A Generalized Likelihood Ratio Approach to the Detection of Jumps in Linear Systems Min Luo

Outlines Kalman filter GLR Adaptive filtering An example FOR MORE INFO... Alan S.Willsky, “A Generalized Likelihood Ratio Approach to the Detection and Estimation of Jumps in Linear System”, IEEE Transaction on Automatic Control, Feb, 1976

Linear Stochastic Systems With Unknown Jumps represent the presence of the ith type of abrupt Change. --- is the unknown time at which the failure occurs --- denotes the magnitude of the failure ……(1)

Kalman Filter Design a Kalman filter based on normal operation:..….(2)

Kalman Filter are the response if no abrupt change occurs. ……(3)

Kalman Filter ……(4)

Matched Filter --- zero-mean white Gaussian with covariance Thus, we have a standard detection problem in white noise. The solution to this problem involves mathched filtering operation. ……(5)

Generalized Likelihood Ratio Test We say that a test is a generalized likelihood ratio test for testing between hypotheses and when where ……(6)

Generalized Likelihood Ratio Test Where the constant is such that The precise optimal properties of the GLR test in the general case are unknown, but for many special cases, the GLR test is optimal. ……(7)

Online GLR Compute the maximum likelihood estimates based on r(1),…, r(k) and the hypothesis H 1. ……(8)

Online GLR ……(9) Deterministic C: Linear combination of residuals:

Online GLR The MLE is the value that maximizes The decision rule is: …(10) …(11)

Online GLR Data Window At any time, we restrict our optimization over to an interval of the form We now consider the case in which we hypothesize --- an unknown scalar --- a given set of hypothesized “failure directions”

Online GLR is unknown, the GLR for this change isIf If is known, the likelihood of a type i change having occurred at time …(12)

Online GLR …(13) The decision as to the failure: …(14)

GLR Algorithm Summary Kalman Filter Matched Filter Likelihood Calculation GLR Algorithm

Adaptive Filtering Once a jump has been detected by the GLR detector, we can use MLE’s to directly increment our state estimate. --- represents the contribution to x(k) if a jump occurs at time --- represents the response of Kalman filter to the jump prior to the detection …(15)

Direct Compensation Implementation of direct compensation technique

Some Comments of Adaptive Filtering Increase the estimation error covariance to reflect the degradation in the quality of the estimate caused by the jump. is error covariance for …(16)

Final Issue Tradeoff between fast detection and accurate estimation of jump: Different size of finite data window of GLR decides small --> accurate estimation of the jump --> quick detection

Detection Probability Calculation The choice of a decision threshold and a window length requires the tradeoff among detection delay time, the P f of the false alarm, and the of correct detection of a jump of magnitude at time. …(17)

Apply GLR to a Tracking Problem The problem is to design a tracking filter which uses position measurements taken at 30s intervals to track the motion of a vehicle along a straight line.

Example The vehicle is subject to occasional step change of unknown magnitude in either position or velocity. The tracking filter is a Kalman filter operating in steady state and requires min to completely respond to such jumps. The GLR system was implemented with the detection law:

Example Jump identification is made at the first time the above formula is satisfied: The optimization of is constrained to P F =0.005, P D >0.9

(a) Filter residuals for a 1320ft jump in position at 5 min (b) Likelihood ratio for (a) using GLR

Conclusion Develop an adaptive filtering technique for discrete-time linear stochastic systems subject to abrupt jumps in state variables. The estimation system consists of Kalman filter and a detection-compensation system based on GLR testing.

Conclusion Once a jump is detected, we can adjust the filter in one of three ways: – Directly increment the state estimate – Increase the estimation error covariance and thus allow the filter to adjust itself to the jump –Adjust both

Process Analysis and Abnormal Situation Detection: From theory to Practice

Problem Large volumes historical data The data are highly correlated The information stored in one variable is small Measurements are often missing on many variables

Possible Solution PCA PLS (projection to latent structures) Outline: Discusses the use of latent variable models Multivariate statistical process monitoring Abnormal situation detection Fault diagnosis

Multivariate Nature of Fault Detection Univariate chart(Shewhart) Problem: Most of the time the variables are not independent of one another, and none of them adequately defines product quality by itself. Multivariate chart

Separate Control Chart Per Variable

Statistical Process Control Versus Statistical Quality Control Statistical quality control (SQC) One can ignore the hundreds of process variables that are measured much more frequently than the product quality data. Statistical process control (SPC) One must look at all the process data as well.

