LMIs in Control/pages/dt mixed H2 Hinf optimal dynamic output feedback control

WIP, Description in progress

This part shows how to design dynamic outpur feedback control in mixed ${\mathscr{H}}_2$ and ${\mathscr{H}}_{\mathrm{\infty }}$ sense for the discrete time .

Problem

Consider the discrete-time generalized LTI plant $𝒫$ with minimal state-space realization

$x_{k+1}=A_d{x}_k+\left[\begin{array}{cc}{B}_{d1,1}& B_{d1,2}\end{array}\right]\left[\begin{array}{c}{w}_{1,k}\\ w_{2,k}\end{array}\right]+B_{d,2}{u}_k,$

$\left[\begin{array}{c}{z}_{1,k}\\ z_{2,k}\end{array}\right]=\left[\begin{array}{c}{C}_{d1,1}\\ D_{d1,2}\end{array}\right]x_k+\left[\begin{array}{cc}{D}_{d11,11}& D_{d11,12}\\ D_{d11,21}& D_{d11,22}\end{array}\right]\left[\begin{array}{c}{w}_{1,k}\\ w_{2,k}\end{array}\right]+\left[\begin{array}{c}{D}_{12,1}\\ D_{12,2}\end{array}\right]u_k,$

$y_k=C_{d2}{x}_k+\left[\begin{array}{cc}{D}_{21,1}& D_{21,2}\end{array}\right]\left[\begin{array}{c}{w}_{1,k}\\ w_{2,k}\end{array}\right]+D_{d22}{u}_k$

Theorem

A discrete-time dynamic output feedback LTI controller with state-space realization $(A_{dc},B_{dc},C_{dc},D_{dc})$ is to be designed to minimize the ${\mathscr{H}}_2$ norm of the closed loop transfer matrix $T_{11}(z)$ from the exogenous input $w_1,k$ to the performance output $z_1,k$ while ensuring the ${\mathscr{H}}_{\mathrm{\infty }}$ norm of the closed-loop transfer matrix $T_{22}(z)$ from the exogenous input $w_2,k$ to the performance output $z_2,k$ is less than ${\gamma }_d$, where

$T_{11}(z)=C_{d_{CL}1,1}(zI-A_{d_{CL}}{)}^{-1}{B}_{d_{CL}1,1},$

$T_{22}(z)=C_{d_{CL}1,2}(zI-A_{d_{CL}}{)}^{-1}{B}_{d_{CL}1,2}+D_{d_{CL}11,22},$

$A_{d_{CL}}=\left[\begin{array}{cc}{A}_d+B_{d2}{D}_{dc}{\tilde{D}}_d^{-1}{C}_{d2}& B_{d2}(I+D_{dc}{\tilde{D}}_d^{-1}{D}_{d22})C_{dc}\\ B_{dc}{\tilde{D}}_d^{-1}{C}_{d2}& A_{dc}+B_{dc}{\tilde{D}}_d^{-1}{D}_{d22}{C}_{dc}\end{array}\right]$,

$B_{d_{CL}1,1}=\left[\begin{array}{c}{B}_{d1,1}+B_{d2}{D}_{dc}{\tilde{D}}_d^{-1}{D}_{d21,1}\\ B_{dc}{\tilde{D}}_d^{-1}{D}_{d21,1}\end{array}\right]$,

$B_{d_{CL}1,2}=\left[\begin{array}{c}{B}_{d1,2}+B_{d2}{D}_{dc}{\tilde{D}}_d^{-1}{D}_{d21,2}\\ B_{dc}{\tilde{D}}_d^{-1}{D}_{d21,2}\end{array}\right]$,

$C_{d_{CL}1,1}=\left[\begin{array}{cc}{C}_{d1,1}+D_{d12,1}{D}_{dc}{\tilde{D}}_d^{-1}{C}_{d2,1}& D_{d12,1}(I+D_{dc}{\tilde{D}}_d^{-1}{D}_{d22})C_{dc}\end{array}\right]$,

$C_{d_{CL}1,2}=\left[\begin{array}{cc}{C}_{d1,2}+D_{d12,2}{D}_{dc}{\tilde{D}}_d^{-1}{C}_{d2,2}& D_{d12,2}(I+D_{dc}{\tilde{D}}_d^{-1}{D}_{d22})C_{dc}\end{array}\right]$,

$D_{d_{CL}11,22}=D_{d11,22}+D_{d12,2}{D}_{dc}{\tilde{D}}_d^{-1}{D}_{d21,2}$,

and ${\tilde{D}}_d=I-D_{d22}{D}_{dc}$.

