LMIs in Control/Controller Synthesis/Continuous Time/Optimal Dynamic Output Feedback/H-infinity

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 $$w_{k}$$ and performance output $$z_{k}$$.

The System
Continuous-Time LTI System with state space realization $$(A,B,C,D)$$

\begin{align} \dot{x}&=Ax + B_1w + B_2u\\ z&=C_1x + D_{11}w+ D_{12}u\\ y &=C_2x + D_{21}w+ D_{22}u\\ \end{align}$$

The Data
The matrices: System $$ (A,B_{1},B_{2},C_{1},C_{2},D_{11},D_{12},D_{21},D_{22}), X_1,Y_1,Z, ,X_2,Y_2 $$

Controller $$ (A_{c},B_{c},C_{c},D_{c}) $$

The Optimization Problem
The following feasibility problem should be optimized:

$$ \gamma $$ is minimized while obeying the LMI constraints.

The LMI:
Solve for $$A_{n} \in {R^{n_x*n_x}}, B_{n} \in {R^{n_x*n_y}}, C_{n} \in {R^{n_u*n_x}}, D_{n} \in {R^{n_u*n_y}}, X_{1}, Y_{1}\in {S^{n_x}}, and \gamma \in {R_{>0}};$$ that minimize $$\mathcal{J}(\gamma) = \gamma$$ subject to $$X_1>0,Y_1>0,$$ $$\begin{align}

\\ \begin{bmatrix}N_{11} &A+A_n^T+B_2D_nC_2 &B_1+B_2D_nD_{21} &Y_1^TC_1^T+C_n^TD_{12}^T \\
 * &X_1A+A^TX_1+B_nC_2+C_2^T+B_n^T &X_1B_1+B_nD_{21} &C_1^T+C_2^TD_n^TD_{12}^T \\
 * & * &-\gamma \mathbf{1} &D_{11}^T+D_{21}^TD_n^TD{12}^T \\
 * & * &* &-\gamma \mathbf{1}\end{bmatrix}&>0,\\

\begin{bmatrix}X_{1} & \mathbf{1}\\
 * & Y_{1}\end{bmatrix}&>0,\\

\end{align}$$

where $$N_{11}=AY_1+Y_1A^T+B_2C_n+C_n^TB_2^T.$$ The controller is recovered by

$$ \begin{align} &A_{c} = A_{k}-B_{c}(1-D_{22}D_{c})^{-1}D_{22}C_{c}\\ &B_{c} = B_{k}(1-D_{c}D_{22})\\ &C_{c} = (1-D_{c}D_{22})C_{k}\\ &D_{c} = 1+D_{k}D_{22})^{-1}D_{k}\\ \end{align}$$

where, $$ \begin{align} \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{align}$$ and the matrices $$X_2$$ and $$Y_2$$ satisfy $$X_2Y_2^T = 1 - X_{1}Y_{1}$$. If $$D_{22}=0$$, then $$A_c=A_k,B_c=B_k,C_c=C_k,$$ and $$D_c=D_k$$.

Given $$X_1$$ and $$Y_1$$, the matrices $$X_2$$ and $$Y_2$$ can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.

Conclusion:
The Continuous-Time H∞-Optimal Dynamic Output Feedback Controller is the system $$ (A_{dc},B_{dc},C_{dc},D_{dc}) $$

Implementation
The LMI given above can be implemented and solved using a tool such as YALMIP, along with an LMI solver such as MOSEK.

Related LMIs
Discrete Time H∞ Optimal Dynamic Output Feedback Control

Continuous Time H2 Optimal Dynamic Feedback Control