parameters indicative of changes in said measurements, wherein said observing further comprises observing parameters comprising one of (a) autoregressive coefficients of said measurements, (b) variances of said measurements and (c) mean values of said measurements;
• obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b), wherein said obtaining further comprises obtaining a probability of the actual values of the current sampling interval being observed given said system being in said one state, wherein said obtaining further comprises employing a classifier trained to output the instantaneous probability for each state in response to said set of actual values;
• computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• computing from the intermediate probability of the current sampling interval for each one of said states a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and, if said system has transitioned to one of said failure states, issuing an indication corresponding thereto, wherein said computing from the intermediate probability comprises dividing said intermediate probability by an unconditional probability of observing the sets of actual values of the current and previous sampling intervals.

Apparatus for monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said apparatus operable in successive sampling intervals, said apparatus comprising:

• means for defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time, wherein said means for defining plural transition probabilities comprises means for estimating a mean time between failures (MTBF) characteristic of each of said failure states and computing each corresponding transition probability therefrom;
• means for observing a set of actual values of said parameters in a current one of said sampling intervals;
• means for obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b);
• means for computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• means for computing from the intermediate probability for each one of said states of the current sampling interval a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and for determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and issuing an indication corresponding thereto.

Apparatus for monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said apparatus operable in successive sampling intervals, said apparatus comprising:

• means for defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time;
• means for observing a set of actual values of said parameters in a current one of said sampling intervals, wherein said means for observing said parameters comprises means for monitoring measurements of input commands and performance variables of said system and converting said measurements to parameters indicative of changes in said measurements;
• means for obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b), wherein said means for obtaining comprises means for obtaining an instantaneous estimate of the probability that said system is in said one state given said set of actual measurements, divided by an unconditional probability of said system being in said one state;
• means for computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• means for computing from the intermediate probability for each one of said states of the current sampling interval a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and for determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and issuing an indication corresponding thereto, wherein said means for computing from the intermediate probability comprises means for equating said posterior probability with the intermediate probability computed for the current sampling interval.

Apparatus for monitoring a system having a normal working state corresponding to normal operation of said system and a plurality of individual failure states corresponding to different failure modes of said system, said system exhibiting respective sets of measurable parameters corresponding to inputs and behavior symptoms causally related to said inputs, said apparatus operable in successive sampling intervals, said apparatus comprising:

• means for defining plural transition probabilities for plural pairs of said states, each transition probability being related to the probability that said system will change from one to the other of said pairs of states at any time, wherein said means for defining plural transition probabilities comprises means for estimating a mean time between failures (MTBF) characteristic of each of said failure states and computing each corresponding transition probability therefrom;
• means for observing a set of actual values of said parameters in a current one of said sampling intervals;
• means for obtaining an instantaneous probability comprising an estimate of the probability of one of (a) said set of actual values being observed and (b) said system being in said one state, given the other of (a) and (b), wherein said means for obtaining an instantaneous probability comprises means for obtaining a probability of the actual values of the current sampling interval being observed given said system being in said one state, and wherein said means for obtaining and instantaneous probability further comprises a classifier trained to output the instantaneous probability for each state in response to said set of actual values;
• means for computing plural respective intermediate probabilities corresponding to respective ones of said states, each intermediate probability being equal to the corresponding instantaneous probability of said one state multiplied by a sum over plural states of the intermediate probability for a given state computed during the previous sampling interval multiplied by the transition probability between said given state and said one state;
• and
• means for computing from the intermediate probability for each one of said states of the current sampling interval a posterior probability that said system is in the corresponding one of said states given the sets of actual values observed over the current and previous sampling intervals, and for determining from the posterior probabilities of the plural states whether said system has transitioned to one of said failure states and issuing an indication corresponding thereto, wherein said means for computing from the intermediate probability comprises means for dividing said intermediate probability by an unconditional probability of observing the sets of actual values of the current and previous sampling intervals.