(Solved): 5.2-1 Eigenstructure Assignment with Full State Feedback. The short-period approximation of an air ...

5.2-1 Eigenstructure Assignment with Full State Feedback. The short-period approximation of an aircraft with the $$\mathrm{cg}\mathrm{far}$$ aft might be described by $$\stackrel{?}{x}=\left[\begin{array}{ccc}?1.10188& 0.90528& ?0.00212\\ 4.0639& ?0.77013& ?0.169190\\ 0& 0& ?10\end{array}\right]x+\left[\begin{array}{cc}0& 0\\ 0& 1\\ 10& 0\end{array}\right]u\text{, }$$ which includes an elevator actuator of $$10/(\mathrm{s}+10)$$. The state is $$x=$$ $${\left[\begin{array}{lll}a& q& {?}_e\end{array}\right]}^{\mathrm{T}}$$. An extra control input $$u_2$$ has been added to illustrate the extra design freedom available in multivariable systems. (a) Find the poles. (b) To conform to flying qualities specifications, it is desired to assign closedloop short-period eigenvalues $${?}_1$$ and $${?}_2$$ of $$?2\pm j2$$. The actuator pole does not matter but may be assigned to $$s=?15$$ to speed up its response. The desired closed-loop eigenvectors are $$\begin{array}{l}{v}_1=v_2^{?}=[0.20+j0.35?0.98+j0.070{]}^{\mathrm{T}},\\ v_3={\left[\begin{array}{lll}0& 0& 1\end{array}\right]}^{\mathrm{T}}\end{array}$$ Find the state feedback gain $$K$$ in $$u=?Kx$$ to assign the desired eigenstructure.