Measurement noise covariance matrix

Measurement noise covariance matrix R The main problem with the instrumentation system was the randomness of the infrared absorption moisture eontent analyser. A number of measurements were taken from the analyser and eompared with samples taken simultaneously by work laboratory staff. The errors eould be approximated to a normal distribution with a standard deviation of 2.73%, or a varianee of 7.46. 

Measurement noise covariance matrix %Disturbance matrix... 

We use the basic sensor model proposed in [4]. While this has limitations, it is simple and therefore useful as a starting point for discussion of the problem. In this model, the sensor is characterized by a measurement noise covariance matrix which is waveform dependent... 

As was shown, the conventional method for data reconciliation is that of weighted least squares, in which the adjustments to the data are weighted by the inverse of the measurement noise covariance matrix so that the model constraints are satisfied. The main assumption of the conventional approach is that the errors follow a normal Gaussian distribution. When this assumption is satisfied, conventional approaches provide unbiased estimates of the plant states. The presence of gross errors violates the assumptions in the conventional approach and makes the results invalid. 

As expecfed the optimal estimator is indeed a function of both the covariance of the atmospheric turbulence and the measurement noise. A matrix identity can be used to derive an equivalent form of Eq. 16 (Law and Lane, 1996)... 

For the regular filter the observation noise covariance if is a constant matrix determined before state estimation. On the other hand, the measurement noise covariance R(t) may be adjusted to compensate for estimation errors. Using finite-duration impulse response (FIR) filter algorithm the observation noise covariance can be adjusted during the state estimation process. 

Equations (41.15) and (41.19) for the extrapolation and update of system states form the so-called state-space model. The solution of the state-space model has been derived by Kalman and is known as the Kalman filter. Assumptions are that the measurement noise v(j) and the system noise w(/) are random and independent, normally distributed, white and uncorrelated. This leads to the general formulation of a Kalman filter given in Table 41.10. Equations (41.15) and (41.19) account for the time dependence of the system. Eq. (41.15) is the system equation which tells us how the system behaves in time (here in j units). Equation (41.16) expresses how the uncertainty in the system state grows as a function of time (here in j units) if no observations would be made. Q(j - 1) is the variance-covariance matrix of the system noise which contains the variance of w. 

Waveform Scheduling The choice of measurement is made using the control variable n(k). In fact two choices are made at each epoch, the target to be measured and the waveform used. The waveforms impinge on the measurement process through the covariance matrix of the noise In this model, the sensor is characterised... 

The vector nk describes the unknown additive measurement noise, which is assumed in accordance with Kalman filter theory to be a Gaussian random variable with zero mean and covariance matrix R. Instead of the additive noise term nj( in equation (20), the errors of the different measurement values are assumed to be statistically independent and identically Gaussian distributed, so... 

PLS is similar to PCR with the exception that the matrix decomposition for PLS is performed on the covariance matrix of the spectra and the reference concentrations, while for PCR only the spectra are used. PLS and PCR have similar performance if noise in the spectral data and errors in the reference concentration measurements are negligible. Otherwise, PLS generally provides better analysis than PCR.26... 

As demonstrated previously the process noise and the measurement noise parameters directly affect the state vectors estimated by the Kalman filter. Furthermore, the covariance matrix of the state estimation is affected as well. Therefore, accurate estimation of the noise parameters is necessary for good performance of the filter. In this example, the Bayesian approach is applied to select a p and a. Figure 2.32 shows the contours of the likelihood function p V 0, C) together with the actual noise variances 0 = [cr, and its optimal estimate 6. The two contours correspond to 50% and 10% of the peak value. The optimal values of ap = 2.8N and a = 7.1 x 10 m /s are at reasonable distance to the actual values as the actual noise variances are located within the region with significant probability density. Therefore, the Bayesian approach is validated to give accurate estimation for both noise variances for the linear oscillator. 

To account for the measurement noise, consider four Gaussian random variables yi, >>2, ys and y4 with zero mean and the covariance matrix ... 

In this simulated-data example, a twelve-story shear building is considered. It is assumed that this building has uniformly distributed floor mass and uniform stiffness across the height. The mass per floor is taken to be 100 metric tons, while the interstory stiffness is chosen to be k = 202.767 MN/m so that the first five modal frequencies are 0.900,2.686,4.429,6.103 and 7.680 Hz. The covariance matrix is diagonal with the variances corresponding to a 1.0% coefficient of variation of the measurement error of the squared modal frequencies and mode shapes for all modes, a reasonable value based on typical modal test results. For the simulated modal data, a sample of zero-mean Gaussian noise with covariance matrix was added to the exact modal frequencies and mode shapes. 

