LMI for Mixed ${\displaystyle H_{2}/H_{\infty }}$ with desired pole location Controller for perturbed system case

The mixed ${\displaystyle H_{2}/H_{\infty }}$ output feedback control has been known as an example of a multi-objective optimal control problem. In this problem, the control feedback should respond properly to several specifications. In the ${\displaystyle H_{2}/H_{\infty }}$ controller, the ${\displaystyle H_{\infty }}$ channel is used to improve the robustness of the design while the ${\displaystyle H_{2}}$ channel guarantees good performance of the system and additional constraint is used to place poles at desired location.

The System

We consider the following state-space representation for a linear system:

${\displaystyle {\begin{aligned}{\dot {x}}&=(A+\Delta A)x+(B_{1}+\Delta B_{1})u+B_{2}w\\z_{\infty }&=C_{\infty }+D_{\infty 1}u+D_{\infty 2}w\\z_{2}&=C_{2}x+D_{21}u\end{aligned}}}$

where

• ${\displaystyle x\in \mathbb {R} ^{n}}$, ${\displaystyle z_{2},z_{\infty }\in \mathbb {R} ^{m}}$are the state vector and the output vectors, respectively

• ${\displaystyle w\in \mathbb {R} ^{p}}$, ${\displaystyle u\in \mathbb {R} ^{r}}$ are the disturbance vector and the control vector

• ${\displaystyle A}$, ${\displaystyle B_{1}}$,${\displaystyle B_{2}}$, ${\displaystyle C_{\infty }}$,${\displaystyle C_{2}}$,${\displaystyle D_{\infty 1}}$,${\displaystyle D_{\infty 2}}$, and ${\displaystyle D_{21}}$ are the system coefficient matrices of appropriate dimensions.

• ${\displaystyle \Delta A}$ and ${\displaystyle \Delta B_{1}}$ are real valued matrix functions which represent the time varying parameters uncertainities.

Furthermore, the parameter uncertainties ${\displaystyle \Delta A}$ and ${\displaystyle \Delta B_{1}}$ are in the form of ${\displaystyle [\Delta A\,\,\,\,\,\Delta B_{1}]=HF[E_{1}\,\,\,\,\,E_{2}]}$ where

• ${\displaystyle H}$, ${\displaystyle E_{1}}$ and ${\displaystyle E_{2}}$ are known matrices of appropriate dimensions.
• ${\displaystyle F}$ is a matrix containing the uncertainty, which satisfies
  

${\displaystyle F^{T}F<I}$

The Data

We assume that all the four matrices of the plant,${\displaystyle A}$, ${\displaystyle \Delta A}$,${\displaystyle B_{1}}$ ${\displaystyle \Delta B_{1}}$,${\displaystyle B_{2}}$, ${\displaystyle C_{\infty }}$,${\displaystyle C_{2}}$,${\displaystyle D_{\infty 1}}$,${\displaystyle D_{\infty 2}}$, and ${\displaystyle D_{21}}$ are given.

The Optimization Problem

For the system with the following feedback law:
${\displaystyle u=Kx}$
The closed loop system can be obtained as:
${\displaystyle {\begin{aligned}{\dot {x}}&=((A+\Delta A)+(B_{1}+\Delta B_{1})K)x+B_{2}w\\z_{\infty }&=(C_{\infty }+D_{\infty 1}K)x+D_{\infty 2}w\\z_{2}&=(C_{2}+D_{21}K)u\end{aligned}}}$

the transfer function matrices are ${\displaystyle G_{z\infty w}(s)}$ and ${\displaystyle G_{z2w}(s)}$
Thus the ${\displaystyle H_{\infty }}$ performance and the ${\displaystyle H_{2}}$ performance requirements for the system are, respectiverly
${\displaystyle ||G_{z\infty w}(s)||_{\infty }<\gamma _{\infty }}$
and
${\displaystyle ||G_{z2w}(s)||_{2}<\gamma _{2}}$
. For the performance of the system response, we introduce the closed-loop eigenvalue location requirement. Let
${\displaystyle D={s|s\in C,L+sM+sM^{T}<0},}$
It is a region on the complex plane, which can be used to restrain the closed-loop eigenvalue locations. Hence a state feedback control law is designed such that,

• The ${\displaystyle H_{\infty }}$ performance and the ${\displaystyle H_{2}}$ performance are satisfied.
• The closed-loop eigenvalues are all located in ${\displaystyle D}$, that is,

${\displaystyle \,\,\,\,\,}$ ${\displaystyle \lambda (A+B_{1}K)\subset D}$.

The LMI: LMI for mixed ${\displaystyle H_{2}}$/${\displaystyle H_{\infty }}$ with desired Pole locations

The optimization problem discussed above has a solution if there exist scalars ${\displaystyle \alpha ,\,\,\,\beta }$ two symmetric matrices ${\displaystyle X,Z}$ and a matrix ${\displaystyle W}$, satisfying

min ${\displaystyle c_{2}\gamma _{2}^{2}+c_{\infty }\gamma _{\infty }}$
s.t
${\displaystyle {\begin{aligned}&{\begin{bmatrix}\Psi (X,W)&B_{2}&(C_{\infty }X+D_{\infty 1}W)^{T}&(E_{1}X+E_{2}W)^{T}\\B_{2}^{T}&-\gamma _{\infty }I&D_{\infty 2}^{T}&0\\C_{\infty }X+D_{\infty 1}W&D_{\infty 2}&-\gamma _{\infty }I&0\\(E_{1}X+E_{2}W)&0&0&-\alpha I\\\end{bmatrix}}<0\\&{\begin{bmatrix}\langle AX+B_{1}W\rangle +B_{2}B_{2}^{T}+\beta HH^{T}&(E_{1}X+E_{2}W)^{T}\\E_{1}X+E_{2}W&-\beta I\\\end{bmatrix}}<0\\&{\begin{bmatrix}-Z&C_{2}X+D_{21}W\\(C_{2}X+D_{21}W)^{T}&-X\\\end{bmatrix}}>0\\&{\text{trace}}(Z)<\gamma _{2}^{2}\\&L\otimes +M\otimes (AX+B_{1}W)+M^{T}\otimes (AX+B_{1}W)^{T}<0\\\end{aligned}}}$

where ${\displaystyle \Psi (X,W)=\langle AX+B_{1}W\rangle +\alpha HH^{T}}$

${\displaystyle c_{2}>0}$ and ${\displaystyle c_{\infty }>0}$ are the weighting factors.

Conclusion:

The calculated scalars ${\displaystyle \gamma _{\infty }}$ and ${\displaystyle \gamma _{2}}$ are the ${\displaystyle H_{2}}$ and ${\displaystyle H_{\infty }}$ norms of the system, respectively. The controller is extracted as ${\displaystyle K=WX^{-1}}$

Implementation

A link to Matlab codes for this problem in the Github repository:

Related LMIs

Mixed H2 Hinf with desired poles controller

External Links

• 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.

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