User:Margav06/sandbox/Click here to continue/Optimal control systems/H2-optimal Dynamic output Feedback control

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 $$w$$ and performance output $$z$$.

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

\begin{align} \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{align}$$

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

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

The Optimization Problem
The following feasibility problem should be optimized:

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

The LMI:
Solve for $$A_{n} \in {R^{n_x \times n_x}}, B_{n} \in {R^{n_x \times n_y}}, C_{n} \in {R^{n_u \times n_x}}, D_{n} \in {R^{n_u \times n_y}},X_{1}, Y_{1}\in {S^{n_x}}, Z \in {S^{n_z}},$$ and $$\nu \in {R_{>0}}$$ that minimize $$\mathcal{J}(\nu)<\nu$$ subject to $$X_{1}>0, Y_{1}>0,Z>0,$$



\begin{align}

\begin{bmatrix} AY_1+Y_1A^T+B_2C_n+C_n^TB_2^T &A+A_n^T+B_2D_nC_2 &B_1+B_2D_nD_{21}\\
 * &X_1A+A^TX_1+B_nC_2+C_2^TB_n^T &X_1B_1+B_nD_{21}\\
 * & * &-\mathbf{1}\end{bmatrix}&>0,\\

\begin{bmatrix} X_1 &\mathbf{1} &Y_1C_1^T+C_n^TD_{12}^T\\
 * &Y_1 &C_1^T+C_2^TD_n^TD_{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{align}$$

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_{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{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.

If $$D_{11} = 0$$, $$D_{12} \neq 0,$$ and $$D_{21} \neq 0$$, then it is often simplest to choose $$D_{n} = 0$$ in order to satisfy the equality constraint

Conclusion:
The Continuous-Time H2-Optimal Dynamic Output feedback controller is the system $$ (A_{c},B_{c},C_{c},D_{c}) $$

Related LMIs
Discrete Time H2 Optimal Dynamic Output Feedback Control

Continuous Time H∞ Optimal Dynamic Output Feedback Control