Question on designing a state observer for discrete time system

I came through this problem while studying for an exam in control systems:

Consider the following discrete time system

$$\overrightarrow{x}(k+1)=A\overrightarrow{x}(k)+b\overrightarrow{u}(k),\phantom{\rule{thickmathspace}{0ex}}\overrightarrow{y}(k)=c\overrightarrow{x}(k)$$

where $b=(0,1{)}^{T},\phantom{\rule{thickmathspace}{0ex}}c=(1,0),\phantom{\rule{thickmathspace}{0ex}}A=\left[\begin{array}{ccc}2& & 1\\ 0& & -g\end{array}\right]$ for some $g\in \mathbb{R}$

Find a feedback regulation (if there is any) of the form $u(k)=-K\hat{x}(k)$ where $\hat{x}(k)$ is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders $e(k)=\overrightarrow{x}(k)-\hat{x}(k)$ go to zero after a few finite time. layout the kingdom observer and the block diagram.

My method

it is clean that the eigenvalues of the machine are ${\lambda}_{1}=2,{\lambda}_{2}=-g$ (consequently it is not BIBO solid) and that the pair (A,b) is controllable for every fee of g, as nicely a the pair (A,c) is observable for all values of g. consequently we will shift the eigenvalues with the aid of deciding on a benefit matrix okay such that our device is strong, i.e. it has its eigenvalues inside the unit circle $|z|=1$.

The state observer equation is

$$[\overrightarrow{x}(k+1)\phantom{\rule{thickmathspace}{0ex}}\overrightarrow{e}(k+1){]}^{T}=\left[\begin{array}{ccc}A-bK& & Bk\\ O& & A-LC\end{array}\right][\overrightarrow{x}(k)\phantom{\rule{thickmathspace}{0ex}}\overrightarrow{e}(k){]}^{T}$$

With characteristic equation

$$\chi (z)=|zI-A+bK|\phantom{\rule{thickmathspace}{0ex}}|zI-A+LC|={\chi}_{K}(z){\chi}_{L}(z)$$

Also consider

$$K=\left[\begin{array}{ccc}{k}_{1}& & {k}_{2}\\ {k}_{3}& & {k}_{4}\end{array}\right]$$

and let $a={k}_{1}+{k}_{3},\phantom{\rule{thickmathspace}{0ex}}\beta ={k}_{2}+{k}_{4}$

Then ${\chi}_{K}(z)=(z-2)(z+g+\beta )+a$.

So we can select some eigenvalues inside the unit circle and determine $a,\beta $ in terms of g. Choosing e.g. ${\lambda}_{1,2}=\pm 1/2$ we get $a=3g+33/8,\phantom{\rule{thickmathspace}{0ex}}\beta =9/4-g,\phantom{\rule{thickmathspace}{0ex}}g\in \mathbb{R}$

Questions

I want to ask the following:

Is my approach correct? Should I select the eigenvalues myself since I am asked to design the observer or should I just solve the characteristic equation and impose $|{\lambda}_{1,2}|<1$?

Should I determine L matrix as well since the error must also vanish? (because it is not asked)

I came through this problem while studying for an exam in control systems:

Consider the following discrete time system

$$\overrightarrow{x}(k+1)=A\overrightarrow{x}(k)+b\overrightarrow{u}(k),\phantom{\rule{thickmathspace}{0ex}}\overrightarrow{y}(k)=c\overrightarrow{x}(k)$$

where $b=(0,1{)}^{T},\phantom{\rule{thickmathspace}{0ex}}c=(1,0),\phantom{\rule{thickmathspace}{0ex}}A=\left[\begin{array}{ccc}2& & 1\\ 0& & -g\end{array}\right]$ for some $g\in \mathbb{R}$

Find a feedback regulation (if there is any) of the form $u(k)=-K\hat{x}(k)$ where $\hat{x}(k)$ is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders $e(k)=\overrightarrow{x}(k)-\hat{x}(k)$ go to zero after a few finite time. layout the kingdom observer and the block diagram.

My method

it is clean that the eigenvalues of the machine are ${\lambda}_{1}=2,{\lambda}_{2}=-g$ (consequently it is not BIBO solid) and that the pair (A,b) is controllable for every fee of g, as nicely a the pair (A,c) is observable for all values of g. consequently we will shift the eigenvalues with the aid of deciding on a benefit matrix okay such that our device is strong, i.e. it has its eigenvalues inside the unit circle $|z|=1$.

The state observer equation is

$$[\overrightarrow{x}(k+1)\phantom{\rule{thickmathspace}{0ex}}\overrightarrow{e}(k+1){]}^{T}=\left[\begin{array}{ccc}A-bK& & Bk\\ O& & A-LC\end{array}\right][\overrightarrow{x}(k)\phantom{\rule{thickmathspace}{0ex}}\overrightarrow{e}(k){]}^{T}$$

With characteristic equation

$$\chi (z)=|zI-A+bK|\phantom{\rule{thickmathspace}{0ex}}|zI-A+LC|={\chi}_{K}(z){\chi}_{L}(z)$$

Also consider

$$K=\left[\begin{array}{ccc}{k}_{1}& & {k}_{2}\\ {k}_{3}& & {k}_{4}\end{array}\right]$$

and let $a={k}_{1}+{k}_{3},\phantom{\rule{thickmathspace}{0ex}}\beta ={k}_{2}+{k}_{4}$

Then ${\chi}_{K}(z)=(z-2)(z+g+\beta )+a$.

So we can select some eigenvalues inside the unit circle and determine $a,\beta $ in terms of g. Choosing e.g. ${\lambda}_{1,2}=\pm 1/2$ we get $a=3g+33/8,\phantom{\rule{thickmathspace}{0ex}}\beta =9/4-g,\phantom{\rule{thickmathspace}{0ex}}g\in \mathbb{R}$

Questions

I want to ask the following:

Is my approach correct? Should I select the eigenvalues myself since I am asked to design the observer or should I just solve the characteristic equation and impose $|{\lambda}_{1,2}|<1$?

Should I determine L matrix as well since the error must also vanish? (because it is not asked)