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# dkalman

## PURPOSE

% Construct the linear quadratic estimator (Kalman predictor) for the

## SYNOPSIS

function [Lp, Lf, P, Z] = dkalman (A, G, C, Qw, Rv, S)

## DESCRIPTION

```% -*- texinfo -*-
% @deftypefn {Function File} {[@var{Lp}, @var{Lf}, @var{P}, @var{Z}] =} dkalman (@var{A}, @var{G}, @var{C}, @var{Qw}, @var{Rv}, @var{S})
% Construct the linear quadratic estimator (Kalman predictor) for the
% discrete time system
% @iftex
% @tex
% \$\$
%  x_{k+1} = A x_k + B u_k + G w_k
% \$\$
% \$\$
%  y_k = C x_k + D u_k + v_k
% \$\$
% @end tex
% @end iftex
% @ifinfo
%
% @example
% x[k+1] = A x[k] + B u[k] + G w[k]
%   y[k] = C x[k] + D u[k] + v[k]
% @end example
%
% @end ifinfo
% where @var{w}, @var{v} are zero-mean gaussian noise processes with
% respective intensities @code{@var{Qw} = cov (@var{w}, @var{w})} and
% @code{@var{Rv} = cov (@var{v}, @var{v})}.
%
% If specified, @var{S} is @code{cov (@var{w}, @var{v})}.  Otherwise
% @code{cov (@var{w}, @var{v}) = 0}.
%
% The observer structure is
% @iftex
% @tex
% \$x_{k+1|k} = A x_{k|k-1} + B u_k + L_p (y_k - C x_{k|k-1} - D u_k)\$
% \$x_{k|k} = x_{k|k} + L_f (y_k - C x_{k|k-1} - D u_k)\$
% @end tex
% @end iftex
% @ifinfo
%
% @example
% x[k+1|k] = A x[k|k-1] + B u[k] + LP (y[k] - C x[k|k-1] - D u[k])
% x[k|k] = x[k|k-1] + LF (y[k] - C x[k|k-1] - D u[k])
% @end example
% @end ifinfo
%
% @noindent
% The following values are returned:
%
% @table @var
% @item Lp
% The predictor gain,
% @iftex
% @tex
% \$(A - L_p C)\$.
% @end tex
% @end iftex
% @ifinfo
% (@var{A} - @var{Lp} @var{C})
% @end ifinfo
% is stable.
%
% @item Lf
% The filter gain.
%
% @item P
% The Riccati solution.
% @iftex
% @tex
% \$P = E \{(x - x_{n|n-1})(x - x_{n|n-1})'\}\$
% @end tex
% @end iftex
%
% @ifinfo
% P = E [(x - x[n|n-1])(x - x[n|n-1])']
% @end ifinfo
%
% @item Z
% The updated error covariance matrix.
% @iftex
% @tex
% \$Z = E \{(x - x_{n|n})(x - x_{n|n})'\}\$
% @end tex
% @end iftex
%
% @ifinfo
% Z = E [(x - x[n|n])(x - x[n|n])']
% @end ifinfo
% @end table
% @end deftypefn```

## CROSS-REFERENCE INFORMATION

This function calls:
This function is called by:

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