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{align} \ \dot x(t) &= A_ix(t)+bi+B_iu(t)+B_{\omega_i}\omega(t),\\ \ y(t) &= C_i x(t),\\ \end{align}$$

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{align} \begin{bmatrix} \ S_i+S_i^T+\eta B_{\omega_i}B_{\omega_i}^T & QV_i^T \\ \ C_iQ & -I \\ \end{bmatrix} \ < 0, \end{align} $$

$$Q=Q^T>0,\eta > 0,\mu_i<0 $$

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

where

$$\eta<\gamma^2,

S_i=A_iQ+B_iY_i$$

for $$i=1,...M$$

Conclusion:
The resulting matrices allow us to construct a state feedback controller $$ Y_i=K_iQ$$ 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:

 * - example