Optimal Observers: Kalman Filters
This chapter has the purpose of reviewing the most important design aspects of Kalman filters as well as some of their most important properties. Heuristic derivations are given of the Kalman filter `equations for both continuous time and discrete time dy
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Optimal Observers: Kalman Filters
Abstract This chapter has the purpose of reviewing the most important design aspects of Kalman filters as well as some of their most important properties. Heuristic derivations are given of the Kalman filter ‘equations for both continuous time and discrete time dynamic systems. It is shown that the state mean values propagate according to the same observer equations as given in Chap. 4. Moreover it is shown that the state noise propagates according to the time dependent Lyapunov equation derived in Chap. 6. When measurements are made on the system this equation has to be modified with a term which expresses the decrease of uncertainty which the measurements make possible. The combination of these two results yields the main stochastic design equation for Kalman filters: the Riccati equation. Solving this equation immediately gives the optimal observer gain for a Kalman filter. Combining a Kalman filter with optimal or LQR feedback results in a very robust controller design: the LQG or Linear Quadratic Gaussian regulator.
7.1 Introduction In many practical situations very few of the states or functions of the states of a dynamical system may be measured directly without error. In general this occurs either because the states or the measurements of the states (or most commonly, both) are corrupted by noise. In such cases it is reasonable to consider the problem of finding an optimal estimate of all the states of the system given noisy measurements of some or all of the other states. Here ‘optimal’ may mean optimal in the sense of least squares, minimum variance, or some other optimality criterion. Briefly this can be accomplished by forcing a mathematical model of the system dynamics to follow the states of the plant or control object itself. The effects of the state noise are accounted for by effectively propagating the state noise through the same mathematical model and filtering it from the state estimates with a weight depending on the measurement noise. Recognizing that no analytical solution is actually required, the state and noise differential equations are solved recursively to find the state estimates. One of the most common such mathematical models E. Hendricks et al., Linear Systems Control, DOI: 10.1007/978-3-540-78486-9_7, Springer-Verlag Berlin Heidelberg 2008
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7 Optimal Observers: Kalman Filters
and its associated noise suppression algorithm is called a Kalman filter. Currently the large computational burden which this entails is often placed on a digital computer but in some cases may also be carried using analog methods. It is also possible to apply the basic Kalman filter to parameter estimation and/or nonlinear processes though the sense in which it is optimal is changed. Kalman filters are at present used in many of the control systems which are familiar. Navigation systems for airplanes, ships and spacecraft based on such filters are very common and applications to automotive navigation systems are currently in production in several companies. At
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