📄 hp48g(x) discrete kalman filter simulator.htm
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<H2 align=center>HP48G(X) Discrete Kalman Filter Simulator</H2>
<P align=center><STRONG></STRONG></P>
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<CENTER>
<ADDRESS>Martin Murillo <BR>Engineering - Idaho State
University</CENTER></ADDRESS>
<P>
<HR>
[an error occurred while processing this directive] This program simulates a
discrete Kalman Filter. It estimates the state <IMG height=15
alt=tex2html_wrap108
src="HP48G(X) Discrete Kalman Filter Simulator.files/img1.gif" width=56
align=bottom> of a discrete-time equation of the process modeled by the linear
difference equation:
<P>
<P><IMG height=18 alt=displaymath272
src="HP48G(X) Discrete Kalman Filter Simulator.files/img2.gif" width=321
align=bottom>
<P>
<P>
<P><IMG height=16 alt=displaymath273
src="HP48G(X) Discrete Kalman Filter Simulator.files/img3.gif" width=311
align=bottom>
<P>Where
<P><IMG height=11 alt=tex2html_wrap109
src="HP48G(X) Discrete Kalman Filter Simulator.files/img4.gif" width=17
align=bottom> is a <IMG height=13 alt=tex2html_wrap110
src="HP48G(X) Discrete Kalman Filter Simulator.files/img5.gif" width=43
align=bottom> process state vector
<P><IMG height=17 alt=tex2html_wrap111
src="HP48G(X) Discrete Kalman Filter Simulator.files/img6.gif" width=17
align=bottom> is the <IMG height=10 alt=tex2html_wrap112
src="HP48G(X) Discrete Kalman Filter Simulator.files/img7.gif" width=46
align=bottom> matrix that relates <IMG height=11 alt=tex2html_wrap109
src="HP48G(X) Discrete Kalman Filter Simulator.files/img4.gif" width=17
align=bottom> to <IMG height=13 alt=tex2html_wrap114
src="HP48G(X) Discrete Kalman Filter Simulator.files/img8.gif" width=34
align=bottom>
<P><IMG height=11 alt=tex2html_wrap115
src="HP48G(X) Discrete Kalman Filter Simulator.files/img9.gif" width=20
align=bottom> is a Gaussian white sequence
<P><IMG height=11 alt=tex2html_wrap116
src="HP48G(X) Discrete Kalman Filter Simulator.files/img10.gif" width=14
align=bottom> is the vector measurement at time <IMG height=13
alt=tex2html_wrap117
src="HP48G(X) Discrete Kalman Filter Simulator.files/img11.gif" width=8
align=bottom>
<P><IMG height=15 alt=tex2html_wrap118
src="HP48G(X) Discrete Kalman Filter Simulator.files/img12.gif" width=21
align=bottom> is the matrix giving a noiseless connection between the
measurement and the state vector.
<P><IMG height=15 alt=tex2html_wrap119
src="HP48G(X) Discrete Kalman Filter Simulator.files/img13.gif" width=20
align=bottom> is the measurement error covariance matrix.
<P><IMG height=14 alt=tex2html_wrap120
src="HP48G(X) Discrete Kalman Filter Simulator.files/img14.gif" width=19
align=bottom> is the power spectral density matrix associated with the forcing
function <IMG height=9 alt=tex2html_wrap121
src="HP48G(X) Discrete Kalman Filter Simulator.files/img15.gif" width=10
align=bottom> .
<P><IMG height=11 alt=tex2html_wrap122
src="HP48G(X) Discrete Kalman Filter Simulator.files/img16.gif" width=16
align=bottom> is a <IMG height=13 alt=tex2html_wrap123
src="HP48G(X) Discrete Kalman Filter Simulator.files/img17.gif" width=48
align=bottom> measurement noise
<P><IMG height=11 alt=tex2html_wrap122
src="HP48G(X) Discrete Kalman Filter Simulator.files/img16.gif" width=16
align=bottom> and <IMG height=11 alt=tex2html_wrap115
src="HP48G(X) Discrete Kalman Filter Simulator.files/img9.gif" width=20
align=bottom> are assumed to be uncorrelated white sequences with known
covariance (and 0 mean; they are white noise).
<P>Giving the inputs: <IMG height=18 alt=tex2html_wrap126
src="HP48G(X) Discrete Kalman Filter Simulator.files/img18.gif" width=194
align=bottom> and <IMG height=9 alt=tex2html_wrap127
src="HP48G(X) Discrete Kalman Filter Simulator.files/img19.gif" width=10
align=bottom> it performs the following tasks.
