LMIs in Control/pages/mixedhinfh2desiredpole

LMI for Mixed $H_2/H_{\mathrm{\infty }}$ with desired pole location Controller

The mixed $H_2/H_{\mathrm{\infty }}$ 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 $H_2/H_{\mathrm{\infty }}$ controller, the $H_{\mathrm{\infty }}$ 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:

$\begin{array}{cc}\dot{x}& =Ax+B_{1}u+B_{2}w\\ z_{\mathrm{\infty }}& =C_{\mathrm{\infty }}+D_{\mathrm{\infty }1}u+D_{\mathrm{\infty }2}w\\ z_2& =C_{2}x+D_{21}u\end{array}$

where

• $x\in {ℝ}^n$, $z_2,z_{\mathrm{\infty }}\in {ℝ}^m$are the state vector and the output vectors, respectively

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

• $A$, $B_1$,$B_2$, $C_{\mathrm{\infty }}$,$C_2$,$D_{\mathrm{\infty }1}$,$D_{\mathrm{\infty }2}$, and $D_{21}$ are the system coefficient matrices of appropriate dimensions

The Data

We assume that all the four matrices of the plant,$A$, $B_1$,$B_2$, $C_{\mathrm{\infty }}$,$C_2$,$D_{\mathrm{\infty }1}$,$D_{\mathrm{\infty }2}$, 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:
$\begin{array}{cc}\dot{x}& =(A+B_{1}K)x+B_{2}w\\ z_{\mathrm{\infty }}& =(C_{\mathrm{\infty }}+D_{\mathrm{\infty }1}K)x+D_{\mathrm{\infty }2}w\\ z_2& =(C_2+D_{21}K)u\end{array}$

the transfer function matrices are $G_{z\mathrm{\infty }w}(s)$ and $G_{z2w}(s)$
Thus the $H_{\mathrm{\infty }}$ performance and the $H_2$ performance requirements for the system are, respectiverly
$||G_{z\mathrm{\infty }w}(s)|{|}_{\mathrm{\infty }}<{\gamma }_{\mathrm{\infty }}$
and
$||G_{z2w}(s)|{|}_2<{\gamma }_2$
. For the performance of the system response, we introduce the closed-loop eigenvalue location requirement. Let
$D=s|s\in C,L+sM+s{M}^T<0,$
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 $H_{\mathrm{\infty }}$ 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 $H_2$/$H_{\mathrm{\infty }}$ with desired Pole locations

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

min $c_2{\gamma }_2^2+c_{\mathrm{\infty }}{\gamma }_{\mathrm{\infty }}$
s.t
$\begin{array}{cc}& \left[\begin{array}{ccc}(AX+B_{1}W{)}^T+AX+B_{1}W& B_2& (C_{\mathrm{\infty }}X+D_{\mathrm{\infty }1}W{)}^T\\ B_2^T& -{\gamma }_{\mathrm{\infty }}I& D_{\mathrm{\infty }2}^T\\ C_{\mathrm{\infty }}X+D_{\mathrm{\infty }1}W& D_{\mathrm{\infty }2}& -{\gamma }_{\mathrm{\infty }}I\end{array}\right]<0\\ & AX+B_{1}W+(AX+B_{1}W{)}^T+B_2{B}_2^T<0\\ & \left[\begin{array}{cc}-Z& C_{2}X+D_{21}W\\ (C_{2}X+D_{21}W{)}^T& -X\end{array}\right]>0\\ & \text{trace}(Z)<{\gamma }_2^2\\ & L\otimes +M\otimes (AX+B_{1}W)+M^T\otimes (AX+B_{1}W{)}^T<0\\ \end{array}$

where $c_2>0$ and $c_{\mathrm{\infty }}>0$ are the weighting factors.

Conclusion:

The calculated scalars ${\gamma }_{\mathrm{\infty }}$ and ${\gamma }_2$ are the $H_2$ and $H_{\mathrm{\infty }}$ norms of the system, respectively. The controller is extracted as $K=W{X}^{-1}$

Implementation

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

Related LMIs

Mixed H2 Hinf with desired pole location for perturbed system

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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