LMIs in Control/pages/mixh2hinfdesiredpole4perturbed

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

The mixed $H_{2}/H_{\infty }$ 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 $H_{2}/H_{\infty }$ controller, the $H_{\infty }$ channel is used to improve the robustness of the design while the $H_{2}$ 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:

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

where

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

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

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

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

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

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

$F^{T}F<I$

The Data

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

The Optimization Problem

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

the transfer function matrices are $G_{z\infty w}(s)$ and $G_{z2w}(s)$
Thus the $H_{\infty }$ performance and the $H_{2}$ performance requirements for the system are, respectiverly
$||G_{z\infty w}(s)||_{\infty }<\gamma _{\infty }$
and
$||G_{z2w}(s)||_{2}<\gamma _{2}$
. For the performance of the system response, we introduce the closed-loop eigenvalue location requirement. Let
$D={s|s\in C,L+sM+sM^{T}<0},$
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 $H_{\infty }$ performance and the $H_{2}$ performance are satisfied.
The closed-loop eigenvalues are all located in $D$ , that is,

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

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

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

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

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

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

Conclusion:

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

Implementation

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

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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