clc

clear all

syms x x$1$ x$2$ x$3$ x$4$ u

A $=[-861008083\mathrm{.}4$; $86100-8488$; $1000$; $0100]$;

B $=[1315\mathrm{.}6$;$-1315\mathrm{.}6$;$0$;$0]$;

W$=$ctrb$($A$,$B$)$;

E$=$rank$($W$)$

$[$n$,$ m$]=$ size$($B$)$;

$[$Q$,$ R$]=$ qr$($B$)$;

T $=[$Q$(($m$+1)$:n$,$:$)$; Q$(1$:m$,$:$)]$;

Acap $=$ T $*$ A $*$ inv$($T$)$;

Bcap $=$ T $*$ B;

A$11=$ Acap$(1$:n$-$m$,1$:n$-$m$)$;

A$12=$ Acap$(1$:n$-$m$,$n$-$m$+1$:n$)$;

A$21=$ Acap$($n$-$m$+1$:n$,1$:n$-$m$)$;

A$22=$ Acap$($n$-$m$+1$:n$,$n$-$m$+1$:n$)$;

B$2=$ Bcap$($n$-$m$+1$:n$,1$:m$)$;

P $=[-3-4-5]$;

g $=$ place$($A$11,$ A$12,$ P$)$;

Gcap $=[$g $1]$;

G $=-$Gcap $*$ T;

ueq $=-($inv$($G $*$ B$)*$ G $*$ A$)*[$x$1$; x$2$; x$3$; x$4]$;

s $=0\mathrm{.}1643*$x$1-1\mathrm{.}2499*$x$2+0\mathrm{.}2097*$x$3-16\mathrm{.}7608*$x$4$;

usw $=$ zeros$($size$($s$))$;

tspan $=0$:$0\mathrm{.}01$:$10$;

x$0=[-5$; $5$; $10$; $-3]$;

$[$t$,$ x$]=$ ode$45($@$($t$,$ x$)$ systemODE$($t$,$ x$,$ G$,$ B$,$ A$,$ usw$),$ tspan, x$0)$;

u $=$ zeros$($size$($t$))$;

v$(1)=0\mathrm{.}01$; $\%$ Initial value for z

for i $=1$:length$($t$)$

ueq $=-($inv$($G $*$ B$)*$ G $*$ A$)*$ x$($i$,1$:$4)\text{'}$;

s $=0\mathrm{.}1643*$x$($i$,1)-1\mathrm{.}2499*$x$($i$,2)+0\mathrm{.}2097*$x$($i$,3)+16\mathrm{.}7608*$x$($i$,4)$;

$\%$ Numerical integration of z$\_$dot

v$\_$dot $=-0\mathrm{.}01*$sign$($s$)$;

v$($i$+1)=$ v$($i$)+0\mathrm{.}01*$ v$\_$dot; $\%$ Using Euler's method

$\%$ Calculate u$\_$sw

usw$($i$)=-0\mathrm{.}01*$ s$^0\mathrm{.}5*$sign$($s$)+$ v$($i$+1)$;

$\%$ Total control input

u$($i$)=$ ueq $+$ usw$($i$)$;

end

$\%$ Plot u vs t

figure;

subplot$(2,1,1)$;

plot$($t$,$ u$)$;

xlabel$(\text{'}$Time $($s$)\text{'})$;

ylabel$(\text{'}$u$\text{'})$;

title$(\text{'}$Control Input u vs Time'$)$;

$\%$ Plot states vs t

subplot$(2,1,2)$;

plot$($t$,$ x$)$;

legend$(\text{'}$x$1\text{'},\text{'}$x$2\text{'},\text{'}$x$3\text{'},\text{'}$x$4\text{'})$;

xlabel$(\text{'}$Time $($s$)\text{'})$;

ylabel$(\text{'}$States$\text{'})$;

title$(\text{'}$States x$1,$ x$2,$ x$3,$ x$4$ vs Time'$)$;

$\%$ Define the system of differential equations

function dxdt $=$ systemODE$($t$,$ x$,$ G$,$ B$,$ A$,$ usw$)$

ueq $=-($inv$($G$*$B$)*$G$*$A$)*$x;

s $=0\mathrm{.}1643*$x$(1)-1\mathrm{.}2499*$x$(2)+0\mathrm{.}2097*$x$(3)-16\mathrm{.}7608*$x$(4)$;

u $=$ ueq $+$ usw;

$\%$ State equations

dxdt $=[-861*$x$(1)+8083\mathrm{.}4*$x$(4)+1315\mathrm{.}6*$u; $861*$x$(1)-8488*$x$(4)-1315\mathrm{.}6*$u ; x$(1)$; x$(2)]$;

end.

THis is the matlab code for super twisted sliding mode control to stablize the system. but one of the system state x$3$ is unstable. please debug the code