LMIs in Control/pages/dt mixed H2 Hinf optimal 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 continuous time.

Problem

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

$\dot{x}=Ax+\left[\begin{array}{cc}{B}_{1,1}& B_{1,2}\end{array}\right]\left[\begin{array}{c}{w}_1\\ w_2\end{array}\right]+B_{2}u,$

$\left[\begin{array}{c}{z}_1\\ z_2\end{array}\right]=\left[\begin{array}{c}{C}_{1,1}\\ D_{1,2}\end{array}\right]x_k+\left[\begin{array}{cc}{D}_{11,11}& D_{11,12}\\ D_{11,21}& D_{11,22}\end{array}\right]\left[\begin{array}{c}{w}_1\\ w_2\end{array}\right]+\left[\begin{array}{c}{D}_{12,1}\\ D_{12,2}\end{array}\right]u,$

$y=C_{d2}x+\left[\begin{array}{cc}{D}_{21,1}& D_{21,2}\end{array}\right]\left[\begin{array}{c}{w}_1\\ w_2\end{array}\right]+D_{d22}u$

Theorem

A continuous-time dynamic output feedback LTI controllerwith state-space realization $(A_c,B_c,C_c,D_c)$ is to be designed to minimize the ${\mathscr{H}}_2$ norm of the closed-loop transfer matrix $T_{11}(s)$ from the exogenous input $w_1$ to the performance output $z_1$ while ensuring the H∞ norm of the closed-loop transfer matrix $T_{22}(s)$ from the exogenous input $w_2$ to the performance output $z_2$ is less than ${\gamma }_d$, where

$T_{11}(s)=C_{CL1,1}(sI-A_{CL}{)}^{-1}{B}_{CL1,1},$

$T_{22}(s)=C_{CL1,2}(sI-A_{CL}{)}^{-1}{B}_{CL1,2}+D_{CL11,22},$

$A_{d_{CL}}=\left[\begin{array}{cc}A+B_2{D}_c{\tilde{D}}^{-1}{C}_2& B_2(I+D_c{\tilde{D}}^{-1}{D}_{22})C_c\\ B_c{\tilde{D}}^{-1}{C}_2& A_c+B_c{\tilde{D}}^{-1}{D}_{22}{C}_c\end{array}\right]$,

$B_{CL1,1}=\left[\begin{array}{c}{B}_{1,1}+B_2{D}_c{\tilde{D}}^{-1}{D}_{21,1}\\ B_c{\tilde{D}}^{-1}{D}_{21,1}\end{array}\right]$,

$B_{CL1,2}=\left[\begin{array}{c}{B}_{1,2}+B_2{D}_c{\tilde{D}}^{-1}{D}_{21,2}\\ B_c{\tilde{D}}^{-1}{D}_{21,2}\end{array}\right]$,

$C_{CL1,1}=\left[\begin{array}{cc}{C}_{1,1}+D_{12,1}{D}_c{\tilde{D}}^{-1}{C}_{2,1}& D_{12,1}(I+D_c{\tilde{D}}^{-1}{D}_{22})C_c\end{array}\right]$,

$C_{CL1,2}=\left[\begin{array}{cc}{C}_{1,2}+D_{12,2}{D}_c{\tilde{D}}^{-1}{C}_{2,2}& D_{12,2}(I+D_c{\tilde{D}}^{-1}{D}_{22})C_c\end{array}\right]$,

$D_{CL11,22}=D_{11,22}+D_{12,2}{D}_c{\tilde{D}}^{-1}{D}_{21,2}$,

and $\tilde{D}=I-D_{22}{D}_c$.

Synthesis Method

Solve for $A_n\in {ℝ}^{n_x\times n_x},B_n\in {ℝ}^{n_x\times n_x},C_n\in {ℝ}^{n_u\times n_x},D_n\in {ℝ}^{n_u\times n_y},X_1,Y_1\in {𝕊}^{n_x},Z\in {𝕊}^{n_{Z_1}},$ and $\mu \in {ℝ}_{>0}$ that minimizes $𝒥(\mu )=\mu$ subjects to $X_1>0,Y_1>0Z>0,$

$\left[\begin{array}{ccc}{N}_{11}& A+A_n^T+B_2{D}_n{C}_2& B_{1,1}+B_2{D}_{n}D21,1\\ *& X_{1}A+A^T{X}_1+B_n{C}_2+C_2^T{B}_n^T& X_1{B}_{1,1}+B_n{D}_{21,1}\\ *& *& -I\end{array}\right]<0$,

$\left[\begin{array}{cccc}{N}_{11}& A+A_n^T+B_2{D}_n{C}_2& B_{1,1}+B_2{D}_{n}D21,1& Y_1^T{C}_{1,2}^T+C_n^T{D}_{12,2}^T\\ *& X_{1}A+A^T{X}_1+B_n{C}_2+C_2^T{B}_n^T& X_1{B}_{1,1}+B_n{D}_{21,1}& C_{1,2}^T+C_2^T{D}_n^T{D}_{12,2}^T\\ *& *& -{\gamma }_{d}I& D_{11,22}^T+D_{21,2}^T{D}_n^T{D}_{12,2}^T\\ *& *& *& -{\gamma }_{d}I\end{array}\right]<0$,

$\left[\begin{array}{c}{Y}_{1}I{Y}_1{C}_{1,1}^T+C_n^T{D}_{12,1}^T\\ *& X_1& C_{1,1}^T+C_2^T{D}_n^T{D}_{12,1}^T\\ *& *& Z\end{array}\right]>0$,

$D_{11,11}+D_{12,1}{D}_n{D}_{21,1}=0,$

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

tr$Z<\mu ,$

where $N_{11}=A{Y}_1+Y_1{A}^T+B_2{C}_n+C_n^T{B}_2^T$.

The controller is recovered by

$A_c=A_K-Bc(I-D_{22}{D}_c{)}^{-1}{D}_{22}{C}_c,$

$B_c=B_K(I-D_c{D}_{22}),$

$C_c=(I-D_c{D}_{22})C_K,$

$D_c=(I+D_{K}D22{)}^{-1}{D}_K,where<math>\left[\begin{array}{cc}{A}_K& B_K\\ C_K& D_K\end{array}\right]={\left[\begin{array}{cc}{X}_2& X_1{B}_2\\ 0& I\end{array}\right]}^{-1}(\left[\begin{array}{cc}{A}_n& B_n\\ C_n& D_n\end{array}\right]-\left[\begin{array}{cc}{X}_1{A}_d{Y}_1& 0\\ 0& 0\end{array}\right]){\left[\begin{array}{cc}{Y}_2^T& 0\\ C_2{Y}_1& I\end{array}\right]}^{-1}$, and the matrices $X_2$ and $Y_2$ satisfy $X_2{Y}_2^T=I-X_1{Y}_1$. If $D_{22}=0$, then $A_c=A_K,B_c=BK,C_c=C_K$ 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 $D_{11,11}=0,D_{12,1}\ne 0,\text{ and }{D}_{21,1}\ne 0,$ then it is often simplest to choose $D_n=0$ in order to satisfy the equality constraint $D_{11,11}+D_{12,1}{D}_n{D}_{21,1}=0,$.

WIP, additional references to be added

External Links

A list of references documenting and validating the LMI.