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{align} \dot{x} &= (A + \Delta A) x + (B_1 + \Delta B_1) u + B_2w \\ z_{\infty} &= C_{\infty} + D_{\infty1}u + D_{\infty2}w\\ z_2 &= C_2x+D_{21}u \end{align} $$

where


 * $$x\in \R^n$$, $$z_2,z_{\infty} \in \R^m$$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


 * $$ A $$, $$ B_1$$,$$ B_2$$, $$ C_{\infty}$$,$$ C_2$$,$$ D_{\infty1}$$,$$ D_{\infty2}$$, 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 $$ F^T F < I $$
 * $$ H $$, $$ E_1 $$ and $$ E_2 $$ are known matrices of appropriate dimensions.
 * $$ F $$ is a matrix containing the uncertainty, which satisfies

The Data
We assume that all the four matrices of the plant,$$ A $$, $$ \Delta A $$,$$ B_1 $$ $$ \Delta B_1 $$,$$ B_2$$, $$ C_{\infty}$$,$$ C_2$$,$$ D_{\infty1}$$,$$ D_{\infty2}$$, 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{align} \dot{x} &= ((A+\Delta A) + (B_1 + \Delta B_1)K)x + B_2w \\ z_{\infty} &= (C_{\infty} + D_{\infty1}K)x + D_{\infty2}w\\ z_2 &= (C_2+D_{21}K)u \end{align} $$

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 $$ ||G_{z2 w}(s)||_{2} < \gamma_2 $$. For the performance of the system response, we introduce the closed-loop eigenvalue location requirement. Let $$ D =, $$ 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, $$ \, \, \, \, \, $$ $$ \lambda (A+B_1K) \subset D $$.
 * The $$ H_{\infty} $$ performance and the $$ H_{2} $$ performance are satisfied.
 * The closed-loop eigenvalues are all located in $$ D $$, that is,

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{align}

&\begin{bmatrix} \Psi (X,W) & B_2 & (C_{\infty}X+D_{\infty 1}W)^T & (E_1X + E_2W)^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_1X + E_2W) & 0 & 0 & -\alpha I \\ \end{bmatrix} < 0 \\

&\begin{bmatrix} \langle AX + B_1W \rangle + B_2B_2^T + \beta HH^T & (E_1X + E_2W)^T \\ E_1X + E_2W & -\beta I \\ \end{bmatrix} < 0 \\

&\begin{bmatrix} -Z  & C_2X+D_{21}W \\ (C_2X+D_{21}W)^T & -X\\ \end{bmatrix} > 0 \\ & \text{trace}(Z) < \gamma_{2}^{2} \\ & L \otimes + M \otimes (AX+B_1W) + M^T \otimes (AX+B_1W)^T < 0 \\ \end{align} $$

where $$ \Psi (X,W) = \langle AX + B_1W \rangle + \alpha HH^T $$

$$ c_2 > 0 $$ 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:

Related LMIs
Mixed H2 Hinf  with desired poles controller

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