Question on designing a state observer for discrete time system

by bolzano   Last Updated January 14, 2018 13:20 PM

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

Consider the following discrete time system

$$\vec x (k + 1) = A\vec x(k) + b \vec u(k), \; \vec y(k) = c \vec x(k)$$

where $b = (0,1)^T, \; c = (1, 0), \; A = \begin{bmatrix} 2 && 1 \\ 0 && -g \end{bmatrix}$ for some $g \in \mathbb R$

Find a feedback law (if there is any) of the form $u(k) = -K \hat x(k)$ where $\hat x(k)$ is the state estimation vector that is produced by a linear full-order state observer such that the state of the system and the estimation error $e(k) = \vec x(k) - \hat x(k)$ go to zero after some finite time. Design the state observer and the block diagram.

My approach

It is clear that the eigenvalues of the system are $\lambda_1 = -2, \lambda_2 = -g$ (therefore it is NOT BIBO stable) and that the pair $(A, b)$ is controllable for every value of $g$, as well a the pair $(A, c)$ is observable for all values of $g$. Therefore we can shift the eigenvalues by choosing a gain matrix $K$ such that our system is stable, i.e. it has its eigenvalues inside the unit circle $|z| = 1$.

The state observer equation is

$$[\vec x(k + 1) \; \vec e(k + 1)]^T = \begin {bmatrix} A - bK && Bk \\ O && A - LC\end{bmatrix} [\vec x(k) \; \vec e(k)]^T$$

With characteristic equation $$\chi (z) = | z I - A + bK | \; |zI - A + LC| = \chi_K(z) \chi_L(z)$$

Also consider $$K = \begin {bmatrix} k_1 && k_2 \\ k_3 && k_4 \end{bmatrix}$$ and let $a = k_1 + k3, \; \beta = k_2 + k_4$

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 $a = 3g + 33 / 8, \; \beta 9 /4 - g, \; g \in \mathbb R$

Questions

I want to ask the following:

1. 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 $| \lambda_{1,2} | < 1$?
2. Should I determine $L$ matrix as well since the error must also vanish? (because it is not asked)
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