Filter Kalman

7 Application to Noise Parameters Selection for the Kalman Filter 

In this section the basic principles of the Kalman filter are presented for linear multi-degree-of-freedom (MDOF) systems. More details can be found elsewhere [36,84,128,129], Even though it is not often emphasized and it was not shown explicitly in the original formulation [128,129], the Kalman filter is a Bayesian updating procedure. Consider a second-order 

Note that the matrix exponential can be computed by the function expm in MATLAB [171] At is the sampling time step. In order to simplify the symbols, the following notations are defined  

The excitation F is modeled as discrete stationary Gaussian white noise with zero mean and covariance matrix LpiOp), where is a vector which parameterizes the covariance matrix 

To account for the modeling error and measurement noise, the relationship between the output measurements and the state vector is defined by  

The partitioned VT was fed into CSTEP to obtain a control schedule. It was specified that no more than 1 MINUS, LEQ, or PLUS operation should be scheduled at a time per partition under the assumption that these operations are similar enough to be implemented by the same hardware in the target technology. MULT operations were assumed to be implemented by separate hardware. Therefore a MULT could be scheduled in parallel with any other data operator. 

Partitioning does not have a large effect on the scheduling of this modular design. The partitions indicate that a PLUS operation may execute in the address partition at the same time that another PLUS operation is executing in the data partition, but it is never necessary to schedule simultaneous PLUS operations, so the partitions do not alter the schedule. 

As is the case for scheduling, partitioning does not have a large effect on the data path allocation for this design. This is a result of EMUCS inability to bind objects of different bitwidths to the same module. In the Kalman Filter, the addressing carriers are 4 bits, while the data carriers are 16 bits, so address and data operators will not share hardware. For this case, this restriction to the way that EMUCS can bind exactly reflects the partitions produced by APARTY. 

Results for 3 different runs through EMUCS for the one Kalman Filter schedule are listed in Table 8-5 as Kalman 1, Kalman 2, and Kalman 3. The different runs show the effect of varying input costs to reflect different target technologies. 

This machine has shown how input costs can be varied in EMUCS to allow for a design space exploration and to allow particular technologies to be targeted in the data path allocation phase. 

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Other chemometrics methods to improve caUbration have been advanced. The method of partial least squares has been usehil in multicomponent cahbration (48—51). In this approach the concentrations are related to latent variables in the block of observed instmment responses. Thus PLS regression can solve the colinearity problem and provide all of the advantages discussed earlier. Principal components analysis coupled with multiple regression, often called Principal Component Regression (PCR), is another cahbration approach that has been compared and contrasted to PLS (52—54). Cahbration problems can also be approached using the Kalman filter as discussed (43). 

At the first eonferenee of the International Federation of Automatie Control (IFAC), Kalman (1960) introdueed the dual eoneept of eontrollability and observability. At the same time Kalman demonstrated that when the system dynamie equations are linear and the performanee eriterion is quadratie (LQ eontrol), then the mathematieal problem has an explieit solution whieh provides an optimal eontrol law. Also Kalman and Buey (1961) developed the idea of an optimal filter (Kalman filter) whieh, when eombined with an optimal eontroller, produeed linear-quadratie-Gaussian (LQG) eontrol. 

This work was extended by Kalman and Buey (1961) who designed a state estimation proeess based upon an optimal minimum varianee filter, generally referred to as a Kalman filter. 

The Kalman filter single variable estimation problem... 

The Kalman filter is a eomplementary form of the Weiner filter. Let be a measurement of a parameter x and let its varianee Pa be given by... 

The general form of the Kalman filter usually eontains a diserete model of the system together with a set of reeursive equations that eontinuously update the Kalman gain matrix K and the system eovarianee matrix P. 

Equations (9.71)-(9.76) are illustrated in Figure 9.7 whieh shows the bloek diagram of the Kalman filter. 

The reeursive equations (9.74)-(9.76) that ealeulate the Kalman gain matrix and eovarianee matrix for a Kalman filter are similar to equations (9.29) and (9.30) that... 

A control system that contains a LQ Regulator/Tracking controller together with a Kalman filter state estimator as shown in Figure 9.8 is called a Linear Quadratic Gaussian (LQG) control system. 

The full LQG system, eomprising of the LQ optimal eontroller and Kalman filter was then eonstrueted. Figure 9.17 shows a set of moisture eontent measurements z ikT) together with the estimated moisture eontent x ikT). 

