Kalman equations

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. 

Kalman demonstrated that as integration in reverse time proeeeds, the solutions of F t) eonverge to eonstant values. Should t be infinite, or far removed from to, the matrix Rieeati equations reduee to a set of simultaneous equations... 

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. 

The Kalman gain matrix K is obtained from a set of reeursive equations that eommenee from some initial eovarianee matrix P(/c//c)... 

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... 

Before equations (9.99) can be run, and initial value of P(/c//c) is required. Ideally, they should not be close to the final value, so that convergence can be seen to have taken place. In this instance, P(/c//c) was set to an identity matrix. Figure 9.16 shows the diagonal elements of the Kalman gain matrix during the first 20 steps of the recursive equation (9.99). 

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... 

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... 

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. 

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... 

Here 4 is the target state vector at time index k and Wg contains two random variables which describe the unknown process error, which is assumed to be a Gaussian random variable with expectation zero and covariance matrix Q. In addition to the target dynamic model, a measurement equation is needed to implement the Kalman filter. This measurement equation maps the state vector t. to the measurement domain. In the next section different measurement equations are considered to handle various types of association strategies. 

The target state vector tk measured by the multilateration procedure can be considered directly as a target plot input of the association process. In this case, the input of the Kalman filter describes the same parameters that the internal state vector does. It is characteristic for the plot-to-track association procedure that the measurement equation contains directly the target state vector tk which is influenced by noise ftsk only ... 

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... 

The respective Kalman filter equations for the position correction and prediction steps can now be formulated based on equations (18) and (19), (20) or (21) accordingly for the different mentioned association schemes. Since the measurement equation is nonlinear in case of range-velocity-to-track or frequency-to-track association, the Extended Kalman filter is used for this particular application [16]. 

Equations (8.11) and (8.12) are approximate expressions for propagating the estimate and the error covariance, and in the literature they are referred to as the extended Kalman filter (EKF) propagation equations (Jaswinski, 1970). Other methods for dealing with the same problem are discussed in Gelb (1974) and Anderson and Moore (1979). 

It should be noted that the solution of the minimization problem simplifies to the updating step of a Kalman filter. In fact, if instead of applying the matrix inversion lemma to Eq. (8.19) to produce Eq. (8.20), the inversion is performed on the estimate equation (8.18), the well-known form of the Kaman filter equations is obtained. 

According to the previous section, in order to deal with the state-parameter estimation problem we have to solve a nonlinear set of filtering equations. The extended Kalman filter leads to the following equations (Ursin, 1980) ... 

The main advantage of the stochastic matrix approach is the simplicity for its computer implementation. Equation 17 directly provides the desired result, and Equation 28 is the basis of a validation test which my or may not be performed according to previous experience. In other words, the proposed method is conceptually and practically easier to implement than the Kalman counterpart. The... 

Typically, one specifies the desired response, C(z)/R(z), which yields from equation (12) the required design of the controller, D(z). In practice, however, this design technique results in a controller which requires excessive valve movement, an undesirable situation. Consequently, Kalman (12) developed a Z-transform algorithm which specifies the desired output, C(z), and the desired valve travel, M(z) for a setpoint change. The desired response and valve travel for a unit step change in setpoint is shown in Figure 22. The system response,... 

Therefore, the Kalman designed algorithm is specified by obtaining the modified Z-transform of the process pulse transfer function ot get P(z) and Q(z) and then substituting these values into equation (19). 

V.M. Becerra, P.D. Roberts, and G.W. Griffiths. Applying the extended Kalman filter to systems described by nonlinear differential-algebraic equations. Control Engineering Practice, 9 267-281,2001. 

Kalman filters can be extended to more complex situations with many variables and many responses. The model does not need to be multilinear but, for example, may be exponential (e.g. in kinetics). Although the equations increase considerably in complexity, the basic ideas are the same. 

When a model state is described by nonlinear equations, the extended Kalman filter has been applied using the well-known Kalman filter equations for the linearization of equations. If the state vector is enlarged with the parameter vector (P]j is used because it corresponds to the discrete version of the state model) and if it is considered to be constant or varying slowly, then it is possible to transform the problem of parameters estimation into a problem of state estimation. The P i i = P]j + njj with n]j white noise correction represents the model suggested for... 

The vector can be calculated either with the normal Kalman Filter (KF) which gives Xk for the discrete equation state (F(Xk, Uk, Vk)) or with the extended Kalman filter (KFE) which gives Pk+i in the calculation system. For this estimation, it is also necessary to obtain the state of the system Xk from the next state Xk+j. This estimation is made by block IT (inversion translator) another IT block gives... 

---