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 $\hat{G}(s)$.

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

Where $A_i\in {ℝ}^{n\times n},b_i\in {ℝ}^{n\times k},B_i\in {ℝ}^{n\times m},B_{{\omega }_i}\in {ℝ}^{n\times m},C_i\in {ℝ}^{p\times n}$.

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 $H_{\mathrm{\infty }}$ control theory.

The LMI:

Objective: $\max \eta$.

Subject to::

$\begin{array}{c}\left[\begin{array}{cc}{S}_i+S_i^T+\eta B_{{\omega }_i}{B}_{{\omega }_i}^T& Q{V}_i^T\\ C_{i}Q& -I\end{array}\right]<0,\end{array}$

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

$-l_1\prec Y_i^j\prec l_1$

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.