Discrete-Time H2-Optimal Dynamic Output Feedback Control
A Dynamic Output feedback controller is designed for a Continuous Time system, to minimize the H2 norm of the closed loop system with exogenous input
and performance output
.
Continuous-Time LTI System with state space realization ![{\displaystyle (A,B,C,D)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9512ac0005d92f0726896726d2f735448455f9c)
![{\displaystyle {\begin{aligned}{\dot {x}}&=A_{x}+B_{1}w+B_{2}u\\z&=C_{1}x+D_{11}w_{k}+D_{12}u\\y&=C_{2}x+D_{21}w+D_{22}u\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/228608afe66b4d956d8177f9738d408c42eaa771)
The matrices: System ![{\displaystyle (A,B_{1},B_{2},C_{1},C_{1},D_{11},D_{12},D_{21},D_{22}),X_{1},Y_{1},Z,,X_{2},Y_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c579825eb9cba5e23cec0499387ffe54bc0eb4a5)
Controller
The following feasibility problem should be optimized:
is minimized while obeying the LMI constraints.
Solve for
and
that minimize
subject to
![{\displaystyle {\begin{aligned}{\begin{bmatrix}AY_{1}+Y_{1}A^{T}+B_{2}C_{n}+C_{n}^{T}B_{2}^{T}&A+A_{n}^{T}+B_{2}D_{n}C_{2}&B_{1}+B_{2}D_{n}D_{21}\\*&X_{1}A+A^{T}X_{1}+B_{n}C_{2}+C_{2}^{T}B_{n}^{T}&X_{1}B_{1}+B_{n}D_{21}\\*&*&-\mathbf {1} \end{bmatrix}}&>0,\\{\begin{bmatrix}X_{1}&\mathbf {1} &Y_{1}C_{1}^{T}+C_{n}^{T}D_{12}^{T}\\*&Y_{1}&C_{1}^{T}+C_{2}^{T}D_{n}^{T}D_{12}^{T}\\*&*&Z\end{bmatrix}}&>0,\\D_{11}+D_{12}D_{n}D_{21}=0\\{\begin{bmatrix}X_{1}&\mathbf {1} \\*&Y_{1}\end{bmatrix}}&>0,\\trZ<\nu \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5b62de57a59f51661f519865f159ecfc0a4620)
The controller is recovered by
![{\displaystyle {\begin{aligned}&A_{c}=A_{k}-B_{c}(1-D_{22}D_{c})^{-1}D_{22}C_{c}\\&B_{c}=B_{k}(1-D_{c}D_{22})\\&C_{c}=(1-D_{c}D_{22})C_{k}\\&D_{c}=(1+D_{k}D_{22})^{-1}D_{k}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d8da82dda7bc665c949b7ddc2b9f6a561d6b6a)
where,
![{\displaystyle {\begin{aligned}{\begin{bmatrix}A_{k}&B_{k}\\C_{k}&D_{k}\end{bmatrix}}&={\begin{bmatrix}X_{2}&X_{1}B_{2}\\\mathbf {0} &1\end{bmatrix}}^{-1}({\begin{bmatrix}A_{n}&B_{n}\\C_{n}&D_{n}\end{bmatrix}}-{\begin{bmatrix}X_{1}AY_{1}&\mathbf {0} \\\mathbf {0} &\mathbf {0} \end{bmatrix}}){\begin{bmatrix}Y_{2}^{T}&\mathbf {0} \\CY_{1}&\mathbf {1} \end{bmatrix}}^{-1}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24f8b56c5054990bcc8b1e067253e2e6b2ad530b)
and the matrices
and
satisfy
. If
then
and ![{\displaystyle D_{c}=D_{k}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55c2196d9c9dcc5fd78faa0235dbfcb1b7f55cf4)
Given
and
, the matrices
and
can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.
If
,
and
, then it is often simplest to choose
in order to satisfy the equality constraint
The Continuous-Time H2-Optimal Dynamic Output feedback controller is the system
Discrete Time H2 Optimal Dynamic Output Feedback Control
Continuous Time H∞ Optimal Dynamic Output Feedback Control
A list of references documenting and validating the LMI.