LMIs in Control/Robustness/Continuous Time/DKIteration

This methods uses LMI techniques iteratively to obtain the result.

The System
Given a state-space representation of a system $$

G(s) $$ and an initial estimate of reduced order model $$

\hat{G}(s) $$.



\begin{align} \ G(s) &= C(sI-A)B + D,\\ \ \hat{G}(s) &= \hat{C}(sI-\hat{A})\hat{B} + \hat{D},\\ \end{align}$$

Where $$ A \in \mathbb{R}^{n\times n}, B \in \mathbb{R}^{n\times m}, C \in \mathbb{R}^{p\times n}, D \in \mathbb{R}^{p\times m}, \hat{A} \in \mathbb{R}^{k\times k}, \hat{B} \in \mathbb{R}^{k\times m}, \hat{C} \in \mathbb{R}^{p\times k}$$ and $$\hat{D} \in \mathbb{R}^{p\times m}$$.

The Data
The full order state matrices $$ A,B_1,B_2,C_1,C_2,D_{11},D_{12},D_{21} $$.

The Optimization Problem
Alternate fixing $$K,\Theta$$ in an iterative process to estimate a solution for Dynamic Output Feedback Synthesis with Structured Norm-Bounded Uncertainty.

The LMI: D-K Iteration
Objective: $$ \min \gamma $$.

Subject to:: Initialize: $$\Theta=I$$\\ Define:\\

$$

\begin{align} \ \hat G_{\Theta}(s) &= \begin{bmatrix} \ A &|& B_1\Theta^{-1/2} & B_2 \\ \hline \ \Theta^{1/2}C_1 &|& \Theta^{1/2}D_{11}\Theta^{-1/2} & \Theta^{1/2}D_{12} \\ \ C_2 &|& D_{12}\Theta^{-1/2} & 0\\ \end{bmatrix} \end{align} $$

Step 1:

Fix: $$ \Theta$$ and solve:

$$\inf_k \|\underline{S}(G_\Theta, K)\|_{H_{\infty}} $$

Step 2:

Fix K and minimize $$ \gamma$$ such that there exists $$ \Theta \in \mathbf{P\Theta}$$ and $$ X>0$$ such that

$$ \begin{align} \begin{bmatrix} \ A_{cl}^{T}X + XA_{xl} & XB_{cl} \\ \ B^T_{cl}X & -\Theta \\ \end{bmatrix} + \gamma^{-2} \begin{bmatrix} \ C_{cl}^{T}\\ \ D_{cl}^T \\ \end{bmatrix} \Theta \begin{bmatrix} \ C_{cl} & D_{cl}\\ \end{bmatrix} \ > 0 \end{align}$$ where $$A_{cl},B_{cl},C_{cl},D_{cl}$$ define $$\inf_k \|\underline{S}(G_\Theta, K)\|_{H_{\infty}} $$ (Bisection)

Step 3: Go to Step 1

Conclusion:
This is less of an LMI and more of a heuristic that allows us to solve for time-invariant scalings $$\Theta$$ and controller K. However, there are no guarantees that this process will return an globally optimized result.

Implementation

 * https://github.com/mosmith3asu/WikibooksLMIs/blob/main/DKiteration.m - Example Code

Related LMIs

 * https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Discrete_Time_H%E2%88%9E_Optimal_Dynamic_Output_Feedback_Control - $$H_\infty$$- Dynamic Output Feedback Controller Synthesis