# "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 x vec (k+1)=Ax vec (k)+bu vec (k),y vec (k)=cx vec (k) where b=(0,1)^T,c=(1,0),A [201−g] for some g in R Find a feedback regulation (if there is any) of the form u(okay)=−Kxhat(ok) where xhat(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(ok)=xvec (k)−xhat(okay) go to zero after a few finite time. layout the kingdo observer and the block diagram.

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
$\stackrel{\to }{x}\left(k+1\right)=A\stackrel{\to }{x}\left(k\right)+b\stackrel{\to }{u}\left(k\right),\phantom{\rule{thickmathspace}{0ex}}\stackrel{\to }{y}\left(k\right)=c\stackrel{\to }{x}\left(k\right)$
where $b=\left(0,1{\right)}^{T},\phantom{\rule{thickmathspace}{0ex}}c=\left(1,0\right),\phantom{\rule{thickmathspace}{0ex}}A=\left[\begin{array}{ccc}2& & 1\\ 0& & -g\end{array}\right]$ for some $g\in \mathbb{R}$
Find a feedback regulation (if there is any) of the form $u\left(k\right)=-K\stackrel{^}{x}\left(k\right)$ where $\stackrel{^}{x}\left(k\right)$ 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 $e\left(k\right)=\stackrel{\to }{x}\left(k\right)-\stackrel{^}{x}\left(k\right)$ 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
$\left[\stackrel{\to }{x}\left(k+1\right)\phantom{\rule{thickmathspace}{0ex}}\stackrel{\to }{e}\left(k+1\right){\right]}^{T}=\left[\begin{array}{ccc}A-bK& & Bk\\ O& & A-LC\end{array}\right]\left[\stackrel{\to }{x}\left(k\right)\phantom{\rule{thickmathspace}{0ex}}\stackrel{\to }{e}\left(k\right){\right]}^{T}$
With characteristic equation
$\chi \left(z\right)=|zI-A+bK|\phantom{\rule{thickmathspace}{0ex}}|zI-A+LC|={\chi }_{K}\left(z\right){\chi }_{L}\left(z\right)$
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 ${\chi }_{K}\left(z\right)=\left(z-2\right)\left(z+g+\beta \right)+a$.
So we can select some eigenvalues inside the unit circle and determine $a,\beta$ in terms of g. Choosing e.g. ${\lambda }_{1,2}=±1/2$ 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)
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Kaya Garza
Your approach is right. In short, find a dynamic compensator for the observer and change coordinates so that you have the state and estimation error. That matrix will be upper triangular. Your result looks correct.
In order to design the observer and compensator gains, remember that the eigenvalues of an upper triangular matrix are on the diagonal. For a block-UT matrix, the eigenvalues are the eigenvalues of the diagonal blocks. Thus you only need your $A-bK$ and $A-LC$ to be Schur for the system to satisfy your requirements.
You can do this using pole placement or guess and check.
Since the matrices are $2×2$ either of these will be simple to do.