LMIs in Control/Applications/LMIforPiecewiseLinearHinfControllerSynthesisInIncentoryControl

This problem dealt with the inventory control problem for a deterministic production system with given deterministic demand rate plus an unknown fluctuating demand rate with finite energy whose bound is known.

The System

Given a state-space representation of a system $G(s)$ and an initial estimate of reduced order model ${\hat {G}}(s)$.

${\begin{aligned}\ {\dot {x}}(t)&=A_{i}x(t)+bi+B_{i}u(t)+B_{\omega _{i}}\omega (t),\\\ y(t)&=C_{i}x(t),\\\end{aligned}}$

Where $A_{i}\in \mathbb {R} ^{n\times n},b_{i}\in \mathbb {R} ^{n\times k},B_{i}\in \mathbb {R} ^{n\times m},B_{\omega _{i}}\in \mathbb {R} ^{n\times m},C_{i}\in \mathbb {R} ^{p\times n}$.

The Data

The full order state matrices $A,b_{i},B_{\omega _{i}},B_{i},Ci,D$.

The Optimization Problem

This problem has been modeled as a control problem of a switched (PWA) system and it has been solved using new results on piecewise-linear $H_{\infty }$ control theory.

The LMI:

Objective: $\max \eta$.

Subject to::

${\begin{aligned}{\begin{bmatrix}\ S_{i}+S_{i}^{T}+\eta B_{\omega _{i}}B_{\omega _{i}}^{T}&QV_{i}^{T}\\\ C_{i}Q&-I\\\end{bmatrix}}\ <0,\end{aligned}}$

$Q=Q^{T}>0,\eta >0,\mu _{i}<0$

$-l_{1}\prec Y_{i}^{j}\prec l_{1}$

where

$\eta <\gamma ^{2},S_{i}=A_{i}Q+B_{i}Y_{i}$

for $i=1,...M$

Conclusion:

The resulting matrices allow us to construct a state feedback controller $Y_{i}=K_{i}Q$ that forces the stock level to be kept close to zero (sometimes called a just-in-time policy), even when there are fluctuations in the demand of the product.

Implementation:

• [1] - example

External Links

A list of references documenting and validating the LMI.

• [2] Rodrigues, Boukas. “Piecewise-Linear H ∞ Controller Synthesis with Applications to Inventory Control of Switched Production Systems.” Automatica (Oxford), vol. 42, no. 8, Elsevier Ltd, 2006, pp. 1245–54, doi:10.1016/j.automatica.2006.04.004.