Synthesis Method

Solve for $A_{dn}\in {ℝ}^{n_x\times n_x},B_{dn}\in {ℝ}^{n_x\times n_x},C_{dn}\in {ℝ}^{n_u\times n_x},D_{dn}\in {ℝ}^{n_u\times n_y},X_1,Y_1\in {𝕊}^{n_x},Z\in {𝕊}^{n_{Z_1}},$ and $\mu \in {ℝ}_{>0}$ that minimizes $𝒥(\mu )=\mu$ subjects to $X_1>0,Y_1>0Z>0,$

$\left[\begin{array}{ccccc}{X}_1& I& X_1{A}_d+B_{d_n}{C}_{d_2}& A_{d_n}& X_1{B}_{d1,1}+B{d}_n{D}_{d21,1}\\ *& Y_1& A_d+B_{d2}{C}_{dn}{D}_{d2}& A_d{Y}_1+B_{d2}{C}_{dn}& B_{d1,1}+B_{d2}{D}_{dn}{D}_{d21,1}\\ *& *& X_1& I& 0\\ *& *& *& Y_1& 0\\ *& *& *& *& I\end{array}\right]>0,$

$\left[\begin{array}{cccccc}{X}_1& I& X_1{A}_d+B_{d_n}{C}_{d_2}& A_{d_n}& X_1{B}_{d1,2}+B{d}_n{D}_{d21,2}& 0\\ *& Y_1& A_d+B_{d2}{C}_{dn}{D}_{d2}& A_d{Y}_1+B_{d2}{C}_{dn}& B_{d1,2}+B_{d2}{D}_{dn}{D}_{d21,2}& 0\\ *& *& X_1& I& 0& C_{d1,2}^T+C_{d2}^T{D}_{dn}^T{D}_{d12,2}^T\\ *& *& *& Y_1& 0& Y_1{C}_{d1,2}^T+C_{dn}^T{D}_{d12,2}^T\\ *& *& *& *& -{\gamma }_{d}I& D_{d11,22}^T+D_{d21,2}^T{D}_{dn}^{T}D{d12,2}^T\\ *& *& *& *& *& -{\gamma }_{d}I\end{array}\right]>0$

$\left[\begin{array}{ccc}Z& C_{d1,1}+D_{d12,1}{D}_{dn}{C}_{d2}& C_{d1,1,}{Y}_1^T+D_{d12,1}{C}_{dn}\\ *& X_1& I\\ *& *& Y_1\end{array}\right]>0,$

$D_{d11,11}+D_{d12,1}{D}_{dn}{D}_{d21,1}=0,$

$\left[\begin{array}{cc}{X}_1& I\\ *& Y_1\end{array}\right]>0,$

tr$Z<\mu .$

The controller is recovered by

$A_{dc}=A_{dK}-Bdc(I-D_{d22}{D}_{dc}{)}^{-1}{D}_{d22}{C}_{dc},$

$B_{dc}=B_{dK}(I-D_{dc}{D}_{d22}),$

$C_{dc}=(I-D_{dc}{D}_{d22})C_{dK},$

$D_{dc}=(I+D_{dK}Dd22{)}^{-1}{D}_{dK},where<math>\left[\begin{array}{cc}{A}_{dK}& B_{dK}\\ C_{dK}& D_{dK}\end{array}\right]={\left[\begin{array}{cc}{X}_2& X_1{B}_{d2}\\ 0& I\end{array}\right]}^{-1}(\left[\begin{array}{cc}{A}_{dn}& B_{dn}\\ C_{dn}& D_{dn}\end{array}\right]-\left[\begin{array}{cc}{X}_1{A}_d{Y}_1& 0\\ 0& 0\end{array}\right]){\left[\begin{array}{cc}{Y}_2^T& 0\\ C_{d2}{Y}_1& I\end{array}\right]}^{-1}$, and the matrices $X_2$ and $Y_2$ satisfy $X_2{Y}_2^T=I-X_1{Y}_1$. If $D_{22}=0$, then $A_{dc}=A_{dK},B_{dc}=BdK,C_{dc}=C_{dK}$ and $D_{dc}=D_{dK}$.

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 $D_{d11,11}=0,D_{d12,1}\ne 0,\text{ and }Dd21,1\ne =0,$ then it is often simplest to choose $D_{dn}=0$ in order to satisfy the equality constraint $D_{d11,11}+D_{d12,1}{D}_{dn}{D}_{d21,1}=0,$.

WIP, additional references to be added

External Links

A list of references documenting and validating the LMI.