LMIs in Control/Controller Synthesis/Continuous Time/Optimal Dynamic Output Feedback/H-2

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 $(A,B,C,D)$

$\begin{array}{cc}\dot{x}& =A_x+B_{1}w+B_{2}u\\ z& =C_{1}x+D_{11}{w}_k+D_{12}u\\ y& =C_{2}x+D_{21}w+D_{22}u\\ \end{array}$

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 $𝒥(\nu )<\nu$ subject to $X_1>0,Y_1>0,Z>0,$

$\begin{array}{cc}\left[\begin{array}{ccc}A{Y}_1+Y_1{A}^T+B_2{C}_n+C_n^T{B}_2^T& A+A_n^T+B_2{D}_n{C}_2& B_1+B_2{D}_n{D}_{21}\\ *& X_{1}A+A^T{X}_1+B_n{C}_2+C_2^T{B}_n^T& X_1{B}_1+B_n{D}_{21}\\ *& *& -\mathrm{𝟏}\end{array}\right]& >0,\\ \left[\begin{array}{ccc}{X}_1& \mathrm{𝟏}& Y_1{C}_1^T+C_n^T{D}_{12}^T\\ *& Y_1& C_1^T+C_2^T{D}_n^T{D}_{12}^T\\ *& *& Z\end{array}\right]& >0,\\ D_{11}+D_{12}{D}_n{D}_{21}=0\\ \left[\begin{array}{cc}{X}_1& \mathrm{𝟏}\\ *& Y_1\end{array}\right]& >0,\\ trZ<\nu \end{array}$

The controller is recovered by

$\begin{array}{cc}& A_c=A_k-B_c(1-D_{22}{D}_c{)}^{-1}{D}_{22}{C}_c\\ & B_c=B_k(1-D_c{D}_{22})\\ & C_c=(1-D_c{D}_{22})C_k\\ & D_c=(1+D_k{D}_{22}{)}^{-1}{D}_k\\ \end{array}$

where, $\begin{array}{cc}\left[\begin{array}{cc}{A}_k& B_k\\ C_k& D_k\end{array}\right]& ={\left[\begin{array}{cc}{X}_2& X_1{B}_2\\ \mathrm{𝟎}& 1\end{array}\right]}^{-1}(\left[\begin{array}{cc}{A}_n& B_n\\ C_n& D_n\end{array}\right]-\left[\begin{array}{cc}{X}_{1}A{Y}_1& \mathrm{𝟎}\\ \mathrm{𝟎}& \mathrm{𝟎}\end{array}\right]){\left[\begin{array}{cc}{Y}_2^T& \mathrm{𝟎}\\ C{Y}_1& \mathrm{𝟏}\end{array}\right]}^{-1}\end{array}$
and the matrices $X_2$ and $Y_2$ satisfy $X_2{Y}_2^T=1-X_1{Y}_1$. 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 $D_{11}=0$, $D_{12}\ne 0,$ and $D_{21}\ne 0$, 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 $(A_c,B_c,C_c,D_c)$

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 H2 Optimal Dynamic Output Feedback Control

Continuous Time H∞ Optimal Dynamic Output Feedback Control

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.