Residuals, the difference between measured and model predicted values of process variables, carry valuable information. Unfortunately, this information is blended with measurement noise and prediction errors due to modeling accuracy. Robust FDD techniques are needed to interpret the residuals in spite of noise in data and modeling errors. Three different approaches for computation of residuals and their interpretation are discussed in this section: Parity equations, diagnostic observers and Kalman filters, and robust observers for unknown inputs. Often, statistical tests are conducted to assess the significance level of the magnitudes of residuals to detect or diagnose a fault. Two popular tests, x2 tests and likelihood ratio tests are introduced below.
X2 tests of residuals for fault detection.Fault:!detection!x2 tests of residuals for Statistical testing of residuals for fault detection can be cast as testing for the zero mean hypothesis. The null hypothesis (Wo) is residual mean being zero (or having a nonsignificant magnitude), which indicates lack of evidence for a fault. The alternative hypothesis (Hi) is large nonzero values of the residual mean, indicating the presence of a fault.
where /xr is the mean of the residual vector. Because of limited data, the test is conducted using the sample mean of residuals r instead of fir. The test may be conducted on a single residual at a given time (r(t)), a single residual over a time window I (r(t) = [r(t), ■ ■ ■ ,r(t — l)]T), or an average residual over the window I (r(t, I) = [1/(1 + 1)] Ylj=o r(t~j))- The same tests can be conducted on a vector of residuals where r(t) = [ri(i), ■ ■ • , rn(t)]T, r(t) = [rT(i), • • • , rT(t - l))T, and f (t, I) = [1/(1 + 1)] £¡«0 r(t - j). The tests are designed for a specified false alarm rate a, and Normal distribution and zero mean of residuals is assumed. \2 tests are used for fault detection. They can be developed for scalar or vector residuals. Detailed discussion of scalar and vector residuals tests is given in . The tests for vector residuals are summarized below.
Single observation of vector residual. The joint density function for a single observation of vector residuals r(t) of length n is
where Er is the population covariance matrix of r. The corresponding sample covariance matrix is Sr. The statistic
follows the x2 distribution with n degrees of freedom and can be used to detect faults with the hypotheses
Ti-0 : Pn(t) < Xn,a Wl : Pn(t) > Xl,a no fault fault
Vector residual sequence. The joint density function of the vector sequence is
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