a$)$ Given a reference r$($t$)=$ sin$(0\mathrm{.}316$ t$),$ by observation, determine the value of k at which the system becomes unstable

b$)$ Given the same reference r$($t$)=$ sin$(0\mathrm{.}316$ t$),$ by observation, determine the time delay at which the original system becomes unstable

HINTS :

The described system can be simulated using the following code in

MATLAB.

clear ;

clc ;

close all ;

$\%$ System

K $=1$;

s $=$ tf $([10],1)$ ;

H $=1/$ s $^2$;

PID $=1+0\mathrm{.}1*$ s $+0\mathrm{.}01/$ s ;

G $=$ PID $*$ H ;

$\%$ Simulation

t $=$ linspace $(0,100,100000)$ ;

w$\_$pc $=0\mathrm{.}316$;

r $=$ sin $($ w$\_$pc $*$ t $)$ ;

y $=$ lsim $(($ K $*$ G $)/(1+$ K $*$ G $),$ r $,$ t $)$ ;

$\%$ Figures

figure $()$

clf ;

hold on

plot $($t $,$ r $)$

plot $($t $,$ y $)$

xlabel $(\text{'}$ time $($ t $)\text{'})$

ylabel $(\text{'}$ output $($ y $)\text{'})$

title $("$ Gain margin $-$ time $($ t $)$ vs output $($ y $)")$

legend $(\text{'}$ Reference r $($ t $)\text{'},\text{'}$ Output y $($ t $)\text{'})$

A time delay on the input can be simulated using the following code.

clear ;

clc ;

close all ;

$\%$ System

delay $=0\mathrm{.}001$; $\%$ Parameter to update input delay

K $=1$;

s $=$ tf $([10],1)$ ;

H $=1/$ s $^2$;

H$\_$tau $=$ tf $($ H $.$ Numerator $,$ H $.$ Denominator $,\text{'}$ InputDelay $\text{'},$ delay $)$ ;

PID $=1+0\mathrm{.}1*$ s $+0\mathrm{.}01/$ s ;

G$2\_$prime $=$ H$\_$tau $*$ PID ;

$5$

$\%$ Simulation

t $=$ linspace $(0,100,100000)$ ;

w$\_$pc $=0\mathrm{.}316$;

r$2=$ sin $($ w$\_$pc $*$ t $)$ ;

y$2=$ lsim $($ minreal $($ K $*$ G$2\_$prime $/(1+$ K $*$ G$2\_$prime $)),$ r$2,$ t $)$ ;

$\%$ Figure

figure $()$

clf ;

hold on

plot $($t $,$ r$2)$

plot $($t $,$ y$2)$

xlabel $(\text{'}$ time $($ t $)\text{'})$

ylabel $(\text{'}$ output $($ y $)\text{'})$

title $("$ Phase margin $-$ time $($ t $)$ vs output $($ y $)")$

legend $(\text{'}$ Reference r $($ t $)\text{'},\text{'}$ Output y $($ t $)\text{'})$