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 $$\mathcal{H}_2$$ and $$\mathcal{H}_\infty$$ sense for the discrete time.

Problem
Consider the discrete-time generalized LTI plant $$\mathcal{P}$$ with minimal state-space realization

$$ x_{k+1}=A_dx_k+\begin{bmatrix}B_{d1,1} & B_{d1,2} \end{bmatrix}\begin{bmatrix}w_{1,k} \\ w_{2,k} \end{bmatrix} +B_{d,2}u_k,$$

$$ \begin{bmatrix}z_{1,k} \\ z_{2,k} \end{bmatrix}=\begin{bmatrix}C_{d1,1} \\ D_{d1,2} \end{bmatrix}x_k + \begin{bmatrix}D_{d11,11} & D_{d11,12} \\ D_{d11,21} & D_{d11, 22}\end{bmatrix}\begin{bmatrix}w_{1,k} \\ w_{2,k} \end{bmatrix}+\begin{bmatrix}D_{12,1} \\ D_{12,2} \end{bmatrix}u_k,$$

$$y_k=C_{d2}x_k + \begin{bmatrix}D_{21,1} & D_{21,2}\end{bmatrix}\begin{bmatrix}w_{1,k} \\ w_{2,k} \end{bmatrix}+D_{d22}u_k$$

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 $$\mathcal{H}_2$$ 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 $$\mathcal{H}_\infty$$ 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

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

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

$$A_{d_{CL}} = \begin{bmatrix} A_d+B_{d2}D_{dc}\tilde{D}_d^{-1}C_{d2} & B_{d2}(I+D_{dc}\tilde{D}_d^{-1}D_{d22})C_{dc} \\ B_{dc}\tilde{D}^{-1}_dC_{d2} & A_{dc}+B_{dc}\tilde{D}^{-1}_dD_{d22}C_{dc} \end{bmatrix}$$,

$$B_{d_{CL}1,1} = \begin{bmatrix}B_{d1,1}+B_{d2}D_{dc}\tilde{D}_d^{-1}D_{d21,1} \\ B_{dc}\tilde{D}_d^{-1}D_{d21, 1}\end{bmatrix}$$,

$$B_{d_{CL}1,2} = \begin{bmatrix}B_{d1,2}+B_{d2}D_{dc}\tilde{D}_d^{-1}D_{d21,2} \\ B_{dc}\tilde{D}_d^{-1}D_{d21, 2}\end{bmatrix}$$,

$$C_{d_{CL}1,1} = \begin{bmatrix}C_{d1,1}+D_{d12,1}D_{dc}\tilde{D}_d^{-1}C_{d2,1} & D_{d12,1}(I+D_{dc}\tilde{D}_d^{-1}D_{d22})C_{dc}\end{bmatrix}$$,

$$C_{d_{CL}1,2} = \begin{bmatrix}C_{d1,2}+D_{d12,2}D_{dc}\tilde{D}_d^{-1}C_{d2,2} & D_{d12,2}(I+D_{dc}\tilde{D}_d^{-1}D_{d22})C_{dc}\end{bmatrix}$$,

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

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

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

$$\begin{bmatrix}X_1 & I & X_1A_d + B_{d_n}C_{d_2} & A_{d_n} & X_1B_{d1,1}+B{d_n}D_{d21,1} \\ \end{bmatrix}>0,$$
 * & Y_1 & A_d+B_{d2}C_{dn}D_{d2} & A_dY_1+B_{d2}C_{dn} & B_{d1, 1}+B_{d2}D_{dn}D_{d21, 1} \\
 * & * & X_1 & I & 0 \\
 * & * & * & Y_1 & 0 \\
 * & * & * & * & I

$$\begin{bmatrix}X_1 & I & X_1A_d + B_{d_n}C_{d_2} & A_{d_n} & X_1B_{d1,2}+B{d_n}D_{d21,2} & 0\\ \end{bmatrix}>0$$
 * & Y_1 & A_d+B_{d2}C_{dn}D_{d2} & A_dY_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}^TD_{dn}^TD_{d12,2}^T \\
 * & * & * & Y_1 & 0 & Y_1 C_{d1,2}^T+C_{dn}^TD_{d12,2}^T\\
 * & * & * & * & -\gamma_d I & D^T_{d11,22} + D_{d21,2}^TD_{dn}^TD{d12,2}^T \\
 * & * & * & * & * & -\gamma_d I

$$\begin{bmatrix}Z & C_{d1,1}+D_{d12,1}D_{dn}C_{d2} & C_{d1,1,}Y_1^T+D_{d12,1}C_{dn} \\
 * & X_1 & I \\
 * & * & Y_1\end{bmatrix}>0,$$

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

$$\begin{bmatrix}X_1 & I \\
 * & Y_1\end{bmatrix}>0,$$

tr$$Z<\mu.$$

The controller is recovered by

$$A_{dc} = A_{dK}-B{dc}(I-D_{d22}D_{dc})^{-1}D_{d22}C_{dc}, $$

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

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

D_{dc} = (I+D_{dK}D{d22})^{-1}D_{dK},

where

$$\begin{bmatrix} A_{dK} & B_{dK} \\ C_{dK} & D_{dK} \end{bmatrix} = \begin{bmatrix} X_2 & X_1B_{d2} \\ 0 & I \end{bmatrix}^{-1}(\begin{bmatrix} A_{dn} & B_{dn} \\ C_{dn} & D_{dn} \end{bmatrix}-\begin{bmatrix} X_1A_dY_1 & 0 \\ 0 & 0 \end{bmatrix})\begin{bmatrix} Y_2^T & 0 \\ C_{d2}Y_1 & I \end{bmatrix}^{-1}$$, and the matrices $$X_2$$ and $$Y_2$$ satisfy $$X_2Y_2^T=I-X_1Y_1$$. If $$D_{22}=0$$, then $$A_{dc}=A_{dK}, B_{dc}=B{dK}, C_{dc}=C_{dK}$$ 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 $$D_{d11,11} = 0, D_{d12,1} \neq 0, \text{ and } D{d21,1} \neq = 0,$$ 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