$\%$ Open$-$Loop Fault Detection, Isolation and Recovery System

$\%$

$\%$INITIAL SEGMENT

$\%$

clear all

$\%$ Define model parameters

U$0=30$; $\%$ m$/$s

delta$\_$r$\_$max $=60*$ pi $/180$; $\%$ Maximum rudder deflection $($rad$)$

delta$\_$r$\_$rate$\_$max $=30*$ pi $/180$; $\%$ Maximum rudder rate limit $($rad$)$

$\%$fault$\_$time $=30$; $\%$ Time at which the stepwise fault occurs

$\%$fault$\_$value $=20*$ pi$/180$; $\%$ Fault value to be added to the measurement

$\%$fault$\_$time $=20$; $\%$ Time at which the driftwise fault occurs

$\%$fault$\_$value $=0\mathrm{.}5*$ pi$/180$; $\%$ Fault value to be added to the measurement

$\%$ Initial Conditions of all inputs, states, and state derivatives

u $=[0,0\mathrm{.}43625]\text{'}$; $\%$ Define Initial Simulation Input

x $=[0,0,0,0,0]\text{'}$; $\%$ Define Initial Model States

xdot $=[0,0,0,0,0]\text{'}$; $\%$ Define Initial Model State Derivatives

$\%$ Define parameters for the simulation

stepsize $=0\mathrm{.}01$; $\%$ Simulation step size

comminterval $=0\mathrm{.}1$; $\%$ Communications interval

EndTime $=100$; $\%$ Simulation End Time

i $=0$; $\%$ Initialize counter for data storage

$\%$ END OF INITIAL SEGMENT $-$ all parameters initialized

$\%$ DYNAMIC SEGMENT

$\%$

for time $=0$:stepsize:EndTime

$\%$ STORAGE SECTION

$\%$ Store time, state, and state derivative data every communication interval

if rem$($time$,$ comminterval$)==0$

i $=$ i $+1$; $\%$ Increment counter

tout$($i$)=$ time; $\%$ Store time

xout$($i$,$ :) $=$ x; $\%$ Store states

xdout$($i$,$ :) $=$ xdot; $\%$ Store state derivatives

uout$($i$,$ :) $=$ u; $\%$ Store input

end

$\%$ End of STORAGE SECTION

$\%$ CONTROL SECTION

u$(1)=0$;

if x$(5)>0\mathrm{.}43625\%$ If heading is greater than $25\backslash$deg $,$ set rudder deflection to $-25\backslash$deg

$\%$ u$(2)=-0\mathrm{.}43625$;

u$(2)=$ u$(2)-$ delta$\_$r$\_$rate$\_$max $*$ stepsize;

if u$(2)<-0\mathrm{.}43625$

u$(2)=-0\mathrm{.}43625$;

end

elseif x$(5)<-0\mathrm{.}43625\%$ Otherwise, set rudder deflection to $+25\backslash$deg

$\%$ u$(2)=0\mathrm{.}43625$;

u$(2)=$ u$(2)+$ delta$\_$r$\_$rate$\_$max $*$ stepsize;

if u$(2)>0\mathrm{.}43625$

u$(2)=0\mathrm{.}43625$;

end

else

u$(2)=$ u$(2)$;

end

$\%$ Apply stepwise fault to the sensor measurement

$\%$if time $==$ fault$\_$time

$\%$x$(5)=$ x$(5)+$ fault$\_$value;

$\%$end

$\%$ Apply driftwise fault to the sensor measurement

$\%$if time $>=$ fault$\_$time

$\%$x$(5)=$ x$(5)+$ fault$\_$value;

$\%$end

$\%$ Apply stepwise fault to the rudder measurement

$\%$if time $==$ fault$\_$time

$\%$u$(2)=$ u$(2)+$ fault$\_$value;

$\%$end

$\%$ Apply driftwise fault to the rudder measurement

$\%$if time $>=$ fault$\_$time

$\%$u$(2)=$ u$(2)+$ fault$\_$value;

$\%$end

$\%$ DERIVATIVE SECTION

xdot $=$ FDIRModel$($x$,$ u$)$;

$\%$ INTEGRATION SECTION

x $=$ rk$4$int$(\text{'}$FDIRModel$\text{'},$ stepsize, x$,$ u$)$; $\%$ Fourth Order Runge$-$Kutta

end

$\%$ END OF DYNAMIC SEGMENT

Given the above MATLAB scripts, Develop a kalman filter algorithm in MATLAB script to analyze the system's behaviour to determine when and

where these faults occur during the $$Open$-$Loop$25\backslash$deg $/-25\backslash$deg zig$-$zag manoeuvre