How to handle asymmetric input constraints in robust model predictive control for a system with polytopic uncertainty?

Question

Consider a linear time variant system

$x(k+1) = A(k)x(k)+B(k)u(k) \\ y(k) = Cx(k)$

where $[A(k), B(k)]\in\Omega $

$\Omega$ is a polytope

$\Omega = \text{Co}([A_1, B_1 ] [A_2, B_2]... [A_L, B_L])$ where Co refers to the convex hull.

The above setup is of a system with polytopic uncertainty. For robust model predictive control of such system, Kothare et. al (1996) proposed a method using Linear Matrix inequalities (LMIs).

But this method only handle the symmetric input constraint on the input to the system i. e.

$-u_{max} \leq u \leq u_{max}$

Can anyone help me to find how to handle asymmetric input constraints using LMIs? or is there any other way to deal with polytopic uncertainty in presence of asymmetric input constraints?

Kothare, Mayuresh V.; Balakrishnan, Venkataramanan; Morari, Manfred, Robust constrained model predictive control using linear matrix inequalities, Automatica 32, No.10, 1361-1379 (1996). ZBL0897.93023.

Answer 1

In Model Predictive Control: Theory and Design by Rawlings and Mayne, Chapter 3.5, they apply tube-base robust MPC to linear discrete-time systems with (time-varying) parametric uncertainties contained in some polytope.

The state- and input-constraints are assumed to be polytopic, so it's applicable to your setup.