If the forward velocity of the ship is the state variable u, a best estimate of which is given by the Kalman filter, the gain scheduling controller can be expressed as... 

The plant deseribed in Example 9.8 by equations (9.185) and (9.186) is to be eontrolled by a Linear Quadratie Gaussian (LQG) eontrol seheme that eonsists of a LQ Regulator eombined with the Kalman filter designed in Example 9.8. The... 

This tutorial uses the MATLAB Control System Toolbox for linear quadratie regulator, linear quadratie estimator (Kalman filter) and linear quadratie Gaussian eontrol system design. The tutorial also employs the Robust Control Toolbox for multivariable robust eontrol system design. Problems in Chapter 9 are used as design examples. 

Continuous Linear Quadratic Estimator (Kalman Filter]... 

The script file kalfild.m solves, in forward-time, the discrete solution of the Kalman filter equations, using equations (9.74), (9.75) and (9.76) in a recursive process. The MATLAB command Iqed gives the same result. 

Discrete solution of Kalman filter equations %Init ialize... 

Dove, M.J., Burns, R.S. and Evison, J.L. (1986) The use of Kalman Filters in Navigation Systems - Current Status and Future Possibilities. In Proc. of the Int. Conf. on Computer Aided Design, Manf. and Operation in the Marine and Offshore Industries, Keramidas, G.A. and Murthy, T.K.S. (eds.), Springer-Verlag, Washington, DC, pp. 361-374. 

Grimble, M.J., Patton, R.J. and Wise, D.A. (1979) The design of dynamic ship positioning control systems using extended Kalman filtering techniques, IEEE Conference, Oceans 79, CA, San Diego. 

Pearson, A.R., Sutton, R., Burns, R.S. and Robinson, P. (2000) A Kalman Filter Approach to Fault Tolerance Control in Autonomous Underwater Vehicles. In Proc. 14th International Conference on Systems Engineering, Coventry, 12-14 September, 2, pp. 458 63. 

An important property of a Kalman filter is that during the measurement and estimation process, regions of the measurement range can be identified where the model is invalid. This allows us to take steps to avoid these measurements affecting the accuracy of the estimated parameters. Such a filter is called the adaptive Kalman fdter. An increasing number of applications of the Kalman filter... 

In this chapter we discuss the principles of the Kalman filter with reference to a few examples from analytical chemistry. The discussion is divided into three parts. First, recursive regression is applied to estimate the parameters of a measurement equation without considering a systems equation. In the second part a systems equation is introduced making it necessary to extend the recursive regression to a Kalman filter, and finally the adaptive Kalman filter is discussed. In the concluding section, the features of the Kalman filter are demonstrated on a few applications. 

Before we introduce the Kalman filter, we reformulate the least-squares algorithm discussed in Chapter 8 in a recursive way. By way of illustration, we consider a simple straight line model which is estimated by recursive regression. Firstly, the measurement model has to be specified, which describes the relationship between the independent variable x, e.g., the concentrations of a series of standard solutions, and the dependent variable, y, the measured response. If we assume a straight line model, any response is described by ... 

Kalman filter algorithm equations for time-invariant system states... 

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. 

Equation (41.11) represents the (deterministic) system equation which describes how the concentrations vary in time. In order to estimate the concentrations of the two compounds as a function of time during the reaction, the absorbance of the mixture is measured as a function of wavelength and time. Let us suppose that the pure spectra (absorptivities) of the compounds A and B are known and that at a time t the spectrometer is set at a wavelength giving the absorptivities h (0- The system and measurement equations can now be solved by the Kalman filter given in Table 41.10. By way of illustration we work out a simplified example of a reaction with a true reaction rate constant equal to A , = 0.1 min and an initial concentration a , (0) = 1. The concentrations are spectrophotometrically measured every 5 minutes and at the start of the reaction after 1 minute. Each time a new measurement is performed, the last estimate of the concentration A is updated. By substituting that concentration in the system equation xff) = JC (0)exp(-A i/) we obtain an update of the reaction rate k. With this new value the concentration of A is extrapolated to the point in time that a new measurement is made. The results for three cycles of the Kalman filter are given in Table 41.11 and in Fig. 41.7. The... 

Fig. 41.7. Concentrations of the reactant A (reaction A B) as a function of time (dotted line) (ca = 1, cb = 0) state updates (after a new measurement), O state extrapolations to the next measurement (see Table 41.11 for Kalman filter settings).

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