It is assumed that only the first three x-directional and y-directional modes are measured but not any of the torsional modes. This is done deliberately to simulate a common situation where some of the modes are not excited sufficiently to be able to observe. In the identification process, it is unknown that there are some missing modes. The six measured modes correspond to the 1st (3.432 Hz), 2nd (3.837 Hz), 4th (10.10 Hz), 5th (11.29 Hz), 7th (18.08 Hz) and 9th (21.31 Hz) modes. Sensors are placed on the +y and —y faces of the 1st, 2nd, 5 th and 6th floors, and the -x face of all floors to measure the modal frequencies and mode shape components. The covariance matrix Te is diagonal with 0.5% COV for the modal data. For the simulated modal data, a sample of zero-mean Gaussian noise with covariance matrix Ze was added to the exact modal frequencies and mode shapes. Initial values for all stiffness parameters are taken to be 100 MN/m, which overestimates the values by 100% and 150% for the x and y faces, respectively. 

Thus the prior distribution of f is iV(0, Q) = Ai(0, fflRR ). However, each measurement contains noise, which we assume to be Gaussian with zero mean and variance cr. The vector of data points also has Gaussian distribution E(t) A (0, Q + ffv). We denote the covariance matrix of t by C =Q + ffyl-The distribution of the joint probability of observing tjv+i having previously observed t can be written as... 

The steady-state optimal Kalman filter can be generalized for time-variant systems or time-invariant systems with non-stationary noise covariance. The time-varying Kalman filter is calculated in two steps, filtering and prediction. For the nonlinear model the state estimate may be relinearized to compensate the inadequacies of the linear model. The resulting filter is referred to the extended Kalman filter. If once a new state estimate is obtained, then a corrected reference state trajectory is determined in the estimation process. In this manner the Filter reduces deviations of the estimated state from the reference state trajectory (Kwon and Wozny, 1999 Vankateswarlu and Avantika, 2001). In the first step the state estimate and its covariance matrix are corrected at time by using new measurement values >[Pg.439]

Here b means the IMU body frame e denotes the ECEF frame i indicates the inertial frame Cl is the Direction Cosine Matrix (DCM) from body frame to ECEF frame, ft is the skew-symmetric matrix for angular rate measurements is the vector of acceleration measurements from the accelerometers. F is the system matrix applied in the ECEF frame is the distance from the earth geometric center to the earth surface g is the local gravity is the position of the IMU in ECEF. The noise vector w contains, in the indicated order, gyroscope bias, acceleration bias, acceleration noise, angular rate noise, receiver clock error and receiver clock rate noise. These noise terms are described by the error covariance matrix Q in the Kalman filter routine ... 

Each of the columns of the above matrix corresponds to a sigma point vector x -i e with components corresponding to the state, process, and measurement noise, respectively, and P is the corresponding augmented covariance vector, incorporating the process and observation noise components ... 

The system under consideradmi is described by Eqs. 3 and 4. The noise processes W[jt] and V[jt] are assumed to be zero mean and white, with known covariance matrices Q, R, and S, defined by Eq. 8. Joint input-state estimatimi consists of estimating the forces pj ] and states X[jt] from a set of response measurements d[ t]. A state estimate X, t / is defined as an estimate of X[ tj, given the output sequence d[ ], with n = 0,1,...,/. The corresponding error covariance matrix, denoted as P[ /], is defined in Eq. 10. An input estimate... 

Here is an s x 1 vector of measurements from the s sensors Vk is the /iv x 1 measurement noise term, modeled as a sequence of zero-mean Gaussian random variables with (vkV = where is riy x riy covariance matrix and Hk is a nmilinear function that relates the measurements to system states through a pertinent mathematical model. The measurement noise arises inherently... 

Q(fc) Covariance of noise driving state changes. This matrix controls allowed accelerations and clock frequency and phase variations. Smaller Q(fc) will result in smaller estimates of state error co-variance and less weighting of measurements. Larger Q(fc) will result in faster system response to measurements. Conventional wisdom suggests using larger Q when uncertain of exact model. 

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