# Question: consider the continuous time system given by the state equations...

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consider the continuous time system given by the state equations

x(t)=$\left[\begin{array}{cc}-1.5& 1\\ 1& 0\end{array}\right]$ x(t) +$\left[\begin{array}{c}1\\ 0\end{array}\right]$ u(t)

y1(t)=$\left[\begin{array}{cc}1& 0\end{array}\right]$ x(t)

y2(t)=$\left[\begin{array}{cc}0& -1\end{array}\right]$ x(t)

find the system transfer functions from u to y1 and from u to y2

Design a state feedback control law integral action to achieve robust tracking of step references r in y2(t) that is

u(t)=-Kx(t)-K ${\mathrm{Z}}_{\mathrm{z}}$(t)

z(t)=r-y2(t)

find the matrices K and Kz to place all the closed loop eigenvalues at -2. would it be possible to achieve robust tracking of constant references also in y1(t)? justify your answer.

write the equations of an observer for the system (1) using only y1(t) as measurement and design the observer gain in order to obtain an estimate x(t) of the state with observer eigenvalues at -10