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I question about Kalman Filter:

If I have system state $$ \mathbf{X} = [x_1\ x_2\ x_3\ x_4\ x_5]^T, $$ these state elements are independent. I have measurement from a sensor to correct the predicted state. Normally, the correction will perform on all elements of state $\mathbf{X}$ and produce new covariance matrix $\mathbf{P}$.

If I just want to correct the substate only $x_1$ and $x_2$, and do not do anything with $x_3$, $x_4$ and $x_5$, how can I do that ? and how to compute $\mathbf{P}$?

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It will depend upon the measurement model. If your sensor provides data for all the states i.e. for $x_1,x_2,x_3,x_4,x_5$, then you can update only $x_1$ and $x_2$ by making the noise covariance corresponding to $x_3,x_4,x_5$ very large. Actually, there will still be some corrections in these three states but they will be small. The higher the covariance corresponding to these three states, the lower is the correction in them. Alternatively, you can make the last three rows and last three columns of process covariance very small.

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  • $\begingroup$ @tn0432: Unless I'm isunderstanding, it sounds like you know the values of $x_3$, $x_4$ and $x_5$. If that's the case, then you can set the variances of those terms in the state equation to zero and use their known values as the initial values. Although not a great example, ( since the focus is conversion from arima to state space ) below shows how this done when converting from arima to state space. robjhyndman.com/talks/ABS3.pdf $\endgroup$ – mark leeds Nov 18 '17 at 17:03
  • $\begingroup$ @tn0432: as an addendum to what I said, you made need to make the variances of the known states non-zero but orders of magnitude less than the variances of the unknown states. Otherwise, if you just set them to zero, you may run into problems inverting the matrix. of course, you still may run into problems inverting it but there's a whole literature on that issue also. $\endgroup$ – mark leeds Nov 19 '17 at 7:14
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There is a literature on Constrained Kalman Filters, as an example of one of 625 hits in IEEE Explore:

S. J. Julier and J. J. LaViola, "On Kalman Filtering With Nonlinear Equality Constraints," in IEEE Transactions on Signal Processing, vol. 55, no. 6, pp. 2774-2784, June 2007. doi: 10.1109/TSP.2007.893949 Abstract: The state space description of some physical systems possess nonlinear equality constraints between some state variables. In this paper, we consider the problem of applying a Kalman filter-type estimator in the presence of such constraints. We categorize previous approaches into pseudo-observation and projection methods and identify two types of constraints-those that act on the entire distribution and those that act on the mean of the distribution. We argue that the pseudo-observation approach enforces neither type of constraint and that the projection method enforces the first type of constraint only. We propose a new method that utilizes the projection method twice-once to constrain the entire distribution and once to constrain the statistics of the distribution. We illustrate these algorithms in a tracking system that uses unit quaternions to encode orientation keywords: {Kalman filters;filtering theory;pseudonoise codes;statistical distributions;Kalman filtering;distribution statistics;nonlinear equality constraints;projection method;pseudo-observation;state space description;unit quaternions;Chemical reactors;Computer science;Filtering;Kalman filters;Kinematics;Physics;Probability distribution;Quaternions;State-space methods;Statistical distributions;Kalman filtering;measurement matrix;nonlinear constraints;quaternions}, URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4203082&isnumber=4203033

but probably more useful to you,

https://www.mathworks.com/matlabcentral/linkexchange/links/2191-kalman-filtering-with-state-constraints-a-survey-of-linear-and-nonlinear-algorithms-tutorial

There are many possible approaches.

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