Discrete-Time H∞-Optimal Dynamic Output Feedback Control
In this section, a Dynamic Output feedback controller is designed for a Continuous Time system, to minimize the H∞ 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}}&=Ax+B_{1}w+B_{2}u\\z&=C_{1}x+D_{11}w+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/5f0337c014855f31c7ea0f21bbace8268a96e4d0)
The matrices: System ![{\displaystyle (A,B_{1},B_{2},C_{1},C_{2},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/35b72844f404c8763192cec3fe64e8d1a34a8bcc)
Controller
The following feasibility problem should be optimized:
is minimized while obeying the LMI constraints.
Solve for
that minimize
subject to
where
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/e16a39232e9bf8c5834c3e998109940089b97e70)
where,
![{\displaystyle {\begin{aligned}{\begin{bmatrix}A_{k}&B_{k}\\C_{k}&D_{k}\end{bmatrix}}&={\begin{bmatrix}X_{2}&X_{1}B_{d2}\\0&1\end{bmatrix}}^{-1}({\begin{bmatrix}A_{n}&B_{n}\\C_{n}&D_{n}\end{bmatrix}}-{\begin{bmatrix}X_{1}AY_{1}&0\\0&0\end{bmatrix}}){\begin{bmatrix}Y_{2}^{T}&0\\C_{2}Y_{1}&1\end{bmatrix}}^{-1}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93470e4d80b45d65b457ca7097abf0569ba06db8)
and the matrices
and
satisfy
. If
, then
and
.
Given
and
, the matrices
and
can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.
The Continuous-Time H∞-Optimal Dynamic Output Feedback Controller is the system
The LMI given above can be implemented and solved using a tool such as YALMIP, along with an LMI solver such as MOSEK.
Discrete Time H∞ Optimal Dynamic Output Feedback Control
Continuous Time H2 Optimal Dynamic Feedback Control
A list of references documenting and validating the LMI.