Statistical Process Control Advantage of monitoring process data: Easier to diagnose the source of the problem Quality data may not be available at certain stages of the process

Latent Variables These variables are highly correlated and the effective dimension of the space in which they move is very small (usually less than ten). Consider the historical process data to consist of an (n by k )  matrix of process variable measurements X and a corresponding ( n by k) matrix of product quality data Y.

Latent Variables T is (n by A) matrix of latent variable scores. P(k by A), Q(m by A) are loading matrices that show how latent variables are related to X, Y variables. Advantage: By working in this low-dimensional space of the latent variables, the problems of process analysis, monitoring and optimization are greatly simplified.

Latent Variable Methods PCA PLS Reduced rank regression (RRR) Canonical variate analysis(CCR) or Canonical correlation regression (CVR)

Exploration and Analysis of Process Databases By examining the behavior of the process data in the projection spaces defined by the small number of latent variables, regions of stable operation, sudden changes, or slow process drifts may be readily observed.

Checking Data Quality for Process Modeling Identify outliers, check data for clusters Select data for the training part of multivariate control charts

Process Monitoring and Fault Diagnosis A model is built to relate X and Y using available historical or specially collected data. Monitoring charts are then constructed for future values of X. Two complementary multivariate control charts for process monitoring : 1.Hotelling’s T 2 chart

T 2 Chart is the estimated variance of the corresponding latent variable. This chart will check if a new observation vector of measurements on k process variables projects on the hyperplane within the limits determined by the reference data.

SPE X Chart 2. SPEx chart is computed from the reference PLS or PCA model. This latter plot will detect the occurrence of any new events that cause the process to move away from the hyperplane defined by the reference model.

Fault Diagnosis PLS or PCA models are used to construct the multivariate charts, they provide the user with the capacity for diagnosing assignable causes. Contribution plots are used to detect variables responsible for an out-of-control signal on SPEx,T 2.

Three Charts for Multivariate Process Monitoring

Troubleshooting and Monitoring of Batch Processes – three-dimensional data array X(n by k by L) k process variables are measured at L time intervals for each of n batches.

Multiway Extensions of PCA/ PLS The matrix is unfolded into a two-dimensional array such that each row corresponds to a batch. Mean centering of the variables effectively subtracts the trajectory, thus converting a nonlinear problem to one that can be tackled with linear methods such as PCA and PLS.

Multiway Extensions of PCA/ PLS plot the loadings of each variable, for each time interval,for the first principal component of a PCA analysis where the batch data are unfolded.

the scores of the first two principal components for 61 completed batches

Online SPC Charts When data are available in a historical database on many past normal batches, multivariate PCA and PLS models can be developed to establish online SPC charts for monitoring the progress of each new batch.

Online SPC Charts Online monitoring of batch 56

Startup and Grade Transition Problems Process transitions are very frequent These transitions lead to problems The use of multivariate statistical methods can improve process transitions

Multivariate Sensor and Image Analysis for Online Monitoring Similar banks of multivariate sensors and color imaging cameras are used online to monitor and control industrial processes. How to handle the huge amount of highly correlated data collected from these sensors and how to efficiently extract the subtle information contained in the data.

Observability/Detectability of Faults The model should be tested with known faults to determine the “observability” of these faults. – SPE chart and the Hotelling’s T 2 at A components are used to monitor the process. – If not both models signal the problem. – It needs more represented variable or it requires that certain process variables be given a higher weight in the model.

Frequency of Sampling For model building, it is important that the model is built with data collected with the same sampling frequency as will be used for the online operation of the model. The choice of the monitoring interval also depends on how quickly the faults we are trying to detect manifest themselves.

Soft Sensors/Inferential Models Soft sensors can either replace the hardware sensor or be used in parallel with it to provide redundancy and verify whether the hardware sensor is drifting or has failed. These inferential models are usually built by fitting either empirical or theoretically based models to plant data.

Using empirical models for soft sensors, latent variable models such as PLS offer some important advantages over standard regression models or neural networks.

Concluding Remarks The use of latent variable model for extracting useful information from historical databases. Wide acceptance in industry, particularly for the problems of process analysis, monitoring, and soft sensors.