<P>1) Calculates the state transition matrix <IMG height=17 alt=tex2html_wrap111
src="HP48G(X) Discrete Kalman Filter Simulator.files/img6.gif" width=17
align=bottom> that relates states <IMG height=11 alt=tex2html_wrap109
src="HP48G(X) Discrete Kalman Filter Simulator.files/img4.gif" width=17
align=bottom> to <IMG height=13 alt=tex2html_wrap114
src="HP48G(X) Discrete Kalman Filter Simulator.files/img8.gif" width=34
align=bottom>
<P>2) Calculates the covariance matrix <IMG height=18 alt=tex2html_wrap131
src="HP48G(X) Discrete Kalman Filter Simulator.files/img20.gif" width=20
align=bottom> associated with <IMG height=11 alt=tex2html_wrap115
src="HP48G(X) Discrete Kalman Filter Simulator.files/img9.gif" width=20
align=bottom>
<P>3) Generates white noise for the state and output stages
<P>4) Simulates a Kalman system
<P>5) Graphs the noisy state,
<P>6) Graphs the estimated state,
<P>6) Graphs the Kalman gain,
<P>6) Graphs the noisy output.
<P>Some of the programs found in the directory are:
<P>" <IMG height=18 alt=tex2html_wrap133
src="HP48G(X) Discrete Kalman Filter Simulator.files/img21.gif" width=37
align=bottom> " Calculates the <IMG height=18 alt=tex2html_wrap134
src="HP48G(X) Discrete Kalman Filter Simulator.files/img22.gif" width=13
align=bottom> matrix and stores it in "Q"
<P>" <IMG height=15 alt=tex2html_wrap135
src="HP48G(X) Discrete Kalman Filter Simulator.files/img23.gif" width=63
align=bottom> " Generates the state white sequence <IMG height=11
alt=tex2html_wrap115
src="HP48G(X) Discrete Kalman Filter Simulator.files/img9.gif" width=20
align=bottom>
<P>" <IMG height=15 alt=tex2html_wrap137
src="HP48G(X) Discrete Kalman Filter Simulator.files/img24.gif" width=66
align=bottom> " Generates the output white sequence <IMG height=11
alt=tex2html_wrap122
src="HP48G(X) Discrete Kalman Filter Simulator.files/img16.gif" width=16
align=bottom>
<P>" <IMG height=15 alt=tex2html_wrap139
src="HP48G(X) Discrete Kalman Filter Simulator.files/img25.gif" width=65
align=bottom> " Calls subroutines and stores results to be ploted
<P>" <IMG height=15 alt=tex2html_wrap140
src="HP48G(X) Discrete Kalman Filter Simulator.files/img26.gif" width=71
align=bottom> " Plots the estimated states <IMG height=13 alt=tex2html_wrap141
src="HP48G(X) Discrete Kalman Filter Simulator.files/img27.gif" width=10
align=bottom>
<P>" <IMG height=15 alt=tex2html_wrap142
src="HP48G(X) Discrete Kalman Filter Simulator.files/img28.gif" width=54
align=bottom> " Plots the noisy states <IMG height=9 alt=tex2html_wrap127
src="HP48G(X) Discrete Kalman Filter Simulator.files/img19.gif" width=10
align=bottom>
<P>" <IMG height=15 alt=tex2html_wrap144
src="HP48G(X) Discrete Kalman Filter Simulator.files/img29.gif" width=54
align=bottom> " Plots the Kalman gain <IMG height=15 alt=tex2html_wrap145
src="HP48G(X) Discrete Kalman Filter Simulator.files/img30.gif" width=13
align=bottom>
<P>" <IMG height=15 alt=tex2html_wrap146
src="HP48G(X) Discrete Kalman Filter Simulator.files/img31.gif" width=51
align=bottom> " Plots the noisy output <IMG height=9 alt=tex2html_wrap147
src="HP48G(X) Discrete Kalman Filter Simulator.files/img32.gif" width=8
align=bottom>
<P>The directory also contains other subroutines that make the state and output
calculations, manipulate lists, etc.
<P>Every single program in the directory must be present.
<P><B>Example</B>
<P>Assume that we have a stationary Gaussian process with the following spectral
density function:
<P><IMG height=44 alt=displaymath274
src="HP48G(X) Discrete Kalman Filter Simulator.files/img33.gif" width=347
align=bottom>
<P>Discrete noisy measurements of <IMG height=13 alt=tex2html_wrap148
src="HP48G(X) Discrete Kalman Filter Simulator.files/img34.gif" width=9
align=bottom> are available at <IMG height=17 alt=tex2html_wrap149
src="HP48G(X) Discrete Kalman Filter Simulator.files/img35.gif" width=118
align=bottom> . Measurement errors are uncorrelated and have a variance <IMG
height=16 alt=tex2html_wrap150
src="HP48G(X) Discrete Kalman Filter Simulator.files/img36.gif" width=17
align=bottom> of 0.2.
<P>
<P>The <IMG height=13 alt=tex2html_wrap151
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