LMIs in Control/Discrete-Time Systems/Discrete-Time H2-Optimal Observer

The System


\begin{align}

x_{k+1}&=A_{d}x_{k}+B_{d1}w_{k},\\

y_{k}&=C_{c2}x_{k} + D_{d21}w_{k}\\ \end{align} $$

Where it is assumed that :$$(A_{d}, C_{d2})$$ is detectable.

The Data
The matrices $$ A_{d}, B_{d1}, C_{cd2}, C_{cd1}, D_{d21} $$.

The Optimization Problem
An observer of the form:



\begin{align}

\hat x_{k+1}&=A_{d}\hat x_{k}+ L_{d}(y_{k}-\hat y_{k}),\\

\hat y_{k}&=C_{d2} \hat x_{k} \\ \end{align} $$

is to be designed, where $$ L_{d} \in R^{n_{x} \times n_{y}}$$ is the observer gain.

Defining the error state $$ e_{k} = x_{k} - \hat x_{k}$$, the error dynamics are found to be

$$e_{k+1} = (A_{d} - L_{d}C_{d2} e_{k} + (B_{d1} - L_{d}D_{d21})w_{k}$$,

and the performance output is defined as

$$z_{k} = C_{d1}e_{k}$$.

The observer gain $$L_{d}$$ is to be designed such that the $$H_{2}$$ of the transfer matrix from $$w_{k}$$ to $$z_{k}$$, given by

$$L_{d} \begin{align} T(z) &= C_{d1}(z1 - (A_{d} - L_{d}C_{d2}))^{-1}(B_{d1} - L_{d}D_{d21}),\\ \end{align} $$

is minimized.

The LMI: Discrete-Time H2-Optimal Observer
The discrete-time $$H_{2}$$-optimal observer gain is synthesized by solving for $$P \in S^{n_{x}}$$, $$Z \in S^{n_{z}}$$, $$G_{d} \in R^{n_{x} \times n_{y}}$$, and $$v \in R_{>0}$$ that minimize $$J(v) = v$$ subject to $$P>0, Z>0$$,



\begin{align}

\begin{bmatrix}P & PA_{d}-G_{d}C_{d2} & PB_{d1}-G_{d}D_{d21}\\
 * & P & 0\\
 * & * & 1\end{bmatrix}&>0,\\

\begin{bmatrix} Z & PC_{d1}\\
 * & P\end{bmatrix}&>0,\\

\operatorname{tr}Z<v

\end{align}$$

Conclusion:
The $$H_{2}$$-optimal observer gain is recovered by $$L_{d} = P^{-1}G_{d}$$ and the $$H_{2}$$ norm of $$T(z)$$ is $$\mu = \sqrt{v}$$.

Implementation
https://github.com/rezajamesahmed/LMImatlabcode/blob/master/Discrete_Time_H2_Optimal_Observer_LMIs_Wikibook_Example.m

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